# Three Dimensional Geometry Questions

We provide three dimensional geometry practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on three dimensional geometry skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.

#### List of three dimensional geometry Questions

Question NoQuestionsClass
1Perimeter of triangle whose vertices are (0,4,0),(3,4,0) and ( (0,4,4), ) is
A . 10
B. 12
c. 25
D. 15
12
287.
(a) 315 (b) 256 (c) 84
The length of the projection of the line segment joining the
points (5,-1, 4) and (4,-1,3) on the plane, x+y+z=
JEE M 2018]
WIN
12
3Prove that the points ( boldsymbol{A}= ) ( (1,2,3), B(3,4,7), C(-3,-2,-5) ) are
collinear ( & ) find the ratio in which ( B )
divides ( boldsymbol{A C} )
A .2: 5
B . 2: 3
c. 2: 8
D. 2: 7
12
4In geometry, we take a point, a line and a plane as undefined terms.
A. True
B. False
c. Ambiguous
D. Data Insufficient
12
5Find the shortest distance between the
skew lines:
( l_{1}: frac{x-1}{2}=frac{y+1}{1}=frac{z-2}{4} )
( l_{2}: frac{x+2}{4}=frac{y-0}{-3}=frac{z+1}{1} )
12
6The direction ratios of the line joining the points (4,3,-5) and (-2,1,-8) are
A ( cdot frac{6}{7}, frac{2}{7}, frac{3}{7} )
в. 6,2,3
c. 5,8,0
D. 3,7,9
12
7If a point ( boldsymbol{P} ) from where line drawn cuts
coordinates axes at ( A ) and ( B ) (with ( A ) on
( x-text { axis and } B text { on } y-text { axis }) ) satisfies ( alpha frac{x^{2}}{P B^{2}}+beta frac{y^{2}}{P A^{2}}=1, ) then ( alpha+beta ) is
( A cdot 1 )
B. 2
( c .3 )
D. 4
12
8The planes ( 2 x-y+4 z=5 ) and ( 5 x- )
( 2.5 y+10 z=6 ) are
A. Parallel
B. Perpendicular
c. Intersect
D. intersect ( x ) axis
12
9The number of straight line that are
equally inclined to the three dimensional co- ordinate axes, is
12
10( boldsymbol{A}=(mathbf{1}, mathbf{2}, mathbf{3}), boldsymbol{B}=(mathbf{4}, mathbf{5}, mathbf{7}), boldsymbol{C}= )
( (-4,3,-6), D=(2, k, 2) ) are four
points. If the lines ( A B ) and ( C D ) are
parallel, then ( k= )
( A cdot 0 )
в. -9
( c .9 )
D. 2
12
11Algebraic sum of intercepts made by the plane ( x+3 y-4 z+6=0 ) on the axes is
A. 7
B. 0
( c cdot frac{13}{2} )
( D cdot-frac{13}{2} )
12
12The following lines are ( hat{boldsymbol{r}}=(hat{boldsymbol{i}}+hat{boldsymbol{j}})+ )
( lambda(hat{i}+2 hat{j}-hat{k})+mu(-hat{i}+hat{j}-2 hat{k}) )
A. collinear
B. skew-lines
c. co-planar lines
D. parallel lines
12
13Find the point on Z-axis which are at a distance ( sqrt{21} ) unit from the point
(1,2,3)
12
14If ( boldsymbol{A}=(mathbf{2},-mathbf{3}, mathbf{1}), boldsymbol{B}=(mathbf{3},-mathbf{4}, mathbf{6}) ) and ( boldsymbol{C} )
is a point of trisection of ( A B, ) then ( C_{y}= )
A ( cdot frac{11}{3} )
B. -11
c. ( frac{10}{3} )
D. ( frac{-11}{3} )
12
15The distance between the circumcentre
and the ortho centre of the triangle formed by the points (2,1,5),(3,2,3) and (4,0,4) is
A ( cdot sqrt{6} )
B. ( frac{sqrt{6}}{2} )
c. ( 2 sqrt{6} )
D.
12
16Find the coordinates of a point
equidistant from four points ( boldsymbol{O}(mathbf{0}, mathbf{0}, mathbf{0}), boldsymbol{A}(ell, mathbf{0}, mathbf{0}), boldsymbol{B}(mathbf{0}, boldsymbol{m}, boldsymbol{0}) ) and
( boldsymbol{C}(mathbf{0}, mathbf{0}, boldsymbol{n}) )
12
17Find the direction cosines of a line
which makes equal angles with the coordinate axes.
12
18The projections of a directed line segment on the coordinate axes are
12,4,3 respectively.

What are the direction cosines of the
line segment?
A ( cdot(12 / 13,4 / 13,3 / 13) )
B . ( (12 / 13,-4 / 13,3 / 13) )
c. ( (12 / 13,-4 / 13,-3 / 13) )
D. ( (-12 / 13,-4 / 13,3 / 13) )

