Maths

### Show the is a solution of

The given equation is $x^{2}+6 x+9=0$ Putting $x=-3$ in the given equation, we get $L H S=(-3)^{2}+6 \times(-3)+9=9-18+9=0=R H S$ $\therefore x=-3$ is a solution of the given equation.

### Find dy/dx in each of the following:

differentiating the equation on both sides with respect to x, we get,

### A train covers a distance of at a uniform speed. If the speed had been less then it would have taken 3 hours more to cover the same distance. Find the usual speed of the train.

Let the usual speed of the train be $x \mathrm{~km} / \mathrm{h}$. $\therefore$ Reduced speed of the train $=(x-8) \mathrm{km} / \mathrm{h}$ Total distance to be covered $=480 \mathrm{~km}$ Time...

### The sum of a natural number and its square is Find the number.

Let the required natural number be $x$. According to the given condition, $x+x^{2}=156$ $\Rightarrow x^{2}+x-156=0$ $\Rightarrow x^{2}+13 x-12 x-156=0$ $\Rightarrow x(x+13)-12(x+13)=0$...

### Choose the correct statement. In conductors (A).valence band and conduction band overlap each other. (B) valence band and conduction band are separated by a large energy gap. (C) very small number of electrons are available for electrical conduction. (D) valence band and conduction band are separated by a small energy gap.

CORRECT OPTION IS OPTION (A) Valence and conduction band overlap with each other. Means electrons can easily jump from valence to conduction band. Hence, the conductivity of a conductor is highest...

### Locate the following points: (i) (1, – 1, 3), (ii) (– 1, 2, 4)

Solution: (i) $(1, – 1, 3)$:- 4th octant, (ii) $(– 1, 2, 4)$:- 2nd octant,

### Find the roots of the given equation:

$\begin{array}{l} x^{2}-6 x+3=0 \\ \Rightarrow x^{2}-6 x=-3 \\ \Rightarrow x^{2}-2 \times x \times 3+3^{2}=-3+3^{2} \\ \Rightarrow(x-3)^{2}=-3+9=6 \\ \Rightarrow x-3=\pm \sqrt{6} \end{array}$...

### Find all the zeroes of , if it is given that two of its zeroes are and

Let f(x)=x4+x3-23x2-3x+60 \text { Let } f(x)=x^{4}+x^{3}-23 x^{2}-3 x+60 Since $\sqrt{3}$ and $-\sqrt{3}$ are the zeroes of $f(x)$, it follows that each one of $(x-\sqrt{3})$ and...

### If f(x) =     is divided by g(x)=

Quotient $q(x)=x^{2}+x-3$ Remainder $r(x)=8$

### Find the intercepts cut off by the plane 2x + y – z = 5.

Solution: It is given that The plane $2 x+y-z=5$ Let us express the equation of the plane in intercept form $x / a+y / b+z / c=1$ Where $a, b, c$ are the intercepts cut-off by the plane at $x, y$...

### Find the vector and Cartesian equations of the planes (a) that passes through the point and the normal to the plane is (b) that passes through the point and the normal vector to the plane is

Solution: (a) That passes through the point $(1,0,-2)$ and the normal to the plane is $\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$ Let's say that the position vector of the point $(1,0,-2)$...

### One card is drawn from a well shuffled deck of     cards. Find the probability of getting:     a diamond or a spade

Solution: $\left( v \right)$ Number of favorable outcomes for a diamond or a spade $=\text{ }13\text{ }+\text{ }13\text{ }=\text{ }26$ So, number of favorable outcomes $=\text{ }26$ Hence,...

### One card is drawn from a well shuffled deck of     cards. Find the probability of getting:     the jack or the queen of the hearts     a diamond

Solution: $\left( iii \right)$ Favorable outcomes for jack or queen of hearts $=\text{ }1\text{ }jack\text{ }+\text{ }1\text{ }queen$ So, the number of favorable outcomes $=\text{ }2$ Hence,...

