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<image>A research institution conducted a statistical analysis of two linearly related variables $$x$$ and $$y$$, and obtained the following data: From the data in the table, the linear regression equation of $$y$$ with respect to $$x$$ is $$\hat y=-0.7\hat x+a$$. Then, $$a$$ equals ( ).
A. $$10.5$$
B. $$5.25$$
C. $$5.2$$
D. $$5.15$$
|
B
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<image>Among the following geometric solids, the one that has two identical views and another different one is ( )
A. 1.2.
B. 2.3.
C. 2.4.
D. 3.4.
|
B
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|
<image>It is known that the advertising expenditure $$x$$ (unit: ten thousand yuan) and sales revenue $$y$$ (unit: ten thousand yuan) of a certain product have the following corresponding data: According to all the data provided in the table, the linear regression equation of $$y$$ and $$x$$ obtained by the method of least squares is $$\hat{y}=6.5x+17.5$$. Then the value of $$m$$ in the table is ( )
A. $$45$$
B. $$50$$
C. $$55$$
D. $$60$$
|
D
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<image>Students have played with kaleidoscopes, which are made up of three equal-length glass pieces. As shown in the figure, it is a pattern seen in a kaleidoscope. All the small triangles in the figure are congruent equilateral triangles. The rhombus $AEFG$ can be considered as the result of rotating rhombus $ABCD$ around point $A$ ( ).
A. 60° clockwise
B. 120° clockwise
C. 60° counterclockwise
D. 120° counterclockwise
|
B
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<image>If \sqrt{a^{3}+3 a^{2}} = -a , then the range of values for a is ()
A. -3≤a≤0
B. ≤0
C. ＜0
D. ≥-3
|
A
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|
<image>To understand the height situation of 300 male students in a middle school, a random sample of male students was selected for height measurement. After organizing the data, a frequency distribution histogram was drawn (as shown in the figure). Estimate the number of male students in the school whose height is between 169.5cm and 174.5cm.
A. 12
B. 48
C. 72
D. 96
|
C
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<image>As shown in the figure, $\text{AB} \bot \text{b}$, $\text{DC} \bot \text{b}$, $\text{CA} \bot \text{a}$, $\text{ED} \bot \text{b}$, then the number of line segments whose lengths can represent the distance from a point to a line is ()
A. 4
B. 6
C. 7
D. 8
|
D
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<image>A store is preparing to order a type of packaging bag for rice. After market research, the following statistical chart was made. Which packaging is the most suitable? ( ).
A. $$2{\rm Kg}/$$ per bag
B. $$3{\rm Kg}/$$ per bag
C. $$4{\rm Kg}/$$ per bag
D. $$5{\rm Kg}/$$ per bag
|
A
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<image>The graphs of the functions $y=a^x$ ($a > 0$, $a \ne 1$) and $y=x^b$ are shown in the figure. Which of the following inequalities must be true?
A. $b^a > 0$
B. $a+b > 0$
C. $ab > 1$
D. $\log_a 2 > b$
|
D
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|
<image>As shown in the figure, the area of the small square ABCD is 1. By extending each side by one time its length, a new square A$_{1}$B$_{1}$C$_{1}$D$_{1}$ is formed; by extending the sides of square A$_{1}$B$_{1}$C$_{1}$D$_{1}$ by one time their length, square A$_{2}$B$_{2}$C$_{2}$D$_{2}$ is formed; and so on... What is the area of square A$_{2019}$B$_{2019}$C$_{2019}$D$_{2019}$?
A. 5$^{2017}$
B. 5$^{2018}$
C. 5$^{2019}$
D. 5$^{2020}$
|
C
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<image>As shown in the figure, AB∥CD. According to the angles marked in the figure, which of the following relationships is true?
A. ∠1=∠3
B. ∠2+∠3=180°
C. ∠2+∠4<180°
D. ∠3+∠5=180°
|
D
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|
<image>The output of the program shown in the figure is ( )
A. $2$, $1$
B. $2$, $2$
C. $1$, $2$
D. $1$, $1$
|
A
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|
<image>In the figure, in parallelogram ABCD, the angle bisector of ∠ABC intersects side CD at point E, and ∠A = 130°. Then the measure of ∠BEC is ( )
A. 20°
B. 25°
C. 30°
D. 50°
|
B
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|
<image>In the figure, the number of consecutive interior angles is ( ).
A. $$1$$ pair
B. $$2$$ pairs
C. $$3$$ pairs
D. $$4$$ pairs
|
D
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|
<image>Given that the real numbers $$a$$ and $$b$$ correspond to points on the number line as shown in the figure, which of the following expressions is correct?
A. $$a \cdot b > 0$$
B. $$a - b > 0$$
C. $$a < -b$$
D. $$|a| < |b|$$
|
B
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<image>As shown in the figure, the line y=kx+b passes through points A(3,1) and B(6,0). The solution set of the inequality system 0 < kx + b < 13x is ()
A. 3 < x < 6
B. x > 3
C. x < 6
D. x > 3 or x < 6
|
A
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|
<image>In a survey about the relationship between gender and lying, the following data was obtained. Based on the data in the table, which of the following conclusions is correct? ( )
A. In this survey, there is a 95% confidence that lying is related to gender
B. In this survey, there is a 99% confidence that lying is related to gender
C. In this survey, there is a 99.5% confidence that lying is related to gender
D. In this survey, there is no sufficient evidence to show that lying is related to gender
|
D
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<image>The figure below shows a $6\times 6$ grid. In $\Delta ABC$, $\Delta {A}'{B}'{C}'$, and $\Delta {{A}'}'{{B}'}'{{C}'}'$, the number of right-angled triangles is ( )
A. $0$
B. $1$
C. $2$
D. $3$
|
C
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|
<image>In the following program, when the input value of $$x$$ is $$\pi$$, the computation result is ( ).
