SynthRL
Collection
Models & Datasets of SynthRL
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10 items
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Updated
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4
Dataset Viewer
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math_7057 | <image>
As shown in the figure, the geometric solid is composed of a cylinder and a cone with the same base radius. The height of the cylinder is equal to its base radius. If the lateral surface areas of the cylinder and the cone are equal, then what is the ratio of the height of the cone to the height of the cylinder? | \sqrt{3} | 16 | false |
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math_2079 | <image>
As shown in the figure, in the Cartesian coordinate system, point $$A$$ is the intersection of the parabola $$y=a(x-1)^{2}+b$$ and the $$y$$-axis, point $$B$$ is another point on this parabola, and $$AB \parallel x$$-axis. Then the perimeter of the equilateral triangle $$ABC$$ with $$AB$$ as one side is ___. | 6 | 16 | false |
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math_1693 | <image>
As shown in the figure, the right-angle vertex of the triangle ruler is on the line $$l$$. If $$\angle 1=40^{°}$$, then the degree measure of $$\angle 2$$ is ___ degrees. | 50 | 3 | false |
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math_1610 | <image>
In the figure, in $\Delta ABC$, $AD \perp AB, {DC}^\rightharpoonup = 2 {BD}^\rightharpoonup , \left| {AD}^\rightharpoonup \right| = 3$. Find the value of $\overrightarrow{AC} \cdot \overrightarrow{AD}$. | 27 | 9 | false |
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math_5733 | <image>
If the plane net in the figure is to be folded into a cube such that the sum of the numbers on opposite faces is 6, then the value of $x+y+z$ is. | 4 | 16 | false |
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math_2571 | <image>
As shown in the figure, in the Cartesian coordinate system, the graph of the function $$y=\dfrac{k}{x}$$ ($$x>0$$, constant $$k>0$$) passes through the points $$A\left ( 1,2\right )$$ and $$B\left ( m,n\right ) \left ( m>1\right )$$. A perpendicular line from point $$B$$ to the y-axis meets the y-axis at point $$C$$. If the area of $$\triangle ABC$$ is $$2$$, then the coordinates of point $$B$$ are ___. | \left ( 3,\dfrac{2}{3}\right ) | 0 | false |
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math_4529 | <image>
As shown in the figure, in $$\triangle ABC$$, $$\angle BAC=33^{ \circ }$$. If $$\triangle ABC$$ is rotated clockwise around point $$A$$ by $$50^{ \circ }$$, the corresponding triangle obtained is $$\triangle AB'C'$$. Then the degree measure of $$\angle B'AC$$ is ___ degrees. | 17 | 1 | false |
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math_3984 | <image>
Xiao Ming hangs a 3×3 square grid paper board on the wall to play a dart game (each dart always lands on the paper board, and the probability of landing on any point on the paper board is equal). What is the probability that the dart lands in the shaded area? | \frac{4}{9} | 1 | false |
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math_7362 | <image>
Run the algorithm shown in the figure, then the output result is. | 16 | 16 | false |
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math_6932 | <image>
Define the operation $$\otimes$$, where the principle of the operation $$a\otimes b=S$$ is shown in the following program, then the expression $$5\otimes 3+2\otimes 4=$$ ___. | 32 | 16 | false |
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math_3854 | <image>
In the figure, CE bisects ∠ACB, and CE is perpendicular to BD. DA = DB, and it is given that AC = 18, the perimeter of △CDB is 28. What is the length of BE? | 4 | 0 | false |
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math_6219 | <image>
As shown in the figure, a student wants to use a measuring rod to measure the height of a large tree. If the height of the measuring rod EC is 1.8 m, and it is measured that AC=0.9 m, AB=2.1 m, then the height of the tree DB is m. | 4.2 | 15 | false |
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math_2942 | <image>
The function of current intensity $$I$$ (Amps) varying with time $$t$$ (seconds) is given by $$I = A \sin\left(\omega t + \frac{\pi}{6}\right) \left(A > 0, \omega > 0\right)$$. The graph of the function is shown below. What is the current intensity when $$t = \frac{1}{50}$$? | 5 | 1 | false |