### One card is drawn from a well shuffled deck of     cards. Find the probability of getting:     a queen of red color     a black face card

Solution: We have, Total possible outcomes $=\text{ }52$ $\left( i \right)$Number queens of red color $=\text{ }2$ Number of favorable outcomes$~=\text{ }2$ Hence, P(queen of red color)...

### A game consists of spinning arrow which comes to rest pointing at one of the numbers     as shown below. If the outcomes are equally likely, find the probability that the pointer will point at:     a number less than or equal to         a number between     and

Solution: $\left( v \right)$ Favorable outcomes for a number less than or equal to $9\text{ }are\text{ }1,\text{ }2,\text{ }3,\text{ }4,\text{ }5,\text{ }6,\text{ }7,\text{ }8,\text{ }9$ So,...

### A game of chance consists of spinning an arrow which is equally likely to come to rest pointing to one of the number, as shown in figure. What is the probability that it will point to:(iii) a number which is multiple of ? (iv) an even number?

(iii) So, Favorable outcomes i.e. to get a multiple of $3$ are $3,6,9,$ and $12$ Therefore, total number of favorable outcomes i.e. to get a multiple of $3$ is $4$ We know that the Probability =...

### A game consists of spinning arrow which comes to rest pointing at one of the numbers     as shown below. If the outcomes are equally likely, find the probability that the pointer will point at:     a prime number     a number greater than

Solution: $\left( iii \right)$Favorable outcomes for a prime number are $2,\text{ }3,\text{ }5,\text{ }7,\text{ }11$ So, number of favorable outcomes$~=\text{ }5$ Hence, P(the pointer will be...

### Five cards are given– ten, jack, queen, king, and an ace of diamonds are shuffled face downwards. One card is picked at random. Then (i) What is the probability that the card is a queen? (ii) If a king is drawn first and put aside, then what is the probability that the second card picked up is the (a) ace? (b) king?

Given that Five cards-ten, jack, queen, king and Ace of diamond are shuffled face downwards. to find: Probability of following Total number of cards is $5$ (i) Now Total number of cards which is a...

Solution: We have, Total number of possible outcomes $=\text{ }12$ $\left( i \right)$ Number of favorable outcomes for $6\text{ }=\text{ }1$6 Hence, $P\left( the\text{ }pointer\text{... read more ### A bag contains twenty Rs coins, fifty Rs coins and thirty Re coins. If it is equally likely that one of the coins will fall down when the bag is turned upside down, what is the probability that the coin: will neither be a Rs coin nor be a Re coin? Solution: \[\left( iii \right)$ Number of favourable outcomes for neither Re $1$nor Rs $5$coins $=$ Number of favourable outcomes for Rs$~2$ coins $=\text{ }50\text{ }=\text{ }n\left( E... read more ### A bag contains red balls and black balls. If A ball is drawn at random from the bag. Then What is the probability that the ball drawn is: (i) Red (ii) Back Given that A bag contains 3 red, and 5 black balls. A ball is drawn at random to find: Probability of getting a (i) red ball (ii) white ball So, Total number of balls 3+5=8 (i) we know that... read more ### A bag contains twenty Rs coins, fifty Rs coins and thirty Re coins. If it is equally likely that one of the coins will fall down when the bag is turned upside down, what is the probability that the coin: will be a Re coin? will not be a Rs coin? Solution: We have, Total number of coins \[=\text{ }20\text{ }+\text{ }50\text{ }+\text{ }30\text{ }=\text{ }100$ So, the total possible outcomes $=\text{ }100\text{ }=\text{ }n\left( S \right)$...

Solution: $\left( iii \right)$ Number of favorable outcomes for white or green ball $=\text{ }16\text{ }+\text{ }8\text{ }=\text{ }24\text{ }=\text{ }n\left( E \right)$ Hence, probability for...

### A bag contains     red balls,     white balls and     green balls. A ball is drawn out of the bag at random. What is the probability that the ball drawn will be:     not red?     neither red nor green?

