SynthRL
Collection
Models & Datasets of SynthRL
•
10 items
•
Updated
•
4
Dataset Viewer
id
stringlengths 6
9
| images
images listlengths 1
1
| problem
stringlengths 14
1.27k
| answer
stringlengths 1
79
| pass_rates
int32 0
16
| is_evolved
bool 1
class |
|---|---|---|---|---|---|
math_4480 | <image>
Please look at the Pascal's Triangle (1) on the left and observe the equation (2) on the right: Write down the coefficient of the term containing x^196 in the expansion of ${(x+\frac{1}{x})}^{200}$. | 19900 | 16 | false |
|
math_4022 | <image>
As shown in the figure, according to the pattern of the shapes and numbers, the 10th number is ______. | 55 | 15 | false |
|
math_7366 | <image>
The content executed by the program flowchart shown in the figure is. | 2+4+6+\cdot \cdot \cdot +2000=? | 16 | false |
|
math_2687 | <image>
In △ABC, points D, E, and F are on AB, AC, and BC respectively, with DE∥BC, EF∥AB, AD:BD = 5:3, and CF = 6. Find the length of DE. | 10 | 2 | false |
|
math_1804 | <image>
In July 2015, the 45th 'Top 500 World Supercomputers' list was released, and the 'Tianhe-2' supercomputer, developed by the National University of Defense Technology in China, won the championship for the fifth time with a floating-point operation speed of \(3386 \times 10^{13}\) operations per second. If \(3386 \times 10^{13}\) is expressed in scientific notation as \(a \times 10^n\), what is the value of \(n\)? | 16 | 6 | false |
|
math_1496 | <image>
As shown in the figure, if P is the circumcenter of △ABC, and $\overrightarrow{PA}$ + $\overrightarrow{PB}$ = $\overrightarrow{PC}$, then ∠ACB = ___ degrees? | 120 | 1 | false |
|
math_592 | <image>
As shown in the figure, the first two scales are in balance. To keep the third scale in balance, if only circles are placed on the right side, how many circles should be placed? | 3 | 5 | false |
|
math_4175 | <image>
As shown in the figure, the diagonal $AC$ of quadrilateral $ABCD$ is the perpendicular bisector of $BD$, and $AB=5$, $BC=3$. What is the perimeter of quadrilateral $ABCD$? | 16 | 15 | false |
|
math_7886 | <image>
In the cube $$ABCD-A_{1}B_{1}C_{1}D_{1}$$, $$E$$ and $$F$$ are the midpoints of $$AB_{1}$$ and $$BC_1$$ respectively. There are the following conclusions: 1. $$EF$$ is perpendicular to $$CC_{1}$$; 2. $$EF$$ is perpendicular to $$BD$$; 3. $$EF$$ and $$A_{1}C_{1}$$ are skew lines; 4. $$EF$$ and $$AD_{1}$$ are skew lines. The incorrect conclusion number is ___. | 3 | 0 | false |
|
math_6846 | <image>
As shown in the figure, the radius of circle O is $1$, and hexagon $ABCDEF$ is a regular hexagon inscribed in circle O. If two points are randomly selected from $A, B, C, D, E, F$ and connected to form a line segment, what is the probability that the length of the line segment is $\sqrt{3}$? | \frac{2}{5} | 12 | false |
|
math_6506 | <image>
As shown in the figure, one side of $$\triangle ABC$$, $$AB$$, is the diameter of circle $$⊙O$$. What condition can be added to make $$BC$$ a tangent to $$⊙O$$? | \angle ABC=90 | 10 | false |
|
math_5650 | <image>
The students of the third year of junior high school in a certain middle school are conducting a practical activity to measure the height of objects. They need to measure the height of a building $$AB$$. As shown in the figure, they first measure the angle of elevation to the top point $$A$$ of the building $$AB$$ from point $$C$$ to be $$30^{\circ}$$, then they move forward $$10\,\text{m}$$ to point $$D$$ and measure the angle of elevation to point $$A$$ again, which is $$60^{\circ}$$. What is the height of the building $$AB$$ in meters? | 5 \sqrt{3} | 11 | false |