12
19If ( boldsymbol{A}=(mathbf{4}, mathbf{1}, mathbf{5}) ) and ( boldsymbol{B}=(mathbf{3}, mathbf{4}, mathbf{5}) ) The
direction ratios of ( overline{A B} ) are
12
20x-2 y. 2+2
56. Let the line –
lie in the plane
3 -5 2 .
x+3y-az+B=0. Then (a, b) equals
[2009]
(a) (-6,7)
(b) (5,-15)
(c) (-5,5)
(d) (6,-17)
12
21If ( boldsymbol{A}=(mathbf{5},-mathbf{1}, mathbf{1}), boldsymbol{B}=(mathbf{7},-mathbf{4}, mathbf{7}), boldsymbol{C}= )
( (1,-6,10), D=(-1,-3,4) . ) Then
( A B C D ) is a
A . square
B. rectangle
c. rhombus
D. none of these
12
22Find the ratio in which ( 2 x+3 y+5 z= )
1 divides the line joining the points
(1,0,-3) and (1,-5,7)
A .1: 2
B . 2: 1
c. 3: 2
D. 2: 3
12
23A plane mirror is placed at the origin so that the direction ratios of its normal
( operatorname{are}(1,-1,1) . ) A ray of light, coming along the positive direction of the ( x ) axis, strikes the mirror. The direction ( operatorname{cosines} ) of the reflected ray are
A ( cdot frac{1}{3}, frac{2}{3}, frac{2}{3} )
B. ( -frac{1}{3}, frac{2}{3}, frac{2}{3} )
c. ( -frac{1}{3},-frac{2}{3},-frac{2}{3} )
D. ( -frac{1}{3},-frac{2}{3}, frac{2}{3} )
12
24Find the square of the distance between
the points whose cartesian coordinates
are:
(-1,1,3),(0,5,6)
12
25Consider three vectors ( vec{P}=hat{i}+widehat{j}+ ) ( widehat{k} ; overrightarrow{boldsymbol{q}}=2 widehat{hat{boldsymbol{i}}}+4 widehat{boldsymbol{j}}-widehat{boldsymbol{k}} ) and ( overrightarrow{boldsymbol{r}}=boldsymbol{2} hat{boldsymbol{i}}+boldsymbol{4} widehat{boldsymbol{j}}+ )
3 ( widehat{k} ). If ( vec{p}, vec{q} ) and ( vec{r} ) denotes the position vector of three non-collinear points, then the equation of the plane containing these points is
A. ( 2 x-3 y+1=0 )
B. ( x-3 y+2 z=0 )
c. ( 3 x-y+z-3=0 )
D. ( 3 x-y-2=0 )
12
26The d.r’s of the line of intersection of the
planes ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}-mathbf{1}=mathbf{0} ) and ( mathbf{2} boldsymbol{x}+ )
( 3 y+4 z-7=0 ) are
A .1,2,-3
в. 2,1,-3
c. 4,2,-6
D. 1,-2,1
12
27Find the distance of point (1,-2,3)
from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} )
measured parallel to the line whose direction cosines are
proportional to 2,3,-6
12
28Let two planes ( p_{1}: 2 x-y+z=2, ) and
( boldsymbol{p}_{2}: boldsymbol{x}+boldsymbol{2} boldsymbol{y}-boldsymbol{z}=boldsymbol{3} ) are given. The
image of plane ( P_{1} ) in the plane mirror
( P_{2} ) is
A. ( x+7 y-4 z+5=0 )
B. ( 3 x+4 y-5 z+9=0 )
c. ( 7 x-y+2 z-9=0 )
0
D. ( 7 x+y+9 z+9=0 )
12
29If the points ( (h, 3,-4),(0,-7,10) ) and ( (1, k, 3) ) are collinear, then ( h+k ) is
( mathbf{A} cdot mathbf{4} )
B.
c. -4
D. 14
12
30Find the equation of the plane containing the line 2x -y+z
-3 = 0, 3x +y+z= 5 and at a distance of Ta from the point
(2,1,-1).
(2005 – 2 Marks)
12
31If ( P(x, y, z) ) moves such that ( x=0, z= )
( 0, ) then the locus of ( P ) is the line whose
d.cs are
A . ( y ) -axis
B. 1,0,0
c. 0,1,0
D. 0,0,0
12
32In three dimensions, the coordinate
axes of a rectangular cartesian coordinate system are
A. three mutually parallel lines
B. three mutually perpendicular lines
c. two mutually perpendicular lines and any two parallel
D. None of these
12
33If line ( frac{boldsymbol{x}-mathbf{2}}{mathbf{3}}=frac{boldsymbol{y}-mathbf{4}}{mathbf{4}}=frac{boldsymbol{z}+mathbf{2}}{mathbf{1}} ) is
parallel to planes ( mu x+3 y-2 z+d= )
0 and ( x-2 lambda y+z=0, ) then value of ( lambda )
and ( mu ) are
A ( cdotleft(mu=4, lambda=-frac{2}{3}right. )
B. ( mu=-6, lambda=-2 )
c. ( _{mu}=frac{-10}{3}, lambda=frac{-1}{2} )
D. ( mu=frac{-10}{3}, lambda=frac{1}{2} )
12
34Vectors ( vec{A}, vec{B} ) and ( vec{C} ) are such that ( vec{A} ) ( vec{B}=0 . ) Then the vector parallel to ( vec{A} ) is
A. ( vec{B} ) and ( vec{C} )
в. ( vec{A} times vec{B} )
c. ( vec{B}+vec{C} )
D. ( vec{B} times vec{C} )
12
35Find the equation of the plane passing through (2,0,1) and (3,-3,4) and perpendicular to ( boldsymbol{x}-mathbf{2} boldsymbol{y}+boldsymbol{z}=mathbf{6} )12
36Number of points having positive integral co-ordinate lying on the plane ( x+2 y+3 z=15 ) is ( n, ) then ( frac{n}{2} ) is equal
to
( mathbf{A} cdot mathbf{6} )
B. 8
c. 9
D.
12
37The vector equation of the plane passes through the points ( A & B ) with position vector ( 2 hat{i}+hat{j}-hat{k} &-hat{i}+3 hat{j}+4 hat{k} )
respectively ( & ) Ler to the plane ( bar{r} cdot(hat{i}-2 hat{j}+4 hat{k})=10 ) is
A ( cdot bar{r} cdot(18 hat{i}+17 hat{j}-3 hat{k})=49 )
B . ( bar{r} .(18 hat{i}-17 hat{j}-3 hat{k})+22=0 )
c. ( bar{r} .(18 hat{i}+17 hat{j}+4 hat{k})=25 )
D・ ( bar{r} .(18 hat{i}+17 hat{j}+4 hat{k})=24 )
12
38The vector equation of the line ( frac{x-2}{2}= ) ( frac{2 y-5}{-3}, z=-1 ) is ( vec{r}= )
( left(2 hat{i}+frac{5}{2} hat{j}-hat{k}right)+lambdaleft(2 hat{i}-frac{3}{2} hat{j}+x hat{k}right) )
where ( x ) is equal to
( mathbf{A} cdot mathbf{0} )
B.
c. 2
D. 3
12
39If ( P(x, y, z) ) is a point on the line segment joining ( Q(2,2,4) ) and ( R(3,5,6) ) such that the projection of ( overrightarrow{O P} ) on the axes are ( frac{13}{5}, frac{19}{5}, frac{26}{5} )
respectively, then ( P ) divides ( Q R ) in ratio
A .1: 3
B. 2: 3
c. 3: 2
D. 3: 1
12
40f ( (p, q, r) ) is equidistant from
(1,2,-3),(2,-3,1) and ( (-3,1,2), ) then
( boldsymbol{p}+boldsymbol{q}+boldsymbol{r}= )
A . -1
B.
c. 0
D.
12
41The acute angle between two lines such
that the direction cosines ( l, boldsymbol{m}, boldsymbol{n} ) of each of them satisfy the equation ( l+ )
( boldsymbol{m}+boldsymbol{n}=mathbf{0} ) and ( l^{2}+boldsymbol{m}^{2}-boldsymbol{n}^{2}=mathbf{0} ) is
( A cdot 30 )
B . 45
( c cdot 60 )
D. ( 15^{circ} )
12
42Plane ( a x+b y+c z=1 ) intersect axes ( operatorname{in} A, B, C ) respectively. If ( Gleft(frac{1}{6},-frac{1}{3}, 1right) )
is a centroid of ( triangle A B C ) then ( a+b+ )
( 3 c=-1 )
A ( cdot frac{4}{3} )
B. 4
( c cdot 2 )
D.
12
43The equation of the plane passing through the straight line ( frac{x-1}{2}= ) ( frac{boldsymbol{y}+mathbf{1}}{mathbf{- 1}}=frac{boldsymbol{z}-mathbf{3}}{mathbf{4}} ) and perpendicular to
plane ( boldsymbol{x}+mathbf{2} boldsymbol{y}+boldsymbol{z}=mathbf{1 2} ) is:
A. ( 9 x+2 y-5 z+8=0 )
в. ( 9 x+2 y-5 z+10=0 )
c. ( 9 x-2 y+5 z+6=0 )
D. ( 9 x-2 y-5 z+4=0 )
12
44Derive the equation of the locus of a point equivalent from the points (1,-2,3) and (-3,4,2)12
45If vector ( vec{a}=4 hat{i}+5 hat{j}-3 hat{k} ) and ( vec{b}=5 hat{i}+ )
( 3 hat{j}+8 hat{k} ) then value of ( frac{text { projection of vector b on a }}{text { projection of vector a on b }} ) is :
A ( cdot frac{7}{5} )
B. ( frac{2}{7} )
( c cdot frac{5}{8} )
D. None of these.
12
46Find ( a, b, c ) if ( a(1,3,2)+b(1,-5,6)+ )
( c(2,1,-2)=(4,10,-8) )
12
47A variable plane at a distance of 1 unit from the origin cuts the co-ordinate axes at ( A, B ) and ( C . ) If the centroid ( D(x, y, z) ) of triangle ( A B C ) satisfies the relation ( frac{1}{x^{2}}+ ) ( frac{1}{y^{2}}+frac{1}{z^{2}}=k, ) then the value of ( k ) is
A . 3
B. 1
c. ( 1 / 3 )
D.
12
48The points (-5,12),(-2,-3),(9,-10),(6,5)
taken in order, form
A. Parallelogram
B. rectangle
c. rhombus
D. square
12
49The distance from the origin to the centroid of the tetrahedron formed by
the points ( (0,0,0),(a, 0,0),(0, b, 0),(0,0, c) ) is:
A ( cdot frac{sqrt{a+b+c}}{4} )
B. ( frac{sqrt{a+b+c}}{3} )
c. ( frac{sqrt{a^{2}+b^{2}+c^{2}}}{16} )
D. ( frac{sqrt{a^{2}+b^{2}+c^{2}}}{4} )
12
50A point ( P ) lies on a line whose ends are ( A(1,2,3) ) and ( B(2,10,1) . ) If ( z )
component of ( boldsymbol{P} ) is ( boldsymbol{7}, ) then the
coordinates of ( boldsymbol{P} ) are
A ( cdot(-1,-14,7) )
B. (1,-14,7)
c. (-1,14,7)
D. (1,14,7)
12
51A parallelopiped ‘S’ has base points A, B, C and D and
upper face points A’, B’, C and D’. This parallelopiped is
compressed by upper face A’B’C’D’ to form a new
parallelopiped ‘T” having upper face points A”, B”, C” and
D”. Volume of parallelopiped Tis 90 percent of the volume
of parallelopiped S. Prove that the locus of ‘A”?, is a plane
12
52If 5,7,6 are the sums of the ( x, y )
intercepts; ( boldsymbol{y}, boldsymbol{z} ) intercepts, ( boldsymbol{z}, boldsymbol{x} )
intercepts respectively of a plane then the perpendicular distance from the origin to that plane is
A ( cdot frac{144}{61} )
в. ( frac{12}{sqrt{61}} )
c. ( frac{sqrt{61}}{12} )
D. ( frac{61}{144} )
12
53Find the angle between the following pairs of lines:
( frac{x-1}{2}=frac{y-2}{3}=frac{z-3}{-3} ) and ( frac{x+3}{-1}= )
( frac{boldsymbol{y}-mathbf{5}}{mathbf{8}}=frac{boldsymbol{z}-mathbf{1}}{mathbf{4}} )
12
54The vector equation of the plane which is at a distance of ( frac{3}{sqrt{14}} ) from the origin and the normal from the origin is ( 2 hat{i}- ) ( mathbf{3} hat{boldsymbol{j}}+hat{boldsymbol{k}} ) is
A ( . vec{r} .(2 hat{i}-3 hat{j}+hat{k})=3 )
B . ( vec{r} .(hat{i}+hat{j}+hat{k})=9 )
c. ( vec{r} .(hat{i}+2 hat{j})=3 )
D. ( vec{r} .(2 hat{i}+hat{k})=3 )
12
55If a line makes angles ( alpha, beta, gamma ) with axes
of co-ordinates, then ( cos 2 alpha+cos 2 beta+ )
( cos 2 gamma ) is equla to
A . -2
B. –
c. 1
D. 2
12
56Name the octants in which the following
points lie:
( (1,2,3),(4,-2,3)(4,-2,-5),(4,2,-5) )
12
57The perpendicular distance of ( vec{A}(1,4,-2) ) from the segment BC where ( vec{B} ) (2,1,-2) and ( vec{C}(0,-5,1) ) is
( A cdot frac{3}{7} sqrt{26} )
B ( cdot frac{6}{7} sqrt{26} )
( mathbf{c} cdot frac{4}{7} sqrt{26} )
( D cdot frac{2}{7} sqrt{26} )
12
58Find the direction cosines of
perpendicular from the origin to the plane ( bar{r}(2 hat{i}+3 hat{j}+6 hat{k})+7=0 )
12
59Perpendiculars ( A P, A Q ) and ( A R ) are drawn to the ( x-, y- ) and ( z- ) axes, respectively
from the point ( boldsymbol{A}(mathbf{1},-mathbf{1}, mathbf{2}) . ) The A.M. of
( A P^{2}, A Q^{2} ) and ( A R^{2} ) is
A .4
B. 5
( c .3 )
D.
12
60If ( boldsymbol{A}(boldsymbol{6},-mathbf{7}, mathbf{0}), boldsymbol{B}(mathbf{1 6},-mathbf{1 9},-mathbf{4}) )
( C(0,3,-6) ) and ( D(2,-5,10) ) are four points in space, then the point of intersection of the lines ( A B ) and ( C D ) is
A ( cdot(2,1,-1) )
в. (1,1,2)
D. does not exist as the lines are skew
12
61A plane meet the co-ordinate axes in
( A, B, C ) such that the centroid of
triangle ( A B C ) is ( (a, b, c) . ) If equation of plane ( frac{x}{a}+frac{y}{b}+frac{z}{c}=k(k neq 0), ) then the
value of ( k ) equals
A . 2
B. 3
( c cdot 4 )
D. 5
12
62Two system of rectangular axes have the same origin. If a plane cuts them at
distances, ( a, b, c ) and ( a_{1}, b_{1}, c_{1} ) from the
origin, then
A ( cdot frac{1}{a^{2}}+frac{1}{b^{2}}+frac{1}{c^{2}}=frac{1}{a_{1}^{2}}+frac{1}{b_{1}^{2}}+frac{1}{c_{1}^{2}} )
B. ( frac{1}{a^{2}}-frac{1}{b^{2}}+frac{1}{c^{2}}=frac{1}{a_{1}^{2}}-frac{1}{b_{1}^{2}}+frac{1}{c_{1}^{2}} )
C ( cdot a^{2}+b^{2}+c^{2}=a_{1}^{2}+b_{1}^{2}+c_{1}^{2} )
D cdot ( a^{2}-b^{2}+c^{2}=a_{1}^{2}-b_{1}^{2}+c_{1}^{2} )
12
63ox, oy are positive x-axis, positive ( y ) axis respectively where ( boldsymbol{O}=(mathbf{0}, mathbf{0}, mathbf{0}) )
The ( d . c . s ) of the llne which bisects ( angle x o y )
are
( mathbf{A} cdot 1,1,0 )
B. ( frac{1}{sqrt{2}}, frac{1}{sqrt{2}}, 0 )
c. ( frac{1}{sqrt{2}}, 0, frac{1}{sqrt{2}} )
D. 0,0,1
12
64Find the equation of the set of points which are equidistant from the points (1,2,3) and (3,2,-1)12
65The ratio in which the plane ( bar{r} .(bar{i}-2 bar{j}+ ) ( mathbf{3} bar{k})=17 ) divides the line joining the points ( -2 bar{i}+4 bar{j}+7 bar{k} ) and ( 3 bar{i}-5 bar{j}+8 bar{k} )
is
A. 1: 10
B. 3: 10
( c .3: 5 )
D. 1: 5
12
66Consider three planes ( 2 x+p y+6 z= )
( mathbf{8}, boldsymbol{x}+mathbf{2} boldsymbol{y}+boldsymbol{q} boldsymbol{z}=mathbf{5} ) and ( boldsymbol{x}+boldsymbol{y}+mathbf{3} boldsymbol{z}=mathbf{4} )
These planes do not have any common point of intersection if-
A. ( p=2, q neq 3 )
B . ( p neq 2, q neq 3 )
c. ( p neq 2, q=3 )
D. ( p=2, q=3 )
12
67The ratio in which the surface ( x^{2}+ )
( y^{2}+z^{2}=25 ) divides the line joining
(0,1,2) and (3,4,5) is ( frac{a pm sqrt{b}}{c} ) then ( boldsymbol{a}+boldsymbol{b}+boldsymbol{c}= )
12
68The direction cosine of a line which is
perpendicular to both the lines whose direction ratios are 1,2,2 and 0,2,1 are
A ( cdot frac{-2}{3}, frac{1}{3}, frac{2}{3} )
B. ( frac{2}{3}, frac{-1}{3}, frac{2}{3} )
c. ( frac{2}{3}, frac{1}{3}, frac{-2}{3} )
D. ( frac{2}{3}, frac{-1}{3}, frac{-2}{3} )
12
69Three lines are given by ( vec{r}=lambda hat{i}, lambda epsilon R ) ( overrightarrow{boldsymbol{r}}=boldsymbol{mu}(hat{boldsymbol{i}}+hat{boldsymbol{j}}), boldsymbol{n} boldsymbol{epsilon} boldsymbol{R} ) and ( overrightarrow{boldsymbol{r}}=boldsymbol{v}(hat{boldsymbol{i}}+hat{boldsymbol{j}}+ )
( hat{boldsymbol{k}}), boldsymbol{v} boldsymbol{epsilon} boldsymbol{R} )
Let the lines cut the plane ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}= )
1 at he points ( A, B ) and ( C ) respectively.
If the area of the triangle ( A B C ) is ( triangle )
then the value of ( (6 triangle)^{2} ) equals
A . 0.75
в. ( 0 . )
( c .0 .85 )
D. 0.65
12
70Find the distance of a point (3,-5) from the line ( 3 x-4 y-5=0 )12
71The direction cosines of a vector ( hat{boldsymbol{i}}+ ) ( hat{boldsymbol{j}}+sqrt{mathbf{2}} hat{boldsymbol{k}} ) are
A ( cdot frac{1}{2}, frac{1}{2}, 1 )
B. ( frac{1}{sqrt{2}}, frac{1}{sqrt{2}}, frac{1}{2} )
c. ( frac{1}{2}, frac{1}{2}, frac{1}{sqrt{2}} )
D. ( frac{1}{sqrt{2}}, frac{1}{sqrt{2}}, frac{1}{sqrt{2}} )
12
72Find the equation of plane with intercepts 2,3 and 4 on the ( x, y ) and ( z ) axis respectively.12
73If the extremities of a diagonal of a square are (1,-2,3) and (4,2,3) then the area of the square is
A . 25
B. 50
c. ( frac{25}{2} )
D. ( sqrt{50} )
12
74The points ( (10,7,0),(6,6-1) ) and
(6,9,-4) form a
A. Right -angled triangle
B. Isosceles triangle
( c cdot ) Both (1)( &(2) )
D. Equilateral triangle
12
75If ( bar{a}, bar{b} ) and ( bar{c} ) are non-zero non collinear vectors and ( theta(neq 0, pi) ) is the angle between ( bar{b} ) and ( bar{c} ) if ( (bar{a} times bar{b}) times bar{c}=frac{1}{2}|bar{b}| bar{c} mid bar{a} )
then ( sin theta= )
A ( cdot sqrt{frac{2}{3}} )
B. ( frac{sqrt{3}}{2} )
( c cdot frac{4 sqrt{2}}{3} )
D. ( frac{2 sqrt{2}}{3} )
12
76The point which is equidistant from the points (-1,1,3),(2,1,2),(0,5,6) and
(3,2,2) is
A ( cdot(-1,3,4) )
в. (3,1,4)
c. (1,3,4)
D. (4,1,3)
12
77Find the coordinates of the point where the line ( frac{boldsymbol{x}+mathbf{1}}{mathbf{2}}=frac{boldsymbol{y}+boldsymbol{2}}{mathbf{3}}=frac{boldsymbol{z}+mathbf{3}}{mathbf{4}} ) meets
the plane ( boldsymbol{x}+boldsymbol{y}+boldsymbol{4} boldsymbol{z}=boldsymbol{6} )
12
78Number of lines is space which are equally inclined to three co-ordinate
axes are?
A .2
B. 4
( c .6 )
D. 8
12
79A cube of side 5 has one vertex at the
point ( (1,0,-1), ) and the three edges from this vertex are, respectively, parallel to the negative ( x ) and ( y ) axes and positive z-axis. Find the coordinates of the other
vertices of the cube.
A. (1,0,1)
B. (0,-1,0)
c. (0,0,-1)
D. (1,0,0)
12
8045.
The two lines x = ay+b, z= cy+d; and x = a’y+b’,
[2006||
z=c’y+d’ are perpendicular to each other if
(a) aa’+cc’=-1 (b) aa’t.cc’ = 1
(c) 9+ =-1 (d) 9+6=1
12
81Distance between two parrallel lines, ( overline{boldsymbol{r}}=overline{boldsymbol{a}}_{1}+boldsymbol{lambda} overline{boldsymbol{b}} ) and ( overline{boldsymbol{r}}=overline{boldsymbol{a}}_{2}+boldsymbol{mu} overline{boldsymbol{b}}, ) is given
by
A ( cdot d=mid frac{left(bar{a}_{2}-bar{a}_{1}right)}{hat{b}} )
B . ( d=midleft(bar{a}_{2}-bar{a}_{1}right) times hat{b} )
c. ( d=midleft(bar{a}_{2}+bar{a}_{1}right) times hat{b} )
D . ( d=midleft(bar{a}_{2}-bar{a}_{1}right) )
12
82If a line ( O P ) of length ( r ) (Where ‘ ( O ) ‘ is the origin) makes an angle ( alpha ) with ( x ) -axis
and lies on the xz-plane, then what are the coordinates of ( P ? )
A ( cdot(r cos alpha, 0, r sin alpha) )
B . ( (0,0, r sin alpha) )
( mathbf{c} cdot(r cos alpha, 0,0) )
D ( cdot(0,0, r cos alpha) )
12
83Find the equation of the plane passing through the points (2,3,-4) and (1,-1,3) and parallel to the ( x- ) axis.12
84Find the angle between the line whose
direction cosines are given by ( l+m+ )
( boldsymbol{n}=mathbf{0} ) and ( l^{2}+boldsymbol{m}^{2}=boldsymbol{n}^{2} )
12
8563.
If the angle between the line x=>
lin _y-1
Z-3
– and the plane
2
x +2y + 33 =4 is cos” (193), then aequals
x + 2y + 3z=4 is cos-1
, then a equals
12011
[2011]
12
86The intercepts of the plane ( 2 x-3 y+ )
( mathbf{5} z-mathbf{3 0}=mathbf{0} ) are
A. 15,-10,6
в. 5,10,6
c. ( 1 / 8,-1 / 6,1 / 4 )
D. 3,-4,6
12
87If ( vec{P}(1,5,4) ) and ( vec{Q}(4,-1,-2), ) find the direction ratio of ( overrightarrow{P Q} )12
8839. Equation of the plane containing the straight line
and perpendicular to the plane containing the
(2010)
x y z
straight lines – ===
3 4 2
(a) x+2y – 2z=0
(c) x-2y+z=0
x y z
is
2
(b) 3x + 2y – 2z=0
(d) 5x + 2y – 4z=0
12
89( A ) point ( C ) with position vector ( frac{3 a+4 b-5 c}{3} )
(where ( a, b ) and ( c ) are non co-planar
vectors) divides the line joining ( A ) and ( B ) in the ratio ( 2: 1 . ) If the position vector
of ( A ) is ( a-2 b+3 c, ) then the position
vector of ( boldsymbol{B} ) is
A ( .2 a+3 b-4 c )
B . ( 2 a-3 b+4 c )
c. ( 2 a+3 b+4 c )
D. ( a+3 b-4 c )
12
90The coordinates of a point which is equidistant from the point ( (0,0,0),(a, 0,0),(0, b, 0) ) and ( (0,0, c) )
are given by
( ^{mathbf{A}} cdotleft(frac{a}{2}, frac{b}{2}, frac{c}{2}right) )
в. ( left(frac{-a}{2}, frac{-b}{2}, frac{c}{2}right) )
( ^{mathrm{c}}left(frac{a}{2}, frac{-b}{2}, frac{-c}{2}right) )
D ( cdotleft(frac{-a}{2}, frac{b}{2}, frac{-c}{2}right) )
12
91Find the magnitude of the shortage
distance between the lines ( , frac{x-8}{3}= )
( frac{y+9}{-16}=frac{z-10}{7} ; frac{x-15}{3}=frac{y-29}{8}=frac{z-5}{-5} )
12
92A line making angles ( 45^{circ} ) and ( 60^{circ} ) with the positive direction of ( x- ) axis and
( y- ) axis respectively. Then the angle made by the line with positive direction
of ( z- ) axis is
A ( .60^{circ} )
B. ( 120^{circ} )
( mathbf{c} cdot 60^{circ} ) or ( 120^{circ} )
D. None of these
12
93The equation to the altitude of the
altitude triangle formed by
( (1,1,1) cdot(1,2,3),(2,-1,1) ) through
(1,1,1) is
A ( cdot bar{r}=(bar{i}+bar{j}+bar{k})+t(bar{i}-bar{j}-2 bar{k}) )
B ( cdot bar{r}=(bar{i}-bar{j}+bar{k})+t(bar{i}+bar{j}-2 bar{k}) )
( mathbf{c} cdot bar{r}=(bar{i}+bar{j}+bar{k})+t(bar{i}-bar{j}+2 bar{k}) )
D ( cdot bar{r}=(bar{i}-bar{j}-bar{k})+t(bar{i}+bar{j}-2 bar{k}) )
12
94Cartesian equation of the plane ( bar{r}= ) ( (1+lambda-mu) bar{i}+(2-lambda) bar{j}+(3-2 lambda+ )
( 2 mu) bar{k} ) is :
A ( .2 x+y=5 )
в. ( 2 x-y=5 )
c. ( 2 x+z=5 )
D. ( 2 x-mathrm{z}=5 )
12
95Unit vector perpendicular to the plane passing through the points ( hat{mathbf{i}}-hat{mathbf{j}}+ ) ( 2 hat{k}, 2 hat{i}-hat{k} ) and ( 2 hat{j}+hat{k} ) is
A ( cdot frac{2 hat{i}-hat{j}+hat{k}}{sqrt{6}} )
( ^{text {В } cdot frac{2 hat{i}+hat{j}+hat{k}}{sqrt{6}}} )
c. ( frac{2 hat{i}+hat{j}-hat{k}}{sqrt{6}} )
D. None of these
12
96Find the direction cosines of two lines
which are connected by the relations ( l+m+n=0 ) and ( m n-2 n l-2 l m= )
( mathbf{0} )
12
97Line ( overrightarrow{boldsymbol{r}}=(hat{boldsymbol{i}}-hat{boldsymbol{j}}+hat{boldsymbol{k}})+boldsymbol{t}(boldsymbol{2} hat{boldsymbol{i}}-hat{boldsymbol{j}}+hat{boldsymbol{k}}) )
contained in a plane to which vector ( vec{n}=3 hat{i}-2 hat{j}+lambda hat{k} ) is normal. Find the
value of ( lambda ). Also find the vector equation
of the plane.
12
98Three vertices of a tetrahedron are
(0,0,0),(6,-5,-1) and ( (-4,1,3) . ) If the centroid of the tetrahedron be (1,-2,5)
then the fourth vertex is
A. (2,-4,18)
в. (2,-4,-18)
( ^{c} cdotleft(frac{3}{4}, frac{-3}{2}, frac{7}{4}right) )
D. none of these
12
99Assertion
If a line makes angles ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} ) with
( O X, O Y, O Z ) respectively, then ( sin ^{2} alpha+sin ^{2} beta+sin ^{2} gamma=2 )
Reason
If ( l=cos alpha, m=cos beta, n=cos gamma, ) are
direction cosines of a line, then ( l^{2}+ )
( boldsymbol{m}^{2}+boldsymbol{n}^{2}=mathbf{1} )
A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
B. Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion
C. Assertion is true but Reason is false
D. Assertion is false but Reason is true
12
100The plane ( x=0 ) divides the joinning of (-2,3,4) and (1,-2,3) in the ratio
A .2: 1
B. 1: 2
c. 3: 2
D. -4: 3
12
101The image of the line ( frac{boldsymbol{x}-mathbf{1}}{mathbf{3}}=frac{boldsymbol{y}-mathbf{3}}{mathbf{1}}= )
( frac{z-4}{-5} ) in the plane ( 2 x-y+z+3=0 )
is the line?
A. ( frac{x-3}{3}=frac{y+5}{1}=frac{z-2}{-5} )
в. ( frac{x-3}{-3}=frac{y+5}{-1}=frac{z-2}{5} )
c. ( frac{x+3}{3}=frac{y-5}{1}=frac{z-2}{-5} )
D. ( frac{x+3}{-3}=frac{y-5}{-1}=frac{z+2}{5} )
12
102If a line makes angles ( alpha, beta, gamma ) with the
coordinate axes, then the value of
( cos 2 alpha+cos 2 beta+cos 2 gamma ) is
( A cdot 3 )
B. – –
( c cdot 2 )
D. –
12
103( begin{array}{ll}text { List I } & text { List II } \ text { 1) d.c’s of } x-text { axis } & text { a) }(1,1,1) \ text { 2) d.c’s of } y-text { axis } & text { b) } \ text { 3) d.c’s of } z-text { axis } & left(frac{1}{sqrt{3}} frac{1}{sqrt{3}}, frac{1}{sqrt{3}}right) \ begin{array}{l}text { 4) d.c’s of a line makes equal } \ text { angles with axes }end{array} & text { c) }(1,0,0) \ begin{array}{l}text { d) }(0,1,0) \ text { e) }(0,0,1)end{array}end{array} ) The correct order for 1,2,3,4 is
( mathbf{A} cdot c, d, e, b )
B. ( a, b, c, e )
( mathbf{C} cdot c, d, a, b )
D. ( b, c, a, e )
12
104If the centroid of the tetrahedron
( O A B C, ) where ( A, B, C ) are given by ( (alpha, 5,6),(1, beta, 4),(3,2, gamma) ) respectively
be ( 1,-1,2, ) then value of ( alpha^{2}+beta^{2}+gamma^{2} )
equals
A ( cdot alpha^{2}+beta^{2} )
B. ( gamma^{2}+beta^{2} )
c. ( alpha^{2}+gamma^{2} )
D. None of these
12
105The shortest distance between the lines ( frac{boldsymbol{x}-mathbf{5}}{mathbf{4}}=frac{boldsymbol{y}-mathbf{7}}{-mathbf{5}}=frac{boldsymbol{z}+mathbf{3}}{-mathbf{5}} ) and ( frac{boldsymbol{x}-mathbf{8}}{mathbf{4}}= )
( frac{y-7}{-5}=frac{z-5}{-5} ) is
A . 45
B . 46
c. 47
D. 48
12
106If ( overrightarrow{mathbf{A}} times overrightarrow{mathbf{B}}=overrightarrow{mathbf{B}} times overrightarrow{mathbf{A}}, ) then the angle
between ( A ) and ( B ) is
A . ( pi )
в. ( pi / 3 )
c. ( pi / 2 )
D . ( pi / 4 )
12
107( boldsymbol{A}=(-1,2,-mathbf{3}), boldsymbol{B}=(mathbf{5}, mathbf{0},-mathbf{6}), boldsymbol{C}= )
(0,4,-1) are the vertices of a triangle.
The d.c’s of the internal bisector of ( angle mathrm{BAC} )
are?
( ^{mathbf{A}} cdotleft(frac{25}{sqrt{714}}, frac{-8}{sqrt{714}}, frac{-5}{sqrt{714}}right) )
в. ( left(frac{5}{sqrt{74}}, frac{6}{sqrt{74}}, frac{8}{sqrt{74}}right) )
( ^{mathbf{C}} cdotleft(frac{25}{sqrt{714}}, frac{8}{sqrt{714}}, frac{5}{sqrt{714}}right) )
D. ( left(frac{-5}{sqrt{74}}, frac{6}{sqrt{74}}, frac{-8}{sqrt{74}}right) )
12
108Equation of plane parallel to ( 3 x+4 y+ )
( mathbf{5} boldsymbol{z}-mathbf{6}=mathbf{0}, mathbf{6} boldsymbol{x}+mathbf{8} boldsymbol{y}+mathbf{1 0} boldsymbol{z}-mathbf{1 6}=mathbf{0} )
and equidistant from them is
A. ( 3 x+4 y+5 z=7 )
B. ( 3 x+4 y+5 z=10 )
c. ( 6 x+8 y+10 z=0 )
D. ( 6 x+8 y+10 z=3 )
12
109Assertion (A): The points ( boldsymbol{A}(mathbf{2}, mathbf{9}, mathbf{1 2}), boldsymbol{B}(mathbf{1}, mathbf{8}, mathbf{8}), boldsymbol{C}(mathbf{2}, mathbf{1 1}, mathbf{8}) boldsymbol{D}(mathbf{1}, mathbf{1 2} )
are the vertices of a rhombus
Reason ( (mathrm{R}): A B=B C=C D=D A )
and ( boldsymbol{A C}=boldsymbol{B D} )
A. Both A and R are individually true and R is the correct explanation of
B. Both A and R individually true but R is not the correct explanation of A
c. ( A ) is true but ( R ) is false
D. Both A and R false
12
110Two equat ions ( vec{r} . vec{n}_{1}=q_{1} ) and ( vec{r} . vec{n}_{2}=q_{2} )
represent two perpendicular planes, where ( vec{n}_{1} ) and ( vec{n}_{2} ) are two unit vectors.
One of these plane is rotated through an angle 45 about line of intersection of
two given planes then equation of plane in new position can be This question has multiple correct options
A ( cdot vec{r} cdotleft(vec{n}_{1}+vec{n}_{2}right)=q_{1}-q_{2} )
В ( cdot vec{r} cdot(overrightarrow{n_{1}}+overrightarrow{n_{2}})=q_{1}+q_{2} )
c. ( vec{r} .left(vec{n}_{1}-vec{n}_{2}right)=q_{1}+q_{2} )
D ( cdot vec{r} cdot(overrightarrow{n_{1}}-overrightarrow{n_{2}})=q_{1}-q_{2} )
12
11113.
The shortest distance from the plane 12x+4y+3z =327
to the sphere x2 + y2 + z2 + 4x – 2y – 6z =155 is
(2) 39
(b) 26
(c) 11
(d) 13
12
11225.
-3
y-k _Z intersect, then
If the lines
x-1 y+1 Z-1
2 3 4 ”
the value of k is
(a) 3/2 (b) 9/2
and
2
1
(c) – 2/9
(2004S)
(d) – 3/2
12
113( boldsymbol{P}(mathbf{0}, mathbf{5}, mathbf{6}), boldsymbol{Q}(mathbf{1}, mathbf{4}, mathbf{7}), boldsymbol{R}(mathbf{2}, mathbf{3}, mathbf{7}) ) and
( S(3,5,16) ) are four points in the space.
The point nearest to the origin ( boldsymbol{O}(mathbf{0}, mathbf{0}, mathbf{0}) )
is
A. ( P )
B. ( Q )
( c . R )
D. ( S )
12
114Show that the lines whose d.c’s are
given by ( 2 l+2 m-n=0, m n+n l+ )
( l m=0 ) are perpendicular to each other
12
115The coordinates of any point, which lies
on ( x ) axis are
A ( .(0, x, 0) )
в. ( (x, 0,0) )
c. ( (x, x, 0) )
D. ( (x, x, x) )
12
116If ( boldsymbol{A}=(mathbf{1}, mathbf{2}, mathbf{3}), boldsymbol{B}=(mathbf{2}, mathbf{3}, mathbf{4}) ) and ( boldsymbol{A} boldsymbol{B} ) is
produced upto ( C ) such that ( 2 A B=B C )
then ( C= )
A. (5,4,6)
в. (6,2,4)
c. (4,5,6)
D. (6,4,5)
12
117If ( P(x, y, z) ) is a point on the line
segment joining ( Q(2,2,4) ) and
( boldsymbol{R}(boldsymbol{3}, boldsymbol{5}, boldsymbol{6}) ) such that the projection of
( O P ) on the axes are ( frac{13}{5}, frac{19}{5}, frac{26}{5} )
respectively, then ( P ) divides ( Q R ) in the ratio
A .1: 2
B. 3: 2
( c cdot 2: 3 )
D. 1: 3
12
118Find the intersection of the line ( x- )
( mathbf{2} boldsymbol{y}+mathbf{4} boldsymbol{z}+mathbf{4}=mathbf{0}, boldsymbol{x}+boldsymbol{y}+boldsymbol{z}-mathbf{8}=mathbf{0} )
with the plane ( boldsymbol{x}-boldsymbol{y}+mathbf{2} boldsymbol{z}+mathbf{1}=mathbf{0} )
12
119The coordinates of a point ( mathrm{P} ) are (3,12,4) w.r.t origin ( 0, ) then the direction cosines of ( O P ) are
( mathbf{A} cdot 3,12,4 )
B. ( frac{1}{4}, frac{1}{3}, frac{1}{2} )
c. ( frac{3}{sqrt{13}}, frac{1}{sqrt{13}}, frac{2}{sqrt{13}} )
D. ( frac{3}{13}, frac{12}{13}, frac{4}{13} )
12
120The angle between vectors ( (bar{M} times bar{N}) ) and ( (overline{boldsymbol{N}} times overline{boldsymbol{M}}) ) is then
( mathbf{A} cdot 0^{circ} )
B . ( 60^{circ} )
( c .90^{circ} )
D. ( 180^{circ} )
12
121If the plane ( 3 x+2 y+6 z=6 )
intersects the coordinate axes at
( A, B, C ) then the area of the ( Delta A B C ) is
( mathbf{A} cdot 49 )
B. 7
( c cdot frac{7}{2} )
D. ( frac{11}{2} )
12
122Arrange the points: ( mathbf{A}(1,2- ) ( mathbf{3}), mathbf{B}(-mathbf{1}, mathbf{2},-mathbf{3}), mathbf{C}(-mathbf{1},-mathbf{2}-mathbf{3}) ) and
( mathbf{D}(mathbf{1},-mathbf{2},-mathbf{3}) ) in the increasing order of
their octant numbers:
A. ( A, B, C, D )
в. ( B, C, D, A )
c. ( C, D, A, B )
D. ( D, C, B, A )
12
123Distance of the point ( boldsymbol{P}(overrightarrow{boldsymbol{p}}) ) from the line ( vec{r}=vec{a}+lambda vec{b} ) is –
A ( cdot(vec{a}-vec{p})+frac{((vec{p}-vec{a}) cdot vec{b}) vec{b}}{|vec{b}|^{2}} mid )
B. ( (vec{b}-vec{p})+frac{((vec{p}-vec{a}) cdot vec{b}) vec{b}}{|vec{b}|^{2}} mid )
c. ( quadleft|(vec{a}-vec{p})+frac{((vec{p}-vec{b}) cdot vec{b}) vec{b}}{|vec{b}|^{2}}right| )
D. None of these.
12
124A line makes angle ( theta_{1}, theta_{2}, theta_{3}, theta_{4} ) with the
diagonals of the cube. Show that ( cos ^{2} theta_{1}+cos ^{2} theta_{2}+cos ^{2} theta_{3}+cos ^{2} theta_{4}= )
( frac{4}{3} ? )
12
125if a line makes angles ( alpha, beta, gamma, delta ) with
four diagonals a cube then value of
( sin ^{2} alpha+sin ^{2} beta+sin ^{2} gamma+sin ^{2} delta ) equals
( A cdot 2 )
B. ( frac{4}{3} )
( c cdot frac{8}{3} )
D.
12
126Derive the coordinates of the points ( R(x, y, z) ) dividing the line joining the
points ( boldsymbol{P}left(boldsymbol{x}_{1}, boldsymbol{y}_{1}, boldsymbol{z}_{1}right) ) and ( boldsymbol{Q}left(boldsymbol{x}_{2}, boldsymbol{y}_{2}, boldsymbol{z}_{2}right) )
internally in the ratio ( m: n )
12
127Find the value of ( lambda ) for which the four
points ( A, B, C, D ) with position vectors ( -widehat{boldsymbol{j}}-widehat{boldsymbol{k}} ; boldsymbol{4} hat{boldsymbol{i}}+boldsymbol{5} hat{boldsymbol{j}}+boldsymbol{lambda} hat{boldsymbol{k}} ; boldsymbol{3} hat{boldsymbol{i}}+boldsymbol{9} hat{boldsymbol{j}}+boldsymbol{4} widehat{boldsymbol{k}} ) and
( -4 hat{i}+4 widehat{j}+4 widehat{k} ) are coplanar.
12
128If direction cosines of two lines are
proportional to (2,3,-6) and (3,-4,5) then the acute angle between them is
( ^{mathbf{A}} cdot cos ^{-1}left(frac{49}{36}right) )
B. ( cos ^{-1}left(frac{18 sqrt{2}}{35}right) )
( c cdot 96^{circ} )
D. ( cos ^{-1}left(frac{18}{35}right) )
12
129A non-zero vector ( vec{a} ) is parallel to the line
of intersection of the plane determined by the vectors ( hat{i}, hat{i}+hat{j} ) and the plane determined by the vectors ( hat{i}-hat{j}, hat{i}+hat{k} ) The angle between ( vec{a} ) and ( hat{i}-2 hat{j}+2 hat{k} ) is
A.
в.
c.
D. ( frac{pi}{2} )
12
130The intercept made by the plane ( vec{r} cdot vec{n}= ) ( q ) on the ( x ) -axis is
A ( cdot frac{q}{hat{i} cdot vec{n}} )
в. ( frac{hat{i} cdot vec{n}}{q} )
( c cdot frac{hat{i} cdot q}{n} )
D. ( frac{q}{|vec{n}|} )
12
131The point ( P ) is the intersection of the
straight line joining the points ( Q(2,3,5) ) and ( R(1,-1,4) ) with the plane
( 5 x-4 y-z=1 . ) If ( S ) is the foot of the
perpendicular drawn from the point
( T(2,1,4) ) to ( Q R, ) then the length of the
line segment ( P S ) is
A ( cdot frac{1}{sqrt{2}} )
B. ( sqrt{2} )
( c cdot 2 )
D. ( 2 sqrt{2} )
12
132Equation of the plane containing the ( operatorname{lines} overline{boldsymbol{r}}=(overline{boldsymbol{i}}-boldsymbol{2} overline{boldsymbol{j}}+overline{boldsymbol{k}})+boldsymbol{t}(overline{boldsymbol{i}}+mathbf{2} overline{boldsymbol{j}}-overline{boldsymbol{k}}) )
( boldsymbol{boldsymbol { r }}=(overline{boldsymbol{i}}+mathbf{2} overline{boldsymbol{j}}-overline{boldsymbol{k}})+boldsymbol{s}(overline{boldsymbol{i}}+overline{boldsymbol{j}}+mathbf{3} overline{boldsymbol{k}}) ) is
A. ( bar{r}(7 bar{i}-4 bar{j}-bar{k})=14 )
В. ( bar{r}(bar{i}+2 bar{j}-bar{k})=10 )
c. ( bar{r}(bar{i}+bar{j}+3 bar{k})=20 )
D. ( bar{r}(bar{i}-2 bar{j}+bar{k})=27 )
12
13318.
A tetrahedron has vertices at O(0, 0, 0), A(1,2,1) B(2,1,3)
and C(-1,1,2). Then the angle between the faces OAB and
ABC will be
[2003]
(a)
90°
(b) cos-
(c) cos-1( 37
(d) 30°
31
12
134Find the length and foot of the perpendicular from the point ( (mathbf{7}, mathbf{1 4}, mathbf{5}) ) to the plane ( 2 x+4 y-z=2 )12
135The circum centre of the triangle formed by the points (2,5,1),(1,4,-3) and (-2,7,-3) is
A. (6,0,1)
()
В. (0,6,-1)
c. (-1,6,2)
D. (6,1,-2)
12
136Graph ( x^{2}+y^{2}=4 ) in ( 3 D ) looks like
A . Circle
B. Cylinder
c. Hemisphere
D. sphere
12
137Write the equations for the ( x ) -and ( y )
axes.
12
13816.
The image of the point (-1,3,4) in the plane x-2y = 0 is
(
17
19
41
(b) (15,11,4)
(2006)
10
U
(d) None of these
12
139Find the image of :
(-2,3,4) in the ( y z ) -plane
12
140The distance of the point (1,-2,3) from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} ) measured
parallel to the line ( frac{x}{2}=frac{y}{3}=frac{z-1}{-6} ) is
A .
B . 2
( c cdot 4 )
D. None of these
12
141ff ( left(x_{1}, y_{1}, z_{1}right) ) and ( Bleft(x_{2}, y_{2}, z_{2}right) ) are
two points such that the direction ( operatorname{cosines} ) of ( A B ) are ( l, m, n ) then
( l=frac{x_{2}-x_{1}}{|A B|}, m=frac{y_{2}-y_{1}}{|A B|}, n=frac{z_{2}-z_{1}}{|A B|} )
12
142Four vertices of a tetrahedron are
(0,0,0),(4,0,0),(0,-8,0) and
( (0,0,12) . ) Its centroid has the
coordinates
A ( cdotleft(frac{4}{3},-frac{8}{3}, 4right) )
В. (2,-4,6)
c. (1,-2,3)
a 5
D. none of these
12
143If ( boldsymbol{A}=(-2,3,4), B=(1,2,3) ) are two
points and ( P ) is the point of intersection
of ( A B ) and ( z x ) -plane, then ( P_{x}+P_{y}+ )
( boldsymbol{P}_{z}= )
( mathbf{A} cdot mathbf{6} )
B. -8
c. 8
D.
12
144If the points ( (1,1, p) ) and (-3,0,1) be equidistant from the plane ( vec{r} .(3 hat{i}+ ) ( 4 hat{j}-12 hat{k})+13=0, ) then find the value
of ( p )
12
145Find the ratio in which YZ-plane divides the line joining ( A(2,4,5) ) and ( B(3,5,-4) . ) Also find the point of
intersection.
12
146( operatorname{Points} boldsymbol{A}(boldsymbol{3}, boldsymbol{2}, boldsymbol{4}), boldsymbol{B}left(frac{boldsymbol{3} boldsymbol{3}}{boldsymbol{5}}, frac{boldsymbol{2} boldsymbol{8}}{boldsymbol{5}}, frac{boldsymbol{3} boldsymbol{8}}{boldsymbol{5}}right), ) and
( C(9,8,10) ) are given. The ratio in which
( B ) divides ( overline{A C} ) is
A . 5: 3
B . 2: 1
c. 1: 3
D. 3: 2
12
147The image of the point (-1,3,4) in the
plane ( boldsymbol{x}-mathbf{2} boldsymbol{y}=mathbf{0} ) is
A ( cdot(15,11,4) )
в. ( left(frac{9}{5},-frac{13}{5}, 4right) )
( c cdot(8,4,4) )
D. None of these
12
14828. P, and P, are planes passing through origin. L, and L, are
two line on P, and P2 respectively such that their
intersection is origin. Show that there exists points A, B, C,
whose permutation A’, B’, C can be chosen such that (i) Ais
on L,, B on P, but not on L, and C not on P, (ii) A’ is on L,,
B’ on P, but not on L and C not on P2
12
14926.
Find the equation of plane passing through (1,1,1) & parallel
to the lines L,, L, having direction ratios (1,0,-1), (1,-1,0)
Find the volume of tetrahedron formed by origin and the
points where these planes intersect the coordinate axes.
(2001.2 Mau
12
1506.
A line makes the same angle , with each of the x and z axis.
If the angle ß, which it makes with y-axis, is such that
sin B = 3 sin?e, then cos2e equals
[2004]
т.
12
151Find the coordinates of the point ( boldsymbol{P} )
which divides the join of ( boldsymbol{A}(-2,5) ) and ( B(3,-5) ) in the ratio 2: 3
12
152If a line makes an angle of ( frac{pi}{4} ) with the positive direction of each of ( x ) -axis and
( boldsymbol{y} ) -axis, then the angle that the line
makes with the positive direction of ( z )
axis is-
A ( cdot frac{pi}{3} )
в.
c. ( frac{pi}{2} )
D.
12
153The eartesian equations of the line are
( mathbf{3} boldsymbol{x}+mathbf{1}=mathbf{6} boldsymbol{y}-mathbf{2}=mathbf{1}-boldsymbol{z} . ) Find its
equation in vector form and find direction ratios of the line.
12
154The projections of a line segment on ( x, y ) and ( z ) axes are respectively ( sqrt{2}, 3,5 ) The length of the line segment is
( mathbf{A} cdot mathbf{6} )
B. 11
c. 8
D. 5
12
155( operatorname{Given} boldsymbol{A}(mathbf{1},-mathbf{1}, mathbf{0}) ; boldsymbol{B}(mathbf{3}, mathbf{1}, mathbf{2}) )
( C(2,-2,4) ) and ( D(-1,1,-1) ) which of
the following points neither lie on ( boldsymbol{A B} )
nor on ( C D ? )
A ( .(2,2,4) )
В. (2,-2,4)
c. (2,0,1)
D. (0,-2,-1)
12
156Directions ratio of two lines are ( 3,-2, k )
and ( -2, k, 4 . ) Find ( k ) if the lines are perpendicular to each other.
12
157Find the distance of the point ( P(3,4,4) ) from the point, where the line joining the point ( A(3,-4,-5) ) and ( B(2,-3,1) ) intersects the plane ( 2 x+y+z=7 )12
158Two vectors ( vec{A} ) and ( vec{B} ) inclined at an
angle ( theta ) have a resultant ( vec{R} ) which makes an angle ( alpha ) with ( vec{A} ) and angle ( beta ) with ( vec{B} ). Let the magnitudes of the vectors ( vec{A}, vec{B} ) and ( vec{R} ) be represented by ( A ) B and R respectively. Which of the following relations is not correct?
A . Asinalpha ( =B sin beta )
B. ( R sin alpha=B sin (alpha+beta) )
c. ( R sin beta=operatorname{Asin}(alpha+beta) )
D. None of these
12
159( A=(2,3,0) ) and ( B=(2,1,2) ) are two
points. If the points ( P, Q ) are on the line
( A B ) such that ( A P=P Q=Q B, ) then
( boldsymbol{P Q}= )
( A cdot 2 sqrt{2} )
B. ( 6 sqrt{2} )
( c cdot sqrt{frac{8}{9}} )
D. ( sqrt{2} )
12
160Point, Plane: ( (0,0,0), 3 x-4 y+12 z= )
3
12
161The angle between two diagonals of a cube is.
A ( .30^{circ} )
B . ( 45^{circ} )
c. ( cos ^{-1}left(frac{1}{3}right) )
D. ( cos ^{-1}left(frac{1}{sqrt{3}}right) )
12
162If direction numbers of two lines are
( a, b, c ) and ( b-c, c-a, a-b ) prove that
they are perpendicular to each other.
12
163The direction cosines of a line equally
inclined to three mutually perpendicular lines having D.C.’s as
( ell_{1} m_{1} n_{1}: ell_{2} m_{2} n_{2}: ell_{3} m_{3} n_{3} ) are
A. ( l_{1}+l_{2}+l_{3}, m_{1}+m_{2}+m_{3}, n_{1}+n_{2}+n_{3} )
B ( cdotleft(pm frac{1}{sqrt{3}}, pm frac{1}{sqrt{3}}, pm frac{1}{sqrt{3}}right) )
( ^{mathbf{c}} cdotleft(pm frac{1}{sqrt{2}}, pm frac{1}{sqrt{3}}, pm frac{1}{sqrt{4}}right) )
D. none of these
12
164The direction cosines of the line which
is perpendicular to the lines with direction cosines proportional to
(1,-2,-2)( &(0,2,1) ) are
( ^{mathrm{A}} cdotleft(frac{2}{3},-frac{1}{3}, frac{2}{3}right) )
в. ( left(frac{2}{3}, frac{1}{3}, frac{2}{3}right) )
c. ( left(frac{2}{3}, frac{1}{3}, frac{-2}{3}right) )
D. ( left(frac{-2}{3}, frac{1}{3}, frac{2}{3}right) )
12
165A plane ( pi ) makes intercept 3 and 4
respectively on z-axis and x-axis. If ( pi ) is
parallel to y-axis, then its equation is
A. ( 3 x+4 z=12 )
B. ( 3 z+4 x=12 )
c. ( 3 y+4 z=12 )
D. ( 3 z+4 y=12 )
12
166Distance between ( boldsymbol{A}(mathbf{4}, mathbf{5}, mathbf{6}) ) from origin
( boldsymbol{O} ) is
A ( cdot 25 sqrt{3} )
B. ( sqrt{77} )
c. ( 3 sqrt{5} )
D. Data Insufficient
12
167If ( vec{A}=-4 hat{i}+3 hat{j} ) and ( vec{B}=2 hat{i}+5 hat{j} ) and
( vec{C}=vec{A} times vec{B} ) then ( vec{C} ) makes an angle of :
A ( cdot 45^{0} ) with ( mathrm{x} ) -axis
B. ( 180^{circ} ) with ( Y ) -axis
( c cdot 0^{0} ) with ( mathrm{z} ) -axis
D. ( 180^{circ} ) with ( z ) -axis
12
168Show that ( A(3,-2) ) is a point trisection of the line segment joining the points
(2,1) and (5,-8)
Also find the co-ordinates of the other
points of trisections.
12
169A point on the line ( frac{boldsymbol{x}-mathbf{1}}{mathbf{1}}=frac{boldsymbol{y}-mathbf{2}}{mathbf{2}}= )
( frac{z+1}{3} ) at a distance ( sqrt{6} ) from the origin
is
This question has multiple correct options
( mathbf{A} cdotleft(frac{-5}{7}, frac{-10}{7}, frac{13}{7}right) )
в. (1,2,-1)
( ^{mathbf{C}} cdotleft(frac{5}{7}, frac{10}{7}, frac{-13}{7}right) )
D. (-1,-2,1)
12
170Let the co – ordinates of the point where the line joining the points
(2,-3,1),(3,-4,-5) cuts the plane
( mathbf{2} boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=mathbf{7} ) be ( (boldsymbol{x}, boldsymbol{y}, boldsymbol{z}) . ) Find ( boldsymbol{x}+ )
( boldsymbol{y}+boldsymbol{z} ? )
12
171The plane ( X O Z ) divides the join of (1,-1,5) and (2,3,4) in the ratio ( lambda: 1 )
then ( lambda ) is
A . -3
B. ( -1 / 3 )
( c cdot 3 )
D. ( 1 / 3 )
12
172The distance of the point ( ,(-1,-5,-10) ) from the point intersection of the line,
( frac{x-2}{3}=frac{y+1}{4}=frac{z-2}{12} ) and the plane ( x- )
( boldsymbol{y}+boldsymbol{z}=mathbf{5}, ) is
( A cdot 13 )
B. 1
c. 12
D. none of these
12
173If ( overrightarrow{P O}+overrightarrow{O Q}=overrightarrow{Q O}+overrightarrow{O R}, ) prove that
the points ( P, Q, R ) are collinear.
12
17417. The radius of the circle in which the sphere
x2 + y2 + 2? + 2x – 2y – 42 -19 = 0 is cut by the plane
x +2y + 2z+ 7 = 0 is
[2003]
(a) 4 (6) 1 (c) 2 (d) 3
12
175Let the vector ( vec{a}, vec{b}, vec{c} ) and ( vec{d} ) be such ( operatorname{that}(overrightarrow{boldsymbol{a}} times overrightarrow{boldsymbol{b}}) times(overrightarrow{boldsymbol{c}} times overrightarrow{boldsymbol{d}})=mathbf{0} cdot operatorname{Let} boldsymbol{P}_{1} )
and ( P_{2} ) be planes determined by the pairs of vectors ( vec{a}, vec{b} ) and ( vec{c}, vec{d} )
respectively then the angle between ( P_{1} )
and ( P_{2} ) is
A. 0
в. ( frac{pi}{4} )
c. ( frac{pi}{3} )
D. ( frac{pi}{2} )
12
176Find the shortest distance between the
skew lines ( r=(6 i+2 j+2 k)+t(i- )
( 2 j+2 k) ) and ( F=(-4 i-k)+s(3 i- )
( 2 j-2 k) ) where s,t are scalars.
12
177Find the direction cosines ( l, m, n ) of a line which are connected by the relation ( l+m-n=0 ) and ( 2 m l-2 m n+n l= )
0
This question has multiple correct options
A ( cdot frac{-2}{sqrt{6}}, frac{1}{sqrt{6}}, frac{-1}{sqrt{6}} )
в. ( frac{2}{sqrt{6}}, frac{-1}{sqrt{6}}, frac{1}{sqrt{6}} )
c. ( frac{-2}{sqrt{6}}, frac{-1}{sqrt{6}}, frac{-1}{sqrt{6}} )
D. ( frac{2}{sqrt{6}}, frac{1}{sqrt{6}}, frac{1}{sqrt{6}} )
12
178Verify the following
(i) (0,7,-10),(1,6,-6) and (4,9,-6) are the vertices of an isosceles triangle
(ii) (0,7,10),(-1,6,6) and (-4,9,6) are the vertices of a right angled triangle
(iii) (-1,2,1),(1,-2,5),(4,-7,8) and (2,-3,4) are the vertices of a parallelogram
12
179Show that the points ( boldsymbol{A}(mathbf{0}, mathbf{1}, mathbf{2}), boldsymbol{B}(mathbf{2},-mathbf{1}, mathbf{3}) ) and ( boldsymbol{C}(mathbf{1},-mathbf{3}, mathbf{1}) )
are vertices of an isosceles right-angled triangle.
12
18047. The equation of the plane passing through the point (1,1,1)
and perpendicular to the planes 2x+y-2z=5 and 3x – 6y-
2z=7, is
(a) 14x +2y-15z=1 (b) 14x – 2y + 15z=27
(c) 14x +2y+ 15z=31 (d) -14x + 2y + 15z=3
12
181The distance of point ( A(-2,3,1) ) from
the PQ through ( P(-3,5,2), ) which makes equal angles with the axes is-
A ( cdot frac{2}{sqrt{3}} )
в. ( sqrt{frac{14}{3}} )
c. ( frac{16}{sqrt{3}} )
D. ( frac{5}{sqrt{3}} )
12
182The cartesian from of equation a line passing through the point position vector ( 2 hat{i}-hat{j}+2 hat{k} ) and is in the direction of ( -2 hat{i}+hat{j}+hat{k}, ) is
A ( frac{x-2}{-2}=frac{y+1}{1}=frac{z-2}{1} )
B. ( frac{x+4}{-2}=frac{y-1}{1}=frac{z+2}{1} )
c. ( frac{x+2}{4}=frac{y-1}{-1}=frac{z-1}{2} )
D. None of these
12
183Find the vector equation of the plane whose cartesian form of equation is ( 3 x- ) ( 4 y+2 z=5 )12
184Column I shows some vector equations. Match Column I with the value of angle between ( vec{A} ) and ( vec{B} ) given in Column II
Column I Column I
12
185The point ( P(x, y, z) ) lies in the first octant and its distance from the origin
is 12 units. If the position vector of ( P )
make ( 45^{circ} ) and ( 60^{circ} ) with the ( x ) -axis and ( y )
axis respectively, then the coordinates of ( boldsymbol{P} ) are
A ( cdot(3 sqrt{3}, 6,3 sqrt{2}) )
B. ( (4 sqrt{3}, 8,4 sqrt{2}) )
c. ( (6 sqrt{2}, 6,6,) )
D. ( (6,6,6 sqrt{2}) )
E ( .(4 sqrt{2}, 8,4 sqrt{3}) )
12
186The projection of a line segment joining
the points ( boldsymbol{P}left(boldsymbol{x}_{1}, boldsymbol{y}_{1}, boldsymbol{z}_{1},right) ) and
( Qleft(x_{1}, y_{1}, z_{1},right) ) on another line whose
DC’s are ( l, m, n ) is given by This question has multiple correct options
A ( cdot lleft(x_{1}+x_{2}right)+mleft(y_{2}+y_{2}right)+nleft(z_{1}+z_{2}right) )
B. ( 2left[frac{left(l x_{2}+m y_{2}+n z_{2}right)}{2}-frac{left(l x_{1}+m y_{1}+n z_{1}right)}{2}right. )
c. ( lleft(x_{2}-x_{1}right)+mleft(y_{2}-y_{1}right)+nleft(z_{2}-z_{1}right) )
D. ( frac{x_{2}-x_{1}}{l}+frac{y_{2}-y_{1}}{m}+frac{z_{2}-z_{1}}{n} )
12
187The point ( P ) is on the ( y ) -axis. If ( P ) is equidistant from (1,2,3) and (2,3,4)
then ( boldsymbol{P}_{boldsymbol{y}}= )
A ( cdot frac{15}{2} )
B. 15
c. 30
D. ( frac{3}{2} )
12
188If the planes ( boldsymbol{x}-boldsymbol{b} boldsymbol{z}=mathbf{0}, boldsymbol{c} boldsymbol{x}-boldsymbol{y}+=mathbf{0} )
and ( b x+a y-z=0, ) pass through a
line, then find the value of ( a^{2}+b^{2}+ )
( c^{2}+2 a b c )
A.
B.
c. -1
D. ( frac{1}{2} )
12
189The vector ( vec{P} ) makes ( 120^{circ} ) with the ( x- )
axis and the vector ( vec{Q} ) makes ( 30^{circ} ) with ( boldsymbol{y}- ) axis. What is the resultant vector?
A. ( P+Q )
в. ( P-Q )
c. ( sqrt{P^{2}+Q^{2}} )
D. ( sqrt{P^{2}-Q^{2}} )
12
19090.
The equation of the line passing through (-4, 2,
to the plane x + 2y – Z-5 = 0 and intersecting the
passing through (-4, 3, 1), parallel
-2.
x+l
-3 1 |
y-3
2..
(JEEM 2019-9 Jan (M)
-1
)
12
1
4
() 4 +3 2+1 (Jet 2
() | 13 1
12
191The variable plane ( (2 lambda+1) x+ )
( (3-lambda) y+z=4 ) always passes
through the line
A ( cdot frac{x}{0}=frac{y}{0}=frac{x+4}{1} )
в. ( frac{x}{1}=frac{y}{2}=frac{z}{-3} )
c. ( frac{x}{1}=frac{y}{2}=frac{z-4}{-7} )
D. none of these
12
192Find the point where the line of
intersection of the planes ( x-2 y+z= )
1 and ( x+2 y-2 z=5 ) intersects the
plane ( 3 x+2 y+z+6=0 )
A ( cdot P(1,-2,-4) )
B . ( P(1,2,-4) )
c. ( P(1,-2,4) )
D. None of these
12
19326. The intersection of the spheres
x2 + y2 + z2 + 7x-2y-z = 13 and
x2 + y2 + 22 – 3x + 3y + 4z = 8
is the same as the intersection of one of the sphere and the
plane
[2004]
(a) 2x – y – z=1 (b) x-2y-z=1
(©) x-y–2z=1 (d) x-y-z = 1
12
194The d.rs of the lines ( boldsymbol{x}=boldsymbol{a} boldsymbol{y}+boldsymbol{b}, boldsymbol{z}= )
( boldsymbol{c} boldsymbol{y}+boldsymbol{d} ) are:
A. ( 1, a, c )
B. ( a, 1, c )
c. ( b, 1, c )
D. ( c, a, 1 )
12
195The circum radius of the triangle
formed by the points (1,2,-3),(2,-3,1) and (-3,1,2) is:
A ( cdot sqrt{14} )
B. 14
c. ( sqrt{13} )
D.
12
196If ( hat{i}, hat{j} ) and ( hat{k} ) represents unit vectors
along the ( x, y ) and ( z- ) axes respectively
then find the value of angle ( theta ) between the vectors ( hat{i}+hat{j}+hat{k} ) and ( hat{i}+hat{j} )
12
197Find the shortest distance between the
line ( boldsymbol{x}=mathbf{1}+boldsymbol{t}, boldsymbol{y}=mathbf{1}+boldsymbol{6} boldsymbol{t}, boldsymbol{z}=boldsymbol{2} boldsymbol{t}, boldsymbol{t} in boldsymbol{R} )
and ( boldsymbol{x}=mathbf{1}+mathbf{2 k}, boldsymbol{y}=mathbf{5}+mathbf{1 5 k}, boldsymbol{z}= )
( -2+6 k, k in R )
12
19829
(0) Tu, ,
A plane which is perpendicular to two planes 2x – 2y+z=0
and x-y + 2z= 4, passes through (1, -2, 1). The distance of
the plane from the point (1,2,2) is (2006 – 3M, -1)
(a) o (b) 1 (c) √ (d) 252
12
199A line makes an angle ( theta ) with each of the
( x- ) and ( z^{-} ) axes. If the angle ( beta, ) which it makes with the ( y ) -axis, is such that
( sin ^{2} beta=3 sin ^{2} theta, ) then ( cos ^{2} theta ) equals-
A ( cdot frac{2}{3} )
B. ( frac{1}{5} )
( c cdot frac{3}{5} )
D. ( frac{2}{5} )
12
200Find the vector equation of line joining the points (2,1,3) and (-4,3,-1)
( mathbf{A} cdot bar{r}=2(1-3 lambda) bar{i}-(1+2 lambda) bar{j}-(3-4 lambda) bar{k} )
B . ( bar{r}=2(1-3 lambda) bar{i}-(1+2 lambda) bar{j}+(3-4 lambda) bar{k} )
C ( . bar{r}=2(1-3 lambda) bar{i}+(1+2 lambda) bar{j}+(3-4 lambda) bar{k} )
D. ( bar{r}=2(1+3 lambda) bar{i}+(1+2 lambda) bar{j}+(3+4 lambda) bar{k} )
12
201Show that LHS=RHS i.e. ( frac{2}{9} times 3=frac{2}{9} times ) ( frac{3}{1}=frac{2 times 3}{9 times 1}=frac{6}{9}=frac{2}{3} )12
202A plane meets the axes in ( A, B ) and ( C )
such that centroid of the ( triangle A B C ) is
( (1,2,3) . ) The equation of the plane is
A ( cdot x+frac{y}{2}+frac{z}{3}=1 )
В ( cdot frac{x}{3}+frac{y}{6}+frac{z}{9}=1 )
c. ( x+2 y+3 z=1 )
D. None of these
12
203If the projections of the line segment ( A B ) on the coordinate axes are 2,3,6
then the square of the sine of the angle made by ( boldsymbol{A B} ) with ( boldsymbol{x}=mathbf{0}, ) is
A ( cdot frac{3}{7} )
B. ( frac{3}{49} )
( c cdot frac{4}{7} )
D. ( frac{40}{49} )
12
204Find the equation of the plane in scalar product form ( hat{r}=(2 hat{i}+hat{k})+lambda hat{i}+ )
( mu(hat{i}+2 hat{j}-3 hat{k}) )
12
205If ( boldsymbol{A}=(mathbf{1}, mathbf{2}, mathbf{3}), boldsymbol{B}=(mathbf{2}, mathbf{1 0}, mathbf{1}), boldsymbol{Q} ) are
collinear points and ( Q_{x}=-1, ) then
( boldsymbol{Q}_{z}= )
A . -3
B. 7
( c cdot-14 )
D. –
12
206The extremities of a diagonal of a rectangular parallelopiped whose faces are parallel to the reference planes are (-2,4,6) and ( (3,16,6) . ) The length of the base diagonal is
A . 13
B. ( sqrt{13} )
c. ( 2 sqrt{13} )
D. 169
12
207The coordinates of the points in which the line joining the points (2,5,-7) and (-3,-1,8) are intersected by the ( y-z )
plane are
A ( cdotleft(0, frac{13}{5},-1right) )
B ( cdotleft(0, frac{-13}{5},-2right) )
D. ( left(0, frac{13}{5}, frac{2}{5}right) )
12
208Find the vector and cartesian equations of the plane passing through the points ( A(1,1,-2), B(1,2,1) ) and ( C(2,-1,1) )12
209The projections of a directed line segment on the coordinate axes 12,4,3 The direction cosines of the line are
A ( cdot frac{12}{13},-frac{4}{13}, frac{3}{13} )
B. ( -frac{12}{13},-frac{4}{13}, frac{3}{13} )
c. ( frac{12}{13}, frac{4}{13}, frac{3}{13} )
D. none of these
12
210The direction cosines to two lines at
right angles are (1,2,3) and ( left(-2, frac{1}{2}, frac{1}{3}right), ) then
the direction cosine perpendicular to both given lines are:
A ( cdot sqrt{frac{25}{219}} cdot sqrt{frac{19}{2198}} cdot sqrt{frac{729}{21988}} )
B. ( sqrt{frac{24}{2198} cdot sqrt{frac{38}{2198}}} sqrt{frac{730}{21988}} )
c. ( frac{1}{3},-2, frac{-7}{2} )
D. None of the above
12
211If the points whose position vectors are ( mathbf{2} overline{mathbf{i}}+overline{boldsymbol{j}}+overline{boldsymbol{k}}, mathbf{6} overline{mathbf{i}}-overline{boldsymbol{j}}+mathbf{2} overline{mathbf{k}} ) and ( mathbf{1 4} overline{mathbf{i}}-mathbf{5} overline{mathbf{j}}+ )
( p bar{k} ) are collinear then the value of ( mathbf{p} ) is
( A cdot 2 )
B. 4
( c cdot 6 )
D. 8
12
212A point ( P ) lies on the line whose end
points are ( boldsymbol{A}(mathbf{1}, mathbf{2}, mathbf{3}) ) and ( boldsymbol{B}(mathbf{2}, mathbf{1 0}, mathbf{1}) )
If ( z ) -co-ordinate of ( P ) is ( 7, ) find sum of its
other co – ordinates.
12
213( A=(2,4,5) ) and ( B=(3,5,-4) ) are two
points. If the ( x y ) -plane, ( y z ) -plane divide ( A B ) in the ratios ( a: b, p: q ) respectively then ( frac{a}{b}+frac{p}{q}= )
A. ( frac{7}{15} )
в. ( frac{-7}{12} )
c. ( frac{7}{12} )
D. ( frac{22}{25} )
12
214The ratio in which the line joining points (2,4,5) and (3,5,-4) divide YZ -plane is
( mathbf{A} cdot-2: 3 )
B. 2: 3
c. -3: 2
D. 3: 2
12
215On a plane are two points ( A ) and ( B ) at a
distance of 5 units apart. The number of straight lines in this plane which are at distance of 2 units from ( A ) and 3 units
from ( mathrm{B} ) are:
A . 1
B. 2
( c .3 )
D. 4
12
216Find the equation of the plane passing through the point (1,-2,1) and perpendicular to the line joining the points ( boldsymbol{A}(mathbf{3}, mathbf{2}, mathbf{1}) ) and ( boldsymbol{B}(mathbf{1}, mathbf{4}, mathbf{2}) )12
217Find the shortest distance between the line ( frac{x-3}{3}=frac{y-8}{-1}=frac{z-3}{1} ) and the
line of intersection of the planes ( 2 x+ )
( 5 y-z+47=0 ) and ( 2 x+y+z+7= )
( mathbf{0} )
12
218The projection of a directed line segment on the co-ordinate axes are
( 12,4,3, ) the DC’s of the line are
A ( cdot frac{-12}{13}, frac{-4}{13}, frac{-3}{13} )
B. ( frac{12}{13}, frac{4}{13}, frac{3}{13} )
c. ( frac{12}{13}, frac{-4}{13}, frac{3}{13} )
D. ( frac{12}{13}, frac{4}{13}, frac{-3}{13} )
12
219The plane ( 2 x-(1+lambda) y+3 z=0 )
passes through the intersection of the
planes
A. ( 2 x-y=0 ) and ( y+3 z=0 )
в. ( 2 x-y=0 ) and ( y-3 z=0 )
c. ( 2 x+3 z=0 ) and ( y=0 )
D. None of the above
12
220If ( x y ) -plane and ( y z ) -plane divides the line segment joining ( A(2,4,5) ) and ( B(3,5,-4) ) in the ratio a:b and p:q respectively then value of ( left(frac{a}{b}, frac{p}{q}right) ) may be
A ( cdot frac{23}{12} )
B. ( frac{7}{5} )
( c cdot frac{7}{12} )
D. ( frac{21}{10} )
12
221The vector equation of the plane through the point (1,-2,-3) and
parallel to the vectors (2,-1,3) and
(2,3,-6) is ( bar{r}= )
A ( cdot(1+2 t+2 s) bar{i}-(2+t-3 s) bar{j}-(3-3 t+6 s) bar{k} )
B. ( (1+2 t+2 s) bar{i}+(2+t+3 s) bar{j}-(3+3 t+6 s) bar{k} )
c. ( (1+2 t+2 s) bar{i}+(2+t+3 s) bar{j}+(3+3 t+6 s) bar{k} )
D. ( (1+2 t+2 s) bar{i}+(2+t-3 s) bar{j}+(3+3 t+6 s) bar{k} )
12
222If ( O ) is origin ( O P=3 ) with direction
ratios proportional to -1,2,-2 then what are the coordinates of ( P ? )
12
223Direction ratios of the line which is perpendicular to the lines with direction ratios -1,2,2 and 0,2,1 are
( mathbf{A} cdot 1,1,2 )
B. 2,-1,2
c. -2,1,2
D. 2,1,-2
12
224( boldsymbol{A}=(mathbf{1},-mathbf{2}, mathbf{3}), boldsymbol{B}=(2,1,3), boldsymbol{C}=(4,2, )
1) and ( G=(-1,3,5) ) is the centroid of
the tetrahedron ( A B C D . ) Then the fourth
coordinate is
A. (11,11,13)
В. (-11,11,45)
c. (-11,11,13)
D. (11,13,11)
12
225A line d.c’s proportional to (2,1,2)
meets each of the lines ( boldsymbol{x}=boldsymbol{y}+boldsymbol{a}=boldsymbol{z} )
and ( x+a=2 y=2 z . ) Then the
coordinates of each of the points of
intersection are given by
A . ( (3 a, 2 a, 3 a) ;(a, a, 2 a) )
в. ( (3 a, 2 a, 3 a) ;(a, a, a) )
C. ( (3 a, 3 a, 3 a) ;(a, a, a) )
D. ( (2 a, 3 a, 3 a) ;(2 a, a, a) )
12
226If the plane a ( 2 x-3 y+5 z-2=0 )
divides the line segment joining (1,2,3) and ( (2,1, k) ) in the ratio 9: 11
then ( k ) is
A .
в. -2
c. -10
D. ( -frac{1}{2} )
12
22737. A line with positive direction cosines passes through
point P(2,-1,2) and makes equal angles with the coordinata
axes. The line meets the plane
2x+y+z=9
at point Q. The length of the line segment PQ equals
(2009)
(a) 1 (b) √ (c) √3 (d) 2
12
228The distance of origin from the image of (1,2,3) in plane ( x-y+z=5 ) is
A ( cdot sqrt{17} )
B. ( sqrt{29} )
c. ( sqrt{34} )
D. ( sqrt{41} )
12
229If ( |vec{A} times vec{B}|=sqrt{3} vec{A} cdot vec{B} ) then the value of
( |vec{A}+vec{B}| ) is:
( ^{A} cdotleft(A^{2}+B^{2}+frac{A B}{sqrt{3}}right)^{1 / 2} )
в. ( A+B )
c. ( left(A^{2}+B^{2}+sqrt{3} A Bright)^{1 / 2} )
D. ( left(A^{2}+B^{2}+A Bright)^{1 / 2} )
12
230Find the vector equation of the line through ( A(3,4,-7) ) and ( B(6,-1,1) )12
231If the orthocentre, circumcentre of a
triangle are (-3,5,2),(6,2,5)
respectively then the centroid of the triangle is
( mathbf{A} cdot(3,3,4) )
В. ( left(frac{3}{2}, frac{7}{2}, frac{9}{2}right) )
c. (9,9,12)
D. ( left(frac{9}{2} frac{-3}{2}, frac{3}{2}right) )
12
232The Cartesian equation of line ( 6 x- ) ( mathbf{2}=mathbf{3} boldsymbol{y}+mathbf{1}=mathbf{2} z-mathbf{2} ) is given by
( ^{text {A } cdot frac{3 x-1}{3}}=frac{3 y+1}{6}=frac{z-1}{3} )
B. ( frac{3 x+1}{3}=frac{3 y-1}{6}=frac{z-1}{3} )
c. ( frac{3 x-1}{3}=frac{3 y-1}{6}=frac{z-1}{3} )
D. ( frac{3 x-1}{6}=frac{3 y-1}{3}=frac{z-1}{3} )
12
233The direction ratios of the line ( boldsymbol{x}-boldsymbol{y}+ )
( z-5=0=x-3 y-6 ) are
A. 3,1,-2
в. 2,-4,1
c. ( frac{3}{sqrt{14}}, frac{1}{sqrt{14}}, frac{-2}{sqrt{14}} )
D. ( frac{2}{sqrt{14}}, frac{-4}{sqrt{14}}, frac{1}{sqrt{14}} )
12
234A plane is at a distance of 5 units from the origin and perpendicular to the vector ( 2 hat{i}+hat{j}+2 hat{k} . ) The equation of the
plane is
A ( . vec{r} .(2 hat{imath}+hat{j}-2 hat{k})=15 )
в. ( vec{r} .(2 hat{i}+hat{j}-hat{k})=15 )
c. ( vec{r} .(2 hat{i}+hat{j}+2 hat{k})=15 )
D. ( vec{r} .(hat{i}+hat{j}+2 hat{k})=15 )
E ( . vec{r} .(2 hat{i}-hat{j}+2 hat{k})=15 )
12
235Find vector equation for the line passing through the points ( 3 bar{i}+4 bar{j}- ) ( mathbf{7} bar{k}, overline{boldsymbol{i}}-overline{boldsymbol{j}}+mathbf{6} overline{boldsymbol{k}} )
A ( . bar{r}=(3-2 lambda) bar{i}+(4-5 lambda) bar{j}+(-7+13 lambda) bar{k} )
B. ( bar{r}=(2 lambda) bar{i}+(4+5 lambda) bar{j}+(-7-13 lambda) bar{k} )
c. ( bar{r}=(3-2 lambda) bar{i}-(4-5 lambda) bar{j}+(-7+13 lambda) bar{k} )
D. ( bar{r}=(3-2 lambda) bar{i}+(4-5 lambda) bar{j}-(-7+13 lambda) bar{k} )
12
236(-1,-5,-7) lies in Octant
( A )
B. VII
( c cdot v )
D. II
12
237If ( boldsymbol{A}(cos boldsymbol{alpha}, sin boldsymbol{alpha}, boldsymbol{0}), boldsymbol{B}(cos boldsymbol{beta}, sin beta, boldsymbol{0}) )
( C(cos gamma, sin gamma, 0) ) are vertices of ( Delta A B C )
and let
[
begin{array}{l}
cos alpha+cos beta+cos gamma=3 a, sin alpha+ \
sin beta+sin gamma=3 b, text { then correct }
end{array}
]
matching of the following is:
List:1
begin{tabular}{ll}
A. Circumcentre & ( 1 .(3 a, 3 b, 0) ) \
cline { 0 }
end{tabular}
B. Centroid
[
2 .(0,0,0)
]
c. Ortho centre
[
3 .(a, b, 0)
]
A . 432
в. 231
c. 123
D. 234
12
238If ( bar{a}, bar{b} ) are the position vectors of ( A ) and
( B ) then one of the following points lie on
( A B )
A ( cdot frac{2(bar{a}+bar{b})}{3} )
в. ( frac{(bar{a}-bar{b})}{3} )
c. ( frac{(bar{a}+bar{b})}{3} )
( ^{mathrm{D}} cdot frac{2 bar{a}+2 bar{b}}{3} )
E. None of these
12
239The perimeter of triangle with vertices at (1,0,0),(0,1,0) and (0,0,1) is :
( mathbf{A} cdot mathbf{3} )
B . 2
( c cdot 2 sqrt{2} )
D. ( 3 sqrt{2} )
12
240Show that the points are collinear
(1)
( boldsymbol{A}(mathbf{3}, mathbf{2},-mathbf{4}), boldsymbol{B}(mathbf{9}, mathbf{8},-mathbf{1 0}), boldsymbol{C}(-mathbf{2},-mathbf{3}, mathbf{1}) )
12
241The points ( A ) and ( B ) have co-ordinate (1,2,4) and (-1,3,5) respectively. Find ( A B ) and its magnitude.12
242Prove the ( boldsymbol{A}(-mathbf{5}, mathbf{4}), boldsymbol{B}(-mathbf{1},-mathbf{2}), boldsymbol{C}(mathbf{5}, mathbf{2}) )
are the vertices of ( n ) isosceles right
angled triangle?
12
243If ( P(x, y, z) ) is a point on the line segment joining ( A(2,2,4) ) and ( B(3,5,6) ) such that projection of ( overline{O P} ) on axes are ( frac{13}{5}, frac{19}{5}, frac{26}{5} ) respectively, then ( P ) divide ( A B ) in the ratio
A .3: 2
B. 2: 3
c. 1: 2
D. 1: 3
12
244The direction cosine of a line equally inclined to the axes are
A ( cdot frac{1}{3}, frac{1}{3}, frac{1}{3} )
B. ( -frac{1}{3},-frac{1}{3},-frac{1}{3} )
c. ( frac{1}{sqrt{3}}, frac{1}{sqrt{3}}, frac{1}{sqrt{3}} )
D. none of these
12
245If ( l, m, n ) are d.c’s of vector ( overline{O P} ) then
maximum value of ( l ) mn is
A ( cdot frac{1}{sqrt{3}} )
B. ( frac{1}{2 sqrt{3}} )
c. ( frac{1}{3 sqrt{3}} )
D. ( frac{2}{sqrt{3}} )
12
246The lines ( frac{x-2}{1}=frac{y-3}{1}=frac{z-4}{-k} & )
( frac{boldsymbol{x}-mathbf{1}}{boldsymbol{k}}=frac{boldsymbol{y}-boldsymbol{4}}{boldsymbol{2}}=frac{boldsymbol{z}-mathbf{5}}{mathbf{1}} ) are coplanar if
A ( . k=0 ) or -1
B. ( k=1 ) or -1
c. ( k=0 ) or -3
D. ( k=3 ) or -3
12
247If ( A B perp B C, ) then the value of ( lambda ) equal
where ( boldsymbol{A}(mathbf{2 k}, mathbf{2}, mathbf{3}), boldsymbol{B}(boldsymbol{k}, mathbf{1}, mathbf{5}), boldsymbol{C}(boldsymbol{3}+ )
( k, 2,1) )
( mathbf{A} cdot mathbf{3} )
B.
( c .-3 )
D. ( -frac{1}{3} )
12
248The equation of altitude through ( B ) to
side ( A C ) is
A. ( r=k+t(7 i-10+2 k) )
в. ( r=k+t(-9 i+6 j-2 k) )
c. ( r=k+t(7 i-10 j-2 k) )
D. ( r=k+t(7 i+10 j+2 k) )
12
249Equation of the plane passing through the point of intersection of ( x+2 y+=0 ) and ( 2 x+y=0 ) and which is perpendicular to
( 5 x+6 z=0 )
This question has multiple correct options
A. ( x=0 )
B. ( y=0 )
( c cdot z=0 )
D. x+y=0
E. ( x+z=0 )
12
250Solve: ( sqrt{mathbf{2}} boldsymbol{x}+sqrt{mathbf{3}} boldsymbol{y}=mathbf{0} )
( sqrt{mathbf{3}} boldsymbol{x}-sqrt{mathbf{8}} boldsymbol{y}=mathbf{0} )
12
251Find the vector equation of the plane through the points (2,1,-1) and (-1,3,4) and perpendicular to the
plane ( boldsymbol{x}-mathbf{2} boldsymbol{y}+mathbf{4} boldsymbol{z}=mathbf{1 0} )
12
252Write the direction cosines of the line
whose cartesian equations are ( 2 x= )
( mathbf{3} boldsymbol{y}=-boldsymbol{z} )
12
253If the angles made by a straight line with the coordinate axes are ( alpha, frac{pi}{2}- ) ( boldsymbol{alpha}, boldsymbol{beta} ) then ( boldsymbol{beta}= )
A . 0
в. ( frac{pi}{6} )
c. ( frac{pi}{2} )
( D )
12
254Find the coordinates of the point which
divides the line segment joining the points (-2,3,5) and (1,-4,6) in the ratio
(i) 2: 3 internally
(ii) 2: 3 externally
12
255If the distance between a point ( boldsymbol{P} ) and the point (1,1,1) on the line ( frac{x-1}{3}= ) ( frac{boldsymbol{y}-mathbf{1}}{mathbf{4}}=frac{boldsymbol{z}-mathbf{1}}{mathbf{1 2}} ) is ( mathbf{1 3}, ) then the
coordinates of ( boldsymbol{P} ) are
A. (3,4,12)
в. ( left(frac{3}{13}, frac{4}{13}, frac{12}{13}right) )
c. (4,5,13)
D. (40,53,157)
12
256If a line makes angles ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} ) with
positive directions of ( mathrm{X}, mathrm{Y}, mathrm{Z} ) -axes, what is the value of ( sin ^{2} alpha+sin ^{2} beta+sin ^{2} gamma )
( ? )
12
257( l=m=n=1 ) represents the direction
cosines of
A. ( x ) -axis
B. ( y ) -axis
c. ( z ) -axis
D. none of these
12
258The coordinates of the foot of the
perpendicular drawn from of the origin to a plane are ( (12,-4,3) . ) Find the equation of the plane.
12
259What is the angle between ( vec{P} times vec{Q} ) and ( vec{P}+vec{Q} ? )
( mathbf{A} cdot mathbf{0} )
в. ( frac{pi}{2} )
( c . pi )
D. ( frac{3 pi}{2} )
12
260If 0 is the origin and the coordinates of ( P ) is ( (1,2,-3), ) then find the equation of the plane passing through P and perpendicular to OP.
A. ( x-2 y-3 z=-15 )
в. ( x+2 y-3 z=14 )
c. ( x-2 y+3 z=15 )
D. ( x-2 y-3 z=15 )
12
261If ( 4 x+4 y-k z=0 ) is the equation of
the plane through the origin that contains the line ( frac{boldsymbol{x}-mathbf{1}}{mathbf{2}}=frac{boldsymbol{y}+mathbf{1}}{mathbf{3}}=frac{boldsymbol{z}}{mathbf{4}} )
then ( boldsymbol{k}= )
A . 1
B. 3
( c .5 )
D.
12
262The sum of the intercepts on the coordinate axes of the plane passing through the point (-2,-2,2) and containing the line joining the points (1,-1,2) and (1,1,1) is?
A . 12
B. -8
( c .-4 )
D. 4
12
263If a line makes angles ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} ) with
positive axes, then the range of ( sin alpha sin beta+sin beta sin gamma+sin gamma sin alpha ) is
( ^{mathrm{A}} cdotleft(frac{-1}{2}, 1right) )
в. ( left(frac{1}{2}, 2right) )
c. (-1,2)
D. (-1,2]
12
264Let the line ( frac{boldsymbol{x}-mathbf{2}}{mathbf{3}}=frac{boldsymbol{y}-mathbf{1}}{-mathbf{5}}=frac{boldsymbol{z}+mathbf{2}}{mathbf{2}} ) lie
in the plane ( x+3 y-alpha z+beta=0 . ) Then
( (alpha, beta) ) equals:
A ( cdot(-6,7) )
B ( cdot(5,-15) )
c. (-5,5)
D. (6,-17)
12
265Find the ratio in which the ( X Y ) – plane
divides ( A B ) if is (1,2,3) and ( B ) is (-3,4,-5)
Also find the positive vector of the point of division.
12
266Find the measure of the angle between two lines if their direction cosines
( ell, boldsymbol{m}, boldsymbol{n} ) satisfy ( ell+boldsymbol{m}-boldsymbol{n}=mathbf{0}, ell^{2}+ )
( boldsymbol{m}^{2}-boldsymbol{n}^{2}=mathbf{0} )
12
267What are the DR’s of vector parallel to
(2,-1,1) and (3,4,-1)( ? )
A ( cdot(1,5,-2) )
B ( cdot(-2,-5,2) )
( mathbf{c} cdot(-1,5,2) )
D ( cdot(-1,-5,-2) )
12
268The equation of motion of a rocket are:
( boldsymbol{x}=mathbf{2} boldsymbol{t}, boldsymbol{y}=-boldsymbol{4} boldsymbol{t}, boldsymbol{z}=boldsymbol{4} boldsymbol{t}, ) where the time
( t ) is given in seconds and the coordinate
of a moving point in kilometers. At what distance will the rocket be from the
starting point ( O(0,0,0) ) in 10 seconds?
( mathbf{A} cdot 60 mathrm{km} )
B. ( 30 mathrm{km} )
c. ( 45 mathrm{km} )
D. None of these
12
269Point ( D ) has coordinates as (3,4,5)
Referring to the given figure, find the
coordinates of point ( boldsymbol{E} )
( mathbf{B} cdot(0,4,5) )
C. (0,5,4)
D. (0,3,4)
12
270The point (0,-2,5) lies on the
( A cdot z ) axis
B. x axis
c. xy plane
D. yz plane
E . xz plane
12
271( left(cos ^{-1} l+cos ^{-1} m+cos ^{-1} nright) ) is equal
to
A . ( 90^{circ} )
B. ( 50^{circ} )
( c cdot 180^{circ} )
D. None of these
12
272Find the distance of the point (2,3,5) from the ( x y- ) plane12
273Find the equation of the plane passing through the point (2,-1,1) and through the line of intersection of the planes ( vec{r} ). ( (2 hat{i}-3 hat{j}+hat{k})=3 ) and ( vec{r} cdot(hat{i}+5 hat{j}- )
( hat{boldsymbol{k}})=mathbf{4 . 4} )
12
274Find the direction cosines of the line:
( frac{x-1}{2}=-y=frac{z+1}{2} )
12
275Find the equation to the plane through the point (-1,3,2) and perpendicular to the planes ( x+2 y+2 z=11 ) and
( 3 x+3 y+2 z=15 )
12
276figure bounded by non coplanar triangular planes. So, a tetrahedron has four non-coplanar points as its vertices. Suppose a tetrehedron has points
A,B,C,D as its vertices which have
coordinates ( left(boldsymbol{x} mathbf{1}, boldsymbol{y} mathbf{1}, boldsymbol{z}_{1}right)left(boldsymbol{x}_{2}, boldsymbol{y}_{2}, boldsymbol{z}_{2}right) )
( left(x_{3}, y_{3}, z_{3}right) ) and ( left(x_{4}, y_{4}, z_{4}right), ) respectively
in a rectangular three dimensional
space. Then, the coordinates of
its centroid are ( left[frac{boldsymbol{x}_{1}+boldsymbol{x}_{2}+boldsymbol{x}_{3}+boldsymbol{x}_{4}}{boldsymbol{4}}, frac{boldsymbol{y}_{1}+boldsymbol{y}_{2}+boldsymbol{y}_{3}+boldsymbol{y}_{4}}{boldsymbol{4}}right. )
Let a tetrahedron have three of its
vertices represented by the points (0,0,0),(6,5,1) and (4,1,3) and
its centroid lies at the point (1,2,5) Now, answer the following question. The coordinate of the fourth vertex of the
tetrahedron is:
A ( cdot(-6,2,16) )
B. (1,-2,13)
c. (-2,4,-2)
D. (1,-1,1)
12
277The distance of the point ( 3 hat{i}+5 hat{k} ) from the line parallel to the vector ( 6 hat{i}+hat{j}- ) ( 2 hat{k} ) and passing through the point ( 8 hat{i}+ ) ( mathbf{3} hat{boldsymbol{j}}+hat{boldsymbol{k}} ) is
A . 1
B . 2
( c .3 )
D.
12
278If ( z=cos frac{pi}{6}+i sin frac{pi}{6}, ) then
A ( cdot|z|=1, arg z=frac{pi}{4} )
в ( cdot|z|=1, arg z=frac{pi}{6} )
с. ( quad|z|=frac{sqrt{3}}{2}, arg z=frac{5 pi}{24} )
D. ( quad|z|=frac{sqrt{3}}{2}, arg z=tan ^{-1} frac{1}{sqrt{2}} )
12
279The vertices of a triangle are ( (2,3,5),(-1,3,2),(3,5,-2), ) then the
angles are
( ^{mathrm{B}} cos ^{-1}left(frac{1}{sqrt{5}}right), 90^{circ}, cos ^{-1}left(frac{sqrt{5}}{sqrt{3}}right) )
c. ( 30^{circ}, 60^{circ}, 90^{circ} )
D ( cdot cos ^{-1}left(frac{1}{sqrt{3}}right), 90^{circ}, cos ^{-1}(sqrt{frac{2}{3}}) )
12
280An equation of a plane containing the
lines ( r=a_{1}+t b_{1} ) and ( r=a_{2}+t b_{2} )
where ( left[boldsymbol{a}_{2}-boldsymbol{a}_{1}, boldsymbol{b}_{1}, boldsymbol{b}_{2}right]=boldsymbol{0} ) is
A ( cdotleft[r-a_{1}, b_{1}, b_{2}right]=0 )
B . ( left[r-a_{2}, b_{1}, b_{2}right]=0 )
c. ( left[r-a_{2}, a_{1}, b_{2}right]=0 )
D . ( left[r-a, a_{2}, b_{2}right]=0 )
12
281f ( A(2,1,3), B(2,-3,4), C(-1,2,7) ) are the
three points ; find a point D such that ( |overrightarrow{A B}|=frac{1}{2}|overrightarrow{C D}| ) and ( A B|| C D )
12
282The distance between the line ( r=2 hat{i}- ) ( 2 hat{j}+3 hat{k}+lambda(hat{i}-hat{j}+4 hat{k}) ) and the plane
( r cdot(hat{i}+5 hat{j}+hat{k})=5 ) is
A ( cdot frac{10}{9} )
в. ( frac{10}{3 sqrt{3}} )
c. ( frac{10}{3} )
D. None of these
12
283The name of the figure formed by the points (-1,-3,4),(5,-1,1),(7,-4,7)
and (1,-6,10) is a
A. square
B. rhombus
c. parallelogram
D. rectangle
12
284The ratio in which the plane ( 2 x+3 y- )
( mathbf{2} z+mathbf{7}=mathbf{0} ) divides the line segment
joining the points (-1,1,3),(2,3,5) is
A .3: 5
B. 7: 5
( mathbf{c} cdot 9: 11 )
D. 1: 5 externally
12
285State the following statement is True or
False
If two distinct lines are intersecting each other in a plane then they cannot
have more than one point in common.
A. True
B. False
12
286The direction cosines of the ray
( boldsymbol{P}(1,-2,4) ) and ( Q(-1,1,-2) ) are
A ( cdot(-2,-3,-6) )
B ( cdot(2,-3,-6) )
( ^{C} cdotleft(frac{2}{7}, frac{3}{7}, frac{6}{7}right) )
D ( cdotleft(-frac{2}{7}, frac{3}{7},-frac{6}{7}right) )
12
287Find the equation of the plane through the line
( boldsymbol{P}=boldsymbol{a} boldsymbol{x}+boldsymbol{b} boldsymbol{y}+boldsymbol{c} boldsymbol{z}+boldsymbol{d}=mathbf{0} )
( boldsymbol{Q}=boldsymbol{a}^{prime} boldsymbol{x}+boldsymbol{b}^{prime} boldsymbol{y}+boldsymbol{c}^{prime} boldsymbol{z}+boldsymbol{d}^{prime}=mathbf{0} )
and parallel to the line ( frac{x}{l}=frac{y}{m}=frac{z}{n} )
A ( cdot Pleft(a^{prime} l+b^{prime} m+c^{prime} nright)+Q(a l+b m+c n)=0 )
B – ( Pleft(a^{prime} l+b^{prime} m+c^{prime} nright)-Q(a l+b m+c n)=0 )
c. ( Qleft(a^{prime} l+b^{prime} m+c^{prime} nright)-P(a l+b m+c n)=0 )
D. ( Qleft(a^{prime} l+b^{prime} m+c^{prime} nright)+P(a l+b m+c n)=0 )
12
28869.
t-3
y
=
and
If the line X-1 y +1 Z-1
1 2 3 4
intersect, then k is equal to:
[2012]
(a)
-1
12
289find the equation of a line passing
through the point (1,2,-4) and
perpendicular to two lines. ( vec{r}= ) ( (8 hat{i}-19 hat{j}+10 hat{k})+ )
( boldsymbol{lambda}(boldsymbol{3} hat{boldsymbol{i}}-boldsymbol{1 6} hat{boldsymbol{j}}+boldsymbol{7} hat{boldsymbol{k}}) ) and ( overrightarrow{boldsymbol{r}}= )
( (15 hat{i}+29 hat{j}+5 hat{k})+ )
( boldsymbol{mu}(mathbf{3} hat{boldsymbol{i}}+quad boldsymbol{8} hat{boldsymbol{j}}-boldsymbol{5} hat{boldsymbol{k}}) )
12
290The line passing through the points ( (5,1, a) ) and ( (3, b, 1) ) crosses the ( y z ) plane at the point ( left(0, frac{17}{2}, frac{-13}{2}right) . ) Then
A ( cdot a=2, b=8 )
В. ( a=4, b=6 )
c. ( a=6, b=4 )
D. ( a=8, b=2 )
2
12
291ff ( y ) varies directly as ( x ) and ( y=12 )
when ( x=4, ) then find the linear
equation.
12
292Cartesian equation of a line is ( frac{x-5}{3}= ) ( frac{boldsymbol{y}+boldsymbol{4}}{mathbf{7}}=frac{boldsymbol{z}-boldsymbol{6}}{boldsymbol{2}} . ) Write it in vector form12
293Name three undefined terms.
A. Point
B. Line
c. Plane
D. All of the above
12
294Find a unit vector normal to the plane
through the points (1,1,1),(-1,2,3) and (2,-1,3)
12
295If the line joining the points (-1,2,3),(2,-1,4) is perpendicular to the line joining the points ( (x,-2,4),(1,2,3) ) then ( x= )
( mathbf{A} cdot mathbf{3} )
B. 10
( c cdot frac{-3}{10} )
D. ( frac{-10}{3} )
12
296The equation of the plane passing through the straight line ( frac{x-1}{2}= ) ( frac{boldsymbol{y}+mathbf{1}}{mathbf{- 1}}=frac{boldsymbol{z}-mathbf{3}}{mathbf{4}} ) and perpendicular to
the plane ( boldsymbol{x}+mathbf{2} boldsymbol{y}+boldsymbol{z}=mathbf{1 2} ) is
A. ( 9 x+2 y-5 z+4=0 )
В. ( 9 x-2 y-5 z+4=0 )
c. ( 9 x+2 y+5 z+4=0 )
D. None of these
12
29792.
x-1 y+1 z-2
If the line,
meets the plane, x+2y+
2 3 4
3z= 15 at a point P, then the distance of P from the origin in
[JEEM 2019-9 April (M)
(a) V5/2
(b) 215
(c) 9/2
(d) 7/2
12
298Let ( boldsymbol{O} ) be the origin and ( boldsymbol{A} ) be the point ( (64,0) . ) If ( P ) and ( Q ) divide ( O A ) in the ratio
( mathbf{1}: mathbf{2}: mathbf{3}, ) then the point ( boldsymbol{P} ) is
( ^{mathbf{A}} cdotleft(frac{32}{3}, 0right) )
в. (32,0)
( ^{mathbf{c}} cdotleft(frac{64}{3}, 0right) )
D. (16,0)
E ( cdotleft(frac{16}{3}, 0right) )
12
299Calculate the distance between the
points (-3,6,7) and (2,-1,4) in ( 3 D )
space.
A . 4.36
в. 5.92
c. 7.91
D. 9.11
E . 22.25
12
300Find the equation of the plane which
passes through the point (3,2,0) and contains the line ( frac{boldsymbol{x}-mathbf{3}}{mathbf{1}}=frac{boldsymbol{y}-mathbf{6}}{mathbf{5}}= )
( frac{z-4}{4} ? )
12
301The direction ratios of the line, given by the planes ( x-y+z-5=0, x-3 y-6=0 )
are
A ( .(3,1,-2) )
в. (2,-4,1)
c. (1,-1,1)
D. (0,2,1)
12
302Using section formula prove that the point ( (0.7 .-7),(1,4,-5) ) and
(-1,10,-9) are collinear.
12
303Find the vector equation of the line joining points ( 2 hat{i}+hat{j}+3 hat{k} ) and ( -4 hat{i}+ ) ( mathbf{3} hat{boldsymbol{j}}-hat{boldsymbol{k}} )12
304Assertion
The point ( boldsymbol{A}(boldsymbol{3}, boldsymbol{1}, boldsymbol{6}) ) is the mirror image
of the point ( B(1,3,4) ) in the plane ( x- )
( boldsymbol{y}+boldsymbol{z}=mathbf{5} )
Reason
The plane ( x-y+z=5 ) bisects the line
segment joining ( boldsymbol{A}(mathbf{3}, mathbf{1}, boldsymbol{6}) ) and
( boldsymbol{B}(mathbf{1}, boldsymbol{3}, boldsymbol{4}) )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
305Show that the plane whose vector equation is ( vec{r} .(hat{i}+2 hat{j}-hat{k})=3 ) contains the line ( vec{r}=hat{i}+hat{j}+lambda(2 hat{i}+hat{j}+4 hat{k}) )12
306A unit vector parallel to the intersection of the planes ( vec{r} cdot(hat{i}-hat{j}+hat{k})=5 ) and ( vec{r} )
( (2 hat{i}+hat{j}-3 hat{k})=4 ) can be
This question has multiple correct options
( frac{2 hat{i}+5 hat{j}+3 hat{k}}{sqrt{38}} )
B. ( frac{2 hat{i}-5 hat{j}+3 hat{k}}{sqrt{38}} )
c. ( frac{-2 hat{i}-5 hat{j}-3 hat{k}}{sqrt{38}} )
D. ( frac{-2 hat{i}+5 hat{j}-3 hat{k}}{sqrt{38}} )
12
307The length of the perpendicular from the
vertex ( D ) on the opposite face is
A ( cdot frac{14}{sqrt{6}} )
в. ( frac{2}{sqrt{6}} )
c. ( frac{3}{sqrt{6}} )
D. none of these
12
308The distance of the point (1,-2,3) from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} ) measured
parallel to the line ( frac{x}{2}=frac{y}{3}=frac{z-1}{-6} ) is
A .
B . 2
( c cdot 4 )
D. none of these
12
309The circum radius of the triangle formed by the points (0,0,0),(0,0,12) and (3,4,0) is
A. ( sqrt{156} )
B. 13
c. ( frac{13}{2} )
D.
12
310What is the equation of the plane containing the parallel lines
( boldsymbol{r}=boldsymbol{a}+boldsymbol{t c}, boldsymbol{r}=boldsymbol{b}+boldsymbol{p} boldsymbol{c} )
( mathbf{A} cdot r cdot(b-a) times c=[a b c] )
B cdot ( r cdot(a-b) times c=[a b c] )
C ( cdot r cdot(b-a) times c=-[a b c] )
D ( cdot r c times(b-a) times c=[a b c] )
12
311The direction cosines of the line joining the points (2,3,-1) and (3,-2,1) are
в. ( frac{1}{sqrt{30}},-sqrt{frac{5}{6}} cdot sqrt{frac{2}{15}} )
c. ( frac{-1}{30}, frac{1}{6},-frac{1}{15} )
D. none of these
12
312The ratio in which the line joining (3,4,-7) and (4,2,1) is dividing the ( x y ) plane
A .3: 4
B . 2: 1
( c cdot 7: 1 )
D. 4: 3
12
313( G(1,1,-2) ) is the centroid of the
triangle ( A B C ) and ( D ) is the mid point of
( boldsymbol{B C} cdot ) If ( boldsymbol{A}=(-1,1,-4), ) then ( boldsymbol{D}= )
( ^{mathbf{A}} cdotleft(frac{1}{2}, 1, frac{-5}{2}right) )
в. (5,1,2)
c. (-5,-1,-2)
D. (2,1,-1)
12
314Let ( S ) be the set of all real values of ( lambda )
such that a plane passing through the points ( left(-lambda^{2}, 1,1right),left(1,-lambda^{2}, 1right) ) and
( left(1,1,-lambda^{2}right) ) also passes through the point ( (-1,-1,1) . ) Then ( S ) is equal to:
A ( cdot(sqrt{3}) )
B. ( {sqrt{3}-sqrt{3}} )
begin{tabular}{l}
c. {1,-1} \
hline
end{tabular}
D. {3,-3}
12
315Find the Cartesian equation of ( vec{r} ) ( (2 hat{i}+3 hat{j}-4 hat{k})=1 )12
316A point on XOZ-plane divides the join of (5,-3,-2) and (1,2,-2) at
( ^{mathrm{A}} cdotleft(frac{13}{5}, 0,-2right) )
в. ( left(frac{13}{5}, 0,2right) )
c. (5,0,2)
D. (5,0,-2)
12
317A swimmer can swim ( 2 mathrm{km} ) in 15
minutes in a lake and in a river he can
swim a distance of ( 4 mathrm{km} ) in 20 minutes
along the stream. If a paper boat is put in the river, then the distance covered by it in ( 2 frac{1}{2} 2 ) hours will be
( mathbf{A} cdot 18 mathrm{km} )
B. ( 12 mathrm{km} )
c. ( 8 mathrm{km} )
D. ( 10 mathrm{km} )
12
318The points ( A(-1,3,0), B(2,2,1) ) and
( C(1,1,3) ) determine a plane. The distance of the plane ( A, B, C ) from the
point ( D(5,7,8) ) is
A . ( sqrt{66} )
B. ( sqrt{71} )
c. ( sqrt{73} )
D. ( sqrt{76} )
12
319If ( l_{1}, m_{1}, n_{1} ) and ( l_{2}, m_{2}, n_{2} ) are DCs of the two lines inclined to each other at an
angle ( theta ), then the DCs of the internal bisector of the angle between these lines are
A ( cdot frac{l_{1}+l_{2}}{2 sin frac{theta}{2}}, frac{m_{1}+m_{2}}{2 sin frac{theta}{2}}, frac{n_{1}+n_{2}}{2 sin frac{theta}{2}} )
в. ( frac{l_{1}+l_{2}}{2 cos frac{theta}{2}}, frac{m_{1}+m_{2}}{2 cos frac{theta}{2}}, frac{n_{1}+n_{2}}{2 cos frac{theta}{2}} )
c. ( frac{l_{1}-l_{2}}{2 sin frac{theta}{2}}, frac{m_{1}-m_{2}}{2 sin frac{theta}{2}}, frac{n_{1}-n_{2}}{2 sin frac{theta}{2}} )
D. ( frac{l_{1}-l_{2}}{2 cos frac{theta}{2}}, frac{m_{1}-m_{2}}{2 cos frac{theta}{2}}, frac{n_{1}-n_{2}}{2 cos frac{theta}{2}} )
12
320Find the equation of the line in vector and in cartesion form that passes through the point with position vector ( mathbf{2} hat{mathbf{i}}-hat{mathbf{j}}+mathbf{4} hat{boldsymbol{k}} ) and is in the direction ( hat{mathbf{i}}+ )
( mathbf{2} hat{boldsymbol{j}}-hat{boldsymbol{k}} )
12
321The number of octants in which ( Z )
coordinate is positive is
A . 2
B. 3
( c cdot 4 )
D. 1
12
322The length of the normal from origin to
the plane ( x+2 y-2 z=9 ) is equal to
A . 2 units
B. 3 units
c. 4 units
D. 5 units
12
32316.
Consider three points
P=(-sin(-a), -cos ),
Q = (cos(-a), sin B)
and
R=(cos(B-a +0), sin(6-0)), where
0<a,ß,e<I
(2008
Then,
(a) Plies on the line segment RQ
(b) Q lies on the line segment PR
(C) R lies on the line segment OP
(d) P, Q, R are non-collinear
12
324A plane passes through (1,-2,1) and is
perpendicular to the planes ( 2 x-2 y+ )
( z=0 ) and ( x-y+2 z=4 . ) Then the
distance of the plane from the point
(1,2,2) is
( mathbf{A} cdot mathbf{0} )
B.
( c cdot sqrt{2} )
( D cdot 2 sqrt{2} )
12
325If the distance between the plane,
( 23 x-10 y-2 z+48=0 ) and the plane containing the lines ( frac{boldsymbol{x}+mathbf{1}}{mathbf{2}}=frac{boldsymbol{y}-boldsymbol{3}}{boldsymbol{4}}= )
( frac{z+1}{3} ) and ( frac{x+3}{2}=frac{y+2}{6}= )
( frac{z-1}{lambda}(lambda epsilon R) ) is equal to ( frac{k}{sqrt{633}} ) then ( k ) is
equal to
12
326Prove that ( (vec{A} times vec{B})^{2}=A^{2} B^{2}-(vec{A} times vec{B})^{2} )12
327A line makes an angle ( alpha, beta, gamma ) with the
( X, Y, Z ) axes. Then ( sin ^{2} alpha+sin ^{2} beta+ )
( sin ^{2} gamma= )
A .
B. 2
( c cdot frac{3}{2} )
D.
12
328Show that the points (1,2,3),(7,0,1)
and (-2,3,4) are collinear.
12
329Find the direction cosines of the sides
of the triangles whose vertices are (3,5,-4),(-1,1,2) and (-5,-5,-2)
12
330The image of the point ( boldsymbol{P}(boldsymbol{alpha}, boldsymbol{beta}, gamma) ) by
the plane ( l x+m y+n z=0 ) is the point ( boldsymbol{Q}left(boldsymbol{alpha}^{prime}, boldsymbol{beta}^{prime}, boldsymbol{gamma}^{prime}right) . ) Then
A ( cdot alpha^{2}+beta^{2}+gamma^{2}=l^{2}+m^{2}+n^{2} )
B . ( alpha^{2}+beta^{2}+gamma^{2}=alpha^{2}+beta^{2}+gamma^{2} )
( mathbf{c} cdot alpha alpha^{prime}+beta beta^{prime}+gamma gamma^{prime}=0 )
D ( cdot lleft(alpha-alpha^{prime}right)+mleft(beta-beta^{prime}right)+nleft(gamma-gamma^{prime}right)=0 )
12
331Find the vector equation of the plane passing through the points (2,5,-3),(-2,-3,5) and (5,3,-3)12
332The distance of the point (1,-2,3) from
the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} ) measured
parallel to the line ( frac{x}{2}=frac{y}{3}=frac{z-1}{-6} ) is
( mathbf{A} cdot mathbf{1} )
B . 2
( c cdot 4 )
D. None of these
12
333Find the vector equation of the plane passing through (1,2,3) and perpendicular to the plane
( vec{r} cdot(hat{i}+2 hat{j}-5 hat{k})+9=0 )
12
334The distance of the point (1,1,1) from
the plane passing through the points (2,1,1),(1,2,1) and (1,1,2) is
A ( cdot frac{1}{sqrt{3}} )
B.
( c cdot sqrt{3} )
D. None of these
12
335If the d.c’s of a line are ( left(frac{1}{c}, frac{1}{c}, frac{1}{c}right), ) find ( c . )12
336The position vectors ( vec{a}, vec{b}, vec{c} ) of three points satisfy the relation ( 2 vec{a}+7 vec{b}+ )
( mathbf{5} vec{c}=overrightarrow{0} . ) Are these points collinear?
12
337If a line makes angles ( 90^{circ} ) and ( 60^{circ} )
respectively with the positive directions of ( x ) and ( y ) axes, find the angle which it
makes with the positive direction of ( z- )
axis.
12
338( operatorname{Given} boldsymbol{A}(mathbf{1},-mathbf{1}, mathbf{0}) ; boldsymbol{B}(mathbf{3}, mathbf{1}, mathbf{2}) ; boldsymbol{C}(mathbf{2},-mathbf{2}, mathbf{4}) )
and ( D(-1,1,-1) ) which of the following
points neither lie on ( A B ) nor on ( C D )
A ( .(2,2,4) )
В. (2,-2,4)
c. (2,0,1)
D. (0,-2,-1)
12
339Find the direction cosines of the line
which is perpendicular to the lines which direction cosines proportional to
1,-2,-2 and 0,2,1
12
340If ( A, B ) are the feet of the perpendiculars
from (2,4,5) to the ( x ) -axis, ( y ) -axis
respectively, then the distance ( A B ) is
A ( .2 sqrt{5} )
B. ( sqrt{29} )
c. ( sqrt{41} )
D. ( 3 sqrt{5} )
12
341Find the distance of the point (2,12,5) from the point of intersection of the line ( vec{r}=2 hat{i}-4 hat{j}+2 hat{k}+lambda(3 hat{i}+4 hat{j}+2 hat{k}) )
and the plane ( vec{r}(hat{i}-2 hat{j}+hat{k})=0 )
12
342The distance of point (-1,-5,-10) from the point of intersection of ( frac{x-2}{3}=frac{y+1}{4}=frac{-2}{12} ) and plane ( x )
( boldsymbol{y}+boldsymbol{z}=mathbf{5} ) is :
A . 10
B. 8
( c cdot 2 )
D. 13
12
343The line ( frac{x-2}{3}=frac{y+1}{2}=frac{z-1}{-1} )
intersects the curve ( boldsymbol{x} boldsymbol{y}=boldsymbol{c}^{2}, boldsymbol{z}=mathbf{0}, ) if
( boldsymbol{c}= )
A. ( 5 sqrt{5} )
B. ( 4 sqrt{5} )
c. ( sqrt{5} )
D. ( 2 sqrt{5} )
12
344Find the ratio in which the YZ-plane divides the line segment formed by joining the points (-2,4,7) and (3,-5,8)12
345Find the direction cosines of the vector
( overrightarrow{boldsymbol{r}}=(boldsymbol{6} hat{boldsymbol{i}}+boldsymbol{2} hat{boldsymbol{j}}-boldsymbol{3} hat{boldsymbol{k}}) )
12
346Four vertices of a tetrahedron are
(0,0,0),(4,0,0),(0,-8,0) and
( (0,0,12), ) Its centroid has the coordinates
A ( cdotleft(frac{4}{3},-frac{8}{3}, 4right) )
В. (2,-4,6)
c. (1,-2,3)
a 5
D. none of these
12
347Find the equation of a plane which is
parallel to the plane ( x-2 y+2 z=5 )
and whose distance from the point (1,2,3) is 1
12
348Let the direction – cosines of the line
which is equally inclined to the axis be ( pm frac{1}{sqrt{k}} . ) Find ( k ? )
A .2
B. 3
c. 5
D. 6
12
349If the extremities of a diagonal of a square are (1,-2,3) and (2,-3,5) then the length of its side is:
A ( cdot sqrt{6} )
B. ( sqrt{3} )
c. ( sqrt{5} )
D. ( sqrt{7} )
12
350( operatorname{can} frac{1}{sqrt{3}}, frac{2}{sqrt{3}}, frac{-2}{sqrt{3}} ) be the direction
cosines of any directed line?
A. Yes
B. No
c. cannot say
D. None of these
12
(3,4,5),(2,5,9),(5,2,8),(2,5,2)
12
352The distance between (0,1,-1) and the point of intersection of the line ( frac{x}{2}= ) ( frac{boldsymbol{y}-mathbf{1}}{mathbf{3}}=frac{boldsymbol{z}+mathbf{1}}{mathbf{4}} ) and the plane ( boldsymbol{x}+boldsymbol{y}+ )
( z=9 ) is
A. ( sqrt{29} )
в. ( frac{1}{2} sqrt{29} )
( c cdot frac{4}{9} )
D. ( frac{2}{9} sqrt{29} )
12
353If the d.rs of two lines are 1,-2,3 and ( 2,0,1, ) then the d.rs of the line
perpendicular to both the given lines is
A. -2,5,4
в. 2,-5,4
c. 2,5,-4
D. 1,5,-4
12
354The direction cosines of ( A B ) are -2,2,1
If ( boldsymbol{A} equiv(mathbf{4}, mathbf{1}, mathbf{5}) ) and ( l(boldsymbol{A} boldsymbol{B})=mathbf{6 u n i t s} )
find the coordinates of ( boldsymbol{B} )
12
355Match the statements/expressions in
List 1 with the values given in List 2
12
356If ( O A B C ) is a tetrahedron such that the
( boldsymbol{O} boldsymbol{A}^{2}+boldsymbol{B} boldsymbol{C}^{2}=boldsymbol{O} boldsymbol{B}^{2}+boldsymbol{C} boldsymbol{A}^{2}=boldsymbol{O} boldsymbol{C}^{2}+ )
( A B^{2}, ) then which of the following is/are
correct
A. ( A B perp O C )
в. ( O B neq C A )
c. ( O C=A B )
D. ( A B perp B C )
12
357Line through origin and parallel to ( Y ) –
axis is ( dots dots )
A ( cdot frac{x}{1}=frac{y}{0}=frac{2}{0} )
B. ( frac{x}{0}=frac{y}{1}=frac{z}{0} )
( mathbf{c} cdot frac{x}{1}=frac{y}{0}=frac{z}{1} )
D ( cdot frac{x}{1}=frac{y}{1}=frac{z}{0} frac{z}{0} )
12
358A lines makes angles ( frac{boldsymbol{alpha}}{2}, frac{boldsymbol{beta}}{2}, frac{gamma}{2} ) with
positive direction of coordinate axes, then ( cos alpha+cos beta+cos gamma ) is equal to
A . -1
B. 1
( c cdot 2 )
D. 3
12
359The cartesian equation of a line is ( frac{boldsymbol{x}+mathbf{3}}{mathbf{2}}=frac{boldsymbol{y}-mathbf{5}}{mathbf{4}}=frac{boldsymbol{z}+mathbf{6}}{mathbf{2}} ) find the vector
equation of the line?
12
360Let ( A(-1,0) ) and ( B(2,0) ) be two points.
A point ( M ) moves in the plane in such a
way that ( angle M B A=2 angle M A B ). Then the
point ( M ) moves along
A. A straight line
B. A parabola
c. An ellipse
D. A hyperbola
12
361The equation of the plane passing through the intersection of the planes ( mathbf{3} boldsymbol{x}-boldsymbol{y}+mathbf{2} boldsymbol{z}-mathbf{4}=mathbf{0} ) and ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}- )
( mathbf{2}=mathbf{0} ) and passing through the point
( A(2,2,1) ) is given by?
A. ( 7 x+5 y-4 z-8=0 )
в. ( 7 x-5 y+4 z-8=0 )
c. ( 5 x-7 y+4 z-8=0 )
D. ( 5 x+7 y-4 z+8=0 )
12
362Show that ( boldsymbol{A}(mathbf{1}, mathbf{1}, mathbf{2}), boldsymbol{B}(mathbf{2}, mathbf{1}, mathbf{3}) ) and
( C(1,3,5) ) are not collinear.
12
363Find the coordinates of the point, which divides the line segment joining the points ( boldsymbol{A}(mathbf{2},-mathbf{6}, mathbf{8}) ) and ( boldsymbol{B}(-mathbf{1}, mathbf{3},-mathbf{4}) )
externally in the ratio 1: 3
12
364Name the octants in which the following
points lie: ( (mathbf{5}, mathbf{2}, mathbf{3}) )
12
365The ratio in which the plane ( vec{r} cdot(vec{i}- ) ( mathbf{2} overrightarrow{boldsymbol{j}}+boldsymbol{3} overrightarrow{boldsymbol{k}})=17 ) divides the line joining
the points ( -2 vec{i}+4 vec{j}+7 vec{k} ) and ( 3 vec{i}- ) ( mathbf{5 j}+mathbf{8 k} ) is
A .1: 5
B. 1: 10
( c .3: 5 )
D. 3: 10
12
366Find the distance between the point (7,2,4) and the plane determined by
the points ( boldsymbol{A}(mathbf{2}, mathbf{5},-mathbf{3}), boldsymbol{B}(-mathbf{2},-mathbf{3}, mathbf{5}) )
and (5,3,-3)
12
367The distance from the origin to the centroid of the tetrahedron formed by
the points (0,0,0),(3,0,0),(0,4,0),(0,0,5) is
A. ( frac{sqrt{3+4+5}}{4} )
B. ( frac{sqrt{3+4+5}}{3} )
c. ( frac{sqrt{3^{2}+4^{2}+5^{2}}}{16} )
D. ( frac{sqrt{3^{2}+4^{2}+5^{2}}}{4} )
12
368The value(s) of ( lambda ), for which the triangle with vertices ( (mathbf{6}, mathbf{1 0}, mathbf{1 0}),(mathbf{1}, mathbf{0},-mathbf{5}) ) and
( (6,-10, lambda) ) will be a right angled
triangle is/ are
A . 1
в. ( frac{70}{3}, 0 )
c. 35
D. ( 0,-frac{70}{3} )
12
369Find the ratio in which the yz-plane
divides the join of the points (-2,4,7)
and (3,-5,8) and also find the ( c o )
ordinates of the point of intersection of
this line with the ( y z ) – plane.
A ( cdot lambda=frac{2}{3} ) and ( left(0, frac{2}{5}, frac{37}{5}right) )
B. ( lambda=frac{1}{3} ) and ( left(frac{-3}{4}, frac{7}{4}, frac{29}{4}right) )
c. ( lambda=frac{2}{3} ) and ( left(frac{-3}{4}, frac{7}{4}, frac{29}{4}right) )
D ( lambda=frac{1}{3} ) and ( left(0, frac{2}{5}, frac{37}{5}right) )
12
370The direction cosines of the normal to
the plane ( 5 y+4=0 ) are?
A ( cdot 0, frac{-4}{5}, 0 )
в. 0,1,0
c. 0,-1,0
D. None of these
12
371The equation of the plane that passes through the points (1,0,2),(-1,1,2),(5,0,3) is
A. ( x+2 y-4 z+7=0 )
B. ( x+2 y-3 z+7=0 )
c. ( x-2 y+4 z+7=0 )
D. ( 2 y-4 z-7+x=0 )
12
372The extremities of a diagonal of a rectangular parallelopiped whose faces are parallel to the reference planes are (-2,4,6) and ( (3,16,6) . ) The length of the base diagonal is
( A cdot 7 )
B. 10
c. 11
D. 13
12
373The equation of a line is ( 5 x-3= )
( mathbf{1 5 y}+mathbf{7}=mathbf{3}-mathbf{1 0 z} . ) Write the direction
( operatorname{cosines} ) of the line.
12
374Find the direction cosines of a line that
pass through the point ( boldsymbol{P}(mathbf{1}, mathbf{4}, boldsymbol{6}) ) and
( Q(5,1,11) ) and is so directed that it
make an acute angle with the positive direction of ( boldsymbol{y}-boldsymbol{a} boldsymbol{x} boldsymbol{i} boldsymbol{s} )
12
375The distance of point ( boldsymbol{P}left(boldsymbol{x}_{1}, boldsymbol{y}_{1}, boldsymbol{z}_{1}right) ) to
the plane ( a x+b y+c z+D=0 ) is
given by
A ( cdot frac{left|a x_{1}+b y_{1}+c z_{1}-dright|}{sqrt{a+b+c}} )
B. ( frac{left|a x_{1}+b y_{1}+c z_{1}right|}{sqrt{a^{2}+b^{2}+c^{2}}} )
c. ( frac{left|a x_{1}+b y_{1}+c z_{1}-dright|}{sqrt{a^{2}+b^{2}+c^{2}}} )
D. None of these
12
376A plane passes through (1,-2,1) and is
perpendicular to two planes ( 2 x-2 y+ )
( z=0 ) and ( x-y+2 z=4 . ) The distance
of the plane from the point is (1,2,2)
( A cdot 0 )
B.
( c cdot sqrt{2} )
D. ( 2 sqrt{2} )
12
377intb, z = c.ytd
2003
14. The two lines x=ay+b,z=cy+d and x=a’y+
will be perpendicular, if and only if
(a) aa’+cc’+1 = 0
(b) aa’ + bb’ + cc’ + 1 = 0
(c) aa’+bb’ +cc’ = 0
(d) (a+a’)(b+b) +c+c’)=0.
12
378A line passes through the points (6,-7,-1) and ( (2,-3,1) . ) The direction cosines of the line so directed that the
angle made by it with the positive direction of x-axis is acute, is?
A ( cdot frac{2}{3},-frac{2}{3},-frac{1}{3} )
B. ( -frac{2}{3}, frac{2}{3}, frac{1}{3} )
c. ( frac{2}{3},-frac{2}{3}, frac{1}{3} )
D. ( frac{2}{3}, frac{2}{3}, frac{1}{3} )
12
379x – 4
The va
The value of k such that
y-2
z-k
22.
lies in the
plane 2x – 4y +z=7, is
(2003)
(a)
7
(b) -7
(d) 4
(c) no real value
.
12
380A plane makes intercept 1,2,3 on the co-ordinate axes. If the distance from
origin is ( p ) then find the value of ( p )
12
381The equation of plane containing the lines ( vec{gamma}=vec{alpha}+overrightarrow{lambda beta} ) and ( vec{gamma}=vec{beta}+mu vec{alpha} )
( mathbf{A} cdotleft[begin{array}{lll}vec{gamma} & vec{alpha} & vec{beta}end{array}right]=vec{alpha} cdot vec{beta} )
B ( cdotleft[begin{array}{lll}vec{gamma} & vec{alpha} & vec{beta}end{array}right]=0 )
( mathbf{c} cdotleft[begin{array}{lll}vec{alpha} & vec{beta} & vec{gamma}end{array}right]=vec{alpha} cdot vec{beta} )
D. None of these
12
382is a
sa
mu on
19. Consider a pyramid OPQRS located in the first octant (x20.
y=0,220) with O as origin, and OP and OR along the x-axis
and the y-axis, respectively. The base OPQR of the pyramid
is a square with OP=3. The point S is directly above the
mid-point, T of diagonal OQ such that TS=3. Then
(a) the acute angle between OQ and OS is
(b)
the equation of the plane containing the triangle OOS
is x-y=0
the length of the perpendicular from P to the plane
(c)
containing the triangle OQS is 5
the perpendicular distance from 0 to the straight lin
(d)
containing RS is
12
383Show that the points ( boldsymbol{A}(-mathbf{3} hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}+mathbf{5} hat{boldsymbol{k}}), boldsymbol{B}(hat{boldsymbol{i}}+mathbf{2} hat{boldsymbol{j}}+mathbf{3} hat{boldsymbol{k}}), boldsymbol{C} )
are collinear.
12
384The points ( (boldsymbol{k}-mathbf{1}, boldsymbol{k}+mathbf{2}),(boldsymbol{k}, boldsymbol{k}+ )
1) ( ,(k+1, k) ) are collinear for
A. any value of ( k )
B. ( k=-frac{1}{2} ) only
c. no value of ( k )
D. integral values of ( k ) only
12
385If ( boldsymbol{A} ) is ( (2,4,5), ) and ( B ) is (-7,-2,8) then which of the following is collinear with ( A ) and ( B ) is
A ( .(1,2,6) )
В. (2,-1,6)
c. (-1,2,6)
D. (2,6,-1)
12
386A given unit vector is orthogonal to ( mathbf{5} hat{mathbf{i}}+mathbf{2} hat{mathbf{j}}+mathbf{6} hat{boldsymbol{k}} ) and coplanar with ( hat{mathbf{i}}-hat{boldsymbol{j}}+ )
( hat{k} ) and ( 2 hat{i}+hat{j}+hat{k} ) then the vector is?
A ( cdot frac{3 hat{j}-hat{k}}{sqrt{10}} )
( ^{text {В } cdot frac{6 hat{i}-5 hat{k}}{sqrt{61}}} )
c. ( frac{2 hat{i}-5 hat{k}}{sqrt{29}} )
( frac{2 hat{i}+hat{j}-2 hat{k}}{3} )
12
387to the plane
68. A equation of a plane parallel to the
x-2y + 2z-5 = 0 and at a unit distance from the origin
2012
(a) x-2y+ 2z – 3 =0
(c) x2y + 2z – 1 =0
(b) x-2y + 2z+1=0
(d) x-2y + 2z+5=0
12
388The direction ratios of the joining ( A(1,2,1) ) and (2,1,2) are
A .3,3,3
B. -1,1,-1
c. 3,1,3
D. ( frac{1}{sqrt{3}}, frac{1}{sqrt{3}}, frac{1}{sqrt{3}} )
12
389( boldsymbol{A}=(mathbf{2}, mathbf{4}, mathbf{5}) ) and ( boldsymbol{B}=(mathbf{3}, mathbf{5},-mathbf{4}) ) are two
points. If the ( X Y ) -plane, ( Y Z ) -plane divide ( A B ) in the ratio ( a: b ) and ( p: q )
respectively, then ( frac{a}{b}+frac{p}{q}= )
A ( cdot frac{23}{12} )
B. ( frac{-7}{12} )
c. ( frac{7}{12} )
D. ( frac{-22}{15} )
12
390The pairs ( bar{a}, bar{b} ) and ( bar{c}, bar{d} ) each determine a plane. Then the planes are parallel if ( (overline{boldsymbol{a}} times overline{boldsymbol{c}}) times(overline{boldsymbol{b}} times overline{boldsymbol{d}})= )
( mathbf{0} 2)(overline{boldsymbol{a}} times overline{boldsymbol{c}}) cdot(overline{boldsymbol{b}} times overline{boldsymbol{d}})=mathbf{0} )
( (bar{a} times bar{b}) times(bar{c} times bar{d})= )
( mathbf{0} 4)(bar{a} times bar{b}) cdot(bar{c} times bar{d})=0 )
12
391If a ray makes angles ( alpha, beta, gamma ) and ( delta ) with the four diagonals of a cube and
( mathbf{A}: cos ^{2} boldsymbol{alpha}+cos ^{2} boldsymbol{beta}+cos ^{2} boldsymbol{gamma}+cos ^{2} boldsymbol{delta} )
( mathbf{B}: sin ^{2} boldsymbol{alpha}+sin ^{2} boldsymbol{beta}+sin ^{2} boldsymbol{gamma}+sin ^{2} boldsymbol{delta} )
( mathbf{C}: cos 2 boldsymbol{alpha}+cos 2 boldsymbol{beta}+cos 2 gamma+cos 2 boldsymbol{delta} )
Arrange ( A, B, C ) in descending order
A. ( B, A, C )
в. ( A, B, C )
c. ( C, A, B )
D. ( B, C, A )
12
392If ( R ) divides the line segment joining ( P(2, )
3, 4) and ( Q(4,5,6) ) in the ratio -3: 2 then the value of the parameter which
represents R is
в. ( =(10,9,8) )
( mathbf{c} .=(10,8,9) )
( mathbf{D} .=(9,10,8) )
12
393Find the coordinates of a point on ( y ) -axis which are at a distance of ( 5 sqrt{2} ) from the
point ( boldsymbol{P}(boldsymbol{3},-boldsymbol{2}, boldsymbol{5}) )
12
394Find the equation of the set of points ( P ) the sum of whose distances from
( boldsymbol{A}(mathbf{4}, mathbf{0}, mathbf{0}) ) and ( boldsymbol{B}(-mathbf{4}, mathbf{0}, mathbf{0}) ) is equal to
10
12
395Distance between the points (12,4,7) and (10,5,3) is
A ( cdot sqrt{21} )
B. ( sqrt{5} )
c. ( sqrt{17} )
D. none of these
12
396The distances of the point ( boldsymbol{P}(mathbf{1}, mathbf{2}, mathbf{3}) )
from the coordinates axes are:
A ( cdot sqrt{13}, sqrt{10}, sqrt{5} )
в. ( sqrt{11}, sqrt{10}, sqrt{5} )
c. ( sqrt{13}, sqrt{20}, sqrt{15} )
D. ( sqrt{23}, sqrt{10}, sqrt{5} )
12
397If a line makes an angle of ( pi / 4 ) with the positive direction of each of ( x ) -axis and
( boldsymbol{y} ) -axis, then the angle that the line makes with the positive direction of the ( z ) -axis is
A ( cdot frac{pi}{6} )
в.
c.
D.
12
398Find the vector equation of the line passing through the point (3,1,2) and perpendicular to the plane ( vec{r} cdot(2 hat{i}-hat{j}+widehat{k})=8 ) Also find the point
of intersection of line and plane.
12
399Which of the following is true for a
plane?
This question has multiple correct options
A. A locus is called a plane if the line joining any two arbitrary points on the locus is also a part of the locus.
B. Value of ( y ) in a ( z x ) plane is non-zero.
C. Value of ( z ) in a ( x y ) plane is zero.
D. None of the above
12
400The point which divides the line joining the points (1,3,4) and (4,3,1) internally in the ratio ( 2: 1, ) is
A. (2,-3,3)
в. (2,3,3)
c. ( left(frac{5}{2}, 3, frac{5}{2}right) )
D. (-3,3,2)
E ( .(3,3,2) )
12
401If ( G ) is centroid of ( triangle A B C, ) then
A ( cdot vec{G}=vec{a}+vec{b}+vec{c} )
B. ( vec{G}=frac{vec{a}+vec{b}+vec{c}}{2} )
c. ( 3 vec{G}=vec{a}+vec{b}+vec{c} )
” ( quad 3 vec{G}=frac{vec{a}+vec{b}+vec{c}}{2} )
12
402A line OP where ( 0=(0,0,0) ) makes
equal angles with ox, oy, oz. The point on OP, which is at a distance of 6 units
from 0 is:
A. ( left(frac{6}{sqrt{3}}, frac{6}{sqrt{3}}, frac{6}{sqrt{3}}right) )
B . ( (2 sqrt{3},-2 sqrt{3}, 2 sqrt{3}) )
( c cdot-(2 sqrt{3}, 2 sqrt{3}, 2 sqrt{3}) )
12
403Show that the points ( boldsymbol{O}(mathbf{0}, mathbf{0}), boldsymbol{A}(mathbf{2},-mathbf{3}, mathbf{3}), boldsymbol{B}(-mathbf{2}, mathbf{3},-mathbf{3}) ) are
collinear. Find the ratio in which each point divides the segment joining the other two.
12
404Find the equation of the line passing through the points ( A(3,2,-1) ) and
( boldsymbol{B}(mathbf{4},-mathbf{1}, mathbf{3}) )
12
405A line makes the same angle ( theta ) with
each of the ( X ) and ( Z ) -axes. If the angle ( beta )
which it makes with ( Y ) -axis, is such
that ( sin ^{2} beta=3 sin ^{2} theta, ) then ( cos ^{2} theta ) equals
A ( cdot frac{2}{5} )
в. ( frac{1}{5} )
( c cdot frac{3}{5} )
D. ( frac{2}{3} )
12
406The foot of the perpendicular from the
point ( boldsymbol{A}(mathbf{7}, mathbf{1 4}, mathbf{5}) ) to the plane ( mathbf{2} boldsymbol{x}+mathbf{4} boldsymbol{y}- )
( z=2 ) is?
A. (3,1,8)
в. (1,2,8)
c. (3,-3,5)
D. (5,-3,-4)
12
407The ratio in which the line segment
joining the points whose position vectors are ( 2 hat{i}-4 hat{j}-7 hat{k} ) and ( -3 hat{i}+ ) ( mathbf{5} hat{boldsymbol{j}}-boldsymbol{8} hat{boldsymbol{k}} ) is divided by the plane whose
equation is ( hat{r} cdot(hat{i}-2 hat{j}+3 hat{k})=13 ) is-
A. 13: 12 internally
B. 12: 25 externally
c. 13: 25 internally
D. 37: 25 internally
12
408Distance between plane ( 3 x+4 y- )
( mathbf{2 0}=mathbf{0} ) and point ( (mathbf{0}, mathbf{0},-mathbf{7}) ) is
A. 4 units
B. 3 units
c. 2 units
D. 1 units
12
409The scalar product and the magnitude of vector products of two vectors are ( 48 sqrt{3} ) and 144 respectively. Then the angle between the two vectors is
A. 54.7
B. 60
c. 90
D. 120
12
410If a line has the direction ratios
4,-12,18 then find its direction
cosines.
A ( cdot-frac{2}{11},-frac{6}{11},-frac{9}{11} )
B. ( -frac{2}{11}, frac{6}{11},-frac{9}{11} )
c. ( frac{2}{11},-frac{6}{11}, frac{9}{11} )
D. ( frac{2}{11}, frac{6}{11}, frac{9}{11} )
12
411The locus of a point ( P ) which moves such
that ( P A^{2}-P B^{2}=2 k^{2} ) where ( A ) and ( B )
( operatorname{are}(3,4,5) ) and (-1,3,-7) respectively
is
A. ( 8 x+2 y+24 z-9+2 k^{2}=0 )
B. ( 8 x+2 y+24 z-2 k^{2}=0 )
c. ( 8 x+2 y+24 z+9+2 k^{2}=0 )
D. ( 8 x-2 y+24 z-2 k^{2}=0 )
12
412Assertion
In each of the three planes determined by two of the lines ( O A, O B, O C ) ( ( O ) being the origin), a straight line is drawn through ( O ) perpendicular to the third line.
The three lines so determined are
coplanar.
Reason
( (a times b) times c+(b times c) times a+(c times a) times )
( b=0, ) where ( O A=a, O B=b ) and
( boldsymbol{O} boldsymbol{C}=boldsymbol{c} )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
413Prove that
( boldsymbol{A}(mathbf{1}, mathbf{2}, mathbf{0}), boldsymbol{B}(mathbf{3}, mathbf{1}, mathbf{1}), boldsymbol{C}(mathbf{7},-mathbf{1}, mathbf{3}) ) are
collinear.
12
414The d.r. of normal to the plane through (1, 0, 0), (0, 1.0
which makes an angle /4 with plane x+y=3 are [20021
(a) 1,12,1
(b) 1,1, 2
(c) 1,1,2
(d) √2,1,1
12
415If the projections of the line segment ( A B ) on the coordinate axes are 2,3,6
then the square of the sine of the angle made by ( boldsymbol{A B} ) with ( boldsymbol{x}=mathbf{0}, ) is
A ( cdot frac{3}{7} )
B. ( frac{3}{49} )
( c cdot frac{4}{7} )
D. ( frac{40}{49} )
12
416The angle between any two faces is
( A cdot cos ^{-1}(1 / 3) )
B. ( cos ^{-1}(1 / 4) )
c. ( pi / 3 )
( mathbf{D} cdot cos ^{-1}(1 / 2) )
12
417The point equidistant from the point ( boldsymbol{O}(mathbf{0}, mathbf{0}, mathbf{0}), boldsymbol{A}(boldsymbol{a}, mathbf{0}, mathbf{0}), boldsymbol{B}(mathbf{0}, boldsymbol{b}, mathbf{0}) ) and
( C(0,0, c) ) has the coordinates
( mathbf{A} cdot(a, b, c) )
B. ( (a / 2, b / 2, c / 2) )
c. ( (a / 3, b / 3, c / 3) )
D. ( (a / 4, b / 4, c / 4) )
12
418The distance between (5,1,3) and the
line ( x=3, y=7+t, z=1+t ) is
A . 4
B. 2
( c cdot 6 )
D. 8
12
419The point of intersection of the lines ( frac{x-5}{3}=frac{y-7}{-1}=frac{z+2}{1} ) and ( frac{x+3}{-36}= )
( frac{y-3}{2}=frac{z-6}{4} ) is
A ( cdotleft(21, frac{5}{3}, frac{10}{3}right) )
в. (2,10,4)
c. (-3,3,6)
D. (5,7,-2)
12
420Assertion(A): If centroid and
circumcentre of a triangle are known its orthocentre can be found.