Solution: Total number of possible outcomes $=\text{ }10\text{ }+\text{ }16\text{ }+\text{ }8\text{ }=\text{ }34$ balls So, $n\left( S \right)\text{ }=\text{ }34$ $\left( i \right)$ Favorable...

### Which of the following cannot be the probability of an event?

Solution $\left( iii \right)\text{ }As\text{ }0\text{ }\le \text{ }37\text{ }%\text{ }=\text{ }\left( 37/100 \right)\text{ }\le \text{ }1$ Thus, $37\text{ }%$ can be a probability of an event....

### In a bundle of     shirts,     are good,     have minor defects and     have major defects. What is the probability that:     it is acceptable to a trader who accepts only a good shirt?     it is acceptable to a trader who rejects only a shirt with major defects?

Solution: We have, Total number of shirts $=\text{ }50$ Total number of elementary events $=\text{ }50\text{ }=\text{ }n\left( S \right)$ $\left( i \right)$ As, trader accepts only good shirts...

### In a musical chairs game, a person has been advised to stop playing the music at any time within     seconds after its start. What is the probability that the music will stop within the first     seconds?

Solution: Total result $=\text{ }0\text{ }sec\text{ }to\text{ }40\text{ }sec$ Total possible outcomes $=\text{ }40$ So$,\text{ }n\left( S \right)\text{ }=\text{ }40$ Favourable results...

### In a match between A and B:     the probability of winning of A is     . What is the probability of winning of B?     the probability of losing the match is     for B. What is the probability of winning of A?

Solution: $\left( i \right)$We know that, The probability of winning of A $+$Probability of losing of A $=\text{ }1$ And, Probability of losing of A $=$ Probability of winning of B...

### From a well shuffled deck of     cards, one card is drawn. Find the probability that the card drawn is:     a card with number less than         a card with number between     and

Solution: $\left( v \right)$ Numbers less than $8\text{ }=\text{ }\left\{ \text{ }2,\text{ }3,\text{ }4,\text{ }5,\text{ }6,\text{ }7 \right\}$ Event of drawing a card with number less than...

### If two coins are tossed once, what is the probability of getting: (i) both heads. (ii) at least one head.

Solution: We know that, when two coins are tossed together possible number of outcomes = {HH, TH, HT, TT} So, $n\left( S \right)\text{ }=\text{ }4$ $\left( i \right)$E = event of getting both...

### A pair of dice is thrown. Find the probability of getting a sum of     or more, if     appears on the first die

Solution: In throwing a dice, total possible outcomes $=\text{ }\left\{ 1,\text{ }2,\text{ }3,\text{ }4,\text{ }5,\text{ }6 \right\}$ So$,\text{ }n\left( S \right)\text{ }=\text{ }6$ For two...

### From     identical cards,     numbered one card is drawn at random. Find the probability that the number on the card drawn is a multiple of

Solution: We know that, there are $25$ cards from which one card is drawn. So, the total number of elementary events $=\text{ }n\left( S \right)\text{ }=\text{ }25$ $\left( i \right)$From...

### multiple Nine cards (identical in all respects) are numbered . A card is selected from them at random. Find the probability that the card selected will be:     an even number and a multiple of         an even number or a of

Solution: $\left( iii \right)$ From numbers $2\text{ }to\text{ }10$, there is one number which is an even number as well as multiple of $3\text{ }i.e.\text{ }6$ So, favorable number of events...

### Nine cards (identical in all respects) are numbered     . A card is selected from them at random. Find the probability that the card selected will be:     an even number     a multiple of

Solution: We know that, there are totally $9$ cards from which one card is drawn. Total number of elementary events $=\text{ }n\left( S \right)\text{ }=\text{ }9$ $\left( i \right)$ From...