|
math_7733 | <image>
Execute the following program flowchart. If the input is $$a=-1$$, then the output $$S=$$ ___. | 3 | 16 | false |
|
math_949 | <image>
If the function $$f(x)=\frac{bx+c}{x^2+ax+1} (a,b,c \in \bf{R})$$, and its graph is shown in the figure, then $$a+b+c=$$______． | 4 | 0 | false |
|
math_3669 | <image>
Given the flowchart of a certain algorithm as shown in the figure, if the blank box is $A=\frac{1}{1+2A}$, then the double of the output result is. | 1 | 16 | false |
|
math_551 | <image>
As shown in the figure, lines $$a$$ and $$b$$ are intersected by line $$c$$. If $$a \parallel b$$, which of the following conditions can be added? (Fill in one only.) | \angle 1=\angle 3 | 4 | false |
|
math_3510 | <image>
As shown in the figure, the main view area of the geometric body composed of three small cubes, each with an edge length of 1 cm, is $\text{cm}^2$. | 3 | 9 | false |
|
math_5597 | <image>
As shown in the figure, $\angle A=\alpha$, the angle bisectors of $\angle ABC$ and $\angle ACD$ intersect at point ${{P}_{1}}$, the angle bisectors of $\angle {{P}_{1}}BC$ and $\angle {{P}_{1}}CD$ intersect at point ${{P}_{2}}$, the angle bisectors of $\angle {{P}_{2}}BC$ and $\angle {{P}_{2}}CD$ intersect at point ${{P}_{3}}$... and so on. The measure of $\angle {{P}_{n}}$ is (expressed as an algebraic expression containing $n$ and $\alpha$). | {{\left( \frac{1}{2} \right)}^{n}}\alpha | 11 | false |
|
math_1425 | <image>
Fold △ABC along EF and DE, such that vertices A and C both land on point M, and CE coincides with AE on segment EM. If ∠FMD = 145°, then the measure of ∠B is ___ degrees. | 35 | 3 | false |
|
math_4316 | <image>
As shown in the figure, the diagonal BD of rectangle ABCD passes through the origin of the coordinate system, and the sides of the rectangle are parallel to the coordinate axes. Point C lies on the graph of the inverse proportion function y=$\frac{k}{x}$. If the coordinates of point A are (﹣2, ﹣2), then the value of k is. | 4 | 13 | false |
|
math_4713 | <image>
As shown in the figure, in the isosceles right triangle $$\text{Rt} \triangle OAA_{1}$$, $$\angle OAA_{1}=90^{\circ}$$, $$OA=1$$. Using $$OA_{1}$$ as the leg, construct the isosceles right triangle $$\text{Rt} \triangle OA_{1}A_{2}$$, and using $$OA_{2}$$ as the leg, construct the isosceles right triangle $$\text{Rt} \triangle OA_{2}A_{3}$$, $$\cdots$$, then the length of $$OA_{6}$$ is ___. | 8 | 0 | false |
|
math_7497 | <image>
Look at the picture and list the equation. ______ | 2+4=6 | 0 | false |
|
math_6814 | <image>
As shown in the figure, point P is on the side AD of square ABCD, connect PB, and draw a ray from point B intersecting the extension of side DC at point Q, such that ∠QBE=∠PBC, where E is a point on the extension of side AB. Connect PQ. If PQ²=PB²+PD²+3, then the area of △PAB is. | \frac{3}{4} | 0 | false |
|
math_7507 | <image>
Execute the following program, the output value is ___. | -1 | 16 | false |
|
math_748 | <image>
As shown in the figure, the diameter of $$\odot O$$ is $$20cm$$, chord $$AB=16cm$$, and $$OD\bot AB$$, with the foot of the perpendicular at $$D$$. Then, $$AB$$ must be translated ___$$cm$$ along the ray $$OD$$ to be tangent to $$\odot O$$. | 4 | 15 | false |
|
math_2369 | <image>