A. $$-2$$
B. $$1$$
C. $$\pi$$
D. $$2$$
|
A
|
|
<image>The tide of a thousand years has not receded, and the wind rises to set sail again. To achieve the 'two centenary goals' and the great rejuvenation of the Chinese nation's Chinese Dream, Harbin No.3 High School actively responds to the national call, continuously increasing the cultivation of outstanding talents. According to incomplete statistics: Based on the table, the regression equation $\hat{y}=\hat{b}x+\hat{a}$ has $\hat{b}$ as 1.35. The number of students from the class of 2018 who won provincial first-class awards or higher in subject competitions is 63. According to this model, the predicted number of students from our school who will be admitted to prestigious world universities such as Tsinghua and Peking University this year is ( )
A. 111
B. 115
C. 117
D. 123
|
C
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<image>If the program flowchart shown in the figure is executed, then the value of the output $S$ is ( )
A. -8
B. -23
C. -44
D. -71
|
C
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|
<image>If the distribution of the random variable $$\eta$$ is as follows: then when $$P\left (\eta< x \right ) =0.8$$, the range of the real number $$x$$ is （ ）
A. $$x\leqslant 1$$
B. $$1\leqslant x\leqslant 2$$
C. $$1< x \leqslant 2$$
D. $$1\leqslant x< 2$$
|
C
|
|
<image>The solution sets of two inequalities in a system of inequalities are shown in the figure. The solution set of this system of inequalities is ( )
A. $-2 \le x < 1$
B. $x \le 2$
C. $-2 < x \le 1$
D. $x \ge 1$
|
A
|
|
<image>As shown in the figure below, in the Cartesian coordinate system $xOy$, the terminal side of angle $\alpha$ intersects the unit circle at point $A$. If the y-coordinate of point $A$ is $\frac{4}{5}$, then the value of $\sin \alpha$ is
A. $\frac{3}{5}$
B. $\frac{4}{5}$
C. $-\frac{3}{5}$
D. $\frac{4}{3}$
|
B
|
|
<image>As shown in Figure 1, it is a right-angled triangular paper piece, $$\angle A=30{}^\circ $$, $$BC=4\text{cm}$$, and it is folded so that point $$C$$ lands on point $${{C}^{\prime }}$$ on the hypotenuse, with the fold line being $$BD$$, as shown in Figure 2. Then, Figure 2 is folded along $$DE$$, so that point $$A$$ lands on point $${{A}^{\prime }}$$ on the extension of $$D{{C}^{\prime }}$$, as shown in Figure 3. What is the length of the fold line $$DE$$?
A. $$\frac{8}{3}\text{cm}$$
B. $$2\sqrt{3}\text{cm}$$
C. $$2\sqrt{2}\text{cm}$$
D. $$3\text{cm}$$
|
A
|
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<image>As shown in the figure, use an unmarked straightedge to construct a parallelogram with a pair of opposite sides of length $$\sqrt{5}$$ in a square grid. In a $$1\times 3$$ square grid, at most $$2$$ can be constructed; in a $$1 \times 4$$ square grid, at most $$6$$ can be constructed; in a $$1\times 5$$ square grid, at most $$12$$ can be constructed. Therefore, in a $$1\times 8$$ square grid, the maximum number that can be constructed is ( ).
A. $$28$$
B. $$42$$
C. $$21$$
D. $$56$$
|
B
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|
<image>Execute the program flowchart as shown in the figure. If the output result is 36, then what should be filled in the judgment box?
A. $i < 7?$
B. $i < 8?$
C. $i < 9?$
D. $i < 10?$
|
C
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<image>As shown in the figure, the parabola y=ax$^{2}$+bx+c intersects the x-axis at point A (-1, 0), with the vertex at (1, n), and intersects the y-axis between (0, 2) and (0, 3) (inclusive). Which of the following conclusions are correct: 1. When x > 3, y < 0; 2. 3a + b > 0; 3. $-1 \le a \le -\frac{2}{3}$; 4. 3 ≤ n ≤ 4?
A. 1.2.
B. 3.4.
C. 1.4.
D. 1.3.
|
D
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<image>As shown in the figure, AB is the diameter of circle O, CD is a chord, and AB is perpendicular to CD at point E. Which of the following conclusions is incorrect?
A. $CE=DE$
B. $AE=OE$
C. $\overset\frown{BC}=\overset\frown{BD}$
D. $\angle COB=\angle DOB$
|
B
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|
<image>In the figure, AB is the diameter of circle O, CD is a chord, AB ⊥ CD, and the foot of the perpendicular is E. Point P is on circle O, and BP, PD, and BC are connected. If CD = $\frac{24}{5}$, sin P = $\frac{3}{5}$, then the diameter of circle O is ( )
A. 8
B. 6 C.5
C. $\frac{16}{5}$
|
C
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<image>For the quadratic function $$y=ax^2+bx+c$$ (where $$a$$, $$b$$, and $$c$$ are constants, and $$a\not=0$$), the partial correspondence between the independent variable $$x$$ and the function value $$y$$ is shown in the table below. It is also given that when $$x=-{1\over2}$$, the corresponding function value $$y>0$$. The following conclusions are: 1. $$abc>0$$; 2. $$-2$$ and $$3$$ are the roots of the equation $$ax^2+bx+c=t$$; 3. $$0<m+n<{20\over3}$$. The number of correct conclusions is ( ).