Reason (R) : Centriod, orthocentre and
circumcentre of a triangle are collinear
A. Both A and R are individually true and R is the correct explanation of
B. Both A and R individually true but R is not the correct explanation of ( A )
c. ( A ) is true but ( R ) is false
D. A is false but R is true

12
421( mathbf{I f A}=(mathbf{1}, mathbf{2}, mathbf{3}), mathbf{B}=(mathbf{2}, mathbf{3}, mathbf{4}) ) and ( mathbf{C} ) is a
point of trisection of AB such that ( mathbf{C}_{mathbf{x}}+ ) ( mathbf{C}_{mathbf{y}}=frac{mathbf{1 3}}{mathbf{3}} ) then ( mathbf{C}_{mathbf{z}}= )
A ( cdot frac{10}{3} )
B. ( frac{11}{3} )
c. ( frac{11}{2} )
D. 11
12
422+1
33.
x
If the angel e between the line
– 1 -2
22
– and
the plane 2x-y+ Váz+4 = 0 is such that sin 0=
then
the value of 1 is
[2005]
12
423Find the distance between the points ( R(-3,0), Sleft(0, frac{5}{2}right) )12
424Find the angle between the two lines
whose direction cosines are given by
equations ( l+m+n=0 ) and ( l^{2}+ )
( boldsymbol{m}^{2}-boldsymbol{n}^{2}=mathbf{0} )
A ( cdot frac{pi}{6} )
в.
( c cdot frac{pi}{2} )
D.
12
425The projection of the line segment joining (0,0,0) and (5,2,4) on the line whose direction ratios are 2,-3,6 is
A . 28
B. 4
c. ( frac{40}{7} )
D. ( sqrt{45} )
12
426Find the co-ordinates of the points of trisection of the line joining the points (-3,0) and (6,6)12
427If the lines ( frac{x-1}{2}=frac{y+1}{3}=frac{z-1}{4} ) and
( frac{boldsymbol{x}-mathbf{3}}{mathbf{1}}=frac{boldsymbol{y}-boldsymbol{k}}{mathbf{2}}=frac{boldsymbol{z}}{mathbf{1}} ) intersect, then the
value of ( k ) is
A ( cdot frac{3}{2} )
B. ( frac{9}{2} )
( c cdot-frac{2}{9} )
D. ( -frac{3}{2} )
12
428If the distance of the point ( P(4,3,5) ) from the Y-axis is ( lambda ), then the value of
( 7 lambda^{2} ) is
A .287
B. ( 7 sqrt{41} )
c. 63
D. 21
12
429The shortest distance between z-axis
and the line
( boldsymbol{x}+boldsymbol{y}+mathbf{2} boldsymbol{z}-mathbf{3}=mathbf{0}=mathbf{2} boldsymbol{x}+mathbf{3} boldsymbol{y}+mathbf{4} boldsymbol{z} )
( 4, ) is
( mathbf{A} cdot mathbf{1} )
B . 2
( c cdot 4 )
D. 3
12
430A line makes angles ( alpha, beta, gamma ) with the
positive directions of the axes of
reference. The value of ( cos 2 alpha+ )
( cos 2 beta+cos 2 gamma ) is
( mathbf{A} cdot mathbf{1} )
B. 2
( c cdot-1 )
D. 0
12
43117.
The radius of the circle in which the sphere
x2 + y2 + z2 + 2x – 2y – 42 – 19=0 is cut by the plane
x +2y + 2z +7= 0 is
[2003]
(a) 4 (6) (c) 2 (d) 3
12
432A line with direction cosines
proportional to 2,1,2 meets each of the line ( boldsymbol{x}=boldsymbol{y}+boldsymbol{a}=boldsymbol{z} ) and ( boldsymbol{x}+boldsymbol{a}=boldsymbol{2} boldsymbol{y}= )
2 ( z ). The co-ordinates of each of the
points of intersection are given by:
A. ( (3 a, 3 a, 3 a),(a, a, a) )
в. ( (3 a, 2 a, 3 a),(a, a, a) )
c. ( (3 a, 2 a, 3 a),(a, a, 2 a) )
D. ( (2 a, 3 a, 3 a),(2 a, a, a) )
12
433If ( vec{A} times vec{B}=vec{C}, ) which of the following
options is wrong?
( A cdot vec{C} ) is ( perp ) to ( vec{A} )
B cdot ( vec{C} ) is ( perp ) to ( vec{B} )
c. ( vec{C} ) is ( perp(vec{A}+vec{B}) )
D ( cdot vec{C} ) is ( perp ) to ( (vec{A} times vec{B}) )
12
434Find the shortest distance between the
following pair of lines. ( overline{boldsymbol{r}}=(overline{boldsymbol{i}}+mathbf{2} overline{boldsymbol{j}}+overline{boldsymbol{k}})+boldsymbol{lambda}(mathbf{2} overline{boldsymbol{i}}-overline{boldsymbol{j}}+mathbf{3} overline{boldsymbol{k}}) & )
( overline{boldsymbol{r}}=(overline{boldsymbol{i}}-mathbf{3} overline{boldsymbol{j}}-overline{boldsymbol{k}})+boldsymbol{mu}(mathbf{3} overline{boldsymbol{i}}+mathbf{2} hat{boldsymbol{j}}-mathbf{5} overline{boldsymbol{k}}) )
12
435Show that the points (2,-1,3),(4,3,1) and (3,1,2) are collinear.12
436f a point is in the ( X Z ) -plane. What can
you say about its ( y ) -coordinate?
12
437The plane ( x+2 y-z=4 ) cuts the
sphere ( boldsymbol{x}^{2}+boldsymbol{y}^{2}+boldsymbol{z}^{2}-boldsymbol{x}+boldsymbol{z}-boldsymbol{2}=mathbf{0} ) in
A . 1
B. 3
( c cdot sqrt{2} )
D.
12
438To find the vector and the Cartesian
equation in symmetric form of line passing through the points, (2,0,-3) and ( (mathbf{7}, mathbf{3},-mathbf{1 0}) )
12
439If ( theta ) is the angle between two lines
whose d.c.s are ( l_{1}, m_{1}, n_{1} ) and ( l_{2}, m_{2}, n_{2} )
then the d.cs of one of the angular bisectors of the two lines are
( ^{text {A }} cdot frac{l_{1}+l_{2}}{2}, frac{m_{1}+m_{2}}{2}, frac{n_{1}+n_{2}}{2} )
B. ( frac{l_{1}+l_{2}}{2 cos left(frac{theta}{2}right)}, frac{m_{1}+m_{2}}{2 cos left(frac{theta}{2}right)}, frac{n_{1}+n_{2}}{2 cos left(frac{theta}{2}right)} )
c. ( frac{l_{1}+l_{2}}{cos left(frac{theta}{2}right)}, frac{m_{1}+m_{2}}{cos left(frac{theta}{2}right)}, frac{n_{1}+n_{2}}{cos left(frac{theta}{2}right)} )
D. ( frac{l_{1}+l_{2}}{2 sin left(frac{theta}{2}right)} frac{m_{1}+m_{2}}{2 sin left(frac{theta}{2}right)} frac{n_{1}+n_{2}}{2 sin left(frac{theta}{2}right)} )
12
440The image of the point with position vector ( hat{i}+3 hat{k} ) in the plane ( r cdot(hat{i}+hat{j}+ )
( hat{boldsymbol{k}})=mathbf{1} ) is
( mathbf{A} cdot hat{i}+2 hat{j}+hat{k} )
B ( cdot hat{i}-2 hat{j}+hat{k} )
c. ( -hat{i}-2 hat{j}+hat{k} )
D. ( hat{i}+2 hat{j}-hat{k} )
12
441The vector form of the equation of the
line passing through points (3,4,7) and (5,1,6) is
( mathbf{A} cdot vec{r}=(3 hat{i}+4 hat{j}-7 hat{k})+lambda(2 hat{i}-3 hat{j}+13 hat{k}) )
B ( cdot vec{r}=(3 hat{i}+4 hat{j}-7 hat{k})+lambda(8 hat{i}+5 hat{j}-hat{k}) )
C ( . vec{r}=(3 hat{i}+4 hat{j}+7 hat{k})+lambda(2 hat{i}-3 hat{j}-hat{k}) )
D ( cdot vec{r}=(3 hat{i}+4 hat{j}-7 hat{k})+lambda(2 hat{i}-3 hat{j}-13 hat{k}) )
12
442The angle between any two diagonals of
cube are:
A ( cdot cos ^{-1}left(frac{1}{2}right) )
B. ( cos ^{-1}left(frac{1}{3}right) )
( ^{mathbf{C}} cdot cos ^{-1}left(frac{1}{sqrt{3}}right) )
( mathrm{D} cdot cos ^{-1}left(frac{1}{sqrt{2}}right) )
12
443Find the point of intersection of the following pair of lines, assuming that the vectors ( vec{a} ) and ( vec{b} ) are not parallel. ( vec{r}=gamma(vec{b}+vec{a}), vec{r}=mu(vec{b}-vec{a}) )
A . origin
в. ( vec{b}+vec{a} )
( c cdot 2 b )
D. no intersection point
12
444Find the equation of the line in vector and in Cartesian form that passes through the point with position vector ( 2 hat{i}-hat{j}+4 hat{k} ) and is in the direction ( hat{i}+ ) ( mathbf{2} hat{boldsymbol{j}}-hat{boldsymbol{k}} )12
445Find the equation of the plane passing through the points (0,0,0) and (3,-1,2) are parallel to the line ( frac{boldsymbol{x}-mathbf{4}}{mathbf{1}}=frac{boldsymbol{y}+mathbf{3}}{mathbf{- 4}}=frac{boldsymbol{z}+mathbf{1}}{mathbf{7}} )12
446If a line has direction ratio 2,-1,-2
determine its direction cosines.
12
44710
80
8
12
448The angle between the lines whose
direction cosines satisfy the equations ( l+m+n=0 ) and ( l^{2}+m^{2}+n^{2} ) is
A ( cdot frac{pi}{2} )
в.
c.
D. ( frac{pi}{6} )
12
449Find the vector and Cartesian equations
of the plane passing through the points with position vectors ( 3 vec{i}+4 vec{jmath}+ ) ( 2 vec{k}, 2 vec{i}-2 vec{j}-vec{k} ) and ( 7 vec{i}+vec{k} )
12
450A straight line ( L ) on the ( x y ) -plane
bisects the angle between ( O X ) and ( O Y )
What are the direction cosines of ( L ? )
A ( cdot(1 / sqrt{2}, 1 / sqrt{2}, 0) )
в. ( (1 / 2, sqrt{3} / 2,0) )
c. (0,0,1)
D. ( (2 / 3,2 / 3,1 / 3) )
12
451( mathbf{A}=(mathbf{1}, mathbf{1}, mathbf{4}) ) and ( mathbf{B}=(mathbf{5},-mathbf{3}, mathbf{4}) ) are two
points. If the points ( P, Q ) are on the line
( A B ) such that ( A P=P Q=Q B ) then ( P Q= )
A ( cdot 2 sqrt{2} )
в. 4
c. ( sqrt{frac{32}{9}} )
D. ( sqrt{2} )
12
452The angle between the lines, whose direction ratios are 1,1,2 and ( sqrt{3}- ) ( 1,-sqrt{3}-1,4, ) is
A . ( 45^{circ} )
B. ( 30^{circ} )
( c cdot 60^{circ} )
D. ( 90^{circ} )
12
453Show that the points
( boldsymbol{A}(mathbf{1}, mathbf{1}, mathbf{1}), boldsymbol{B}(mathbf{1}, mathbf{2}, mathbf{3}) ) and ( boldsymbol{C}(mathbf{2},-mathbf{1}, mathbf{1}) ) are
vertices of an isosceles triangle.
12
454Find the vector and Cartesian equation of the line that passes through the points (3,-2,-5) and (3,-2,6)12
455If two vertices of an equilateral triangle ( operatorname{are}(2,1,5) ) and ( (3,2,3), ) then its third vertex is:
A. (1,2,4)
в. (4,0,4)
c. (0,-4,4)
a 5
D. (4,4,1)
12
456Given that ( boldsymbol{P}(boldsymbol{3}, boldsymbol{2},-boldsymbol{4}), boldsymbol{Q}(boldsymbol{5}, boldsymbol{4},-boldsymbol{6}) ) and
( R(9,8,-10) ) are collinear. Find the ratio in which ( Q ) divides ( P R )
12
457If the lines ( frac{x-0}{1}=frac{y+1}{2}=frac{z-1}{-1} ) and ( frac{x+1}{k}= )
( frac{y-3}{-2}=frac{z-2}{1} ) are at right angles, then the
value of k is
( mathbf{A} cdot mathbf{5} )
B.
( c cdot 3 )
D. –
12
4582.
The equation x +2y + 2z = 1 and 2x + 4y + 4z=9 have
(a) Only one solution
ono (1979)
(b) Only two solutions
(c) Infinite number of solutions i n the
(d) None of these.
12
459Find the angles at which the normal
vector to the plane ( 4 x+8 y+z=5 ) is
inclined to the coordinate axes.
12
460Find ( a ), if the distance between the
points ( P(11,-2) ) and ( Q(a, 1) ) is 5 units.
12
461In what ratio, the line joining (-1,1) and (5,7) is divided by the line ( x+y= )
( 4 ? )
12
462If the origin is the centroid of the triangle whose vertices are ( boldsymbol{A}(mathbf{2}, boldsymbol{p},-mathbf{3}), boldsymbol{B}(boldsymbol{q},-mathbf{2}, mathbf{5}) ) and ( boldsymbol{R}(-mathbf{5}, mathbf{1}, boldsymbol{r}) )
then find the values of ( boldsymbol{p}, boldsymbol{q}, boldsymbol{r} )
12
463The equation of the plane passing through the point (1,1,1) and perpendicular to the planes ( 2 x+y- )
( 2 z=5 ) and ( 3 x-6 y-2 z=7 ) is?
A. ( 14 x+2 y-15 z=1 )
B. ( -14 x+2 y+15 z=3 )
c. ( 14 x-2 y+15 z=27 )
D. ( 14 x+2 y+15 z=31 )
12
464f points (1,2),(3,5) and ( (0, b) ) are collinear the value of b is
A ( cdot frac{1}{2} )
в. ( frac{7}{2} )
( c cdot 2 )
D. –
12
465Two distinct lines in a plane
A. always intersect
B. always either intersect or are parallel
C . always have two common points
D. none of these
12
466h the origin.
ant distance
P.: 2x-y+z
17. In R’, let L be a straight line passing through the ori
Suppose that all the points on L are at a constant dista
from the two planes P, :x+2y-z+1=0 and P, : 2x –
-1= 0. Let M be the locus of the feet of the perpendicula,
drawn from the points on L to the plane P,. Which of th
following points lie (s) on M?
(b)
12
467( boldsymbol{L} ) and ( boldsymbol{M} ) are two points with position vectors ( 2 bar{a}-bar{b} ) and ( a+2 bar{b} ) respectively.
The position vector of the point ( N ) which
divides the line segment ( L M ) in the ratio 2: 1 externally is
( mathbf{A} cdot 3 bar{b} )
в. ( 4 bar{b} )
( c .5 bar{b} )
D. ( 3 bar{a}+4 bar{b} bar{hline} bar{b}+4+4 bar{a} bar{a} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} )
12
468A non-zero vector ( vec{a} ) is parallel to the
line of intersection of plane ( boldsymbol{p}_{mathbf{1}} )
determined by ( hat{i}+hat{j} ) and ( hat{i} ) and plane ( P_{2} ) determined by vectors ( hat{mathbf{i}}-hat{mathbf{j}} ) and ( hat{mathbf{i}}+hat{boldsymbol{k}} )
then angle between a and vector ( hat{mathbf{i}}- ) ( 2 hat{j}+2 hat{k} ) is
This question has multiple correct options
A ( cdot frac{pi}{4} )
B. ( frac{pi}{2} )
c. ( frac{pi}{3} )
D. ( frac{3 pi}{4} )
12
469Find the coordinates of the points of trisection of the line segment joining the points ( A(-4,3) ) and ( B(2,-1) )12
470The position vectors of three points are ( mathbf{2} overrightarrow{boldsymbol{a}}-overrightarrow{boldsymbol{b}}+mathbf{3} overrightarrow{boldsymbol{c}}, overrightarrow{boldsymbol{a}}-mathbf{2} overrightarrow{boldsymbol{b}}+boldsymbol{lambda} overrightarrow{boldsymbol{c}} ) and ( boldsymbol{mu} overrightarrow{boldsymbol{a}}-mathbf{5} overrightarrow{mathbf{b}} )
where ( vec{a}, vec{b}, vec{c} ) are non coplanar vectors then the points are collinear when
A ( cdot lambda=-2, mu=frac{9}{4} )
B. ( lambda=-frac{9}{4}, mu=2 )
c. ( lambda=frac{9}{4}, mu=-2 )
D. None of these
12
471If the projections ofthe line segment ( A B )
on the ( y z ) -plane, ( z x ) -plane, ( x y ) -plane are ( sqrt{160}, sqrt{153}, 5 ) respectively, then the projection of ( A B ) on the ( z ) -axis is
A ( cdot sqrt{12} )
B. ( sqrt{13} )
c. 12
D. 144
12
472Find the distance of the point (-1,-5,-10) from the point of intersection of the line
( boldsymbol{r}=mathbf{2} overline{boldsymbol{i}}-overline{boldsymbol{j}}+mathbf{2} overline{boldsymbol{k}}+overline{boldsymbol{lambda}}(mathbf{3} overline{boldsymbol{i}}+mathbf{4} overline{boldsymbol{j}}+mathbf{2} overline{boldsymbol{k}}) ) and
the plane ( bar{r} .(bar{i}-bar{j}+bar{k})=5 )
12
473The coordinates of a point which divides the line joining the points ( P(2,3,1) ) and ( Q(5,0,4) ) in the ratio 1: 2 are
( ^{mathbf{A}} cdotleft(frac{7}{3}, 1, frac{5}{3}right) )
в. (4,1,3)
c. (3,2,2)
D. (1,-1,1)
12
474Find the value of ( k ) if the pts ( A equiv )
( (1,2,-1), B equiv(4,-2,4) ) and ( C equiv )
( (0,0, k) ) form a triangle right angled at
( boldsymbol{C} )
12
475If ( P(3,2,-4), Q(5,4,-6) ) and ( R )
(9,8,-10) are collinear, then ( R ) divides PQ in the ratio
A. 3: 2 internally
B. 3: 2 externally
c. 2: 1 internally
D. 2: 1 externally
12
476Are the points (1,1),(2,3) and (8,11) collinear?
A. collinear
B. Non collinear
c. coplaner
D. None of above
12
477The distance between the parallel planes given by the equations, ( vec{r} .(2 hat{i}- ) ( mathbf{2} hat{mathbf{j}}+hat{boldsymbol{k}})+mathbf{3}=mathbf{0} ) and ( vec{r} cdot(mathbf{4} hat{mathbf{i}}-mathbf{4} hat{mathbf{j}}+ )
( mathbf{2} hat{boldsymbol{k}})+mathbf{5}=mathbf{0} ) is –
A ( cdot 1 / 2 )
в. ( 1 / 3 )
c. ( 1 / 4 )
D. ( 1 / 6 )
12
478The equation of a plane which passes through the point of intersection of lines ( frac{x-1}{3}=frac{y-2}{1}=frac{z-3}{2}, ) and ( frac{x-3}{1}= )
( frac{boldsymbol{y}-mathbf{1}}{mathbf{2}}=frac{boldsymbol{z}-mathbf{2}}{mathbf{3}} ) and at greatest
distance from point (0,0,0) is-
A. ( 4 x+3 y+5 z=25 )
B. ( 4 x+3 y+5 z=50 )
c. ( 3 x+4 y+5 z=49 )
D. ( x+7 y-5 z=2 )
12
479( boldsymbol{A}=(mathbf{1},-mathbf{1}, mathbf{2}) ) and ( boldsymbol{B}=(mathbf{2}, mathbf{3}, mathbf{7}) ) are two
points. If ( boldsymbol{P}, boldsymbol{O} ) divide ( boldsymbol{A B} ) in the ratios
2: 3,-2: 3 respectively then ( P_{x}+ )
( boldsymbol{Q}_{boldsymbol{y}}= )
A ( cdot frac{-38}{5} )
в. ( frac{38}{5} )
c. ( frac{-2}{5} )
D. ( frac{-47}{6} )
12
480A line makes the same angle ( theta ) with
each of the ( x ) and ( z ) -axes. If the angle ( beta )
which it makes with ( y ) -axis, is such that ( sin ^{2} beta=3 sin ^{2} theta, ) then ( cos ^{2} theta ) is equal to
A ( cdot frac{2}{3} )
B. ( frac{1}{5} )
( c cdot frac{3}{5} )
D.
12
481If
( boldsymbol{A}(1,2,-1), B(4,0,-3), C(1,2,-1), D( )
find the distance between ( A B ) and ( C D )
12
48290.
The equation of the line passing through (4,3,1), parallel
to the plane x + 2y – Z-5 = 0 and intersecting the line
x +1 y-3 2-2
[JEE M 2019-9 Jan (M)
-3 2

1.
is:
x

4
y
+3
z+1
2
1
12
483The image of the point ( P(1,2,3) ) in the
plane ( 2 x-y+z+3=0 ) is
A ( cdot(-3,4,1) )
в. (3,5,2)
c. (-3,5,2)
D. (3,-5,2)
12
484If ( A ) and ( B ) be the points (3,4,5) and (-1,3,-7) respectively. Find the equation of the set of points ( boldsymbol{P} ) such
that ( boldsymbol{P} boldsymbol{A}^{2}+boldsymbol{P} boldsymbol{B}^{2}=boldsymbol{K}^{2}, ) where ( boldsymbol{K} ) is a
constant
12
485If sum of the perpendicular distances of a variable point ( boldsymbol{P}(boldsymbol{x}, boldsymbol{y}) ) from the lines
( boldsymbol{x}+boldsymbol{y}-mathbf{5}=mathbf{0} ) and ( mathbf{3} boldsymbol{x}-mathbf{2} boldsymbol{y}+mathbf{7}=mathbf{0} ) is
always ( 10 . ) Show that ( P ) must move on a
line.
12
48614. The image of the line *<!-Y-3-2-4 in the plane
3
2x – y +2+3= 0 is the line:
[JEE M 2014)
x-3
y + 5
z-2
X-3 y +5 2-2
-3 –
15
x +3
15
Z
-2
x+3y-5z +2
-3 -1
3
-5
12
487Two planes intersect each other to form
( mathbf{a}: )
A. plane
B. point
c. straight line
D. angle
12
488Find the direction cosines of vector ( vec{r} )
which is equally inclined to ( O X, O Y )
and ( O Z ). Find total number of such
vectors.
A ( cdot frac{1}{sqrt{3}}, frac{1}{sqrt{3}}, frac{1}{sqrt{3}} ; 6 )
B. ( frac{1}{sqrt{3}}, pm frac{1}{sqrt{3}}, frac{1}{sqrt{3}} ; 8 )
( ^{mathbf{C}} pm frac{1}{sqrt{3}}, pm frac{1}{sqrt{3}}, pm frac{1}{sqrt{3}} ; 8 )
D. None of these
12
489If the projection of a line segment on
( x, y ) and ( z ) axes are respectively 3,4 and
( 5, ) then the length of the line segment is
begin{tabular}{l}
A ( 3 sqrt{2} ) \
hline
end{tabular}
B. ( 5 sqrt{2} )
c. ( 6 sqrt{2} )
D. None of these
12
490Show that the points with position vectors ( vec{a}+vec{b}, vec{a}-vec{b} ) and ( vec{a}+k vec{b} ) are
collinear for all values of ( k )
12
491The equation of the plane through the points (2,3,1) and (4,-5,3) and parallel to ( x ) -axis is
A. ( x-z-1=0 )
в. ( 4 x+y-11=0 )
c. ( y+4 z-7=0 )
D. None of these
12
492If the d.rs of ( O A ) and ( O B ) are 1,-1,-1
and ( 2,-1,1, ) then the d.cs of the line perpendicular to both ( boldsymbol{O} boldsymbol{A} ) and ( boldsymbol{O B} ) are
в. -2,-3,1
c. ( frac{-2}{sqrt{14}}, frac{-3}{sqrt{14}}, frac{1}{sqrt{14}} )
D. ( frac{2}{sqrt{41}}, frac{3}{sqrt{41}}, frac{-1}{sqrt{41}} )
12
493Point ( D ) has coordinates as (3,4,5)
Find the coordinates of point ( G )
( mathbf{A} cdot(0,3,5) )
B. (3,0,4)
( mathbf{C} cdot(3,5,4) )
( mathbf{D} cdot(3,0,5) )
12
494The number of straight lines that are equally inclined to the threedimensional coordinate axes, is
A .2
B. 4
( c .6 )
D. 8
12
495( frac{x-2}{1}=frac{y-3}{1}=frac{z-4}{-1} & frac{x-1}{k}= )
( frac{boldsymbol{y}-boldsymbol{4}}{boldsymbol{2}}=frac{boldsymbol{z}-boldsymbol{5}}{boldsymbol{2}} ) are coplanar then ( mathbf{k}=? )
A. any value
B. exactly one value
c. exactly 2 values
D. exactly 3 values
12
496Determine if the points (1,5)(2,3) and (-2,-11) are collinear.
A. True
B. False
12
497The direction cosines of a line whose
equations are ( frac{x-1}{2}=frac{y+3}{4}=frac{z-2}{-3} )
A ( cdot frac{1}{sqrt{14}}, frac{-3}{sqrt{14}}, frac{2}{sqrt{14}} )
В ( cdot frac{2}{sqrt{29}}, frac{4}{sqrt{29}}, frac{-3}{sqrt{29}} )
c. ( frac{1}{sqrt{29}}, frac{-3}{sqrt{29}}, frac{2}{sqrt{29}} )
D. 2,4,-3
12
498Solve the following differential equation ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=boldsymbol{x}-mathbf{1} )
A ( cdot y=x^{2}+x )
B . ( y=x^{2} )
c. ( y=x^{2}-x )
D. None of the above
12
499If the foot of the perpendicular from
(0,0,0) to a plane is ( P(1,2,2) . ) Then, the equation of the plane is
A. ( -x+2 y+8 z-9=0 )
B. ( x+2 y+2 z-9=0 )
c. ( x+y+z-5=0 )
D. ( x+2 y-3 z+3=0 )
12
500The ratio in which ( x y- ) plane divides the line joining the points (1,0,-3) and (1,-5,7) is given by
A . 7: 3
в. 3: 7
( c .3: 4 )
D. 4: 7
12
501( vec{A} cdot(vec{A} times vec{B}) )
( A cdot vec{A} cdot vec{B} )
( mathbf{B} cdot underset{A}{longrightarrow} times underset{B}{longrightarrow} )
( c cdot 0 )
D. 1
12
502The distance of the point (1,-2,3) from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} ) measured
parallel to the line. ( frac{x}{2}=frac{y}{3}=frac{z}{-6}, quad ) is :
( A )
B. 6/7
( c cdot 7 / 6 )
D. ( 1 / 6 )
12
503Find the vector equation of the line through ( boldsymbol{A}(boldsymbol{3}, boldsymbol{4},-boldsymbol{7}) ) and ( boldsymbol{B}(boldsymbol{6},-1,1) )
Also find the cartesian form.
12
504One of the rectangular components of a
force of ( 40 N ) is ( 20 N . ) Find the angle it
makes with this component and
magnitude of other component.
12
505Find the distance of the point (1,2,-1) from the plane ( boldsymbol{x}-mathbf{2} boldsymbol{y}+mathbf{4} boldsymbol{z}-mathbf{1 0}= )
( mathbf{0} )
12
506Consider three vectors ( overrightarrow{boldsymbol{p}}=boldsymbol{i}+boldsymbol{j}+ )
( boldsymbol{k}, overrightarrow{boldsymbol{q}}=2 boldsymbol{i}+boldsymbol{4} boldsymbol{j}-boldsymbol{k} ) and ( overrightarrow{boldsymbol{r}}=boldsymbol{i}+boldsymbol{j}+boldsymbol{3} boldsymbol{k} . )
( p, q ) and ( r ) denotes the position vector of three non-collinear points, then the equation of the plane containing these points is
A. ( 2 x-3 y+1=0 )
B. ( x-3 y+2 z=0 )
c. ( 3 x-y+z-3=0 )
D. ( 3 x-y-2=0 )
12
507A line makes equal angles with the coordinate axis. The direction cosines of this line are
A ( cdotleft(frac{1}{3}, frac{1}{3}, frac{1}{3}right) )
B ( cdotleft(frac{1}{sqrt{3}}, frac{1}{sqrt{3}}, frac{1}{sqrt{3}}right) )
( ^{mathbf{c}} cdotleft(frac{1}{sqrt{3}}, frac{1}{3}, frac{1}{3}right) )
D ( cdotleft(frac{1}{sqrt{2}}, frac{1}{sqrt{2}}, frac{1}{sqrt{2}}right) )
12
508[2004]
25.
If the straight lines
1+t, z=2-t,
x=1+s, y=-3-as,z=1+as and x
with parameters s and t respectively, are co-planar, then a
equals.
(a) 0 (6) 1 (c) — 1 (d) -2
1.
1
.
12
509The expression in the vector form for the
point ( vec{r}_{1} ) of intersection of the plane ( vec{r}_{1} cdot vec{n}=d ) and the perpendicular line
( vec{r}=vec{r}_{0}+t vec{n} ) where ( t ) is a parameter
given by
A ( cdot overrightarrow{r_{1}}=overrightarrow{r_{0}}+left(frac{d-overrightarrow{r_{0}} cdot vec{n}}{vec{n}^{2}}right) vec{n} )
B ( cdot overrightarrow{r_{1}}=overrightarrow{r_{0}}+left(frac{overrightarrow{r_{0}} cdot vec{n}}{vec{n}^{2}}right) vec{n} )
C・ ( _{overrightarrow{r_{1}}}=overrightarrow{r_{0}}+left(frac{overrightarrow{r_{0}} cdot vec{n}-d}{|vec{n}|}right) vec{n} )
D ( overrightarrow{r_{1}}=overrightarrow{r_{0}}+left(frac{overrightarrow{r_{0}} cdot vec{n}}{|vec{n}|}right) vec{n} )
12
510what is the distance of the point (p,q,r) from the ( x ) -axis ?12
511( A B C D ) is a parallelogram. ( L ) is a point on ( B C ) which divides ( B C ) in the ratio
( mathbf{1}: mathbf{2} . boldsymbol{A} boldsymbol{L} ) intersects ( boldsymbol{B} boldsymbol{D} ) at ( boldsymbol{P} . boldsymbol{M} ) is a
point on ( D C ) which divides ( D C ) in the
ratio 1: 2 and ( A M ) intersects ( B D ) in ( Q )
Point ( P ) divides ( A L ) in the ratio
A . 1: 2
B. 1: 3
c. 3: 1
D. 2: 1
12
512For waht value of ( lambda ), the three numbers
( 2 lambda-1, frac{1}{4}, lambda-frac{1}{2} ) can be the direction
cosines of a straight line?
A ( cdot frac{1}{2} pm frac{sqrt{3}}{4} )
B. ( frac{3}{4} )
( c cdot pm frac{3}{4} )
D. ( frac{sqrt{3}}{2} pm frac{1}{4} )
12
513If the sum of the squares of the distance of a point from the three
coordinate axes be ( 36, ) then its distance
from the origin is
A. 6 units
B. ( 3 sqrt{2} ) units
c. ( 2 sqrt{3} ) units
D. none of these
12
514If (1,-1,0),(-2,1,8) and (-1,2,7) are three consecutive vertices of a
parallelogram then the fourth vertex is
A. (2,0,-1)
В. (1,0,-1)
c. (1,-2,0)
a 5
D. (0,-2,1)
12
515In ( triangle A B C ) the mid points of the sides
( A B, B C ) and ( C A ) are respectively
( (l, 0,0),(0, m, 0) ) and ( (0,0, n) . ) Then
( frac{A B^{2}+B C^{2}+C A^{2}}{l^{2}+m^{2}+n^{2}} ) is equal to
( A cdot 2 )
B. 4
c. 8
D. 16
12
516A point at which all the three perpendicular coordinate axes meets is known as
A. Meeting point
B. Origin
c. Triple point
D. None of these
12
517Find the coordinate of the point ( boldsymbol{P} ) where the line through ( A(3,-4,-5) ) and ( B(2,-3,1) ) crosses the plane passing through three points ( boldsymbol{L}(mathbf{2}, mathbf{2}, mathbf{1}), boldsymbol{M}(mathbf{3}, mathbf{0}, mathbf{1}) ) and ( boldsymbol{N}(mathbf{4},-mathbf{1}, mathbf{0}) )
Also, find the ratio in which ( P ) divides
the line segment ( boldsymbol{A B} )
12
51835.
If the plane 2ax – 3ay + 4az + 6 = 0 passes through the
midpoint of the line joining the centres of the spheres
x2 + y2 +z2 + 6x – 8y – 2z = 13 and
x2 + y2 +22 – 10x + 4y – 2z = 8 then a equals [2005]
(a) -1 (b) 1 (c) – (d) 2
12
519The ratio in which the plane ( vec{r} .(hat{i}-2 hat{j}+ ) ( mathbf{3} hat{k})=17 ) divides the line joining the points ( (-2 hat{i}+4 hat{j}+7 hat{k}) ) and ( (3 hat{i}-5 hat{j}+ )
( mathbf{8} hat{boldsymbol{k}}) ) is
A .1: 5
B. 1: 10
( c .3: 5 )
D. 3: 10
12
520Find the equation of line of intersection of planes ( vec{r} cdot(3 vec{i}-vec{j}+vec{k})=1 ) and ( vec{r} )
( (3 vec{i}+4 vec{j}-2 vec{k})=2 )
12
521Find he equation of the line passing through (1,2,3) and perpendicular to the two lines ( frac{x}{1}=frac{y}{2}=frac{z}{-1} ) and ( frac{x-1}{3}=frac{y}{2}=frac{z}{6} )12
522Plane passing through the points ( A(2,1 )
3), ( mathrm{B}(-1,2,4) ) and ( mathrm{C}(0,2,1) ). Determine its
point of intersection with the line ( r= )
( boldsymbol{j}+boldsymbol{k}+boldsymbol{t}(boldsymbol{2} boldsymbol{i}+boldsymbol{k}) )
A ( cdot(7,+1,4) )
B. (9,+1,-2)
c. (7,-1,4)
D. (9,-1,2)
12
523If ( theta ) is the angle between two lines
whose d.cs are ( l_{1}, m_{1}, n_{1} ) and ( l_{2}, m_{2}, n_{2} )
then
( frac{Sigmaleft(l_{1}+l_{2}right)^{2}}{4 cos ^{2}left(frac{theta}{2}right)}+frac{Sigmaleft(l_{mathrm{I}}-l_{2}right)^{2}}{4 sin ^{2}left(frac{theta}{2}right)}= )
( A )
B.
c. -1
( D )
12
524( mathbf{2} boldsymbol{x}-boldsymbol{y}+mathbf{3} boldsymbol{z}-mathbf{1}=mathbf{0} ) and ( mathbf{2} boldsymbol{x}-boldsymbol{y}+ )
( mathbf{3} boldsymbol{z}+mathbf{3}=mathbf{0} )
12
525Find the direction cosines of a line
which is perpendicular to the lines whose direction ratios are (1,-1,2) and (2,1,-1)
12
526The line passes through the points ( (5,1, a) &(3, b, 1) ) crosses the ( y z ) plane at the point ( left(0, frac{17}{2},-frac{13}{2}right), ) then
A ( . a=4, b=6 )
B. ( a=6, b=4 )
c. ( a=8, b=2 )
D. ( a=2, b=8 )
8
12
527The point in the ( x y- ) plane which is
equidistant from (2,0,3),(0,3,2) and (0,0,1) is
A ( .(1,2,3) )
В. (-3,2,0)
c. (3,-2.0)
D. (3,2,0)
E ( .(3,2,1) )
12
528The equation of the plane, which bisects the line joining the points (1,2,3) and (3,4,5) at right angles is?
A. ( x+y+z=0 )
B. ( x+y-z=9 )
c. ( x+y+z=9 )
D. ( x+y-z+9=0 )
12
529Find vector equation of line passing through the point whose position vector
is ( mathbf{3} hat{mathbf{i}}-mathbf{4} hat{mathbf{j}}+hat{boldsymbol{k}} ) and parallel to the vector
( 2 hat{i}+hat{j}-3 hat{k} . ) Also write the equation in
Cartesian form.
12
530Find the coordinates of the point ( boldsymbol{P} ) where the line through ( boldsymbol{A}(mathbf{3},-mathbf{4},-mathbf{5}) )
and ( B(2,-3,1) ) crosses the plane
passing through three points ( boldsymbol{L}(mathbf{2}, mathbf{2}, mathbf{1}), boldsymbol{M}(mathbf{3}, mathbf{0}, mathbf{1}) ) and ( boldsymbol{N}(mathbf{4},-mathbf{1}, mathbf{0}) )
Also, find the ratio in which ( P ) divides
the line segment ( boldsymbol{A B} )
12
531The distance between two points (1,1) and ( left(frac{2 t^{2}}{1+t^{2}}, frac{(1-t)^{2}}{1+t^{2}}right) ) is
( A cdot 4 t )
B. 3t
( c )
D. none of these
12
532The coordinates of the foot of the
perpendicular drawn from the point ( A(1,0,3) ) to the join of the points
( B(4,7,1) ) and ( C(3,5,3) ) are
A ( cdot(5,7,17) )
B ( cdotleft(frac{-5}{7}, frac{7}{3}, frac{-17}{3}right) )
( ^{mathbf{c}} cdotleft(frac{5}{7}, frac{-7}{3}, frac{17}{3}right) )
D. ( left(frac{5}{7}, frac{7}{3}, frac{17}{3}right) )
12
5331. The angles which a vector i + j + 2 k makes with X, Y
and Z axes respectively are
(a) 60°, 60°, 60° (b) 45°, 45°, 45°
(c) 60°, 60°, 45°
(d) 45°, 45°, 60°
12
534The angle between the line ( 2 x=3 y= )
( -z ) and ( 6 x=-y=-4 z ) is
A ( .90^{circ} )
B . ( 0^{circ} )
( c cdot 30^{0} )
D. ( 45^{circ} )
12
535The equation of a line passing through
(2,-3) and inclined at an angle of ( 135^{circ} )
with the positive direction of ( x ) -axis is.
A. ( x+y-1=0 )
в. ( x+y+1=0 )
c. ( x-y-1=0 )
D. ( -x+y+1=0 )
12
536A ray makes angles ( frac{pi}{3}, frac{pi}{3} ) with ( overrightarrow{O X} ) and ( boldsymbol{O Y} ) respectively. Find the angle made by it with ( overrightarrow{O Z} )12
53749.
If (2,3,5) is one end of a diameter of the sphere x2 + y2 +22
– 6x – 12y – 2z + 20 = 0, then the cooordinates of the other
end of the diameter are
[2007]
(a) (4,3,5)
(b) (4,3,-3)
(c) (4,9, -3)
(d) (4, -3,3).
12
538If ( boldsymbol{A} times boldsymbol{B}=boldsymbol{B} times boldsymbol{A}, ) then the angle
between ( A ) and ( B ) is
A. ( pi )
в. ( pi / 3 )
c. ( pi / 2 )
D. ( pi / 4 )
12
539The chord of contact of tangents from a
point ( boldsymbol{P} ) to a circle passes through ( boldsymbol{q} ). If
( l_{1} ) and ( l_{2} ) are the lengths of the tangents
from ( boldsymbol{P} ) and ( boldsymbol{Q} ) to the circle, then ( boldsymbol{P} boldsymbol{Q} ) is
equal to
A. ( frac{l_{1}+l_{2}}{2} )
в. ( frac{l_{1}-l_{2}}{2} )
C ( . sqrt{left|l_{1}^{2}-l_{2}^{2}right|} )
D. ( sqrt[2]{l_{1}^{2}+l_{2}^{2}} )
12
540In what ratio does the plane ( 2 x+y- )
( z=3 ) divide line segment joining the
point ( boldsymbol{a}=(mathbf{2}, mathbf{1}, mathbf{3}), boldsymbol{b}=(mathbf{9},-mathbf{2}, mathbf{5}) )
12
541If (1,-2,-2) and (0,2,1) are direction ratios of two lines, then the direction
cosines of a perpendicular to both the
lines are
( ^{mathrm{A}} cdotleft(frac{1}{3},-frac{1}{3}, frac{2}{3}right) )
в. ( left(frac{2}{3},-frac{1}{3}, frac{2}{3}right) )
( ^{c} cdotleft(-frac{2}{3},-frac{1}{3}, frac{2}{3}right) )
D. ( left(frac{2}{sqrt{14}},-frac{1}{sqrt{14}}, frac{3}{sqrt{14}}right) )
12
542If the vectors ( 3 bar{p}+bar{q} ; 5 bar{p}-3 bar{q} ) and ( 2 bar{p}+ )
( bar{q} ; 4 bar{p}-2 bar{q} ) are pairs of mutually
perpendicular vectors then ( sin (theta) ) is ( (theta ) is the angle between ( overline{boldsymbol{p}} ) and ( overline{boldsymbol{q}} )
A ( cdot sqrt{55} / 4 )
B . ( sqrt{55} / 8 )
c. ( 3 / 16 )
D. ( sqrt{247} / 16 )
12
543A hall has dimensions ( 24 m times 8 m times )
6 ( m ). The length of the longest pole which
can be accommodated in the hall is
A. 26
B. 28 m
( c cdot 30 m )
D. 36 m
12
544Let ( vec{a}=x^{2} hat{i}+2 hat{j}-2 hat{k}, vec{b}=hat{i}-hat{j}+hat{k} )
and ( vec{c}=x^{2} hat{i}+5 hat{j}-4 hat{k} ) be three vectors.
Find the values of ( x ) for which the angle between ( vec{a} ) and ( vec{b} ) is acute and the angle between ( vec{b} ) and ( vec{c} ) is obtuse.
A ( cdot(-3,-2) cup(2,3) )
B . (-3,-1)( cup(1,3) )
c. (-3,-1)( cap(1,3) )
D. (-3,-2)( cap(2,3) )
12
54591.
The plane through the intersection of the planes
x+y+z=1 and 2x+3y-z+4= 0 and parallel to y-axis also
passes through the point:
JEEM 2019-9 Jan (MI
(a) (-3,0,-1)
(b) (-3,1,1)
(c) (3,3,-1)
(d) (3, 2, 1)
12
546Find ( x ) so that the point (6,5,-3) is at a distance of 13 from the point ( (x,-7,0) )12
547Find the locus of the point, the sum of the squares of whose distances from
the planes ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=mathbf{0}, boldsymbol{x}-boldsymbol{y}=mathbf{0} )
( x+y-2 z=0 ) is 7
12
548Find the equation of line passing through ( (5,0,5) &(2,1,3) . ) Also show that ( (5,0,5),(2,1,3) &(-4,3,-1) ) are
collinear.
12
549Let the equation of the plane through
the points (-1,1,1) and (1,-1,1) and
perpendicular to the plane ( boldsymbol{x}+mathbf{2} boldsymbol{y}+ )
( mathbf{2} z=mathbf{7} ) be ( boldsymbol{k} boldsymbol{x}+boldsymbol{m} boldsymbol{y}-boldsymbol{n} boldsymbol{z}+boldsymbol{p}=mathbf{0} . ) Find
( boldsymbol{k}+boldsymbol{m}+boldsymbol{n}+boldsymbol{p} ? )
12
550( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=mathbf{1} )12
551The distance of the point (1,-5,9) from the planer. ( (hat{mathbf{i}}-hat{mathbf{j}}+hat{boldsymbol{k}})=mathbf{5} )
measured long the line ( r=hat{i}+hat{j}+hat{k} ) is
A ( 3 sqrt{5} )
5
B. ( 10 sqrt{3} )
( c cdot 5 sqrt{3} )
D. ( 3 sqrt{10} )
12
552Find the direction cosines of the unit
vector perpendicular to the plane ( vec{r} cdot(6 hat{i}-3 hat{j}-2 hat{k})+1=0 )
12
553The coordinates of any point, which lies in ( boldsymbol{y} boldsymbol{z} ) plane, are
This question has multiple correct options
A ( .(x, y, y) )
в. ( (0, y, y) )
c. ( (0, y, x) )
D. ( (x, y, z) )
12
55434.
The line passing through the points (5, 1, a) and (3, b, 1)
crosses the yz-plane at the point (0,7 -13). Then
(a) a=2, b=8
(b) a=4, b=6
(c) a=6, b=4
(d) a=8, b=2
12
555A line passes through the point (6,-7,-1) and ( (2,-3,1) . ) if the angle ( alpha ) which the line makes with the positive
direction of ( x ) -axis is acute, the
direction cosines of the line are,
A ( .2 / 3,-2 / 3,-1 / 3 )
в. ( 2 / 3,2 / 3,-1 / 3 )
c. ( 2 / 3,-2 / 3,1 / 3 )
D. ( 2 / 3,2 / 3,1 / 3 )
12
556Find the equations to the straight lines which are conjugate to the coordinate
axes with respect to the conic ( A x^{2}+ )
( 2 H x y+B y^{2}=1 )
Find the condition that they may
coincide, and interpret the result.
12
55780.
Z+4
3
lies in the plane, kx +my-z=9,
x-3 y +2
If the line,
2 -1
then 12 + m2 is equal to :
(a) 5 (b) 2
(c) 26
[JEE M 2016]
(d) 18
12
558x

1
V
-4
z
5
1
1
-k
15. The lines *-2 =973 – 3-4 and **=
coplanar if
(a) k=3 or-2
b) k=0 or -1
(c) k=1 or-1
1) k=0 or-3
=-7 are
[2003]
12
559Let ( boldsymbol{A}=(mathbf{1}, mathbf{2}, mathbf{3}) boldsymbol{B}= )
( (-1,-2,-1) C=(2,3,2) ) and ( D= )
( (4,7,6) . ) Then ( A B C D ) is a
A. rectangle
B. square
c. parallelogram
D. none of these
12
560If the projections of the line segment ( A B ) on the coordinate axes are ( 12,3, k )
and ( A B=13, ) then ( k^{2}-2 k+3 ) is equal
to:
This question has multiple correct options
A . 0
B.
c. 11
D. 27
12
561Find the coordinate of the points which trisect the line segment joining the points ( boldsymbol{A}(mathbf{2}, mathbf{1},-mathbf{3}) ) and ( boldsymbol{B}(mathbf{5},-mathbf{8}, mathbf{3}) )12
562If ( mathbf{A}=(-mathbf{1}, mathbf{6}, mathbf{6}), mathbf{B}=(-mathbf{4}, mathbf{9}, mathbf{6}), mathbf{G}= )
( frac{1}{3}(-5,22,22) ) and ( G ) is the centroid of
the ( Delta A B C ) then the name of the
triangle ( mathbf{A B C} ) is
A. an isosceles triangle
B. a right angled triangle
c. an equilateral triangle
D. a right-angled isosceles triangle
12
563Find ( x, ) if ( triangle A B C ) is right-angled at ( A )
where ( boldsymbol{A} equiv(mathbf{4}, mathbf{2}, mathbf{3}), boldsymbol{B} equiv(mathbf{3}, mathbf{1}, mathbf{8}), boldsymbol{C} equiv )
( (x,-1,2) )
12
564A plane ( pi ) makes intercepts 3 and 4
respectively on ( z- ) axis. If ( pi ) is parallel to ( boldsymbol{y}- ) axis, then its equation is ?
A. ( 3 x+4 z=12 )
B. ( 3 z+4 x=12 )
c. ( 3 y+4 z=12 )
D. ( 3 z+4 y=12 )
12
565Determine whether the points are
collinear.
( boldsymbol{P}(-mathbf{2}, mathbf{3}), boldsymbol{B}(mathbf{1}, mathbf{2}), boldsymbol{C}(mathbf{4}, mathbf{1}) )
12
566If ( P(x, y, z) ) is a point on the line segment joining ( boldsymbol{A}(mathbf{2}, mathbf{2}, mathbf{4}) ) and
( B(3,5,6) ) such that projection of ( overrightarrow{O P} ) on axes are ( frac{13}{5}, frac{19}{5}, frac{26}{5} ) respectively, then ( P ) divide AB in the ratio
A .3: 2
B. 2: 3
c. 1: 2
D. 1: 3
12
567Find the direction cosines of the
line passing through the two points (-2,4,-5) and (1,2,3)
12
568Which one of the following is best condition for the plane ( a x+b y+c z+ )
( d=0 ) to intersect the ( x ) and ( y ) axes at
equal angle
A ( cdot|a|=|b| )
( b mid )
B . ( a=-b )
c. ( a=b )
D. ( a^{2}+b^{2}=1 )
12
569The point of intersection of the line
joining the points (-3,4,-8) and (5,-6,4) with the ( X Y ) -plane is
( mathbf{A} cdotleft(frac{7}{3},-frac{8}{3}, 0right) )
в. ( left(-frac{7}{3},-frac{8}{3}, 0right) )
( ^{mathbf{c}} cdotleft(-frac{7}{3}, frac{8}{3}, 0right) )
D. ( left(frac{7}{3}, frac{8}{3}, 0right) )
12
570If ( P(x, y, x) ) is a point on the line
segment joining ( Q(2,2,4) ) and ( R(3,5,6) ) such that the projection of ( O P ) on the axis are ( frac{13}{5}, frac{19}{5}, frac{26}{5} )
respectively, then ( P ) divides ( Q R ) in the
ratio
A . 1: 2
B. 3: 2
( c cdot 2: 3 )
D. 1: 3
12
571The xy-plane divides the line joining the points (-1,3,4) and (2,-5,6)
A. internally in the ratio 2: 3
B. externally in the ratio 2: 3
c. internally in the ratio 3: 2
D. externally in the ratio 3: 2
12
572Point ( (boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma}) ) lies on the plane ( boldsymbol{x}+boldsymbol{y}+ )
( z=2 . ) Let ( vec{a}=alpha hat{i}+beta hat{j}+gamma hat{k} ) and ( hat{k} times )
( (hat{k} times vec{a})=0 ) then ( gamma= )
A .
B. 1
c. 2
D.
12
573Write the abscissa of the following
point
(0,5)
12
574The ratio in which the line joining
(2,-4,3) and (-4,5,-6) is divided by the plane ( 3 x+2 y+z-4=0 ) is
A .2: 1
B . 4: 3
( c cdot-1: 4 )
D. 2: 3
12
575If the point ( P(a, b, c), ) with reference to Eq. ( (i) ) lies on the plane ( 2 x+y+z=1 )
then the value of ( 7 a+b+c ) is
( mathbf{A} cdot mathbf{0} )
B. 12
( c cdot 7 )
D. 6
12
576The image of the point ( 3 hat{i}-2 hat{j}+hat{k} ) in the plane ( bar{r} .(3 hat{i}-hat{j}+4 hat{k})=2 )
( mathbf{A} cdot-hat{j}+3 hat{k} )
B . ( hat{j}-3 hat{k} )
( mathbf{c} .-hat{j}-3 hat{k} )
D ( .-2 hat{j}-3 hat{k} )
12
577Given planes are ( boldsymbol{P}_{1}: boldsymbol{c} boldsymbol{y}+boldsymbol{b} boldsymbol{z}=boldsymbol{x} quad boldsymbol{P}_{2}: boldsymbol{a} boldsymbol{z}+boldsymbol{c} boldsymbol{x}=boldsymbol{y} quad boldsymbol{P}_{3} )
( P_{1}, P_{2} ) and ( P_{3} ) pass through one line, if
A ( cdot a^{2}+b^{2}+c^{2}=a b+b c+c a )
B . ( a^{2}+b^{2}+c^{2}+2 a b c=1 )
( mathbf{c} cdot a^{2}+b^{2}+c^{2}=1 )
D. ( a^{2}+b^{2}+c^{2}+2 a b+2 b c+2 c a+2 a b c=1 )
12
578The plane which bisects the line
segment joining the points (-3,-3,4) and (3,7,6) at right angles, passes through which one of the following points?
A ( .(4,1,7) )
В. (4,1,-2)
c. (2,3,5)
D. (2,1,3)
12
579The point (3,0,-4) lies on the
A. Y-axis
B. z-axis
c. XY-plane
D. xz-plane
E. YZ-plane
12
580The point of intersection of the line ( frac{x-1}{3}=frac{y+2}{4}=frac{z-3}{-2} ) and plane
( 2 x-y+3 z-1=0 ) is.
B . (10,10,-3)
c. (-10,10,3)
D. None of these
12
581Show that the points ( boldsymbol{A}(mathbf{1}, mathbf{2}, mathbf{3}) )
( boldsymbol{B}(-1,-2,-3), C(2,3,2) ) and ( D(4,7,3) )
are the vertices of a parallelogram.
12
582The distance of the point (1,-2,3)
from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} )
measured parallel to the line whose direction cosines are proportional to
2,3,-5 is
A ( cdot frac{9}{7} )
B. ( frac{11}{7} )
c. ( frac{15}{7} )
D. None of these
12
583Find the distance between the following pairs of points:
(i) (2,3,5) and (4,3,1)
( (i i)(-3,7,2) ) and ((2,4,-1)
(iii) (-1,3,-4) and (1,-3,4)
(iv) (2,-1,3) and (-2,1,3)
12
584In the ( Delta A B C, ) if ( A B=sqrt{2} ; A C= )
( sqrt{mathbf{2 0}}, boldsymbol{B}=(mathbf{3}, mathbf{2}, mathbf{0}) ) and ( boldsymbol{C}=(mathbf{0}, mathbf{1}, mathbf{4}) )
then the length of the median passing
through ( boldsymbol{A} ) is
A ( cdot frac{3}{2} )
B. ( frac{9}{2} )
c. ( frac{3}{sqrt{2}} )
D. ( frac{sqrt{3}}{2} )
12
585A point on the line ( frac{boldsymbol{x}+mathbf{2}}{mathbf{1}}=frac{boldsymbol{y}-mathbf{3}}{-mathbf{4}}= )
( frac{z-1}{2 sqrt{2}} ) at a distance 6 from the point ( (2, )
3, 1) is
A ( cdot(4-21,1+12 sqrt{2}) )
В. ( left(frac{-4}{5}, frac{-9}{5}, 1right) )
C ( cdotleft(frac{-16}{5}, frac{39}{5}, frac{5-12 sqrt{2}}{5}right) )
D. ( left(frac{-16}{5},-21,1+12 sqrt{2}right) )
12
586The coordinates of the point where the line segment joining ( boldsymbol{A}(mathbf{5}, mathbf{1}, boldsymbol{6}) ) and ( B(3,4,1) ) crosses the yz plane are
A ( cdotleft(0, frac{17}{2}, frac{13}{2}right) )
в. ( left(0,-frac{17}{2}, frac{13}{2}right) )
c. ( left(0, frac{17}{2},-frac{13}{2}right) )
D. ( left(0,-frac{17}{2},-frac{13}{2}right) )
12
587Show that the points ( boldsymbol{A}(boldsymbol{3}, boldsymbol{2},-boldsymbol{4}), boldsymbol{B}(boldsymbol{5}, boldsymbol{4},-boldsymbol{6}) ) and
( C(9,8,-10) ) are collinear, find the ratio in which ( B ) divides ( overline{A C} ).
12
588Find the direction cosines of the sides
of the triangle whose vertices are (3,5,-4),(-1,1,2) and (-5,-5,-2)
12
589The distance between the X-axis and
the point (3,12,5) is
A. 3
B. 13
( c cdot 14 )
D. 12
E. 5
12
590The line ( frac{x-2}{3}=frac{y+1}{2}=frac{z-1}{-1} )
intersects the curve ( x y=c^{2}, z=0 ) if ( c ) is
equal to:
( A cdot pm 1 )
B. ( pm frac{1}{3} )
( mathrm{c} cdot pm sqrt{5} )
D. None of these
12
591If the points ( (h, 3,-4),(0,-7,10) ) and ( (1, k, 3) ) are collinear, then ( h+k ) is
( mathbf{A} cdot mathbf{4} )
B.
c. -4
D. 14
12
592The direction cosines of the line
passing through ( mathbf{P}(mathbf{2}, mathbf{3},-mathbf{1}) ) and the origin are
A ( cdot frac{2}{sqrt{14}}, frac{3}{sqrt{14}}, frac{1}{sqrt{14}} )
B. ( frac{2}{sqrt{14}}, frac{-3}{sqrt{14}}, frac{1}{sqrt{14}} )
c. ( frac{-2}{sqrt{14}}, frac{-3}{sqrt{14}}, frac{1}{sqrt{14}} )
D. ( frac{2}{sqrt{14}}, frac{-3}{sqrt{14}}, frac{-1}{sqrt{14}} )
12
593The vertices of a triangle are 2,3,5)( ,(-1,3,2),(3,5,-2), ) then the
angles are
( mathbf{A} cdot 30^{circ}, 30^{circ}, 30^{circ} )
( ^{mathrm{B}} cos ^{-1}left(frac{1}{sqrt{5}}right), 90^{circ}, cos ^{-1}left(frac{sqrt{5}}{sqrt{3}}right) )
c. ( 30^{circ}, 60^{circ}, 90^{circ} )
D ( cdot cos ^{-1}left(frac{1}{sqrt{3}}right), 90^{circ}, cos ^{-1}left(frac{sqrt{2}}{sqrt{3}}right) )
12
594Determine the three planes through the intersection of the planes ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=mathbf{1} )
and ( 2 x+3 y-z+4=0 ) which are
parallel to the three co – ordinate axes.
Also find the equation of the plane
perpendicular to the ( y z- ) plane and
passing through the point (2,3,1) and
(4,-5,3)
A ( cdot y-3 z-6=0 ; x-4 z=7 ) and ( 3 x-4 y-3=0 )
( y+4 z+7=0 )
B. ( y-3 z+6=0 ; x+4 z=7 ) and ( 3 x+4 y+3=0 )
( y-4 z+7=0 )
c. ( y+3 z+6=0 ; x+4 z=7 ) and ( 3 x+4 y+3=0 )
( y-4 z-7=0 )
D. ( y-3 z+6=0 ; x+4 z=7 ) and ( 3 x+4 y+3=0 )
( y+4 z-7=0 )
12
595Find the angle between the planes
whose vector equations are ( vec{r} cdot(2 hat{i}+2 hat{j}-3 hat{k})=5 ) and
( vec{r} cdot(3 hat{i}-3 hat{j}+5 hat{k})=3 )
12
596If a line makes an angle ( theta_{1}, theta_{2}, theta_{3} ) which
the axis respectively, then ( cos 2 theta_{1}+ )
( cos 2 theta_{2}+cos 2 theta_{3}=? )
A . -4
B . 2
( c .3 )
D. –
12
597The projection of the join of the two points (1,4,5),(6,7,2) on the line whose d.s’s are (4,5,6) is
A ( cdot frac{17}{sqrt{77}} )
B. ( frac{7}{6} )
c. 21
D. ( frac{7}{9} )
12
598( boldsymbol{A}(boldsymbol{3}, boldsymbol{2}, boldsymbol{0}), boldsymbol{B}(boldsymbol{5}, boldsymbol{3}, boldsymbol{2}), boldsymbol{C}(-boldsymbol{9}, boldsymbol{6},-boldsymbol{3}) ) are
three points forming a triangle. If ( boldsymbol{A} boldsymbol{D} )
the bisector of ( angle B A C ) meets ( B C ) in ( D )
then coordinates of ( D ) are
( ^{mathrm{A}} cdotleft(-frac{19}{8}, frac{57}{16}, frac{17}{16}right) )
В ( cdotleft(frac{19}{8},-frac{57}{16}, frac{17}{16}right) )
( ^{mathrm{C}} cdotleft(frac{19}{8}, frac{57}{16}, frac{17}{16}right) )
D. None of these
12
599Let ( P(3,2,6) ) be point in space and ( Q ) be appoint on the line ( vec{r}=(hat{i}-hat{j}+2 hat{k})+ )
( mu(-3 hat{i}+hat{j}+5 hat{k}) . ) Then the value of ( mu ) for which the vector ( overrightarrow{P Q} ) is parallel to the
plane ( boldsymbol{x}-mathbf{4} boldsymbol{y}+mathbf{3} boldsymbol{z}=mathbf{1} ) is
A. ( 1 / 4 )
B. – 1/4
c. ( 1 / 8 )
D. ( -1 / 8 )
12
600Find the direction cosines of the vector
( hat{mathbf{i}}+mathbf{2} hat{mathbf{j}}+mathbf{3} hat{boldsymbol{k}} )
12
601If the planes ( boldsymbol{x}-boldsymbol{c} boldsymbol{y}-boldsymbol{b} boldsymbol{z}=boldsymbol{0}, boldsymbol{c} boldsymbol{x}-boldsymbol{y}+ )
( boldsymbol{a} boldsymbol{z}=mathbf{0} ) and ( boldsymbol{b} boldsymbol{x}+boldsymbol{a} boldsymbol{y}-boldsymbol{z}=mathbf{0} ) pass
through a stright line,then the value of ( a^{2}+b^{2}+c^{2}+2 a b c ) is:
A . 1
B. 2
( c .3 )
D. none of these
12
602The equations of two planes are ( P_{1} )
( 2 x-y+z=2, ) and ( P_{2}: x+2 y-z= )
3. The equation of the plane which
passes through the point (-1,3,2) and is perpendicular to each of the planes
( P_{1} ) and ( P_{2} ) is
A ( cdot x+3 y-5 z+2=0 )
B. ( x+3 y+5 z-18=0 )
c. ( x-3 y-5 z+20=0 )
D. ( x-3 y+5 z=0 )
12
603Let the equation of the plane through the intersection of the planes ( x+2 y+ )
( mathbf{3} boldsymbol{z}-mathbf{4}=mathbf{0} ) and ( mathbf{2} boldsymbol{x}+boldsymbol{y}-boldsymbol{z}+mathbf{5}=mathbf{0} ) and
perpendicular to the plane ( 5 x+3 y+ )
( mathbf{6} z+mathbf{8}=mathbf{0} ) be ( boldsymbol{k} boldsymbol{x}+mathbf{1 5} boldsymbol{y}+boldsymbol{m} boldsymbol{z}+mathbf{1 7 3}= )
0. Find ( k+m )
12
604The ( x ) -coordinate of a point on the line joining the points ( P(2,2,1) ) and ( Q(5,1,-2) ) is ( 4 . ) Find its z-coordinate.
A . -1
B. -2
( c .1 )
D.
12
605The Cartesian equation of the plane ( overrightarrow{boldsymbol{r}}=(1+boldsymbol{lambda}-boldsymbol{mu}) hat{boldsymbol{i}}+(boldsymbol{2}-boldsymbol{lambda}) hat{boldsymbol{j}}+(boldsymbol{3}- )
( 2 lambda+2 mu) hat{k} ) is-
( mathbf{A} cdot 2 x+y=5 )
В . ( 2 x-y=5 )
c. ( 2 x+z=5 )
D. ( 2 x-z=5 )
12
606The projection of the join of the points (3,4,2),(5,1,8) on the line whose d.c’s ( operatorname{are}left(frac{2}{7}, frac{3}{7}, frac{6}{7}right) ) is
A. 7
B. ( frac{31}{71} )
( mathbf{c} cdot frac{42}{13} )
( D cdot frac{38}{138} )
12
607Find the value of ( p ) for which the points ( (-5,1),(1, p) ) and (4,-2) are collinear
( mathbf{A} cdot mathbf{1} )
B.
( c .-1 )
D. 2
12
608Find the shortest distance between the ( operatorname{lines} bar{r}=4 bar{i}-bar{j}+lambda(bar{i}+2 bar{j}-5 bar{k}) ) and
( overline{boldsymbol{r}}=overline{boldsymbol{i}}-overline{boldsymbol{j}}+2 overline{boldsymbol{k}}+boldsymbol{mu}(overline{boldsymbol{i}}+mathbf{2} overline{boldsymbol{j}}-mathbf{5} overline{boldsymbol{k}}) )
A. ( sqrt{220} )
B. ( frac{sqrt{221}}{sqrt{30}} )
c. 432
D. ( sqrt{33} )
12
609For what value of ( boldsymbol{m}, ) the points ( (boldsymbol{3}, mathbf{5}) ) ( (m, 6) ) and ( left(frac{1}{2}, frac{15}{2}right) ) are collinear?
( A cdot 9 )
B. 5
( c .3 )
D.
12
610The plane ( 2 x+3 y+k z-7=0 ) is
parallel to the line whose direction
ratios are (2,-3,1) then ( k= )
( mathbf{A} cdot mathbf{5} )
B. 8
c. 1
( D )
12
611Image of point ( mathrm{P}(1,2,3) ) with respect to plane ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=mathbf{1 2}, ) is
A. (5,4,3)
)
B. (9,6,3)
c. (5,6,7)
D. (3,4,5)
12
612The point of intersection of the lines ( overrightarrow{boldsymbol{r}}=(-overrightarrow{boldsymbol{i}}+mathbf{2} overrightarrow{boldsymbol{j}}+boldsymbol{3} overrightarrow{boldsymbol{k}})+boldsymbol{t}(-boldsymbol{2} overrightarrow{boldsymbol{i}}+overrightarrow{boldsymbol{j}}+overrightarrow{boldsymbol{k}}) )
and ( vec{r}=(2 vec{i}+3 vec{j}+5 vec{k})+s(vec{i}+2 vec{j}+ )
( mathbf{3} overrightarrow{boldsymbol{k}}) ) is:
A ( .(1,1,2) )
в. (2,1,1)
c. (1,1,1)
D. (1,2,1)
12
613The equation of the plane which is parallel to the ( x y- ) plane is
A. ( x=y )
B. ( z=c )
( mathbf{c} cdot y=c )
D. ( z=x y )
12
614The sum of the intercepts on the coordinate axes of the plane passing
through the point (-2,-2,2) and
containing the line joining the points
(1,-1,2) and ( (1,1,1), ) is
A . 4
B. -4
c. 12
D. -8
12
615Determine if the points (5,-1,1),(7,-4,7),(1,-8,10) and
(-1,-3,4) are the vertices of a rhombus or a square
12
616Assertion ( (A) ). The direction ratios of the line joining origin and point ( (x, y, z) ) must be ( x, y, z )

Reason (R): If ( P(x, y, z) ) is a point in space and ( |O P|=r, ) then the direction cosines of ( O P ) are ( frac{x}{r}, frac{y}{r}, frac{z}{r} )
A. Both A and R are individually true and R is the correct explanation of
B. Both A and R individually true but R is not the correct explanation of A
c. A is true but R is false
D. A is false but R is true