Each basket of waxberries is based on a standard of 5 kilograms. Excess kilograms are recorded as positive numbers, and deficiencies are recorded as negative numbers, as shown in the figure. What is the total weight of the 4 baskets of waxberries in kilograms? | 20.1 | 12 | false |
|
math_5167 | <image>
As shown in the figure, DE is the midsegment of △ABC, and the altitude AM of △ABC intersects DE at N. Then the value of $\frac{AN}{AM}$ is. | \frac{1}{2} | 16 | false |
|
math_6991 | <image>
Execute the program flowchart shown in the figure. If the input is $$t \in [-2,2]$$, then the output $$S$$ belongs to ___. | [-3,6] | 0 | false |
|
math_5298 | <image>
During the 'Great Motherland, Beautiful Hometown' themed publicity week, a school launched five travel routes: A, B, C, D, and E. The school's photography club randomly selected some students to participate in a 'Favorite Travel Route' voting activity, where each participant chose one route they loved the most. The club then compiled the votes and created the following incomplete bar chart and pie chart. Among the 2400 students in the school, estimate the number of students who chose route 'C'. | 600 | 0 | false |
|
math_3999 | <image>
As shown in the figure, in the Cartesian coordinate system, the graphs of the functions $$y=\dfrac{k}{x}$$ ($$x>0$$) and $$y=x-1$$ intersect at point $$P(a,b)$$. Given that $$\dfrac{1}{a}-\dfrac{1}{b}=-\dfrac{1}{4}$$, find the value of $$k$$. | 4 | 15 | false |
|
math_4110 | <image>
As shown in the figure, let the universal set be $$U=\mathbf{R}$$, $$A=\{x|x(x-2) < 0\}$$, $$B=\{x|y= \ln (1-x)\}$$, then what set is represented by the shaded area in the figure? | \{x|1 \leqslant x < 2\} | 0 | false |
|
math_945 | <image>
As shown in the figure, given the planar quadrilateral $$ABCD$$, $$AB \perp BC$$, $$AB = BC = AD = 2$$, $$CD = 3$$, and $$AC$$ intersects $$BD$$ at point $$O$$. Let $$I_{1} = \overrightarrow{OA} \cdot \overrightarrow{OB}$$, $$I_{2} = \overrightarrow{OB} \cdot \overrightarrow{OC}$$, $$I_{3} = \overrightarrow{OC} \cdot \overrightarrow{OD}$$. The relationship in magnitude of $$I_{1}$$, $$I_{2}$$, and $$I_{3}$$ is ___. (Arrange from smallest to largest) | I_{3} < I_{1} < I_{2} | 0 | false |
|
math_8049 | <image>
As shown in the figure, the diameter $$AB \perp CD$$ at point $$E$$, and $$\angle COB= \alpha$$. Then $$\dfrac{AB}{BE} \sin ^{2}\dfrac{ \alpha }{2}=$$ ___. | 1 | 1 | false |
|
math_2159 | <image>
What is the running result of the algorithm represented by the following program flowchart? | 6 \sqrt{6} | 16 | false |
|
math_4647 | <image>
In 5 physical fitness tests, the scores of person A and person B are shown in the table below, where ● represents a digit that has been stained. The probability that the average score of A exceeds the average score of B is. | \frac{4}{5} | 0 | false |
|
math_1081 | <image>
The point representing the real number $$a$$ on the number line is shown in the figure. Simplify $$\sqrt{\left(a-5\right)^2} +|a-2|$$, the result is ___. | 3 | 16 | false |
|
math_7207 | <image>
As shown in the figure, line AB intersects line CD at point E, EF is perpendicular to AB, and the foot of the perpendicular is E, ∠1 = 130°, then ∠2 = ___ degrees. | 40 | 0 | false |
|
math_2900 | <image>
As shown in the figure, in the rectangular coordinate system $xOy$, the terminal side of angle $\alpha$ intersects the unit circle at point $A$ in the second quadrant, where $\cos \alpha =-\frac{3}{5}$. What are the coordinates of point $A$? | \left( -\frac{3}{5}, \frac{4}{5} \right) | 16 | false |