A. $$0$$
B. $$1$$
C. $$2$$
D. $$3$$
|
C
|
|
<image>As shown in the figure, after rotating △AOB counterclockwise around point O by $45{}^\circ$, we get △COD. If $\angle AOB=15{}^\circ$, then the degree measure of $\angle AOD$ is （）
A. $75{}^\circ$
B. $60{}^\circ$
C. $45{}^\circ$
D. $30{}^\circ$
|
B
|
|
<image>In the figure, in $\vartriangle ABC$, $\angle B=90{}^\circ $, $BC=2AB$, then $sinC=$ ( )
A. $\frac{\sqrt{5}}{2}$
B. $\frac{1}{2}$
C. $\frac{2\sqrt{5}}{2}$
D. $\frac{\sqrt{5}}{5}$
|
D
|
|
<image>As shown in the figure, in the right triangle $$ABC$$, $$AB=9$$, $$BC=6$$, $$\angle B=90{{}^\circ}$$, the triangle $$ABC$$ is folded so that point $$A$$ coincides with the midpoint $$D$$ of $$BC$$. The fold line is $$MN$$. What is the length of the segment $$BN$$?
A. $$\frac{5}{3}$$
B. $$\frac{5}{2}$$
C. $$4$$
D. $$5$$
|
C
|
|
<image>As shown in the figure, in quadrilateral ABCD, E, F, G, and H are the midpoints of AB, BD, CD, and AC, respectively. For quadrilateral EFGH to be a rhombus, quadrilateral ABCD only needs to satisfy one condition ( )
A. AD = BC
B. AC = BD
C. AB = CD
D. AD = CD
|
A
|
|
<image>A die is a small cube with the numbers 1, 2, 3, 4, 5, 6 written on its six faces, such that the sum of the numbers on any two opposite faces is 7. Several such identical dice are arranged to form a geometric solid where the product of the numbers on the touching faces is 6. The three views of this solid are shown in the figure 3-36. Given that some of the numbers on the faces are marked in the figure, the number represented by '※' is ( ).
A. 2
B. 4
C. 5
D. 6
|
B
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|
<image>In △ABC, point D is a point on side BC. AD is connected, and P is the midpoint of AD. BP and CP are connected. If the area of △ABC is 4cm², then the area of △BPC is ( )
A. 4cm²
B. 3cm²
C. 2cm²
D. 1cm²
|
C
|
|
<image>As shown in Figure 1, there is a quadrilateral paper piece $ABCD$, where $\angle B=120^{\circ}$ and $\angle D=50^{\circ}$. If the lower right corner is folded inward to form $\triangle PCR$ as shown in Figure 2, such that $CP$∥$AB$ and $RC$∥$AD$, then the measure of $\angle C$ is ( ).
A. $105^{\circ}$
B. $100^{\circ}$
C. $95^{\circ}$
D. $90^{\circ}$
|
C
|
|
<image>As shown in the figure, given AD//BC, which of the following conclusions is correct?
A. $\angle 1 = \angle 2$
B. $\angle 2 = \angle 3$
C. $\angle 1 = \angle 4$
D. $\angle 3 = \angle 4$
|
B
|
|
<image>The three views of a geometric solid are shown in the figure. According to the data in the figure, the volume of the solid is ( )
A. $$\dfrac{4}{3}\pi $$
B. $$\dfrac{\sqrt{15}}{3}\pi $$
C. $$\dfrac{4}{3}\pi -\dfrac{\sqrt{15}}{3}\pi $$
D. $$\dfrac{4}{3}\pi +\dfrac{\sqrt{15}}{3}\pi $$
|
D
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|
<image>As shown in the figure, in $$\triangle ABC$$, point $$D$$ is on $$AB$$, $$BD=2AD$$, $$DE\parallel BC$$ intersects $$AC$$ at $$E$$. Which of the following conclusions is incorrect?
A. $$BC=3DE$$
B. $$\dfrac{BD}{BA}=\dfrac{CE}{CA}$$
C. $$\triangle ADE\sim \triangle ABC$$
D. $$S_{\triangle ADE}=\dfrac{1}{3}S_{\triangle ABC}$$
|
D
|
|
<image>The profit statistics of Supermarket A and Supermarket B from January to August are shown in the chart below. Which of the following conclusions is incorrect?
A. The profit of Supermarket A decreases month by month
B. The profit of Supermarket B increases month by month from January to April
C. The profits of both supermarkets are the same in August
D. The profit of Supermarket B in September will definitely exceed that of Supermarket A
|
D
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|
<image>Xiao Cong and Xiao Ming start from two places, Jia and Yi, which are 30 kilometers apart, and travel towards each other at the same time. Xiao Cong rides a motorcycle to Yi and immediately returns to Jia, while Xiao Ming rides a bicycle directly from Yi to Jia. The function graphs $y_{1}\left(km\right)$ and $y_{2}\left(km\right)$ represent the distances from Jia and Yi, respectively, to the time $t\left(h\right)$ that has passed. As shown in the figure, the following statements are: 1. The broken line segment $OAB$ represents Xiao Cong's function graph $y_{1}$, and the line segment $OC$ represents Xiao Ming's function graph $y_{2}$; 2. Xiao Cong's average speed to Yi and back to Jia is the same; 3. The two meet for the first time 80 minutes after departure; 4. Xiao Ming's average speed on the bicycle is $15km/h$. The number of incorrect statements is ( )
A. $0$
B. $ 1$
C. $ 2$
D. $ 3$
|
B
|
|
<image>As shown in the figure, if the rational numbers corresponding to points A and B on the number line are a and b, respectively, then which of the following expressions is correct?