### From a well shuffled deck of     cards, one card is drawn. Find the probability that the card drawn will:(v) be a face card of red colour

Solution: $\left( v \right)$There are $26$ red cards in a deck, and $6$ of these cards are face cards ($2$ kings, $2$queens and $2$jacks). The number of favourable outcomes for the event...

### From a well shuffled deck of     cards, one card is drawn. Find the probability that the card drawn will:     be a red card.     be a face card

Solution: $\left( iii \right)$ Number of red cards in a deck $=\text{ }26$ The number of favourable outcomes for the event of drawing a red card $=\text{ }26$ Then, probability of drawing a...

### From a well shuffled deck of 52 cards, one card is drawn. Find the probability that the card drawn will:     be a black card.     not be a red card

Solution: We know that, Total number of cards $=\text{ }52$ So, the total number of outcomes $=\text{ }52$ There are $13$ cards of each type. The cards of heart and diamond are red in colour....

### In a single throw of a die, find the probability that the number:     will be an odd number

Solution: $\left( iii \right)$ If $E\text{ }=$event of getting an odd number $=\text{ }\left\{ 1,\text{ }3,\text{ }5 \right\}$ So$,\text{ }n\left( E \right)\text{ }=\text{ }3$ Then,...

### In a single throw of a die, find the probability that the number:     will be an even number.     will not be an even number.

Solution: Here, the sample space $=\text{ }\left\{ 1,\text{ }2,\text{ }3,\text{ }4,\text{ }5,\text{ }6 \right\}$ $n\left( s \right)\text{ }=\text{ }6$ $\left( i \right)$ If $E\text{ }=$event...

### A bag contains     white,     black and     red balls, all of the same shape and size. A ball is drawn from the bag without looking into it, find the probability that the ball drawn is:     not a black ball.

Solution: $\left( v \right)$ There are $3\text{ }+\text{ }2\text{ }=\text{ }5$ balls which are not black So, the number of favourable outcomes $=\text{ }5$ Thus, P(getting a white ball)...

### Draw a rough sketch of the curve y = √(x – 1) in the interval [1, 5]. Find the area under the curve and between the lines x = 1 and x = 5.

The curve is $\begin{array}{*{35}{l}} y~=\text{ }\surd \left( x~-\text{ }1 \right) \\ \Rightarrow \text{ }{{y}^{2}}~=\text{ }x\text{ }-\text{ }1 \\ \end{array}$ Plotting the curve and finding...

### If the lines 3x – 4y + 4 = 0 and 6x – 8y – 7 = 0 are tangents to a circle, then find the radius of the circle.

Given lines are 6x – 8y + 8 = 0 and 6x – 8y – 7 = 0. Distance d between two parallel lines y = mx + c1 and y = mx + c2 is given by d = |C1–C2|/√(A2 + B2 ) These parallel lines are tangent to a...

### Find the equation of the circle which touches the both axes in first quadrant and whose radius is a.

The circle touches both the x and y axes in the first quadrant and the radius is a. For a circle of radius a, the centre is (a, a). The equation of a circle having centre (h, k), having radius as r...
### Content Structure:
The text passage contains multiple mathematical questions and solutions, primarily from the fields of algebra and probability. Each question is followed by a detailed solution.

### Analysis:
multiple_questions_with_answers

### explanation:
1. **multiple questions**: the passage contains several distinct mathematical questions, each dealing with different topics such as solving equations, finding derivatives, probability, and geometry.
2. **answers provided**: for each question, a detailed solution is provided, showing the step-by-step process to arrive at the answer.

### detailed breakdown:
1. **solving equations**:
   - **question**: show that \( x = -3 \) is a solution of the equation \( x^2 + 6x + 9 = 0 \).
   - **answer**: the solution verifies that substituting \( x = -3 \) into the equation results in \( 0 = 0 \), confirming that \( x = -3 \) is indeed a solution.

2. **differentiation**:
   - **question**: find \( \frac{dy}{dx} \) for a given equation.
   - **answer**: the solution involves differentiating the equation with respect to \( x \).