|
math_505 | <image>
As shown in the figure, in square ABCD, point E is the midpoint of AD. Connect EC, and draw EF⊥EC, intersecting AB at point F. Then tan∠ECF=． | \frac{1}{2} | 4 | false |
|
math_2746 | <image>
If $$\triangle ADE\cong \triangle ACB$$, and $$\dfrac{AD}{AC}=\dfrac{2}{3}$$, $$DE=10$$, then $$BC=$$___. | 15 | 6 | false |
|
math_2163 | <image>
As shown in the figure, in the measuring tool $$ABC$$ for measuring the diameter of a small glass tube, the length of $$AB$$ is $$10\,\text{mm}$$, and $$AC$$ is divided into $$60$$ equal parts. If the diameter of the small tube $$DE$$ is exactly aligned with the $$30$$th part of the measuring tool $$(DE \parallel AB)$$, then the length of the small tube diameter $$DE$$ is ___ $$\text{mm}$$. | 5 | 16 | false |
|
math_1846 | <image>
As shown in the figure, in quadrilateral $$ABCD$$, diagonal $$AC \bot BD$$, with the foot of the perpendicular at $$O$$. Points $$E$$, $$F$$, $$G$$, and $$H$$ are the midpoints of sides $$AD$$, $$AB$$, $$BC$$, and $$CD$$, respectively. If $$AC=8$$ and $$BD=6$$, then the area of quadrilateral $$EFGH$$ is ___. . | 12 | 15 | false |
|
math_3815 | <image>
The three views of a geometric solid composed of a cuboid and two $$\dfrac{1}{4}$$ cylinders are shown in the figure. What is the volume of this geometric solid? | 2+\dfrac{\pi }{2} | 10 | false |
|
math_1887 | <image>
In △ABC, points D and E are on AB and AC respectively, with DE∥BC, AD = 1, BD = 3. The ratio of the area of △ADE to the area of △ABC is. | \frac{1}{16} | 16 | false |
|
math_7376 | <image>
Run the pseudocode as shown in the figure, the result is ___. | 17 | 14 | false |
|
math_4230 | <image>
In the 'Luo Shu' of our country, the world's oldest magic square is recorded: Fill the numbers 1 to 9 into a $3\times 3$ grid so that the sum of the three numbers in each row, each column, and each diagonal is equal. In the magic square shown in the figure, the number represented by the letter $m$ is. | 4 | 1 | false |
|
math_7038 | <image>
As shown in the figure, points A and B are both on the graph of the inverse proportion function y = $\frac{k}{x}$ in the second quadrant, and triangle OAB is an equilateral triangle. If AB = 6, then the value of k is. | -9 | 1 | false |
|
math_4128 | <image>
As shown in the figure, the number of times statement 1 is executed in the program flowchart is. | 34 | 0 | false |
|
math_7519 | <image>
The output of the following program is ___. | 21 | 5 | false |
|
math_956 | <image>
In summer, lotus flowers are in full bloom. To help visitors enjoy the beautiful scenery of 'people walking on the bridge as if they were walking in the river,' a scenic spot plans to build a small bridge over the rectangular lotus pond as shown in the figure. If the perimeter of the pond is 280 meters and the width of the bridge is negligible, then the total length of the bridge is ___ meters. | 140 | 13 | false |
|
math_6423 | <image>
To understand the growth of an economic forest, the base circumference (unit: cm) of 60 trees was randomly sampled, and all the data fall within the range [80, 130]. The frequency distribution histogram is shown in the figure. In the 60 sampled trees, there are trees with a base circumference less than 100 cm. | 24 | 14 | false |
|
math_6724 | <image>