12
617A line separates a plane into three parts namely the two half-planes and the line itself
A. True
B. False
12
618If the line passing through the origin
makes angles ( theta_{1}, theta_{2}, theta_{3} ) with the planes
( X O Y, X O Z ) and ( Z O X ) respectively
then prove that ( cos ^{2} theta_{1}+cos ^{2} theta_{2}+ )
( cos ^{2} theta_{3}=2 )
12
619A triangle ( A B C ) is placed so that the
mid-points of the sides are on the ( x, y, z )
axes. Lengths of the intercepts made by the plane containing the triangle on
these axes are respectively ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} )
Coordinates of the centroid of the
triangle ( boldsymbol{A B C} ) are
A ( cdot(-alpha / 3, beta / 3, gamma / 3) )
B. ( (alpha / 3,-beta / 3, gamma / 3) )
c. ( (alpha / 3, beta / 3,-gamma / 3) )
D. ( (alpha / 3, beta / 3, gamma / 3) )
12
620The point on the line ( frac{boldsymbol{x}-mathbf{1}}{mathbf{1}}=frac{boldsymbol{y}+mathbf{3}}{-mathbf{2}}= )
( frac{z+5}{-2} ) at a distance of 6 from the point
(1,-3,-5) is
в. (3,-7,-9)
D. (-3,5,3)
12
621Find the direction cosines of the vector
joining the points ( A(1,2,-3) ) and ( B(-1,-2,1) ) directed from ( A ) to ( B )
12
622The ordinate of the point which divides the lines joining the origin and the point (1,2) externally in the ratio of 3: 2 is
A . -2
B. ( frac{3}{5} )
( c cdot frac{2}{5} )
D. 6
12
623A tangent to the curve ( y=f(x) ) at
( boldsymbol{p}(boldsymbol{x}, boldsymbol{y}) ) meets ( boldsymbol{x}-boldsymbol{a} boldsymbol{x} boldsymbol{i} boldsymbol{s} ) at ( boldsymbol{A} ) and ( boldsymbol{y}- )
axis at ( B . ) If ( overline{A P}: overline{B P}=1: 3 ) and
( f(1)=1 ) then the curve also passes
through the point.
( ^{mathbf{A}} cdotleft(frac{1}{2}, 4right) )
B ( cdotleft(frac{1}{3}, 24right) )
c. ( left(2, frac{1}{8}right) )
D. ( left(3, frac{1}{28}right) )
12
624The equation of the plane passing through ( (a, b, c) ) and parallel to the plane ( r cdot(hat{i}+hat{j}+hat{k})=2 ) is
A. ( x+y+z=1 )
B . ( a x+b y+c z=1 )
c. ( x+y+z=a+b+c )
D. None of these
12
625If a unit vector ( vec{a} ) makes angles ( frac{pi}{3} ) with ( hat{i} )
( frac{pi}{4} ) with ( hat{j} ) and an acute angle ( theta ) with ( hat{k} )
then find ( theta ) and hence, the components of ( overrightarrow{boldsymbol{a}} )
12
626The three planes divides the space into
A. four parts
B. six parts
c. eight parts
D. sixteen parts
12
627Assertion
The points (1,1)( &(-1,-1) ) lie on the
same side of the line ( boldsymbol{x}-boldsymbol{y}+mathbf{1}=mathbf{0} )
Reason
The algebraic perpendicular distances from the given points to the line have
same sign
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
6284.
A vector Ă has components A,, A2, A, in a right-handed
rectangular Cartesian coordinate system oxyz. The
coordinate system is rotated about the x-axis through an
T
angle =. Find the components of A in the new coordinate
system, in terms of A,, A2, Az.
(1983 – 2 Marks)
12
629The three point ( boldsymbol{A}(mathbf{0}, mathbf{0}, mathbf{0}), boldsymbol{B}(mathbf{2},-mathbf{3}, mathbf{3}), boldsymbol{C}(-mathbf{2}, mathbf{3},-mathbf{3}) ) are
collinear. Find in what ratio each point
divides the segment joining other two
12
630The ratio in which the join of (1,-2,4)
and (4,2,-1) divided by the ( X-Y )
plane is
A . 1: 3
B. 3: 1
c. 4: 1
D. 1: 4
12
631The graph of the equation ( y^{2}+z^{2}=0 )
in three dimensional space is
A. x- axis
B. y- axis
c. z- axis
D. yz-plane
12
632If ( frac{1}{2}, frac{1}{2}, n(n<0) ) are the dos of a line,
then the angle made by that line with ( boldsymbol{O} boldsymbol{Z} ) where ( boldsymbol{O}=(mathbf{0}, mathbf{0}, mathbf{0}) ) is
A ( cdot frac{-1}{sqrt{2}} )
B . ( 45^{circ} )
( c cdot 60^{circ} )
D. ( 135^{circ} )
12
633Find the co-ordinates of the points on the join of (-3,7,-13) and (-6,1,-10) which is nearest to the intersection of the planes: ( 3 x-y- )
( mathbf{3} z+mathbf{3 2}=mathbf{0} ) and ( mathbf{3} boldsymbol{x}+mathbf{2} boldsymbol{y}-mathbf{1 5 z}-mathbf{8}= )
0
12
634The equation of plane passing through (-1,0,-1) parallel to ( x z ) plane is
В. ( y=0 )
c. ( -x-z=0 )
D. None of the above
12
635The condition that the line ( frac{x-alpha^{prime}}{l}= ) ( frac{boldsymbol{y}-boldsymbol{beta}^{prime}}{boldsymbol{m}}=frac{boldsymbol{z}-boldsymbol{gamma}^{prime}}{boldsymbol{n}} ) in the plane ( boldsymbol{A} boldsymbol{x}+ )
( boldsymbol{B} boldsymbol{y}+boldsymbol{C} boldsymbol{z}+boldsymbol{D}=mathbf{0} ) is
( mathbf{A} cdot A alpha^{prime}+B beta^{prime}+C gamma^{prime}+D=0 ) and ( A l+B m+C n neq 0 )
B ( cdot A alpha^{prime}+B beta^{prime}+C gamma^{prime}+D neq 0 ) and ( A l+B m+C n=0 )
C ( cdot A alpha^{prime}+B beta^{prime}+C gamma^{prime}+D=0 ) and ( A l+B m+C n=0 )
D ( cdot A alpha^{prime}+B beta^{prime}+C gamma^{prime}=0 ) and ( A l+B m+C n=0 )
12
636A rectangular parallelopiped is formed by drawing planes through the point (-1,2,5) and (1,-1,-1) and paralle to the coordinates planes. The length of the diagonal of the parallelopiped is12
63782.
The distance of the point (1, -5,9) from the plane x-y+z=5
measured along the line x=y=zis: [JEE M 2016]
10
a)
TT
(b)
(c) 3/10
(d) 103
12
638If the points (-1,3,2),(-4,2,-2) and ( (5,5, lambda) ) are collinear, then ( lambda ) is equal to
A . -10
B. 5
( c .-5 )
D. 10
12
639The equations of the line of intersection
of the planes ( x+y+z=2 ) and ( 3 x- )
( boldsymbol{y}+mathbf{2} z=mathbf{5} ) in symmetric form are
A ( frac{x-frac{7}{4}}{4}=frac{y-frac{1}{4}}{-1}=frac{z}{-3} )
c. ( frac{x}{1}=frac{3 y+1}{1}=frac{3 z-7}{-4} )
D. none of these
12
64023.
(1)
Find
Find the equation of the plane passing through the
points (2,1,0), (5,0, 1) and (4,1,1).
If P is the point (2,1, 6) then find the point Q such that
PQ is perpendicular to the plane in (i) and the mid point
of PQ lies on it.
(2003 – 4 Marks)
12
641Find the angle between the following pair of lines:
(i) ( vec{r}=2 hat{i}-5 hat{j}+hat{k}+lambda(3 hat{i}-2 hat{j}+6 hat{k}) )
and ( vec{r}=mathbf{7} hat{boldsymbol{i}}-boldsymbol{6} hat{boldsymbol{k}}+boldsymbol{mu}(hat{boldsymbol{i}}+mathbf{2} hat{boldsymbol{j}}+mathbf{2} hat{boldsymbol{k}}) )
(ii) ( vec{r}=3 hat{i}+hat{j}-2 hat{k}+lambda(hat{i}-hat{j}-2 hat{k}) )
and ( vec{r}=2 hat{i}-hat{j}-56 hat{k}+ )
( mu(hat{mathbf{3}} mathbf{i}-mathbf{5} hat{mathbf{j}}-mathbf{4} hat{boldsymbol{k}}) )
12
6422.
Let : 2x + y – z = 3 and P2 : x +2y +z = 2 be two planes.
Then, which of the following statement(s) is (are) TRUE?
(a) The line of intersection of P, and P, has direction
ratios 1, 2, -1
3x -4 1-3yZ
(b) The line . -= =
is perpendicular to the line of intersection of P, and
P2
c)
(d)
The acute angle between P, and P, is 60°.
If P, is the plane passing through the point (4, 2, -2)
and perpendicular to the line of intersection of P, and
P, then the distance of the point (2, 1, 1) from the
plane Pz is 7
12
64314
The equation of a plane passing through the line of
intersection of the planes x + 2y + 3z=2 and x-y+z=3 and
at a distance
from the point (3,1,-1) is
(2012)
(a) 5x-1ly+z=17
(c) x+y+z= 13
(b) V2x+y = 3/2 – 1
(d) x-V2y =1-2
12
644Find the vector equation of the line joining (1,2,3) and (-3,4,3) and show pependicular to the z-axis12
645In the given figure, co-ordinates of the
midpoint of ( boldsymbol{A B} ) are
A . (0,2)
в. (0,3)
c. (1,2)
D. (3,1)
12
646The centroid of triangle ( boldsymbol{A}(mathbf{3}, mathbf{4}, mathbf{5}) ; boldsymbol{B}(mathbf{6}, mathbf{7}, mathbf{2}) ; boldsymbol{C}(mathbf{0},-mathbf{5}, mathbf{2}) ) is
A ( .(3,2,3) )
в. (5,2,1)
c. (2,5,1)
D. (3,4,1)
12
647Find the coordinates of point which
divides the line joining the points (3,4) and (6,1) in the ratio of 1: 2
12
648( bar{a}, bar{b}, bar{c} ) are three non-zero vectors such that any two of them are non-collinear. If ( bar{a}+bar{b} ) is collinear with ( bar{c} ) and ( bar{b}+bar{c} ) is
collinear with ( bar{a} ), then what is their sum?
A . -1
B. 0
c. 1
D. 2
12
649If a line makes angles ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} ) with
coordinate axes, find
( cos ^{2} alpha+cos ^{2} beta+cos ^{2} gamma+1 )
12
650The straight lines ( frac{boldsymbol{x}-mathbf{1}}{mathbf{1}}=frac{boldsymbol{y}-mathbf{2}}{mathbf{2}}= )
( frac{z-3}{3} ) and ( frac{x-1}{1}=frac{y-2}{2}=frac{z-3}{3} ) are
A. Parallel lines
B. Intersecting at ( 60^{circ} )
c. skew lines
D. Intersecting at right angle
12
651Find the angles between the lines,
whose direction cosines are give by the equation ( l^{2}-m^{2}+n^{2}=0, l+m+ )
( boldsymbol{n}=mathbf{0} )
A. 0
в. ( frac{pi}{6} )
( c cdot frac{pi}{4} )
D.
12
652Find the equation of the line passing through (1,2,-4) and perpendicular to both the lines ( frac{x-1}{2}=frac{y+2}{-3}=frac{z-4}{4} )
and ( frac{boldsymbol{x}-mathbf{3}}{mathbf{5}}=frac{boldsymbol{y}+mathbf{6}}{mathbf{1}}=frac{boldsymbol{z}+mathbf{1 0}}{mathbf{2}} )
12
653If the ( z x ) -plane divides the line segment joining (1,-1,5) and (2,3,4) in the ratio ( p: 1, ) then ( p+1= )
A ( cdot frac{1}{3} )
B. 1: 3
( c cdot frac{3}{4} )
D. ( frac{4}{3} )
12
654The image of the point (-1,3,4) in the
plane ( boldsymbol{x}-mathbf{2} boldsymbol{y}=mathbf{0} ) is
( ^{mathbf{A}} cdotleft(-frac{17}{3},-frac{19}{3}, 4right) )
В. (15,11,4)
( ^{mathbf{C}} cdotleft(-frac{17}{3},-frac{19}{3}, 1right) )
D. ( left(frac{9}{5},-frac{13}{5}, 4right) )
12
65575. The angle between the lines whose direction cosines satisfy
the equations 1+m+n= 0 and 12 = m² +nis
[JEE M 2014
wa
12
656f a plane passes through the point (1,1,1) and is perpendicular to the line ( frac{x-1}{3}=frac{y-1}{0}=frac{z-1}{4} ) then its
perpendicular distance from the origin is
A ( cdot frac{3}{4} )
B. ( frac{4}{3} )
( c cdot frac{7}{5} )
D.
12
657If the lines ( frac{x-1}{2}=frac{y+1}{3}=frac{z-1}{4} ) and
( frac{boldsymbol{x}-mathbf{3}}{mathbf{1}}=frac{boldsymbol{y}-boldsymbol{k}}{mathbf{2}}=frac{boldsymbol{z}}{mathbf{1}} ) intersect, then ( boldsymbol{k}= )
A. 0
B. 3
( c cdot frac{7}{2} )
D. –
12
658If the vertices of a triangle are (-1,6,-4),(2,1,1) and (5,-1,0) then
the centroid of the triangle is
В. (2,2,-1)
( ^{mathbf{c}} cdotleft(3,3,-frac{3}{2}right) )
D. none of these
12
659An equation of sphere with centre at
origin and radius ( r ) can be represented
as
A ( cdot x^{2}+y^{2}+z^{2}=r )
B . ( x^{2}+y^{2}+z^{2}=r^{2} )
c. ( x^{2}+y^{2}+z^{2}=2 r^{2} )
D. None of the above
12
660Obtain the equation of the line passing through (1,1,2) and (2,1,2) in the vector form.12
661The reflection of the plane ( 2 x+3 y+ )
( 4 z-3=0 ) in the plane ( x-y+z- )
( mathbf{3}=mathbf{0} ) is the plane
A. ( 4 x-3 y+2 z-15=0 )
B. ( x-3 y+2 z-15=0 )
c. ( 4 x+3 y-2 z+15=0 )
D. none of these
12
662A normal to the plane ( x=2 ) is…
A ( .(0,1,1) )
в. (2,0,2)
c. (1,0,0)
D. (0,1,0)
12
663An ordered triplet corresponds to in three dimensional
space.
A. three points
B. a unique point
c. a point in each octant
D. infinite number of points
12
664A point at a distance of ( sqrt{6} ) from the origin which lies on the straight line ( frac{x-1}{1}=frac{y-2}{2}=frac{z+1}{3} ) will be
( mathbf{A} cdot(1,-1,2) )
В. (1,2,-1)
C ( cdotleft(frac{5}{7}, frac{10}{7}, frac{-13}{7}right) )
D. ( left(frac{5}{7}, frac{2}{7}, frac{-6}{7}right) )
12
665If ( (3, lambda, mu) ) is a point on the line then
( 2 x+y+z=0=x-2 y+z-1 ) then
A ( cdot lambda=frac{-8}{3}, mu=-frac{1}{3} )
B. ( lambda=frac{-1}{3}, mu=-frac{8}{3} )
c. ( lambda=frac{-4}{3} mu=frac{-14}{3} )
D. ( lambda=-5, mu=-1 )
12
666Find the point of intersection of the plane ( bar{r} .(1,1,1)=2 ) and the line ( bar{r}= ) ( (4,5,3)+k(2,2,1), k in R )12
667The equation of the plane which is equidistant from the two parallel planes ( 2 x-2 y+z+3=0 ) and ( 4 x-4 y+ )
( mathbf{2} z+mathbf{9}=mathbf{0} ) is :
A. ( 8 x-8 y+2 z+15=0 )
B. ( 8 x-8 y+4 z+15=0 )
c. ( 8 x-8 y+4 z+3=0 )
D. ( 8 x-8 y+4 z-3=0 )
E ( .8 x-8 y+4 z+4=0 )
12
668The distance between the line ( r=2 hat{i}- ) ( 2 hat{j}+3 hat{k}+lambda(hat{i}-hat{j}+4 hat{k}) ) and the plane
( r cdot(hat{i}+5 hat{j}+hat{k})=5 ) is
A ( cdot frac{10}{9} )
в. ( frac{10}{3 sqrt{3}} )
c. ( frac{10}{3} )
D. None of these
12
669If the distance between a point ( P ) and
the point (1,1,1) on the line ( frac{x-1}{3}= )
( frac{y-1}{4}=frac{z-1}{12} ) is ( 13, ) then the coordinates
of ( P ) are
A. (3,4,12)
В. ( left(frac{3}{13}, frac{4}{13}, frac{12}{13}right) )
c. (4,5,12)
D. (40, 53, 157)
12
670The equation of the plane through the
intersection of ( p_{1} & p_{2} ) containing the
point (1,1,2) is
A. ( 5 x-6 y+4 z=4 )
в. ( 5 x+6 y-4 z=3 )
c. ( 3 x-2 y+4 z=9 )
D. Nonoe of these
12
671If a plane passes through a fixed point
(2,3,4) and meets the axes of reference
in ( A, B ) and ( C, ) the point of intersection
of the planes through ( A, B, C ) parallel to the coordinate planes can be
This question has multiple correct options
A ( cdot(6,9,12) )
B ( cdot(4,12,16) )
( mathbf{c} cdot(1,1,-1) )
D. (2,3,-4)
12
672( boldsymbol{L}_{1}: frac{boldsymbol{x}-mathbf{1}}{mathbf{2}}=frac{boldsymbol{y}-mathbf{2}}{mathbf{3}}=frac{boldsymbol{z}-mathbf{3}}{mathbf{4}} )
( L_{2}: frac{x-2}{3}=frac{y-4}{2}=frac{z-5}{5} ) be two
given lines, point P lies on ( L_{1} ) and Q lies
on ( L_{2} ) then distance between ( P ) and ( Q )
can be
This question has multiple correct options
A ( cdot frac{1}{3} )
B.
c. 15
D. 30
12
673Let ( X ) and ( Y ) be two related variables.
The two regression lines are given by ( boldsymbol{x}-boldsymbol{y}+mathbf{1}=mathbf{0} ) and ( mathbf{2} boldsymbol{x}-boldsymbol{y}+mathbf{4}=mathbf{0} . ) The
two regression lines pass through the point:
A ( cdot(-4,-3) )
в. (-6,-5)
c. (3,-2)
D. (-3,-2)
12
674The values of a for which
( (8,-7, a),(5,2,4) ) and (6,-1,2) are
collinear, is given by?
A .2
B. -2
c. -1
D.
12
675If ( z_{1} ) and ( z_{2} ) are ( z ) co-ordinates of the
points of trisection of the segment joining the points ( boldsymbol{A}(mathbf{2}, mathbf{1}, mathbf{4}), boldsymbol{B}(-mathbf{1}, mathbf{3}, mathbf{6}) )
then ( z_{1}+z_{2}= )
A . 1
B. 4
( c .5 )
D. 10
12
676If ( boldsymbol{P}=(mathbf{0}, mathbf{0}, mathbf{0}), boldsymbol{Q}=(mathbf{3}, mathbf{6}, mathbf{9}) ) and ( boldsymbol{R} ) is a
point of trisection of ( boldsymbol{P Q}, ) then ( boldsymbol{R}_{boldsymbol{y}}= )
( A cdot frac{4}{3} )
B . 2
( c .3 )
D.
12
677Find the image of (1,5,1) in the plane
( boldsymbol{x}-mathbf{2} boldsymbol{y}+boldsymbol{z}+mathbf{5}=mathbf{0} )
12
678If the centroid of tetrahedron ( O A B C )
where ( A, B, C ) are given by ( (a, 2,3),(1, b, 2) ) and ( (2,1, c) )
respectively is ( (1,2,-2), ) then distance of ( boldsymbol{P}(boldsymbol{a}, boldsymbol{b}, boldsymbol{c}) ) from origin is
( mathbf{A} cdot sqrt{195} )
в. ( sqrt{14} )
c. ( sqrt{frac{107}{14}} )
D. ( sqrt{13} )
12
6791.
From a point O inside a triangle ABC, perpendiculars OD,
OE, OF are drawn to the sides BC, CA, AB respectively.
Prove that the perpendiculars from A, B, C to the sides EF,
FD, DE are concurrent.
(1978)
hone
12
680A parallelepiped is formed by planes
drawn through the point ( P(6,8,10) ) and
( Q(3,4,8) ) parallel to the coordinate
planes. Find the length of edges and edges and diagonals of the
parallelepiped.
12
681The angle between the lines whose
direction cosines satisfy the equations ( l+m+n=0 ) and ( l^{2}+m^{2}+n^{2} ) is
A ( cdot frac{pi}{2} )
в.
c.
D. ( frac{pi}{6} )
12
682The line passing through the points ( 10 hat{i}+3 hat{j}, 12 hat{i}+5 hat{j} ) also passes through
the point ( a hat{i}+11 hat{j}, ) then ( a= )
A . -8
B. 4
c. 18
D. 12
12
683Direction ratio of line given by ( frac{x-1}{3}= ) ( frac{6-2 y}{10}=frac{1-z}{-7} ) are:
( A cdot )
в. ( )
c. ( )
D. ( )
12
684Point ( D ) has coordinates as (3,4,5)
Referring to the given figure, find the
coordinates of point ( boldsymbol{B} )
A ( cdot(3,0,4) )
( mathbf{B} cdot(4,3,0) )
C. (4,0,3)
D. (3,4,0)
12
685Which of the following are equations for
the plane passing through the points
( P(1,1,-1), Q(3,0,2) ) and ( R(-2,1,0) ? )
A ( cdot(2 hat{i}-3 hat{j}+3 hat{k}) cdot((x+2) hat{i}+(y-1) hat{j}+z hat{k})=0 )
B . ( x=3-t, y=-11 t, z=2-3 t )
( mathbf{c} cdot(x+2)+11(y-1)=3 x )
D. ( (2 hat{i}-hat{j}+3 hat{k}) times(-3 hat{i}+hat{k}) cdot((x+2) hat{i}+(y-1) hat{j}+z hat{k})= )
0
12
686The direction ratios of a vector are
( 2,-3,4 . ) Find its direction cosines
12
687If ( A=5 ) units, ( B=6 ) units and ( |vec{A} times vec{B}|= ) 15 units, then the angle between ( vec{A} ) and ( vec{B} )
is:
This question has multiple correct options
( A cdot 30^{circ} )
B. ( 60^{circ} )
c. ( 90^{circ} )
D. ( 150^{circ} )
12
688Find the shortest distance between the
skew lines:
( l_{1}: frac{x-1}{2}=frac{y+1}{1}=frac{z-2}{4} )
( l_{2}: frac{x+2}{4}=frac{y-0}{-3}=frac{z+1}{1} )
12
689Consider the plane ( (boldsymbol{x}, boldsymbol{y}, boldsymbol{z})= )
( (0,1,1)+lambda(1,-1,1)+mu(2,-1,0) . ) The
distance of this plane from the origin is:
A ( cdot frac{1}{3} )
B. ( frac{sqrt{3}}{2} )
( c cdot sqrt{frac{3}{2}} )
D. ( frac{2}{sqrt{3}} )
12
69071. Distance between two parallel planes 2x + y + 2z = 8 and
4x + 2y + 4z +5=0 is
[JEE M 2013]
12
691Vector equation of line ( frac{mathbf{3}-boldsymbol{x}}{mathbf{3}}= )
( frac{2 y-3}{5}=frac{z}{2} ) is ( _{–}——-k in R )
A ( cdot bar{r}=(3,5,2)+k(3,3,0) )
B ( cdot quad bar{r}=left(3, frac{3}{2}, 0right)+k(-6,5,4) )
c. ( bar{r}=(3,3,0)+k(3,5,2) )
D ( cdot_{bar{r}}=(-6,5,4)+kleft(3, frac{3}{2}, 0right) )
12
692If ( (1,1, a) ) is the centroid of the triangle formed by the points ( (1,2,-3),(mathrm{b}, 0,1) ) and (-1,1,-4) then ( a-b= )
A . -5
B. -7
( c .5 )
( D )
12
693Show that the lines ( frac{x-1}{2}=frac{y-2}{3}= ) ( frac{z-3}{4} ) and ( 4 x-3 y+1=0=5 x- )
( 3 z+2 ) are interesting lines. Also find
point of intersection.
12
69488. If L, is the line of intersection of the planes
2x-2y+3z-2=0, x-y+z+1=0 and L, is the line of
intersection of the planes x+2y-z-3=0,
3x-y+2z-1=0, then the distance of the origin from the
plane, containing the lines L, and L,, is: [JEEM 2018]
(a) ZNZ
(6) 212
12
695What is the direction cosine of angle which the vector ( sqrt{2 hat{i}}+hat{j}+ )
( hat{k} ) makes with ( y- ) axis ( ? )
12
6962. If a vector P making angles a, b, and yrespectively with
the X, Y and Z axes respectively.
Then sin? a + sin2 B + sin2 y=
(a) (b) 1 (c) 2 (d) 3
12
697Point ( D ) has coordinates as (3,4,5)
Find the coordinates of the point ( boldsymbol{F} )
( mathbf{A} cdot(0,4,0) )
B. (0,0,4)
( mathbf{C} cdot(0,0,5) )
( mathbf{D} cdot(0,5,0) )
12
698If ( overline{O A}=3 bar{i}+bar{j}-bar{k},|overline{A B}|=2 sqrt{6} ) and ( A B )
has the direction ratios 1,-1,2 then
( |boldsymbol{O} boldsymbol{B}|= )
A . ( sqrt{35} )
B. ( sqrt{41} )
c. ( sqrt{26} )
D. ( sqrt{55} )
12
699Find the equation of the plane through the points ( boldsymbol{A}(mathbf{2}, mathbf{2}-mathbf{1}), boldsymbol{B}(mathbf{3}, mathbf{4}, mathbf{2}) ) and
( boldsymbol{C}(boldsymbol{7}, boldsymbol{0}, boldsymbol{6}) )
A. ( 5 x+2 y-3 z=17 )
B. ( 5 x+2 y+3 z=17 )
c. ( 5 x+y-3 z=7 )
D. ( 5 x+y+3 z=7 )
12
700Show that the three lines with direction
( operatorname{cosines} )
( frac{12}{13}, frac{-3}{13}, frac{-4}{13}: frac{4}{13}, frac{12}{13}, frac{3}{13} ; frac{-4}{13}, frac{12}{13} ) are mutually
perpendicular
12
701If the projection of point ( boldsymbol{P}(overrightarrow{boldsymbol{p}}) ) on the
plane ( vec{r} cdot vec{n}=q ) is the point ( S(vec{s}), ) then?
A ( cdot vec{s}=frac{(q-vec{p} cdot vec{n}) vec{n}}{|vec{n}|^{2}} )
В ( cdot vec{s}=vec{p}+frac{(vec{p} cdot vec{n}-q)}{|vec{n}|^{2}} vec{n} )
C・ ( _{vec{s}}=vec{p}-frac{(vec{p} cdot vec{n}) vec{n}}{|vec{n}|^{2}} )
D ( vec{s}=vec{p}-frac{(q-vec{p} cdot vec{n}) vec{n}}{|vec{n}|^{2}} )
12
702Two systems of rectangular axes have
the same origin. If a plane cuts them at
distance ( a, b, c ) and ( d, b^{prime}, c^{prime} ) from the
origin, then
A ( cdot frac{1}{a^{2}}-frac{1}{b^{2}}-frac{1}{c^{2}}-frac{1}{a^{2}}-frac{1}{b^{2}}-frac{1}{c^{2}}=0 )
B ( cdot frac{1}{a^{2}}-frac{1}{b^{2}}-frac{1}{c^{2}}-frac{1}{a^{2}}-frac{1}{b^{2}}+frac{1}{c^{2}}=0 )
c. ( frac{1}{a^{2}}+frac{1}{b^{2}}+frac{1}{c^{2}}-frac{1}{a^{2}}-frac{1}{b^{2}}-frac{1}{c^{2}}=0 )
D ( frac{1}{a^{2}}+frac{1}{b^{2}}+frac{1}{c^{2}}+frac{1}{a^{2}}+frac{1}{b^{2}}+frac{1}{c^{2}}=0 )
12
703Find the equation of the line joining the points (-1,3) and (4,-2)12
704If the position vectors of the points ( boldsymbol{A}, boldsymbol{B} ) and ( C ) be ( i+j, i-j ) and ( a i+b j+c k )
respective;y, then the points ( A, B ) and
( C ) are collinear if:
A ( . a=b=c=1 )
B. ( a=1, b ) and ( c ) are arbitrary scalars
C. ( a=b=c=0 )
0
12
705A point ( R ) with ( x ) -coordinate 4 lies on the line segment joining the points
( P(2,-3,4) ) and ( Q(8,0,10) . ) Find the
coordinates of the point ( boldsymbol{R} )
12
706Fill in the blanks:
(i) The ( x ) -axis and ( y ) -axis taken together
determine a plane known as
(ii) The coordinates of points in the ( boldsymbol{X} boldsymbol{Y} )
-plane are of the form
(iii) Coordinate planes divide the space
into octants
12
707The Cartesian equation of a line is
( frac{boldsymbol{x}-mathbf{5}}{mathbf{3}}=frac{boldsymbol{y}+mathbf{4}}{mathbf{7}}=frac{boldsymbol{z}-mathbf{6}}{mathbf{2}} . ) Write its vector
form.
12
708The direction cosines of the lines
bisecting the internal angle ( theta ) between the lines whose direction cosines are
( l_{1}, m_{1}, n_{1} ) and ( l_{2}, m_{2}, n_{2} ) are
( mathbf{A} cdot )
B. ( )
c. ( )
D. none of these
12
709Find the distance of the point (-6,0,0)
from the plane ( 2 x-3 y+6 z=2 )
12
710Let ( boldsymbol{A}=(mathbf{1}, mathbf{2}, mathbf{2}), boldsymbol{B}=(mathbf{2}, mathbf{3}, mathbf{6}) ) and ( boldsymbol{C}= )
( (3,4,12) . ) The direction cosines of a line
equally inclined with ( O A, O B ) and ( O C )
where ( boldsymbol{O} ) is the origin, are
A ( cdot frac{1}{sqrt{2}}, frac{-1}{sqrt{2}}, 0 )
B. ( frac{1}{sqrt{2}}, frac{1}{sqrt{2}}, 0 )
c. ( frac{1}{sqrt{3}}, frac{-1}{sqrt{3}}, frac{1}{sqrt{3}} )
D. ( frac{1}{sqrt{3}}, frac{-1}{sqrt{3}}, frac{-1}{sqrt{3}} )
12
711Find the coordinates of the point on the ( x ) -axis that is equidistant from
( boldsymbol{P}(mathbf{4}, mathbf{3}, mathbf{1}) ) and ( boldsymbol{Q}(-mathbf{2},-mathbf{6},-mathbf{2}) )
A ( cdotleft(frac{3}{2}, 0,0right) )
B ( cdotleft(-frac{3}{2}, 0,0right) )
c. ( left(0,-frac{3}{2}, 0right) )
D. ( left(0, frac{3}{2}, 0right) )
12
712In the triangle with vertices ( boldsymbol{A}(mathbf{1},-mathbf{1}, mathbf{2}), boldsymbol{B}(mathbf{5},-mathbf{6}, mathbf{2}) ) and ( boldsymbol{C}(mathbf{1}, mathbf{3},-mathbf{1}) )
find the altitude ( n=|B D| )
( A cdot 5 )
B. 10
c. ( 5 sqrt{2} )
D. ( frac{10}{sqrt{2}} )
12
713If the origin is the centroid of the
triangle ( P Q R ) with vertices ( boldsymbol{P}(mathbf{2} boldsymbol{a}, mathbf{2}, mathbf{6}), boldsymbol{Q}(-mathbf{4}, mathbf{3} boldsymbol{b},-mathbf{1 0}) ) and
( boldsymbol{R}(mathbf{8}, mathbf{1 4}, mathbf{2 c}), ) then find the values of ( boldsymbol{a}, boldsymbol{b} )
and ( c )
12
714The points with the co-ordinates
( (2 a, 3 a),(3 b, 2 b) &(c, c) ) are collinear.
12
715Using section formula show that the
points ( boldsymbol{A}(mathbf{2},-mathbf{3}, mathbf{4}), boldsymbol{B}(-mathbf{1}, mathbf{2}, mathbf{1}) ) and
( Cleft(0, frac{1}{3}, 2right) ) are collinear.
12
716If the line joining ( boldsymbol{A}(mathbf{1}, mathbf{3}, mathbf{4}) ) and ( boldsymbol{B} ) is divided by the point (-2,3,5) in the ratio ( 1: 3, ) then ( B ) is
A. (-11,3,8)
(年) (-1,3,8)
в. (-11,3,-8)
D. (13,6,-13)
12
71716. In R}, consider the planes P, :y=0 and P2: x+z=1. Let P
be the plane, different from P, and P2, which passes through
the intersection of P, and P,. If the distance of the point
(0, 1, 0) from P, is 1 and the distance of a point (a, b, y) from
Pz is 2, then which of the following relations is (are) true
(a) 2a+3+2y+2=0
(c) 2a+B-2y-10=0
(b) 20-3+2y+4=0
(d) 2a-B+2y-8=0
12
718Find the distance between the points
(3,4,-2),(1,0,7)
12
719W
J
84. If the image of the point P(1, -2, 3) in the plane,
X
Y
Z
2x + 3y– 4z + 22=0 measured parallel to line, = = is
Q, then PQ is equal to :
JJEE M 2017]
(a) 615
(b) 315
(c) 2742
(d) 142
12
720The points ( boldsymbol{A}(mathbf{1}, mathbf{2},-mathbf{1}), boldsymbol{B}(mathbf{2}, mathbf{5},-mathbf{2}), boldsymbol{C}(mathbf{4}, mathbf{4},-mathbf{3}) )
and ( D(3,1,-2) ) are
A. collinear
B. vertices of a rectangle
c. vertices of a square
D. vertices of a rhombus
12
721Find the directions cosines of ( x, y ) and ( z )
axis.
12
722The perpendicular distance from the
point (3,1,1) on the plane passing through the point (1,2,3) and containing the line, ( vec{r}=hat{i}+hat{j}+ ) ( lambda(2 hat{i}+hat{j}+4 hat{k}), ) is:
( ^{A} cdot frac{1}{sqrt{11}} )
в. ( frac{4}{sqrt{41}} )
( c .0 )
D. ( frac{3}{sqrt{11}} )
12
723If ( (0, b, 0) ) is the centroid of the triangle
formed by the points (4,2,-3) ( (a,-5,1) ) and ( (2,-6,2) . ) If ( a, b ) are the
roots of the quadratic equation ( x^{2}+ )
( boldsymbol{p} boldsymbol{x}+boldsymbol{q}=mathbf{0}, ) then ( boldsymbol{p}, boldsymbol{q} ) are
A .9,18
в. -9,-18
c. 3,-18
D. -3,18
12
724For two vectors ( overrightarrow{boldsymbol{A}} ) and ( overrightarrow{boldsymbol{B}}, overrightarrow{boldsymbol{A}}+overrightarrow{boldsymbol{B}}=overrightarrow{boldsymbol{C}} )
and ( |vec{A}|+|vec{B}|=|vec{C}| . ) The angle between
two vectors is:
A .
в. ( pi / 3 )
c. ( pi / 2 )
D.
12
725A plane meets the co-ordinate axes in A,B,C such that the centroid of the
triangle ( A B C ) is the point ( (p, q, r) . ) The
equation of the plane is
A ( cdot frac{x}{p}+frac{y}{q}+frac{z}{r}=0 )
В ( cdot frac{x}{p}+frac{y}{q}+frac{z}{r}=1 )
c. ( frac{x}{p}+frac{y}{q}+frac{z}{r}=2 )
D. none of these
12
726What is the sum of the squares of direction cosines of the line joining the points (1,2,-3) and (-2,3,1)( ? )
( A cdot O )
B.
( c cdot 3 )
D. ( frac{2}{sqrt{26}} )
12
727The image of: (-4,0,0) in the ( x z ) – plane is (4,0,0)
A. True
B. False
12
728The point which is equidistant from the points ( (boldsymbol{a}, boldsymbol{0}, boldsymbol{0}),(boldsymbol{0}, boldsymbol{b}, boldsymbol{0}),(boldsymbol{0}, boldsymbol{0}, boldsymbol{c}) ) and
(0,0,0) is:
( mathbf{A} cdot(a, b, c) )
B . ( (sqrt{a}, sqrt{b}, sqrt{c}) )
c. ( (2 a, 2 b, 2 c) )
D. ( left(frac{a}{2}, frac{b}{2}, frac{c}{2}right) )
12
729Find distance of a point (3,4) from the origin.12
73042.
The plane x +2y-z=4 cuts the sphere x + y< +22-x+,
– 2 = 0 in a circle of radius
(a) 3 (6) 1 (c) 2 (d) &
[2005]
12
73134.
The angle between the lines 2x = 3y = – z and
6x=-y=-4z is
[2005]
(a) O
). (b) 90°
(c) 45°
(d) 30°
12
732The number of lines which are equally
inclined to the axes is
( A cdot 2 )
B. 4
( c cdot 6 )
D. 8
12
733VEU
43.
point P is the intersection of the straight line joining
points (2,3,5) and R(1,-1, 4) with the plane 5x – 4y
1. If S is the foot of the perpendicular drawn from the
point T(2, 1, 4) to QR, then the length of the line segment PS
(2012)
7
(b) √2
(c) 2
(2) 252
12
734( operatorname{lines} frac{x-1}{2}=frac{y-1}{2}=frac{z-2}{3} ) and
( frac{x-1}{2}=frac{y-2}{2}=frac{z-3}{-2} )
Check whether the lines are parallel, mutually perpendicular or intersecting in acute angle)
12
735A line is perpendicular to the plane ( x+ )
( 2 y+2 z=0 ) and passes through
( (0,1,0) . ) The perpendicular distance of this line from the origin is
A ( frac{sqrt{5}}{3} )
B. ( frac{sqrt{7}}{3} )
( c cdot frac{2}{3} )
D. 3
12
736The name of the figure formed by the points (0,0,0),(1,0,1) and (0,1,1) is
A . a straight line
B. an isosceles triangle
c. an equilateral triangle
D. a scalene triangle
12
737The cartesian equation of the plane ( overline{boldsymbol{r}}=(mathbf{1}+boldsymbol{s}-boldsymbol{t}) hat{boldsymbol{i}}+(boldsymbol{2}-boldsymbol{s}) hat{boldsymbol{j}}+ )
( (3-2 s+2 t) hat{k} )
A. ( 2 x-y=5 )
B. ( 2 x+z=5 )
c. ( 2 x+y=5 )
D. ( 2 x-z=5 )
12
73827.
A variable plane at a distance of the one unit from the
origin cuts the coordinates axes at A, B and C. If the
centroid D (x, y, z) of triangle ABC satisfies the relation
2
then the value k is
(2005)
y
(a) 3
(b) 1
(c)
(d) 9
12
73921. Let ✓ = 2i +1 -k and W = i +3k . If Ū is a unit vector,
then the maximum value of the scalar triple product |ŪVW | is
(a) -1
(b) V10 + V6 (2002)
(c) 159
(d) 160
12
740Two opposite vertices of a square are (2,-3,4) and ( (4,1,-2) . ) The length of the side of the square is
A ( cdot sqrt{58} )
B. ( 2 sqrt{7} )
c. ( sqrt{14} )
D. ( sqrt{7} )
12
741If the angle between the planes ( boldsymbol{r} cdot(boldsymbol{m} hat{boldsymbol{i}}-hat{boldsymbol{j}}+boldsymbol{2} hat{boldsymbol{k}})+boldsymbol{3}=boldsymbol{0} ) and ( overline{boldsymbol{r}} cdot(boldsymbol{2} hat{boldsymbol{i}}- )
( m hat{j}-hat{k})-5=0 ) is ( frac{pi}{3} ) then ( m= )
( A cdot 2 )
B. ±3
( c cdot 3 )
D. – –
12
742Find the ratio in which (the plane) ( 2 x+ ) ( 3 y+5 z=1 ) divides the line joining the
points (1,0,-3) and (1,-5,7)
A .1: 2
B . 2: 3
( c .3: 1 )
D. None of these
12
743Let ( O ) be the origin and ( P ) be the point at a distance 3 units from origin. If d.x.s’ of OP are ( 1,-2,-2, ) then coordinates of ( P ) is
given by
A. 1,-2,-2
в. 3,-6,-6
c. ( frac{1}{3},-frac{2}{3},-frac{2}{3} )
D. ( frac{1}{9},-frac{2}{9},-frac{2}{9} )
12
744Write the direction cosines of ( x ) -axis12
745The product of the d.cs of the line which makes equal angles with ( o x, o y, o z ) is
( mathbf{A} cdot mathbf{1} )
B. ( sqrt{3} )
c. ( frac{1}{3 sqrt{3}} )
D. ( frac{1}{sqrt{3}} )
12
746If a line makes ( theta_{1}, theta_{2}, theta_{3} ) angles with the
co-ordinates axes, then prove that
( cos 2 theta_{1}+cos 2 theta_{2}+cos 2 theta_{3}+1=0 )
12
747Verify the following:
(0,7,-10),(1,6,-6) and (4,9,-6) are
the vertices of an isosceles triangle.
12
748If ( vec{A} times vec{B}=vec{B} times vec{A}, ) then the angle between ( vec{A} ) and ( vec{B} ) is
A . ( pi )
в.
( c cdot frac{pi}{2} )
D.
12
749Show that the lines whose d.c.s are
given by ( l+m+n=0,2 m n+3 l n- )
( 5 l m=0 ) are perpendicular to each
other.
12
750Show that the following set of point are collinear?
(2,3,-4),(-1,0,5),(3,4,-7)
12
751Equation of a plane making X-intercept
4, Y-intercept ( (-6), mathrm{Z} ) -intercept 3 is
A. ( 3 x-4 y+6 z=12 )
B. ( 3 x-2 y+4 z=12 )
c. ( 4 x-6 y+3 z=1 )
D. ( 4 x-3 y+2 z=12 )
12
752Find the distance between (12,3,4) and (4,5,2)
A ( cdot sqrt{72} )
B. ( sqrt{62} )
( c cdot sqrt{64} )
D. None of these
12
753The general equation of plane which is
parallel to x-axis is
( mathbf{A} cdot a x+b y+c z+d=0, a neq 0, b neq 0, c neq 0 )
B. ( b y+a x+d=0, a neq 0, b neq 0 )
c. ( a x+c z+d=0, a neq 0 . c neq 0 )
D. ( b y+c z+d=0, b neq 0, c neq 0 )
12
754If from the point ( boldsymbol{P}(boldsymbol{f}, boldsymbol{g}, boldsymbol{h}) )
perpendiculars ( P L, P M ) be drawn to ( y z )
and ( z x ) planes, then the equation to the
plane ( boldsymbol{O} boldsymbol{L} boldsymbol{M} ) is
A ( cdot frac{x}{f}+frac{y}{g}-frac{z}{h}=0 )
В ( cdot frac{x}{f}+frac{y}{g}+frac{z}{h}=0 )
c. ( frac{x}{f}-frac{y}{g}+frac{z}{h}=0 )
D. ( -frac{x}{f}+frac{y}{g}+frac{z}{h}=0 )
12
755The position vectors of point ( A ) and ( B ) ( operatorname{are} hat{boldsymbol{i}}-hat{boldsymbol{j}}+boldsymbol{3} hat{boldsymbol{k}} ) and ( boldsymbol{3} hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}+boldsymbol{3} hat{boldsymbol{k}} )
respectively. The equation of a plane is
( r cdot(5 hat{i}+2 hat{j}-7 hat{k})+9=0 . ) The point ( A )
and ( B )
A. lie on the plane
B. are on the same side of the plane
c. are on the opposite side of the plane
D. None of these
12
756Find the distance between the following
pairs of points
(-5,7) and (-1,3)
12
757The equation of the plane which is parallel to ( x y ) plane and cuts intercept of length 3 from the z-axis
( mathbf{A} cdot x=3 )
B. ( y=3 )
( mathbf{c} cdot z=3 )
D. ( x+y+z=3 )
12
758The coordinate of any point, which lies
in ( boldsymbol{x} boldsymbol{y} ) plane, is
( mathbf{A} cdot(x, 0, y) )
в. ( (x, x, 0) )
c. ( (x, 0, x) )
D. ( (y, 0, x) )
12
759What is the angle between the lines
( frac{boldsymbol{x}-mathbf{2}}{mathbf{1}}=frac{boldsymbol{y}+mathbf{1}}{-mathbf{2}}=frac{boldsymbol{z}+mathbf{2}}{mathbf{1}} ) and ( frac{boldsymbol{x}-mathbf{1}}{mathbf{1}}= )
( frac{2 y+3}{3}=frac{z+5}{2}=? )
( ^{A} cdot frac{pi}{2} )
в.
c. ( frac{pi}{6} )
D. None of the above
12
760The distance between the origin and the centroid of the tetrahedron whose vertices are ( (mathbf{0}, mathbf{0}, mathbf{0}) )
( (a, 0,0),(0, b, 0),(0,0, c) ) is?
A ( cdot sqrt{a^{2}+b^{2}+c^{2}} )
B. ( frac{sqrt{a^{2}+b^{2}+c^{2}}}{2} )
c. ( frac{sqrt{a^{2}+b^{2}+c^{2}}}{4} )
D. ( 4 sqrt{a^{2}+b^{2}+c^{2}} )
12
761A parallelepiped is formed by planes
drawn through the points ( boldsymbol{P}(boldsymbol{6}, boldsymbol{8}, boldsymbol{1} boldsymbol{0}) )
and ( Q(3,4,8) ) parallel to the coordinate
planes. Find the length of edges and diagonals of the parallelepiped.
12
762Find the equation of the plane passing through the points ( A=(2.3,-1), B=(4,5, )
2), ( C=(3,6,5) )
12
763The cartesian equation of plane ( bar{r} cdot(2,-3,4)=5 ) is
A. ( 3 y-2 x-4 z+5=0 )
B. ( 2 x-3 y+4 z=0 )
c. ( 2 x-3 y+4 z+5=0 )
D. ( frac{x-1}{2}=frac{y-1}{-3}=frac{z-1}{4} )
12
764A plane which passes through the point (3,2,0) and the line ( frac{x-3}{1}=frac{y-7}{5}= )
( frac{z-4}{4} ) is?
A. ( x-y+z=1 )
в. ( x+y+z=5 )
c. ( x+2 y-z=1 )
D. ( 2 x-y+z=5 )
12
76555. If the straight lines *;?v=2=27
and
k
2
=intersect at a point, then the integer k
2
=
3 k
is equal to
(a) 5
[2008]
(b) 5
(c) 2
(d) -2
12
766The line ( frac{x-3}{2}=frac{y-4}{5}=frac{z-6}{7} )
A . lies in ( 3 x+2 y+4 z-6=0 )
B. is parallel to ( 2 x-5 y+3 z=0 )
( mathbf{c} cdot ) is ( perp ) to ( 2 x-5 y+3 z=0 )
D. passing through (1,2,3)
12
767Direction cosines of ray from
( boldsymbol{P}(mathbf{1},-mathbf{2}, mathbf{4}) ) to ( boldsymbol{Q}(-mathbf{1}, mathbf{1},-mathbf{2}) ) are
в. 2,-3,6
( mathbf{c} cdot 2,3,6 )
D. ( frac{-2}{7}, frac{3}{7}, frac{-6}{7} )
12
768If the line, ( frac{x-1}{2}=frac{y+1}{3}=frac{z-1}{4} ) and
( frac{boldsymbol{x}-mathbf{3}}{mathbf{1}}=frac{boldsymbol{y}-boldsymbol{k}}{mathbf{2}}=frac{boldsymbol{z}}{mathbf{1}} ) intersect, then find
the value of ( k )
12
769Point of intersection of the point
(2,4,5)(3,6,-4)
12
770Find the position vector of a point ( boldsymbol{R} )
which divides the line joining two
points ( P ) and ( Q ) whose position vectors ( operatorname{are}(2 vec{a}+vec{b}) ) and ( (vec{a}-3 vec{b}) ) externally in
the ratio ( 1: 2 . ) Also, show that ( P ) is the
mid point of the line segment ( boldsymbol{R} boldsymbol{Q} )
12
77113. Two lines L, : x=5, 3a = – and Ly :x=a, -2-a
are coplanar. Then a can take value(s) (JEE Adv. 2013)
(a) 1 (b) 2 (c) 3 (d) 4
12
772The distance between the points ( (cos theta, sin theta) ) and ( (sin theta-cos theta) ) is
A. ( sqrt{3} )
B. ( sqrt{2} )
( c cdot 2 )
D.
12
773Find the coordinates of those points on the line ( frac{boldsymbol{x}+mathbf{1}}{mathbf{2}}=frac{boldsymbol{y}+mathbf{2}}{mathbf{3}}=frac{boldsymbol{z}-mathbf{3}}{mathbf{6}} ) which
is at a distance of 3 units from the point
(1,-2,3)
12
774Find ( m ) if the point on the ( x ) -axis which
is equidistant from (7,6) and (3,4) is ( left(frac{m}{2}, 0right) )
12
77593. A plane passing through the points (0,–1, 0) and (0,0,1)
and making an angle with the plane y-z+5 = 0, also
passes through the point: [JEEM 2019-9 April (M)
(a) (-,1,-4) (b) (12,-1,4)
(C) (12,-1,-4) (d) (V2,1,4)
12
776The distance between the line ( vec{r}=2 vec{i}- ) ( mathbf{2} overrightarrow{boldsymbol{j}}+boldsymbol{3} overrightarrow{boldsymbol{k}}+boldsymbol{lambda}(overrightarrow{boldsymbol{i}}-overrightarrow{boldsymbol{j}}+boldsymbol{4} overrightarrow{boldsymbol{k}}) ) and the plane
( vec{r} cdot(vec{i}+5 vec{j}+vec{k})=5 ) is
A ( cdot frac{10}{3 sqrt{3}} )
в. ( frac{10}{9} )
c. ( frac{10}{3} )
D.
12
777If ( vec{a}, vec{b} ) and ( vec{c} ) are mutually perpendicular vectors of equal magnitudes, If the angles which the vector ( 2 vec{a}+vec{b}+2 vec{c} ) makes with the vectors ( vec{a} ) is ( cos ^{-1} sqrt{frac{2}{m}} ) Find ( boldsymbol{m} )12
778Find the equation of following planes:12
779The plane ( a x+b y+c z+d=0 ) divides
the line joining the points ( left(x_{1}, y_{1}, z_{1}right) )
and ( left(x_{2}, y_{2}, z_{2}right) ) in the ratio
A ( cdot frac{-left(a x_{1}+b y_{1}+c z_{1}+dright)}{left(a x_{2}+b y_{2}+c z_{2}+dright)} )
B. ( frac{left(a x_{1}+b y_{1}+c z_{1}+dright)}{left(a x_{2}+b y_{2}+c z_{2}+dright)} )
c. ( frac{a x_{1} x_{2}+b y_{1} y_{2}+c z_{1} z_{2}}{d^{2}} )
D. None of these
12
780The vector equation ( boldsymbol{r}=boldsymbol{i}-boldsymbol{2} boldsymbol{j}-boldsymbol{k}+ )
( t(6 j-k) ) represents a straight line
passing through the points:
A. (0,6,-1) and (1,-2,-1)
в. (0,6,-1) and (-1,-4,-2)
c. (1,-2,-1) and (1,4,-2)
D. (1,-2,-1) and (0,-6,1)
12
781If the distance of a point ( (a, a, a) ) from the origin is ( sqrt{108}, ) then the value of ( a ) is
This question has multiple correct options
( A cdot 9 )
B. 6
( c .-9 )
D. – 6
12
782Equation of a plane12
783In which ratio the plane ( Y Z ) divides the
lines joining the points (2,1,2) and (-6,3,4)
12
784A straight line passes through (1,-2,3) and perpendicular to the plane ( 2 x+ ) ( 3 y-z=7 . ) Find the direction ratios of
normal to plane
( A cdot )
В. ( )
c. ( )
D. None of the above
12
785( X O Z ) plane divides the join of (2,3,1) and (6,7,1) in the ratio
A . 3: 7
B. 2: 7
c. -3: 7
D. -2: 7
12
78685. The distance of the point (1,3, -7) from the plane passing
through the point (1, -1, -1), having normal perpendicular
to both the lines
Z-4
X-2
y+1
Z+7
,
X-1
1
y +2
– 2
and 1-2
3
2
-1 , is:
2
JEE M 2017]
S
12
787If ( P(x, y, z) ) is point in the space at a
distance ( r ) from the origin ( O ), then
direction cosines of the line ( O P ) are
12
788If the extremities of a diagonal of a
square are (1,-2,3) and (2,-3,5)
then area of the square is
( A cdot 6 )
B. 3
( c cdot frac{3}{2} )
D. ( sqrt{3} )
12
789The equation of the plane passing through the intersection of the planes ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=boldsymbol{6} ) and ( boldsymbol{2} boldsymbol{x}+boldsymbol{3} boldsymbol{y}+boldsymbol{4} boldsymbol{z}+boldsymbol{5}= )
( 0, ) and the point (1,1,1) is
A. ( 20 x+23 y+26 z-69=0 )
B. ( 20 x+23 y+26 z+69=0 )
c. ( 23 x+20 y+26 z-69=0 )
D. None of these
12
790What are the direction ratios of the line
if it passes through the intersection of the planes ( x=3 z+4 ) and ( y=2 z-3 ? )
A ( .(1,2,3) )
в. (2,1,3)
c. (3,2,1)
D. (1,3,2)
12
791The ratio in which the joint of
(2,1,5),(3,4,3) is divided by the plane ( 2 x+2 y-2 z-1=0 )
( mathbf{A} cdot 5: 12 )
B. 12: 5
( c .5: 7 )
D. 7: 5
12
792Three vertices of a tetrahedron are
(0,0,0),(6,-5,-1) and ( (-4,1,3) . )
the centroid of the tetrahedron be
(1,-2,5) then the fourth vertex is
A. (2,-4,18)
в. (1,-4,18)
( ^{c} cdotleft(frac{3}{2}, frac{-3}{2}, frac{7}{4}right) )
D. none of these
12
793If ( boldsymbol{R} ) divides the line segment joining ( P(2,3,4) ) and ( Q(4,5,6) ) in the ratio
( -3: 2, ) then the parameter which
represent ( boldsymbol{R} ) is
( A cdot 3 )
B . 2
( c . )
D. –
12
794If ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} in[mathbf{0}, boldsymbol{2} boldsymbol{pi}], ) then the sum of all
possible values of ( alpha, beta, gamma ) if ( sin alpha= ) ( -frac{1}{sqrt{2}}, cos beta=-frac{1}{2}, tan gamma=-sqrt{3}, ) is
A ( cdot frac{22 pi}{3} )
B. ( frac{21 pi}{3} )
c. ( frac{20 pi}{3} )
D. ( 8 pi )
12
795A plane intersects the co ordinate axes
at ( A, B, C . ) If ( O=(0,0,0) ) and (1,1,1) is
the centroid of the tetrahedron ( O A B C )
then the sum of the reciprocals of the
intercepts of the plane
( mathbf{A} cdot 12 )
B. ( frac{4}{3} )
c. 1
( D cdot frac{3}{4} )
12
796The plane ( a x+b y+c z+(-3)=0 )
meet the co-ordinate axes in ( A, B, C )
The centroid of the triangle is
B ( cdotleft(frac{3}{a} cdot frac{3}{b}, frac{3}{c}right) )
c. ( left(frac{a}{3} cdot frac{b}{3}, frac{c}{3}right) )
D. ( left(frac{1}{a} cdot frac{1}{b}, frac{1}{c}right) )
12
797The area of triangle whose vertices are (1,2,3),(2,5,-1) and (-1,1,2) is
A . 150 sq.units
B. 145 sq.units
c. ( sqrt{155} / 2 ) sq.units
D. ( 155 / 2 ) sq.units
12
798Find the shortest distance between
lines: ( frac{x-1}{1}=frac{y-2}{3}=frac{z-3}{2} ) and
( frac{x-4}{2}=frac{y-5}{3}=frac{z-6}{1} )
A . ( sqrt{6} )
B. ( sqrt{5} )
( c cdot sqrt{3} )
D. 6
12
799Let the equation of the plane through the points
(-2,-2,2),(1,1,1),(1,-1,2) be ( k x+ )
( boldsymbol{m} boldsymbol{y}+boldsymbol{n} boldsymbol{z}+boldsymbol{p} . ) Find ( boldsymbol{k}+boldsymbol{m}+boldsymbol{n}+boldsymbol{p} )
( A cdot 7 )
B.
( c cdot 4 )
D. 6
12
800Find the distance between the following pairs of points (-2,-3) and (3,2)12
801If ( C_{1}: x^{2}+y^{2}-20 x+64=0 ) and
( C_{2}: x^{2}+y^{2}+30 x+144=0 . ) Then the
length of the shortest line segment ( boldsymbol{P Q} )
which touches ( C_{1} ) at ( P ) and to ( C_{2} ) at ( Q )
is
A . 10
B. 15
( c cdot 22 )
D. 27
12
802If ( x y- ) plane and ( y z- ) plane divides the line segment joining ( A(2,4,5) ) and ( B(3,5,-4) ) in the ratio ( a: b ) and ( p: q ) respectively then value of ( left(frac{a}{b}+frac{p}{q}right) ) may be
A ( cdot frac{23}{12} )
в. ( frac{7}{5} )
( c cdot frac{7}{12} )
D. ( frac{21}{10} )
12
803The image of the point (2,-1,1) by the
plane ( 3 x+4 y-5 z=0 ) is
A ( cdot(-2,1,-1) )
в. ( left(frac{2}{3}, frac{-1}{4}, frac{-1}{5}right) )
( ^{mathbf{C}} cdotleft(frac{59}{25}, frac{-13}{25}, frac{2}{5}right) )
D. none of these
12
804Distance between ( vec{r}=hat{i}+lambda(hat{j}+hat{k}) )
and ( vec{r}=hat{j}+mu(hat{j}+hat{k}) ) is equal to
12
805( operatorname{Let} boldsymbol{A}(mathbf{2} hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}+mathbf{5} hat{boldsymbol{k}}) boldsymbol{B}(-hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}+2 hat{boldsymbol{k}}) )
and ( C(lambda hat{i}+5 hat{j}+mu hat{k}) ) are vertices of ( a ) triangle and its median through ( A ) is equally inclined to the positive directions of the axes. The value of ( lambda+ )
( mu ) is equal to
A . -7
B . 2
c. 7
D. 17
12
806Direction cosines of the line ( frac{x+2}{2}= )
( frac{2 y-5}{3}, z=-1 ) are
A ( cdot frac{4}{5}, frac{3}{5}, 0 )
B. ( frac{3}{5^{prime}}, frac{4}{5^{prime}} frac{1}{5} )
c. ( quad-frac{3}{5}, frac{4}{5}, 0 )
D. ( frac{4}{5},-frac{2}{5}, frac{1}{5} )
12
807The plane through the intersection of the planes ( x+y+z=1 ) and ( 2 x+ )
( 3 y-z+4=0 ) and parallel to ( y ) -axis
also passes through the point.
A. (-3,0,1)
В. (3,3,-1)
c. (3,2,1)
D. (-3,1,1)
12
808Find the equation of the plane passing through the points
( (mathbf{1}, mathbf{1}, mathbf{1}),(mathbf{3},-mathbf{1}, mathbf{2}),(-mathbf{3}, mathbf{5},-mathbf{4}) )
12
809Find the equation of the plane bisecting the line segment joining the points
(-3,-2,1) and (1,6,-5)
perpendicularly.
12
810is
46. Let P be the image of the point (3,1,7) with respect to the
plane x-y+z=3. Then the equation of the plane passing
x y z .
through P and containing the straight line =
1 z 1
(a) x+y-3z=0
(b) 3x+z=0
(c) X-4y+z=0
(d) 2x-y=0
hub the point (1 1 1
12
811The plane ( a x+b y+c z+(-3)=0 )
meet the co-ordinate axes in ( A, B, C . ) Then
centroid
of the triangle is
A. ( (3 a, 3 b, 3 c) )
в. ( left(frac{3}{a} frac{3}{b}, frac{3}{c}right) )
( ^{mathbf{C}} cdotleft(frac{a}{3}, frac{b}{3}, frac{c}{3}right) )
D. ( left(frac{1}{a}, frac{1}{b}, frac{1}{c}right) )
12
812If the points ( boldsymbol{A}(mathbf{3},-mathbf{2}, mathbf{4}), boldsymbol{B}(mathbf{1}, mathbf{1}, mathbf{1}) ) and
( C(-1,4,-2) ) are collinear, then the ratio in which ( C ) divides ( A B ) is
A .1: 2
B . -2: 1
c. -1: 2
D. 4: 0
12
813Find the direction cosines (d.cs) of
directed line ( O P ) if coordinates of ( P ) is
( (2,3,7), O ) being the origin.
12
814Find the co-ordinates of a point lying on the line ( frac{boldsymbol{x}-mathbf{2}}{mathbf{3}}=frac{boldsymbol{y}+mathbf{3}}{mathbf{4}}=frac{boldsymbol{z}-mathbf{1}}{mathbf{7}} ) which
is at a distance 10 units from (2,-3,1)
begin{tabular}{l}
A. (32,37,71) \
hline
end{tabular}
в. (-28,-43,-69)
c. (-32,-37,-71)
D. None of these
12
815The direction angles of the line ( x= )
( mathbf{4} z+mathbf{3}, boldsymbol{y}=mathbf{2}-mathbf{3} z ) are ( boldsymbol{alpha}, boldsymbol{beta} ) and ( gamma, ) then
( cos alpha+cos beta+cos gamma= )
A ( cdot frac{2}{sqrt{26}} )
B. ( frac{8}{sqrt{26}} )
c. 1
D. 2
12
816The equation of a plane passing through the point ( A(2,-3,7) ) and making equal intercepts on the axes,
is?
A. ( x+y+z=3 )
в. ( x+y+z=6 )
c. ( x+y+z=9 )
D. ( x+y+z=4 )
12
817If the dr’s the line are ( (1+lambda, 1-lambda, 2) )
and it makes an angle ( 60^{circ} ) with the ( Y ) –
axis then ( lambda ) is
A ( .1 pm sqrt{3} )
B. ( 4 pm sqrt{5} )
c. ( 2 pm 2 sqrt{3} )
D. ( 2 pm sqrt{5} )
12
818If direction ratios of the normal of the
plane which contains the lines ( frac{x-2}{3}= ) ( frac{boldsymbol{y}-boldsymbol{4}}{boldsymbol{2}}=frac{boldsymbol{z}-boldsymbol{1}}{boldsymbol{1}} & frac{boldsymbol{x}-boldsymbol{6}}{boldsymbol{3}}=frac{boldsymbol{y}+boldsymbol{2}}{boldsymbol{2}}= )
( frac{z-2}{1} ) are ( (a, 1,-26), ) then ( a ) is equal
to
A . 5
B. 6
( c cdot 7 )
D.
12
819What is the distance of the point
( (p, q, r) ) from the ( x- ) axis.
12
820What is the angle between ( vec{A} ) and the resultant of ( (overrightarrow{boldsymbol{A}}+overrightarrow{boldsymbol{B}}) ) and ( (overrightarrow{boldsymbol{A}}-overrightarrow{boldsymbol{B}}) )
( mathbf{A} cdot 0^{circ} )
B ( cdot tan ^{-1}left(frac{A}{B}right) )
( ^{mathbf{c}} cdot tan ^{-1}left(frac{B}{A}right) )
D. ( tan ^{-1}left(frac{A-B}{A+B}right) )
12
821Show that angles between any two ( operatorname{diagonals} boldsymbol{theta}=cos ^{-1}left(frac{1}{3}right) )12
822A triangle ( A B C ) is placed so that the
midpoints of its sides are on the ( boldsymbol{x}, boldsymbol{y} )
and ( z ) axes respectively. Lengths of the
intercepts made by the plane containing the triangle on these axes
( operatorname{are} ) respectively ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma}, ) then
the coordinates of the centroid of the
triangle ( A B C ) are
A ( cdotleft(-frac{alpha}{3}, frac{beta}{3}, frac{gamma}{3}right) )
В ( cdotleft(frac{alpha}{3},-frac{beta}{3}, frac{gamma}{3}right) )
( ^{mathbf{C}} cdotleft(frac{alpha}{3}, frac{beta}{3},-frac{gamma}{3}right) )
D ( cdotleft(frac{alpha}{3}, frac{beta}{3}, frac{gamma}{3}right) )
12
823Using vectors, find the value of ( lambda ) such
that the points ( (boldsymbol{lambda},-mathbf{1 0}, mathbf{3}),(mathbf{1},-mathbf{1}, mathbf{3}) )
and (3,5,3) are collinear.
12
824Find the equation of the plane through
the intersection of the planes ( 3 x- )
( 4 y+5 z=10 ) and ( 2 x+2 y-3 z=4 )
and parallel to the line ( boldsymbol{x}=mathbf{2} boldsymbol{y}=mathbf{3} boldsymbol{z} )
12
825Find the direction cosines of the line PQ
joining the points ( P(2,3,4) ) and ( Q(2,1,1) )
12
826A symmetrical form of the line of intersection of the planes ( boldsymbol{x}=boldsymbol{a} boldsymbol{y}+boldsymbol{b} )
and ( boldsymbol{z}=boldsymbol{c} boldsymbol{y}+boldsymbol{d} ) is :
A. ( frac{x-b}{a}=frac{y-1}{1}=frac{z-d}{c} )
( ^{text {В }} cdot frac{x-b-a}{a}=frac{y-1}{1}=frac{z-d-c}{c} )
c. ( frac{x-a}{b}=frac{y-0}{1}=frac{z-c}{d} )
D. ( frac{x-b-a}{b}=frac{y-1}{0}=frac{z-d-c}{d} )
12
827Find the distance of the point ( boldsymbol{P}(boldsymbol{3}, boldsymbol{4}, boldsymbol{4},) )
from the point, where the line joining
the points ( boldsymbol{A}(boldsymbol{3},-boldsymbol{4},-boldsymbol{5}) ) and ( boldsymbol{B}(boldsymbol{2},-boldsymbol{3}, boldsymbol{1}) )
intersected the plane ( 2 x+y+z=7 )
12
828The distance of the point (1,-5,9) from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} ) measured
along the line ( x=y=z ) is:
( A cdot 3 sqrt{10} )
в. ( 10 sqrt{3} )
c. ( frac{10}{sqrt{3}} )
D. ( frac{20}{3} )
12
829Find the equation of the plane passing through the intresection of the planes ( x-2 y+z=1 ) and ( 2 x+y+z=8 )
and parallel to the line with direction ratio proportional to ( 1,2,1, ) find also the perpendicular distance of (1,1,1) from this plane.
12
830A rectangular parallelopiped is formed by drawing planes through the points (-1,2,5) and (1,-1,-1) and paralle to the coordinate planes. the length of the diagonal of the parallelopiped is
( A cdot 2 )
B. 3
c. 6
D.
12
831Derive the equation of the locus of a point twice as far from (-2,3,4) as from
(3,-1,-2)
12
832Prove that 1,1,1 cannot be direction cosines of a straight line12
833The acute angle between two lines such
that the direction cosines ( I, m, n ) of each
of them satisfy the equations ( l+m+ )
( boldsymbol{n}=mathbf{0} ) and ( l^{2}+boldsymbol{m}^{2}-boldsymbol{n}^{2}=mathbf{0} ) is :
A . 30
B . 45
c. 60
D. 15
12
834Vector components of the vector with
initial points (2,1) and terminal point (-5,7) are
( mathbf{A} cdot-6 hat{i}+7 widehat{j} )
B. ( -7 hat{i}+6 hat{j} )
c. ( -6 hat{i}-7 widehat{j} )
D. None
12
835A mirror and a source of light are situated at the origin ( mathrm{O} ) and at a point on ( mathrm{O} X, ) respectively. A ray of light from the source strikes the mirror and is
reflected. If the direction ratios of the
normal to the plane are ( 1,-1,1, ) then find the DCs of the reflected ray.
A ( cdot frac{1}{3}, frac{2}{3}, frac{2}{3} )
в. ( -frac{1}{3}, frac{2}{3}, frac{2}{3} )
( c cdot-frac{1}{3},-frac{2}{3},-frac{2}{3} )
D. ( -frac{1}{3},-frac{2}{3}, frac{2}{3} )
12
836The points (2,5) and (5,1) are the two opposite vertices of a rectangle. If the other two vertices are points on the straight line ( y=2 x+k, ) then the value
of k is
( A cdot 4 )
B. 3
( c cdot-4 )
( D cdot-3 )
( E )
12
837Let ( vec{A}, vec{B} ) and ( vec{C} ) be unit vectors. Suppose that ( vec{A} cdot vec{B}=vec{A} cdot vec{C}=0 ) and that the
angle between ( vec{B} ) and ( vec{C} ) is ( frac{pi}{6} ) then ( overrightarrow{boldsymbol{A}}= )
A ( cdot pm 2(vec{B} times vec{C}) )
B ( cdot pm(vec{B} times vec{C}) )
c. ( pm 2(vec{B}+vec{C}) )
D. ( pm(vec{B}+vec{C}) )
12
838If the lines through the points (4,1,2) and ( (5, k, 0) ) is parallel to the line through the points (2,1,1) and (3,3,1)
find ( k )
12
839The coordinates of the foot of the
perpendicular from the point (1,-2,1) on the plane containing the lines, ( frac{x+1}{6}=frac{y-1}{7}=frac{z-3}{8} ) and ( frac{x-1}{3}= )
( frac{boldsymbol{y}-boldsymbol{2}}{mathbf{5}}=frac{boldsymbol{z}-boldsymbol{3}}{boldsymbol{7}}, ) is :
A ( cdot(2,-4,2) )
в. (-1,2,-1
c. (0,0,0)
D. (1,1,1)
12
840If the points ( boldsymbol{A}(1,2,-1), B(2,6,2) ) and
( C(lambda,-2,-4) ) are collinear, then ( lambda ) is
( mathbf{A} cdot mathbf{0} )
B.
( c .-2 )
( D )
12
841Find the centroid of a triangle, midpoints of whose sides are
(1,2,-3),(3,0,1) and (-1,1,-4)
12
842If the points ( (a, 1),(1,2) ) and ( (0, b+1) )
are collinear, then show that ( frac{1}{a}+frac{1}{b}=1 )
12
843The ratio of ( y z ) -plane divide the line
joining the points ( A(3,1,-5), B(1,4,-6) ) is
A . 3: 1
B. -1: 3
c. 1: 3
D ( .-3: 1 )
12
844( P(1,1,1) ) and ( Q(lambda, lambda, lambda) ) are two points in the space such that ( P Q=sqrt{27} ), then the value(s) of ( lambda ) can be
A . -4
в. -2,4
( c cdot 2 )
D. 4,3
12
845The line ( boldsymbol{x}-mathbf{2} boldsymbol{y}+mathbf{4} boldsymbol{z}+mathbf{4}=mathbf{0}, boldsymbol{x}+boldsymbol{y}+ )
( z-8=0 ) intersects the plane ( x-y+ )
( mathbf{2} z+mathbf{1}=mathbf{0} ) at the point
A ( cdot(3,2,3) )
B . (5,2,1)
c. (2,5,1)
D . (3,4,1)
12
846In which ratio does the ( Y Z ) plane divide
the line joining the points (-2,4,7) and (3,-5,8)
12
847A sphere of constant radius ( 2 k ) passes
through the origin and meets the axes in ( A, B, C . ) The locus of the centroid of
the tetrahedron ( boldsymbol{O} boldsymbol{A} boldsymbol{B} boldsymbol{C} ) is
A ( cdot x^{2}+y^{2}+z^{2}=4 k^{2} )
B cdot ( 9left(x^{2}+y^{2}+z^{2}right)=4 k^{2} )
C . ( x^{2}+y^{2}+z^{2}=k^{2} )
D. None of these
12
848The point lying on angle bisector of the
planes ( boldsymbol{x}+mathbf{2} boldsymbol{y}+mathbf{2} boldsymbol{z}-mathbf{6}=mathbf{0} ) and ( mathbf{2} boldsymbol{x}- )
( boldsymbol{y}+boldsymbol{4}=mathbf{0} ) is
A. (2,4,0)
B. (-1,3,2)
D. (-2,4,0)
12
849Find the values of ( a ) and ( b ) so that the
points ( (boldsymbol{a}, boldsymbol{b}, mathbf{3}),(mathbf{2}, mathbf{0},-mathbf{1}) ) and
(1,-1,-3) are collinear.
12
850If ( |overrightarrow{boldsymbol{A}} times overrightarrow{boldsymbol{B}}|=overrightarrow{boldsymbol{A}} cdot overrightarrow{boldsymbol{B}}, ) then ( |overrightarrow{boldsymbol{A}}+overrightarrow{boldsymbol{B}}| ) is:
( mathbf{A} cdot|vec{A}|+|vec{B}| )
B. ( sqrt{|vec{A}|^{2}+|vec{B}|^{2}} )
( ^{mathrm{c}} cdot sqrt{|vec{A}|^{2}+|vec{B}|^{2}+frac{|vec{A}||vec{B}|}{sqrt{2}}} )
D cdot ( sqrt{|vec{A}|^{2}+|vec{B}|^{2}+sqrt{2}|vec{A}||vec{B}|} )
12
85161. Statement-1: The point A(3, 1, 6) is the mirror image of the
point B(1, 3, 4) in the plane x-y +z=5.
Statement-2: The plane x-y+z=5 bisects the line segment
joining A(3, 1,6) and B(1,3,4).
[2010]
(a) Statement -1 is true, Statement -2 is true; Statement-2
is not a correct explanation for Statement-1.
(6) Statement -1 is true, Statement -2 is false.
(c) Statement-1 is false, Statement -2 is true.
(d) Statement – 1 is true, Statement 2 is true; Statement-2
is a correct explanation for Statement -1.
12
852Write the vector equation of the line equation ( frac{x-1}{2}=frac{y-2}{3}=frac{z-3}{4} )12
853The name of the figure formed by the points (3,-5,1),(-1,0,8) and
(7,-10,-6) is
A. a triangle
B. a straight line
c. an isosceles triangle
D. an equilateral triangle
12
854The lines ( frac{x-1}{2}=frac{y+1}{3}=frac{z-1}{4} ) and
( frac{boldsymbol{x}-mathbf{3}}{mathbf{1}}=frac{boldsymbol{y}-boldsymbol{k}}{mathbf{2}}=frac{boldsymbol{z}}{mathbf{1}} ) intersect if ( boldsymbol{K} )
equals
( A cdot frac{3}{2} )
в. ( frac{9}{2} )
c. ( frac{-2}{9} )
D. ( frac{-3}{2} )
12
855Find unit vector perpendicular to the plane passing through the points (1,2,3),(2,-1,1) and (1,2,-4)12
856If ( P ) is a point ( (x, y) ) on the line ( y=-3 x )
such that ( boldsymbol{P} ) and ( boldsymbol{Q}(boldsymbol{3}, boldsymbol{4}) ) are on opposite
side of the line ( 3 x-4 y=8, ) then:
A ( cdot x>frac{8}{5}, yfrac{8}{15}, y<-frac{8}{5} )
c. ( x=frac{8}{15}, y=frac{-8}{5} )
D. ( x=2, y=-2 )
12
857( P ) is a variable points which moves
such that ( 3 P A=2 P B . ) If ( A= )
(-2,2,3) and ( B=(13,-3,13) ) prove
that ( P ) satisfies the equation ( x^{2}+y^{2}+ )
( z^{2}+28 x-12 y+10 z-247=0 )
12
858For what value of ( mathrm{m}, ) the points (3,5)
( (m, 6) ) and ( left(frac{1}{2}, frac{15}{2}right) ) are collinear?
12
859Find the direction cosines of the line
( frac{boldsymbol{x}+mathbf{2}}{mathbf{2}}=frac{mathbf{2} boldsymbol{y}-mathbf{5}}{mathbf{3}} ; boldsymbol{z}=-mathbf{1} )
12
860Find the co-ordinates of a point, which is at a distance of 21 units from the
point ( boldsymbol{A}=(mathbf{1},-mathbf{3}, mathbf{4}) ) in the direction of vectors ( 2 hat{i}-3 hat{j}-6 hat{k} )
12
86172.
22. If the lines *-72 173
If the lines
Z
y-3
1
-4
-ka
and
-1
and

k
z-5
– are coplanar, then k can have
[JEE M 2013]
(a) any value
(c) exactly two values
(b) exactly one value
(d) exactly three values
12
862If two vertices of a triangle ( A B C ) are ( A(-1,2,4) ) and ( B(2,-3,0), ) and the
centroid is (2,0,2) then the vertex ( C ) has the coordinates
A. (5,1,2)
в. ( left(1,-frac{1}{3}, frac{7}{3}right) )
( ^{mathbf{C}} cdotleft(3,-frac{2}{3}, frac{5}{3}right) )
D. none of these
12
863Find the distance of the point (2,12,5) from the point of intersection of the line ( overrightarrow{boldsymbol{r}}=2 hat{hat{boldsymbol{i}}}-mathbf{4} hat{boldsymbol{j}}+mathbf{2} hat{boldsymbol{k}}+boldsymbol{lambda}(boldsymbol{3} hat{boldsymbol{i}}+boldsymbol{4} hat{boldsymbol{j}}+boldsymbol{2} hat{boldsymbol{k}}) )
and the plane ( vec{r} cdot(hat{boldsymbol{i}}-boldsymbol{2} hat{boldsymbol{j}}+hat{boldsymbol{k}})=mathbf{0} )
12
864Find the direction ratio of the line
( frac{x-1}{2}=3 y=frac{2 z+3}{4} )
12
865The equation of line passing through (1,-2,3) and having ( operatorname{drs}(2,3,1) ) is
( ^{text {A }} cdot frac{x-1}{2}=frac{y+2}{3}=frac{z-3}{1} )
B. ( frac{x+1}{2}=frac{y+2}{3}=frac{z-3}{1} )
c. ( frac{x-1}{2}=frac{y-2}{3}=frac{z-3}{1} )
D. none of these
12
866If a point ( P ) in the space such that ( overline{O P} )
is inclined to ( O X ) at 45 and ( O Z ) to 60
then ( overline{O P} ) inclined to ( O Y ) is
A ( .75^{circ} )
B . ( 75^{circ} ) or ( 105^{circ} )
C. ( 60^{circ} ) or ( 120^{circ} )
D. None of these
12
867Find the root of the perpendicular from point (2,3,2) to the line ( frac{4-x}{2}=frac{y}{6}= ) ( frac{1-z}{3} ) also find perpendicular distance
from the point to the line.
12
86811.
– and
25
x-1 y+1
x +1
Z.
Ifthe straight lines
4+1 Z
1 2 k
5 2 k
are coplanar, then the plane (s) containing these two lines is
(are)
(2012)
(a) y + 2z=-1
(b) y+z=-1
(c) y-z=-1 .
(d) y-2z=-1
12
869The projection of a directed line segment on the co-ordinate axes are
( 12,4,3, ) then the direction cosines of
the line are
A ( cdot frac{-12}{13}, frac{-4}{13}, frac{-3}{13} )
В. ( frac{12}{13}, frac{4}{13}, frac{3}{13} )
c. ( frac{12}{13}, frac{-4}{13}, frac{3}{13} )
D. ( frac{12}{13}, frac{4}{13}, frac{-3}{13} )
12
870If the point ( (x, y) ) is equidistant from the points ( (a+b, b-a) ) and ( (a- )
( b, a+b), ) then ( b x=a y )
A. True
B. False
12
871If the points ( a(1,2,-1), B(2,6,2) ) and ( c(lambda,-2,-4) ) are collinear then ( lambda ) is
( mathbf{A} cdot mathbf{0} )
B . 2
( c .-2 )
D.
12
872The image of the point (1,2,3) through
the plane ( boldsymbol{x}+boldsymbol{y}+boldsymbol{4} boldsymbol{z}=boldsymbol{0} ) is
A ( cdotleft(frac{-2}{3}, frac{1}{3}, frac{-11}{3}right) )
в. ( left(frac{3}{7},-frac{6}{7}, frac{19}{7}right) )
( ^{mathbf{c}} cdotleft(frac{2}{3}, frac{1}{3}, frac{11}{3}right) )
D. ( left(frac{-2}{3}, frac{-1}{3}, frac{11}{3}right) )
12
873Show that the points ( (3,3),(h, 0) ) and
( (0, k) ) are collinear, if ( frac{1}{n}+frac{1}{k}=frac{1}{3} )
12
874Angle between lines whose direction
( operatorname{cosine} operatorname{satisfy} l+m+n=0, l^{2}+ )
( boldsymbol{m}^{2}-boldsymbol{n}^{2}=mathbf{0} )
12
875a b c a b c
23. Distance between two parallel planes
2x+y+2z=8 and 4x + 2y +4z +5=0 is
[2004]
(a)
2
min
12
876( operatorname{can} frac{2}{sqrt{3}}, frac{-2}{sqrt{3}}, frac{-1}{sqrt{3}} ) be the direction ratios
of any directed line? Justify your
12
877A vector equation of the line of
intersection of the planes ( boldsymbol{r}=boldsymbol{b}+ )
( boldsymbol{lambda}_{1}(boldsymbol{b}-boldsymbol{a})+boldsymbol{mu}_{1}(boldsymbol{a}+boldsymbol{c}) )
( boldsymbol{r}=boldsymbol{c}+boldsymbol{lambda}_{2}(boldsymbol{b}-boldsymbol{c})+boldsymbol{mu}_{1}(boldsymbol{a}+boldsymbol{b}) boldsymbol{a}, boldsymbol{b}, boldsymbol{c} )
being non-coplanar vectors is.
A ( cdot r=a+mu_{1}(b+c) )
B . ( r=b+mu_{1}(a+2 c) )
C . ( r=a+mu_{1}(b+2 c) )
D. ( r=b+mu_{1}(a+c) )
12
878The length of the perpendicular from the origin to the plane passing through the point ( a ) and containing the line ( r=bar{b}+ )
( lambda bar{c} ) is
A ( cdot frac{[a b c]}{|a times b+b times c+c times a|} )
В. ( frac{[a b c]}{|a times b+b times c|} )
c. ( frac{[a b c]}{|b times c+c times a|} )
D. ( frac{[a b c]}{|a times b+c times a|} )
12
879Let the equation of the plane which contains the line ( boldsymbol{x}=frac{boldsymbol{y}-boldsymbol{3}}{boldsymbol{2}}=frac{boldsymbol{z}-boldsymbol{5}}{boldsymbol{3}} )
and which is perpendicular to the plane ( 2 x+7 y-3 z=1 . ) be ( k x-m y-z+ )
( boldsymbol{p}=mathbf{0} . ) Find ( boldsymbol{p}-boldsymbol{k}-boldsymbol{m} ? )
12
880Find direction cosine line ( boldsymbol{x}=mathbf{3} boldsymbol{z}+ )
( mathbf{2}, boldsymbol{y}=mathbf{2}-mathbf{5} boldsymbol{z} )
12
881Find the coordinates of the points which divides the line joining the points (2,-4,3),(-4,5,-6) in the ratio
( (i) 1:-4 )
( (i i) 2: 1 )
12
882Direction ratio of two lines are ( l_{1}, boldsymbol{m}_{1}, boldsymbol{n}_{1} )
and ( l_{2}, m_{2}, n_{2} ) then direction ratios of the line perpendicular to both the lines
are
A ( . l_{1}-l_{2}, m_{1}-m_{2}, n_{1}-n_{2} )
в. ( l_{1}+l_{2}, m_{1}+m_{2}, n_{1}+n_{2} )
c. ( m_{1} n_{2}-n_{1} m_{2}, n_{1} l_{2}-n_{2} l_{1}, l_{1} m_{2}-m_{1} l_{2} )
D. ( m_{1} n_{2}-n_{1} m_{2}, n_{1} l_{2}-n_{1} l_{1}, l_{1} m_{2}-m_{1} l_{2} )
12
883The points (3,2,0),(5,3,2) and ( (-9,6,-3), ) are the vertices of a triangle ( A B C . A D ) is the internal
bisector of ( angle B A C ) which meets ( B C ) at
D. Then the co-ordinates of ( D ), are
A ( cdotleft[frac{17}{16}, frac{57}{16}, frac{19}{8}right] )
В. ( left[frac{19}{8}, frac{57}{16}, frac{17}{16}right] )
( ^{mathbf{c}} cdotleft[0,0, frac{17}{16}right] )
D. ( left[frac{17}{16}, 0,0right] )
12

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# Three Dimensional Geometry Questions

We provide three dimensional geometry practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on three dimensional geometry skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.