3. **train
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Maths

### Show the is a solution of

The given equation is $x^{2}+6 x+9=0$ Putting $x=-3$ in the given equation, we get $L H S=(-3)^{2}+6 \times(-3)+9=9-18+9=0=R H S$ $\therefore x=-3$ is a solution of the given equation.

### Find dy/dx in each of the following:

differentiating the equation on both sides with respect to x, we get,

### A train covers a distance of at a uniform speed. If the speed had been less then it would have taken 3 hours more to cover the same distance. Find the usual speed of the train.

Let the usual speed of the train be $x \mathrm{~km} / \mathrm{h}$. $\therefore$ Reduced speed of the train $=(x-8) \mathrm{km} / \mathrm{h}$ Total distance to be covered $=480 \mathrm{~km}$ Time...

### The sum of a natural number and its square is Find the number.

Let the required natural number be $x$. According to the given condition, $x+x^{2}=156$ $\Rightarrow x^{2}+x-156=0$ $\Rightarrow x^{2}+13 x-12 x-156=0$ $\Rightarrow x(x+13)-12(x+13)=0$...

### Choose the correct statement. In conductors (A).valence band and conduction band overlap each other. (B) valence band and conduction band are separated by a large energy gap. (C) very small number of electrons are available for electrical conduction. (D) valence band and conduction band are separated by a small energy gap.

CORRECT OPTION IS OPTION (A) Valence and conduction band overlap with each other. Means electrons can easily jump from valence to conduction band. Hence, the conductivity of a conductor is highest...

### Locate the following points: (i) (1, – 1, 3), (ii) (– 1, 2, 4)

Solution: (i) $(1, – 1, 3)$:- 4th octant, (ii) $(– 1, 2, 4)$:- 2nd octant,

### Find the roots of the given equation:

$\begin{array}{l} x^{2}-6 x+3=0 \\ \Rightarrow x^{2}-6 x=-3 \\ \Rightarrow x^{2}-2 \times x \times 3+3^{2}=-3+3^{2} \\ \Rightarrow(x-3)^{2}=-3+9=6 \\ \Rightarrow x-3=\pm \sqrt{6} \end{array}$...

### Find all the zeroes of , if it is given that two of its zeroes are and

Let f(x)=x4+x3-23x2-3x+60 \text { Let } f(x)=x^{4}+x^{3}-23 x^{2}-3 x+60 Since $\sqrt{3}$ and $-\sqrt{3}$ are the zeroes of $f(x)$, it follows that each one of $(x-\sqrt{3})$ and...

### If f(x) =     is divided by g(x)=

Quotient $q(x)=x^{2}+x-3$ Remainder $r(x)=8$

### Find the intercepts cut off by the plane 2x + y – z = 5.

Solution: It is given that The plane $2 x+y-z=5$ Let us express the equation of the plane in intercept form $x / a+y / b+z / c=1$ Where $a, b, c$ are the intercepts cut-off by the plane at $x, y$...

### Find the vector and Cartesian equations of the planes (a) that passes through the point and the normal to the plane is (b) that passes through the point and the normal vector to the plane is

Solution: (a) That passes through the point $(1,0,-2)$ and the normal to the plane is $\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$ Let's say that the position vector of the point $(1,0,-2)$...

### One card is drawn from a well shuffled deck of     cards. Find the probability of getting:     a diamond or a spade

Solution: $\left( v \right)$ Number of favorable outcomes for a diamond or a spade $=\text{ }13\text{ }+\text{ }13\text{ }=\text{ }26$ So, number of favorable outcomes $=\text{ }26$ Hence,...

### One card is drawn from a well shuffled deck of     cards. Find the probability of getting:     the jack or the queen of the hearts     a diamond

Solution: $\left( iii \right)$ Favorable outcomes for jack or queen of hearts $=\text{ }1\text{ }jack\text{ }+\text{ }1\text{ }queen$ So, the number of favorable outcomes $=\text{ }2$ Hence,...