As shown in the figure, points D and E are on the sides AB and AC of △ABC respectively, with DE∥BC, $AD:AB=2:5$. If the vector $\overrightarrow{BC}=\overrightarrow{a}$, then $\overrightarrow{DE}=$. | \frac{2}{5}\overrightarrow{a} | 16 | false |
|
math_7716 | <image>
In the figure, in $\Delta ABC$, points E and F are on sides $AB$ and $AC$ respectively, with $EF//BC$, and $\frac{AE}{EB}=\frac{AF}{FC}=\frac{1}{2}$. If the area of $\Delta AEF$ is 1, then the area of quadrilateral $EBCF$ is. | 8 | 16 | false |
|
math_3807 | <image>
In order to set labor hour quotas, a workshop needs to determine the time spent on processing parts. For this purpose, five experiments were conducted. Based on the collected data (as shown in the table below), the regression equation obtained by the least squares method is $\hat{y}=0.67x+54.9$. It is now found that one of the data points in the table is blurry and unreadable. Please infer the value of this data point. | 68 | 2 | false |
|
math_3663 | <image>
As shown in the figure, the composite shape is made up of a circle, a triangle, and a rectangle. If we color these shapes using red and blue, with each shape being colored with only one color, what is the probability that all three shapes are colored the same? | \dfrac{1}{4} | 16 | false |
|
math_85 | <image>
The results of a germination test for a certain type of rapeseed under the same conditions are shown in the table below: The probability of this type of rapeseed germinating is ___ (result rounded to $$0.01$$). | 0.95 | 14 | false |
|
math_3530 | <image>
As shown in the figure, F is the midpoint of side DC of parallelogram ABCD. If the areas of triangles EFC, ABE, and AFD are 3 square centimeters, 4 square centimeters, and 5 square centimeters respectively, and the area of parallelogram ABCD is an integer. What is the area of triangle AEF in square centimeters? | 8 | 0 | false |
|
math_6389 | <image>
Execute the program flowchart as shown. If the output result is 5, then the value of the integer m is _____. | 5 | 7 | false |
|
math_5990 | <image>
As shown in the figure, the edge length of the cube $ABCD-{{A}_{1}}{{B}_{1}}{{C}_{1}}{{D}_{1}}$ is 2. Point $E$ is the midpoint of line segment ${{A}_{1}}{{B}_{1}}$, and points $F$ and $G$ are moving points on line segments ${{A}_{1}}D$ and $B{{C}_{1}}$, respectively. When the area of the top view of the tetrahedron $E-FGC$ is maximized, the area of the front view of the tetrahedron is . | 2 | 2 | false |
|
math_5257 | <image>
Among the following four figures, the one that cannot be the graph of the function $$y=f(x)$$ is ___. | 4 | 12 | false |
|
math_3590 | <image>
As shown in the figure, the plane figure is composed of rays $$AB$$, $$BC$$, $$CD$$, $$DE$$, and $$EA$$. What is the value of $$∠1+∠2+∠3+∠4+∠5$$ in degrees? | 360 | 4 | false |
|
math_5957 | <image>
As shown in the figure, in Figure 1, A1, B1, C1 are the midpoints of sides BC, CA, AB of △ABC, respectively. In Figure 2, A2, B2, C2 are the midpoints of sides B1C1, C1A1, A1B1 of △A1B1C1, respectively, and so on. Following this pattern, the number of parallelograms in the nth figure is ______. | 3n | 2 | false |
|
math_2066 | <image>
Run the flowchart shown in the figure. If the range of the output value $$y$$ is $$[0,10]$$, then the range of the input value $$x$$ is ___. | [-7,9] | 2 | false |
|
math_56 | <image>