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math_2947 | <image>
As shown in the figure, lines $$AB$$ and $$CD$$ intersect at point $$O$$, and $$\angle AOC + \angle BOD = 210^{\circ}$$, then $$\angle BOC =$$ ___ $$^{\circ}$$. | 75 | 10 | false |
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math_7616 | <image>
Fill in the appropriate digit, there are ______ ways to fill it? | 2 | 0 | false |
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math_250 | <image>
As shown in the figure, given that a, b, and c are the lengths of the three sides of the right triangle ABC, with ∠C = 90°, the linear function of the form y = $\frac{a}{c}x + \frac{b}{c}$ is called a 'Pythagorean linear function'. If point P (1, $\frac{3\sqrt{5}}{5}$) lies on the graph of the 'Pythagorean linear function', and the area of the right triangle ABC is 5, then the value of c is. | 5 | 16 | false |
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math_3855 | <image>
In the right triangle ABC, CD is the altitude on the hypotenuse AB. If AC = $\sqrt{5}$ and DB = 4, then the length of AD is. | 1 | 12 | false |
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math_2306 | <image>
According to the flowchart below, if $$x=5$$, the program should run ___ times before it stops. | 4 | 16 | false |
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math_2430 | <image>
As shown in the figure, the vertices C and A of rectangle OABC are on the x-axis and y-axis, respectively. The coordinates of point A are (0, 3), and the inverse proportion function $y=\frac{6}{x}$ passes through point B. A line BD is drawn parallel to AC and intersects the x-axis at point D. What are the coordinates of point D? | (4, 0) | 15 | false |
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math_4632 | <image>
Use the bisection method to find a zero of the function $$f(x)=3^{x}-x-4$$. The reference data is shown in the following table: From the table, an approximate solution to the equation $$3^{x}-x-4=0$$ is ___ (accurate to $$\number{0.01}$$). | 1.56 | 14 | false |
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math_5653 | <image>
As shown in the figure, point $$E$$ is inside the square $$ABCD$$, $$\angle AEB=90\degree$$, $$AE=6$$, $$BE=8$$. What is the area of the shaded part? | 76 | 15 | false |
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math_5292 | <image>
As shown in the graph of the quadratic function $y=ax^2+bx+c (a \ne 0)$, the following five pieces of information are observed: 1. $abc < 0$; 2. $a+b+c < 0$; 3. $b+2c > 0$; 4. $a-2b+4c > 0$; 5. $a=\frac{3}{2}b$. How many of these pieces of information do you think are correct? | 4 | 6 | false |
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math_180 | <image>
In the figure, the diagonals $AC$ and $BD$ of parallelogram $ABCD$ intersect at point $O$. Line $EF$ passes through point $O$ and intersects $AD$ and $BC$ at points $E$ and $F$, respectively. It is known that the area of parallelogram $ABCD$ is $20cm^2$. What is the area of the shaded region in cm^2? | 5 | 4 | false |
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math_4498 | <image>
As shown in the figure, the side length of equilateral triangle ABC is 16. The three midlines of △ABC form △A$_{1}$B$_{1}$C$_{1}$, the three midlines of △A$_{1}$B$_{1}$C$_{1}$ form △A$_{2}$B$_{2}$C$_{2}$, and so on. The perimeter of △A$_{4}$B$_{4}$C$_{4}$ is | 3 | 16 | false |
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math_7873 | <image>
As shown in the figure, in $$\triangle ABC$$, $$AC=6cm$$, $$BC=8cm$$, $$AB=10cm$$. Points $$D$$, $$E$$, and $$F$$ are the midpoints of $$AB$$, $$BC$$, and $$CA$$, respectively. What is the area of $$\triangle DEF$$ in cm^2? | 6 | 15 | false |
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math_2773 | <image>
As shown in the figure, the vertices A and C of rectangle OABC are on the positive y-axis and x-axis, respectively. D is the midpoint of AB, and the graph of the inverse proportion function y = k/x (k > 0) passes through point D and intersects BC at point E. Connecting OD, OE, and DE, if the area of ΔODE is 3, then the value of k is. | 4 | 5 | false |
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math_6894 | <image>