### One card is drawn from a well shuffled deck of     cards. Find the probability of getting:     a queen of red color     a black face card

Solution: We have, Total possible outcomes $=\text{ }52$ $\left( i \right)$Number queens of red color $=\text{ }2$ Number of favorable outcomes$~=\text{ }2$ Hence, P(queen of red color)...

### A game consists of spinning arrow which comes to rest pointing at one of the numbers     as shown below. If the outcomes are equally likely, find the probability that the pointer will point at:     a number less than or equal to         a number between     and

Solution: $\left( v \right)$ Favorable outcomes for a number less than or equal to $9\text{ }are\text{ }1,\text{ }2,\text{ }3,\text{ }4,\text{ }5,\text{ }6,\text{ }7,\text{ }8,\text{ }9$ So,...

### A game of chance consists of spinning an arrow which is equally likely to come to rest pointing to one of the number, as shown in figure. What is the probability that it will point to:(iii) a number which is multiple of ? (iv) an even number?

(iii) So, Favorable outcomes i.e. to get a multiple of $3$ are $3,6,9,$ and $12$ Therefore, total number of favorable outcomes i.e. to get a multiple of $3$ is $4$ We know that the Probability =...

### A game consists of spinning arrow which comes to rest pointing at one of the numbers     as shown below. If the outcomes are equally likely, find the probability that the pointer will point at:     a prime number     a number greater than

Solution: $\left( iii \right)$Favorable outcomes for a prime number are $2,\text{ }3,\text{ }5,\text{ }7,\text{ }11$ So, number of favorable outcomes$~=\text{ }5$ Hence, P(the pointer will be...

### Five cards are given– ten, jack, queen, king, and an ace of diamonds are shuffled face downwards. One card is picked at random. Then (i) What is the probability that the card is a queen? (ii) If a king is drawn first and put aside, then what is the probability that the second card picked up is the (a) ace? (b) king?

Given that Five cards-ten, jack, queen, king and Ace of diamond are shuffled face downwards. to find: Probability of following Total number of cards is $5$ (i) Now Total number of cards which is a...

Solution: We have, Total number of possible outcomes $=\text{ }12$ $\left( i \right)$ Number of favorable outcomes for $6\text{ }=\text{ }1$6 Hence, $P\left( the\text{ }pointer\text{... read more ### A bag contains twenty Rs coins, fifty Rs coins and thirty Re coins. If it is equally likely that one of the coins will fall down when the bag is turned upside down, what is the probability that the coin: will neither be a Rs coin nor be a Re coin? Solution: \[\left( iii \right)$ Number of favourable outcomes for neither Re $1$nor Rs $5$coins $=$ Number of favourable outcomes for Rs$~2$ coins $=\text{ }50\text{ }=\text{ }n\left( E... read more ### A bag contains red balls and black balls. If A ball is drawn at random from the bag. Then What is the probability that the ball drawn is: (i) Red (ii) Back Given that A bag contains 3 red, and 5 black balls. A ball is drawn at random to find: Probability of getting a (i) red ball (ii) white ball So, Total number of balls 3+5=8 (i) we know that... read more ### A bag contains twenty Rs coins, fifty Rs coins and thirty Re coins. If it is equally likely that one of the coins will fall down when the bag is turned upside down, what is the probability that the coin: will be a Re coin? will not be a Rs coin? Solution: We have, Total number of coins \[=\text{ }20\text{ }+\text{ }50\text{ }+\text{ }30\text{ }=\text{ }100$ So, the total possible outcomes $=\text{ }100\text{ }=\text{ }n\left( S \right)$...

Solution: $\left( iii \right)$ Number of favorable outcomes for white or green ball $=\text{ }16\text{ }+\text{ }8\text{ }=\text{ }24\text{ }=\text{ }n\left( E \right)$ Hence, probability for...

### A bag contains     red balls,     white balls and     green balls. A ball is drawn out of the bag at random. What is the probability that the ball drawn will be:     not red?     neither red nor green?