#### List of three dimensional geometry Questions

Question NoQuestionsClass
1Perimeter of triangle whose vertices are (0,4,0),(3,4,0) and ( (0,4,4), ) is
A . 10
B. 12
c. 25
D. 15
12
287.
(a) 315 (b) 256 (c) 84
The length of the projection of the line segment joining the
points (5,-1, 4) and (4,-1,3) on the plane, x+y+z=
JEE M 2018]
WIN
12
3Prove that the points ( boldsymbol{A}= ) ( (1,2,3), B(3,4,7), C(-3,-2,-5) ) are
collinear ( & ) find the ratio in which ( B )
divides ( boldsymbol{A C} )
A .2: 5
B . 2: 3
c. 2: 8
D. 2: 7
12
4In geometry, we take a point, a line and a plane as undefined terms.
A. True
B. False
c. Ambiguous
D. Data Insufficient
12
5Find the shortest distance between the
skew lines:
( l_{1}: frac{x-1}{2}=frac{y+1}{1}=frac{z-2}{4} )
( l_{2}: frac{x+2}{4}=frac{y-0}{-3}=frac{z+1}{1} )
12
6The direction ratios of the line joining the points (4,3,-5) and (-2,1,-8) are
A ( cdot frac{6}{7}, frac{2}{7}, frac{3}{7} )
в. 6,2,3
c. 5,8,0
D. 3,7,9
12
7If a point ( boldsymbol{P} ) from where line drawn cuts
coordinates axes at ( A ) and ( B ) (with ( A ) on
( x-text { axis and } B text { on } y-text { axis }) ) satisfies ( alpha frac{x^{2}}{P B^{2}}+beta frac{y^{2}}{P A^{2}}=1, ) then ( alpha+beta ) is
( A cdot 1 )
B. 2
( c .3 )
D. 4
12
8The planes ( 2 x-y+4 z=5 ) and ( 5 x- )
( 2.5 y+10 z=6 ) are
A. Parallel
B. Perpendicular
c. Intersect
D. intersect ( x ) axis
12
9The number of straight line that are
equally inclined to the three dimensional co- ordinate axes, is
12
10( boldsymbol{A}=(mathbf{1}, mathbf{2}, mathbf{3}), boldsymbol{B}=(mathbf{4}, mathbf{5}, mathbf{7}), boldsymbol{C}= )
( (-4,3,-6), D=(2, k, 2) ) are four
points. If the lines ( A B ) and ( C D ) are
parallel, then ( k= )
( A cdot 0 )
в. -9
( c .9 )
D. 2
12
11Algebraic sum of intercepts made by the plane ( x+3 y-4 z+6=0 ) on the axes is
A. 7
B. 0
( c cdot frac{13}{2} )
( D cdot-frac{13}{2} )
12
12The following lines are ( hat{boldsymbol{r}}=(hat{boldsymbol{i}}+hat{boldsymbol{j}})+ )
( lambda(hat{i}+2 hat{j}-hat{k})+mu(-hat{i}+hat{j}-2 hat{k}) )
A. collinear
B. skew-lines
c. co-planar lines
D. parallel lines
12
13Find the point on Z-axis which are at a distance ( sqrt{21} ) unit from the point
(1,2,3)
12
14If ( boldsymbol{A}=(mathbf{2},-mathbf{3}, mathbf{1}), boldsymbol{B}=(mathbf{3},-mathbf{4}, mathbf{6}) ) and ( boldsymbol{C} )
is a point of trisection of ( A B, ) then ( C_{y}= )
A ( cdot frac{11}{3} )
B. -11
c. ( frac{10}{3} )
D. ( frac{-11}{3} )
12
15The distance between the circumcentre
and the ortho centre of the triangle formed by the points (2,1,5),(3,2,3) and (4,0,4) is
A ( cdot sqrt{6} )
B. ( frac{sqrt{6}}{2} )
c. ( 2 sqrt{6} )
D.
12
16Find the coordinates of a point
equidistant from four points ( boldsymbol{O}(mathbf{0}, mathbf{0}, mathbf{0}), boldsymbol{A}(ell, mathbf{0}, mathbf{0}), boldsymbol{B}(mathbf{0}, boldsymbol{m}, boldsymbol{0}) ) and
( boldsymbol{C}(mathbf{0}, mathbf{0}, boldsymbol{n}) )
12
17Find the direction cosines of a line
which makes equal angles with the coordinate axes.
12
18The projections of a directed line segment on the coordinate axes are
12,4,3 respectively.

What are the direction cosines of the
line segment?
A ( cdot(12 / 13,4 / 13,3 / 13) )
B . ( (12 / 13,-4 / 13,3 / 13) )
c. ( (12 / 13,-4 / 13,-3 / 13) )
D. ( (-12 / 13,-4 / 13,3 / 13) )