As shown in the figure, in $$\triangle ABC$$, $$D$$ is a point on the side $$BC$$, and $$BD=2DC$$. If $$\overrightarrow{AC}=m\overrightarrow{AB}+n\overrightarrow{AD}$$ ($$m$$, $$n \in \mathbf{R}$$), then $$m-n=$$ ___. | -2 | 6 | false |
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math_2382 | <image>
The positions of real numbers a, b, and c on the number line are shown in the figure. Simplify the expression $\sqrt{(a+c)^2} - \sqrt{(c-b)^2} - |a-2b|$. | -3b | 14 | false |
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math_41 | <image>
The German mathematician Leibniz discovered a unit fraction triangle (a unit fraction refers to a fraction with a numerator of $1$ and a denominator that is a positive integer), known as the Leibniz triangle. According to the pattern in the first $6$ rows, the third number from the left in the $7$th row is. | \frac{1}{105} | 0 | false |
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math_66 | <image>
In the flowchart shown, if the input $$x \in [-1,4]$$, then the probability that the output $$y \in (0,1]$$ is ___. | \dfrac{1}{5} | 8 | false |
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math_5763 | <image>
Given, as shown in the figure, $$AC=AE$$, $$\angle1=\angle2$$, and $$AB=AD$$. If $$\angle D=25^{\circ}$$, then the measure of $$\angle B$$ is ___ degrees. | 25 | 4 | false |
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math_5479 | <image>
As shown in the figure, the side length of square $$ABCD$$ is $$4$$. Points $$M$$ and $$N$$ are moving points on $$BC$$ and $$CD$$, respectively, and it is always true that $$AM \bot MN$$. When $$BM=$$___, the area of quadrilateral $$ABCN$$ is maximized. | 2 | 13 | false |
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math_6199 | <image>
As shown in the figure, the side length of the square $$ABCD$$ is $$2$$. If line segment $$BD$$ is rotated around point $$B$$, point $$D$$ lands on point $$D'$$ on the extension of $$CB$$. Then, what is the value of $$tan\angle BAD'$$? | \sqrt{2} | 0 | false |
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math_1949 | <image>
As shown in the figure, if the circumference of the circle is $$62.8\rm cm$$, then the perimeter of the square is ______ $$cm$$ (take the value of π as $$3.14$$). | 80 | 16 | false |
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math_5055 | <image>
As shown in the figure, line $MN$ is the axis of symmetry of quadrilateral $AMBN$, and point $P$ is a point on line $MN$. Among the following conclusions: 1. $AM = BM$; 2. $AP = BN$; 3. $\angle MAP = \angle MBP$; 4. $\angle ANP = \angle BNM$, which statement is incorrect? | 2 | 14 | false |
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math_255 | <image>
As shown in the figure, in trapezoid ABCD, AD∥BC, BD⊥DC. If AD=2, BC=4, then the maximum value of the area of trapezoid ABCD is. | 6 | 3 | false |
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math_2308 | <image>
As shown in the figure, the graph of the linear function $$y=kx+b$$ (where $$k, b$$ are constants and $$k \neq 0$$) intersects the graph of the inverse proportion function $$y=\dfrac{4}{x}$$ (for $$x > 0$$) at points $$A$$ and $$B$$. Using the graph, directly write the solution set of the inequality $$\dfrac{4}{x} < kx+b$$. | 1 < x < 4 | 9 | false |
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math_2523 | <image>
Congcong likes reading. The chart below shows his reading situation. Congcong read ______ pages in a week. | 63 | 16 | false |
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math_287 | <image>
As shown in the figure, in the Cartesian coordinate system $xOy$, the curves ${{C}_{1}}$, ${{C}_{2}}$, and ${{C}_{3}}$ are the graphs of $y=2{{\log }_{2}}x$, $y={{\log }_{2}}x$, and $y=k{{\log }_{2}}x$ respectively, where $k$ is a constant, $0 < k < 1$. Point $A$ is a point on curve ${{C}_{1}}$ located in the first quadrant. Lines parallel to the $x$-axis and $y$-axis through $A$ intersect curve ${{C}_{2}}$ at points $B$ and $D$ respectively. A line parallel to the $y$-axis through point $B$ intersects curve ${{C}_{3}}$ at point $C$. If quadrilateral $ABCD$ is a rectangle, then the value of $k$ is. | \frac{1}{2} | 16 | false |
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math_7777 | <image>
The front view and left view of a geometric solid made up of several identical small cubes are shown in the figure. The maximum number of small cubes that make up this geometric solid is ___. | 5 | 4 | false |