As shown in the figure, there is a right-angled triangular piece of paper, with the two perpendicular sides AC=6cm and BC=8cm. Now, the right-angled side is folded along the line AD, so that it falls on the hypotenuse AB and coincides with AE. What is the length of CD in cm? | 3 | 2 | false |
|
math_3648 | <image>
As shown in the figure, after $$\triangle ABC$$ is folded along $$DE$$, point $$A$$ lands on point $$A'$$ on $$BC$$. If point $$D$$ is the midpoint of $$AB$$, and $$\angle B=50^{\circ}$$, then the degree measure of $$\angle BDA'$$ is ___ degrees. | 80 | 3 | false |
|
math_2489 | <image>
As shown in the figure, △ABC and △DCE are both equilateral triangles with side lengths of 4. Points B, C, and E lie on the same straight line. Connecting BD, what is the length of BD? | 4 \sqrt{3} | 3 | false |
|
math_5696 | <image>
As shown in the figure, $$\angle A$$ is the inscribed angle of circle $$\odot O$$, $$\angle OBC=55^{\circ}$$, then $$\angle A=$$ ___ degrees. | 35 | 6 | false |
|
math_6783 | <image>
As shown in the figure, in $$\triangle ABC$$, $$\angle C = 25^{\circ}$$, $$\angle B = 85^{\circ}$$, a circle passing through points $$A$$ and $$B$$ intersects sides $$AC$$ and $$BC$$ at points $$E$$ and $$D$$, respectively. Then $$\angle EDC = $$___ degrees. | 70 | 0 | false |
|
math_4219 | <image>
As shown in the figure, in the rectangular prism $$ABCD-A_{1}B_{1}C_{1}D_{1}$$, $$AB=AD=3\ \unit{cm}$$, $$AA_{1}=2\ \unit{cm}$$, the volume of the quadrilateral pyramid $$A-BB_{1}D_{1}D$$ is ___$$\unit{cm^{3}}$$. | 6 | 11 | false |
|
math_3850 | <image>
As shown in the figure, $AB$ is the diameter of $\odot O$, $CD$ is a chord of $\odot O$, and $\angle DCB = 32^\circ$. What is the measure of $\angle ABD$ in degrees? | 58 | 7 | false |
|
math_6833 | <image>
As shown in the figure, a square ABCD with a side length of $$\dfrac{1}{2}$$ is placed inside a semicircle with a radius of $$1$$. If a point is randomly thrown into the semicircle, what is the probability that the point lands inside the square? | \dfrac{1}{2\pi } | 16 | false |
|
math_5589 | <image>
In an experiment, four sets of data for $$(x,y)$$ were measured as follows: According to the table, the regression equation is $$\widehat{y}=-5x+\hat{a}$$, based on this model, the predicted value of $$y$$ when $$x=20$$ is ___. | 26.5 | 10 | false |
|
math_1988 | <image>
As shown in the figure, it is known that $\vartriangle ABC \sim \vartriangle ACP$, $AB=5$, $AC=2$. The ratio of the perimeters of $\vartriangle ABC$ and $\vartriangle ACP$ is. | 5:2 | 16 | false |
|
math_3986 | <image>
As shown in the diagram, this is a schematic designed by Xiao Ming to measure the height of a city wall using a laser pointer. A flat mirror is placed horizontally at point $$P$$. The light beam starts from point $$A$$, reflects off the flat mirror, and precisely hits the top of the city wall $$CD$$ at point $$C$$. It is known that $$AB \perp BD$$, $$CD \perp BD$$, $$AB=1.2$$ meters, $$BP=1.8$$ meters, $$PD=12$$ meters. Therefore, the height of the city wall $$CD=$$___ meters. | 8 | 16 | false |
|
math_2469 | <image>
In the figure, in parallelogram ABCD, AE bisects ∠BAD. If ∠B = 52°, then the measure of ∠AEC is ___ degrees? | 116 | 8 | false |
|
math_271 | <image>
In the figure, in triangle ABC, E is a point on BC such that EC = 2BE, and point D is the midpoint of AC. Let the areas of triangles ABC, ADF, and BEF be S$_{△ABC}$, S$_{△ADF}$, and S$_{△BEF}$, respectively, and S$_{△ABC}$ = 12. Then S$_{△ADF}$ - S$_{△BEF}$ =. | 2 | 3 | false |
|
math_4813 | <image>