12
19If ( boldsymbol{A}=(mathbf{4}, mathbf{1}, mathbf{5}) ) and ( boldsymbol{B}=(mathbf{3}, mathbf{4}, mathbf{5}) ) The
direction ratios of ( overline{A B} ) are
12
20x-2 y. 2+2
56. Let the line –
lie in the plane
3 -5 2 .
x+3y-az+B=0. Then (a, b) equals
[2009]
(a) (-6,7)
(b) (5,-15)
(c) (-5,5)
(d) (6,-17)
12
21If ( boldsymbol{A}=(mathbf{5},-mathbf{1}, mathbf{1}), boldsymbol{B}=(mathbf{7},-mathbf{4}, mathbf{7}), boldsymbol{C}= )
( (1,-6,10), D=(-1,-3,4) . ) Then
( A B C D ) is a
A . square
B. rectangle
c. rhombus
D. none of these
12
22Find the ratio in which ( 2 x+3 y+5 z= )
1 divides the line joining the points
(1,0,-3) and (1,-5,7)
A .1: 2
B . 2: 1
c. 3: 2
D. 2: 3
12
23A plane mirror is placed at the origin so that the direction ratios of its normal
( operatorname{are}(1,-1,1) . ) A ray of light, coming along the positive direction of the ( x ) axis, strikes the mirror. The direction ( operatorname{cosines} ) of the reflected ray are
A ( cdot frac{1}{3}, frac{2}{3}, frac{2}{3} )
B. ( -frac{1}{3}, frac{2}{3}, frac{2}{3} )
c. ( -frac{1}{3},-frac{2}{3},-frac{2}{3} )
D. ( -frac{1}{3},-frac{2}{3}, frac{2}{3} )
12
24Find the square of the distance between
the points whose cartesian coordinates
are:
(-1,1,3),(0,5,6)
12
25Consider three vectors ( vec{P}=hat{i}+widehat{j}+ ) ( widehat{k} ; overrightarrow{boldsymbol{q}}=2 widehat{hat{boldsymbol{i}}}+4 widehat{boldsymbol{j}}-widehat{boldsymbol{k}} ) and ( overrightarrow{boldsymbol{r}}=boldsymbol{2} hat{boldsymbol{i}}+boldsymbol{4} widehat{boldsymbol{j}}+ )
3 ( widehat{k} ). If ( vec{p}, vec{q} ) and ( vec{r} ) denotes the position vector of three non-collinear points, then the equation of the plane containing these points is
A. ( 2 x-3 y+1=0 )
B. ( x-3 y+2 z=0 )
c. ( 3 x-y+z-3=0 )
D. ( 3 x-y-2=0 )
12
26The d.r’s of the line of intersection of the
planes ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}-mathbf{1}=mathbf{0} ) and ( mathbf{2} boldsymbol{x}+ )
( 3 y+4 z-7=0 ) are
A .1,2,-3
в. 2,1,-3
c. 4,2,-6
D. 1,-2,1
12
27Find the distance of point (1,-2,3)
from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} )
measured parallel to the line whose direction cosines are
proportional to 2,3,-6
12
28Let two planes ( p_{1}: 2 x-y+z=2, ) and
( boldsymbol{p}_{2}: boldsymbol{x}+boldsymbol{2} boldsymbol{y}-boldsymbol{z}=boldsymbol{3} ) are given. The
image of plane ( P_{1} ) in the plane mirror
( P_{2} ) is
A. ( x+7 y-4 z+5=0 )
B. ( 3 x+4 y-5 z+9=0 )
c. ( 7 x-y+2 z-9=0 )
0
D. ( 7 x+y+9 z+9=0 )
12
29If the points ( (h, 3,-4),(0,-7,10) ) and ( (1, k, 3) ) are collinear, then ( h+k ) is
( mathbf{A} cdot mathbf{4} )
B.
c. -4
D. 14
12
30Find the equation of the plane containing the line 2x -y+z
-3 = 0, 3x +y+z= 5 and at a distance of Ta from the point
(2,1,-1).
(2005 – 2 Marks)
12
31If ( P(x, y, z) ) moves such that ( x=0, z= )
( 0, ) then the locus of ( P ) is the line whose
d.cs are
A . ( y ) -axis
B. 1,0,0
c. 0,1,0
D. 0,0,0
12
32In three dimensions, the coordinate
axes of a rectangular cartesian coordinate system are
A. three mutually parallel lines
B. three mutually perpendicular lines
c. two mutually perpendicular lines and any two parallel
D. None of these
12
33If line ( frac{boldsymbol{x}-mathbf{2}}{mathbf{3}}=frac{boldsymbol{y}-mathbf{4}}{mathbf{4}}=frac{boldsymbol{z}+mathbf{2}}{mathbf{1}} ) is
parallel to planes ( mu x+3 y-2 z+d= )
0 and ( x-2 lambda y+z=0, ) then value of ( lambda )
and ( mu ) are
A ( cdotleft(mu=4, lambda=-frac{2}{3}right. )
B. ( mu=-6, lambda=-2 )
c. ( _{mu}=frac{-10}{3}, lambda=frac{-1}{2} )
D. ( mu=frac{-10}{3}, lambda=frac{1}{2} )
12
34Vectors ( vec{A}, vec{B} ) and ( vec{C} ) are such that ( vec{A} ) ( vec{B}=0 . ) Then the vector parallel to ( vec{A} ) is
A. ( vec{B} ) and ( vec{C} )
в. ( vec{A} times vec{B} )
c. ( vec{B}+vec{C} )
D. ( vec{B} times vec{C} )
12
35Find the equation of the plane passing through (2,0,1) and (3,-3,4) and perpendicular to ( boldsymbol{x}-mathbf{2} boldsymbol{y}+boldsymbol{z}=mathbf{6} )12
36Number of points having positive integral co-ordinate lying on the plane ( x+2 y+3 z=15 ) is ( n, ) then ( frac{n}{2} ) is equal
to
( mathbf{A} cdot mathbf{6} )
B. 8
c. 9
D.
12
37The vector equation of the plane passes through the points ( A & B ) with position vector ( 2 hat{i}+hat{j}-hat{k} &-hat{i}+3 hat{j}+4 hat{k} )
respectively ( & ) Ler to the plane ( bar{r} cdot(hat{i}-2 hat{j}+4 hat{k})=10 ) is
A ( cdot bar{r} cdot(18 hat{i}+17 hat{j}-3 hat{k})=49 )
B . ( bar{r} .(18 hat{i}-17 hat{j}-3 hat{k})+22=0 )
c. ( bar{r} .(18 hat{i}+17 hat{j}+4 hat{k})=25 )
D・ ( bar{r} .(18 hat{i}+17 hat{j}+4 hat{k})=24 )
12
38The vector equation of the line ( frac{x-2}{2}= ) ( frac{2 y-5}{-3}, z=-1 ) is ( vec{r}= )
( left(2 hat{i}+frac{5}{2} hat{j}-hat{k}right)+lambdaleft(2 hat{i}-frac{3}{2} hat{j}+x hat{k}right) )
where ( x ) is equal to
( mathbf{A} cdot mathbf{0} )
B.
c. 2
D. 3
12
39If ( P(x, y, z) ) is a point on the line segment joining ( Q(2,2,4) ) and ( R(3,5,6) ) such that the projection of ( overrightarrow{O P} ) on the axes are ( frac{13}{5}, frac{19}{5}, frac{26}{5} )
respectively, then ( P ) divides ( Q R ) in ratio
A .1: 3
B. 2: 3
c. 3: 2
D. 3: 1
12
40f ( (p, q, r) ) is equidistant from
(1,2,-3),(2,-3,1) and ( (-3,1,2), ) then
( boldsymbol{p}+boldsymbol{q}+boldsymbol{r}= )
A . -1
B.
c. 0
D.
12
41The acute angle between two lines such
that the direction cosines ( l, boldsymbol{m}, boldsymbol{n} ) of each of them satisfy the equation ( l+ )
( boldsymbol{m}+boldsymbol{n}=mathbf{0} ) and ( l^{2}+boldsymbol{m}^{2}-boldsymbol{n}^{2}=mathbf{0} ) is
( A cdot 30 )
B . 45
( c cdot 60 )
D. ( 15^{circ} )
12
42Plane ( a x+b y+c z=1 ) intersect axes ( operatorname{in} A, B, C ) respectively. If ( Gleft(frac{1}{6},-frac{1}{3}, 1right) )
is a centroid of ( triangle A B C ) then ( a+b+ )
( 3 c=-1 )
A ( cdot frac{4}{3} )
B. 4
( c cdot 2 )
D.
12
43The equation of the plane passing through the straight line ( frac{x-1}{2}= ) ( frac{boldsymbol{y}+mathbf{1}}{mathbf{- 1}}=frac{boldsymbol{z}-mathbf{3}}{mathbf{4}} ) and perpendicular to
plane ( boldsymbol{x}+mathbf{2} boldsymbol{y}+boldsymbol{z}=mathbf{1 2} ) is:
A. ( 9 x+2 y-5 z+8=0 )
в. ( 9 x+2 y-5 z+10=0 )
c. ( 9 x-2 y+5 z+6=0 )
D. ( 9 x-2 y-5 z+4=0 )
12
44Derive the equation of the locus of a point equivalent from the points (1,-2,3) and (-3,4,2)12
45If vector ( vec{a}=4 hat{i}+5 hat{j}-3 hat{k} ) and ( vec{b}=5 hat{i}+ )
( 3 hat{j}+8 hat{k} ) then value of ( frac{text { projection of vector b on a }}{text { projection of vector a on b }} ) is :
A ( cdot frac{7}{5} )
B. ( frac{2}{7} )
( c cdot frac{5}{8} )
D. None of these.
12
46Find ( a, b, c ) if ( a(1,3,2)+b(1,-5,6)+ )
( c(2,1,-2)=(4,10,-8) )
12
47A variable plane at a distance of 1 unit from the origin cuts the co-ordinate axes at ( A, B ) and ( C . ) If the centroid ( D(x, y, z) ) of triangle ( A B C ) satisfies the relation ( frac{1}{x^{2}}+ ) ( frac{1}{y^{2}}+frac{1}{z^{2}}=k, ) then the value of ( k ) is
A . 3
B. 1
c. ( 1 / 3 )
D.
12
48The points (-5,12),(-2,-3),(9,-10),(6,5)
taken in order, form
A. Parallelogram
B. rectangle
c. rhombus
D. square
12
49The distance from the origin to the centroid of the tetrahedron formed by
the points ( (0,0,0),(a, 0,0),(0, b, 0),(0,0, c) ) is:
A ( cdot frac{sqrt{a+b+c}}{4} )
B. ( frac{sqrt{a+b+c}}{3} )
c. ( frac{sqrt{a^{2}+b^{2}+c^{2}}}{16} )
D. ( frac{sqrt{a^{2}+b^{2}+c^{2}}}{4} )
12
50A point ( P ) lies on a line whose ends are ( A(1,2,3) ) and ( B(2,10,1) . ) If ( z )
component of ( boldsymbol{P} ) is ( boldsymbol{7}, ) then the
coordinates of ( boldsymbol{P} ) are
A ( cdot(-1,-14,7) )
B. (1,-14,7)
c. (-1,14,7)
D. (1,14,7)
12
51A parallelopiped ‘S’ has base points A, B, C and D and
upper face points A’, B’, C and D’. This parallelopiped is
compressed by upper face A’B’C’D’ to form a new
parallelopiped ‘T” having upper face points A”, B”, C” and
D”. Volume of parallelopiped Tis 90 percent of the volume
of parallelopiped S. Prove that the locus of ‘A”?, is a plane
12
52If 5,7,6 are the sums of the ( x, y )
intercepts; ( boldsymbol{y}, boldsymbol{z} ) intercepts, ( boldsymbol{z}, boldsymbol{x} )
intercepts respectively of a plane then the perpendicular distance from the origin to that plane is
A ( cdot frac{144}{61} )
в. ( frac{12}{sqrt{61}} )
c. ( frac{sqrt{61}}{12} )
D. ( frac{61}{144} )
12
53Find the angle between the following pairs of lines:
( frac{x-1}{2}=frac{y-2}{3}=frac{z-3}{-3} ) and ( frac{x+3}{-1}= )
( frac{boldsymbol{y}-mathbf{5}}{mathbf{8}}=frac{boldsymbol{z}-mathbf{1}}{mathbf{4}} )
12
54The vector equation of the plane which is at a distance of ( frac{3}{sqrt{14}} ) from the origin and the normal from the origin is ( 2 hat{i}- ) ( mathbf{3} hat{boldsymbol{j}}+hat{boldsymbol{k}} ) is
A ( . vec{r} .(2 hat{i}-3 hat{j}+hat{k})=3 )
B . ( vec{r} .(hat{i}+hat{j}+hat{k})=9 )
c. ( vec{r} .(hat{i}+2 hat{j})=3 )
D. ( vec{r} .(2 hat{i}+hat{k})=3 )
12
55If a line makes angles ( alpha, beta, gamma ) with axes
of co-ordinates, then ( cos 2 alpha+cos 2 beta+ )
( cos 2 gamma ) is equla to
A . -2
B. –
c. 1
D. 2
12
56Name the octants in which the following
points lie:
( (1,2,3),(4,-2,3)(4,-2,-5),(4,2,-5) )
12
57The perpendicular distance of ( vec{A}(1,4,-2) ) from the segment BC where ( vec{B} ) (2,1,-2) and ( vec{C}(0,-5,1) ) is
( A cdot frac{3}{7} sqrt{26} )
B ( cdot frac{6}{7} sqrt{26} )
( mathbf{c} cdot frac{4}{7} sqrt{26} )
( D cdot frac{2}{7} sqrt{26} )
12
58Find the direction cosines of
perpendicular from the origin to the plane ( bar{r}(2 hat{i}+3 hat{j}+6 hat{k})+7=0 )
12
59Perpendiculars ( A P, A Q ) and ( A R ) are drawn to the ( x-, y- ) and ( z- ) axes, respectively
from the point ( boldsymbol{A}(mathbf{1},-mathbf{1}, mathbf{2}) . ) The A.M. of
( A P^{2}, A Q^{2} ) and ( A R^{2} ) is
A .4
B. 5
( c .3 )
D.
12
60If ( boldsymbol{A}(boldsymbol{6},-mathbf{7}, mathbf{0}), boldsymbol{B}(mathbf{1 6},-mathbf{1 9},-mathbf{4}) )
( C(0,3,-6) ) and ( D(2,-5,10) ) are four points in space, then the point of intersection of the lines ( A B ) and ( C D ) is
A ( cdot(2,1,-1) )
в. (1,1,2)
D. does not exist as the lines are skew
12
61A plane meet the co-ordinate axes in
( A, B, C ) such that the centroid of
triangle ( A B C ) is ( (a, b, c) . ) If equation of plane ( frac{x}{a}+frac{y}{b}+frac{z}{c}=k(k neq 0), ) then the
value of ( k ) equals
A . 2
B. 3
( c cdot 4 )
D. 5
12
62Two system of rectangular axes have the same origin. If a plane cuts them at
distances, ( a, b, c ) and ( a_{1}, b_{1}, c_{1} ) from the
origin, then
A ( cdot frac{1}{a^{2}}+frac{1}{b^{2}}+frac{1}{c^{2}}=frac{1}{a_{1}^{2}}+frac{1}{b_{1}^{2}}+frac{1}{c_{1}^{2}} )
B. ( frac{1}{a^{2}}-frac{1}{b^{2}}+frac{1}{c^{2}}=frac{1}{a_{1}^{2}}-frac{1}{b_{1}^{2}}+frac{1}{c_{1}^{2}} )
C ( cdot a^{2}+b^{2}+c^{2}=a_{1}^{2}+b_{1}^{2}+c_{1}^{2} )
D cdot ( a^{2}-b^{2}+c^{2}=a_{1}^{2}-b_{1}^{2}+c_{1}^{2} )
12
63ox, oy are positive x-axis, positive ( y ) axis respectively where ( boldsymbol{O}=(mathbf{0}, mathbf{0}, mathbf{0}) )
The ( d . c . s ) of the llne which bisects ( angle x o y )
are
( mathbf{A} cdot 1,1,0 )
B. ( frac{1}{sqrt{2}}, frac{1}{sqrt{2}}, 0 )
c. ( frac{1}{sqrt{2}}, 0, frac{1}{sqrt{2}} )
D. 0,0,1
12
64Find the equation of the set of points which are equidistant from the points (1,2,3) and (3,2,-1)12
65The ratio in which the plane ( bar{r} .(bar{i}-2 bar{j}+ ) ( mathbf{3} bar{k})=17 ) divides the line joining the points ( -2 bar{i}+4 bar{j}+7 bar{k} ) and ( 3 bar{i}-5 bar{j}+8 bar{k} )
is
A. 1: 10
B. 3: 10
( c .3: 5 )
D. 1: 5
12
66Consider three planes ( 2 x+p y+6 z= )
( mathbf{8}, boldsymbol{x}+mathbf{2} boldsymbol{y}+boldsymbol{q} boldsymbol{z}=mathbf{5} ) and ( boldsymbol{x}+boldsymbol{y}+mathbf{3} boldsymbol{z}=mathbf{4} )
These planes do not have any common point of intersection if-
A. ( p=2, q neq 3 )
B . ( p neq 2, q neq 3 )
c. ( p neq 2, q=3 )
D. ( p=2, q=3 )
12
67The ratio in which the surface ( x^{2}+ )
( y^{2}+z^{2}=25 ) divides the line joining
(0,1,2) and (3,4,5) is ( frac{a pm sqrt{b}}{c} ) then ( boldsymbol{a}+boldsymbol{b}+boldsymbol{c}= )
12
68The direction cosine of a line which is
perpendicular to both the lines whose direction ratios are 1,2,2 and 0,2,1 are
A ( cdot frac{-2}{3}, frac{1}{3}, frac{2}{3} )
B. ( frac{2}{3}, frac{-1}{3}, frac{2}{3} )
c. ( frac{2}{3}, frac{1}{3}, frac{-2}{3} )
D. ( frac{2}{3}, frac{-1}{3}, frac{-2}{3} )
12
69Three lines are given by ( vec{r}=lambda hat{i}, lambda epsilon R ) ( overrightarrow{boldsymbol{r}}=boldsymbol{mu}(hat{boldsymbol{i}}+hat{boldsymbol{j}}), boldsymbol{n} boldsymbol{epsilon} boldsymbol{R} ) and ( overrightarrow{boldsymbol{r}}=boldsymbol{v}(hat{boldsymbol{i}}+hat{boldsymbol{j}}+ )
( hat{boldsymbol{k}}), boldsymbol{v} boldsymbol{epsilon} boldsymbol{R} )
Let the lines cut the plane ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}= )
1 at he points ( A, B ) and ( C ) respectively.
If the area of the triangle ( A B C ) is ( triangle )
then the value of ( (6 triangle)^{2} ) equals
A . 0.75
в. ( 0 . )
( c .0 .85 )
D. 0.65
12
70Find the distance of a point (3,-5) from the line ( 3 x-4 y-5=0 )12
71The direction cosines of a vector ( hat{boldsymbol{i}}+ ) ( hat{boldsymbol{j}}+sqrt{mathbf{2}} hat{boldsymbol{k}} ) are
A ( cdot frac{1}{2}, frac{1}{2}, 1 )
B. ( frac{1}{sqrt{2}}, frac{1}{sqrt{2}}, frac{1}{2} )
c. ( frac{1}{2}, frac{1}{2}, frac{1}{sqrt{2}} )
D. ( frac{1}{sqrt{2}}, frac{1}{sqrt{2}}, frac{1}{sqrt{2}} )
12
72Find the equation of plane with intercepts 2,3 and 4 on the ( x, y ) and ( z ) axis respectively.12
73If the extremities of a diagonal of a square are (1,-2,3) and (4,2,3) then the area of the square is
A . 25
B. 50
c. ( frac{25}{2} )
D. ( sqrt{50} )
12
74The points ( (10,7,0),(6,6-1) ) and
(6,9,-4) form a
A. Right -angled triangle
B. Isosceles triangle
( c cdot ) Both (1)( &(2) )
D. Equilateral triangle
12
75If ( bar{a}, bar{b} ) and ( bar{c} ) are non-zero non collinear vectors and ( theta(neq 0, pi) ) is the angle between ( bar{b} ) and ( bar{c} ) if ( (bar{a} times bar{b}) times bar{c}=frac{1}{2}|bar{b}| bar{c} mid bar{a} )
then ( sin theta= )
A ( cdot sqrt{frac{2}{3}} )
B. ( frac{sqrt{3}}{2} )
( c cdot frac{4 sqrt{2}}{3} )
D. ( frac{2 sqrt{2}}{3} )
12
76The point which is equidistant from the points (-1,1,3),(2,1,2),(0,5,6) and
(3,2,2) is
A ( cdot(-1,3,4) )
в. (3,1,4)
c. (1,3,4)
D. (4,1,3)
12
77Find the coordinates of the point where the line ( frac{boldsymbol{x}+mathbf{1}}{mathbf{2}}=frac{boldsymbol{y}+boldsymbol{2}}{mathbf{3}}=frac{boldsymbol{z}+mathbf{3}}{mathbf{4}} ) meets
the plane ( boldsymbol{x}+boldsymbol{y}+boldsymbol{4} boldsymbol{z}=boldsymbol{6} )
12
78Number of lines is space which are equally inclined to three co-ordinate
axes are?
A .2
B. 4
( c .6 )
D. 8
12
79A cube of side 5 has one vertex at the
point ( (1,0,-1), ) and the three edges from this vertex are, respectively, parallel to the negative ( x ) and ( y ) axes and positive z-axis. Find the coordinates of the other
vertices of the cube.
A. (1,0,1)
B. (0,-1,0)
c. (0,0,-1)
D. (1,0,0)
12
8045.
The two lines x = ay+b, z= cy+d; and x = a’y+b’,
[2006||
z=c’y+d’ are perpendicular to each other if
(a) aa’+cc’=-1 (b) aa’t.cc’ = 1
(c) 9+ =-1 (d) 9+6=1
12
81Distance between two parrallel lines, ( overline{boldsymbol{r}}=overline{boldsymbol{a}}_{1}+boldsymbol{lambda} overline{boldsymbol{b}} ) and ( overline{boldsymbol{r}}=overline{boldsymbol{a}}_{2}+boldsymbol{mu} overline{boldsymbol{b}}, ) is given
by
A ( cdot d=mid frac{left(bar{a}_{2}-bar{a}_{1}right)}{hat{b}} )
B . ( d=midleft(bar{a}_{2}-bar{a}_{1}right) times hat{b} )
c. ( d=midleft(bar{a}_{2}+bar{a}_{1}right) times hat{b} )
D . ( d=midleft(bar{a}_{2}-bar{a}_{1}right) )
12
82If a line ( O P ) of length ( r ) (Where ‘ ( O ) ‘ is the origin) makes an angle ( alpha ) with ( x ) -axis
and lies on the xz-plane, then what are the coordinates of ( P ? )
A ( cdot(r cos alpha, 0, r sin alpha) )
B . ( (0,0, r sin alpha) )
( mathbf{c} cdot(r cos alpha, 0,0) )
D ( cdot(0,0, r cos alpha) )
12
83Find the equation of the plane passing through the points (2,3,-4) and (1,-1,3) and parallel to the ( x- ) axis.12
84Find the angle between the line whose
direction cosines are given by ( l+m+ )
( boldsymbol{n}=mathbf{0} ) and ( l^{2}+boldsymbol{m}^{2}=boldsymbol{n}^{2} )
12
8563.
If the angle between the line x=>
lin _y-1
Z-3
– and the plane
2
x +2y + 33 =4 is cos” (193), then aequals
x + 2y + 3z=4 is cos-1
, then a equals
12011
[2011]
12
86The intercepts of the plane ( 2 x-3 y+ )
( mathbf{5} z-mathbf{3 0}=mathbf{0} ) are
A. 15,-10,6
в. 5,10,6
c. ( 1 / 8,-1 / 6,1 / 4 )
D. 3,-4,6
12
87If ( vec{P}(1,5,4) ) and ( vec{Q}(4,-1,-2), ) find the direction ratio of ( overrightarrow{P Q} )12
8839. Equation of the plane containing the straight line
and perpendicular to the plane containing the
(2010)
x y z
straight lines – ===
3 4 2
(a) x+2y – 2z=0
(c) x-2y+z=0
x y z
is
2
(b) 3x + 2y – 2z=0
(d) 5x + 2y – 4z=0
12
89( A ) point ( C ) with position vector ( frac{3 a+4 b-5 c}{3} )
(where ( a, b ) and ( c ) are non co-planar
vectors) divides the line joining ( A ) and ( B ) in the ratio ( 2: 1 . ) If the position vector
of ( A ) is ( a-2 b+3 c, ) then the position
vector of ( boldsymbol{B} ) is
A ( .2 a+3 b-4 c )
B . ( 2 a-3 b+4 c )
c. ( 2 a+3 b+4 c )
D. ( a+3 b-4 c )
12
90The coordinates of a point which is equidistant from the point ( (0,0,0),(a, 0,0),(0, b, 0) ) and ( (0,0, c) )
are given by
( ^{mathbf{A}} cdotleft(frac{a}{2}, frac{b}{2}, frac{c}{2}right) )
в. ( left(frac{-a}{2}, frac{-b}{2}, frac{c}{2}right) )
( ^{mathrm{c}}left(frac{a}{2}, frac{-b}{2}, frac{-c}{2}right) )
D ( cdotleft(frac{-a}{2}, frac{b}{2}, frac{-c}{2}right) )
12
91Find the magnitude of the shortage
distance between the lines ( , frac{x-8}{3}= )
( frac{y+9}{-16}=frac{z-10}{7} ; frac{x-15}{3}=frac{y-29}{8}=frac{z-5}{-5} )
12
92A line making angles ( 45^{circ} ) and ( 60^{circ} ) with the positive direction of ( x- ) axis and
( y- ) axis respectively. Then the angle made by the line with positive direction
of ( z- ) axis is
A ( .60^{circ} )
B. ( 120^{circ} )
( mathbf{c} cdot 60^{circ} ) or ( 120^{circ} )
D. None of these
12
93The equation to the altitude of the
altitude triangle formed by
( (1,1,1) cdot(1,2,3),(2,-1,1) ) through
(1,1,1) is
A ( cdot bar{r}=(bar{i}+bar{j}+bar{k})+t(bar{i}-bar{j}-2 bar{k}) )
B ( cdot bar{r}=(bar{i}-bar{j}+bar{k})+t(bar{i}+bar{j}-2 bar{k}) )
( mathbf{c} cdot bar{r}=(bar{i}+bar{j}+bar{k})+t(bar{i}-bar{j}+2 bar{k}) )
D ( cdot bar{r}=(bar{i}-bar{j}-bar{k})+t(bar{i}+bar{j}-2 bar{k}) )
12
94Cartesian equation of the plane ( bar{r}= ) ( (1+lambda-mu) bar{i}+(2-lambda) bar{j}+(3-2 lambda+ )
( 2 mu) bar{k} ) is :
A ( .2 x+y=5 )
в. ( 2 x-y=5 )
c. ( 2 x+z=5 )
D. ( 2 x-mathrm{z}=5 )
12
95Unit vector perpendicular to the plane passing through the points ( hat{mathbf{i}}-hat{mathbf{j}}+ ) ( 2 hat{k}, 2 hat{i}-hat{k} ) and ( 2 hat{j}+hat{k} ) is
A ( cdot frac{2 hat{i}-hat{j}+hat{k}}{sqrt{6}} )
( ^{text {В } cdot frac{2 hat{i}+hat{j}+hat{k}}{sqrt{6}}} )
c. ( frac{2 hat{i}+hat{j}-hat{k}}{sqrt{6}} )
D. None of these
12
96Find the direction cosines of two lines
which are connected by the relations ( l+m+n=0 ) and ( m n-2 n l-2 l m= )
( mathbf{0} )
12
97Line ( overrightarrow{boldsymbol{r}}=(hat{boldsymbol{i}}-hat{boldsymbol{j}}+hat{boldsymbol{k}})+boldsymbol{t}(boldsymbol{2} hat{boldsymbol{i}}-hat{boldsymbol{j}}+hat{boldsymbol{k}}) )
contained in a plane to which vector ( vec{n}=3 hat{i}-2 hat{j}+lambda hat{k} ) is normal. Find the
value of ( lambda ). Also find the vector equation
of the plane.
12
98Three vertices of a tetrahedron are
(0,0,0),(6,-5,-1) and ( (-4,1,3) . ) If the centroid of the tetrahedron be (1,-2,5)
then the fourth vertex is
A. (2,-4,18)
в. (2,-4,-18)
( ^{c} cdotleft(frac{3}{4}, frac{-3}{2}, frac{7}{4}right) )
D. none of these
12
99Assertion
If a line makes angles ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} ) with
( O X, O Y, O Z ) respectively, then ( sin ^{2} alpha+sin ^{2} beta+sin ^{2} gamma=2 )
Reason
If ( l=cos alpha, m=cos beta, n=cos gamma, ) are
direction cosines of a line, then ( l^{2}+ )
( boldsymbol{m}^{2}+boldsymbol{n}^{2}=mathbf{1} )
A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
B. Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion
C. Assertion is true but Reason is false
D. Assertion is false but Reason is true
12
100The plane ( x=0 ) divides the joinning of (-2,3,4) and (1,-2,3) in the ratio
A .2: 1
B. 1: 2
c. 3: 2
D. -4: 3
12
101The image of the line ( frac{boldsymbol{x}-mathbf{1}}{mathbf{3}}=frac{boldsymbol{y}-mathbf{3}}{mathbf{1}}= )
( frac{z-4}{-5} ) in the plane ( 2 x-y+z+3=0 )
is the line?
A. ( frac{x-3}{3}=frac{y+5}{1}=frac{z-2}{-5} )
в. ( frac{x-3}{-3}=frac{y+5}{-1}=frac{z-2}{5} )
c. ( frac{x+3}{3}=frac{y-5}{1}=frac{z-2}{-5} )
D. ( frac{x+3}{-3}=frac{y-5}{-1}=frac{z+2}{5} )
12
102If a line makes angles ( alpha, beta, gamma ) with the
coordinate axes, then the value of
( cos 2 alpha+cos 2 beta+cos 2 gamma ) is
( A cdot 3 )
B. – –
( c cdot 2 )
D. –
12
103( begin{array}{ll}text { List I } & text { List II } \ text { 1) d.c’s of } x-text { axis } & text { a) }(1,1,1) \ text { 2) d.c’s of } y-text { axis } & text { b) } \ text { 3) d.c’s of } z-text { axis } & left(frac{1}{sqrt{3}} frac{1}{sqrt{3}}, frac{1}{sqrt{3}}right) \ begin{array}{l}text { 4) d.c’s of a line makes equal } \ text { angles with axes }end{array} & text { c) }(1,0,0) \ begin{array}{l}text { d) }(0,1,0) \ text { e) }(0,0,1)end{array}end{array} ) The correct order for 1,2,3,4 is
( mathbf{A} cdot c, d, e, b )
B. ( a, b, c, e )
( mathbf{C} cdot c, d, a, b )
D. ( b, c, a, e )
12
104If the centroid of the tetrahedron
( O A B C, ) where ( A, B, C ) are given by ( (alpha, 5,6),(1, beta, 4),(3,2, gamma) ) respectively
be ( 1,-1,2, ) then value of ( alpha^{2}+beta^{2}+gamma^{2} )
equals
A ( cdot alpha^{2}+beta^{2} )
B. ( gamma^{2}+beta^{2} )
c. ( alpha^{2}+gamma^{2} )
D. None of these
12
105The shortest distance between the lines ( frac{boldsymbol{x}-mathbf{5}}{mathbf{4}}=frac{boldsymbol{y}-mathbf{7}}{-mathbf{5}}=frac{boldsymbol{z}+mathbf{3}}{-mathbf{5}} ) and ( frac{boldsymbol{x}-mathbf{8}}{mathbf{4}}= )
( frac{y-7}{-5}=frac{z-5}{-5} ) is
A . 45
B . 46
c. 47
D. 48
12
106If ( overrightarrow{mathbf{A}} times overrightarrow{mathbf{B}}=overrightarrow{mathbf{B}} times overrightarrow{mathbf{A}}, ) then the angle
between ( A ) and ( B ) is
A . ( pi )
в. ( pi / 3 )
c. ( pi / 2 )
D . ( pi / 4 )
12
107( boldsymbol{A}=(-1,2,-mathbf{3}), boldsymbol{B}=(mathbf{5}, mathbf{0},-mathbf{6}), boldsymbol{C}= )
(0,4,-1) are the vertices of a triangle.
The d.c’s of the internal bisector of ( angle mathrm{BAC} )
are?
( ^{mathbf{A}} cdotleft(frac{25}{sqrt{714}}, frac{-8}{sqrt{714}}, frac{-5}{sqrt{714}}right) )
в. ( left(frac{5}{sqrt{74}}, frac{6}{sqrt{74}}, frac{8}{sqrt{74}}right) )
( ^{mathbf{C}} cdotleft(frac{25}{sqrt{714}}, frac{8}{sqrt{714}}, frac{5}{sqrt{714}}right) )
D. ( left(frac{-5}{sqrt{74}}, frac{6}{sqrt{74}}, frac{-8}{sqrt{74}}right) )
12
108Equation of plane parallel to ( 3 x+4 y+ )
( mathbf{5} boldsymbol{z}-mathbf{6}=mathbf{0}, mathbf{6} boldsymbol{x}+mathbf{8} boldsymbol{y}+mathbf{1 0} boldsymbol{z}-mathbf{1 6}=mathbf{0} )
and equidistant from them is
A. ( 3 x+4 y+5 z=7 )
B. ( 3 x+4 y+5 z=10 )
c. ( 6 x+8 y+10 z=0 )
D. ( 6 x+8 y+10 z=3 )
12
109Assertion (A): The points ( boldsymbol{A}(mathbf{2}, mathbf{9}, mathbf{1 2}), boldsymbol{B}(mathbf{1}, mathbf{8}, mathbf{8}), boldsymbol{C}(mathbf{2}, mathbf{1 1}, mathbf{8}) boldsymbol{D}(mathbf{1}, mathbf{1 2} )
are the vertices of a rhombus
Reason ( (mathrm{R}): A B=B C=C D=D A )
and ( boldsymbol{A C}=boldsymbol{B D} )
A. Both A and R are individually true and R is the correct explanation of
B. Both A and R individually true but R is not the correct explanation of A
c. ( A ) is true but ( R ) is false
D. Both A and R false
12
110Two equat ions ( vec{r} . vec{n}_{1}=q_{1} ) and ( vec{r} . vec{n}_{2}=q_{2} )
represent two perpendicular planes, where ( vec{n}_{1} ) and ( vec{n}_{2} ) are two unit vectors.
One of these plane is rotated through an angle 45 about line of intersection of
two given planes then equation of plane in new position can be This question has multiple correct options
A ( cdot vec{r} cdotleft(vec{n}_{1}+vec{n}_{2}right)=q_{1}-q_{2} )
В ( cdot vec{r} cdot(overrightarrow{n_{1}}+overrightarrow{n_{2}})=q_{1}+q_{2} )
c. ( vec{r} .left(vec{n}_{1}-vec{n}_{2}right)=q_{1}+q_{2} )
D ( cdot vec{r} cdot(overrightarrow{n_{1}}-overrightarrow{n_{2}})=q_{1}-q_{2} )
12
11113.
The shortest distance from the plane 12x+4y+3z =327
to the sphere x2 + y2 + z2 + 4x – 2y – 6z =155 is
(2) 39
(b) 26
(c) 11
(d) 13
12
11225.
-3
y-k _Z intersect, then
If the lines
x-1 y+1 Z-1
2 3 4 ”
the value of k is
(a) 3/2 (b) 9/2
and
2
1
(c) – 2/9
(2004S)
(d) – 3/2
12
113( boldsymbol{P}(mathbf{0}, mathbf{5}, mathbf{6}), boldsymbol{Q}(mathbf{1}, mathbf{4}, mathbf{7}), boldsymbol{R}(mathbf{2}, mathbf{3}, mathbf{7}) ) and
( S(3,5,16) ) are four points in the space.
The point nearest to the origin ( boldsymbol{O}(mathbf{0}, mathbf{0}, mathbf{0}) )
is
A. ( P )
B. ( Q )
( c . R )
D. ( S )
12
114Show that the lines whose d.c’s are
given by ( 2 l+2 m-n=0, m n+n l+ )
( l m=0 ) are perpendicular to each other
12
115The coordinates of any point, which lies
on ( x ) axis are
A ( .(0, x, 0) )
в. ( (x, 0,0) )
c. ( (x, x, 0) )
D. ( (x, x, x) )
12
116If ( boldsymbol{A}=(mathbf{1}, mathbf{2}, mathbf{3}), boldsymbol{B}=(mathbf{2}, mathbf{3}, mathbf{4}) ) and ( boldsymbol{A} boldsymbol{B} ) is
produced upto ( C ) such that ( 2 A B=B C )
then ( C= )
A. (5,4,6)
в. (6,2,4)
c. (4,5,6)
D. (6,4,5)
12
117If ( P(x, y, z) ) is a point on the line
segment joining ( Q(2,2,4) ) and
( boldsymbol{R}(boldsymbol{3}, boldsymbol{5}, boldsymbol{6}) ) such that the projection of
( O P ) on the axes are ( frac{13}{5}, frac{19}{5}, frac{26}{5} )
respectively, then ( P ) divides ( Q R ) in the ratio
A .1: 2
B. 3: 2
( c cdot 2: 3 )
D. 1: 3
12
118Find the intersection of the line ( x- )
( mathbf{2} boldsymbol{y}+mathbf{4} boldsymbol{z}+mathbf{4}=mathbf{0}, boldsymbol{x}+boldsymbol{y}+boldsymbol{z}-mathbf{8}=mathbf{0} )
with the plane ( boldsymbol{x}-boldsymbol{y}+mathbf{2} boldsymbol{z}+mathbf{1}=mathbf{0} )
12
119The coordinates of a point ( mathrm{P} ) are (3,12,4) w.r.t origin ( 0, ) then the direction cosines of ( O P ) are
( mathbf{A} cdot 3,12,4 )
B. ( frac{1}{4}, frac{1}{3}, frac{1}{2} )
c. ( frac{3}{sqrt{13}}, frac{1}{sqrt{13}}, frac{2}{sqrt{13}} )
D. ( frac{3}{13}, frac{12}{13}, frac{4}{13} )
12
120The angle between vectors ( (bar{M} times bar{N}) ) and ( (overline{boldsymbol{N}} times overline{boldsymbol{M}}) ) is then
( mathbf{A} cdot 0^{circ} )
B . ( 60^{circ} )
( c .90^{circ} )
D. ( 180^{circ} )
12
121If the plane ( 3 x+2 y+6 z=6 )
intersects the coordinate axes at
( A, B, C ) then the area of the ( Delta A B C ) is
( mathbf{A} cdot 49 )
B. 7
( c cdot frac{7}{2} )
D. ( frac{11}{2} )
12
122Arrange the points: ( mathbf{A}(1,2- ) ( mathbf{3}), mathbf{B}(-mathbf{1}, mathbf{2},-mathbf{3}), mathbf{C}(-mathbf{1},-mathbf{2}-mathbf{3}) ) and
( mathbf{D}(mathbf{1},-mathbf{2},-mathbf{3}) ) in the increasing order of
their octant numbers:
A. ( A, B, C, D )
в. ( B, C, D, A )
c. ( C, D, A, B )
D. ( D, C, B, A )
12
123Distance of the point ( boldsymbol{P}(overrightarrow{boldsymbol{p}}) ) from the line ( vec{r}=vec{a}+lambda vec{b} ) is –
A ( cdot(vec{a}-vec{p})+frac{((vec{p}-vec{a}) cdot vec{b}) vec{b}}{|vec{b}|^{2}} mid )
B. ( (vec{b}-vec{p})+frac{((vec{p}-vec{a}) cdot vec{b}) vec{b}}{|vec{b}|^{2}} mid )
c. ( quadleft|(vec{a}-vec{p})+frac{((vec{p}-vec{b}) cdot vec{b}) vec{b}}{|vec{b}|^{2}}right| )
D. None of these.
12
124A line makes angle ( theta_{1}, theta_{2}, theta_{3}, theta_{4} ) with the
diagonals of the cube. Show that ( cos ^{2} theta_{1}+cos ^{2} theta_{2}+cos ^{2} theta_{3}+cos ^{2} theta_{4}= )
( frac{4}{3} ? )
12
125if a line makes angles ( alpha, beta, gamma, delta ) with
four diagonals a cube then value of
( sin ^{2} alpha+sin ^{2} beta+sin ^{2} gamma+sin ^{2} delta ) equals
( A cdot 2 )
B. ( frac{4}{3} )
( c cdot frac{8}{3} )
D.
12
126Derive the coordinates of the points ( R(x, y, z) ) dividing the line joining the
points ( boldsymbol{P}left(boldsymbol{x}_{1}, boldsymbol{y}_{1}, boldsymbol{z}_{1}right) ) and ( boldsymbol{Q}left(boldsymbol{x}_{2}, boldsymbol{y}_{2}, boldsymbol{z}_{2}right) )
internally in the ratio ( m: n )
12
127Find the value of ( lambda ) for which the four
points ( A, B, C, D ) with position vectors ( -widehat{boldsymbol{j}}-widehat{boldsymbol{k}} ; boldsymbol{4} hat{boldsymbol{i}}+boldsymbol{5} hat{boldsymbol{j}}+boldsymbol{lambda} hat{boldsymbol{k}} ; boldsymbol{3} hat{boldsymbol{i}}+boldsymbol{9} hat{boldsymbol{j}}+boldsymbol{4} widehat{boldsymbol{k}} ) and
( -4 hat{i}+4 widehat{j}+4 widehat{k} ) are coplanar.
12
128If direction cosines of two lines are
proportional to (2,3,-6) and (3,-4,5) then the acute angle between them is
( ^{mathbf{A}} cdot cos ^{-1}left(frac{49}{36}right) )
B. ( cos ^{-1}left(frac{18 sqrt{2}}{35}right) )
( c cdot 96^{circ} )
D. ( cos ^{-1}left(frac{18}{35}right) )
12
129A non-zero vector ( vec{a} ) is parallel to the line
of intersection of the plane determined by the vectors ( hat{i}, hat{i}+hat{j} ) and the plane determined by the vectors ( hat{i}-hat{j}, hat{i}+hat{k} ) The angle between ( vec{a} ) and ( hat{i}-2 hat{j}+2 hat{k} ) is
A.
в.
c.
D. ( frac{pi}{2} )
12
130The intercept made by the plane ( vec{r} cdot vec{n}= ) ( q ) on the ( x ) -axis is
A ( cdot frac{q}{hat{i} cdot vec{n}} )
в. ( frac{hat{i} cdot vec{n}}{q} )
( c cdot frac{hat{i} cdot q}{n} )
D. ( frac{q}{|vec{n}|} )
12
131The point ( P ) is the intersection of the
straight line joining the points ( Q(2,3,5) ) and ( R(1,-1,4) ) with the plane
( 5 x-4 y-z=1 . ) If ( S ) is the foot of the
perpendicular drawn from the point
( T(2,1,4) ) to ( Q R, ) then the length of the
line segment ( P S ) is
A ( cdot frac{1}{sqrt{2}} )
B. ( sqrt{2} )
( c cdot 2 )
D. ( 2 sqrt{2} )
12
132Equation of the plane containing the ( operatorname{lines} overline{boldsymbol{r}}=(overline{boldsymbol{i}}-boldsymbol{2} overline{boldsymbol{j}}+overline{boldsymbol{k}})+boldsymbol{t}(overline{boldsymbol{i}}+mathbf{2} overline{boldsymbol{j}}-overline{boldsymbol{k}}) )
( boldsymbol{boldsymbol { r }}=(overline{boldsymbol{i}}+mathbf{2} overline{boldsymbol{j}}-overline{boldsymbol{k}})+boldsymbol{s}(overline{boldsymbol{i}}+overline{boldsymbol{j}}+mathbf{3} overline{boldsymbol{k}}) ) is
A. ( bar{r}(7 bar{i}-4 bar{j}-bar{k})=14 )
В. ( bar{r}(bar{i}+2 bar{j}-bar{k})=10 )
c. ( bar{r}(bar{i}+bar{j}+3 bar{k})=20 )
D. ( bar{r}(bar{i}-2 bar{j}+bar{k})=27 )
12
13318.
A tetrahedron has vertices at O(0, 0, 0), A(1,2,1) B(2,1,3)
and C(-1,1,2). Then the angle between the faces OAB and
ABC will be
[2003]
(a)
90°
(b) cos-
(c) cos-1( 37
(d) 30°
31
12
134Find the length and foot of the perpendicular from the point ( (mathbf{7}, mathbf{1 4}, mathbf{5}) ) to the plane ( 2 x+4 y-z=2 )12
135The circum centre of the triangle formed by the points (2,5,1),(1,4,-3) and (-2,7,-3) is
A. (6,0,1)
()
В. (0,6,-1)
c. (-1,6,2)
D. (6,1,-2)
12
136Graph ( x^{2}+y^{2}=4 ) in ( 3 D ) looks like
A . Circle
B. Cylinder
c. Hemisphere
D. sphere
12
137Write the equations for the ( x ) -and ( y )
axes.
12
13816.
The image of the point (-1,3,4) in the plane x-2y = 0 is
(
17
19
41
(b) (15,11,4)
(2006)
10
U
(d) None of these
12
139Find the image of :
(-2,3,4) in the ( y z ) -plane
12
140The distance of the point (1,-2,3) from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} ) measured
parallel to the line ( frac{x}{2}=frac{y}{3}=frac{z-1}{-6} ) is
A .
B . 2
( c cdot 4 )
D. None of these
12
141ff ( left(x_{1}, y_{1}, z_{1}right) ) and ( Bleft(x_{2}, y_{2}, z_{2}right) ) are
two points such that the direction ( operatorname{cosines} ) of ( A B ) are ( l, m, n ) then
( l=frac{x_{2}-x_{1}}{|A B|}, m=frac{y_{2}-y_{1}}{|A B|}, n=frac{z_{2}-z_{1}}{|A B|} )
12
142Four vertices of a tetrahedron are
(0,0,0),(4,0,0),(0,-8,0) and
( (0,0,12) . ) Its centroid has the
coordinates
A ( cdotleft(frac{4}{3},-frac{8}{3}, 4right) )
В. (2,-4,6)
c. (1,-2,3)
a 5
D. none of these
12
143If ( boldsymbol{A}=(-2,3,4), B=(1,2,3) ) are two
points and ( P ) is the point of intersection
of ( A B ) and ( z x ) -plane, then ( P_{x}+P_{y}+ )
( boldsymbol{P}_{z}= )
( mathbf{A} cdot mathbf{6} )
B. -8
c. 8
D.
12
144If the points ( (1,1, p) ) and (-3,0,1) be equidistant from the plane ( vec{r} .(3 hat{i}+ ) ( 4 hat{j}-12 hat{k})+13=0, ) then find the value
of ( p )
12
145Find the ratio in which YZ-plane divides the line joining ( A(2,4,5) ) and ( B(3,5,-4) . ) Also find the point of
intersection.
12
146( operatorname{Points} boldsymbol{A}(boldsymbol{3}, boldsymbol{2}, boldsymbol{4}), boldsymbol{B}left(frac{boldsymbol{3} boldsymbol{3}}{boldsymbol{5}}, frac{boldsymbol{2} boldsymbol{8}}{boldsymbol{5}}, frac{boldsymbol{3} boldsymbol{8}}{boldsymbol{5}}right), ) and
( C(9,8,10) ) are given. The ratio in which
( B ) divides ( overline{A C} ) is
A . 5: 3
B . 2: 1
c. 1: 3
D. 3: 2
12
147The image of the point (-1,3,4) in the
plane ( boldsymbol{x}-mathbf{2} boldsymbol{y}=mathbf{0} ) is
A ( cdot(15,11,4) )
в. ( left(frac{9}{5},-frac{13}{5}, 4right) )
( c cdot(8,4,4) )
D. None of these
12
14828. P, and P, are planes passing through origin. L, and L, are
two line on P, and P2 respectively such that their
intersection is origin. Show that there exists points A, B, C,
whose permutation A’, B’, C can be chosen such that (i) Ais
on L,, B on P, but not on L, and C not on P, (ii) A’ is on L,,
B’ on P, but not on L and C not on P2
12
14926.
Find the equation of plane passing through (1,1,1) & parallel
to the lines L,, L, having direction ratios (1,0,-1), (1,-1,0)
Find the volume of tetrahedron formed by origin and the
points where these planes intersect the coordinate axes.
(2001.2 Mau
12
1506.
A line makes the same angle , with each of the x and z axis.
If the angle ß, which it makes with y-axis, is such that
sin B = 3 sin?e, then cos2e equals
[2004]
т.
12
151Find the coordinates of the point ( boldsymbol{P} )
which divides the join of ( boldsymbol{A}(-2,5) ) and ( B(3,-5) ) in the ratio 2: 3
12
152If a line makes an angle of ( frac{pi}{4} ) with the positive direction of each of ( x ) -axis and
( boldsymbol{y} ) -axis, then the angle that the line
makes with the positive direction of ( z )
axis is-
A ( cdot frac{pi}{3} )
в.
c. ( frac{pi}{2} )
D.
12
153The eartesian equations of the line are
( mathbf{3} boldsymbol{x}+mathbf{1}=mathbf{6} boldsymbol{y}-mathbf{2}=mathbf{1}-boldsymbol{z} . ) Find its
equation in vector form and find direction ratios of the line.
12
154The projections of a line segment on ( x, y ) and ( z ) axes are respectively ( sqrt{2}, 3,5 ) The length of the line segment is
( mathbf{A} cdot mathbf{6} )
B. 11
c. 8
D. 5
12
155( operatorname{Given} boldsymbol{A}(mathbf{1},-mathbf{1}, mathbf{0}) ; boldsymbol{B}(mathbf{3}, mathbf{1}, mathbf{2}) )
( C(2,-2,4) ) and ( D(-1,1,-1) ) which of
the following points neither lie on ( boldsymbol{A B} )
nor on ( C D ? )
A ( .(2,2,4) )
В. (2,-2,4)
c. (2,0,1)
D. (0,-2,-1)
12
156Directions ratio of two lines are ( 3,-2, k )
and ( -2, k, 4 . ) Find ( k ) if the lines are perpendicular to each other.
12
157Find the distance of the point ( P(3,4,4) ) from the point, where the line joining the point ( A(3,-4,-5) ) and ( B(2,-3,1) ) intersects the plane ( 2 x+y+z=7 )12
158Two vectors ( vec{A} ) and ( vec{B} ) inclined at an
angle ( theta ) have a resultant ( vec{R} ) which makes an angle ( alpha ) with ( vec{A} ) and angle ( beta ) with ( vec{B} ). Let the magnitudes of the vectors ( vec{A}, vec{B} ) and ( vec{R} ) be represented by ( A ) B and R respectively. Which of the following relations is not correct?
A . Asinalpha ( =B sin beta )
B. ( R sin alpha=B sin (alpha+beta) )
c. ( R sin beta=operatorname{Asin}(alpha+beta) )
D. None of these
12
159( A=(2,3,0) ) and ( B=(2,1,2) ) are two
points. If the points ( P, Q ) are on the line
( A B ) such that ( A P=P Q=Q B, ) then
( boldsymbol{P Q}= )
( A cdot 2 sqrt{2} )
B. ( 6 sqrt{2} )
( c cdot sqrt{frac{8}{9}} )
D. ( sqrt{2} )
12
160Point, Plane: ( (0,0,0), 3 x-4 y+12 z= )
3
12
161The angle between two diagonals of a cube is.
A ( .30^{circ} )
B . ( 45^{circ} )
c. ( cos ^{-1}left(frac{1}{3}right) )
D. ( cos ^{-1}left(frac{1}{sqrt{3}}right) )
12
162If direction numbers of two lines are
( a, b, c ) and ( b-c, c-a, a-b ) prove that
they are perpendicular to each other.
12
163The direction cosines of a line equally
inclined to three mutually perpendicular lines having D.C.’s as
( ell_{1} m_{1} n_{1}: ell_{2} m_{2} n_{2}: ell_{3} m_{3} n_{3} ) are
A. ( l_{1}+l_{2}+l_{3}, m_{1}+m_{2}+m_{3}, n_{1}+n_{2}+n_{3} )
B ( cdotleft(pm frac{1}{sqrt{3}}, pm frac{1}{sqrt{3}}, pm frac{1}{sqrt{3}}right) )
( ^{mathbf{c}} cdotleft(pm frac{1}{sqrt{2}}, pm frac{1}{sqrt{3}}, pm frac{1}{sqrt{4}}right) )
D. none of these
12
164The direction cosines of the line which
is perpendicular to the lines with direction cosines proportional to
(1,-2,-2)( &(0,2,1) ) are
( ^{mathrm{A}} cdotleft(frac{2}{3},-frac{1}{3}, frac{2}{3}right) )
в. ( left(frac{2}{3}, frac{1}{3}, frac{2}{3}right) )
c. ( left(frac{2}{3}, frac{1}{3}, frac{-2}{3}right) )
D. ( left(frac{-2}{3}, frac{1}{3}, frac{2}{3}right) )
12
165A plane ( pi ) makes intercept 3 and 4
respectively on z-axis and x-axis. If ( pi ) is
parallel to y-axis, then its equation is
A. ( 3 x+4 z=12 )
B. ( 3 z+4 x=12 )
c. ( 3 y+4 z=12 )
D. ( 3 z+4 y=12 )
12
166Distance between ( boldsymbol{A}(mathbf{4}, mathbf{5}, mathbf{6}) ) from origin
( boldsymbol{O} ) is
A ( cdot 25 sqrt{3} )
B. ( sqrt{77} )
c. ( 3 sqrt{5} )
D. Data Insufficient
12
167If ( vec{A}=-4 hat{i}+3 hat{j} ) and ( vec{B}=2 hat{i}+5 hat{j} ) and
( vec{C}=vec{A} times vec{B} ) then ( vec{C} ) makes an angle of :
A ( cdot 45^{0} ) with ( mathrm{x} ) -axis
B. ( 180^{circ} ) with ( Y ) -axis
( c cdot 0^{0} ) with ( mathrm{z} ) -axis
D. ( 180^{circ} ) with ( z ) -axis
12
168Show that ( A(3,-2) ) is a point trisection of the line segment joining the points
(2,1) and (5,-8)
Also find the co-ordinates of the other
points of trisections.
12
169A point on the line ( frac{boldsymbol{x}-mathbf{1}}{mathbf{1}}=frac{boldsymbol{y}-mathbf{2}}{mathbf{2}}= )
( frac{z+1}{3} ) at a distance ( sqrt{6} ) from the origin
is
This question has multiple correct options
( mathbf{A} cdotleft(frac{-5}{7}, frac{-10}{7}, frac{13}{7}right) )
в. (1,2,-1)
( ^{mathbf{C}} cdotleft(frac{5}{7}, frac{10}{7}, frac{-13}{7}right) )
D. (-1,-2,1)
12
170Let the co – ordinates of the point where the line joining the points
(2,-3,1),(3,-4,-5) cuts the plane
( mathbf{2} boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=mathbf{7} ) be ( (boldsymbol{x}, boldsymbol{y}, boldsymbol{z}) . ) Find ( boldsymbol{x}+ )
( boldsymbol{y}+boldsymbol{z} ? )
12
171The plane ( X O Z ) divides the join of (1,-1,5) and (2,3,4) in the ratio ( lambda: 1 )
then ( lambda ) is
A . -3
B. ( -1 / 3 )
( c cdot 3 )
D. ( 1 / 3 )
12
172The distance of the point ( ,(-1,-5,-10) ) from the point intersection of the line,
( frac{x-2}{3}=frac{y+1}{4}=frac{z-2}{12} ) and the plane ( x- )
( boldsymbol{y}+boldsymbol{z}=mathbf{5}, ) is
( A cdot 13 )
B. 1
c. 12
D. none of these
12
173If ( overrightarrow{P O}+overrightarrow{O Q}=overrightarrow{Q O}+overrightarrow{O R}, ) prove that
the points ( P, Q, R ) are collinear.
12
17417. The radius of the circle in which the sphere
x2 + y2 + 2? + 2x – 2y – 42 -19 = 0 is cut by the plane
x +2y + 2z+ 7 = 0 is
[2003]
(a) 4 (6) 1 (c) 2 (d) 3
12
175Let the vector ( vec{a}, vec{b}, vec{c} ) and ( vec{d} ) be such ( operatorname{that}(overrightarrow{boldsymbol{a}} times overrightarrow{boldsymbol{b}}) times(overrightarrow{boldsymbol{c}} times overrightarrow{boldsymbol{d}})=mathbf{0} cdot operatorname{Let} boldsymbol{P}_{1} )
and ( P_{2} ) be planes determined by the pairs of vectors ( vec{a}, vec{b} ) and ( vec{c}, vec{d} )
respectively then the angle between ( P_{1} )
and ( P_{2} ) is
A. 0
в. ( frac{pi}{4} )
c. ( frac{pi}{3} )
D. ( frac{pi}{2} )
12
176Find the shortest distance between the
skew lines ( r=(6 i+2 j+2 k)+t(i- )
( 2 j+2 k) ) and ( F=(-4 i-k)+s(3 i- )
( 2 j-2 k) ) where s,t are scalars.
12
177Find the direction cosines ( l, m, n ) of a line which are connected by the relation ( l+m-n=0 ) and ( 2 m l-2 m n+n l= )
0
This question has multiple correct options
A ( cdot frac{-2}{sqrt{6}}, frac{1}{sqrt{6}}, frac{-1}{sqrt{6}} )
в. ( frac{2}{sqrt{6}}, frac{-1}{sqrt{6}}, frac{1}{sqrt{6}} )
c. ( frac{-2}{sqrt{6}}, frac{-1}{sqrt{6}}, frac{-1}{sqrt{6}} )
D. ( frac{2}{sqrt{6}}, frac{1}{sqrt{6}}, frac{1}{sqrt{6}} )
12
178Verify the following
(i) (0,7,-10),(1,6,-6) and (4,9,-6) are the vertices of an isosceles triangle
(ii) (0,7,10),(-1,6,6) and (-4,9,6) are the vertices of a right angled triangle
(iii) (-1,2,1),(1,-2,5),(4,-7,8) and (2,-3,4) are the vertices of a parallelogram
12
179Show that the points ( boldsymbol{A}(mathbf{0}, mathbf{1}, mathbf{2}), boldsymbol{B}(mathbf{2},-mathbf{1}, mathbf{3}) ) and ( boldsymbol{C}(mathbf{1},-mathbf{3}, mathbf{1}) )
are vertices of an isosceles right-angled triangle.
12
18047. The equation of the plane passing through the point (1,1,1)
and perpendicular to the planes 2x+y-2z=5 and 3x – 6y-
2z=7, is
(a) 14x +2y-15z=1 (b) 14x – 2y + 15z=27
(c) 14x +2y+ 15z=31 (d) -14x + 2y + 15z=3
12
181The distance of point ( A(-2,3,1) ) from
the PQ through ( P(-3,5,2), ) which makes equal angles with the axes is-
A ( cdot frac{2}{sqrt{3}} )
в. ( sqrt{frac{14}{3}} )
c. ( frac{16}{sqrt{3}} )
D. ( frac{5}{sqrt{3}} )
12
182The cartesian from of equation a line passing through the point position vector ( 2 hat{i}-hat{j}+2 hat{k} ) and is in the direction of ( -2 hat{i}+hat{j}+hat{k}, ) is
A ( frac{x-2}{-2}=frac{y+1}{1}=frac{z-2}{1} )
B. ( frac{x+4}{-2}=frac{y-1}{1}=frac{z+2}{1} )
c. ( frac{x+2}{4}=frac{y-1}{-1}=frac{z-1}{2} )
D. None of these
12
183Find the vector equation of the plane whose cartesian form of equation is ( 3 x- ) ( 4 y+2 z=5 )12
184Column I shows some vector equations. Match Column I with the value of angle between ( vec{A} ) and ( vec{B} ) given in Column II
Column I Column I
12
185The point ( P(x, y, z) ) lies in the first octant and its distance from the origin
is 12 units. If the position vector of ( P )
make ( 45^{circ} ) and ( 60^{circ} ) with the ( x ) -axis and ( y )
axis respectively, then the coordinates of ( boldsymbol{P} ) are
A ( cdot(3 sqrt{3}, 6,3 sqrt{2}) )
B. ( (4 sqrt{3}, 8,4 sqrt{2}) )
c. ( (6 sqrt{2}, 6,6,) )
D. ( (6,6,6 sqrt{2}) )
E ( .(4 sqrt{2}, 8,4 sqrt{3}) )
12
186The projection of a line segment joining
the points ( boldsymbol{P}left(boldsymbol{x}_{1}, boldsymbol{y}_{1}, boldsymbol{z}_{1},right) ) and
( Qleft(x_{1}, y_{1}, z_{1},right) ) on another line whose
DC’s are ( l, m, n ) is given by This question has multiple correct options
A ( cdot lleft(x_{1}+x_{2}right)+mleft(y_{2}+y_{2}right)+nleft(z_{1}+z_{2}right) )
B. ( 2left[frac{left(l x_{2}+m y_{2}+n z_{2}right)}{2}-frac{left(l x_{1}+m y_{1}+n z_{1}right)}{2}right. )
c. ( lleft(x_{2}-x_{1}right)+mleft(y_{2}-y_{1}right)+nleft(z_{2}-z_{1}right) )
D. ( frac{x_{2}-x_{1}}{l}+frac{y_{2}-y_{1}}{m}+frac{z_{2}-z_{1}}{n} )
12
187The point ( P ) is on the ( y ) -axis. If ( P ) is equidistant from (1,2,3) and (2,3,4)
then ( boldsymbol{P}_{boldsymbol{y}}= )
A ( cdot frac{15}{2} )
B. 15
c. 30
D. ( frac{3}{2} )
12
188If the planes ( boldsymbol{x}-boldsymbol{b} boldsymbol{z}=mathbf{0}, boldsymbol{c} boldsymbol{x}-boldsymbol{y}+=mathbf{0} )
and ( b x+a y-z=0, ) pass through a
line, then find the value of ( a^{2}+b^{2}+ )
( c^{2}+2 a b c )
A.
B.
c. -1
D. ( frac{1}{2} )
12
189The vector ( vec{P} ) makes ( 120^{circ} ) with the ( x- )
axis and the vector ( vec{Q} ) makes ( 30^{circ} ) with ( boldsymbol{y}- ) axis. What is the resultant vector?
A. ( P+Q )
в. ( P-Q )
c. ( sqrt{P^{2}+Q^{2}} )
D. ( sqrt{P^{2}-Q^{2}} )
12
19090.
The equation of the line passing through (-4, 2,
to the plane x + 2y – Z-5 = 0 and intersecting the
passing through (-4, 3, 1), parallel
-2.
x+l
-3 1 |
y-3
2..
(JEEM 2019-9 Jan (M)
-1
)
12
1
4
() 4 +3 2+1 (Jet 2
() | 13 1
12
191The variable plane ( (2 lambda+1) x+ )
( (3-lambda) y+z=4 ) always passes
through the line
A ( cdot frac{x}{0}=frac{y}{0}=frac{x+4}{1} )
в. ( frac{x}{1}=frac{y}{2}=frac{z}{-3} )
c. ( frac{x}{1}=frac{y}{2}=frac{z-4}{-7} )
D. none of these
12
192Find the point where the line of
intersection of the planes ( x-2 y+z= )
1 and ( x+2 y-2 z=5 ) intersects the
plane ( 3 x+2 y+z+6=0 )
A ( cdot P(1,-2,-4) )
B . ( P(1,2,-4) )
c. ( P(1,-2,4) )
D. None of these
12
19326. The intersection of the spheres
x2 + y2 + z2 + 7x-2y-z = 13 and
x2 + y2 + 22 – 3x + 3y + 4z = 8
is the same as the intersection of one of the sphere and the
plane
[2004]
(a) 2x – y – z=1 (b) x-2y-z=1
(©) x-y–2z=1 (d) x-y-z = 1
12
194The d.rs of the lines ( boldsymbol{x}=boldsymbol{a} boldsymbol{y}+boldsymbol{b}, boldsymbol{z}= )
( boldsymbol{c} boldsymbol{y}+boldsymbol{d} ) are:
A. ( 1, a, c )
B. ( a, 1, c )
c. ( b, 1, c )
D. ( c, a, 1 )
12
195The circum radius of the triangle
formed by the points (1,2,-3),(2,-3,1) and (-3,1,2) is:
A ( cdot sqrt{14} )
B. 14
c. ( sqrt{13} )
D.
12
196If ( hat{i}, hat{j} ) and ( hat{k} ) represents unit vectors
along the ( x, y ) and ( z- ) axes respectively
then find the value of angle ( theta ) between the vectors ( hat{i}+hat{j}+hat{k} ) and ( hat{i}+hat{j} )
12
197Find the shortest distance between the
line ( boldsymbol{x}=mathbf{1}+boldsymbol{t}, boldsymbol{y}=mathbf{1}+boldsymbol{6} boldsymbol{t}, boldsymbol{z}=boldsymbol{2} boldsymbol{t}, boldsymbol{t} in boldsymbol{R} )
and ( boldsymbol{x}=mathbf{1}+mathbf{2 k}, boldsymbol{y}=mathbf{5}+mathbf{1 5 k}, boldsymbol{z}= )
( -2+6 k, k in R )
12
19829
(0) Tu, ,
A plane which is perpendicular to two planes 2x – 2y+z=0
and x-y + 2z= 4, passes through (1, -2, 1). The distance of
the plane from the point (1,2,2) is (2006 – 3M, -1)
(a) o (b) 1 (c) √ (d) 252
12
199A line makes an angle ( theta ) with each of the
( x- ) and ( z^{-} ) axes. If the angle ( beta, ) which it makes with the ( y ) -axis, is such that
( sin ^{2} beta=3 sin ^{2} theta, ) then ( cos ^{2} theta ) equals-
A ( cdot frac{2}{3} )
B. ( frac{1}{5} )
( c cdot frac{3}{5} )
D. ( frac{2}{5} )
12
200Find the vector equation of line joining the points (2,1,3) and (-4,3,-1)
( mathbf{A} cdot bar{r}=2(1-3 lambda) bar{i}-(1+2 lambda) bar{j}-(3-4 lambda) bar{k} )
B . ( bar{r}=2(1-3 lambda) bar{i}-(1+2 lambda) bar{j}+(3-4 lambda) bar{k} )
C ( . bar{r}=2(1-3 lambda) bar{i}+(1+2 lambda) bar{j}+(3-4 lambda) bar{k} )
D. ( bar{r}=2(1+3 lambda) bar{i}+(1+2 lambda) bar{j}+(3+4 lambda) bar{k} )
12
201Show that LHS=RHS i.e. ( frac{2}{9} times 3=frac{2}{9} times ) ( frac{3}{1}=frac{2 times 3}{9 times 1}=frac{6}{9}=frac{2}{3} )12
202A plane meets the axes in ( A, B ) and ( C )
such that centroid of the ( triangle A B C ) is
( (1,2,3) . ) The equation of the plane is
A ( cdot x+frac{y}{2}+frac{z}{3}=1 )
В ( cdot frac{x}{3}+frac{y}{6}+frac{z}{9}=1 )
c. ( x+2 y+3 z=1 )
D. None of these
12
203If the projections of the line segment ( A B ) on the coordinate axes are 2,3,6
then the square of the sine of the angle made by ( boldsymbol{A B} ) with ( boldsymbol{x}=mathbf{0}, ) is
A ( cdot frac{3}{7} )
B. ( frac{3}{49} )
( c cdot frac{4}{7} )
D. ( frac{40}{49} )
12
204Find the equation of the plane in scalar product form ( hat{r}=(2 hat{i}+hat{k})+lambda hat{i}+ )
( mu(hat{i}+2 hat{j}-3 hat{k}) )
12
205If ( boldsymbol{A}=(mathbf{1}, mathbf{2}, mathbf{3}), boldsymbol{B}=(mathbf{2}, mathbf{1 0}, mathbf{1}), boldsymbol{Q} ) are
collinear points and ( Q_{x}=-1, ) then
( boldsymbol{Q}_{z}= )
A . -3
B. 7
( c cdot-14 )
D. –
12
206The extremities of a diagonal of a rectangular parallelopiped whose faces are parallel to the reference planes are (-2,4,6) and ( (3,16,6) . ) The length of the base diagonal is
A . 13
B. ( sqrt{13} )
c. ( 2 sqrt{13} )
D. 169
12
207The coordinates of the points in which the line joining the points (2,5,-7) and (-3,-1,8) are intersected by the ( y-z )
plane are
A ( cdotleft(0, frac{13}{5},-1right) )
B ( cdotleft(0, frac{-13}{5},-2right) )
D. ( left(0, frac{13}{5}, frac{2}{5}right) )
12
208Find the vector and cartesian equations of the plane passing through the points ( A(1,1,-2), B(1,2,1) ) and ( C(2,-1,1) )12
209The projections of a directed line segment on the coordinate axes 12,4,3 The direction cosines of the line are
A ( cdot frac{12}{13},-frac{4}{13}, frac{3}{13} )
B. ( -frac{12}{13},-frac{4}{13}, frac{3}{13} )
c. ( frac{12}{13}, frac{4}{13}, frac{3}{13} )
D. none of these
12
210The direction cosines to two lines at
right angles are (1,2,3) and ( left(-2, frac{1}{2}, frac{1}{3}right), ) then
the direction cosine perpendicular to both given lines are:
A ( cdot sqrt{frac{25}{219}} cdot sqrt{frac{19}{2198}} cdot sqrt{frac{729}{21988}} )
B. ( sqrt{frac{24}{2198} cdot sqrt{frac{38}{2198}}} sqrt{frac{730}{21988}} )
c. ( frac{1}{3},-2, frac{-7}{2} )
D. None of the above
12
211If the points whose position vectors are ( mathbf{2} overline{mathbf{i}}+overline{boldsymbol{j}}+overline{boldsymbol{k}}, mathbf{6} overline{mathbf{i}}-overline{boldsymbol{j}}+mathbf{2} overline{mathbf{k}} ) and ( mathbf{1 4} overline{mathbf{i}}-mathbf{5} overline{mathbf{j}}+ )
( p bar{k} ) are collinear then the value of ( mathbf{p} ) is
( A cdot 2 )
B. 4
( c cdot 6 )
D. 8
12
212A point ( P ) lies on the line whose end
points are ( boldsymbol{A}(mathbf{1}, mathbf{2}, mathbf{3}) ) and ( boldsymbol{B}(mathbf{2}, mathbf{1 0}, mathbf{1}) )
If ( z ) -co-ordinate of ( P ) is ( 7, ) find sum of its
other co – ordinates.
12
213( A=(2,4,5) ) and ( B=(3,5,-4) ) are two
points. If the ( x y ) -plane, ( y z ) -plane divide ( A B ) in the ratios ( a: b, p: q ) respectively then ( frac{a}{b}+frac{p}{q}= )
A. ( frac{7}{15} )
в. ( frac{-7}{12} )
c. ( frac{7}{12} )
D. ( frac{22}{25} )
12
214The ratio in which the line joining points (2,4,5) and (3,5,-4) divide YZ -plane is
( mathbf{A} cdot-2: 3 )
B. 2: 3
c. -3: 2
D. 3: 2
12
215On a plane are two points ( A ) and ( B ) at a
distance of 5 units apart. The number of straight lines in this plane which are at distance of 2 units from ( A ) and 3 units
from ( mathrm{B} ) are:
A . 1
B. 2
( c .3 )
D. 4
12
216Find the equation of the plane passing through the point (1,-2,1) and perpendicular to the line joining the points ( boldsymbol{A}(mathbf{3}, mathbf{2}, mathbf{1}) ) and ( boldsymbol{B}(mathbf{1}, mathbf{4}, mathbf{2}) )12
217Find the shortest distance between the line ( frac{x-3}{3}=frac{y-8}{-1}=frac{z-3}{1} ) and the
line of intersection of the planes ( 2 x+ )
( 5 y-z+47=0 ) and ( 2 x+y+z+7= )
( mathbf{0} )
12
218The projection of a directed line segment on the co-ordinate axes are
( 12,4,3, ) the DC’s of the line are
A ( cdot frac{-12}{13}, frac{-4}{13}, frac{-3}{13} )
B. ( frac{12}{13}, frac{4}{13}, frac{3}{13} )
c. ( frac{12}{13}, frac{-4}{13}, frac{3}{13} )
D. ( frac{12}{13}, frac{4}{13}, frac{-3}{13} )
12
219The plane ( 2 x-(1+lambda) y+3 z=0 )
passes through the intersection of the
planes
A. ( 2 x-y=0 ) and ( y+3 z=0 )
в. ( 2 x-y=0 ) and ( y-3 z=0 )
c. ( 2 x+3 z=0 ) and ( y=0 )
D. None of the above
12
220If ( x y ) -plane and ( y z ) -plane divides the line segment joining ( A(2,4,5) ) and ( B(3,5,-4) ) in the ratio a:b and p:q respectively then value of ( left(frac{a}{b}, frac{p}{q}right) ) may be
A ( cdot frac{23}{12} )
B. ( frac{7}{5} )
( c cdot frac{7}{12} )
D. ( frac{21}{10} )
12
221The vector equation of the plane through the point (1,-2,-3) and
parallel to the vectors (2,-1,3) and
(2,3,-6) is ( bar{r}= )
A ( cdot(1+2 t+2 s) bar{i}-(2+t-3 s) bar{j}-(3-3 t+6 s) bar{k} )
B. ( (1+2 t+2 s) bar{i}+(2+t+3 s) bar{j}-(3+3 t+6 s) bar{k} )
c. ( (1+2 t+2 s) bar{i}+(2+t+3 s) bar{j}+(3+3 t+6 s) bar{k} )
D. ( (1+2 t+2 s) bar{i}+(2+t-3 s) bar{j}+(3+3 t+6 s) bar{k} )
12
222If ( O ) is origin ( O P=3 ) with direction
ratios proportional to -1,2,-2 then what are the coordinates of ( P ? )
12
223Direction ratios of the line which is perpendicular to the lines with direction ratios -1,2,2 and 0,2,1 are
( mathbf{A} cdot 1,1,2 )
B. 2,-1,2
c. -2,1,2
D. 2,1,-2
12
224( boldsymbol{A}=(mathbf{1},-mathbf{2}, mathbf{3}), boldsymbol{B}=(2,1,3), boldsymbol{C}=(4,2, )
1) and ( G=(-1,3,5) ) is the centroid of
the tetrahedron ( A B C D . ) Then the fourth
coordinate is
A. (11,11,13)
В. (-11,11,45)
c. (-11,11,13)
D. (11,13,11)
12
225A line d.c’s proportional to (2,1,2)
meets each of the lines ( boldsymbol{x}=boldsymbol{y}+boldsymbol{a}=boldsymbol{z} )
and ( x+a=2 y=2 z . ) Then the
coordinates of each of the points of
intersection are given by
A . ( (3 a, 2 a, 3 a) ;(a, a, 2 a) )
в. ( (3 a, 2 a, 3 a) ;(a, a, a) )
C. ( (3 a, 3 a, 3 a) ;(a, a, a) )
D. ( (2 a, 3 a, 3 a) ;(2 a, a, a) )
12
226If the plane a ( 2 x-3 y+5 z-2=0 )
divides the line segment joining (1,2,3) and ( (2,1, k) ) in the ratio 9: 11
then ( k ) is
A .
в. -2
c. -10
D. ( -frac{1}{2} )
12
22737. A line with positive direction cosines passes through
point P(2,-1,2) and makes equal angles with the coordinata
axes. The line meets the plane
2x+y+z=9
at point Q. The length of the line segment PQ equals
(2009)
(a) 1 (b) √ (c) √3 (d) 2
12
228The distance of origin from the image of (1,2,3) in plane ( x-y+z=5 ) is
A ( cdot sqrt{17} )
B. ( sqrt{29} )
c. ( sqrt{34} )
D. ( sqrt{41} )
12
229If ( |vec{A} times vec{B}|=sqrt{3} vec{A} cdot vec{B} ) then the value of
( |vec{A}+vec{B}| ) is:
( ^{A} cdotleft(A^{2}+B^{2}+frac{A B}{sqrt{3}}right)^{1 / 2} )
в. ( A+B )
c. ( left(A^{2}+B^{2}+sqrt{3} A Bright)^{1 / 2} )
D. ( left(A^{2}+B^{2}+A Bright)^{1 / 2} )
12
230Find the vector equation of the line through ( A(3,4,-7) ) and ( B(6,-1,1) )12
231If the orthocentre, circumcentre of a
triangle are (-3,5,2),(6,2,5)
respectively then the centroid of the triangle is
( mathbf{A} cdot(3,3,4) )
В. ( left(frac{3}{2}, frac{7}{2}, frac{9}{2}right) )
c. (9,9,12)
D. ( left(frac{9}{2} frac{-3}{2}, frac{3}{2}right) )
12
232The Cartesian equation of line ( 6 x- ) ( mathbf{2}=mathbf{3} boldsymbol{y}+mathbf{1}=mathbf{2} z-mathbf{2} ) is given by
( ^{text {A } cdot frac{3 x-1}{3}}=frac{3 y+1}{6}=frac{z-1}{3} )
B. ( frac{3 x+1}{3}=frac{3 y-1}{6}=frac{z-1}{3} )
c. ( frac{3 x-1}{3}=frac{3 y-1}{6}=frac{z-1}{3} )
D. ( frac{3 x-1}{6}=frac{3 y-1}{3}=frac{z-1}{3} )
12
233The direction ratios of the line ( boldsymbol{x}-boldsymbol{y}+ )
( z-5=0=x-3 y-6 ) are
A. 3,1,-2
в. 2,-4,1
c. ( frac{3}{sqrt{14}}, frac{1}{sqrt{14}}, frac{-2}{sqrt{14}} )
D. ( frac{2}{sqrt{14}}, frac{-4}{sqrt{14}}, frac{1}{sqrt{14}} )
12
234A plane is at a distance of 5 units from the origin and perpendicular to the vector ( 2 hat{i}+hat{j}+2 hat{k} . ) The equation of the
plane is
A ( . vec{r} .(2 hat{imath}+hat{j}-2 hat{k})=15 )
в. ( vec{r} .(2 hat{i}+hat{j}-hat{k})=15 )
c. ( vec{r} .(2 hat{i}+hat{j}+2 hat{k})=15 )
D. ( vec{r} .(hat{i}+hat{j}+2 hat{k})=15 )
E ( . vec{r} .(2 hat{i}-hat{j}+2 hat{k})=15 )
12
235Find vector equation for the line passing through the points ( 3 bar{i}+4 bar{j}- ) ( mathbf{7} bar{k}, overline{boldsymbol{i}}-overline{boldsymbol{j}}+mathbf{6} overline{boldsymbol{k}} )
A ( . bar{r}=(3-2 lambda) bar{i}+(4-5 lambda) bar{j}+(-7+13 lambda) bar{k} )
B. ( bar{r}=(2 lambda) bar{i}+(4+5 lambda) bar{j}+(-7-13 lambda) bar{k} )
c. ( bar{r}=(3-2 lambda) bar{i}-(4-5 lambda) bar{j}+(-7+13 lambda) bar{k} )
D. ( bar{r}=(3-2 lambda) bar{i}+(4-5 lambda) bar{j}-(-7+13 lambda) bar{k} )
12
236(-1,-5,-7) lies in Octant
( A )
B. VII
( c cdot v )
D. II
12
237If ( boldsymbol{A}(cos boldsymbol{alpha}, sin boldsymbol{alpha}, boldsymbol{0}), boldsymbol{B}(cos boldsymbol{beta}, sin beta, boldsymbol{0}) )
( C(cos gamma, sin gamma, 0) ) are vertices of ( Delta A B C )
and let
[
begin{array}{l}
cos alpha+cos beta+cos gamma=3 a, sin alpha+ \
sin beta+sin gamma=3 b, text { then correct }
end{array}
]
matching of the following is:
List:1
begin{tabular}{ll}
A. Circumcentre & ( 1 .(3 a, 3 b, 0) ) \
cline { 0 }
end{tabular}
B. Centroid
[
2 .(0,0,0)
]
c. Ortho centre
[
3 .(a, b, 0)
]
A . 432
в. 231
c. 123
D. 234
12
238If ( bar{a}, bar{b} ) are the position vectors of ( A ) and
( B ) then one of the following points lie on
( A B )
A ( cdot frac{2(bar{a}+bar{b})}{3} )
в. ( frac{(bar{a}-bar{b})}{3} )
c. ( frac{(bar{a}+bar{b})}{3} )
( ^{mathrm{D}} cdot frac{2 bar{a}+2 bar{b}}{3} )
E. None of these
12
239The perimeter of triangle with vertices at (1,0,0),(0,1,0) and (0,0,1) is :
( mathbf{A} cdot mathbf{3} )
B . 2
( c cdot 2 sqrt{2} )
D. ( 3 sqrt{2} )
12
240Show that the points are collinear
(1)
( boldsymbol{A}(mathbf{3}, mathbf{2},-mathbf{4}), boldsymbol{B}(mathbf{9}, mathbf{8},-mathbf{1 0}), boldsymbol{C}(-mathbf{2},-mathbf{3}, mathbf{1}) )
12
241The points ( A ) and ( B ) have co-ordinate (1,2,4) and (-1,3,5) respectively. Find ( A B ) and its magnitude.12
242Prove the ( boldsymbol{A}(-mathbf{5}, mathbf{4}), boldsymbol{B}(-mathbf{1},-mathbf{2}), boldsymbol{C}(mathbf{5}, mathbf{2}) )
are the vertices of ( n ) isosceles right
angled triangle?
12
243If ( P(x, y, z) ) is a point on the line segment joining ( A(2,2,4) ) and ( B(3,5,6) ) such that projection of ( overline{O P} ) on axes are ( frac{13}{5}, frac{19}{5}, frac{26}{5} ) respectively, then ( P ) divide ( A B ) in the ratio
A .3: 2
B. 2: 3
c. 1: 2
D. 1: 3
12
244The direction cosine of a line equally inclined to the axes are
A ( cdot frac{1}{3}, frac{1}{3}, frac{1}{3} )
B. ( -frac{1}{3},-frac{1}{3},-frac{1}{3} )
c. ( frac{1}{sqrt{3}}, frac{1}{sqrt{3}}, frac{1}{sqrt{3}} )
D. none of these
12
245If ( l, m, n ) are d.c’s of vector ( overline{O P} ) then
maximum value of ( l ) mn is
A ( cdot frac{1}{sqrt{3}} )
B. ( frac{1}{2 sqrt{3}} )
c. ( frac{1}{3 sqrt{3}} )
D. ( frac{2}{sqrt{3}} )
12
246The lines ( frac{x-2}{1}=frac{y-3}{1}=frac{z-4}{-k} & )
( frac{boldsymbol{x}-mathbf{1}}{boldsymbol{k}}=frac{boldsymbol{y}-boldsymbol{4}}{boldsymbol{2}}=frac{boldsymbol{z}-mathbf{5}}{mathbf{1}} ) are coplanar if
A ( . k=0 ) or -1
B. ( k=1 ) or -1
c. ( k=0 ) or -3
D. ( k=3 ) or -3
12
247If ( A B perp B C, ) then the value of ( lambda ) equal
where ( boldsymbol{A}(mathbf{2 k}, mathbf{2}, mathbf{3}), boldsymbol{B}(boldsymbol{k}, mathbf{1}, mathbf{5}), boldsymbol{C}(boldsymbol{3}+ )
( k, 2,1) )
( mathbf{A} cdot mathbf{3} )
B.
( c .-3 )
D. ( -frac{1}{3} )
12
248The equation of altitude through ( B ) to
side ( A C ) is
A. ( r=k+t(7 i-10+2 k) )
в. ( r=k+t(-9 i+6 j-2 k) )
c. ( r=k+t(7 i-10 j-2 k) )
D. ( r=k+t(7 i+10 j+2 k) )
12
249Equation of the plane passing through the point of intersection of ( x+2 y+=0 ) and ( 2 x+y=0 ) and which is perpendicular to
( 5 x+6 z=0 )
This question has multiple correct options
A. ( x=0 )
B. ( y=0 )
( c cdot z=0 )
D. x+y=0
E. ( x+z=0 )
12
250Solve: ( sqrt{mathbf{2}} boldsymbol{x}+sqrt{mathbf{3}} boldsymbol{y}=mathbf{0} )
( sqrt{mathbf{3}} boldsymbol{x}-sqrt{mathbf{8}} boldsymbol{y}=mathbf{0} )
12
251Find the vector equation of the plane through the points (2,1,-1) and (-1,3,4) and perpendicular to the
plane ( boldsymbol{x}-mathbf{2} boldsymbol{y}+mathbf{4} boldsymbol{z}=mathbf{1 0} )
12
252Write the direction cosines of the line
whose cartesian equations are ( 2 x= )
( mathbf{3} boldsymbol{y}=-boldsymbol{z} )
12
253If the angles made by a straight line with the coordinate axes are ( alpha, frac{pi}{2}- ) ( boldsymbol{alpha}, boldsymbol{beta} ) then ( boldsymbol{beta}= )
A . 0
в. ( frac{pi}{6} )
c. ( frac{pi}{2} )
( D )
12
254Find the coordinates of the point which
divides the line segment joining the points (-2,3,5) and (1,-4,6) in the ratio
(i) 2: 3 internally
(ii) 2: 3 externally
12
255If the distance between a point ( boldsymbol{P} ) and the point (1,1,1) on the line ( frac{x-1}{3}= ) ( frac{boldsymbol{y}-mathbf{1}}{mathbf{4}}=frac{boldsymbol{z}-mathbf{1}}{mathbf{1 2}} ) is ( mathbf{1 3}, ) then the
coordinates of ( boldsymbol{P} ) are
A. (3,4,12)
в. ( left(frac{3}{13}, frac{4}{13}, frac{12}{13}right) )
c. (4,5,13)
D. (40,53,157)
12
256If a line makes angles ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} ) with
positive directions of ( mathrm{X}, mathrm{Y}, mathrm{Z} ) -axes, what is the value of ( sin ^{2} alpha+sin ^{2} beta+sin ^{2} gamma )
( ? )
12
257( l=m=n=1 ) represents the direction
cosines of
A. ( x ) -axis
B. ( y ) -axis
c. ( z ) -axis
D. none of these
12
258The coordinates of the foot of the
perpendicular drawn from of the origin to a plane are ( (12,-4,3) . ) Find the equation of the plane.
12
259What is the angle between ( vec{P} times vec{Q} ) and ( vec{P}+vec{Q} ? )
( mathbf{A} cdot mathbf{0} )
в. ( frac{pi}{2} )
( c . pi )
D. ( frac{3 pi}{2} )
12
260If 0 is the origin and the coordinates of ( P ) is ( (1,2,-3), ) then find the equation of the plane passing through P and perpendicular to OP.
A. ( x-2 y-3 z=-15 )
в. ( x+2 y-3 z=14 )
c. ( x-2 y+3 z=15 )
D. ( x-2 y-3 z=15 )
12
261If ( 4 x+4 y-k z=0 ) is the equation of
the plane through the origin that contains the line ( frac{boldsymbol{x}-mathbf{1}}{mathbf{2}}=frac{boldsymbol{y}+mathbf{1}}{mathbf{3}}=frac{boldsymbol{z}}{mathbf{4}} )
then ( boldsymbol{k}= )
A . 1
B. 3
( c .5 )
D.
12
262The sum of the intercepts on the coordinate axes of the plane passing through the point (-2,-2,2) and containing the line joining the points (1,-1,2) and (1,1,1) is?
A . 12
B. -8
( c .-4 )
D. 4
12
263If a line makes angles ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} ) with
positive axes, then the range of ( sin alpha sin beta+sin beta sin gamma+sin gamma sin alpha ) is
( ^{mathrm{A}} cdotleft(frac{-1}{2}, 1right) )
в. ( left(frac{1}{2}, 2right) )
c. (-1,2)
D. (-1,2]
12
264Let the line ( frac{boldsymbol{x}-mathbf{2}}{mathbf{3}}=frac{boldsymbol{y}-mathbf{1}}{-mathbf{5}}=frac{boldsymbol{z}+mathbf{2}}{mathbf{2}} ) lie
in the plane ( x+3 y-alpha z+beta=0 . ) Then
( (alpha, beta) ) equals:
A ( cdot(-6,7) )
B ( cdot(5,-15) )
c. (-5,5)
D. (6,-17)
12
265Find the ratio in which the ( X Y ) – plane
divides ( A B ) if is (1,2,3) and ( B ) is (-3,4,-5)
Also find the positive vector of the point of division.
12
266Find the measure of the angle between two lines if their direction cosines
( ell, boldsymbol{m}, boldsymbol{n} ) satisfy ( ell+boldsymbol{m}-boldsymbol{n}=mathbf{0}, ell^{2}+ )
( boldsymbol{m}^{2}-boldsymbol{n}^{2}=mathbf{0} )
12
267What are the DR’s of vector parallel to
(2,-1,1) and (3,4,-1)( ? )
A ( cdot(1,5,-2) )
B ( cdot(-2,-5,2) )
( mathbf{c} cdot(-1,5,2) )
D ( cdot(-1,-5,-2) )
12
268The equation of motion of a rocket are:
( boldsymbol{x}=mathbf{2} boldsymbol{t}, boldsymbol{y}=-boldsymbol{4} boldsymbol{t}, boldsymbol{z}=boldsymbol{4} boldsymbol{t}, ) where the time
( t ) is given in seconds and the coordinate
of a moving point in kilometers. At what distance will the rocket be from the
starting point ( O(0,0,0) ) in 10 seconds?
( mathbf{A} cdot 60 mathrm{km} )
B. ( 30 mathrm{km} )
c. ( 45 mathrm{km} )
D. None of these
12
269Point ( D ) has coordinates as (3,4,5)
Referring to the given figure, find the
coordinates of point ( boldsymbol{E} )
( mathbf{B} cdot(0,4,5) )
C. (0,5,4)
D. (0,3,4)
12
270The point (0,-2,5) lies on the
( A cdot z ) axis
B. x axis
c. xy plane
D. yz plane
E . xz plane
12
271( left(cos ^{-1} l+cos ^{-1} m+cos ^{-1} nright) ) is equal
to
A . ( 90^{circ} )
B. ( 50^{circ} )
( c cdot 180^{circ} )
D. None of these
12
272Find the distance of the point (2,3,5) from the ( x y- ) plane12
273Find the equation of the plane passing through the point (2,-1,1) and through the line of intersection of the planes ( vec{r} ). ( (2 hat{i}-3 hat{j}+hat{k})=3 ) and ( vec{r} cdot(hat{i}+5 hat{j}- )
( hat{boldsymbol{k}})=mathbf{4 . 4} )
12
274Find the direction cosines of the line:
( frac{x-1}{2}=-y=frac{z+1}{2} )
12
275Find the equation to the plane through the point (-1,3,2) and perpendicular to the planes ( x+2 y+2 z=11 ) and
( 3 x+3 y+2 z=15 )
12
276figure bounded by non coplanar triangular planes. So, a tetrahedron has four non-coplanar points as its vertices. Suppose a tetrehedron has points
A,B,C,D as its vertices which have
coordinates ( left(boldsymbol{x} mathbf{1}, boldsymbol{y} mathbf{1}, boldsymbol{z}_{1}right)left(boldsymbol{x}_{2}, boldsymbol{y}_{2}, boldsymbol{z}_{2}right) )
( left(x_{3}, y_{3}, z_{3}right) ) and ( left(x_{4}, y_{4}, z_{4}right), ) respectively
in a rectangular three dimensional
space. Then, the coordinates of
its centroid are ( left[frac{boldsymbol{x}_{1}+boldsymbol{x}_{2}+boldsymbol{x}_{3}+boldsymbol{x}_{4}}{boldsymbol{4}}, frac{boldsymbol{y}_{1}+boldsymbol{y}_{2}+boldsymbol{y}_{3}+boldsymbol{y}_{4}}{boldsymbol{4}}right. )
Let a tetrahedron have three of its
vertices represented by the points (0,0,0),(6,5,1) and (4,1,3) and
its centroid lies at the point (1,2,5) Now, answer the following question. The coordinate of the fourth vertex of the
tetrahedron is:
A ( cdot(-6,2,16) )
B. (1,-2,13)
c. (-2,4,-2)
D. (1,-1,1)
12
277The distance of the point ( 3 hat{i}+5 hat{k} ) from the line parallel to the vector ( 6 hat{i}+hat{j}- ) ( 2 hat{k} ) and passing through the point ( 8 hat{i}+ ) ( mathbf{3} hat{boldsymbol{j}}+hat{boldsymbol{k}} ) is
A . 1
B . 2
( c .3 )
D.
12
278If ( z=cos frac{pi}{6}+i sin frac{pi}{6}, ) then
A ( cdot|z|=1, arg z=frac{pi}{4} )
в ( cdot|z|=1, arg z=frac{pi}{6} )
с. ( quad|z|=frac{sqrt{3}}{2}, arg z=frac{5 pi}{24} )
D. ( quad|z|=frac{sqrt{3}}{2}, arg z=tan ^{-1} frac{1}{sqrt{2}} )
12
279The vertices of a triangle are ( (2,3,5),(-1,3,2),(3,5,-2), ) then the
angles are
( ^{mathrm{B}} cos ^{-1}left(frac{1}{sqrt{5}}right), 90^{circ}, cos ^{-1}left(frac{sqrt{5}}{sqrt{3}}right) )
c. ( 30^{circ}, 60^{circ}, 90^{circ} )
D ( cdot cos ^{-1}left(frac{1}{sqrt{3}}right), 90^{circ}, cos ^{-1}(sqrt{frac{2}{3}}) )
12
280An equation of a plane containing the
lines ( r=a_{1}+t b_{1} ) and ( r=a_{2}+t b_{2} )
where ( left[boldsymbol{a}_{2}-boldsymbol{a}_{1}, boldsymbol{b}_{1}, boldsymbol{b}_{2}right]=boldsymbol{0} ) is
A ( cdotleft[r-a_{1}, b_{1}, b_{2}right]=0 )
B . ( left[r-a_{2}, b_{1}, b_{2}right]=0 )
c. ( left[r-a_{2}, a_{1}, b_{2}right]=0 )
D . ( left[r-a, a_{2}, b_{2}right]=0 )
12
281f ( A(2,1,3), B(2,-3,4), C(-1,2,7) ) are the
three points ; find a point D such that ( |overrightarrow{A B}|=frac{1}{2}|overrightarrow{C D}| ) and ( A B|| C D )
12
282The distance between the line ( r=2 hat{i}- ) ( 2 hat{j}+3 hat{k}+lambda(hat{i}-hat{j}+4 hat{k}) ) and the plane
( r cdot(hat{i}+5 hat{j}+hat{k})=5 ) is
A ( cdot frac{10}{9} )
в. ( frac{10}{3 sqrt{3}} )
c. ( frac{10}{3} )
D. None of these
12
283The name of the figure formed by the points (-1,-3,4),(5,-1,1),(7,-4,7)
and (1,-6,10) is a
A. square
B. rhombus
c. parallelogram
D. rectangle
12
284The ratio in which the plane ( 2 x+3 y- )
( mathbf{2} z+mathbf{7}=mathbf{0} ) divides the line segment
joining the points (-1,1,3),(2,3,5) is
A .3: 5
B. 7: 5
( mathbf{c} cdot 9: 11 )
D. 1: 5 externally
12
285State the following statement is True or
False
If two distinct lines are intersecting each other in a plane then they cannot
have more than one point in common.
A. True
B. False
12
286The direction cosines of the ray
( boldsymbol{P}(1,-2,4) ) and ( Q(-1,1,-2) ) are
A ( cdot(-2,-3,-6) )
B ( cdot(2,-3,-6) )
( ^{C} cdotleft(frac{2}{7}, frac{3}{7}, frac{6}{7}right) )
D ( cdotleft(-frac{2}{7}, frac{3}{7},-frac{6}{7}right) )
12
287Find the equation of the plane through the line
( boldsymbol{P}=boldsymbol{a} boldsymbol{x}+boldsymbol{b} boldsymbol{y}+boldsymbol{c} boldsymbol{z}+boldsymbol{d}=mathbf{0} )
( boldsymbol{Q}=boldsymbol{a}^{prime} boldsymbol{x}+boldsymbol{b}^{prime} boldsymbol{y}+boldsymbol{c}^{prime} boldsymbol{z}+boldsymbol{d}^{prime}=mathbf{0} )
and parallel to the line ( frac{x}{l}=frac{y}{m}=frac{z}{n} )
A ( cdot Pleft(a^{prime} l+b^{prime} m+c^{prime} nright)+Q(a l+b m+c n)=0 )
B – ( Pleft(a^{prime} l+b^{prime} m+c^{prime} nright)-Q(a l+b m+c n)=0 )
c. ( Qleft(a^{prime} l+b^{prime} m+c^{prime} nright)-P(a l+b m+c n)=0 )
D. ( Qleft(a^{prime} l+b^{prime} m+c^{prime} nright)+P(a l+b m+c n)=0 )
12
28869.
t-3
y
=
and
If the line X-1 y +1 Z-1
1 2 3 4
intersect, then k is equal to:
[2012]
(a)
-1
12
289find the equation of a line passing
through the point (1,2,-4) and
perpendicular to two lines. ( vec{r}= ) ( (8 hat{i}-19 hat{j}+10 hat{k})+ )
( boldsymbol{lambda}(boldsymbol{3} hat{boldsymbol{i}}-boldsymbol{1 6} hat{boldsymbol{j}}+boldsymbol{7} hat{boldsymbol{k}}) ) and ( overrightarrow{boldsymbol{r}}= )
( (15 hat{i}+29 hat{j}+5 hat{k})+ )
( boldsymbol{mu}(mathbf{3} hat{boldsymbol{i}}+quad boldsymbol{8} hat{boldsymbol{j}}-boldsymbol{5} hat{boldsymbol{k}}) )
12
290The line passing through the points ( (5,1, a) ) and ( (3, b, 1) ) crosses the ( y z ) plane at the point ( left(0, frac{17}{2}, frac{-13}{2}right) . ) Then
A ( cdot a=2, b=8 )
В. ( a=4, b=6 )
c. ( a=6, b=4 )
D. ( a=8, b=2 )
2
12
291ff ( y ) varies directly as ( x ) and ( y=12 )
when ( x=4, ) then find the linear
equation.
12
292Cartesian equation of a line is ( frac{x-5}{3}= ) ( frac{boldsymbol{y}+boldsymbol{4}}{mathbf{7}}=frac{boldsymbol{z}-boldsymbol{6}}{boldsymbol{2}} . ) Write it in vector form12
293Name three undefined terms.
A. Point
B. Line
c. Plane
D. All of the above
12
294Find a unit vector normal to the plane
through the points (1,1,1),(-1,2,3) and (2,-1,3)
12
295If the line joining the points (-1,2,3),(2,-1,4) is perpendicular to the line joining the points ( (x,-2,4),(1,2,3) ) then ( x= )
( mathbf{A} cdot mathbf{3} )
B. 10
( c cdot frac{-3}{10} )
D. ( frac{-10}{3} )
12
296The equation of the plane passing through the straight line ( frac{x-1}{2}= ) ( frac{boldsymbol{y}+mathbf{1}}{mathbf{- 1}}=frac{boldsymbol{z}-mathbf{3}}{mathbf{4}} ) and perpendicular to
the plane ( boldsymbol{x}+mathbf{2} boldsymbol{y}+boldsymbol{z}=mathbf{1 2} ) is
A. ( 9 x+2 y-5 z+4=0 )
В. ( 9 x-2 y-5 z+4=0 )
c. ( 9 x+2 y+5 z+4=0 )
D. None of these
12
29792.
x-1 y+1 z-2
If the line,
meets the plane, x+2y+
2 3 4
3z= 15 at a point P, then the distance of P from the origin in
[JEEM 2019-9 April (M)
(a) V5/2
(b) 215
(c) 9/2
(d) 7/2
12
298Let ( boldsymbol{O} ) be the origin and ( boldsymbol{A} ) be the point ( (64,0) . ) If ( P ) and ( Q ) divide ( O A ) in the ratio
( mathbf{1}: mathbf{2}: mathbf{3}, ) then the point ( boldsymbol{P} ) is
( ^{mathbf{A}} cdotleft(frac{32}{3}, 0right) )
в. (32,0)
( ^{mathbf{c}} cdotleft(frac{64}{3}, 0right) )
D. (16,0)
E ( cdotleft(frac{16}{3}, 0right) )
12
299Calculate the distance between the
points (-3,6,7) and (2,-1,4) in ( 3 D )
space.
A . 4.36
в. 5.92
c. 7.91
D. 9.11
E . 22.25
12
300Find the equation of the plane which
passes through the point (3,2,0) and contains the line ( frac{boldsymbol{x}-mathbf{3}}{mathbf{1}}=frac{boldsymbol{y}-mathbf{6}}{mathbf{5}}= )
( frac{z-4}{4} ? )
12
301The direction ratios of the line, given by the planes ( x-y+z-5=0, x-3 y-6=0 )
are
A ( .(3,1,-2) )
в. (2,-4,1)
c. (1,-1,1)
D. (0,2,1)
12
302Using section formula prove that the point ( (0.7 .-7),(1,4,-5) ) and
(-1,10,-9) are collinear.
12
303Find the vector equation of the line joining points ( 2 hat{i}+hat{j}+3 hat{k} ) and ( -4 hat{i}+ ) ( mathbf{3} hat{boldsymbol{j}}-hat{boldsymbol{k}} )12
304Assertion
The point ( boldsymbol{A}(boldsymbol{3}, boldsymbol{1}, boldsymbol{6}) ) is the mirror image
of the point ( B(1,3,4) ) in the plane ( x- )
( boldsymbol{y}+boldsymbol{z}=mathbf{5} )
Reason
The plane ( x-y+z=5 ) bisects the line
segment joining ( boldsymbol{A}(mathbf{3}, mathbf{1}, boldsymbol{6}) ) and
( boldsymbol{B}(mathbf{1}, boldsymbol{3}, boldsymbol{4}) )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
305Show that the plane whose vector equation is ( vec{r} .(hat{i}+2 hat{j}-hat{k})=3 ) contains the line ( vec{r}=hat{i}+hat{j}+lambda(2 hat{i}+hat{j}+4 hat{k}) )12
306A unit vector parallel to the intersection of the planes ( vec{r} cdot(hat{i}-hat{j}+hat{k})=5 ) and ( vec{r} )
( (2 hat{i}+hat{j}-3 hat{k})=4 ) can be
This question has multiple correct options
( frac{2 hat{i}+5 hat{j}+3 hat{k}}{sqrt{38}} )
B. ( frac{2 hat{i}-5 hat{j}+3 hat{k}}{sqrt{38}} )
c. ( frac{-2 hat{i}-5 hat{j}-3 hat{k}}{sqrt{38}} )
D. ( frac{-2 hat{i}+5 hat{j}-3 hat{k}}{sqrt{38}} )
12
307The length of the perpendicular from the
vertex ( D ) on the opposite face is
A ( cdot frac{14}{sqrt{6}} )
в. ( frac{2}{sqrt{6}} )
c. ( frac{3}{sqrt{6}} )
D. none of these
12
308The distance of the point (1,-2,3) from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} ) measured
parallel to the line ( frac{x}{2}=frac{y}{3}=frac{z-1}{-6} ) is
A .
B . 2
( c cdot 4 )
D. none of these
12
309The circum radius of the triangle formed by the points (0,0,0),(0,0,12) and (3,4,0) is
A. ( sqrt{156} )
B. 13
c. ( frac{13}{2} )
D.
12
310What is the equation of the plane containing the parallel lines
( boldsymbol{r}=boldsymbol{a}+boldsymbol{t c}, boldsymbol{r}=boldsymbol{b}+boldsymbol{p} boldsymbol{c} )
( mathbf{A} cdot r cdot(b-a) times c=[a b c] )
B cdot ( r cdot(a-b) times c=[a b c] )
C ( cdot r cdot(b-a) times c=-[a b c] )
D ( cdot r c times(b-a) times c=[a b c] )
12
311The direction cosines of the line joining the points (2,3,-1) and (3,-2,1) are
в. ( frac{1}{sqrt{30}},-sqrt{frac{5}{6}} cdot sqrt{frac{2}{15}} )
c. ( frac{-1}{30}, frac{1}{6},-frac{1}{15} )
D. none of these
12
312The ratio in which the line joining (3,4,-7) and (4,2,1) is dividing the ( x y ) plane
A .3: 4
B . 2: 1
( c cdot 7: 1 )
D. 4: 3
12
313( G(1,1,-2) ) is the centroid of the
triangle ( A B C ) and ( D ) is the mid point of
( boldsymbol{B C} cdot ) If ( boldsymbol{A}=(-1,1,-4), ) then ( boldsymbol{D}= )
( ^{mathbf{A}} cdotleft(frac{1}{2}, 1, frac{-5}{2}right) )
в. (5,1,2)
c. (-5,-1,-2)
D. (2,1,-1)
12
314Let ( S ) be the set of all real values of ( lambda )
such that a plane passing through the points ( left(-lambda^{2}, 1,1right),left(1,-lambda^{2}, 1right) ) and
( left(1,1,-lambda^{2}right) ) also passes through the point ( (-1,-1,1) . ) Then ( S ) is equal to:
A ( cdot(sqrt{3}) )
B. ( {sqrt{3}-sqrt{3}} )
begin{tabular}{l}
c. {1,-1} \
hline
end{tabular}
D. {3,-3}
12
315Find the Cartesian equation of ( vec{r} ) ( (2 hat{i}+3 hat{j}-4 hat{k})=1 )12
316A point on XOZ-plane divides the join of (5,-3,-2) and (1,2,-2) at
( ^{mathrm{A}} cdotleft(frac{13}{5}, 0,-2right) )
в. ( left(frac{13}{5}, 0,2right) )
c. (5,0,2)
D. (5,0,-2)
12
317A swimmer can swim ( 2 mathrm{km} ) in 15
minutes in a lake and in a river he can
swim a distance of ( 4 mathrm{km} ) in 20 minutes
along the stream. If a paper boat is put in the river, then the distance covered by it in ( 2 frac{1}{2} 2 ) hours will be
( mathbf{A} cdot 18 mathrm{km} )
B. ( 12 mathrm{km} )
c. ( 8 mathrm{km} )
D. ( 10 mathrm{km} )
12
318The points ( A(-1,3,0), B(2,2,1) ) and
( C(1,1,3) ) determine a plane. The distance of the plane ( A, B, C ) from the
point ( D(5,7,8) ) is
A . ( sqrt{66} )
B. ( sqrt{71} )
c. ( sqrt{73} )
D. ( sqrt{76} )
12
319If ( l_{1}, m_{1}, n_{1} ) and ( l_{2}, m_{2}, n_{2} ) are DCs of the two lines inclined to each other at an
angle ( theta ), then the DCs of the internal bisector of the angle between these lines are
A ( cdot frac{l_{1}+l_{2}}{2 sin frac{theta}{2}}, frac{m_{1}+m_{2}}{2 sin frac{theta}{2}}, frac{n_{1}+n_{2}}{2 sin frac{theta}{2}} )
в. ( frac{l_{1}+l_{2}}{2 cos frac{theta}{2}}, frac{m_{1}+m_{2}}{2 cos frac{theta}{2}}, frac{n_{1}+n_{2}}{2 cos frac{theta}{2}} )
c. ( frac{l_{1}-l_{2}}{2 sin frac{theta}{2}}, frac{m_{1}-m_{2}}{2 sin frac{theta}{2}}, frac{n_{1}-n_{2}}{2 sin frac{theta}{2}} )
D. ( frac{l_{1}-l_{2}}{2 cos frac{theta}{2}}, frac{m_{1}-m_{2}}{2 cos frac{theta}{2}}, frac{n_{1}-n_{2}}{2 cos frac{theta}{2}} )
12
320Find the equation of the line in vector and in cartesion form that passes through the point with position vector ( mathbf{2} hat{mathbf{i}}-hat{mathbf{j}}+mathbf{4} hat{boldsymbol{k}} ) and is in the direction ( hat{mathbf{i}}+ )
( mathbf{2} hat{boldsymbol{j}}-hat{boldsymbol{k}} )
12
321The number of octants in which ( Z )
coordinate is positive is
A . 2
B. 3
( c cdot 4 )
D. 1
12
322The length of the normal from origin to
the plane ( x+2 y-2 z=9 ) is equal to
A . 2 units
B. 3 units
c. 4 units
D. 5 units
12
32316.
Consider three points
P=(-sin(-a), -cos ),
Q = (cos(-a), sin B)
and
R=(cos(B-a +0), sin(6-0)), where
0<a,ß,e<I
(2008
Then,
(a) Plies on the line segment RQ
(b) Q lies on the line segment PR
(C) R lies on the line segment OP
(d) P, Q, R are non-collinear
12
324A plane passes through (1,-2,1) and is
perpendicular to the planes ( 2 x-2 y+ )
( z=0 ) and ( x-y+2 z=4 . ) Then the
distance of the plane from the point
(1,2,2) is
( mathbf{A} cdot mathbf{0} )
B.
( c cdot sqrt{2} )
( D cdot 2 sqrt{2} )
12
325If the distance between the plane,
( 23 x-10 y-2 z+48=0 ) and the plane containing the lines ( frac{boldsymbol{x}+mathbf{1}}{mathbf{2}}=frac{boldsymbol{y}-boldsymbol{3}}{boldsymbol{4}}= )
( frac{z+1}{3} ) and ( frac{x+3}{2}=frac{y+2}{6}= )
( frac{z-1}{lambda}(lambda epsilon R) ) is equal to ( frac{k}{sqrt{633}} ) then ( k ) is
equal to
12
326Prove that ( (vec{A} times vec{B})^{2}=A^{2} B^{2}-(vec{A} times vec{B})^{2} )12
327A line makes an angle ( alpha, beta, gamma ) with the
( X, Y, Z ) axes. Then ( sin ^{2} alpha+sin ^{2} beta+ )
( sin ^{2} gamma= )
A .
B. 2
( c cdot frac{3}{2} )
D.
12
328Show that the points (1,2,3),(7,0,1)
and (-2,3,4) are collinear.
12
329Find the direction cosines of the sides
of the triangles whose vertices are (3,5,-4),(-1,1,2) and (-5,-5,-2)
12
330The image of the point ( boldsymbol{P}(boldsymbol{alpha}, boldsymbol{beta}, gamma) ) by
the plane ( l x+m y+n z=0 ) is the point ( boldsymbol{Q}left(boldsymbol{alpha}^{prime}, boldsymbol{beta}^{prime}, boldsymbol{gamma}^{prime}right) . ) Then
A ( cdot alpha^{2}+beta^{2}+gamma^{2}=l^{2}+m^{2}+n^{2} )
B . ( alpha^{2}+beta^{2}+gamma^{2}=alpha^{2}+beta^{2}+gamma^{2} )
( mathbf{c} cdot alpha alpha^{prime}+beta beta^{prime}+gamma gamma^{prime}=0 )
D ( cdot lleft(alpha-alpha^{prime}right)+mleft(beta-beta^{prime}right)+nleft(gamma-gamma^{prime}right)=0 )
12
331Find the vector equation of the plane passing through the points (2,5,-3),(-2,-3,5) and (5,3,-3)12
332The distance of the point (1,-2,3) from
the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} ) measured
parallel to the line ( frac{x}{2}=frac{y}{3}=frac{z-1}{-6} ) is
( mathbf{A} cdot mathbf{1} )
B . 2
( c cdot 4 )
D. None of these
12
333Find the vector equation of the plane passing through (1,2,3) and perpendicular to the plane
( vec{r} cdot(hat{i}+2 hat{j}-5 hat{k})+9=0 )
12
334The distance of the point (1,1,1) from
the plane passing through the points (2,1,1),(1,2,1) and (1,1,2) is
A ( cdot frac{1}{sqrt{3}} )
B.
( c cdot sqrt{3} )
D. None of these
12
335If the d.c’s of a line are ( left(frac{1}{c}, frac{1}{c}, frac{1}{c}right), ) find ( c . )12
336The position vectors ( vec{a}, vec{b}, vec{c} ) of three points satisfy the relation ( 2 vec{a}+7 vec{b}+ )
( mathbf{5} vec{c}=overrightarrow{0} . ) Are these points collinear?
12
337If a line makes angles ( 90^{circ} ) and ( 60^{circ} )
respectively with the positive directions of ( x ) and ( y ) axes, find the angle which it
makes with the positive direction of ( z- )
axis.
12
338( operatorname{Given} boldsymbol{A}(mathbf{1},-mathbf{1}, mathbf{0}) ; boldsymbol{B}(mathbf{3}, mathbf{1}, mathbf{2}) ; boldsymbol{C}(mathbf{2},-mathbf{2}, mathbf{4}) )
and ( D(-1,1,-1) ) which of the following
points neither lie on ( A B ) nor on ( C D )
A ( .(2,2,4) )
В. (2,-2,4)
c. (2,0,1)
D. (0,-2,-1)
12
339Find the direction cosines of the line
which is perpendicular to the lines which direction cosines proportional to
1,-2,-2 and 0,2,1
12
340If ( A, B ) are the feet of the perpendiculars
from (2,4,5) to the ( x ) -axis, ( y ) -axis
respectively, then the distance ( A B ) is
A ( .2 sqrt{5} )
B. ( sqrt{29} )
c. ( sqrt{41} )
D. ( 3 sqrt{5} )
12
341Find the distance of the point (2,12,5) from the point of intersection of the line ( vec{r}=2 hat{i}-4 hat{j}+2 hat{k}+lambda(3 hat{i}+4 hat{j}+2 hat{k}) )
and the plane ( vec{r}(hat{i}-2 hat{j}+hat{k})=0 )
12
342The distance of point (-1,-5,-10) from the point of intersection of ( frac{x-2}{3}=frac{y+1}{4}=frac{-2}{12} ) and plane ( x )
( boldsymbol{y}+boldsymbol{z}=mathbf{5} ) is :
A . 10
B. 8
( c cdot 2 )
D. 13
12
343The line ( frac{x-2}{3}=frac{y+1}{2}=frac{z-1}{-1} )
intersects the curve ( boldsymbol{x} boldsymbol{y}=boldsymbol{c}^{2}, boldsymbol{z}=mathbf{0}, ) if
( boldsymbol{c}= )
A. ( 5 sqrt{5} )
B. ( 4 sqrt{5} )
c. ( sqrt{5} )
D. ( 2 sqrt{5} )
12
344Find the ratio in which the YZ-plane divides the line segment formed by joining the points (-2,4,7) and (3,-5,8)12
345Find the direction cosines of the vector
( overrightarrow{boldsymbol{r}}=(boldsymbol{6} hat{boldsymbol{i}}+boldsymbol{2} hat{boldsymbol{j}}-boldsymbol{3} hat{boldsymbol{k}}) )
12
346Four vertices of a tetrahedron are
(0,0,0),(4,0,0),(0,-8,0) and
( (0,0,12), ) Its centroid has the coordinates
A ( cdotleft(frac{4}{3},-frac{8}{3}, 4right) )
В. (2,-4,6)
c. (1,-2,3)
a 5
D. none of these
12
347Find the equation of a plane which is
parallel to the plane ( x-2 y+2 z=5 )
and whose distance from the point (1,2,3) is 1
12
348Let the direction – cosines of the line
which is equally inclined to the axis be ( pm frac{1}{sqrt{k}} . ) Find ( k ? )
A .2
B. 3
c. 5
D. 6
12
349If the extremities of a diagonal of a square are (1,-2,3) and (2,-3,5) then the length of its side is:
A ( cdot sqrt{6} )
B. ( sqrt{3} )
c. ( sqrt{5} )
D. ( sqrt{7} )
12
350( operatorname{can} frac{1}{sqrt{3}}, frac{2}{sqrt{3}}, frac{-2}{sqrt{3}} ) be the direction
cosines of any directed line?
A. Yes
B. No
c. cannot say
D. None of these
12
(3,4,5),(2,5,9),(5,2,8),(2,5,2)
12
352The distance between (0,1,-1) and the point of intersection of the line ( frac{x}{2}= ) ( frac{boldsymbol{y}-mathbf{1}}{mathbf{3}}=frac{boldsymbol{z}+mathbf{1}}{mathbf{4}} ) and the plane ( boldsymbol{x}+boldsymbol{y}+ )
( z=9 ) is
A. ( sqrt{29} )
в. ( frac{1}{2} sqrt{29} )
( c cdot frac{4}{9} )
D. ( frac{2}{9} sqrt{29} )
12
353If the d.rs of two lines are 1,-2,3 and ( 2,0,1, ) then the d.rs of the line
perpendicular to both the given lines is
A. -2,5,4
в. 2,-5,4
c. 2,5,-4
D. 1,5,-4
12
354The direction cosines of ( A B ) are -2,2,1
If ( boldsymbol{A} equiv(mathbf{4}, mathbf{1}, mathbf{5}) ) and ( l(boldsymbol{A} boldsymbol{B})=mathbf{6 u n i t s} )
find the coordinates of ( boldsymbol{B} )
12
355Match the statements/expressions in
List 1 with the values given in List 2
12
356If ( O A B C ) is a tetrahedron such that the
( boldsymbol{O} boldsymbol{A}^{2}+boldsymbol{B} boldsymbol{C}^{2}=boldsymbol{O} boldsymbol{B}^{2}+boldsymbol{C} boldsymbol{A}^{2}=boldsymbol{O} boldsymbol{C}^{2}+ )
( A B^{2}, ) then which of the following is/are
correct
A. ( A B perp O C )
в. ( O B neq C A )
c. ( O C=A B )
D. ( A B perp B C )
12
357Line through origin and parallel to ( Y ) –
axis is ( dots dots )
A ( cdot frac{x}{1}=frac{y}{0}=frac{2}{0} )
B. ( frac{x}{0}=frac{y}{1}=frac{z}{0} )
( mathbf{c} cdot frac{x}{1}=frac{y}{0}=frac{z}{1} )
D ( cdot frac{x}{1}=frac{y}{1}=frac{z}{0} frac{z}{0} )
12
358A lines makes angles ( frac{boldsymbol{alpha}}{2}, frac{boldsymbol{beta}}{2}, frac{gamma}{2} ) with
positive direction of coordinate axes, then ( cos alpha+cos beta+cos gamma ) is equal to
A . -1
B. 1
( c cdot 2 )
D. 3
12
359The cartesian equation of a line is ( frac{boldsymbol{x}+mathbf{3}}{mathbf{2}}=frac{boldsymbol{y}-mathbf{5}}{mathbf{4}}=frac{boldsymbol{z}+mathbf{6}}{mathbf{2}} ) find the vector
equation of the line?
12
360Let ( A(-1,0) ) and ( B(2,0) ) be two points.
A point ( M ) moves in the plane in such a
way that ( angle M B A=2 angle M A B ). Then the
point ( M ) moves along
A. A straight line
B. A parabola
c. An ellipse
D. A hyperbola
12
361The equation of the plane passing through the intersection of the planes ( mathbf{3} boldsymbol{x}-boldsymbol{y}+mathbf{2} boldsymbol{z}-mathbf{4}=mathbf{0} ) and ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}- )
( mathbf{2}=mathbf{0} ) and passing through the point
( A(2,2,1) ) is given by?
A. ( 7 x+5 y-4 z-8=0 )
в. ( 7 x-5 y+4 z-8=0 )
c. ( 5 x-7 y+4 z-8=0 )
D. ( 5 x+7 y-4 z+8=0 )
12
362Show that ( boldsymbol{A}(mathbf{1}, mathbf{1}, mathbf{2}), boldsymbol{B}(mathbf{2}, mathbf{1}, mathbf{3}) ) and
( C(1,3,5) ) are not collinear.
12
363Find the coordinates of the point, which divides the line segment joining the points ( boldsymbol{A}(mathbf{2},-mathbf{6}, mathbf{8}) ) and ( boldsymbol{B}(-mathbf{1}, mathbf{3},-mathbf{4}) )
externally in the ratio 1: 3
12
364Name the octants in which the following
points lie: ( (mathbf{5}, mathbf{2}, mathbf{3}) )
12
365The ratio in which the plane ( vec{r} cdot(vec{i}- ) ( mathbf{2} overrightarrow{boldsymbol{j}}+boldsymbol{3} overrightarrow{boldsymbol{k}})=17 ) divides the line joining
the points ( -2 vec{i}+4 vec{j}+7 vec{k} ) and ( 3 vec{i}- ) ( mathbf{5 j}+mathbf{8 k} ) is
A .1: 5
B. 1: 10
( c .3: 5 )
D. 3: 10
12
366Find the distance between the point (7,2,4) and the plane determined by
the points ( boldsymbol{A}(mathbf{2}, mathbf{5},-mathbf{3}), boldsymbol{B}(-mathbf{2},-mathbf{3}, mathbf{5}) )
and (5,3,-3)
12
367The distance from the origin to the centroid of the tetrahedron formed by
the points (0,0,0),(3,0,0),(0,4,0),(0,0,5) is
A. ( frac{sqrt{3+4+5}}{4} )
B. ( frac{sqrt{3+4+5}}{3} )
c. ( frac{sqrt{3^{2}+4^{2}+5^{2}}}{16} )
D. ( frac{sqrt{3^{2}+4^{2}+5^{2}}}{4} )
12
368The value(s) of ( lambda ), for which the triangle with vertices ( (mathbf{6}, mathbf{1 0}, mathbf{1 0}),(mathbf{1}, mathbf{0},-mathbf{5}) ) and
( (6,-10, lambda) ) will be a right angled
triangle is/ are
A . 1
в. ( frac{70}{3}, 0 )
c. 35
D. ( 0,-frac{70}{3} )
12
369Find the ratio in which the yz-plane
divides the join of the points (-2,4,7)
and (3,-5,8) and also find the ( c o )
ordinates of the point of intersection of
this line with the ( y z ) – plane.
A ( cdot lambda=frac{2}{3} ) and ( left(0, frac{2}{5}, frac{37}{5}right) )
B. ( lambda=frac{1}{3} ) and ( left(frac{-3}{4}, frac{7}{4}, frac{29}{4}right) )
c. ( lambda=frac{2}{3} ) and ( left(frac{-3}{4}, frac{7}{4}, frac{29}{4}right) )
D ( lambda=frac{1}{3} ) and ( left(0, frac{2}{5}, frac{37}{5}right) )
12
370The direction cosines of the normal to
the plane ( 5 y+4=0 ) are?
A ( cdot 0, frac{-4}{5}, 0 )
в. 0,1,0
c. 0,-1,0
D. None of these
12
371The equation of the plane that passes through the points (1,0,2),(-1,1,2),(5,0,3) is
A. ( x+2 y-4 z+7=0 )
B. ( x+2 y-3 z+7=0 )
c. ( x-2 y+4 z+7=0 )
D. ( 2 y-4 z-7+x=0 )
12
372The extremities of a diagonal of a rectangular parallelopiped whose faces are parallel to the reference planes are (-2,4,6) and ( (3,16,6) . ) The length of the base diagonal is
( A cdot 7 )
B. 10
c. 11
D. 13
12
373The equation of a line is ( 5 x-3= )
( mathbf{1 5 y}+mathbf{7}=mathbf{3}-mathbf{1 0 z} . ) Write the direction
( operatorname{cosines} ) of the line.
12
374Find the direction cosines of a line that
pass through the point ( boldsymbol{P}(mathbf{1}, mathbf{4}, boldsymbol{6}) ) and
( Q(5,1,11) ) and is so directed that it
make an acute angle with the positive direction of ( boldsymbol{y}-boldsymbol{a} boldsymbol{x} boldsymbol{i} boldsymbol{s} )
12
375The distance of point ( boldsymbol{P}left(boldsymbol{x}_{1}, boldsymbol{y}_{1}, boldsymbol{z}_{1}right) ) to
the plane ( a x+b y+c z+D=0 ) is
given by
A ( cdot frac{left|a x_{1}+b y_{1}+c z_{1}-dright|}{sqrt{a+b+c}} )
B. ( frac{left|a x_{1}+b y_{1}+c z_{1}right|}{sqrt{a^{2}+b^{2}+c^{2}}} )
c. ( frac{left|a x_{1}+b y_{1}+c z_{1}-dright|}{sqrt{a^{2}+b^{2}+c^{2}}} )
D. None of these
12
376A plane passes through (1,-2,1) and is
perpendicular to two planes ( 2 x-2 y+ )
( z=0 ) and ( x-y+2 z=4 . ) The distance
of the plane from the point is (1,2,2)
( A cdot 0 )
B.
( c cdot sqrt{2} )
D. ( 2 sqrt{2} )
12
377intb, z = c.ytd
2003
14. The two lines x=ay+b,z=cy+d and x=a’y+
will be perpendicular, if and only if
(a) aa’+cc’+1 = 0
(b) aa’ + bb’ + cc’ + 1 = 0
(c) aa’+bb’ +cc’ = 0
(d) (a+a’)(b+b) +c+c’)=0.
12
378A line passes through the points (6,-7,-1) and ( (2,-3,1) . ) The direction cosines of the line so directed that the
angle made by it with the positive direction of x-axis is acute, is?
A ( cdot frac{2}{3},-frac{2}{3},-frac{1}{3} )
B. ( -frac{2}{3}, frac{2}{3}, frac{1}{3} )
c. ( frac{2}{3},-frac{2}{3}, frac{1}{3} )
D. ( frac{2}{3}, frac{2}{3}, frac{1}{3} )
12
379x – 4
The va
The value of k such that
y-2
z-k
22.
lies in the
plane 2x – 4y +z=7, is
(2003)
(a)
7
(b) -7
(d) 4
(c) no real value
.
12
380A plane makes intercept 1,2,3 on the co-ordinate axes. If the distance from
origin is ( p ) then find the value of ( p )
12
381The equation of plane containing the lines ( vec{gamma}=vec{alpha}+overrightarrow{lambda beta} ) and ( vec{gamma}=vec{beta}+mu vec{alpha} )
( mathbf{A} cdotleft[begin{array}{lll}vec{gamma} & vec{alpha} & vec{beta}end{array}right]=vec{alpha} cdot vec{beta} )
B ( cdotleft[begin{array}{lll}vec{gamma} & vec{alpha} & vec{beta}end{array}right]=0 )
( mathbf{c} cdotleft[begin{array}{lll}vec{alpha} & vec{beta} & vec{gamma}end{array}right]=vec{alpha} cdot vec{beta} )
D. None of these
12
382is a
sa
mu on
19. Consider a pyramid OPQRS located in the first octant (x20.
y=0,220) with O as origin, and OP and OR along the x-axis
and the y-axis, respectively. The base OPQR of the pyramid
is a square with OP=3. The point S is directly above the
mid-point, T of diagonal OQ such that TS=3. Then
(a) the acute angle between OQ and OS is
(b)
the equation of the plane containing the triangle OOS
is x-y=0
the length of the perpendicular from P to the plane
(c)
containing the triangle OQS is 5
the perpendicular distance from 0 to the straight lin
(d)
containing RS is
12
383Show that the points ( boldsymbol{A}(-mathbf{3} hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}+mathbf{5} hat{boldsymbol{k}}), boldsymbol{B}(hat{boldsymbol{i}}+mathbf{2} hat{boldsymbol{j}}+mathbf{3} hat{boldsymbol{k}}), boldsymbol{C} )
are collinear.
12
384The points ( (boldsymbol{k}-mathbf{1}, boldsymbol{k}+mathbf{2}),(boldsymbol{k}, boldsymbol{k}+ )
1) ( ,(k+1, k) ) are collinear for
A. any value of ( k )
B. ( k=-frac{1}{2} ) only
c. no value of ( k )
D. integral values of ( k ) only
12
385If ( boldsymbol{A} ) is ( (2,4,5), ) and ( B ) is (-7,-2,8) then which of the following is collinear with ( A ) and ( B ) is
A ( .(1,2,6) )
В. (2,-1,6)
c. (-1,2,6)
D. (2,6,-1)
12
386A given unit vector is orthogonal to ( mathbf{5} hat{mathbf{i}}+mathbf{2} hat{mathbf{j}}+mathbf{6} hat{boldsymbol{k}} ) and coplanar with ( hat{mathbf{i}}-hat{boldsymbol{j}}+ )
( hat{k} ) and ( 2 hat{i}+hat{j}+hat{k} ) then the vector is?
A ( cdot frac{3 hat{j}-hat{k}}{sqrt{10}} )
( ^{text {В } cdot frac{6 hat{i}-5 hat{k}}{sqrt{61}}} )
c. ( frac{2 hat{i}-5 hat{k}}{sqrt{29}} )
( frac{2 hat{i}+hat{j}-2 hat{k}}{3} )
12
387to the plane
68. A equation of a plane parallel to the
x-2y + 2z-5 = 0 and at a unit distance from the origin
2012
(a) x-2y+ 2z – 3 =0
(c) x2y + 2z – 1 =0
(b) x-2y + 2z+1=0
(d) x-2y + 2z+5=0
12
388The direction ratios of the joining ( A(1,2,1) ) and (2,1,2) are
A .3,3,3
B. -1,1,-1
c. 3,1,3
D. ( frac{1}{sqrt{3}}, frac{1}{sqrt{3}}, frac{1}{sqrt{3}} )
12
389( boldsymbol{A}=(mathbf{2}, mathbf{4}, mathbf{5}) ) and ( boldsymbol{B}=(mathbf{3}, mathbf{5},-mathbf{4}) ) are two
points. If the ( X Y ) -plane, ( Y Z ) -plane divide ( A B ) in the ratio ( a: b ) and ( p: q )
respectively, then ( frac{a}{b}+frac{p}{q}= )
A ( cdot frac{23}{12} )
B. ( frac{-7}{12} )
c. ( frac{7}{12} )
D. ( frac{-22}{15} )
12
390The pairs ( bar{a}, bar{b} ) and ( bar{c}, bar{d} ) each determine a plane. Then the planes are parallel if ( (overline{boldsymbol{a}} times overline{boldsymbol{c}}) times(overline{boldsymbol{b}} times overline{boldsymbol{d}})= )
( mathbf{0} 2)(overline{boldsymbol{a}} times overline{boldsymbol{c}}) cdot(overline{boldsymbol{b}} times overline{boldsymbol{d}})=mathbf{0} )
( (bar{a} times bar{b}) times(bar{c} times bar{d})= )
( mathbf{0} 4)(bar{a} times bar{b}) cdot(bar{c} times bar{d})=0 )
12
391If a ray makes angles ( alpha, beta, gamma ) and ( delta ) with the four diagonals of a cube and
( mathbf{A}: cos ^{2} boldsymbol{alpha}+cos ^{2} boldsymbol{beta}+cos ^{2} boldsymbol{gamma}+cos ^{2} boldsymbol{delta} )
( mathbf{B}: sin ^{2} boldsymbol{alpha}+sin ^{2} boldsymbol{beta}+sin ^{2} boldsymbol{gamma}+sin ^{2} boldsymbol{delta} )
( mathbf{C}: cos 2 boldsymbol{alpha}+cos 2 boldsymbol{beta}+cos 2 gamma+cos 2 boldsymbol{delta} )
Arrange ( A, B, C ) in descending order
A. ( B, A, C )
в. ( A, B, C )
c. ( C, A, B )
D. ( B, C, A )
12
392If ( R ) divides the line segment joining ( P(2, )
3, 4) and ( Q(4,5,6) ) in the ratio -3: 2 then the value of the parameter which
represents R is
в. ( =(10,9,8) )
( mathbf{c} .=(10,8,9) )
( mathbf{D} .=(9,10,8) )
12
393Find the coordinates of a point on ( y ) -axis which are at a distance of ( 5 sqrt{2} ) from the
point ( boldsymbol{P}(boldsymbol{3},-boldsymbol{2}, boldsymbol{5}) )
12
394Find the equation of the set of points ( P ) the sum of whose distances from
( boldsymbol{A}(mathbf{4}, mathbf{0}, mathbf{0}) ) and ( boldsymbol{B}(-mathbf{4}, mathbf{0}, mathbf{0}) ) is equal to
10
12
395Distance between the points (12,4,7) and (10,5,3) is
A ( cdot sqrt{21} )
B. ( sqrt{5} )
c. ( sqrt{17} )
D. none of these
12
396The distances of the point ( boldsymbol{P}(mathbf{1}, mathbf{2}, mathbf{3}) )
from the coordinates axes are:
A ( cdot sqrt{13}, sqrt{10}, sqrt{5} )
в. ( sqrt{11}, sqrt{10}, sqrt{5} )
c. ( sqrt{13}, sqrt{20}, sqrt{15} )
D. ( sqrt{23}, sqrt{10}, sqrt{5} )
12
397If a line makes an angle of ( pi / 4 ) with the positive direction of each of ( x ) -axis and
( boldsymbol{y} ) -axis, then the angle that the line makes with the positive direction of the ( z ) -axis is
A ( cdot frac{pi}{6} )
в.
c.
D.
12
398Find the vector equation of the line passing through the point (3,1,2) and perpendicular to the plane ( vec{r} cdot(2 hat{i}-hat{j}+widehat{k})=8 ) Also find the point
of intersection of line and plane.
12
399Which of the following is true for a
plane?
This question has multiple correct options
A. A locus is called a plane if the line joining any two arbitrary points on the locus is also a part of the locus.
B. Value of ( y ) in a ( z x ) plane is non-zero.
C. Value of ( z ) in a ( x y ) plane is zero.
D. None of the above
12
400The point which divides the line joining the points (1,3,4) and (4,3,1) internally in the ratio ( 2: 1, ) is
A. (2,-3,3)
в. (2,3,3)
c. ( left(frac{5}{2}, 3, frac{5}{2}right) )
D. (-3,3,2)
E ( .(3,3,2) )
12
401If ( G ) is centroid of ( triangle A B C, ) then
A ( cdot vec{G}=vec{a}+vec{b}+vec{c} )
B. ( vec{G}=frac{vec{a}+vec{b}+vec{c}}{2} )
c. ( 3 vec{G}=vec{a}+vec{b}+vec{c} )
” ( quad 3 vec{G}=frac{vec{a}+vec{b}+vec{c}}{2} )
12
402A line OP where ( 0=(0,0,0) ) makes
equal angles with ox, oy, oz. The point on OP, which is at a distance of 6 units
from 0 is:
A. ( left(frac{6}{sqrt{3}}, frac{6}{sqrt{3}}, frac{6}{sqrt{3}}right) )
B . ( (2 sqrt{3},-2 sqrt{3}, 2 sqrt{3}) )
( c cdot-(2 sqrt{3}, 2 sqrt{3}, 2 sqrt{3}) )
12
403Show that the points ( boldsymbol{O}(mathbf{0}, mathbf{0}), boldsymbol{A}(mathbf{2},-mathbf{3}, mathbf{3}), boldsymbol{B}(-mathbf{2}, mathbf{3},-mathbf{3}) ) are
collinear. Find the ratio in which each point divides the segment joining the other two.
12
404Find the equation of the line passing through the points ( A(3,2,-1) ) and
( boldsymbol{B}(mathbf{4},-mathbf{1}, mathbf{3}) )
12
405A line makes the same angle ( theta ) with
each of the ( X ) and ( Z ) -axes. If the angle ( beta )
which it makes with ( Y ) -axis, is such
that ( sin ^{2} beta=3 sin ^{2} theta, ) then ( cos ^{2} theta ) equals
A ( cdot frac{2}{5} )
в. ( frac{1}{5} )
( c cdot frac{3}{5} )
D. ( frac{2}{3} )
12
406The foot of the perpendicular from the
point ( boldsymbol{A}(mathbf{7}, mathbf{1 4}, mathbf{5}) ) to the plane ( mathbf{2} boldsymbol{x}+mathbf{4} boldsymbol{y}- )
( z=2 ) is?
A. (3,1,8)
в. (1,2,8)
c. (3,-3,5)
D. (5,-3,-4)
12
407The ratio in which the line segment
joining the points whose position vectors are ( 2 hat{i}-4 hat{j}-7 hat{k} ) and ( -3 hat{i}+ ) ( mathbf{5} hat{boldsymbol{j}}-boldsymbol{8} hat{boldsymbol{k}} ) is divided by the plane whose
equation is ( hat{r} cdot(hat{i}-2 hat{j}+3 hat{k})=13 ) is-
A. 13: 12 internally
B. 12: 25 externally
c. 13: 25 internally
D. 37: 25 internally
12
408Distance between plane ( 3 x+4 y- )
( mathbf{2 0}=mathbf{0} ) and point ( (mathbf{0}, mathbf{0},-mathbf{7}) ) is
A. 4 units
B. 3 units
c. 2 units
D. 1 units
12
409The scalar product and the magnitude of vector products of two vectors are ( 48 sqrt{3} ) and 144 respectively. Then the angle between the two vectors is
A. 54.7
B. 60
c. 90
D. 120
12
410If a line has the direction ratios
4,-12,18 then find its direction
cosines.
A ( cdot-frac{2}{11},-frac{6}{11},-frac{9}{11} )
B. ( -frac{2}{11}, frac{6}{11},-frac{9}{11} )
c. ( frac{2}{11},-frac{6}{11}, frac{9}{11} )
D. ( frac{2}{11}, frac{6}{11}, frac{9}{11} )
12
411The locus of a point ( P ) which moves such
that ( P A^{2}-P B^{2}=2 k^{2} ) where ( A ) and ( B )
( operatorname{are}(3,4,5) ) and (-1,3,-7) respectively
is
A. ( 8 x+2 y+24 z-9+2 k^{2}=0 )
B. ( 8 x+2 y+24 z-2 k^{2}=0 )
c. ( 8 x+2 y+24 z+9+2 k^{2}=0 )
D. ( 8 x-2 y+24 z-2 k^{2}=0 )
12
412Assertion
In each of the three planes determined by two of the lines ( O A, O B, O C ) ( ( O ) being the origin), a straight line is drawn through ( O ) perpendicular to the third line.
The three lines so determined are
coplanar.
Reason
( (a times b) times c+(b times c) times a+(c times a) times )
( b=0, ) where ( O A=a, O B=b ) and
( boldsymbol{O} boldsymbol{C}=boldsymbol{c} )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
413Prove that
( boldsymbol{A}(mathbf{1}, mathbf{2}, mathbf{0}), boldsymbol{B}(mathbf{3}, mathbf{1}, mathbf{1}), boldsymbol{C}(mathbf{7},-mathbf{1}, mathbf{3}) ) are
collinear.
12
414The d.r. of normal to the plane through (1, 0, 0), (0, 1.0
which makes an angle /4 with plane x+y=3 are [20021
(a) 1,12,1
(b) 1,1, 2
(c) 1,1,2
(d) √2,1,1
12
415If the projections of the line segment ( A B ) on the coordinate axes are 2,3,6
then the square of the sine of the angle made by ( boldsymbol{A B} ) with ( boldsymbol{x}=mathbf{0}, ) is
A ( cdot frac{3}{7} )
B. ( frac{3}{49} )
( c cdot frac{4}{7} )
D. ( frac{40}{49} )
12
416The angle between any two faces is
( A cdot cos ^{-1}(1 / 3) )
B. ( cos ^{-1}(1 / 4) )
c. ( pi / 3 )
( mathbf{D} cdot cos ^{-1}(1 / 2) )
12
417The point equidistant from the point ( boldsymbol{O}(mathbf{0}, mathbf{0}, mathbf{0}), boldsymbol{A}(boldsymbol{a}, mathbf{0}, mathbf{0}), boldsymbol{B}(mathbf{0}, boldsymbol{b}, mathbf{0}) ) and
( C(0,0, c) ) has the coordinates
( mathbf{A} cdot(a, b, c) )
B. ( (a / 2, b / 2, c / 2) )
c. ( (a / 3, b / 3, c / 3) )
D. ( (a / 4, b / 4, c / 4) )
12
418The distance between (5,1,3) and the
line ( x=3, y=7+t, z=1+t ) is
A . 4
B. 2
( c cdot 6 )
D. 8
12
419The point of intersection of the lines ( frac{x-5}{3}=frac{y-7}{-1}=frac{z+2}{1} ) and ( frac{x+3}{-36}= )
( frac{y-3}{2}=frac{z-6}{4} ) is
A ( cdotleft(21, frac{5}{3}, frac{10}{3}right) )
в. (2,10,4)
c. (-3,3,6)
D. (5,7,-2)
12
420Assertion(A): If centroid and
circumcentre of a triangle are known its orthocentre can be found.