Solution: Total number of possible outcomes $=\text{ }10\text{ }+\text{ }16\text{ }+\text{ }8\text{ }=\text{ }34$ balls So, $n\left( S \right)\text{ }=\text{ }34$ $\left( i \right)$ Favorable...

### Which of the following cannot be the probability of an event?

Solution $\left( iii \right)\text{ }As\text{ }0\text{ }\le \text{ }37\text{ }%\text{ }=\text{ }\left( 37/100 \right)\text{ }\le \text{ }1$ Thus, $37\text{ }%$ can be a probability of an event....

### In a bundle of     shirts,     are good,     have minor defects and     have major defects. What is the probability that:     it is acceptable to a trader who accepts only a good shirt?     it is acceptable to a trader who rejects only a shirt with major defects?

Solution: We have, Total number of shirts $=\text{ }50$ Total number of elementary events $=\text{ }50\text{ }=\text{ }n\left( S \right)$ $\left( i \right)$ As, trader accepts only good shirts...

### In a musical chairs game, a person has been advised to stop playing the music at any time within     seconds after its start. What is the probability that the music will stop within the first     seconds?

Solution: Total result $=\text{ }0\text{ }sec\text{ }to\text{ }40\text{ }sec$ Total possible outcomes $=\text{ }40$ So$,\text{ }n\left( S \right)\text{ }=\text{ }40$ Favourable results...

### In a match between A and B:     the probability of winning of A is     . What is the probability of winning of B?     the probability of losing the match is     for B. What is the probability of winning of A?

Solution: $\left( i \right)$We know that, The probability of winning of A $+$Probability of losing of A $=\text{ }1$ And, Probability of losing of A $=$ Probability of winning of B...

### From a well shuffled deck of     cards, one card is drawn. Find the probability that the card drawn is:     a card with number less than         a card with number between     and

Solution: $\left( v \right)$ Numbers less than $8\text{ }=\text{ }\left\{ \text{ }2,\text{ }3,\text{ }4,\text{ }5,\text{ }6,\text{ }7 \right\}$ Event of drawing a card with number less than...

### If two coins are tossed once, what is the probability of getting: (i) both heads. (ii) at least one head.

Solution: We know that, when two coins are tossed together possible number of outcomes = {HH, TH, HT, TT} So, $n\left( S \right)\text{ }=\text{ }4$ $\left( i \right)$E = event of getting both...

### A pair of dice is thrown. Find the probability of getting a sum of     or more, if     appears on the first die

Solution: In throwing a dice, total possible outcomes $=\text{ }\left\{ 1,\text{ }2,\text{ }3,\text{ }4,\text{ }5,\text{ }6 \right\}$ So$,\text{ }n\left( S \right)\text{ }=\text{ }6$ For two...

### From     identical cards,     numbered one card is drawn at random. Find the probability that the number on the card drawn is a multiple of

Solution: We know that, there are $25$ cards from which one card is drawn. So, the total number of elementary events $=\text{ }n\left( S \right)\text{ }=\text{ }25$ $\left( i \right)$From...

### multiple Nine cards (identical in all respects) are numbered . A card is selected from them at random. Find the probability that the card selected will be:     an even number and a multiple of         an even number or a of

Solution: $\left( iii \right)$ From numbers $2\text{ }to\text{ }10$, there is one number which is an even number as well as multiple of $3\text{ }i.e.\text{ }6$ So, favorable number of events...

### Nine cards (identical in all respects) are numbered     . A card is selected from them at random. Find the probability that the card selected will be:     an even number     a multiple of

Solution: We know that, there are totally $9$ cards from which one card is drawn. Total number of elementary events $=\text{ }n\left( S \right)\text{ }=\text{ }9$ $\left( i \right)$ From...