A. ab > 0
B. |a| < |b|
C. a - b < 0
D. a + b > 0
|
C
|
|
<image>In the parallelogram $ABCD$, the diagonals $AC$ and $BD$ intersect at point $O$. If $AC=4$, $BD=5$, and $BC=3$, then the perimeter of $\Delta BOC$ is ( )
A. $6$
B. $7.5$
C. $8$
D. $12$
|
B
|
|
<image>As shown in the figure, the vertex of the ${60}^{\circ}$ angle of a right-angled triangular ruler is placed on the center $O$ of a circle. The hypotenuse and one of the legs intersect the circle $\odot O$ at points $A$ and $B$, respectively. $P$ is any point on the major arc $AB$ (not coinciding with $A$ or $B$). Then $\angle APB=$ ( )
A. ${15}^{\circ}$
B. ${30}^{\circ}$
C. ${45}^{\circ}$
D. ${60}^{\circ}$
|
B
|
|
<image>As shown in the figure, $CD$ is a plane mirror. A light ray starts from point $A$, reflects at point $E$ on $CD$, and then reaches point $B$. If the angle of incidence is $\alpha$, $AC\bot CD$, $BD\bot CD$, and the feet of the perpendiculars are $C$ and $D$ respectively, with $AC=3$, $BD=6$, and $CD=11$, then the value of $\text{tan}\alpha$ is ( )
A. $\frac{1}{3}$
B. $\frac{3}{11}$
C. $\frac{9}{11}$
D. $\frac{11}{9}$
|
D
|
|
<image>As shown in the figure, in $\vartriangle ABC$, the angle bisectors of $\angle ABC$ and $\angle ACB$ intersect at point $F$. A line $DE \parallel BC$ is drawn through point $F$, intersecting $AB$ at point $D$ and $AC$ at point $E$. If $BD=3$ and $DE=5$, then the length of segment $EC$ is ( )
A. 3
B. 2
C. 4
D. 2.5
|
B
|
|
<image>As shown in the figure, points $$C$$ and $$D$$ are on the line segment $$AB$$, and point $$D$$ is the midpoint of line segment $$AC$$. If $$AB=10\,\rm cm$$ and $$BC=4\,\rm cm$$, then the length of line segment $$DB$$ is ( )
A. $$2\,\rm cm$$
B. $$3\,\rm cm$$
C. $$6\,\rm cm$$
D. $$7\,\rm cm$$
|
D
|
|
<image>As shown in the figure, in the sector $$OAB$$, the radius $$OA=2$$, $$∠AOB=120^{\circ}$$, and $$C$$ is the midpoint of the arc $$\overset{\frown} {AB}$$. Connecting $$AC$$ and $$BC$$, the area of the shaded part in the figure is ( )
A. $$\dfrac{4\pi }{3}-2\sqrt{3}$$
B. $$\dfrac{2\pi }{3}-2$$
C. $$\dfrac{4\sqrt{\pi }}{3}-\sqrt{3}$$
D. $$\dfrac{2\pi }{3}-\sqrt{3}$$
|
A
|
|
<image>The flowchart of a program is shown in the figure below. The function of the program is ( )
A. To calculate $1+2+2^2+2^3+\ldots+2^{63}$
B. To calculate $1+2+2^2+2^3+\ldots+2^{63}+2^{64}$
C. To calculate $2+2^2+2^3+\ldots+2^{63}$
D. To calculate $2+2^2+2^3+\ldots+2^{63}+2^{64}$
|
A
|
|
<image>As shown in the figure, $\Delta AEC \cong \Delta BED$, point $D$ is on the side $AC$, $\angle 1 = \angle 2$, and $AE$ intersects $BD$ at point $O$. Consider the following statements: (1) If $\angle B = \angle A$, then $BE \parallel AC$; (2) If $BE = AC$, then $BE \parallel AC$; (3) If $\Delta ECD \cong \Delta EOD$, $\angle 1 = 36^\circ$, then $BE \parallel AC$. How many of these statements are correct?
A. 3
B. 2
C. 1
D. 0
|
B
|
|
<image>In the figure, in $$\triangle PAB$$, $$PA=PB$$, and $$M$$, $$N$$, $$K$$ are points on $$PA$$, $$PB$$, and $$AB$$ respectively, such that $$AM=BK$$ and $$BN=AK$$. If $$\angle MKN=44^{\circ}$$, then the measure of $$\angle P$$ is ( ).
A. $$44^{\circ}$$
B. $$66^{\circ}$$
C. $$88^{\circ}$$
D. $$92^{\circ}$$
|
D
|
|
<image>A company produces a product that is classified into four grades from highest to lowest: $$A$$, $$B$$, $$C$$, and $$D$$. To increase production and improve quality, the company improved its production process, doubling the total production. To understand the effect of the new production process, the company statistically analyzed the proportion of the four grades of products before and after the improvement, and created the following pie charts: Based on the above information, which of the following inferences is reasonable?