Reason (R) : Centriod, orthocentre and
circumcentre of a triangle are collinear
A. Both A and R are individually true and R is the correct explanation of
B. Both A and R individually true but R is not the correct explanation of ( A )
c. ( A ) is true but ( R ) is false
D. A is false but R is true

12
421( mathbf{I f A}=(mathbf{1}, mathbf{2}, mathbf{3}), mathbf{B}=(mathbf{2}, mathbf{3}, mathbf{4}) ) and ( mathbf{C} ) is a
point of trisection of AB such that ( mathbf{C}_{mathbf{x}}+ ) ( mathbf{C}_{mathbf{y}}=frac{mathbf{1 3}}{mathbf{3}} ) then ( mathbf{C}_{mathbf{z}}= )
A ( cdot frac{10}{3} )
B. ( frac{11}{3} )
c. ( frac{11}{2} )
D. 11
12
422+1
33.
x
If the angel e between the line
– 1 -2
22
– and
the plane 2x-y+ Váz+4 = 0 is such that sin 0=
then
the value of 1 is
[2005]
12
423Find the distance between the points ( R(-3,0), Sleft(0, frac{5}{2}right) )12
424Find the angle between the two lines
whose direction cosines are given by
equations ( l+m+n=0 ) and ( l^{2}+ )
( boldsymbol{m}^{2}-boldsymbol{n}^{2}=mathbf{0} )
A ( cdot frac{pi}{6} )
в.
( c cdot frac{pi}{2} )
D.
12
425The projection of the line segment joining (0,0,0) and (5,2,4) on the line whose direction ratios are 2,-3,6 is
A . 28
B. 4
c. ( frac{40}{7} )
D. ( sqrt{45} )
12
426Find the co-ordinates of the points of trisection of the line joining the points (-3,0) and (6,6)12
427If the lines ( frac{x-1}{2}=frac{y+1}{3}=frac{z-1}{4} ) and
( frac{boldsymbol{x}-mathbf{3}}{mathbf{1}}=frac{boldsymbol{y}-boldsymbol{k}}{mathbf{2}}=frac{boldsymbol{z}}{mathbf{1}} ) intersect, then the
value of ( k ) is
A ( cdot frac{3}{2} )
B. ( frac{9}{2} )
( c cdot-frac{2}{9} )
D. ( -frac{3}{2} )
12
428If the distance of the point ( P(4,3,5) ) from the Y-axis is ( lambda ), then the value of
( 7 lambda^{2} ) is
A .287
B. ( 7 sqrt{41} )
c. 63
D. 21
12
429The shortest distance between z-axis
and the line
( boldsymbol{x}+boldsymbol{y}+mathbf{2} boldsymbol{z}-mathbf{3}=mathbf{0}=mathbf{2} boldsymbol{x}+mathbf{3} boldsymbol{y}+mathbf{4} boldsymbol{z} )
( 4, ) is
( mathbf{A} cdot mathbf{1} )
B . 2
( c cdot 4 )
D. 3
12
430A line makes angles ( alpha, beta, gamma ) with the
positive directions of the axes of
reference. The value of ( cos 2 alpha+ )
( cos 2 beta+cos 2 gamma ) is
( mathbf{A} cdot mathbf{1} )
B. 2
( c cdot-1 )
D. 0
12
43117.
The radius of the circle in which the sphere
x2 + y2 + z2 + 2x – 2y – 42 – 19=0 is cut by the plane
x +2y + 2z +7= 0 is
[2003]
(a) 4 (6) (c) 2 (d) 3
12
432A line with direction cosines
proportional to 2,1,2 meets each of the line ( boldsymbol{x}=boldsymbol{y}+boldsymbol{a}=boldsymbol{z} ) and ( boldsymbol{x}+boldsymbol{a}=boldsymbol{2} boldsymbol{y}= )
2 ( z ). The co-ordinates of each of the
points of intersection are given by:
A. ( (3 a, 3 a, 3 a),(a, a, a) )
в. ( (3 a, 2 a, 3 a),(a, a, a) )
c. ( (3 a, 2 a, 3 a),(a, a, 2 a) )
D. ( (2 a, 3 a, 3 a),(2 a, a, a) )
12
433If ( vec{A} times vec{B}=vec{C}, ) which of the following
options is wrong?
( A cdot vec{C} ) is ( perp ) to ( vec{A} )
B cdot ( vec{C} ) is ( perp ) to ( vec{B} )
c. ( vec{C} ) is ( perp(vec{A}+vec{B}) )
D ( cdot vec{C} ) is ( perp ) to ( (vec{A} times vec{B}) )
12
434Find the shortest distance between the
following pair of lines. ( overline{boldsymbol{r}}=(overline{boldsymbol{i}}+mathbf{2} overline{boldsymbol{j}}+overline{boldsymbol{k}})+boldsymbol{lambda}(mathbf{2} overline{boldsymbol{i}}-overline{boldsymbol{j}}+mathbf{3} overline{boldsymbol{k}}) & )
( overline{boldsymbol{r}}=(overline{boldsymbol{i}}-mathbf{3} overline{boldsymbol{j}}-overline{boldsymbol{k}})+boldsymbol{mu}(mathbf{3} overline{boldsymbol{i}}+mathbf{2} hat{boldsymbol{j}}-mathbf{5} overline{boldsymbol{k}}) )
12
435Show that the points (2,-1,3),(4,3,1) and (3,1,2) are collinear.12
436f a point is in the ( X Z ) -plane. What can
you say about its ( y ) -coordinate?
12
437The plane ( x+2 y-z=4 ) cuts the
sphere ( boldsymbol{x}^{2}+boldsymbol{y}^{2}+boldsymbol{z}^{2}-boldsymbol{x}+boldsymbol{z}-boldsymbol{2}=mathbf{0} ) in
A . 1
B. 3
( c cdot sqrt{2} )
D.
12
438To find the vector and the Cartesian
equation in symmetric form of line passing through the points, (2,0,-3) and ( (mathbf{7}, mathbf{3},-mathbf{1 0}) )
12
439If ( theta ) is the angle between two lines
whose d.c.s are ( l_{1}, m_{1}, n_{1} ) and ( l_{2}, m_{2}, n_{2} )
then the d.cs of one of the angular bisectors of the two lines are
( ^{text {A }} cdot frac{l_{1}+l_{2}}{2}, frac{m_{1}+m_{2}}{2}, frac{n_{1}+n_{2}}{2} )
B. ( frac{l_{1}+l_{2}}{2 cos left(frac{theta}{2}right)}, frac{m_{1}+m_{2}}{2 cos left(frac{theta}{2}right)}, frac{n_{1}+n_{2}}{2 cos left(frac{theta}{2}right)} )
c. ( frac{l_{1}+l_{2}}{cos left(frac{theta}{2}right)}, frac{m_{1}+m_{2}}{cos left(frac{theta}{2}right)}, frac{n_{1}+n_{2}}{cos left(frac{theta}{2}right)} )
D. ( frac{l_{1}+l_{2}}{2 sin left(frac{theta}{2}right)} frac{m_{1}+m_{2}}{2 sin left(frac{theta}{2}right)} frac{n_{1}+n_{2}}{2 sin left(frac{theta}{2}right)} )
12
440The image of the point with position vector ( hat{i}+3 hat{k} ) in the plane ( r cdot(hat{i}+hat{j}+ )
( hat{boldsymbol{k}})=mathbf{1} ) is
( mathbf{A} cdot hat{i}+2 hat{j}+hat{k} )
B ( cdot hat{i}-2 hat{j}+hat{k} )
c. ( -hat{i}-2 hat{j}+hat{k} )
D. ( hat{i}+2 hat{j}-hat{k} )
12
441The vector form of the equation of the
line passing through points (3,4,7) and (5,1,6) is
( mathbf{A} cdot vec{r}=(3 hat{i}+4 hat{j}-7 hat{k})+lambda(2 hat{i}-3 hat{j}+13 hat{k}) )
B ( cdot vec{r}=(3 hat{i}+4 hat{j}-7 hat{k})+lambda(8 hat{i}+5 hat{j}-hat{k}) )
C ( . vec{r}=(3 hat{i}+4 hat{j}+7 hat{k})+lambda(2 hat{i}-3 hat{j}-hat{k}) )
D ( cdot vec{r}=(3 hat{i}+4 hat{j}-7 hat{k})+lambda(2 hat{i}-3 hat{j}-13 hat{k}) )
12
442The angle between any two diagonals of
cube are:
A ( cdot cos ^{-1}left(frac{1}{2}right) )
B. ( cos ^{-1}left(frac{1}{3}right) )
( ^{mathbf{C}} cdot cos ^{-1}left(frac{1}{sqrt{3}}right) )
( mathrm{D} cdot cos ^{-1}left(frac{1}{sqrt{2}}right) )
12
443Find the point of intersection of the following pair of lines, assuming that the vectors ( vec{a} ) and ( vec{b} ) are not parallel. ( vec{r}=gamma(vec{b}+vec{a}), vec{r}=mu(vec{b}-vec{a}) )
A . origin
в. ( vec{b}+vec{a} )
( c cdot 2 b )
D. no intersection point
12
444Find the equation of the line in vector and in Cartesian form that passes through the point with position vector ( 2 hat{i}-hat{j}+4 hat{k} ) and is in the direction ( hat{i}+ ) ( mathbf{2} hat{boldsymbol{j}}-hat{boldsymbol{k}} )12
445Find the equation of the plane passing through the points (0,0,0) and (3,-1,2) are parallel to the line ( frac{boldsymbol{x}-mathbf{4}}{mathbf{1}}=frac{boldsymbol{y}+mathbf{3}}{mathbf{- 4}}=frac{boldsymbol{z}+mathbf{1}}{mathbf{7}} )12
446If a line has direction ratio 2,-1,-2
determine its direction cosines.
12
44710
80
8
12
448The angle between the lines whose
direction cosines satisfy the equations ( l+m+n=0 ) and ( l^{2}+m^{2}+n^{2} ) is
A ( cdot frac{pi}{2} )
в.
c.
D. ( frac{pi}{6} )
12
449Find the vector and Cartesian equations
of the plane passing through the points with position vectors ( 3 vec{i}+4 vec{jmath}+ ) ( 2 vec{k}, 2 vec{i}-2 vec{j}-vec{k} ) and ( 7 vec{i}+vec{k} )
12
450A straight line ( L ) on the ( x y ) -plane
bisects the angle between ( O X ) and ( O Y )
What are the direction cosines of ( L ? )
A ( cdot(1 / sqrt{2}, 1 / sqrt{2}, 0) )
в. ( (1 / 2, sqrt{3} / 2,0) )
c. (0,0,1)
D. ( (2 / 3,2 / 3,1 / 3) )
12
451( mathbf{A}=(mathbf{1}, mathbf{1}, mathbf{4}) ) and ( mathbf{B}=(mathbf{5},-mathbf{3}, mathbf{4}) ) are two
points. If the points ( P, Q ) are on the line
( A B ) such that ( A P=P Q=Q B ) then ( P Q= )
A ( cdot 2 sqrt{2} )
в. 4
c. ( sqrt{frac{32}{9}} )
D. ( sqrt{2} )
12
452The angle between the lines, whose direction ratios are 1,1,2 and ( sqrt{3}- ) ( 1,-sqrt{3}-1,4, ) is
A . ( 45^{circ} )
B. ( 30^{circ} )
( c cdot 60^{circ} )
D. ( 90^{circ} )
12
453Show that the points
( boldsymbol{A}(mathbf{1}, mathbf{1}, mathbf{1}), boldsymbol{B}(mathbf{1}, mathbf{2}, mathbf{3}) ) and ( boldsymbol{C}(mathbf{2},-mathbf{1}, mathbf{1}) ) are
vertices of an isosceles triangle.
12
454Find the vector and Cartesian equation of the line that passes through the points (3,-2,-5) and (3,-2,6)12
455If two vertices of an equilateral triangle ( operatorname{are}(2,1,5) ) and ( (3,2,3), ) then its third vertex is:
A. (1,2,4)
в. (4,0,4)
c. (0,-4,4)
a 5
D. (4,4,1)
12
456Given that ( boldsymbol{P}(boldsymbol{3}, boldsymbol{2},-boldsymbol{4}), boldsymbol{Q}(boldsymbol{5}, boldsymbol{4},-boldsymbol{6}) ) and
( R(9,8,-10) ) are collinear. Find the ratio in which ( Q ) divides ( P R )
12
457If the lines ( frac{x-0}{1}=frac{y+1}{2}=frac{z-1}{-1} ) and ( frac{x+1}{k}= )
( frac{y-3}{-2}=frac{z-2}{1} ) are at right angles, then the
value of k is
( mathbf{A} cdot mathbf{5} )
B.
( c cdot 3 )
D. –
12
4582.
The equation x +2y + 2z = 1 and 2x + 4y + 4z=9 have
(a) Only one solution
ono (1979)
(b) Only two solutions
(c) Infinite number of solutions i n the
(d) None of these.
12
459Find the angles at which the normal
vector to the plane ( 4 x+8 y+z=5 ) is
inclined to the coordinate axes.
12
460Find ( a ), if the distance between the
points ( P(11,-2) ) and ( Q(a, 1) ) is 5 units.
12
461In what ratio, the line joining (-1,1) and (5,7) is divided by the line ( x+y= )
( 4 ? )
12
462If the origin is the centroid of the triangle whose vertices are ( boldsymbol{A}(mathbf{2}, boldsymbol{p},-mathbf{3}), boldsymbol{B}(boldsymbol{q},-mathbf{2}, mathbf{5}) ) and ( boldsymbol{R}(-mathbf{5}, mathbf{1}, boldsymbol{r}) )
then find the values of ( boldsymbol{p}, boldsymbol{q}, boldsymbol{r} )
12
463The equation of the plane passing through the point (1,1,1) and perpendicular to the planes ( 2 x+y- )
( 2 z=5 ) and ( 3 x-6 y-2 z=7 ) is?
A. ( 14 x+2 y-15 z=1 )
B. ( -14 x+2 y+15 z=3 )
c. ( 14 x-2 y+15 z=27 )
D. ( 14 x+2 y+15 z=31 )
12
464f points (1,2),(3,5) and ( (0, b) ) are collinear the value of b is
A ( cdot frac{1}{2} )
в. ( frac{7}{2} )
( c cdot 2 )
D. –
12
465Two distinct lines in a plane
A. always intersect
B. always either intersect or are parallel
C . always have two common points
D. none of these
12
466h the origin.
ant distance
P.: 2x-y+z
17. In R’, let L be a straight line passing through the ori
Suppose that all the points on L are at a constant dista
from the two planes P, :x+2y-z+1=0 and P, : 2x –
-1= 0. Let M be the locus of the feet of the perpendicula,
drawn from the points on L to the plane P,. Which of th
following points lie (s) on M?
(b)
12
467( boldsymbol{L} ) and ( boldsymbol{M} ) are two points with position vectors ( 2 bar{a}-bar{b} ) and ( a+2 bar{b} ) respectively.
The position vector of the point ( N ) which
divides the line segment ( L M ) in the ratio 2: 1 externally is
( mathbf{A} cdot 3 bar{b} )
в. ( 4 bar{b} )
( c .5 bar{b} )
D. ( 3 bar{a}+4 bar{b} bar{hline} bar{b}+4+4 bar{a} bar{a} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} )
12
468A non-zero vector ( vec{a} ) is parallel to the
line of intersection of plane ( boldsymbol{p}_{mathbf{1}} )
determined by ( hat{i}+hat{j} ) and ( hat{i} ) and plane ( P_{2} ) determined by vectors ( hat{mathbf{i}}-hat{mathbf{j}} ) and ( hat{mathbf{i}}+hat{boldsymbol{k}} )
then angle between a and vector ( hat{mathbf{i}}- ) ( 2 hat{j}+2 hat{k} ) is
This question has multiple correct options
A ( cdot frac{pi}{4} )
B. ( frac{pi}{2} )
c. ( frac{pi}{3} )
D. ( frac{3 pi}{4} )
12
469Find the coordinates of the points of trisection of the line segment joining the points ( A(-4,3) ) and ( B(2,-1) )12
470The position vectors of three points are ( mathbf{2} overrightarrow{boldsymbol{a}}-overrightarrow{boldsymbol{b}}+mathbf{3} overrightarrow{boldsymbol{c}}, overrightarrow{boldsymbol{a}}-mathbf{2} overrightarrow{boldsymbol{b}}+boldsymbol{lambda} overrightarrow{boldsymbol{c}} ) and ( boldsymbol{mu} overrightarrow{boldsymbol{a}}-mathbf{5} overrightarrow{mathbf{b}} )
where ( vec{a}, vec{b}, vec{c} ) are non coplanar vectors then the points are collinear when
A ( cdot lambda=-2, mu=frac{9}{4} )
B. ( lambda=-frac{9}{4}, mu=2 )
c. ( lambda=frac{9}{4}, mu=-2 )
D. None of these
12
471If the projections ofthe line segment ( A B )
on the ( y z ) -plane, ( z x ) -plane, ( x y ) -plane are ( sqrt{160}, sqrt{153}, 5 ) respectively, then the projection of ( A B ) on the ( z ) -axis is
A ( cdot sqrt{12} )
B. ( sqrt{13} )
c. 12
D. 144
12
472Find the distance of the point (-1,-5,-10) from the point of intersection of the line
( boldsymbol{r}=mathbf{2} overline{boldsymbol{i}}-overline{boldsymbol{j}}+mathbf{2} overline{boldsymbol{k}}+overline{boldsymbol{lambda}}(mathbf{3} overline{boldsymbol{i}}+mathbf{4} overline{boldsymbol{j}}+mathbf{2} overline{boldsymbol{k}}) ) and
the plane ( bar{r} .(bar{i}-bar{j}+bar{k})=5 )
12
473The coordinates of a point which divides the line joining the points ( P(2,3,1) ) and ( Q(5,0,4) ) in the ratio 1: 2 are
( ^{mathbf{A}} cdotleft(frac{7}{3}, 1, frac{5}{3}right) )
в. (4,1,3)
c. (3,2,2)
D. (1,-1,1)
12
474Find the value of ( k ) if the pts ( A equiv )
( (1,2,-1), B equiv(4,-2,4) ) and ( C equiv )
( (0,0, k) ) form a triangle right angled at
( boldsymbol{C} )
12
475If ( P(3,2,-4), Q(5,4,-6) ) and ( R )
(9,8,-10) are collinear, then ( R ) divides PQ in the ratio
A. 3: 2 internally
B. 3: 2 externally
c. 2: 1 internally
D. 2: 1 externally
12
476Are the points (1,1),(2,3) and (8,11) collinear?
A. collinear
B. Non collinear
c. coplaner
D. None of above
12
477The distance between the parallel planes given by the equations, ( vec{r} .(2 hat{i}- ) ( mathbf{2} hat{mathbf{j}}+hat{boldsymbol{k}})+mathbf{3}=mathbf{0} ) and ( vec{r} cdot(mathbf{4} hat{mathbf{i}}-mathbf{4} hat{mathbf{j}}+ )
( mathbf{2} hat{boldsymbol{k}})+mathbf{5}=mathbf{0} ) is –
A ( cdot 1 / 2 )
в. ( 1 / 3 )
c. ( 1 / 4 )
D. ( 1 / 6 )
12
478The equation of a plane which passes through the point of intersection of lines ( frac{x-1}{3}=frac{y-2}{1}=frac{z-3}{2}, ) and ( frac{x-3}{1}= )
( frac{boldsymbol{y}-mathbf{1}}{mathbf{2}}=frac{boldsymbol{z}-mathbf{2}}{mathbf{3}} ) and at greatest
distance from point (0,0,0) is-
A. ( 4 x+3 y+5 z=25 )
B. ( 4 x+3 y+5 z=50 )
c. ( 3 x+4 y+5 z=49 )
D. ( x+7 y-5 z=2 )
12
479( boldsymbol{A}=(mathbf{1},-mathbf{1}, mathbf{2}) ) and ( boldsymbol{B}=(mathbf{2}, mathbf{3}, mathbf{7}) ) are two
points. If ( boldsymbol{P}, boldsymbol{O} ) divide ( boldsymbol{A B} ) in the ratios
2: 3,-2: 3 respectively then ( P_{x}+ )
( boldsymbol{Q}_{boldsymbol{y}}= )
A ( cdot frac{-38}{5} )
в. ( frac{38}{5} )
c. ( frac{-2}{5} )
D. ( frac{-47}{6} )
12
480A line makes the same angle ( theta ) with
each of the ( x ) and ( z ) -axes. If the angle ( beta )
which it makes with ( y ) -axis, is such that ( sin ^{2} beta=3 sin ^{2} theta, ) then ( cos ^{2} theta ) is equal to
A ( cdot frac{2}{3} )
B. ( frac{1}{5} )
( c cdot frac{3}{5} )
D.
12
481If
( boldsymbol{A}(1,2,-1), B(4,0,-3), C(1,2,-1), D( )
find the distance between ( A B ) and ( C D )
12
48290.
The equation of the line passing through (4,3,1), parallel
to the plane x + 2y – Z-5 = 0 and intersecting the line
x +1 y-3 2-2
[JEE M 2019-9 Jan (M)
-3 2

1.
is:
x

4
y
+3
z+1
2
1
12
483The image of the point ( P(1,2,3) ) in the
plane ( 2 x-y+z+3=0 ) is
A ( cdot(-3,4,1) )
в. (3,5,2)
c. (-3,5,2)
D. (3,-5,2)
12
484If ( A ) and ( B ) be the points (3,4,5) and (-1,3,-7) respectively. Find the equation of the set of points ( boldsymbol{P} ) such
that ( boldsymbol{P} boldsymbol{A}^{2}+boldsymbol{P} boldsymbol{B}^{2}=boldsymbol{K}^{2}, ) where ( boldsymbol{K} ) is a
constant
12
485If sum of the perpendicular distances of a variable point ( boldsymbol{P}(boldsymbol{x}, boldsymbol{y}) ) from the lines
( boldsymbol{x}+boldsymbol{y}-mathbf{5}=mathbf{0} ) and ( mathbf{3} boldsymbol{x}-mathbf{2} boldsymbol{y}+mathbf{7}=mathbf{0} ) is
always ( 10 . ) Show that ( P ) must move on a
line.
12
48614. The image of the line *<!-Y-3-2-4 in the plane
3
2x – y +2+3= 0 is the line:
[JEE M 2014)
x-3
y + 5
z-2
X-3 y +5 2-2
-3 –
15
x +3
15
Z
-2
x+3y-5z +2
-3 -1
3
-5
12
487Two planes intersect each other to form
( mathbf{a}: )
A. plane
B. point
c. straight line
D. angle
12
488Find the direction cosines of vector ( vec{r} )
which is equally inclined to ( O X, O Y )
and ( O Z ). Find total number of such
vectors.
A ( cdot frac{1}{sqrt{3}}, frac{1}{sqrt{3}}, frac{1}{sqrt{3}} ; 6 )
B. ( frac{1}{sqrt{3}}, pm frac{1}{sqrt{3}}, frac{1}{sqrt{3}} ; 8 )
( ^{mathbf{C}} pm frac{1}{sqrt{3}}, pm frac{1}{sqrt{3}}, pm frac{1}{sqrt{3}} ; 8 )
D. None of these
12
489If the projection of a line segment on
( x, y ) and ( z ) axes are respectively 3,4 and
( 5, ) then the length of the line segment is
begin{tabular}{l}
A ( 3 sqrt{2} ) \
hline
end{tabular}
B. ( 5 sqrt{2} )
c. ( 6 sqrt{2} )
D. None of these
12
490Show that the points with position vectors ( vec{a}+vec{b}, vec{a}-vec{b} ) and ( vec{a}+k vec{b} ) are
collinear for all values of ( k )
12
491The equation of the plane through the points (2,3,1) and (4,-5,3) and parallel to ( x ) -axis is
A. ( x-z-1=0 )
в. ( 4 x+y-11=0 )
c. ( y+4 z-7=0 )
D. None of these
12
492If the d.rs of ( O A ) and ( O B ) are 1,-1,-1
and ( 2,-1,1, ) then the d.cs of the line perpendicular to both ( boldsymbol{O} boldsymbol{A} ) and ( boldsymbol{O B} ) are
в. -2,-3,1
c. ( frac{-2}{sqrt{14}}, frac{-3}{sqrt{14}}, frac{1}{sqrt{14}} )
D. ( frac{2}{sqrt{41}}, frac{3}{sqrt{41}}, frac{-1}{sqrt{41}} )
12
493Point ( D ) has coordinates as (3,4,5)
Find the coordinates of point ( G )
( mathbf{A} cdot(0,3,5) )
B. (3,0,4)
( mathbf{C} cdot(3,5,4) )
( mathbf{D} cdot(3,0,5) )
12
494The number of straight lines that are equally inclined to the threedimensional coordinate axes, is
A .2
B. 4
( c .6 )
D. 8
12
495( frac{x-2}{1}=frac{y-3}{1}=frac{z-4}{-1} & frac{x-1}{k}= )
( frac{boldsymbol{y}-boldsymbol{4}}{boldsymbol{2}}=frac{boldsymbol{z}-boldsymbol{5}}{boldsymbol{2}} ) are coplanar then ( mathbf{k}=? )
A. any value
B. exactly one value
c. exactly 2 values
D. exactly 3 values
12
496Determine if the points (1,5)(2,3) and (-2,-11) are collinear.
A. True
B. False
12
497The direction cosines of a line whose
equations are ( frac{x-1}{2}=frac{y+3}{4}=frac{z-2}{-3} )
A ( cdot frac{1}{sqrt{14}}, frac{-3}{sqrt{14}}, frac{2}{sqrt{14}} )
В ( cdot frac{2}{sqrt{29}}, frac{4}{sqrt{29}}, frac{-3}{sqrt{29}} )
c. ( frac{1}{sqrt{29}}, frac{-3}{sqrt{29}}, frac{2}{sqrt{29}} )
D. 2,4,-3
12
498Solve the following differential equation ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=boldsymbol{x}-mathbf{1} )
A ( cdot y=x^{2}+x )
B . ( y=x^{2} )
c. ( y=x^{2}-x )
D. None of the above
12
499If the foot of the perpendicular from
(0,0,0) to a plane is ( P(1,2,2) . ) Then, the equation of the plane is
A. ( -x+2 y+8 z-9=0 )
B. ( x+2 y+2 z-9=0 )
c. ( x+y+z-5=0 )
D. ( x+2 y-3 z+3=0 )
12
500The ratio in which ( x y- ) plane divides the line joining the points (1,0,-3) and (1,-5,7) is given by
A . 7: 3
в. 3: 7
( c .3: 4 )
D. 4: 7
12
501( vec{A} cdot(vec{A} times vec{B}) )
( A cdot vec{A} cdot vec{B} )
( mathbf{B} cdot underset{A}{longrightarrow} times underset{B}{longrightarrow} )
( c cdot 0 )
D. 1
12
502The distance of the point (1,-2,3) from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} ) measured
parallel to the line. ( frac{x}{2}=frac{y}{3}=frac{z}{-6}, quad ) is :
( A )
B. 6/7
( c cdot 7 / 6 )
D. ( 1 / 6 )
12
503Find the vector equation of the line through ( boldsymbol{A}(boldsymbol{3}, boldsymbol{4},-boldsymbol{7}) ) and ( boldsymbol{B}(boldsymbol{6},-1,1) )
Also find the cartesian form.
12
504One of the rectangular components of a
force of ( 40 N ) is ( 20 N . ) Find the angle it
makes with this component and
magnitude of other component.
12
505Find the distance of the point (1,2,-1) from the plane ( boldsymbol{x}-mathbf{2} boldsymbol{y}+mathbf{4} boldsymbol{z}-mathbf{1 0}= )
( mathbf{0} )
12
506Consider three vectors ( overrightarrow{boldsymbol{p}}=boldsymbol{i}+boldsymbol{j}+ )
( boldsymbol{k}, overrightarrow{boldsymbol{q}}=2 boldsymbol{i}+boldsymbol{4} boldsymbol{j}-boldsymbol{k} ) and ( overrightarrow{boldsymbol{r}}=boldsymbol{i}+boldsymbol{j}+boldsymbol{3} boldsymbol{k} . )
( p, q ) and ( r ) denotes the position vector of three non-collinear points, then the equation of the plane containing these points is
A. ( 2 x-3 y+1=0 )
B. ( x-3 y+2 z=0 )
c. ( 3 x-y+z-3=0 )
D. ( 3 x-y-2=0 )
12
507A line makes equal angles with the coordinate axis. The direction cosines of this line are
A ( cdotleft(frac{1}{3}, frac{1}{3}, frac{1}{3}right) )
B ( cdotleft(frac{1}{sqrt{3}}, frac{1}{sqrt{3}}, frac{1}{sqrt{3}}right) )
( ^{mathbf{c}} cdotleft(frac{1}{sqrt{3}}, frac{1}{3}, frac{1}{3}right) )
D ( cdotleft(frac{1}{sqrt{2}}, frac{1}{sqrt{2}}, frac{1}{sqrt{2}}right) )
12
508[2004]
25.
If the straight lines
1+t, z=2-t,
x=1+s, y=-3-as,z=1+as and x
with parameters s and t respectively, are co-planar, then a
equals.
(a) 0 (6) 1 (c) — 1 (d) -2
1.
1
.
12
509The expression in the vector form for the
point ( vec{r}_{1} ) of intersection of the plane ( vec{r}_{1} cdot vec{n}=d ) and the perpendicular line
( vec{r}=vec{r}_{0}+t vec{n} ) where ( t ) is a parameter
given by
A ( cdot overrightarrow{r_{1}}=overrightarrow{r_{0}}+left(frac{d-overrightarrow{r_{0}} cdot vec{n}}{vec{n}^{2}}right) vec{n} )
B ( cdot overrightarrow{r_{1}}=overrightarrow{r_{0}}+left(frac{overrightarrow{r_{0}} cdot vec{n}}{vec{n}^{2}}right) vec{n} )
C・ ( _{overrightarrow{r_{1}}}=overrightarrow{r_{0}}+left(frac{overrightarrow{r_{0}} cdot vec{n}-d}{|vec{n}|}right) vec{n} )
D ( overrightarrow{r_{1}}=overrightarrow{r_{0}}+left(frac{overrightarrow{r_{0}} cdot vec{n}}{|vec{n}|}right) vec{n} )
12
510what is the distance of the point (p,q,r) from the ( x ) -axis ?12
511( A B C D ) is a parallelogram. ( L ) is a point on ( B C ) which divides ( B C ) in the ratio
( mathbf{1}: mathbf{2} . boldsymbol{A} boldsymbol{L} ) intersects ( boldsymbol{B} boldsymbol{D} ) at ( boldsymbol{P} . boldsymbol{M} ) is a
point on ( D C ) which divides ( D C ) in the
ratio 1: 2 and ( A M ) intersects ( B D ) in ( Q )
Point ( P ) divides ( A L ) in the ratio
A . 1: 2
B. 1: 3
c. 3: 1
D. 2: 1
12
512For waht value of ( lambda ), the three numbers
( 2 lambda-1, frac{1}{4}, lambda-frac{1}{2} ) can be the direction
cosines of a straight line?
A ( cdot frac{1}{2} pm frac{sqrt{3}}{4} )
B. ( frac{3}{4} )
( c cdot pm frac{3}{4} )
D. ( frac{sqrt{3}}{2} pm frac{1}{4} )
12
513If the sum of the squares of the distance of a point from the three
coordinate axes be ( 36, ) then its distance
from the origin is
A. 6 units
B. ( 3 sqrt{2} ) units
c. ( 2 sqrt{3} ) units
D. none of these
12
514If (1,-1,0),(-2,1,8) and (-1,2,7) are three consecutive vertices of a
parallelogram then the fourth vertex is
A. (2,0,-1)
В. (1,0,-1)
c. (1,-2,0)
a 5
D. (0,-2,1)
12
515In ( triangle A B C ) the mid points of the sides
( A B, B C ) and ( C A ) are respectively
( (l, 0,0),(0, m, 0) ) and ( (0,0, n) . ) Then
( frac{A B^{2}+B C^{2}+C A^{2}}{l^{2}+m^{2}+n^{2}} ) is equal to
( A cdot 2 )
B. 4
c. 8
D. 16
12
516A point at which all the three perpendicular coordinate axes meets is known as
A. Meeting point
B. Origin
c. Triple point
D. None of these
12
517Find the coordinate of the point ( boldsymbol{P} ) where the line through ( A(3,-4,-5) ) and ( B(2,-3,1) ) crosses the plane passing through three points ( boldsymbol{L}(mathbf{2}, mathbf{2}, mathbf{1}), boldsymbol{M}(mathbf{3}, mathbf{0}, mathbf{1}) ) and ( boldsymbol{N}(mathbf{4},-mathbf{1}, mathbf{0}) )
Also, find the ratio in which ( P ) divides
the line segment ( boldsymbol{A B} )
12
51835.
If the plane 2ax – 3ay + 4az + 6 = 0 passes through the
midpoint of the line joining the centres of the spheres
x2 + y2 +z2 + 6x – 8y – 2z = 13 and
x2 + y2 +22 – 10x + 4y – 2z = 8 then a equals [2005]
(a) -1 (b) 1 (c) – (d) 2
12
519The ratio in which the plane ( vec{r} .(hat{i}-2 hat{j}+ ) ( mathbf{3} hat{k})=17 ) divides the line joining the points ( (-2 hat{i}+4 hat{j}+7 hat{k}) ) and ( (3 hat{i}-5 hat{j}+ )
( mathbf{8} hat{boldsymbol{k}}) ) is
A .1: 5
B. 1: 10
( c .3: 5 )
D. 3: 10
12
520Find the equation of line of intersection of planes ( vec{r} cdot(3 vec{i}-vec{j}+vec{k})=1 ) and ( vec{r} )
( (3 vec{i}+4 vec{j}-2 vec{k})=2 )
12
521Find he equation of the line passing through (1,2,3) and perpendicular to the two lines ( frac{x}{1}=frac{y}{2}=frac{z}{-1} ) and ( frac{x-1}{3}=frac{y}{2}=frac{z}{6} )12
522Plane passing through the points ( A(2,1 )
3), ( mathrm{B}(-1,2,4) ) and ( mathrm{C}(0,2,1) ). Determine its
point of intersection with the line ( r= )
( boldsymbol{j}+boldsymbol{k}+boldsymbol{t}(boldsymbol{2} boldsymbol{i}+boldsymbol{k}) )
A ( cdot(7,+1,4) )
B. (9,+1,-2)
c. (7,-1,4)
D. (9,-1,2)
12
523If ( theta ) is the angle between two lines
whose d.cs are ( l_{1}, m_{1}, n_{1} ) and ( l_{2}, m_{2}, n_{2} )
then
( frac{Sigmaleft(l_{1}+l_{2}right)^{2}}{4 cos ^{2}left(frac{theta}{2}right)}+frac{Sigmaleft(l_{mathrm{I}}-l_{2}right)^{2}}{4 sin ^{2}left(frac{theta}{2}right)}= )
( A )
B.
c. -1
( D )
12
524( mathbf{2} boldsymbol{x}-boldsymbol{y}+mathbf{3} boldsymbol{z}-mathbf{1}=mathbf{0} ) and ( mathbf{2} boldsymbol{x}-boldsymbol{y}+ )
( mathbf{3} boldsymbol{z}+mathbf{3}=mathbf{0} )
12
525Find the direction cosines of a line
which is perpendicular to the lines whose direction ratios are (1,-1,2) and (2,1,-1)
12
526The line passes through the points ( (5,1, a) &(3, b, 1) ) crosses the ( y z ) plane at the point ( left(0, frac{17}{2},-frac{13}{2}right), ) then
A ( . a=4, b=6 )
B. ( a=6, b=4 )
c. ( a=8, b=2 )
D. ( a=2, b=8 )
8
12
527The point in the ( x y- ) plane which is
equidistant from (2,0,3),(0,3,2) and (0,0,1) is
A ( .(1,2,3) )
В. (-3,2,0)
c. (3,-2.0)
D. (3,2,0)
E ( .(3,2,1) )
12
528The equation of the plane, which bisects the line joining the points (1,2,3) and (3,4,5) at right angles is?
A. ( x+y+z=0 )
B. ( x+y-z=9 )
c. ( x+y+z=9 )
D. ( x+y-z+9=0 )
12
529Find vector equation of line passing through the point whose position vector
is ( mathbf{3} hat{mathbf{i}}-mathbf{4} hat{mathbf{j}}+hat{boldsymbol{k}} ) and parallel to the vector
( 2 hat{i}+hat{j}-3 hat{k} . ) Also write the equation in
Cartesian form.
12
530Find the coordinates of the point ( boldsymbol{P} ) where the line through ( boldsymbol{A}(mathbf{3},-mathbf{4},-mathbf{5}) )
and ( B(2,-3,1) ) crosses the plane
passing through three points ( boldsymbol{L}(mathbf{2}, mathbf{2}, mathbf{1}), boldsymbol{M}(mathbf{3}, mathbf{0}, mathbf{1}) ) and ( boldsymbol{N}(mathbf{4},-mathbf{1}, mathbf{0}) )
Also, find the ratio in which ( P ) divides
the line segment ( boldsymbol{A B} )
12
531The distance between two points (1,1) and ( left(frac{2 t^{2}}{1+t^{2}}, frac{(1-t)^{2}}{1+t^{2}}right) ) is
( A cdot 4 t )
B. 3t
( c )
D. none of these
12
532The coordinates of the foot of the
perpendicular drawn from the point ( A(1,0,3) ) to the join of the points
( B(4,7,1) ) and ( C(3,5,3) ) are
A ( cdot(5,7,17) )
B ( cdotleft(frac{-5}{7}, frac{7}{3}, frac{-17}{3}right) )
( ^{mathbf{c}} cdotleft(frac{5}{7}, frac{-7}{3}, frac{17}{3}right) )
D. ( left(frac{5}{7}, frac{7}{3}, frac{17}{3}right) )
12
5331. The angles which a vector i + j + 2 k makes with X, Y
and Z axes respectively are
(a) 60°, 60°, 60° (b) 45°, 45°, 45°
(c) 60°, 60°, 45°
(d) 45°, 45°, 60°
12
534The angle between the line ( 2 x=3 y= )
( -z ) and ( 6 x=-y=-4 z ) is
A ( .90^{circ} )
B . ( 0^{circ} )
( c cdot 30^{0} )
D. ( 45^{circ} )
12
535The equation of a line passing through
(2,-3) and inclined at an angle of ( 135^{circ} )
with the positive direction of ( x ) -axis is.
A. ( x+y-1=0 )
в. ( x+y+1=0 )
c. ( x-y-1=0 )
D. ( -x+y+1=0 )
12
536A ray makes angles ( frac{pi}{3}, frac{pi}{3} ) with ( overrightarrow{O X} ) and ( boldsymbol{O Y} ) respectively. Find the angle made by it with ( overrightarrow{O Z} )12
53749.
If (2,3,5) is one end of a diameter of the sphere x2 + y2 +22
– 6x – 12y – 2z + 20 = 0, then the cooordinates of the other
end of the diameter are
[2007]
(a) (4,3,5)
(b) (4,3,-3)
(c) (4,9, -3)
(d) (4, -3,3).
12
538If ( boldsymbol{A} times boldsymbol{B}=boldsymbol{B} times boldsymbol{A}, ) then the angle
between ( A ) and ( B ) is
A. ( pi )
в. ( pi / 3 )
c. ( pi / 2 )
D. ( pi / 4 )
12
539The chord of contact of tangents from a
point ( boldsymbol{P} ) to a circle passes through ( boldsymbol{q} ). If
( l_{1} ) and ( l_{2} ) are the lengths of the tangents
from ( boldsymbol{P} ) and ( boldsymbol{Q} ) to the circle, then ( boldsymbol{P} boldsymbol{Q} ) is
equal to
A. ( frac{l_{1}+l_{2}}{2} )
в. ( frac{l_{1}-l_{2}}{2} )
C ( . sqrt{left|l_{1}^{2}-l_{2}^{2}right|} )
D. ( sqrt[2]{l_{1}^{2}+l_{2}^{2}} )
12
540In what ratio does the plane ( 2 x+y- )
( z=3 ) divide line segment joining the
point ( boldsymbol{a}=(mathbf{2}, mathbf{1}, mathbf{3}), boldsymbol{b}=(mathbf{9},-mathbf{2}, mathbf{5}) )
12
541If (1,-2,-2) and (0,2,1) are direction ratios of two lines, then the direction
cosines of a perpendicular to both the
lines are
( ^{mathrm{A}} cdotleft(frac{1}{3},-frac{1}{3}, frac{2}{3}right) )
в. ( left(frac{2}{3},-frac{1}{3}, frac{2}{3}right) )
( ^{c} cdotleft(-frac{2}{3},-frac{1}{3}, frac{2}{3}right) )
D. ( left(frac{2}{sqrt{14}},-frac{1}{sqrt{14}}, frac{3}{sqrt{14}}right) )
12
542If the vectors ( 3 bar{p}+bar{q} ; 5 bar{p}-3 bar{q} ) and ( 2 bar{p}+ )
( bar{q} ; 4 bar{p}-2 bar{q} ) are pairs of mutually
perpendicular vectors then ( sin (theta) ) is ( (theta ) is the angle between ( overline{boldsymbol{p}} ) and ( overline{boldsymbol{q}} )
A ( cdot sqrt{55} / 4 )
B . ( sqrt{55} / 8 )
c. ( 3 / 16 )
D. ( sqrt{247} / 16 )
12
543A hall has dimensions ( 24 m times 8 m times )
6 ( m ). The length of the longest pole which
can be accommodated in the hall is
A. 26
B. 28 m
( c cdot 30 m )
D. 36 m
12
544Let ( vec{a}=x^{2} hat{i}+2 hat{j}-2 hat{k}, vec{b}=hat{i}-hat{j}+hat{k} )
and ( vec{c}=x^{2} hat{i}+5 hat{j}-4 hat{k} ) be three vectors.
Find the values of ( x ) for which the angle between ( vec{a} ) and ( vec{b} ) is acute and the angle between ( vec{b} ) and ( vec{c} ) is obtuse.
A ( cdot(-3,-2) cup(2,3) )
B . (-3,-1)( cup(1,3) )
c. (-3,-1)( cap(1,3) )
D. (-3,-2)( cap(2,3) )
12
54591.
The plane through the intersection of the planes
x+y+z=1 and 2x+3y-z+4= 0 and parallel to y-axis also
passes through the point:
JEEM 2019-9 Jan (MI
(a) (-3,0,-1)
(b) (-3,1,1)
(c) (3,3,-1)
(d) (3, 2, 1)
12
546Find ( x ) so that the point (6,5,-3) is at a distance of 13 from the point ( (x,-7,0) )12
547Find the locus of the point, the sum of the squares of whose distances from
the planes ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=mathbf{0}, boldsymbol{x}-boldsymbol{y}=mathbf{0} )
( x+y-2 z=0 ) is 7
12
548Find the equation of line passing through ( (5,0,5) &(2,1,3) . ) Also show that ( (5,0,5),(2,1,3) &(-4,3,-1) ) are
collinear.
12
549Let the equation of the plane through
the points (-1,1,1) and (1,-1,1) and
perpendicular to the plane ( boldsymbol{x}+mathbf{2} boldsymbol{y}+ )
( mathbf{2} z=mathbf{7} ) be ( boldsymbol{k} boldsymbol{x}+boldsymbol{m} boldsymbol{y}-boldsymbol{n} boldsymbol{z}+boldsymbol{p}=mathbf{0} . ) Find
( boldsymbol{k}+boldsymbol{m}+boldsymbol{n}+boldsymbol{p} ? )
12
550( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=mathbf{1} )12
551The distance of the point (1,-5,9) from the planer. ( (hat{mathbf{i}}-hat{mathbf{j}}+hat{boldsymbol{k}})=mathbf{5} )
measured long the line ( r=hat{i}+hat{j}+hat{k} ) is
A ( 3 sqrt{5} )
5
B. ( 10 sqrt{3} )
( c cdot 5 sqrt{3} )
D. ( 3 sqrt{10} )
12
552Find the direction cosines of the unit
vector perpendicular to the plane ( vec{r} cdot(6 hat{i}-3 hat{j}-2 hat{k})+1=0 )
12
553The coordinates of any point, which lies in ( boldsymbol{y} boldsymbol{z} ) plane, are
This question has multiple correct options
A ( .(x, y, y) )
в. ( (0, y, y) )
c. ( (0, y, x) )
D. ( (x, y, z) )
12
55434.
The line passing through the points (5, 1, a) and (3, b, 1)
crosses the yz-plane at the point (0,7 -13). Then
(a) a=2, b=8
(b) a=4, b=6
(c) a=6, b=4
(d) a=8, b=2
12
555A line passes through the point (6,-7,-1) and ( (2,-3,1) . ) if the angle ( alpha ) which the line makes with the positive
direction of ( x ) -axis is acute, the
direction cosines of the line are,
A ( .2 / 3,-2 / 3,-1 / 3 )
в. ( 2 / 3,2 / 3,-1 / 3 )
c. ( 2 / 3,-2 / 3,1 / 3 )
D. ( 2 / 3,2 / 3,1 / 3 )
12
556Find the equations to the straight lines which are conjugate to the coordinate
axes with respect to the conic ( A x^{2}+ )
( 2 H x y+B y^{2}=1 )
Find the condition that they may
coincide, and interpret the result.
12
55780.
Z+4
3
lies in the plane, kx +my-z=9,
x-3 y +2
If the line,
2 -1
then 12 + m2 is equal to :
(a) 5 (b) 2
(c) 26
[JEE M 2016]
(d) 18
12
558x

1
V
-4
z
5
1
1
-k
15. The lines *-2 =973 – 3-4 and **=
coplanar if
(a) k=3 or-2
b) k=0 or -1
(c) k=1 or-1
1) k=0 or-3
=-7 are
[2003]
12
559Let ( boldsymbol{A}=(mathbf{1}, mathbf{2}, mathbf{3}) boldsymbol{B}= )
( (-1,-2,-1) C=(2,3,2) ) and ( D= )
( (4,7,6) . ) Then ( A B C D ) is a
A. rectangle
B. square
c. parallelogram
D. none of these
12
560If the projections of the line segment ( A B ) on the coordinate axes are ( 12,3, k )
and ( A B=13, ) then ( k^{2}-2 k+3 ) is equal
to:
This question has multiple correct options
A . 0
B.
c. 11
D. 27
12
561Find the coordinate of the points which trisect the line segment joining the points ( boldsymbol{A}(mathbf{2}, mathbf{1},-mathbf{3}) ) and ( boldsymbol{B}(mathbf{5},-mathbf{8}, mathbf{3}) )12
562If ( mathbf{A}=(-mathbf{1}, mathbf{6}, mathbf{6}), mathbf{B}=(-mathbf{4}, mathbf{9}, mathbf{6}), mathbf{G}= )
( frac{1}{3}(-5,22,22) ) and ( G ) is the centroid of
the ( Delta A B C ) then the name of the
triangle ( mathbf{A B C} ) is
A. an isosceles triangle
B. a right angled triangle
c. an equilateral triangle
D. a right-angled isosceles triangle
12
563Find ( x, ) if ( triangle A B C ) is right-angled at ( A )
where ( boldsymbol{A} equiv(mathbf{4}, mathbf{2}, mathbf{3}), boldsymbol{B} equiv(mathbf{3}, mathbf{1}, mathbf{8}), boldsymbol{C} equiv )
( (x,-1,2) )
12
564A plane ( pi ) makes intercepts 3 and 4
respectively on ( z- ) axis. If ( pi ) is parallel to ( boldsymbol{y}- ) axis, then its equation is ?
A. ( 3 x+4 z=12 )
B. ( 3 z+4 x=12 )
c. ( 3 y+4 z=12 )
D. ( 3 z+4 y=12 )
12
565Determine whether the points are
collinear.
( boldsymbol{P}(-mathbf{2}, mathbf{3}), boldsymbol{B}(mathbf{1}, mathbf{2}), boldsymbol{C}(mathbf{4}, mathbf{1}) )
12
566If ( P(x, y, z) ) is a point on the line segment joining ( boldsymbol{A}(mathbf{2}, mathbf{2}, mathbf{4}) ) and
( B(3,5,6) ) such that projection of ( overrightarrow{O P} ) on axes are ( frac{13}{5}, frac{19}{5}, frac{26}{5} ) respectively, then ( P ) divide AB in the ratio
A .3: 2
B. 2: 3
c. 1: 2
D. 1: 3
12
567Find the direction cosines of the
line passing through the two points (-2,4,-5) and (1,2,3)
12
568Which one of the following is best condition for the plane ( a x+b y+c z+ )
( d=0 ) to intersect the ( x ) and ( y ) axes at
equal angle
A ( cdot|a|=|b| )
( b mid )
B . ( a=-b )
c. ( a=b )
D. ( a^{2}+b^{2}=1 )
12
569The point of intersection of the line
joining the points (-3,4,-8) and (5,-6,4) with the ( X Y ) -plane is
( mathbf{A} cdotleft(frac{7}{3},-frac{8}{3}, 0right) )
в. ( left(-frac{7}{3},-frac{8}{3}, 0right) )
( ^{mathbf{c}} cdotleft(-frac{7}{3}, frac{8}{3}, 0right) )
D. ( left(frac{7}{3}, frac{8}{3}, 0right) )
12
570If ( P(x, y, x) ) is a point on the line
segment joining ( Q(2,2,4) ) and ( R(3,5,6) ) such that the projection of ( O P ) on the axis are ( frac{13}{5}, frac{19}{5}, frac{26}{5} )
respectively, then ( P ) divides ( Q R ) in the
ratio
A . 1: 2
B. 3: 2
( c cdot 2: 3 )
D. 1: 3
12
571The xy-plane divides the line joining the points (-1,3,4) and (2,-5,6)
A. internally in the ratio 2: 3
B. externally in the ratio 2: 3
c. internally in the ratio 3: 2
D. externally in the ratio 3: 2
12
572Point ( (boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma}) ) lies on the plane ( boldsymbol{x}+boldsymbol{y}+ )
( z=2 . ) Let ( vec{a}=alpha hat{i}+beta hat{j}+gamma hat{k} ) and ( hat{k} times )
( (hat{k} times vec{a})=0 ) then ( gamma= )
A .
B. 1
c. 2
D.
12
573Write the abscissa of the following
point
(0,5)
12
574The ratio in which the line joining
(2,-4,3) and (-4,5,-6) is divided by the plane ( 3 x+2 y+z-4=0 ) is
A .2: 1
B . 4: 3
( c cdot-1: 4 )
D. 2: 3
12
575If the point ( P(a, b, c), ) with reference to Eq. ( (i) ) lies on the plane ( 2 x+y+z=1 )
then the value of ( 7 a+b+c ) is
( mathbf{A} cdot mathbf{0} )
B. 12
( c cdot 7 )
D. 6
12
576The image of the point ( 3 hat{i}-2 hat{j}+hat{k} ) in the plane ( bar{r} .(3 hat{i}-hat{j}+4 hat{k})=2 )
( mathbf{A} cdot-hat{j}+3 hat{k} )
B . ( hat{j}-3 hat{k} )
( mathbf{c} .-hat{j}-3 hat{k} )
D ( .-2 hat{j}-3 hat{k} )
12
577Given planes are ( boldsymbol{P}_{1}: boldsymbol{c} boldsymbol{y}+boldsymbol{b} boldsymbol{z}=boldsymbol{x} quad boldsymbol{P}_{2}: boldsymbol{a} boldsymbol{z}+boldsymbol{c} boldsymbol{x}=boldsymbol{y} quad boldsymbol{P}_{3} )
( P_{1}, P_{2} ) and ( P_{3} ) pass through one line, if
A ( cdot a^{2}+b^{2}+c^{2}=a b+b c+c a )
B . ( a^{2}+b^{2}+c^{2}+2 a b c=1 )
( mathbf{c} cdot a^{2}+b^{2}+c^{2}=1 )
D. ( a^{2}+b^{2}+c^{2}+2 a b+2 b c+2 c a+2 a b c=1 )
12
578The plane which bisects the line
segment joining the points (-3,-3,4) and (3,7,6) at right angles, passes through which one of the following points?
A ( .(4,1,7) )
В. (4,1,-2)
c. (2,3,5)
D. (2,1,3)
12
579The point (3,0,-4) lies on the
A. Y-axis
B. z-axis
c. XY-plane
D. xz-plane
E. YZ-plane
12
580The point of intersection of the line ( frac{x-1}{3}=frac{y+2}{4}=frac{z-3}{-2} ) and plane
( 2 x-y+3 z-1=0 ) is.
B . (10,10,-3)
c. (-10,10,3)
D. None of these
12
581Show that the points ( boldsymbol{A}(mathbf{1}, mathbf{2}, mathbf{3}) )
( boldsymbol{B}(-1,-2,-3), C(2,3,2) ) and ( D(4,7,3) )
are the vertices of a parallelogram.
12
582The distance of the point (1,-2,3)
from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} )
measured parallel to the line whose direction cosines are proportional to
2,3,-5 is
A ( cdot frac{9}{7} )
B. ( frac{11}{7} )
c. ( frac{15}{7} )
D. None of these
12
583Find the distance between the following pairs of points:
(i) (2,3,5) and (4,3,1)
( (i i)(-3,7,2) ) and ((2,4,-1)
(iii) (-1,3,-4) and (1,-3,4)
(iv) (2,-1,3) and (-2,1,3)
12
584In the ( Delta A B C, ) if ( A B=sqrt{2} ; A C= )
( sqrt{mathbf{2 0}}, boldsymbol{B}=(mathbf{3}, mathbf{2}, mathbf{0}) ) and ( boldsymbol{C}=(mathbf{0}, mathbf{1}, mathbf{4}) )
then the length of the median passing
through ( boldsymbol{A} ) is
A ( cdot frac{3}{2} )
B. ( frac{9}{2} )
c. ( frac{3}{sqrt{2}} )
D. ( frac{sqrt{3}}{2} )
12
585A point on the line ( frac{boldsymbol{x}+mathbf{2}}{mathbf{1}}=frac{boldsymbol{y}-mathbf{3}}{-mathbf{4}}= )
( frac{z-1}{2 sqrt{2}} ) at a distance 6 from the point ( (2, )
3, 1) is
A ( cdot(4-21,1+12 sqrt{2}) )
В. ( left(frac{-4}{5}, frac{-9}{5}, 1right) )
C ( cdotleft(frac{-16}{5}, frac{39}{5}, frac{5-12 sqrt{2}}{5}right) )
D. ( left(frac{-16}{5},-21,1+12 sqrt{2}right) )
12
586The coordinates of the point where the line segment joining ( boldsymbol{A}(mathbf{5}, mathbf{1}, boldsymbol{6}) ) and ( B(3,4,1) ) crosses the yz plane are
A ( cdotleft(0, frac{17}{2}, frac{13}{2}right) )
в. ( left(0,-frac{17}{2}, frac{13}{2}right) )
c. ( left(0, frac{17}{2},-frac{13}{2}right) )
D. ( left(0,-frac{17}{2},-frac{13}{2}right) )
12
587Show that the points ( boldsymbol{A}(boldsymbol{3}, boldsymbol{2},-boldsymbol{4}), boldsymbol{B}(boldsymbol{5}, boldsymbol{4},-boldsymbol{6}) ) and
( C(9,8,-10) ) are collinear, find the ratio in which ( B ) divides ( overline{A C} ).
12
588Find the direction cosines of the sides
of the triangle whose vertices are (3,5,-4),(-1,1,2) and (-5,-5,-2)
12
589The distance between the X-axis and
the point (3,12,5) is
A. 3
B. 13
( c cdot 14 )
D. 12
E. 5
12
590The line ( frac{x-2}{3}=frac{y+1}{2}=frac{z-1}{-1} )
intersects the curve ( x y=c^{2}, z=0 ) if ( c ) is
equal to:
( A cdot pm 1 )
B. ( pm frac{1}{3} )
( mathrm{c} cdot pm sqrt{5} )
D. None of these
12
591If the points ( (h, 3,-4),(0,-7,10) ) and ( (1, k, 3) ) are collinear, then ( h+k ) is
( mathbf{A} cdot mathbf{4} )
B.
c. -4
D. 14
12
592The direction cosines of the line
passing through ( mathbf{P}(mathbf{2}, mathbf{3},-mathbf{1}) ) and the origin are
A ( cdot frac{2}{sqrt{14}}, frac{3}{sqrt{14}}, frac{1}{sqrt{14}} )
B. ( frac{2}{sqrt{14}}, frac{-3}{sqrt{14}}, frac{1}{sqrt{14}} )
c. ( frac{-2}{sqrt{14}}, frac{-3}{sqrt{14}}, frac{1}{sqrt{14}} )
D. ( frac{2}{sqrt{14}}, frac{-3}{sqrt{14}}, frac{-1}{sqrt{14}} )
12
593The vertices of a triangle are 2,3,5)( ,(-1,3,2),(3,5,-2), ) then the
angles are
( mathbf{A} cdot 30^{circ}, 30^{circ}, 30^{circ} )
( ^{mathrm{B}} cos ^{-1}left(frac{1}{sqrt{5}}right), 90^{circ}, cos ^{-1}left(frac{sqrt{5}}{sqrt{3}}right) )
c. ( 30^{circ}, 60^{circ}, 90^{circ} )
D ( cdot cos ^{-1}left(frac{1}{sqrt{3}}right), 90^{circ}, cos ^{-1}left(frac{sqrt{2}}{sqrt{3}}right) )
12
594Determine the three planes through the intersection of the planes ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=mathbf{1} )
and ( 2 x+3 y-z+4=0 ) which are
parallel to the three co – ordinate axes.
Also find the equation of the plane
perpendicular to the ( y z- ) plane and
passing through the point (2,3,1) and
(4,-5,3)
A ( cdot y-3 z-6=0 ; x-4 z=7 ) and ( 3 x-4 y-3=0 )
( y+4 z+7=0 )
B. ( y-3 z+6=0 ; x+4 z=7 ) and ( 3 x+4 y+3=0 )
( y-4 z+7=0 )
c. ( y+3 z+6=0 ; x+4 z=7 ) and ( 3 x+4 y+3=0 )
( y-4 z-7=0 )
D. ( y-3 z+6=0 ; x+4 z=7 ) and ( 3 x+4 y+3=0 )
( y+4 z-7=0 )
12
595Find the angle between the planes
whose vector equations are ( vec{r} cdot(2 hat{i}+2 hat{j}-3 hat{k})=5 ) and
( vec{r} cdot(3 hat{i}-3 hat{j}+5 hat{k})=3 )
12
596If a line makes an angle ( theta_{1}, theta_{2}, theta_{3} ) which
the axis respectively, then ( cos 2 theta_{1}+ )
( cos 2 theta_{2}+cos 2 theta_{3}=? )
A . -4
B . 2
( c .3 )
D. –
12
597The projection of the join of the two points (1,4,5),(6,7,2) on the line whose d.s’s are (4,5,6) is
A ( cdot frac{17}{sqrt{77}} )
B. ( frac{7}{6} )
c. 21
D. ( frac{7}{9} )
12
598( boldsymbol{A}(boldsymbol{3}, boldsymbol{2}, boldsymbol{0}), boldsymbol{B}(boldsymbol{5}, boldsymbol{3}, boldsymbol{2}), boldsymbol{C}(-boldsymbol{9}, boldsymbol{6},-boldsymbol{3}) ) are
three points forming a triangle. If ( boldsymbol{A} boldsymbol{D} )
the bisector of ( angle B A C ) meets ( B C ) in ( D )
then coordinates of ( D ) are
( ^{mathrm{A}} cdotleft(-frac{19}{8}, frac{57}{16}, frac{17}{16}right) )
В ( cdotleft(frac{19}{8},-frac{57}{16}, frac{17}{16}right) )
( ^{mathrm{C}} cdotleft(frac{19}{8}, frac{57}{16}, frac{17}{16}right) )
D. None of these
12
599Let ( P(3,2,6) ) be point in space and ( Q ) be appoint on the line ( vec{r}=(hat{i}-hat{j}+2 hat{k})+ )
( mu(-3 hat{i}+hat{j}+5 hat{k}) . ) Then the value of ( mu ) for which the vector ( overrightarrow{P Q} ) is parallel to the
plane ( boldsymbol{x}-mathbf{4} boldsymbol{y}+mathbf{3} boldsymbol{z}=mathbf{1} ) is
A. ( 1 / 4 )
B. – 1/4
c. ( 1 / 8 )
D. ( -1 / 8 )
12
600Find the direction cosines of the vector
( hat{mathbf{i}}+mathbf{2} hat{mathbf{j}}+mathbf{3} hat{boldsymbol{k}} )
12
601If the planes ( boldsymbol{x}-boldsymbol{c} boldsymbol{y}-boldsymbol{b} boldsymbol{z}=boldsymbol{0}, boldsymbol{c} boldsymbol{x}-boldsymbol{y}+ )
( boldsymbol{a} boldsymbol{z}=mathbf{0} ) and ( boldsymbol{b} boldsymbol{x}+boldsymbol{a} boldsymbol{y}-boldsymbol{z}=mathbf{0} ) pass
through a stright line,then the value of ( a^{2}+b^{2}+c^{2}+2 a b c ) is:
A . 1
B. 2
( c .3 )
D. none of these
12
602The equations of two planes are ( P_{1} )
( 2 x-y+z=2, ) and ( P_{2}: x+2 y-z= )
3. The equation of the plane which
passes through the point (-1,3,2) and is perpendicular to each of the planes
( P_{1} ) and ( P_{2} ) is
A ( cdot x+3 y-5 z+2=0 )
B. ( x+3 y+5 z-18=0 )
c. ( x-3 y-5 z+20=0 )
D. ( x-3 y+5 z=0 )
12
603Let the equation of the plane through the intersection of the planes ( x+2 y+ )
( mathbf{3} boldsymbol{z}-mathbf{4}=mathbf{0} ) and ( mathbf{2} boldsymbol{x}+boldsymbol{y}-boldsymbol{z}+mathbf{5}=mathbf{0} ) and
perpendicular to the plane ( 5 x+3 y+ )
( mathbf{6} z+mathbf{8}=mathbf{0} ) be ( boldsymbol{k} boldsymbol{x}+mathbf{1 5} boldsymbol{y}+boldsymbol{m} boldsymbol{z}+mathbf{1 7 3}= )
0. Find ( k+m )
12
604The ( x ) -coordinate of a point on the line joining the points ( P(2,2,1) ) and ( Q(5,1,-2) ) is ( 4 . ) Find its z-coordinate.
A . -1
B. -2
( c .1 )
D.
12
605The Cartesian equation of the plane ( overrightarrow{boldsymbol{r}}=(1+boldsymbol{lambda}-boldsymbol{mu}) hat{boldsymbol{i}}+(boldsymbol{2}-boldsymbol{lambda}) hat{boldsymbol{j}}+(boldsymbol{3}- )
( 2 lambda+2 mu) hat{k} ) is-
( mathbf{A} cdot 2 x+y=5 )
В . ( 2 x-y=5 )
c. ( 2 x+z=5 )
D. ( 2 x-z=5 )
12
606The projection of the join of the points (3,4,2),(5,1,8) on the line whose d.c’s ( operatorname{are}left(frac{2}{7}, frac{3}{7}, frac{6}{7}right) ) is
A. 7
B. ( frac{31}{71} )
( mathbf{c} cdot frac{42}{13} )
( D cdot frac{38}{138} )
12
607Find the value of ( p ) for which the points ( (-5,1),(1, p) ) and (4,-2) are collinear
( mathbf{A} cdot mathbf{1} )
B.
( c .-1 )
D. 2
12
608Find the shortest distance between the ( operatorname{lines} bar{r}=4 bar{i}-bar{j}+lambda(bar{i}+2 bar{j}-5 bar{k}) ) and
( overline{boldsymbol{r}}=overline{boldsymbol{i}}-overline{boldsymbol{j}}+2 overline{boldsymbol{k}}+boldsymbol{mu}(overline{boldsymbol{i}}+mathbf{2} overline{boldsymbol{j}}-mathbf{5} overline{boldsymbol{k}}) )
A. ( sqrt{220} )
B. ( frac{sqrt{221}}{sqrt{30}} )
c. 432
D. ( sqrt{33} )
12
609For what value of ( boldsymbol{m}, ) the points ( (boldsymbol{3}, mathbf{5}) ) ( (m, 6) ) and ( left(frac{1}{2}, frac{15}{2}right) ) are collinear?
( A cdot 9 )
B. 5
( c .3 )
D.
12
610The plane ( 2 x+3 y+k z-7=0 ) is
parallel to the line whose direction
ratios are (2,-3,1) then ( k= )
( mathbf{A} cdot mathbf{5} )
B. 8
c. 1
( D )
12
611Image of point ( mathrm{P}(1,2,3) ) with respect to plane ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=mathbf{1 2}, ) is
A. (5,4,3)
)
B. (9,6,3)
c. (5,6,7)
D. (3,4,5)
12
612The point of intersection of the lines ( overrightarrow{boldsymbol{r}}=(-overrightarrow{boldsymbol{i}}+mathbf{2} overrightarrow{boldsymbol{j}}+boldsymbol{3} overrightarrow{boldsymbol{k}})+boldsymbol{t}(-boldsymbol{2} overrightarrow{boldsymbol{i}}+overrightarrow{boldsymbol{j}}+overrightarrow{boldsymbol{k}}) )
and ( vec{r}=(2 vec{i}+3 vec{j}+5 vec{k})+s(vec{i}+2 vec{j}+ )
( mathbf{3} overrightarrow{boldsymbol{k}}) ) is:
A ( .(1,1,2) )
в. (2,1,1)
c. (1,1,1)
D. (1,2,1)
12
613The equation of the plane which is parallel to the ( x y- ) plane is
A. ( x=y )
B. ( z=c )
( mathbf{c} cdot y=c )
D. ( z=x y )
12
614The sum of the intercepts on the coordinate axes of the plane passing
through the point (-2,-2,2) and
containing the line joining the points
(1,-1,2) and ( (1,1,1), ) is
A . 4
B. -4
c. 12
D. -8
12
615Determine if the points (5,-1,1),(7,-4,7),(1,-8,10) and
(-1,-3,4) are the vertices of a rhombus or a square
12
616Assertion ( (A) ). The direction ratios of the line joining origin and point ( (x, y, z) ) must be ( x, y, z )

Reason (R): If ( P(x, y, z) ) is a point in space and ( |O P|=r, ) then the direction cosines of ( O P ) are ( frac{x}{r}, frac{y}{r}, frac{z}{r} )
A. Both A and R are individually true and R is the correct explanation of
B. Both A and R individually true but R is not the correct explanation of A
c. A is true but R is false
D. A is false but R is true