Given the function $f\left( x \right)=2\sin\left( \omega x+\varphi \right)(\omega > 0,\varphi \in \left[ \frac{\pi }{2},\pi \right])$, part of its graph is shown in the figure, where $f\left( 0 \right)=1$ and $\left| MN \right|=\frac{5}{2}$. Then $f\left( 1 \right)=$． | -1 | 1 | false |
|
math_7085 | <image>
As shown in the figure, AB∥DC, and AC intersects BD at point O. Given $\frac{AO}{CO}=\frac{3}{4}$, and BO＝6, then DO＝. | 8 | 16 | false |
|
math_4516 | <image>
As shown in the figure, points $$A$$, $$B$$, and $$C$$ are on the same straight line, and point $$M$$ is outside $$AC$$. How many circles can be drawn passing through any three of these points? | 3 | 8 | false |
|
math_1498 | <image>
As shown in the figure, the graph of the quadratic function $y=ax^2+bx+c$ intersects the positive half of the y-axis, with its vertex at $\left( \frac{1}{2}, 1 \right)$. Which of the following conclusions are correct: 1. $abc < 0$; 2. $a+b=0$; 3. $4ac-b^2=4a$; 4. $a+b+c < 0$. How many of these are correct? | 3 | 10 | false |
|
math_7928 | <image>
The following shapes are all composed of circles and equilateral triangles of the same size according to a certain pattern. The 1st shape is composed of 8 circles and 1 equilateral triangle, the 2nd shape is composed of 16 circles and 4 equilateral triangles, the 3rd shape is composed of 24 circles and 9 equilateral triangles, … then in which shape do the number of circles and equilateral triangles become equal? | 8 | 12 | false |
|
math_1491 | <image>
As shown in the figure, in the Cartesian coordinate system, point P lies on the graph of the function y = $\frac{6}{x}$ (x > 0). Perpendiculars are drawn from point P to the x-axis and y-axis, with the feet of the perpendiculars being A and B, respectively. Point C is the midpoint of line segment OB, and line segment PC is extended to intersect the x-axis at point D. What is the area of △APD? | 6 | 6 | false |
|
math_2058 | <image>
In the right triangle ABC, ∠ACB = 90°, D, E, and F are the midpoints of AB, BC, and CA, respectively. If CD = 4 cm, then EF = cm. | 4 | 16 | false |
|
math_570 | <image>
As shown in the figure, without adding any auxiliary lines, write a condition that can determine $DE$∥$BC$. | \angle DAB=\angle B | 1 | false |
|
math_6942 | <image>
As shown in the figure, in square $$ABCD$$, $$P$$ is a point on $$BC$$ such that $$BC=4PC$$, and $$Q$$ is the midpoint of $$CD$$. If $$PQ=5$$, then $$AQ=$$____. | 10 | 1 | false |
|
math_1592 | <image>
Point $$P$$ is on the bisector of $$\angle AOB$$, and $$PD \perp OA$$, $$PE \perp OB$$, with the feet of the perpendiculars being $$D$$ and $$E$$ respectively, $$PD=3\ \unit{cm}$$, then $$PE=$$___ cm | 3 | 16 | false |
|
math_6605 | <image>
As shown in the figure, in triangle ABC, AB = AC, ∠A = 36°, and BD is the angle bisector of ∠ABC. What is the value of $\frac{AD}{AC}$? | \frac{-1+\sqrt{5}}{2} | 3 | false |
|
math_4318 | <image>
The top view of a shell is shown in the figure. Point $C$ divides segment $AB$ approximately according to the golden ratio. Given $AB$ = 10 cm, the length of $AC$ is approximately __ cm. (Round the result to 0.1 cm) | 6.2 | 15 | false |
|
math_2574 | <image>
In the right triangle $$\triangle ABC$$, $$\angle ACB = 90°$$, points $$D$$, $$E$$, and $$F$$ are the midpoints of $$AB$$, $$AC$$, and $$BC$$, respectively. If $$CD = 5$$, then the length of $$EF$$ is ___. | 5 | 16 | false |
|
math_2188 | <image>