A. The quantity of $$A$$ grade products did not change after the production process was improved.
B. The quantity of $$B$$ grade products increased by less than one fold after the production process was improved.
C. The quantity of $$C$$ grade products decreased after the production process was improved.
D. The quantity of $$D$$ grade products decreased after the production process was improved.
|
C
|
|
<image>In the right triangle $ABC$, $\angle C=90^{\circ}$, $AB=10$, $AC=8$. Points $E$ and $F$ are the midpoints of $AC$ and $AB$, respectively. Then $EF=\left( \right)$
A. $3$
B. $4$
C. $5$
D. $6$
|
A
|
|
<image>As shown in the figure, $D$ is a point on side $AC$ of equilateral triangle $ABC$, and $E$ is a point outside equilateral triangle $ABC$. If $BD=CE$ and $\angle 1=\angle 2$, then the shape of $\triangle ADE$ is ( )
A. Isosceles triangle
B. Equilateral triangle
C. Right triangle
D. Scalene triangle
|
B
|
|
<image>As shown in the figure, points A, D, C, and F lie on the same straight line, AB = DE, and BC = EF. To prove that △ABC ≌ △DEF, an additional condition is needed ( ).
A. BC ∥ EF
B. ∠B = ∠F
C. AD = CF
D. ∠A = ∠EDF
|
C
|
|
<image>The radishes below represent the $$\dfrac{1}{2}$$, $$\dfrac{1}{3}$$, and $$\dfrac{1}{4}$$ of the radishes pulled by the white rabbit, the gray rabbit, and the black rabbit, respectively. ( ) pulled the most radishes.
A. White Rabbit
B. Gray Rabbit
C. Black Rabbit
|
C
|
|
<image>As shown in the figure, $$\angle BAC = 100\degree$$, point $$D$$ is on the perpendicular bisector of $$AB$$, and point $$E$$ is on the perpendicular bisector of $$AC$$. Then the measure of $$\angle DAE$$ is ( ).
A. $$15\degree$$
B. $$20\degree$$
C. $$25\degree$$
D. $$30\degree$$
|
B
|
|
<image>As shown in the figure, in the cube $ABCD-{{A}_{1}}{{B}_{1}}{{C}_{1}}{{D}_{1}}$, the moving point $P$ is within the side face ${{B}_{1}}BC{{C}_{1}}$, and the distance from point $P$ to edge $AB$ is equal to the distance from $P$ to edge $C{{C}_{1}}$. The figure formed by the movement of point $P$ is ( ).
A. Line segment
B. Circular arc
C. Part of an ellipse
D. Part of a parabola
|
D
|
|
<image>As shown in the figure, for the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b > 0$), the left and right foci are $F_1$ and $F_2$, respectively, and $|F_1F_2| = \sqrt{10}$. Point $P$ is on the positive half of the y-axis, and $PF_1$ intersects the ellipse at point $A$. If $AF_2 \perp PF_1$ and the inradius of $\Delta APF_2$ is $\frac{\sqrt{2}}{2}$, then the eccentricity of the ellipse is ( )
A. $\frac{\sqrt{5}}{4}$
B. $\frac{\sqrt{5}}{3}$
C. $\frac{\sqrt{10}}{4}$
D. $\frac{\sqrt{15}}{4}$
|
B
|
|
<image>If the function $$f(x) = A\sin (\omega x + \varphi)$$ (where $$A > 0$$, $$\omega > 0$$, and $$|\varphi| \leqslant \pi$$) has the local graph shown in the figure, then the expression for the function $$y = f(x)$$ is ( ).
A. $$y = \frac{3}{2} \sin (2x + \frac{\pi}{6})$$
B. $$y = \frac{3}{2} \sin (2x - \frac{\pi}{6})$$
C. $$y = \frac{3}{2} \sin (2x + \frac{\pi}{3})$$
D. $$y = \frac{3}{2} \sin (2x - \frac{\pi}{3})$$
|
D
|
|
<image>As shown in the figure, $\triangle ABC$ is an equilateral triangle paper with side length $a$. Now, a triangle is cut from each corner so that the remaining hexagon $PQRSTU$ is a regular hexagon. What is the perimeter of this regular hexagon? ( )
A. $2a$
B. $3a$
C. $\dfrac{3}{2}a$
D. $\dfrac{9}{4}a$
|
A
|
|
<image>In the figure, in $$\triangle ABC$$, $$AB=AC$$, $$AD\bot BC$$, with the foot of the perpendicular at $$D$$. Which of the following conclusions is incorrect?
A. $$\angle B=\angle C$$
B. $$BA=BC$$
C. $$\angle 1=\angle 2$$
D. $$BD=DC$$
|
B
|
|
<image>If the result of the program given in the flowchart is $$S=90$$, then the judgment condition about $$k$$ that should be filled in the judgment box is ( ).
A. $$k \leqslant 8$$
B. $$k \leqslant 7$$
C. $$k>9$$
D. $$k>8$$
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D
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<image>As shown in the figure, in the sector $$AOB$$, $$\angle AOB=90^\circ$$, the vertex $$C$$ of the square $$CDEF$$ is the midpoint of $$\overparen{AB}$$, point $$D$$ is on $$OB$$, and point $$E$$ is on the extension of $$OB$$. When the side length of the square $$CDEF$$ is $$2\sqrt2$$, the area of the shaded part is ( ).