12
617A line separates a plane into three parts namely the two half-planes and the line itself
A. True
B. False
12
618If the line passing through the origin
makes angles ( theta_{1}, theta_{2}, theta_{3} ) with the planes
( X O Y, X O Z ) and ( Z O X ) respectively
then prove that ( cos ^{2} theta_{1}+cos ^{2} theta_{2}+ )
( cos ^{2} theta_{3}=2 )
12
619A triangle ( A B C ) is placed so that the
mid-points of the sides are on the ( x, y, z )
axes. Lengths of the intercepts made by the plane containing the triangle on
these axes are respectively ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} )
Coordinates of the centroid of the
triangle ( boldsymbol{A B C} ) are
A ( cdot(-alpha / 3, beta / 3, gamma / 3) )
B. ( (alpha / 3,-beta / 3, gamma / 3) )
c. ( (alpha / 3, beta / 3,-gamma / 3) )
D. ( (alpha / 3, beta / 3, gamma / 3) )
12
620The point on the line ( frac{boldsymbol{x}-mathbf{1}}{mathbf{1}}=frac{boldsymbol{y}+mathbf{3}}{-mathbf{2}}= )
( frac{z+5}{-2} ) at a distance of 6 from the point
(1,-3,-5) is
в. (3,-7,-9)
D. (-3,5,3)
12
621Find the direction cosines of the vector
joining the points ( A(1,2,-3) ) and ( B(-1,-2,1) ) directed from ( A ) to ( B )
12
622The ordinate of the point which divides the lines joining the origin and the point (1,2) externally in the ratio of 3: 2 is
A . -2
B. ( frac{3}{5} )
( c cdot frac{2}{5} )
D. 6
12
623A tangent to the curve ( y=f(x) ) at
( boldsymbol{p}(boldsymbol{x}, boldsymbol{y}) ) meets ( boldsymbol{x}-boldsymbol{a} boldsymbol{x} boldsymbol{i} boldsymbol{s} ) at ( boldsymbol{A} ) and ( boldsymbol{y}- )
axis at ( B . ) If ( overline{A P}: overline{B P}=1: 3 ) and
( f(1)=1 ) then the curve also passes
through the point.
( ^{mathbf{A}} cdotleft(frac{1}{2}, 4right) )
B ( cdotleft(frac{1}{3}, 24right) )
c. ( left(2, frac{1}{8}right) )
D. ( left(3, frac{1}{28}right) )
12
624The equation of the plane passing through ( (a, b, c) ) and parallel to the plane ( r cdot(hat{i}+hat{j}+hat{k})=2 ) is
A. ( x+y+z=1 )
B . ( a x+b y+c z=1 )
c. ( x+y+z=a+b+c )
D. None of these
12
625If a unit vector ( vec{a} ) makes angles ( frac{pi}{3} ) with ( hat{i} )
( frac{pi}{4} ) with ( hat{j} ) and an acute angle ( theta ) with ( hat{k} )
then find ( theta ) and hence, the components of ( overrightarrow{boldsymbol{a}} )
12
626The three planes divides the space into
A. four parts
B. six parts
c. eight parts
D. sixteen parts
12
627Assertion
The points (1,1)( &(-1,-1) ) lie on the
same side of the line ( boldsymbol{x}-boldsymbol{y}+mathbf{1}=mathbf{0} )
Reason
The algebraic perpendicular distances from the given points to the line have
same sign
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
6284.
A vector Ă has components A,, A2, A, in a right-handed
rectangular Cartesian coordinate system oxyz. The
coordinate system is rotated about the x-axis through an
T
angle =. Find the components of A in the new coordinate
system, in terms of A,, A2, Az.
(1983 – 2 Marks)
12
629The three point ( boldsymbol{A}(mathbf{0}, mathbf{0}, mathbf{0}), boldsymbol{B}(mathbf{2},-mathbf{3}, mathbf{3}), boldsymbol{C}(-mathbf{2}, mathbf{3},-mathbf{3}) ) are
collinear. Find in what ratio each point
divides the segment joining other two
12
630The ratio in which the join of (1,-2,4)
and (4,2,-1) divided by the ( X-Y )
plane is
A . 1: 3
B. 3: 1
c. 4: 1
D. 1: 4
12
631The graph of the equation ( y^{2}+z^{2}=0 )
in three dimensional space is
A. x- axis
B. y- axis
c. z- axis
D. yz-plane
12
632If ( frac{1}{2}, frac{1}{2}, n(n<0) ) are the dos of a line,
then the angle made by that line with ( boldsymbol{O} boldsymbol{Z} ) where ( boldsymbol{O}=(mathbf{0}, mathbf{0}, mathbf{0}) ) is
A ( cdot frac{-1}{sqrt{2}} )
B . ( 45^{circ} )
( c cdot 60^{circ} )
D. ( 135^{circ} )
12
633Find the co-ordinates of the points on the join of (-3,7,-13) and (-6,1,-10) which is nearest to the intersection of the planes: ( 3 x-y- )
( mathbf{3} z+mathbf{3 2}=mathbf{0} ) and ( mathbf{3} boldsymbol{x}+mathbf{2} boldsymbol{y}-mathbf{1 5 z}-mathbf{8}= )
0
12
634The equation of plane passing through (-1,0,-1) parallel to ( x z ) plane is
В. ( y=0 )
c. ( -x-z=0 )
D. None of the above
12
635The condition that the line ( frac{x-alpha^{prime}}{l}= ) ( frac{boldsymbol{y}-boldsymbol{beta}^{prime}}{boldsymbol{m}}=frac{boldsymbol{z}-boldsymbol{gamma}^{prime}}{boldsymbol{n}} ) in the plane ( boldsymbol{A} boldsymbol{x}+ )
( boldsymbol{B} boldsymbol{y}+boldsymbol{C} boldsymbol{z}+boldsymbol{D}=mathbf{0} ) is
( mathbf{A} cdot A alpha^{prime}+B beta^{prime}+C gamma^{prime}+D=0 ) and ( A l+B m+C n neq 0 )
B ( cdot A alpha^{prime}+B beta^{prime}+C gamma^{prime}+D neq 0 ) and ( A l+B m+C n=0 )
C ( cdot A alpha^{prime}+B beta^{prime}+C gamma^{prime}+D=0 ) and ( A l+B m+C n=0 )
D ( cdot A alpha^{prime}+B beta^{prime}+C gamma^{prime}=0 ) and ( A l+B m+C n=0 )
12
636A rectangular parallelopiped is formed by drawing planes through the point (-1,2,5) and (1,-1,-1) and paralle to the coordinates planes. The length of the diagonal of the parallelopiped is12
63782.
The distance of the point (1, -5,9) from the plane x-y+z=5
measured along the line x=y=zis: [JEE M 2016]
10
a)
TT
(b)
(c) 3/10
(d) 103
12
638If the points (-1,3,2),(-4,2,-2) and ( (5,5, lambda) ) are collinear, then ( lambda ) is equal to
A . -10
B. 5
( c .-5 )
D. 10
12
639The equations of the line of intersection
of the planes ( x+y+z=2 ) and ( 3 x- )
( boldsymbol{y}+mathbf{2} z=mathbf{5} ) in symmetric form are
A ( frac{x-frac{7}{4}}{4}=frac{y-frac{1}{4}}{-1}=frac{z}{-3} )
c. ( frac{x}{1}=frac{3 y+1}{1}=frac{3 z-7}{-4} )
D. none of these
12
64023.
(1)
Find
Find the equation of the plane passing through the
points (2,1,0), (5,0, 1) and (4,1,1).
If P is the point (2,1, 6) then find the point Q such that
PQ is perpendicular to the plane in (i) and the mid point
of PQ lies on it.
(2003 – 4 Marks)
12
641Find the angle between the following pair of lines:
(i) ( vec{r}=2 hat{i}-5 hat{j}+hat{k}+lambda(3 hat{i}-2 hat{j}+6 hat{k}) )
and ( vec{r}=mathbf{7} hat{boldsymbol{i}}-boldsymbol{6} hat{boldsymbol{k}}+boldsymbol{mu}(hat{boldsymbol{i}}+mathbf{2} hat{boldsymbol{j}}+mathbf{2} hat{boldsymbol{k}}) )
(ii) ( vec{r}=3 hat{i}+hat{j}-2 hat{k}+lambda(hat{i}-hat{j}-2 hat{k}) )
and ( vec{r}=2 hat{i}-hat{j}-56 hat{k}+ )
( mu(hat{mathbf{3}} mathbf{i}-mathbf{5} hat{mathbf{j}}-mathbf{4} hat{boldsymbol{k}}) )
12
6422.
Let : 2x + y – z = 3 and P2 : x +2y +z = 2 be two planes.
Then, which of the following statement(s) is (are) TRUE?
(a) The line of intersection of P, and P, has direction
ratios 1, 2, -1
3x -4 1-3yZ
(b) The line . -= =
is perpendicular to the line of intersection of P, and
P2
c)
(d)
The acute angle between P, and P, is 60°.
If P, is the plane passing through the point (4, 2, -2)
and perpendicular to the line of intersection of P, and
P, then the distance of the point (2, 1, 1) from the
plane Pz is 7
12
64314
The equation of a plane passing through the line of
intersection of the planes x + 2y + 3z=2 and x-y+z=3 and
at a distance
from the point (3,1,-1) is
(2012)
(a) 5x-1ly+z=17
(c) x+y+z= 13
(b) V2x+y = 3/2 – 1
(d) x-V2y =1-2
12
644Find the vector equation of the line joining (1,2,3) and (-3,4,3) and show pependicular to the z-axis12
645In the given figure, co-ordinates of the
midpoint of ( boldsymbol{A B} ) are
A . (0,2)
в. (0,3)
c. (1,2)
D. (3,1)
12
646The centroid of triangle ( boldsymbol{A}(mathbf{3}, mathbf{4}, mathbf{5}) ; boldsymbol{B}(mathbf{6}, mathbf{7}, mathbf{2}) ; boldsymbol{C}(mathbf{0},-mathbf{5}, mathbf{2}) ) is
A ( .(3,2,3) )
в. (5,2,1)
c. (2,5,1)
D. (3,4,1)
12
647Find the coordinates of point which
divides the line joining the points (3,4) and (6,1) in the ratio of 1: 2
12
648( bar{a}, bar{b}, bar{c} ) are three non-zero vectors such that any two of them are non-collinear. If ( bar{a}+bar{b} ) is collinear with ( bar{c} ) and ( bar{b}+bar{c} ) is
collinear with ( bar{a} ), then what is their sum?
A . -1
B. 0
c. 1
D. 2
12
649If a line makes angles ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} ) with
coordinate axes, find
( cos ^{2} alpha+cos ^{2} beta+cos ^{2} gamma+1 )
12
650The straight lines ( frac{boldsymbol{x}-mathbf{1}}{mathbf{1}}=frac{boldsymbol{y}-mathbf{2}}{mathbf{2}}= )
( frac{z-3}{3} ) and ( frac{x-1}{1}=frac{y-2}{2}=frac{z-3}{3} ) are
A. Parallel lines
B. Intersecting at ( 60^{circ} )
c. skew lines
D. Intersecting at right angle
12
651Find the angles between the lines,
whose direction cosines are give by the equation ( l^{2}-m^{2}+n^{2}=0, l+m+ )
( boldsymbol{n}=mathbf{0} )
A. 0
в. ( frac{pi}{6} )
( c cdot frac{pi}{4} )
D.
12
652Find the equation of the line passing through (1,2,-4) and perpendicular to both the lines ( frac{x-1}{2}=frac{y+2}{-3}=frac{z-4}{4} )
and ( frac{boldsymbol{x}-mathbf{3}}{mathbf{5}}=frac{boldsymbol{y}+mathbf{6}}{mathbf{1}}=frac{boldsymbol{z}+mathbf{1 0}}{mathbf{2}} )
12
653If the ( z x ) -plane divides the line segment joining (1,-1,5) and (2,3,4) in the ratio ( p: 1, ) then ( p+1= )
A ( cdot frac{1}{3} )
B. 1: 3
( c cdot frac{3}{4} )
D. ( frac{4}{3} )
12
654The image of the point (-1,3,4) in the
plane ( boldsymbol{x}-mathbf{2} boldsymbol{y}=mathbf{0} ) is
( ^{mathbf{A}} cdotleft(-frac{17}{3},-frac{19}{3}, 4right) )
В. (15,11,4)
( ^{mathbf{C}} cdotleft(-frac{17}{3},-frac{19}{3}, 1right) )
D. ( left(frac{9}{5},-frac{13}{5}, 4right) )
12
65575. The angle between the lines whose direction cosines satisfy
the equations 1+m+n= 0 and 12 = m² +nis
[JEE M 2014
wa
12
656f a plane passes through the point (1,1,1) and is perpendicular to the line ( frac{x-1}{3}=frac{y-1}{0}=frac{z-1}{4} ) then its
perpendicular distance from the origin is
A ( cdot frac{3}{4} )
B. ( frac{4}{3} )
( c cdot frac{7}{5} )
D.
12
657If the lines ( frac{x-1}{2}=frac{y+1}{3}=frac{z-1}{4} ) and
( frac{boldsymbol{x}-mathbf{3}}{mathbf{1}}=frac{boldsymbol{y}-boldsymbol{k}}{mathbf{2}}=frac{boldsymbol{z}}{mathbf{1}} ) intersect, then ( boldsymbol{k}= )
A. 0
B. 3
( c cdot frac{7}{2} )
D. –
12
658If the vertices of a triangle are (-1,6,-4),(2,1,1) and (5,-1,0) then
the centroid of the triangle is
В. (2,2,-1)
( ^{mathbf{c}} cdotleft(3,3,-frac{3}{2}right) )
D. none of these
12
659An equation of sphere with centre at
origin and radius ( r ) can be represented
as
A ( cdot x^{2}+y^{2}+z^{2}=r )
B . ( x^{2}+y^{2}+z^{2}=r^{2} )
c. ( x^{2}+y^{2}+z^{2}=2 r^{2} )
D. None of the above
12
660Obtain the equation of the line passing through (1,1,2) and (2,1,2) in the vector form.12
661The reflection of the plane ( 2 x+3 y+ )
( 4 z-3=0 ) in the plane ( x-y+z- )
( mathbf{3}=mathbf{0} ) is the plane
A. ( 4 x-3 y+2 z-15=0 )
B. ( x-3 y+2 z-15=0 )
c. ( 4 x+3 y-2 z+15=0 )
D. none of these
12
662A normal to the plane ( x=2 ) is…
A ( .(0,1,1) )
в. (2,0,2)
c. (1,0,0)
D. (0,1,0)
12
663An ordered triplet corresponds to in three dimensional
space.
A. three points
B. a unique point
c. a point in each octant
D. infinite number of points
12
664A point at a distance of ( sqrt{6} ) from the origin which lies on the straight line ( frac{x-1}{1}=frac{y-2}{2}=frac{z+1}{3} ) will be
( mathbf{A} cdot(1,-1,2) )
В. (1,2,-1)
C ( cdotleft(frac{5}{7}, frac{10}{7}, frac{-13}{7}right) )
D. ( left(frac{5}{7}, frac{2}{7}, frac{-6}{7}right) )
12
665If ( (3, lambda, mu) ) is a point on the line then
( 2 x+y+z=0=x-2 y+z-1 ) then
A ( cdot lambda=frac{-8}{3}, mu=-frac{1}{3} )
B. ( lambda=frac{-1}{3}, mu=-frac{8}{3} )
c. ( lambda=frac{-4}{3} mu=frac{-14}{3} )
D. ( lambda=-5, mu=-1 )
12
666Find the point of intersection of the plane ( bar{r} .(1,1,1)=2 ) and the line ( bar{r}= ) ( (4,5,3)+k(2,2,1), k in R )12
667The equation of the plane which is equidistant from the two parallel planes ( 2 x-2 y+z+3=0 ) and ( 4 x-4 y+ )
( mathbf{2} z+mathbf{9}=mathbf{0} ) is :
A. ( 8 x-8 y+2 z+15=0 )
B. ( 8 x-8 y+4 z+15=0 )
c. ( 8 x-8 y+4 z+3=0 )
D. ( 8 x-8 y+4 z-3=0 )
E ( .8 x-8 y+4 z+4=0 )
12
668The distance between the line ( r=2 hat{i}- ) ( 2 hat{j}+3 hat{k}+lambda(hat{i}-hat{j}+4 hat{k}) ) and the plane
( r cdot(hat{i}+5 hat{j}+hat{k})=5 ) is
A ( cdot frac{10}{9} )
в. ( frac{10}{3 sqrt{3}} )
c. ( frac{10}{3} )
D. None of these
12
669If the distance between a point ( P ) and
the point (1,1,1) on the line ( frac{x-1}{3}= )
( frac{y-1}{4}=frac{z-1}{12} ) is ( 13, ) then the coordinates
of ( P ) are
A. (3,4,12)
В. ( left(frac{3}{13}, frac{4}{13}, frac{12}{13}right) )
c. (4,5,12)
D. (40, 53, 157)
12
670The equation of the plane through the
intersection of ( p_{1} & p_{2} ) containing the
point (1,1,2) is
A. ( 5 x-6 y+4 z=4 )
в. ( 5 x+6 y-4 z=3 )
c. ( 3 x-2 y+4 z=9 )
D. Nonoe of these
12
671If a plane passes through a fixed point
(2,3,4) and meets the axes of reference
in ( A, B ) and ( C, ) the point of intersection
of the planes through ( A, B, C ) parallel to the coordinate planes can be
This question has multiple correct options
A ( cdot(6,9,12) )
B ( cdot(4,12,16) )
( mathbf{c} cdot(1,1,-1) )
D. (2,3,-4)
12
672( boldsymbol{L}_{1}: frac{boldsymbol{x}-mathbf{1}}{mathbf{2}}=frac{boldsymbol{y}-mathbf{2}}{mathbf{3}}=frac{boldsymbol{z}-mathbf{3}}{mathbf{4}} )
( L_{2}: frac{x-2}{3}=frac{y-4}{2}=frac{z-5}{5} ) be two
given lines, point P lies on ( L_{1} ) and Q lies
on ( L_{2} ) then distance between ( P ) and ( Q )
can be
This question has multiple correct options
A ( cdot frac{1}{3} )
B.
c. 15
D. 30
12
673Let ( X ) and ( Y ) be two related variables.
The two regression lines are given by ( boldsymbol{x}-boldsymbol{y}+mathbf{1}=mathbf{0} ) and ( mathbf{2} boldsymbol{x}-boldsymbol{y}+mathbf{4}=mathbf{0} . ) The
two regression lines pass through the point:
A ( cdot(-4,-3) )
в. (-6,-5)
c. (3,-2)
D. (-3,-2)
12
674The values of a for which
( (8,-7, a),(5,2,4) ) and (6,-1,2) are
collinear, is given by?
A .2
B. -2
c. -1
D.
12
675If ( z_{1} ) and ( z_{2} ) are ( z ) co-ordinates of the
points of trisection of the segment joining the points ( boldsymbol{A}(mathbf{2}, mathbf{1}, mathbf{4}), boldsymbol{B}(-mathbf{1}, mathbf{3}, mathbf{6}) )
then ( z_{1}+z_{2}= )
A . 1
B. 4
( c .5 )
D. 10
12
676If ( boldsymbol{P}=(mathbf{0}, mathbf{0}, mathbf{0}), boldsymbol{Q}=(mathbf{3}, mathbf{6}, mathbf{9}) ) and ( boldsymbol{R} ) is a
point of trisection of ( boldsymbol{P Q}, ) then ( boldsymbol{R}_{boldsymbol{y}}= )
( A cdot frac{4}{3} )
B . 2
( c .3 )
D.
12
677Find the image of (1,5,1) in the plane
( boldsymbol{x}-mathbf{2} boldsymbol{y}+boldsymbol{z}+mathbf{5}=mathbf{0} )
12
678If the centroid of tetrahedron ( O A B C )
where ( A, B, C ) are given by ( (a, 2,3),(1, b, 2) ) and ( (2,1, c) )
respectively is ( (1,2,-2), ) then distance of ( boldsymbol{P}(boldsymbol{a}, boldsymbol{b}, boldsymbol{c}) ) from origin is
( mathbf{A} cdot sqrt{195} )
в. ( sqrt{14} )
c. ( sqrt{frac{107}{14}} )
D. ( sqrt{13} )
12
6791.
From a point O inside a triangle ABC, perpendiculars OD,
OE, OF are drawn to the sides BC, CA, AB respectively.
Prove that the perpendiculars from A, B, C to the sides EF,
FD, DE are concurrent.
(1978)
hone
12
680A parallelepiped is formed by planes
drawn through the point ( P(6,8,10) ) and
( Q(3,4,8) ) parallel to the coordinate
planes. Find the length of edges and edges and diagonals of the
parallelepiped.
12
681The angle between the lines whose
direction cosines satisfy the equations ( l+m+n=0 ) and ( l^{2}+m^{2}+n^{2} ) is
A ( cdot frac{pi}{2} )
в.
c.
D. ( frac{pi}{6} )
12
682The line passing through the points ( 10 hat{i}+3 hat{j}, 12 hat{i}+5 hat{j} ) also passes through
the point ( a hat{i}+11 hat{j}, ) then ( a= )
A . -8
B. 4
c. 18
D. 12
12
683Direction ratio of line given by ( frac{x-1}{3}= ) ( frac{6-2 y}{10}=frac{1-z}{-7} ) are:
( A cdot )
в. ( )
c. ( )
D. ( )
12
684Point ( D ) has coordinates as (3,4,5)
Referring to the given figure, find the
coordinates of point ( boldsymbol{B} )
A ( cdot(3,0,4) )
( mathbf{B} cdot(4,3,0) )
C. (4,0,3)
D. (3,4,0)
12
685Which of the following are equations for
the plane passing through the points
( P(1,1,-1), Q(3,0,2) ) and ( R(-2,1,0) ? )
A ( cdot(2 hat{i}-3 hat{j}+3 hat{k}) cdot((x+2) hat{i}+(y-1) hat{j}+z hat{k})=0 )
B . ( x=3-t, y=-11 t, z=2-3 t )
( mathbf{c} cdot(x+2)+11(y-1)=3 x )
D. ( (2 hat{i}-hat{j}+3 hat{k}) times(-3 hat{i}+hat{k}) cdot((x+2) hat{i}+(y-1) hat{j}+z hat{k})= )
0
12
686The direction ratios of a vector are
( 2,-3,4 . ) Find its direction cosines
12
687If ( A=5 ) units, ( B=6 ) units and ( |vec{A} times vec{B}|= ) 15 units, then the angle between ( vec{A} ) and ( vec{B} )
is:
This question has multiple correct options
( A cdot 30^{circ} )
B. ( 60^{circ} )
c. ( 90^{circ} )
D. ( 150^{circ} )
12
688Find the shortest distance between the
skew lines:
( l_{1}: frac{x-1}{2}=frac{y+1}{1}=frac{z-2}{4} )
( l_{2}: frac{x+2}{4}=frac{y-0}{-3}=frac{z+1}{1} )
12
689Consider the plane ( (boldsymbol{x}, boldsymbol{y}, boldsymbol{z})= )
( (0,1,1)+lambda(1,-1,1)+mu(2,-1,0) . ) The
distance of this plane from the origin is:
A ( cdot frac{1}{3} )
B. ( frac{sqrt{3}}{2} )
( c cdot sqrt{frac{3}{2}} )
D. ( frac{2}{sqrt{3}} )
12
69071. Distance between two parallel planes 2x + y + 2z = 8 and
4x + 2y + 4z +5=0 is
[JEE M 2013]
12
691Vector equation of line ( frac{mathbf{3}-boldsymbol{x}}{mathbf{3}}= )
( frac{2 y-3}{5}=frac{z}{2} ) is ( _{–}——-k in R )
A ( cdot bar{r}=(3,5,2)+k(3,3,0) )
B ( cdot quad bar{r}=left(3, frac{3}{2}, 0right)+k(-6,5,4) )
c. ( bar{r}=(3,3,0)+k(3,5,2) )
D ( cdot_{bar{r}}=(-6,5,4)+kleft(3, frac{3}{2}, 0right) )
12
692If ( (1,1, a) ) is the centroid of the triangle formed by the points ( (1,2,-3),(mathrm{b}, 0,1) ) and (-1,1,-4) then ( a-b= )
A . -5
B. -7
( c .5 )
( D )
12
693Show that the lines ( frac{x-1}{2}=frac{y-2}{3}= ) ( frac{z-3}{4} ) and ( 4 x-3 y+1=0=5 x- )
( 3 z+2 ) are interesting lines. Also find
point of intersection.
12
69488. If L, is the line of intersection of the planes
2x-2y+3z-2=0, x-y+z+1=0 and L, is the line of
intersection of the planes x+2y-z-3=0,
3x-y+2z-1=0, then the distance of the origin from the
plane, containing the lines L, and L,, is: [JEEM 2018]
(a) ZNZ
(6) 212
12
695What is the direction cosine of angle which the vector ( sqrt{2 hat{i}}+hat{j}+ )
( hat{k} ) makes with ( y- ) axis ( ? )
12
6962. If a vector P making angles a, b, and yrespectively with
the X, Y and Z axes respectively.
Then sin? a + sin2 B + sin2 y=
(a) (b) 1 (c) 2 (d) 3
12
697Point ( D ) has coordinates as (3,4,5)
Find the coordinates of the point ( boldsymbol{F} )
( mathbf{A} cdot(0,4,0) )
B. (0,0,4)
( mathbf{C} cdot(0,0,5) )
( mathbf{D} cdot(0,5,0) )
12
698If ( overline{O A}=3 bar{i}+bar{j}-bar{k},|overline{A B}|=2 sqrt{6} ) and ( A B )
has the direction ratios 1,-1,2 then
( |boldsymbol{O} boldsymbol{B}|= )
A . ( sqrt{35} )
B. ( sqrt{41} )
c. ( sqrt{26} )
D. ( sqrt{55} )
12
699Find the equation of the plane through the points ( boldsymbol{A}(mathbf{2}, mathbf{2}-mathbf{1}), boldsymbol{B}(mathbf{3}, mathbf{4}, mathbf{2}) ) and
( boldsymbol{C}(boldsymbol{7}, boldsymbol{0}, boldsymbol{6}) )
A. ( 5 x+2 y-3 z=17 )
B. ( 5 x+2 y+3 z=17 )
c. ( 5 x+y-3 z=7 )
D. ( 5 x+y+3 z=7 )
12
700Show that the three lines with direction
( operatorname{cosines} )
( frac{12}{13}, frac{-3}{13}, frac{-4}{13}: frac{4}{13}, frac{12}{13}, frac{3}{13} ; frac{-4}{13}, frac{12}{13} ) are mutually
perpendicular
12
701If the projection of point ( boldsymbol{P}(overrightarrow{boldsymbol{p}}) ) on the
plane ( vec{r} cdot vec{n}=q ) is the point ( S(vec{s}), ) then?
A ( cdot vec{s}=frac{(q-vec{p} cdot vec{n}) vec{n}}{|vec{n}|^{2}} )
В ( cdot vec{s}=vec{p}+frac{(vec{p} cdot vec{n}-q)}{|vec{n}|^{2}} vec{n} )
C・ ( _{vec{s}}=vec{p}-frac{(vec{p} cdot vec{n}) vec{n}}{|vec{n}|^{2}} )
D ( vec{s}=vec{p}-frac{(q-vec{p} cdot vec{n}) vec{n}}{|vec{n}|^{2}} )
12
702Two systems of rectangular axes have
the same origin. If a plane cuts them at
distance ( a, b, c ) and ( d, b^{prime}, c^{prime} ) from the
origin, then
A ( cdot frac{1}{a^{2}}-frac{1}{b^{2}}-frac{1}{c^{2}}-frac{1}{a^{2}}-frac{1}{b^{2}}-frac{1}{c^{2}}=0 )
B ( cdot frac{1}{a^{2}}-frac{1}{b^{2}}-frac{1}{c^{2}}-frac{1}{a^{2}}-frac{1}{b^{2}}+frac{1}{c^{2}}=0 )
c. ( frac{1}{a^{2}}+frac{1}{b^{2}}+frac{1}{c^{2}}-frac{1}{a^{2}}-frac{1}{b^{2}}-frac{1}{c^{2}}=0 )
D ( frac{1}{a^{2}}+frac{1}{b^{2}}+frac{1}{c^{2}}+frac{1}{a^{2}}+frac{1}{b^{2}}+frac{1}{c^{2}}=0 )
12
703Find the equation of the line joining the points (-1,3) and (4,-2)12
704If the position vectors of the points ( boldsymbol{A}, boldsymbol{B} ) and ( C ) be ( i+j, i-j ) and ( a i+b j+c k )
respective;y, then the points ( A, B ) and
( C ) are collinear if:
A ( . a=b=c=1 )
B. ( a=1, b ) and ( c ) are arbitrary scalars
C. ( a=b=c=0 )
0
12
705A point ( R ) with ( x ) -coordinate 4 lies on the line segment joining the points
( P(2,-3,4) ) and ( Q(8,0,10) . ) Find the
coordinates of the point ( boldsymbol{R} )
12
706Fill in the blanks:
(i) The ( x ) -axis and ( y ) -axis taken together
determine a plane known as
(ii) The coordinates of points in the ( boldsymbol{X} boldsymbol{Y} )
-plane are of the form
(iii) Coordinate planes divide the space
into octants
12
707The Cartesian equation of a line is
( frac{boldsymbol{x}-mathbf{5}}{mathbf{3}}=frac{boldsymbol{y}+mathbf{4}}{mathbf{7}}=frac{boldsymbol{z}-mathbf{6}}{mathbf{2}} . ) Write its vector
form.
12
708The direction cosines of the lines
bisecting the internal angle ( theta ) between the lines whose direction cosines are
( l_{1}, m_{1}, n_{1} ) and ( l_{2}, m_{2}, n_{2} ) are
( mathbf{A} cdot )
B. ( )
c. ( )
D. none of these
12
709Find the distance of the point (-6,0,0)
from the plane ( 2 x-3 y+6 z=2 )
12
710Let ( boldsymbol{A}=(mathbf{1}, mathbf{2}, mathbf{2}), boldsymbol{B}=(mathbf{2}, mathbf{3}, mathbf{6}) ) and ( boldsymbol{C}= )
( (3,4,12) . ) The direction cosines of a line
equally inclined with ( O A, O B ) and ( O C )
where ( boldsymbol{O} ) is the origin, are
A ( cdot frac{1}{sqrt{2}}, frac{-1}{sqrt{2}}, 0 )
B. ( frac{1}{sqrt{2}}, frac{1}{sqrt{2}}, 0 )
c. ( frac{1}{sqrt{3}}, frac{-1}{sqrt{3}}, frac{1}{sqrt{3}} )
D. ( frac{1}{sqrt{3}}, frac{-1}{sqrt{3}}, frac{-1}{sqrt{3}} )
12
711Find the coordinates of the point on the ( x ) -axis that is equidistant from
( boldsymbol{P}(mathbf{4}, mathbf{3}, mathbf{1}) ) and ( boldsymbol{Q}(-mathbf{2},-mathbf{6},-mathbf{2}) )
A ( cdotleft(frac{3}{2}, 0,0right) )
B ( cdotleft(-frac{3}{2}, 0,0right) )
c. ( left(0,-frac{3}{2}, 0right) )
D. ( left(0, frac{3}{2}, 0right) )
12
712In the triangle with vertices ( boldsymbol{A}(mathbf{1},-mathbf{1}, mathbf{2}), boldsymbol{B}(mathbf{5},-mathbf{6}, mathbf{2}) ) and ( boldsymbol{C}(mathbf{1}, mathbf{3},-mathbf{1}) )
find the altitude ( n=|B D| )
( A cdot 5 )
B. 10
c. ( 5 sqrt{2} )
D. ( frac{10}{sqrt{2}} )
12
713If the origin is the centroid of the
triangle ( P Q R ) with vertices ( boldsymbol{P}(mathbf{2} boldsymbol{a}, mathbf{2}, mathbf{6}), boldsymbol{Q}(-mathbf{4}, mathbf{3} boldsymbol{b},-mathbf{1 0}) ) and
( boldsymbol{R}(mathbf{8}, mathbf{1 4}, mathbf{2 c}), ) then find the values of ( boldsymbol{a}, boldsymbol{b} )
and ( c )
12
714The points with the co-ordinates
( (2 a, 3 a),(3 b, 2 b) &(c, c) ) are collinear.
12
715Using section formula show that the
points ( boldsymbol{A}(mathbf{2},-mathbf{3}, mathbf{4}), boldsymbol{B}(-mathbf{1}, mathbf{2}, mathbf{1}) ) and
( Cleft(0, frac{1}{3}, 2right) ) are collinear.
12
716If the line joining ( boldsymbol{A}(mathbf{1}, mathbf{3}, mathbf{4}) ) and ( boldsymbol{B} ) is divided by the point (-2,3,5) in the ratio ( 1: 3, ) then ( B ) is
A. (-11,3,8)
(年) (-1,3,8)
в. (-11,3,-8)
D. (13,6,-13)
12
71716. In R}, consider the planes P, :y=0 and P2: x+z=1. Let P
be the plane, different from P, and P2, which passes through
the intersection of P, and P,. If the distance of the point
(0, 1, 0) from P, is 1 and the distance of a point (a, b, y) from
Pz is 2, then which of the following relations is (are) true
(a) 2a+3+2y+2=0
(c) 2a+B-2y-10=0
(b) 20-3+2y+4=0
(d) 2a-B+2y-8=0
12
718Find the distance between the points
(3,4,-2),(1,0,7)
12
719W
J
84. If the image of the point P(1, -2, 3) in the plane,
X
Y
Z
2x + 3y– 4z + 22=0 measured parallel to line, = = is
Q, then PQ is equal to :
JJEE M 2017]
(a) 615
(b) 315
(c) 2742
(d) 142
12
720The points ( boldsymbol{A}(mathbf{1}, mathbf{2},-mathbf{1}), boldsymbol{B}(mathbf{2}, mathbf{5},-mathbf{2}), boldsymbol{C}(mathbf{4}, mathbf{4},-mathbf{3}) )
and ( D(3,1,-2) ) are
A. collinear
B. vertices of a rectangle
c. vertices of a square
D. vertices of a rhombus
12
721Find the directions cosines of ( x, y ) and ( z )
axis.
12
722The perpendicular distance from the
point (3,1,1) on the plane passing through the point (1,2,3) and containing the line, ( vec{r}=hat{i}+hat{j}+ ) ( lambda(2 hat{i}+hat{j}+4 hat{k}), ) is:
( ^{A} cdot frac{1}{sqrt{11}} )
в. ( frac{4}{sqrt{41}} )
( c .0 )
D. ( frac{3}{sqrt{11}} )
12
723If ( (0, b, 0) ) is the centroid of the triangle
formed by the points (4,2,-3) ( (a,-5,1) ) and ( (2,-6,2) . ) If ( a, b ) are the
roots of the quadratic equation ( x^{2}+ )
( boldsymbol{p} boldsymbol{x}+boldsymbol{q}=mathbf{0}, ) then ( boldsymbol{p}, boldsymbol{q} ) are
A .9,18
в. -9,-18
c. 3,-18
D. -3,18
12
724For two vectors ( overrightarrow{boldsymbol{A}} ) and ( overrightarrow{boldsymbol{B}}, overrightarrow{boldsymbol{A}}+overrightarrow{boldsymbol{B}}=overrightarrow{boldsymbol{C}} )
and ( |vec{A}|+|vec{B}|=|vec{C}| . ) The angle between
two vectors is:
A .
в. ( pi / 3 )
c. ( pi / 2 )
D.
12
725A plane meets the co-ordinate axes in A,B,C such that the centroid of the
triangle ( A B C ) is the point ( (p, q, r) . ) The
equation of the plane is
A ( cdot frac{x}{p}+frac{y}{q}+frac{z}{r}=0 )
В ( cdot frac{x}{p}+frac{y}{q}+frac{z}{r}=1 )
c. ( frac{x}{p}+frac{y}{q}+frac{z}{r}=2 )
D. none of these
12
726What is the sum of the squares of direction cosines of the line joining the points (1,2,-3) and (-2,3,1)( ? )
( A cdot O )
B.
( c cdot 3 )
D. ( frac{2}{sqrt{26}} )
12
727The image of: (-4,0,0) in the ( x z ) – plane is (4,0,0)
A. True
B. False
12
728The point which is equidistant from the points ( (boldsymbol{a}, boldsymbol{0}, boldsymbol{0}),(boldsymbol{0}, boldsymbol{b}, boldsymbol{0}),(boldsymbol{0}, boldsymbol{0}, boldsymbol{c}) ) and
(0,0,0) is:
( mathbf{A} cdot(a, b, c) )
B . ( (sqrt{a}, sqrt{b}, sqrt{c}) )
c. ( (2 a, 2 b, 2 c) )
D. ( left(frac{a}{2}, frac{b}{2}, frac{c}{2}right) )
12
729Find distance of a point (3,4) from the origin.12
73042.
The plane x +2y-z=4 cuts the sphere x + y< +22-x+,
– 2 = 0 in a circle of radius
(a) 3 (6) 1 (c) 2 (d) &
[2005]
12
73134.
The angle between the lines 2x = 3y = – z and
6x=-y=-4z is
[2005]
(a) O
). (b) 90°
(c) 45°
(d) 30°
12
732The number of lines which are equally
inclined to the axes is
( A cdot 2 )
B. 4
( c cdot 6 )
D. 8
12
733VEU
43.
point P is the intersection of the straight line joining
points (2,3,5) and R(1,-1, 4) with the plane 5x – 4y
1. If S is the foot of the perpendicular drawn from the
point T(2, 1, 4) to QR, then the length of the line segment PS
(2012)
7
(b) √2
(c) 2
(2) 252
12
734( operatorname{lines} frac{x-1}{2}=frac{y-1}{2}=frac{z-2}{3} ) and
( frac{x-1}{2}=frac{y-2}{2}=frac{z-3}{-2} )
Check whether the lines are parallel, mutually perpendicular or intersecting in acute angle)
12
735A line is perpendicular to the plane ( x+ )
( 2 y+2 z=0 ) and passes through
( (0,1,0) . ) The perpendicular distance of this line from the origin is
A ( frac{sqrt{5}}{3} )
B. ( frac{sqrt{7}}{3} )
( c cdot frac{2}{3} )
D. 3
12
736The name of the figure formed by the points (0,0,0),(1,0,1) and (0,1,1) is
A . a straight line
B. an isosceles triangle
c. an equilateral triangle
D. a scalene triangle
12
737The cartesian equation of the plane ( overline{boldsymbol{r}}=(mathbf{1}+boldsymbol{s}-boldsymbol{t}) hat{boldsymbol{i}}+(boldsymbol{2}-boldsymbol{s}) hat{boldsymbol{j}}+ )
( (3-2 s+2 t) hat{k} )
A. ( 2 x-y=5 )
B. ( 2 x+z=5 )
c. ( 2 x+y=5 )
D. ( 2 x-z=5 )
12
73827.
A variable plane at a distance of the one unit from the
origin cuts the coordinates axes at A, B and C. If the
centroid D (x, y, z) of triangle ABC satisfies the relation
2
then the value k is
(2005)
y
(a) 3
(b) 1
(c)
(d) 9
12
73921. Let ✓ = 2i +1 -k and W = i +3k . If Ū is a unit vector,
then the maximum value of the scalar triple product |ŪVW | is
(a) -1
(b) V10 + V6 (2002)
(c) 159
(d) 160
12
740Two opposite vertices of a square are (2,-3,4) and ( (4,1,-2) . ) The length of the side of the square is
A ( cdot sqrt{58} )
B. ( 2 sqrt{7} )
c. ( sqrt{14} )
D. ( sqrt{7} )
12
741If the angle between the planes ( boldsymbol{r} cdot(boldsymbol{m} hat{boldsymbol{i}}-hat{boldsymbol{j}}+boldsymbol{2} hat{boldsymbol{k}})+boldsymbol{3}=boldsymbol{0} ) and ( overline{boldsymbol{r}} cdot(boldsymbol{2} hat{boldsymbol{i}}- )
( m hat{j}-hat{k})-5=0 ) is ( frac{pi}{3} ) then ( m= )
( A cdot 2 )
B. ±3
( c cdot 3 )
D. – –
12
742Find the ratio in which (the plane) ( 2 x+ ) ( 3 y+5 z=1 ) divides the line joining the
points (1,0,-3) and (1,-5,7)
A .1: 2
B . 2: 3
( c .3: 1 )
D. None of these
12
743Let ( O ) be the origin and ( P ) be the point at a distance 3 units from origin. If d.x.s’ of OP are ( 1,-2,-2, ) then coordinates of ( P ) is
given by
A. 1,-2,-2
в. 3,-6,-6
c. ( frac{1}{3},-frac{2}{3},-frac{2}{3} )
D. ( frac{1}{9},-frac{2}{9},-frac{2}{9} )
12
744Write the direction cosines of ( x ) -axis12
745The product of the d.cs of the line which makes equal angles with ( o x, o y, o z ) is
( mathbf{A} cdot mathbf{1} )
B. ( sqrt{3} )
c. ( frac{1}{3 sqrt{3}} )
D. ( frac{1}{sqrt{3}} )
12
746If a line makes ( theta_{1}, theta_{2}, theta_{3} ) angles with the
co-ordinates axes, then prove that
( cos 2 theta_{1}+cos 2 theta_{2}+cos 2 theta_{3}+1=0 )
12
747Verify the following:
(0,7,-10),(1,6,-6) and (4,9,-6) are
the vertices of an isosceles triangle.
12
748If ( vec{A} times vec{B}=vec{B} times vec{A}, ) then the angle between ( vec{A} ) and ( vec{B} ) is
A . ( pi )
в.
( c cdot frac{pi}{2} )
D.
12
749Show that the lines whose d.c.s are
given by ( l+m+n=0,2 m n+3 l n- )
( 5 l m=0 ) are perpendicular to each
other.
12
750Show that the following set of point are collinear?
(2,3,-4),(-1,0,5),(3,4,-7)
12
751Equation of a plane making X-intercept
4, Y-intercept ( (-6), mathrm{Z} ) -intercept 3 is
A. ( 3 x-4 y+6 z=12 )
B. ( 3 x-2 y+4 z=12 )
c. ( 4 x-6 y+3 z=1 )
D. ( 4 x-3 y+2 z=12 )
12
752Find the distance between (12,3,4) and (4,5,2)
A ( cdot sqrt{72} )
B. ( sqrt{62} )
( c cdot sqrt{64} )
D. None of these
12
753The general equation of plane which is
parallel to x-axis is
( mathbf{A} cdot a x+b y+c z+d=0, a neq 0, b neq 0, c neq 0 )
B. ( b y+a x+d=0, a neq 0, b neq 0 )
c. ( a x+c z+d=0, a neq 0 . c neq 0 )
D. ( b y+c z+d=0, b neq 0, c neq 0 )
12
754If from the point ( boldsymbol{P}(boldsymbol{f}, boldsymbol{g}, boldsymbol{h}) )
perpendiculars ( P L, P M ) be drawn to ( y z )
and ( z x ) planes, then the equation to the
plane ( boldsymbol{O} boldsymbol{L} boldsymbol{M} ) is
A ( cdot frac{x}{f}+frac{y}{g}-frac{z}{h}=0 )
В ( cdot frac{x}{f}+frac{y}{g}+frac{z}{h}=0 )
c. ( frac{x}{f}-frac{y}{g}+frac{z}{h}=0 )
D. ( -frac{x}{f}+frac{y}{g}+frac{z}{h}=0 )
12
755The position vectors of point ( A ) and ( B ) ( operatorname{are} hat{boldsymbol{i}}-hat{boldsymbol{j}}+boldsymbol{3} hat{boldsymbol{k}} ) and ( boldsymbol{3} hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}+boldsymbol{3} hat{boldsymbol{k}} )
respectively. The equation of a plane is
( r cdot(5 hat{i}+2 hat{j}-7 hat{k})+9=0 . ) The point ( A )
and ( B )
A. lie on the plane
B. are on the same side of the plane
c. are on the opposite side of the plane
D. None of these
12
756Find the distance between the following
pairs of points
(-5,7) and (-1,3)
12
757The equation of the plane which is parallel to ( x y ) plane and cuts intercept of length 3 from the z-axis
( mathbf{A} cdot x=3 )
B. ( y=3 )
( mathbf{c} cdot z=3 )
D. ( x+y+z=3 )
12
758The coordinate of any point, which lies
in ( boldsymbol{x} boldsymbol{y} ) plane, is
( mathbf{A} cdot(x, 0, y) )
в. ( (x, x, 0) )
c. ( (x, 0, x) )
D. ( (y, 0, x) )
12
759What is the angle between the lines
( frac{boldsymbol{x}-mathbf{2}}{mathbf{1}}=frac{boldsymbol{y}+mathbf{1}}{-mathbf{2}}=frac{boldsymbol{z}+mathbf{2}}{mathbf{1}} ) and ( frac{boldsymbol{x}-mathbf{1}}{mathbf{1}}= )
( frac{2 y+3}{3}=frac{z+5}{2}=? )
( ^{A} cdot frac{pi}{2} )
в.
c. ( frac{pi}{6} )
D. None of the above
12
760The distance between the origin and the centroid of the tetrahedron whose vertices are ( (mathbf{0}, mathbf{0}, mathbf{0}) )
( (a, 0,0),(0, b, 0),(0,0, c) ) is?
A ( cdot sqrt{a^{2}+b^{2}+c^{2}} )
B. ( frac{sqrt{a^{2}+b^{2}+c^{2}}}{2} )
c. ( frac{sqrt{a^{2}+b^{2}+c^{2}}}{4} )
D. ( 4 sqrt{a^{2}+b^{2}+c^{2}} )
12
761A parallelepiped is formed by planes
drawn through the points ( boldsymbol{P}(boldsymbol{6}, boldsymbol{8}, boldsymbol{1} boldsymbol{0}) )
and ( Q(3,4,8) ) parallel to the coordinate
planes. Find the length of edges and diagonals of the parallelepiped.
12
762Find the equation of the plane passing through the points ( A=(2.3,-1), B=(4,5, )
2), ( C=(3,6,5) )
12
763The cartesian equation of plane ( bar{r} cdot(2,-3,4)=5 ) is
A. ( 3 y-2 x-4 z+5=0 )
B. ( 2 x-3 y+4 z=0 )
c. ( 2 x-3 y+4 z+5=0 )
D. ( frac{x-1}{2}=frac{y-1}{-3}=frac{z-1}{4} )
12
764A plane which passes through the point (3,2,0) and the line ( frac{x-3}{1}=frac{y-7}{5}= )
( frac{z-4}{4} ) is?
A. ( x-y+z=1 )
в. ( x+y+z=5 )
c. ( x+2 y-z=1 )
D. ( 2 x-y+z=5 )
12
76555. If the straight lines *;?v=2=27
and
k
2
=intersect at a point, then the integer k
2
=
3 k
is equal to
(a) 5
[2008]
(b) 5
(c) 2
(d) -2
12
766The line ( frac{x-3}{2}=frac{y-4}{5}=frac{z-6}{7} )
A . lies in ( 3 x+2 y+4 z-6=0 )
B. is parallel to ( 2 x-5 y+3 z=0 )
( mathbf{c} cdot ) is ( perp ) to ( 2 x-5 y+3 z=0 )
D. passing through (1,2,3)
12
767Direction cosines of ray from
( boldsymbol{P}(mathbf{1},-mathbf{2}, mathbf{4}) ) to ( boldsymbol{Q}(-mathbf{1}, mathbf{1},-mathbf{2}) ) are
в. 2,-3,6
( mathbf{c} cdot 2,3,6 )
D. ( frac{-2}{7}, frac{3}{7}, frac{-6}{7} )
12
768If the line, ( frac{x-1}{2}=frac{y+1}{3}=frac{z-1}{4} ) and
( frac{boldsymbol{x}-mathbf{3}}{mathbf{1}}=frac{boldsymbol{y}-boldsymbol{k}}{mathbf{2}}=frac{boldsymbol{z}}{mathbf{1}} ) intersect, then find
the value of ( k )
12
769Point of intersection of the point
(2,4,5)(3,6,-4)
12
770Find the position vector of a point ( boldsymbol{R} )
which divides the line joining two
points ( P ) and ( Q ) whose position vectors ( operatorname{are}(2 vec{a}+vec{b}) ) and ( (vec{a}-3 vec{b}) ) externally in
the ratio ( 1: 2 . ) Also, show that ( P ) is the
mid point of the line segment ( boldsymbol{R} boldsymbol{Q} )
12
77113. Two lines L, : x=5, 3a = – and Ly :x=a, -2-a
are coplanar. Then a can take value(s) (JEE Adv. 2013)
(a) 1 (b) 2 (c) 3 (d) 4
12
772The distance between the points ( (cos theta, sin theta) ) and ( (sin theta-cos theta) ) is
A. ( sqrt{3} )
B. ( sqrt{2} )
( c cdot 2 )
D.
12
773Find the coordinates of those points on the line ( frac{boldsymbol{x}+mathbf{1}}{mathbf{2}}=frac{boldsymbol{y}+mathbf{2}}{mathbf{3}}=frac{boldsymbol{z}-mathbf{3}}{mathbf{6}} ) which
is at a distance of 3 units from the point
(1,-2,3)
12
774Find ( m ) if the point on the ( x ) -axis which
is equidistant from (7,6) and (3,4) is ( left(frac{m}{2}, 0right) )
12
77593. A plane passing through the points (0,–1, 0) and (0,0,1)
and making an angle with the plane y-z+5 = 0, also
passes through the point: [JEEM 2019-9 April (M)
(a) (-,1,-4) (b) (12,-1,4)
(C) (12,-1,-4) (d) (V2,1,4)
12
776The distance between the line ( vec{r}=2 vec{i}- ) ( mathbf{2} overrightarrow{boldsymbol{j}}+boldsymbol{3} overrightarrow{boldsymbol{k}}+boldsymbol{lambda}(overrightarrow{boldsymbol{i}}-overrightarrow{boldsymbol{j}}+boldsymbol{4} overrightarrow{boldsymbol{k}}) ) and the plane
( vec{r} cdot(vec{i}+5 vec{j}+vec{k})=5 ) is
A ( cdot frac{10}{3 sqrt{3}} )
в. ( frac{10}{9} )
c. ( frac{10}{3} )
D.
12
777If ( vec{a}, vec{b} ) and ( vec{c} ) are mutually perpendicular vectors of equal magnitudes, If the angles which the vector ( 2 vec{a}+vec{b}+2 vec{c} ) makes with the vectors ( vec{a} ) is ( cos ^{-1} sqrt{frac{2}{m}} ) Find ( boldsymbol{m} )12
778Find the equation of following planes:12
779The plane ( a x+b y+c z+d=0 ) divides
the line joining the points ( left(x_{1}, y_{1}, z_{1}right) )
and ( left(x_{2}, y_{2}, z_{2}right) ) in the ratio
A ( cdot frac{-left(a x_{1}+b y_{1}+c z_{1}+dright)}{left(a x_{2}+b y_{2}+c z_{2}+dright)} )
B. ( frac{left(a x_{1}+b y_{1}+c z_{1}+dright)}{left(a x_{2}+b y_{2}+c z_{2}+dright)} )
c. ( frac{a x_{1} x_{2}+b y_{1} y_{2}+c z_{1} z_{2}}{d^{2}} )
D. None of these
12
780The vector equation ( boldsymbol{r}=boldsymbol{i}-boldsymbol{2} boldsymbol{j}-boldsymbol{k}+ )
( t(6 j-k) ) represents a straight line
passing through the points:
A. (0,6,-1) and (1,-2,-1)
в. (0,6,-1) and (-1,-4,-2)
c. (1,-2,-1) and (1,4,-2)
D. (1,-2,-1) and (0,-6,1)
12
781If the distance of a point ( (a, a, a) ) from the origin is ( sqrt{108}, ) then the value of ( a ) is
This question has multiple correct options
( A cdot 9 )
B. 6
( c .-9 )
D. – 6
12
782Equation of a plane12
783In which ratio the plane ( Y Z ) divides the
lines joining the points (2,1,2) and (-6,3,4)
12
784A straight line passes through (1,-2,3) and perpendicular to the plane ( 2 x+ ) ( 3 y-z=7 . ) Find the direction ratios of
normal to plane
( A cdot )
В. ( )
c. ( )
D. None of the above
12
785( X O Z ) plane divides the join of (2,3,1) and (6,7,1) in the ratio
A . 3: 7
B. 2: 7
c. -3: 7
D. -2: 7
12
78685. The distance of the point (1,3, -7) from the plane passing
through the point (1, -1, -1), having normal perpendicular
to both the lines
Z-4
X-2
y+1
Z+7
,
X-1
1
y +2
– 2
and 1-2
3
2
-1 , is:
2
JEE M 2017]
S
12
787If ( P(x, y, z) ) is point in the space at a
distance ( r ) from the origin ( O ), then
direction cosines of the line ( O P ) are
12
788If the extremities of a diagonal of a
square are (1,-2,3) and (2,-3,5)
then area of the square is
( A cdot 6 )
B. 3
( c cdot frac{3}{2} )
D. ( sqrt{3} )
12
789The equation of the plane passing through the intersection of the planes ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=boldsymbol{6} ) and ( boldsymbol{2} boldsymbol{x}+boldsymbol{3} boldsymbol{y}+boldsymbol{4} boldsymbol{z}+boldsymbol{5}= )
( 0, ) and the point (1,1,1) is
A. ( 20 x+23 y+26 z-69=0 )
B. ( 20 x+23 y+26 z+69=0 )
c. ( 23 x+20 y+26 z-69=0 )
D. None of these
12
790What are the direction ratios of the line
if it passes through the intersection of the planes ( x=3 z+4 ) and ( y=2 z-3 ? )
A ( .(1,2,3) )
в. (2,1,3)
c. (3,2,1)
D. (1,3,2)
12
791The ratio in which the joint of
(2,1,5),(3,4,3) is divided by the plane ( 2 x+2 y-2 z-1=0 )
( mathbf{A} cdot 5: 12 )
B. 12: 5
( c .5: 7 )
D. 7: 5
12
792Three vertices of a tetrahedron are
(0,0,0),(6,-5,-1) and ( (-4,1,3) . )
the centroid of the tetrahedron be
(1,-2,5) then the fourth vertex is
A. (2,-4,18)
в. (1,-4,18)
( ^{c} cdotleft(frac{3}{2}, frac{-3}{2}, frac{7}{4}right) )
D. none of these
12
793If ( boldsymbol{R} ) divides the line segment joining ( P(2,3,4) ) and ( Q(4,5,6) ) in the ratio
( -3: 2, ) then the parameter which
represent ( boldsymbol{R} ) is
( A cdot 3 )
B . 2
( c . )
D. –
12
794If ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} in[mathbf{0}, boldsymbol{2} boldsymbol{pi}], ) then the sum of all
possible values of ( alpha, beta, gamma ) if ( sin alpha= ) ( -frac{1}{sqrt{2}}, cos beta=-frac{1}{2}, tan gamma=-sqrt{3}, ) is
A ( cdot frac{22 pi}{3} )
B. ( frac{21 pi}{3} )
c. ( frac{20 pi}{3} )
D. ( 8 pi )
12
795A plane intersects the co ordinate axes
at ( A, B, C . ) If ( O=(0,0,0) ) and (1,1,1) is
the centroid of the tetrahedron ( O A B C )
then the sum of the reciprocals of the
intercepts of the plane
( mathbf{A} cdot 12 )
B. ( frac{4}{3} )
c. 1
( D cdot frac{3}{4} )
12
796The plane ( a x+b y+c z+(-3)=0 )
meet the co-ordinate axes in ( A, B, C )
The centroid of the triangle is
B ( cdotleft(frac{3}{a} cdot frac{3}{b}, frac{3}{c}right) )
c. ( left(frac{a}{3} cdot frac{b}{3}, frac{c}{3}right) )
D. ( left(frac{1}{a} cdot frac{1}{b}, frac{1}{c}right) )
12
797The area of triangle whose vertices are (1,2,3),(2,5,-1) and (-1,1,2) is
A . 150 sq.units
B. 145 sq.units
c. ( sqrt{155} / 2 ) sq.units
D. ( 155 / 2 ) sq.units
12
798Find the shortest distance between
lines: ( frac{x-1}{1}=frac{y-2}{3}=frac{z-3}{2} ) and
( frac{x-4}{2}=frac{y-5}{3}=frac{z-6}{1} )
A . ( sqrt{6} )
B. ( sqrt{5} )
( c cdot sqrt{3} )
D. 6
12
799Let the equation of the plane through the points
(-2,-2,2),(1,1,1),(1,-1,2) be ( k x+ )
( boldsymbol{m} boldsymbol{y}+boldsymbol{n} boldsymbol{z}+boldsymbol{p} . ) Find ( boldsymbol{k}+boldsymbol{m}+boldsymbol{n}+boldsymbol{p} )
( A cdot 7 )
B.
( c cdot 4 )
D. 6
12
800Find the distance between the following pairs of points (-2,-3) and (3,2)12
801If ( C_{1}: x^{2}+y^{2}-20 x+64=0 ) and
( C_{2}: x^{2}+y^{2}+30 x+144=0 . ) Then the
length of the shortest line segment ( boldsymbol{P Q} )
which touches ( C_{1} ) at ( P ) and to ( C_{2} ) at ( Q )
is
A . 10
B. 15
( c cdot 22 )
D. 27
12
802If ( x y- ) plane and ( y z- ) plane divides the line segment joining ( A(2,4,5) ) and ( B(3,5,-4) ) in the ratio ( a: b ) and ( p: q ) respectively then value of ( left(frac{a}{b}+frac{p}{q}right) ) may be
A ( cdot frac{23}{12} )
в. ( frac{7}{5} )
( c cdot frac{7}{12} )
D. ( frac{21}{10} )
12
803The image of the point (2,-1,1) by the
plane ( 3 x+4 y-5 z=0 ) is
A ( cdot(-2,1,-1) )
в. ( left(frac{2}{3}, frac{-1}{4}, frac{-1}{5}right) )
( ^{mathbf{C}} cdotleft(frac{59}{25}, frac{-13}{25}, frac{2}{5}right) )
D. none of these
12
804Distance between ( vec{r}=hat{i}+lambda(hat{j}+hat{k}) )
and ( vec{r}=hat{j}+mu(hat{j}+hat{k}) ) is equal to
12
805( operatorname{Let} boldsymbol{A}(mathbf{2} hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}+mathbf{5} hat{boldsymbol{k}}) boldsymbol{B}(-hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}+2 hat{boldsymbol{k}}) )
and ( C(lambda hat{i}+5 hat{j}+mu hat{k}) ) are vertices of ( a ) triangle and its median through ( A ) is equally inclined to the positive directions of the axes. The value of ( lambda+ )
( mu ) is equal to
A . -7
B . 2
c. 7
D. 17
12
806Direction cosines of the line ( frac{x+2}{2}= )
( frac{2 y-5}{3}, z=-1 ) are
A ( cdot frac{4}{5}, frac{3}{5}, 0 )
B. ( frac{3}{5^{prime}}, frac{4}{5^{prime}} frac{1}{5} )
c. ( quad-frac{3}{5}, frac{4}{5}, 0 )
D. ( frac{4}{5},-frac{2}{5}, frac{1}{5} )
12
807The plane through the intersection of the planes ( x+y+z=1 ) and ( 2 x+ )
( 3 y-z+4=0 ) and parallel to ( y ) -axis
also passes through the point.
A. (-3,0,1)
В. (3,3,-1)
c. (3,2,1)
D. (-3,1,1)
12
808Find the equation of the plane passing through the points
( (mathbf{1}, mathbf{1}, mathbf{1}),(mathbf{3},-mathbf{1}, mathbf{2}),(-mathbf{3}, mathbf{5},-mathbf{4}) )
12
809Find the equation of the plane bisecting the line segment joining the points
(-3,-2,1) and (1,6,-5)
perpendicularly.
12
810is
46. Let P be the image of the point (3,1,7) with respect to the
plane x-y+z=3. Then the equation of the plane passing
x y z .
through P and containing the straight line =
1 z 1
(a) x+y-3z=0
(b) 3x+z=0
(c) X-4y+z=0
(d) 2x-y=0
hub the point (1 1 1
12
811The plane ( a x+b y+c z+(-3)=0 )
meet the co-ordinate axes in ( A, B, C . ) Then
centroid
of the triangle is
A. ( (3 a, 3 b, 3 c) )
в. ( left(frac{3}{a} frac{3}{b}, frac{3}{c}right) )
( ^{mathbf{C}} cdotleft(frac{a}{3}, frac{b}{3}, frac{c}{3}right) )
D. ( left(frac{1}{a}, frac{1}{b}, frac{1}{c}right) )
12
812If the points ( boldsymbol{A}(mathbf{3},-mathbf{2}, mathbf{4}), boldsymbol{B}(mathbf{1}, mathbf{1}, mathbf{1}) ) and
( C(-1,4,-2) ) are collinear, then the ratio in which ( C ) divides ( A B ) is
A .1: 2
B . -2: 1
c. -1: 2
D. 4: 0
12
813Find the direction cosines (d.cs) of
directed line ( O P ) if coordinates of ( P ) is
( (2,3,7), O ) being the origin.
12
814Find the co-ordinates of a point lying on the line ( frac{boldsymbol{x}-mathbf{2}}{mathbf{3}}=frac{boldsymbol{y}+mathbf{3}}{mathbf{4}}=frac{boldsymbol{z}-mathbf{1}}{mathbf{7}} ) which
is at a distance 10 units from (2,-3,1)
begin{tabular}{l}
A. (32,37,71) \
hline
end{tabular}
в. (-28,-43,-69)
c. (-32,-37,-71)
D. None of these
12
815The direction angles of the line ( x= )
( mathbf{4} z+mathbf{3}, boldsymbol{y}=mathbf{2}-mathbf{3} z ) are ( boldsymbol{alpha}, boldsymbol{beta} ) and ( gamma, ) then
( cos alpha+cos beta+cos gamma= )
A ( cdot frac{2}{sqrt{26}} )
B. ( frac{8}{sqrt{26}} )
c. 1
D. 2
12
816The equation of a plane passing through the point ( A(2,-3,7) ) and making equal intercepts on the axes,
is?
A. ( x+y+z=3 )
в. ( x+y+z=6 )
c. ( x+y+z=9 )
D. ( x+y+z=4 )
12
817If the dr’s the line are ( (1+lambda, 1-lambda, 2) )
and it makes an angle ( 60^{circ} ) with the ( Y ) –
axis then ( lambda ) is
A ( .1 pm sqrt{3} )
B. ( 4 pm sqrt{5} )
c. ( 2 pm 2 sqrt{3} )
D. ( 2 pm sqrt{5} )
12
818If direction ratios of the normal of the
plane which contains the lines ( frac{x-2}{3}= ) ( frac{boldsymbol{y}-boldsymbol{4}}{boldsymbol{2}}=frac{boldsymbol{z}-boldsymbol{1}}{boldsymbol{1}} & frac{boldsymbol{x}-boldsymbol{6}}{boldsymbol{3}}=frac{boldsymbol{y}+boldsymbol{2}}{boldsymbol{2}}= )
( frac{z-2}{1} ) are ( (a, 1,-26), ) then ( a ) is equal
to
A . 5
B. 6
( c cdot 7 )
D.
12
819What is the distance of the point
( (p, q, r) ) from the ( x- ) axis.
12
820What is the angle between ( vec{A} ) and the resultant of ( (overrightarrow{boldsymbol{A}}+overrightarrow{boldsymbol{B}}) ) and ( (overrightarrow{boldsymbol{A}}-overrightarrow{boldsymbol{B}}) )
( mathbf{A} cdot 0^{circ} )
B ( cdot tan ^{-1}left(frac{A}{B}right) )
( ^{mathbf{c}} cdot tan ^{-1}left(frac{B}{A}right) )
D. ( tan ^{-1}left(frac{A-B}{A+B}right) )
12
821Show that angles between any two ( operatorname{diagonals} boldsymbol{theta}=cos ^{-1}left(frac{1}{3}right) )12
822A triangle ( A B C ) is placed so that the
midpoints of its sides are on the ( boldsymbol{x}, boldsymbol{y} )
and ( z ) axes respectively. Lengths of the
intercepts made by the plane containing the triangle on these axes
( operatorname{are} ) respectively ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma}, ) then
the coordinates of the centroid of the
triangle ( A B C ) are
A ( cdotleft(-frac{alpha}{3}, frac{beta}{3}, frac{gamma}{3}right) )
В ( cdotleft(frac{alpha}{3},-frac{beta}{3}, frac{gamma}{3}right) )
( ^{mathbf{C}} cdotleft(frac{alpha}{3}, frac{beta}{3},-frac{gamma}{3}right) )
D ( cdotleft(frac{alpha}{3}, frac{beta}{3}, frac{gamma}{3}right) )
12
823Using vectors, find the value of ( lambda ) such
that the points ( (boldsymbol{lambda},-mathbf{1 0}, mathbf{3}),(mathbf{1},-mathbf{1}, mathbf{3}) )
and (3,5,3) are collinear.
12
824Find the equation of the plane through
the intersection of the planes ( 3 x- )
( 4 y+5 z=10 ) and ( 2 x+2 y-3 z=4 )
and parallel to the line ( boldsymbol{x}=mathbf{2} boldsymbol{y}=mathbf{3} boldsymbol{z} )
12
825Find the direction cosines of the line PQ
joining the points ( P(2,3,4) ) and ( Q(2,1,1) )
12
826A symmetrical form of the line of intersection of the planes ( boldsymbol{x}=boldsymbol{a} boldsymbol{y}+boldsymbol{b} )
and ( boldsymbol{z}=boldsymbol{c} boldsymbol{y}+boldsymbol{d} ) is :
A. ( frac{x-b}{a}=frac{y-1}{1}=frac{z-d}{c} )
( ^{text {В }} cdot frac{x-b-a}{a}=frac{y-1}{1}=frac{z-d-c}{c} )
c. ( frac{x-a}{b}=frac{y-0}{1}=frac{z-c}{d} )
D. ( frac{x-b-a}{b}=frac{y-1}{0}=frac{z-d-c}{d} )
12
827Find the distance of the point ( boldsymbol{P}(boldsymbol{3}, boldsymbol{4}, boldsymbol{4},) )
from the point, where the line joining
the points ( boldsymbol{A}(boldsymbol{3},-boldsymbol{4},-boldsymbol{5}) ) and ( boldsymbol{B}(boldsymbol{2},-boldsymbol{3}, boldsymbol{1}) )
intersected the plane ( 2 x+y+z=7 )
12
828The distance of the point (1,-5,9) from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} ) measured
along the line ( x=y=z ) is:
( A cdot 3 sqrt{10} )
в. ( 10 sqrt{3} )
c. ( frac{10}{sqrt{3}} )
D. ( frac{20}{3} )
12
829Find the equation of the plane passing through the intresection of the planes ( x-2 y+z=1 ) and ( 2 x+y+z=8 )
and parallel to the line with direction ratio proportional to ( 1,2,1, ) find also the perpendicular distance of (1,1,1) from this plane.
12
830A rectangular parallelopiped is formed by drawing planes through the points (-1,2,5) and (1,-1,-1) and paralle to the coordinate planes. the length of the diagonal of the parallelopiped is
( A cdot 2 )
B. 3
c. 6
D.
12
831Derive the equation of the locus of a point twice as far from (-2,3,4) as from
(3,-1,-2)
12
832Prove that 1,1,1 cannot be direction cosines of a straight line12
833The acute angle between two lines such
that the direction cosines ( I, m, n ) of each
of them satisfy the equations ( l+m+ )
( boldsymbol{n}=mathbf{0} ) and ( l^{2}+boldsymbol{m}^{2}-boldsymbol{n}^{2}=mathbf{0} ) is :
A . 30
B . 45
c. 60
D. 15
12
834Vector components of the vector with
initial points (2,1) and terminal point (-5,7) are
( mathbf{A} cdot-6 hat{i}+7 widehat{j} )
B. ( -7 hat{i}+6 hat{j} )
c. ( -6 hat{i}-7 widehat{j} )
D. None
12
835A mirror and a source of light are situated at the origin ( mathrm{O} ) and at a point on ( mathrm{O} X, ) respectively. A ray of light from the source strikes the mirror and is
reflected. If the direction ratios of the
normal to the plane are ( 1,-1,1, ) then find the DCs of the reflected ray.
A ( cdot frac{1}{3}, frac{2}{3}, frac{2}{3} )
в. ( -frac{1}{3}, frac{2}{3}, frac{2}{3} )
( c cdot-frac{1}{3},-frac{2}{3},-frac{2}{3} )
D. ( -frac{1}{3},-frac{2}{3}, frac{2}{3} )
12
836The points (2,5) and (5,1) are the two opposite vertices of a rectangle. If the other two vertices are points on the straight line ( y=2 x+k, ) then the value
of k is
( A cdot 4 )
B. 3
( c cdot-4 )
( D cdot-3 )
( E )
12
837Let ( vec{A}, vec{B} ) and ( vec{C} ) be unit vectors. Suppose that ( vec{A} cdot vec{B}=vec{A} cdot vec{C}=0 ) and that the
angle between ( vec{B} ) and ( vec{C} ) is ( frac{pi}{6} ) then ( overrightarrow{boldsymbol{A}}= )
A ( cdot pm 2(vec{B} times vec{C}) )
B ( cdot pm(vec{B} times vec{C}) )
c. ( pm 2(vec{B}+vec{C}) )
D. ( pm(vec{B}+vec{C}) )
12
838If the lines through the points (4,1,2) and ( (5, k, 0) ) is parallel to the line through the points (2,1,1) and (3,3,1)
find ( k )
12
839The coordinates of the foot of the
perpendicular from the point (1,-2,1) on the plane containing the lines, ( frac{x+1}{6}=frac{y-1}{7}=frac{z-3}{8} ) and ( frac{x-1}{3}= )
( frac{boldsymbol{y}-boldsymbol{2}}{mathbf{5}}=frac{boldsymbol{z}-boldsymbol{3}}{boldsymbol{7}}, ) is :
A ( cdot(2,-4,2) )
в. (-1,2,-1
c. (0,0,0)
D. (1,1,1)
12
840If the points ( boldsymbol{A}(1,2,-1), B(2,6,2) ) and
( C(lambda,-2,-4) ) are collinear, then ( lambda ) is
( mathbf{A} cdot mathbf{0} )
B.
( c .-2 )
( D )
12
841Find the centroid of a triangle, midpoints of whose sides are
(1,2,-3),(3,0,1) and (-1,1,-4)
12
842If the points ( (a, 1),(1,2) ) and ( (0, b+1) )
are collinear, then show that ( frac{1}{a}+frac{1}{b}=1 )
12
843The ratio of ( y z ) -plane divide the line
joining the points ( A(3,1,-5), B(1,4,-6) ) is
A . 3: 1
B. -1: 3
c. 1: 3
D ( .-3: 1 )
12
844( P(1,1,1) ) and ( Q(lambda, lambda, lambda) ) are two points in the space such that ( P Q=sqrt{27} ), then the value(s) of ( lambda ) can be
A . -4
в. -2,4
( c cdot 2 )
D. 4,3
12
845The line ( boldsymbol{x}-mathbf{2} boldsymbol{y}+mathbf{4} boldsymbol{z}+mathbf{4}=mathbf{0}, boldsymbol{x}+boldsymbol{y}+ )
( z-8=0 ) intersects the plane ( x-y+ )
( mathbf{2} z+mathbf{1}=mathbf{0} ) at the point
A ( cdot(3,2,3) )
B . (5,2,1)
c. (2,5,1)
D . (3,4,1)
12
846In which ratio does the ( Y Z ) plane divide
the line joining the points (-2,4,7) and (3,-5,8)
12
847A sphere of constant radius ( 2 k ) passes
through the origin and meets the axes in ( A, B, C . ) The locus of the centroid of
the tetrahedron ( boldsymbol{O} boldsymbol{A} boldsymbol{B} boldsymbol{C} ) is
A ( cdot x^{2}+y^{2}+z^{2}=4 k^{2} )
B cdot ( 9left(x^{2}+y^{2}+z^{2}right)=4 k^{2} )
C . ( x^{2}+y^{2}+z^{2}=k^{2} )
D. None of these
12
848The point lying on angle bisector of the
planes ( boldsymbol{x}+mathbf{2} boldsymbol{y}+mathbf{2} boldsymbol{z}-mathbf{6}=mathbf{0} ) and ( mathbf{2} boldsymbol{x}- )
( boldsymbol{y}+boldsymbol{4}=mathbf{0} ) is
A. (2,4,0)
B. (-1,3,2)
D. (-2,4,0)
12
849Find the values of ( a ) and ( b ) so that the
points ( (boldsymbol{a}, boldsymbol{b}, mathbf{3}),(mathbf{2}, mathbf{0},-mathbf{1}) ) and
(1,-1,-3) are collinear.
12
850If ( |overrightarrow{boldsymbol{A}} times overrightarrow{boldsymbol{B}}|=overrightarrow{boldsymbol{A}} cdot overrightarrow{boldsymbol{B}}, ) then ( |overrightarrow{boldsymbol{A}}+overrightarrow{boldsymbol{B}}| ) is:
( mathbf{A} cdot|vec{A}|+|vec{B}| )
B. ( sqrt{|vec{A}|^{2}+|vec{B}|^{2}} )
( ^{mathrm{c}} cdot sqrt{|vec{A}|^{2}+|vec{B}|^{2}+frac{|vec{A}||vec{B}|}{sqrt{2}}} )
D cdot ( sqrt{|vec{A}|^{2}+|vec{B}|^{2}+sqrt{2}|vec{A}||vec{B}|} )
12
85161. Statement-1: The point A(3, 1, 6) is the mirror image of the
point B(1, 3, 4) in the plane x-y +z=5.
Statement-2: The plane x-y+z=5 bisects the line segment
joining A(3, 1,6) and B(1,3,4).
[2010]
(a) Statement -1 is true, Statement -2 is true; Statement-2
is not a correct explanation for Statement-1.
(6) Statement -1 is true, Statement -2 is false.
(c) Statement-1 is false, Statement -2 is true.
(d) Statement – 1 is true, Statement 2 is true; Statement-2
is a correct explanation for Statement -1.
12
852Write the vector equation of the line equation ( frac{x-1}{2}=frac{y-2}{3}=frac{z-3}{4} )12
853The name of the figure formed by the points (3,-5,1),(-1,0,8) and
(7,-10,-6) is
A. a triangle
B. a straight line
c. an isosceles triangle
D. an equilateral triangle
12
854The lines ( frac{x-1}{2}=frac{y+1}{3}=frac{z-1}{4} ) and
( frac{boldsymbol{x}-mathbf{3}}{mathbf{1}}=frac{boldsymbol{y}-boldsymbol{k}}{mathbf{2}}=frac{boldsymbol{z}}{mathbf{1}} ) intersect if ( boldsymbol{K} )
equals
( A cdot frac{3}{2} )
в. ( frac{9}{2} )
c. ( frac{-2}{9} )
D. ( frac{-3}{2} )
12
855Find unit vector perpendicular to the plane passing through the points (1,2,3),(2,-1,1) and (1,2,-4)12
856If ( P ) is a point ( (x, y) ) on the line ( y=-3 x )
such that ( boldsymbol{P} ) and ( boldsymbol{Q}(boldsymbol{3}, boldsymbol{4}) ) are on opposite
side of the line ( 3 x-4 y=8, ) then:
A ( cdot x>frac{8}{5}, yfrac{8}{15}, y<-frac{8}{5} )
c. ( x=frac{8}{15}, y=frac{-8}{5} )
D. ( x=2, y=-2 )
12
857( P ) is a variable points which moves
such that ( 3 P A=2 P B . ) If ( A= )
(-2,2,3) and ( B=(13,-3,13) ) prove
that ( P ) satisfies the equation ( x^{2}+y^{2}+ )
( z^{2}+28 x-12 y+10 z-247=0 )
12
858For what value of ( mathrm{m}, ) the points (3,5)
( (m, 6) ) and ( left(frac{1}{2}, frac{15}{2}right) ) are collinear?
12
859Find the direction cosines of the line
( frac{boldsymbol{x}+mathbf{2}}{mathbf{2}}=frac{mathbf{2} boldsymbol{y}-mathbf{5}}{mathbf{3}} ; boldsymbol{z}=-mathbf{1} )
12
860Find the co-ordinates of a point, which is at a distance of 21 units from the
point ( boldsymbol{A}=(mathbf{1},-mathbf{3}, mathbf{4}) ) in the direction of vectors ( 2 hat{i}-3 hat{j}-6 hat{k} )
12
86172.
22. If the lines *-72 173
If the lines
Z
y-3
1
-4
-ka
and
-1
and

k
z-5
– are coplanar, then k can have
[JEE M 2013]
(a) any value
(c) exactly two values
(b) exactly one value
(d) exactly three values
12
862If two vertices of a triangle ( A B C ) are ( A(-1,2,4) ) and ( B(2,-3,0), ) and the
centroid is (2,0,2) then the vertex ( C ) has the coordinates
A. (5,1,2)
в. ( left(1,-frac{1}{3}, frac{7}{3}right) )
( ^{mathbf{C}} cdotleft(3,-frac{2}{3}, frac{5}{3}right) )
D. none of these
12
863Find the distance of the point (2,12,5) from the point of intersection of the line ( overrightarrow{boldsymbol{r}}=2 hat{hat{boldsymbol{i}}}-mathbf{4} hat{boldsymbol{j}}+mathbf{2} hat{boldsymbol{k}}+boldsymbol{lambda}(boldsymbol{3} hat{boldsymbol{i}}+boldsymbol{4} hat{boldsymbol{j}}+boldsymbol{2} hat{boldsymbol{k}}) )
and the plane ( vec{r} cdot(hat{boldsymbol{i}}-boldsymbol{2} hat{boldsymbol{j}}+hat{boldsymbol{k}})=mathbf{0} )
12
864Find the direction ratio of the line
( frac{x-1}{2}=3 y=frac{2 z+3}{4} )
12
865The equation of line passing through (1,-2,3) and having ( operatorname{drs}(2,3,1) ) is
( ^{text {A }} cdot frac{x-1}{2}=frac{y+2}{3}=frac{z-3}{1} )
B. ( frac{x+1}{2}=frac{y+2}{3}=frac{z-3}{1} )
c. ( frac{x-1}{2}=frac{y-2}{3}=frac{z-3}{1} )
D. none of these
12
866If a point ( P ) in the space such that ( overline{O P} )
is inclined to ( O X ) at 45 and ( O Z ) to 60
then ( overline{O P} ) inclined to ( O Y ) is
A ( .75^{circ} )
B . ( 75^{circ} ) or ( 105^{circ} )
C. ( 60^{circ} ) or ( 120^{circ} )
D. None of these
12
867Find the root of the perpendicular from point (2,3,2) to the line ( frac{4-x}{2}=frac{y}{6}= ) ( frac{1-z}{3} ) also find perpendicular distance
from the point to the line.
12
86811.
– and
25
x-1 y+1
x +1
Z.
Ifthe straight lines
4+1 Z
1 2 k
5 2 k
are coplanar, then the plane (s) containing these two lines is
(are)
(2012)
(a) y + 2z=-1
(b) y+z=-1
(c) y-z=-1 .
(d) y-2z=-1
12
869The projection of a directed line segment on the co-ordinate axes are
( 12,4,3, ) then the direction cosines of
the line are
A ( cdot frac{-12}{13}, frac{-4}{13}, frac{-3}{13} )
В. ( frac{12}{13}, frac{4}{13}, frac{3}{13} )
c. ( frac{12}{13}, frac{-4}{13}, frac{3}{13} )
D. ( frac{12}{13}, frac{4}{13}, frac{-3}{13} )
12
870If the point ( (x, y) ) is equidistant from the points ( (a+b, b-a) ) and ( (a- )
( b, a+b), ) then ( b x=a y )
A. True
B. False
12
871If the points ( a(1,2,-1), B(2,6,2) ) and ( c(lambda,-2,-4) ) are collinear then ( lambda ) is
( mathbf{A} cdot mathbf{0} )
B . 2
( c .-2 )
D.
12
872The image of the point (1,2,3) through
the plane ( boldsymbol{x}+boldsymbol{y}+boldsymbol{4} boldsymbol{z}=boldsymbol{0} ) is
A ( cdotleft(frac{-2}{3}, frac{1}{3}, frac{-11}{3}right) )
в. ( left(frac{3}{7},-frac{6}{7}, frac{19}{7}right) )
( ^{mathbf{c}} cdotleft(frac{2}{3}, frac{1}{3}, frac{11}{3}right) )
D. ( left(frac{-2}{3}, frac{-1}{3}, frac{11}{3}right) )
12
873Show that the points ( (3,3),(h, 0) ) and
( (0, k) ) are collinear, if ( frac{1}{n}+frac{1}{k}=frac{1}{3} )
12
874Angle between lines whose direction
( operatorname{cosine} operatorname{satisfy} l+m+n=0, l^{2}+ )
( boldsymbol{m}^{2}-boldsymbol{n}^{2}=mathbf{0} )
12
875a b c a b c
23. Distance between two parallel planes
2x+y+2z=8 and 4x + 2y +4z +5=0 is
[2004]
(a)
2
min
12
876( operatorname{can} frac{2}{sqrt{3}}, frac{-2}{sqrt{3}}, frac{-1}{sqrt{3}} ) be the direction ratios
of any directed line? Justify your
12
877A vector equation of the line of
intersection of the planes ( boldsymbol{r}=boldsymbol{b}+ )
( boldsymbol{lambda}_{1}(boldsymbol{b}-boldsymbol{a})+boldsymbol{mu}_{1}(boldsymbol{a}+boldsymbol{c}) )
( boldsymbol{r}=boldsymbol{c}+boldsymbol{lambda}_{2}(boldsymbol{b}-boldsymbol{c})+boldsymbol{mu}_{1}(boldsymbol{a}+boldsymbol{b}) boldsymbol{a}, boldsymbol{b}, boldsymbol{c} )
being non-coplanar vectors is.
A ( cdot r=a+mu_{1}(b+c) )
B . ( r=b+mu_{1}(a+2 c) )
C . ( r=a+mu_{1}(b+2 c) )
D. ( r=b+mu_{1}(a+c) )
12
878The length of the perpendicular from the origin to the plane passing through the point ( a ) and containing the line ( r=bar{b}+ )
( lambda bar{c} ) is
A ( cdot frac{[a b c]}{|a times b+b times c+c times a|} )
В. ( frac{[a b c]}{|a times b+b times c|} )
c. ( frac{[a b c]}{|b times c+c times a|} )
D. ( frac{[a b c]}{|a times b+c times a|} )
12
879Let the equation of the plane which contains the line ( boldsymbol{x}=frac{boldsymbol{y}-boldsymbol{3}}{boldsymbol{2}}=frac{boldsymbol{z}-boldsymbol{5}}{boldsymbol{3}} )
and which is perpendicular to the plane ( 2 x+7 y-3 z=1 . ) be ( k x-m y-z+ )
( boldsymbol{p}=mathbf{0} . ) Find ( boldsymbol{p}-boldsymbol{k}-boldsymbol{m} ? )
12
880Find direction cosine line ( boldsymbol{x}=mathbf{3} boldsymbol{z}+ )
( mathbf{2}, boldsymbol{y}=mathbf{2}-mathbf{5} boldsymbol{z} )
12
881Find the coordinates of the points which divides the line joining the points (2,-4,3),(-4,5,-6) in the ratio
( (i) 1:-4 )
( (i i) 2: 1 )
12
882Direction ratio of two lines are ( l_{1}, boldsymbol{m}_{1}, boldsymbol{n}_{1} )
and ( l_{2}, m_{2}, n_{2} ) then direction ratios of the line perpendicular to both the lines
are
A ( . l_{1}-l_{2}, m_{1}-m_{2}, n_{1}-n_{2} )
в. ( l_{1}+l_{2}, m_{1}+m_{2}, n_{1}+n_{2} )
c. ( m_{1} n_{2}-n_{1} m_{2}, n_{1} l_{2}-n_{2} l_{1}, l_{1} m_{2}-m_{1} l_{2} )
D. ( m_{1} n_{2}-n_{1} m_{2}, n_{1} l_{2}-n_{1} l_{1}, l_{1} m_{2}-m_{1} l_{2} )
12
883The points (3,2,0),(5,3,2) and ( (-9,6,-3), ) are the vertices of a triangle ( A B C . A D ) is the internal
bisector of ( angle B A C ) which meets ( B C ) at
D. Then the co-ordinates of ( D ), are
A ( cdotleft[frac{17}{16}, frac{57}{16}, frac{19}{8}right] )
В. ( left[frac{19}{8}, frac{57}{16}, frac{17}{16}right] )
( ^{mathbf{c}} cdotleft[0,0, frac{17}{16}right] )
D. ( left[frac{17}{16}, 0,0right] )
12

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