### From a well shuffled deck of     cards, one card is drawn. Find the probability that the card drawn will:(v) be a face card of red colour

Solution: $\left( v \right)$There are $26$ red cards in a deck, and $6$ of these cards are face cards ($2$ kings, $2$queens and $2$jacks). The number of favourable outcomes for the event...

### From a well shuffled deck of     cards, one card is drawn. Find the probability that the card drawn will:     be a red card.     be a face card

Solution: $\left( iii \right)$ Number of red cards in a deck $=\text{ }26$ The number of favourable outcomes for the event of drawing a red card $=\text{ }26$ Then, probability of drawing a...

### From a well shuffled deck of 52 cards, one card is drawn. Find the probability that the card drawn will:     be a black card.     not be a red card

Solution: We know that, Total number of cards $=\text{ }52$ So, the total number of outcomes $=\text{ }52$ There are $13$ cards of each type. The cards of heart and diamond are red in colour....

### In a single throw of a die, find the probability that the number:     will be an odd number

Solution: $\left( iii \right)$ If $E\text{ }=$event of getting an odd number $=\text{ }\left\{ 1,\text{ }3,\text{ }5 \right\}$ So$,\text{ }n\left( E \right)\text{ }=\text{ }3$ Then,...

### In a single throw of a die, find the probability that the number:     will be an even number.     will not be an even number.

Solution: Here, the sample space $=\text{ }\left\{ 1,\text{ }2,\text{ }3,\text{ }4,\text{ }5,\text{ }6 \right\}$ $n\left( s \right)\text{ }=\text{ }6$ $\left( i \right)$ If $E\text{ }=$event...

### A bag contains     white,     black and     red balls, all of the same shape and size. A ball is drawn from the bag without looking into it, find the probability that the ball drawn is:     not a black ball.

Solution: $\left( v \right)$ There are $3\text{ }+\text{ }2\text{ }=\text{ }5$ balls which are not black So, the number of favourable outcomes $=\text{ }5$ Thus, P(getting a white ball)...

### Draw a rough sketch of the curve y = √(x – 1) in the interval [1, 5]. Find the area under the curve and between the lines x = 1 and x = 5.

The curve is $\begin{array}{*{35}{l}} y~=\text{ }\surd \left( x~-\text{ }1 \right) \\ \Rightarrow \text{ }{{y}^{2}}~=\text{ }x\text{ }-\text{ }1 \\ \end{array}$ Plotting the curve and finding...

### If the lines 3x – 4y + 4 = 0 and 6x – 8y – 7 = 0 are tangents to a circle, then find the radius of the circle.

Given lines are 6x – 8y + 8 = 0 and 6x – 8y – 7 = 0. Distance d between two parallel lines y = mx + c1 and y = mx + c2 is given by d = |C1–C2|/√(A2 + B2 ) These parallel lines are tangent to a...

### Find the equation of the circle which touches the both axes in first quadrant and whose radius is a.

The circle touches both the x and y axes in the first quadrant and the radius is a. For a circle of radius a, the centre is (a, a). The equation of a circle having centre (h, k), having radius as r...
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### Content Structure:
The text passage contains multiple mathematical questions and solutions, primarily from the fields of algebra and probability. Each question is followed by a detailed solution.

### Analysis:
TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS

### Explanation:
1. **Multiple Questions**: The passage contains several distinct mathematical questions, each dealing with different topics such as solving equations, finding derivatives, probability, and geometry.
2. **Answers Provided**: For each question, a detailed solution is provided, showing the step-by-step process to arrive at the answer.

### Detailed Breakdown:
1. **Solving Equations**:
   - **Question**: Show that \( x = -3 \) is a solution of the equation \( x^2 + 6x + 9 = 0 \).
   - **Answer**: The solution verifies that substituting \( x = -3 \) into the equation results in \( 0 = 0 \), confirming that \( x = -3 \) is indeed a solution.

2. **Differentiation**:
   - **Question**: Find \( \frac{dy}{dx} \) for a given equation.
   - **Answer**: The solution involves differentiating the equation with respect to \( x \).

3. **Train