A. $$2\mathrm \pi-4$$
B. $$4\mathrm \pi-8$$
C. $$2\mathrm \pi-8$$
D. $$4\mathrm \pi-4$$
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A
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<image>As shown in the figure, the area of parallelogram OABC is 8. Diagonal AC is perpendicular to CO, and AC intersects BO at point E. An exponential function y = a^x (a > 0 and a ≠ 1) passes through points E and B. What is the value of a?
A. √2
B. √3
C. 2
D. 3
|
A
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|
<image>As shown in the figure, AB = AC, AD = AE. To prove that △ABD ≌ △ACE, which additional condition can be used?
A. ∠1 = ∠2
B. ∠B = ∠C
C. ∠D = ∠E
D. ∠BAE = ∠CAD
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A
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<image>As shown in the figure, the main view of the regular triangular prism $ABC-{{A}_{1}}{{B}_{1}}{{C}_{1}}$ is a square with a side length of 4. What is the area of the left view of this regular triangular prism?
A. 16
B. $2\sqrt{3}$
C. $4\sqrt{3}$
D. $8\sqrt{3}$
|
D
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<image>As shown in the figure, $$AD \parallel BC \parallel x$$-axis. Which of the following statements is correct?
A. The x-coordinates of points $$A$$ and $$D$$ are the same
B. The y-coordinates of points $$B$$ and $$C$$ are the same
C. The x-coordinates of points $$C$$ and $$D$$ are the same
D. The y-coordinates of points $$B$$ and $$D$$ are the same
|
B
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<image>Given the values of $$x$$ and $$y$$ as shown in the table: From the scatter plot analysis, $$y$$ is linearly related to $$x$$, and the regression equation is $$\hat y=0.95x+a$$. Then $$a=$$（）.
A. $$1.9$$
B. $$2$$
C. $$2.6$$
D. $$4.5$$
|
C
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<image>As shown in the figure, in quadrilateral $$ABCD$$, diagonals $$AC$$ and $$BD$$ intersect at $$E$$, $$\angle CBD=90\degree$$, $$BC=4$$, $$BE=ED=3$$, $$AC=10$$. What is the area of quadrilateral $$ABCD$$?
A. $$6$$
B. $$12$$
C. $$20$$
D. $$24$$
|
D
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<image>In the figure, in △ABC, points D and E are the midpoints of sides AB and AC, respectively, and BC = 6, then DE = ( )
A. 5
B. 4
C. 3
D. 2
|
C
|
|
<image>As shown in the figure, $$AB$$ is the diameter of circle $$\odot O$$, and points $$C$$, $$D$$ are two trisection points of the semicircular arc $$\overset{\frown} {AB}$$. Given $$\overrightarrow{AB}=\overrightarrow{a}$$ and $$\overrightarrow{AC}=\overrightarrow{b}$$, then $$\overrightarrow{AD}=$$ ( )
A. $$\overrightarrow{a}-\dfrac{1}{2}\overrightarrow{b}$$
B. $$\dfrac{1}{2}\overrightarrow{a}-\overrightarrow{b}$$
C. $$\overrightarrow{a}+\dfrac{1}{2}\overrightarrow{b}$$
D. $$\dfrac{1}{2}\overrightarrow{a}+\overrightarrow{b}$$
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D
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<image>Near point $$O$$ in the park, there are four trees $$E$$, $$F$$, $$G$$, and $$H$$, as shown in the figure (the sides of the small squares in the figure are all equal). It is planned to build a circular pond with $$O$$ as the center and $$OA$$ as the radius. It is required that no trees remain in the pond. Which of the following trees need to be removed?
A. $$E$$, $$F$$, $$G$$
B. $$F$$, $$G$$, $$H$$
C. $$G$$, $$H$$, $$E$$
D. $$H$$, $$E$$, $$F$$
|
A
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<image>As shown in the figure, given that DE∥BC and EF∥AB, which of the following proportional expressions is incorrect?
A. $\frac{\text{AD}}{\text{AB}}=\frac{\text{AE}}{\text{AC}}$
B. $\frac{\text{CE}}{\text{CF}}=\frac{\text{EA}}{\text{FB}}$
C. $\frac{\text{DE}}{\text{BC}}=\frac{\text{AD}}{\text{BD}}$
D. $\frac{\text{EF}}{\text{AB}}=\frac{\text{CF}}{\text{CB}}$
|
C
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<image>As shown in the figure, in the Cartesian coordinate system $xOy$, the vertices of angles $\alpha$ and $\beta$ coincide with the origin, and their initial sides coincide with the non-negative half of the $x$-axis. Their terminal sides intersect the unit circle at points $A$ and $B$, respectively. If the coordinates of points $A$ and $B$ are $(\dfrac{3}{5}, \dfrac{4}{5})$ and $(-\dfrac{4}{5}, \dfrac{3}{5})$, then the value of $\cos(\alpha + \beta)$ is ( ).
A. $-\dfrac{24}{25}$
B. $-\dfrac{7}{25}$
C. $0$
D. $\dfrac{24}{25}$
|
A
|
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<image>In the geometric solid shown in the figure, the shapes seen from above that are the same are ( )
A. 1. 2.
B. 1. 3.
C. 2. 3.
D. 2. 4.
|
C
|
|
<image>In the right-angled triangle paper $\triangle ABC$, it is known that $\angle B=90^{\circ}$, $AB=6$, and $BC=8$. The paper is folded so that side $AB$ overlaps with side $AC$, and point $B$ lands on point $E$. The fold line is $AD$. What is the length of $BD$?