There are 1,000 fibers of a certain cotton variety. A random sample of 50 fibers is taken, and the data on fiber length (unit: mm) is grouped and the frequency of each group is shown in the table. Based on this, estimate the number of fibers out of the 1,000 that have a length of at least 37.5 mm. | 180 | 4 | false |
|
math_454 | <image>
In the right triangle ABC, ∠B = 90°, AB = 6, BC = 8. The triangle ABC is folded so that point B exactly falls on side AC, coinciding with point B'. AE is the fold line. Find EB'. | 3 | 3 | false |
|
math_145 | <image>
As shown in the figure, in $$\triangle ABC$$, it is given that $$\overrightarrow{AN}=\dfrac{1}{2}\overrightarrow{AC}$$, and $$P$$ is a point on $$BN$$. If $$\overrightarrow{AP}=m\overrightarrow{AB}+\dfrac{1}{4}\overrightarrow{AC}$$, then the value of the real number $$m$$ is ___. | \dfrac{1}{2} | 3 | false |
|
math_1561 | <image>
As shown in the figure, the coordinates of the three vertices of $$\triangle ABC$$ are $$A\left (1,2\right )$$, $$B\left ( 3,1\right )$$, and $$C\left ( 2,3\right )$$. On the grid, with the origin $$O$$ as the center of dilation, $$\triangle ABC$$ is enlarged to twice its size to obtain $$\triangle A'B'C'$$. The area of $$\triangle A'B'C'$$ is ___. | 6 | 15 | false |
|
math_7119 | <image>
As shown in the figure, given point $$F(0,p)$$, line $$l$$: $$y=-p$$ (where $$p$$ is a constant and $$p > 0$$), $$M$$ is a moving point in the coordinate plane. A perpendicular line is drawn from $$M$$ to $$l$$, with the foot of the perpendicular being $$N$$. It is also given that $$\overrightarrow{NM}\cdot \overrightarrow{NF}=\overrightarrow{FM}\cdot \overrightarrow{FN}$$. Then the equation of the trajectory $$C$$ of the moving point $$M$$ is ___. | x^{2}=4py | 10 | false |
|
math_7272 | <image>
The result of the following pseudocode is ___. | 6 | 0 | false |
|
math_615 | <image>
Student Qingqing folded a rectangular piece of paper twice, as shown in the figure, so that points A and B both fall on DG, with the creases being DE and DF, respectively. What is the measure of ∠EDF in degrees? | 90 | 12 | false |
|
math_327 | <image>
The following shapes that have stability are ______ (fill in the code). | (2)(3) | 0 | false |
|
math_1293 | <image>
The retail price x (unit: yuan) and the daily sales volume y (unit: pieces) of a product at a retail stall are as follows: From the table, the linear regression equation $\widehat{y}=\widehat{b}x+\widehat{a}$ has $\widehat{b}=-4$. According to this model, predict the daily sales volume when the retail price is 15 yuan. | 49 | 0 | false |
|
math_1953 | <image>
In a class of 50 students, Xiaoli collected the height data of each student and drew the pie chart shown in the figure (where $$x$$ represents height, in centimeters). According to the chart, in what range does the median height of the 50 students fall? | 150 \leqslant x<160 | 16 | false |
|
math_6329 | <image>
Given the quadratic function y = (x - 2a)^2 + (a - 1) (where a is a constant), when a takes different values, its graph forms a 'family of parabolas'. The figure below shows the graphs of the quadratic function when a = -1, a = 0, a = 1, and a = 2. The vertices of these parabolas lie on a straight line. The equation of this line is. | y=0.5x-1 | 15 | false |
End of preview. Expand
in Data Studio
No dataset card yet
- Downloads last month
- 16
Size of downloaded dataset files:
53 MB
Size of the auto-converted Parquet files:
53 MB
Number of rows:
2,048