A. $3$
B. $4$
C. $5$
D. $6$
|
A
|
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<image>As shown in the figure, the side length of square $ABCD$ is $2$. A line segment $QR$ of length $2$ has its ends placed on two adjacent sides of the square and slides along them. If point $Q$ starts from point $A$ and slides in the direction $A\rightarrow B\rightarrow C\rightarrow D\rightarrow A$ until it returns to $A$, and point $R$ starts from point $B$ and slides in the direction $B\rightarrow C\rightarrow D\rightarrow A\rightarrow B$ until it returns to $B$, during this process, the path traced by the midpoint $M$ of line segment $QR$ forms a figure with area denoted as $S$. Point $N$ is any point inside square $ABCD$, and the probability that the distance from $N$ to each of the four vertices $A$, $B$, $C$, and $D$ is not less than $1$ is denoted as $P$. Then $S=\left( \right)$
A. $\left(4-\pi \right)P$
B. $ 4\left(1-P\right)$
C. $ 4P$
D. $ \left(\pi -1\right)P$
|
C
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<image>Execute the program flowchart as shown. If the inputs $$a$$, $$b$$, $$k$$ are $$1$$, $$2$$, $$3$$ respectively, then the output $$M=$$ ( )
A. $$\dfrac{20}{3}$$
B. $$\dfrac{7}{2}$$
C. $$\dfrac{16}{5}$$
D. $$\dfrac{15}{8}$$
|
D
|
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<image>As shown in the figure, in $\Delta ABC$, $CD$ bisects $\angle ACB$ and intersects $AB$ at point $D$. A line through point $D$ parallel to $BC$ intersects $AC$ at point $E$. If $\angle A={{54}^{\circ }},\angle B={{48}^{\circ }}$, then the measure of $\angle CDE$ is ( )
A. ${{44}^{\circ }}$
B. ${{40}^{\circ }}$
C. ${{39}^{\circ }}$
D. ${{38}^{\circ }}$
|
C
|
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<image>A school conducted a sampling survey to understand the height conditions of students. The sample included an equal number of boys and girls, and the groups were divided as follows (unit: cm): A: x < 155, B: 155 ≤ x < 160, C: 160 ≤ x < 165, D: 165 ≤ x < 170, E: x ≥ 170. The following statistical chart was created using the collected data. Based on the information provided by the chart, which of the following statements is correct? A. There are 3 more boys than girls in the height range 155 ≤ x < 160. B. The proportion of boys and girls in Group B is the same. C. More than half of the boys are taller than 165 cm. D. There are 2 girls in Group E.
|
D
|
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<image>The diagram below belongs to ( )
A. Flowchart
B. Structure Chart
C. Program Flowchart
D. Process Flowchart
|
A
|
|
<image>In the figure, in $\triangle ABC$, point $D$ is a point on side $AB$, and point $F$ is a point on the extension of $BC$. $DF$ intersects $AC$ at point $E$. Which of the following conclusions is incorrect?
A. $\angle F + \angle ACF = \angle A + \angle ADF$
B. $ \angle B + \angle ACB < 180^{\circ}$
C. $ \angle DEC > \angle B$
D. $ \angle A > \angle ACF$
|
D
|
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<image>As shown in the figure, which of the following conditions cannot be used to determine that the lines $$l_1 \parallel l_2$$?
A. $$\angle 1 = \angle 3$$
B. $$\angle 2 = \angle 3$$
C. $$\angle 4 = \angle 5$$
D. $$\angle 2 + \angle 4 = 180^\circ$$
|
B
|
|
<image>A school conducts a physical examination for the senior high school students. The weight data (unit: kg) of the senior high school boys has been organized and divided into five groups, and a frequency distribution histogram has been drawn (as shown in the figure). According to general standards, senior high school boys weighing more than 65 kg are considered overweight, and those weighing less than 55 kg are considered underweight. It is known that the frequencies of the first, third, fourth, and fifth groups from left to right in the figure are 0.25, 0.20, 0.10, and 0.05, respectively, and the frequency count of the second group is 400. Therefore, the total number of senior high school boys and the frequency of those with normal weight are ()
A. 1000, 0.50
B. 800, 0.50
C. 800, 0.60
D. 1000, 0.60
|
D
|
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<image>As shown in the figure, in △ABC, arcs are drawn with points A and B as centers and a radius greater than $\frac{1}{2}$AB, intersecting at points M and N. The line MN is drawn, intersecting BC at point D. AD is connected. If the perimeter of △ADC is 8, and AB=6, then the perimeter of △ABC is ( )
A. 20
B. 22
C. 14
D. 16
|
C
|
|
<image>As shown in the figure, parallelogram ABCD is folded so that vertex D lands exactly on point M on side AB, with the fold line being AN. For the conclusions 1. MN∥BC, 2. MN=AM, which of the following statements is correct?
A. Both 1 and 2 are correct
B. Both 1 and 2 are incorrect
C. 1 is correct, 2 is incorrect
D. 1 is incorrect, 2 is correct
|
A
|
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<image>As shown in the figure, this is the process of measuring the volume of an object: (1 ml = 1 cm³) Step one: Pour 300 ml of water into a 500 ml cup. Step two: Place four identical glass balls into the water, and the water does not overflow. Step three: Add one more identical glass ball, and the water overflows. Based on the above process, infer that the volume of one glass ball falls within the following range ( ).
A. More than 10 cm³, less than 20 cm³
B. More than 20 cm³, less than 30 cm³
C. More than 30 cm³, less than 40 cm³
D. More than 40 cm³, less than 50 cm³
|
D
|
|
<image>As shown in the figure, the line $$y=kx+b$$ intersects the $$x$$-axis at the point $$\left(-4,0\right)$$. When $$y>0$$, the range of $$x$$ is ( ).
A. $$x< -4$$
B. $$x>0$$
C. $$x>-4$$
D. $$x< 0$$
|
C
|
|
<image>If the flowchart of a certain program is as shown in the figure, then the value of $i$ output after the program runs is ( )
A. 4
B. 5
C. 6
D. 7
|
C
|
|
<image>The C919 large passenger aircraft is China's first large civil aircraft developed in accordance with the latest international airworthiness standards, with complete independent intellectual property rights, and a maximum seating capacity of 190. In the aircraft logo shown, the angle that can be drawn using a set of triangular rulers is ( )
A. $$angle 1$$
B. $$angle 2$$
C. $$angle 3$$
D. $$angle 4$$
|
A
|
|
<image>As shown in the figure, the 'Zhao Shuang String Diagram' dartboard is made up of four congruent right-angled triangles with legs of lengths $1$ and $2$. Student Xiaoming stands a certain distance from the dartboard and throws darts at it (assuming all thrown darts stick to the dartboard). The probability that a dart lands in the shaded area is ( )
A. $\dfrac{1}{2}$
B. $\dfrac{1}{3}$
C. $\dfrac{1}{4}$
D. $\dfrac{1}{5}$
|
D
|
|
<image>In the ancient Chinese mathematical classic 'Nine Chapters on the Mathematical Art', there is a problem in the 'Surplus and Deficiency' section: 'There is a wall 9 feet high. A melon grows above it, with the vine growing 7 inches per day; a gourd grows below it, with the vine growing 1 foot per day. How many days will it take for them to meet?' Now, this is described using a program flowchart as shown in the figure. What is the output result $$n=$$ ( )?
A. $$4$$
B. $$5$$
C. $$6$$
D. $$7$$
|
C
|
|
<image>As shown in the figure, $$AB$$ and $$AC$$ are tangents to circle $$ \odot O$$, with $$B$$ and $$C$$ being the points of tangency, and $$ \angle A = 50^{ \circ }$$. Point $$P$$ is a moving point on the circle, distinct from $$B$$ and $$C$$, and lies on the arc $$\overset{\frown} {BMC}$$. The measure of $$ \angle BPC$$ is ( ).
A. $$65^{ \circ }$$
B. $$115^{ \circ }$$
C. $$115^{ \circ }$$ or $$65^{ \circ }$$
D. $$130^{ \circ }$$ or $$65^{ \circ }$$
|
A
|
|
<image>2019 was the 70th anniversary of the founding of China and a critical year for building a moderately prosperous society in all respects. To celebrate the 70th birthday of the country, the entire population united to build a moderately prosperous society. A school specially organized a 'Welcoming National Day, Building a Moderately Prosperous Society' knowledge competition. The stem-and-leaf plot below shows the scores of the two groups of contestants. Which of the following statements is correct?
A. The average score of Group A contestants is less than that of Group B contestants
B. The median score of Group A contestants is greater than that of Group B contestants
C. The median score of Group A contestants is equal to that of Group B contestants
D. The variance of the scores of Group A contestants is greater than that of Group B contestants
|
D
|
|
<image>As shown in the figure, quadrilateral $$ABCD$$ is a square, and $$O$$ is the intersection point of the diagonals $$AC$$ and $$BD$$. To transform $$ \triangle COD$$ into $$ \triangle DOA$$, $$ \triangle COD$$ must be rotated around point $$O$$ ( )
A. 90° clockwise
B. 45° clockwise
C. 90° counterclockwise
D. 45° counterclockwise
|
C
|
|
<image>As shown in the figure, the diameter CD of circle O is perpendicular to chord AB at point E, and CE=2, DE=8. Then the length of AB is ( )
A. 2
B. 4
C. 6
D. 8
|
D
|
|
<image>As shown in the diagram, the algorithm flowchart is inspired by the 'method of successive subtraction' from the ancient Chinese mathematical classic 'The Nine Chapters on the Mathematical Art'. If the input values of $a$ and $b$ are 12 and 30 respectively, then the output value of $a$ is ( )
A. 2
B. 4
C. 6
D. 18
|
C
|
End of preview. Expand
in Data Studio
This is the official release of the training data for paper PAPO: Perception-Aware Policy Optimization for Multimodal Reasoning. (arxiv.org/abs/2507.06448)
(Optional) This dataset can be used as the val split of the training dataset for PAPO. You may find the full training dataset at PAPOGalaxy/PAPO_ViRL39K_train.
Data Source
Training
- We adapt the multimodal benchmark TIGER-Lab/ViRL39K to construct our PAPO training dataset.
Validation (Optional)
- (Optional) We use the
testset from FanqingM/MMK12 for validation during training. - Note that this is solely for monitoring. We do not pick checkpoints based on this in our paper.
Dataset Structure
- train: training set consisting of 38870 multimodal reasoning samples
- val: validation set consisting of 2000 multimodal reasoning samples
Data Fields
- id: data id
- data type: String
- problem: input question or statement
- data type: String
- images: input image(s)
- data type: List
- answer: ground-truth answer
- data type: String
Usage
To use the full dataset with both train and val split, you may code as follows:
# Train
train_dataset = load_dataset("PAPOGalaxy/PAPO_ViRL39K_train")
# Val
val_dataset = load_dataset("PAPOGalaxy/PAPO_MMK12_test")
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