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math_1029 | <image>
In the diagram, in $$\triangle ABC$$, $$AD=DE$$, $$AB=BE$$, $$\angle A=85^{\circ}$$, then the measure of $$\angle CED$$ is ___ degrees. | 95 | 3 | false |
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math_6006 | <image>
The partial graph of the function $f(x) = \sin(\omega x + \varphi)$, where $0 < \omega < 3$ and $0 < \varphi < \frac{\pi}{2}$, is shown below. Find the value of $\omega + \varphi$. | \frac{\pi}{6} + 2 | 7 | false |
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math_5165 | <image>
Given the graph of the linear function $$y=kx+b$$ as shown, the range of $$x$$ when $$y < 0$$ is ___. | x<1 | 16 | false |
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math_4169 | <image>
A ship is at point $O$, and it measures that lighthouse $A$ is in the direction of $40^\circ$ north of east, and lighthouse $B$ is in the direction of $60^\circ$ south of east. What is $\angle AOB$ in degrees? | 80 | 0 | false |
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math_2754 | <image>
Given the quadratic function y = -x^2 - 2x + 3, the graph intersects the x-axis at points A and B (A is to the left of B) and the y-axis at point C, with the vertex of the graph being D. Find the equation of the line CD. | y = -x + 3 | 16 | false |
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math_2757 | <image>
The daily maximum temperatures for a week in my city are recorded in the following table: What is the average of the daily maximum temperatures for this week in $$\unit{\degreeCelsius } $$? | 27 | 3 | false |
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math_5011 | <image>
As shown in the figure, in $\Delta ABC$, $AB=AC=16cm$. The perpendicular bisector of $AB$ intersects $AC$ at point $D$. If the perimeter of $\Delta BCD$ is 26cm, then $BC=$cm. | 10 | 15 | false |
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math_6441 | <image>
As shown in the figure, in the Cartesian coordinate system, point A (0, 3). Triangle AOB is translated to the right along the x-axis to obtain triangle A'O'B'. The corresponding point A' of point A lies exactly on the line y = $\frac{3}{2}$x. Then BB' =. | 2 | 12 | false |
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math_6387 | <image>
Let the function f(x) = , then f(f(2)) = ______. | 2 | 16 | false |
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math_7957 | <image>
The figure is a plane shape composed of rays $$AB$$, $$BC$$, $$CD$$, $$DE$$, and $$EA$$. What is the value of $$ \angle 1+ \angle 2+ \angle 3+ \angle 4+ \angle 5$$ in degrees? | 360 | 9 | false |
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math_6782 | <image>
The probability of annual precipitation in a certain region falling within the following ranges is shown in the table: The probability of annual precipitation being in the range $$[150,300)(\unit{mm})$$ is ___. | 0.55 | 12 | false |
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math_5450 | <image>
As shown in the figure, it is known that $$A$$, $$B$$, $$C$$, $$D$$, and $$E$$ are all on circle $$⊙O$$, and $$AC$$ is the diameter of $$⊙O$$. Then, $$∠A + ∠B + ∠C =$$ ___ degrees. | 90 | 0 | false |
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math_4136 | <image>
Given the probability distribution of the random variable $$X$$ as follows: where $$a$$, $$b$$, $$c$$ form an arithmetic sequence, then $$P(|X|=1)=$$ ___. | \dfrac{2}{3} | 16 | false |
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math_7935 | <image>
As shown in the figure, $$AC$$ intersects $$BD$$ at point $$O$$, and $$AB=CD$$. Please add a condition ___ to make $$\triangle ABO\cong \triangle CDO$$. | \angle A=\angle C | 0 | false |
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math_6362 | <image>
The function $$f(x)=A \sin ( \omega x+ \varphi )\left(A > 0, \omega > 0,| \varphi | < \dfrac{ \pi }{2}\right)$$ has a partial graph as shown in the figure. If the graph of $$y=f(x)$$ is shifted to the right by $$\dfrac{ \pi }{6}$$ units, what is the expression for the resulting function $$g(x)$$? | g(x)= \sin \left ( 2x-\dfrac{\pi }{6}\right ) | 0 | false |
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math_4501 | <image>
As shown in the figure, in the Cartesian coordinate system, the coordinates of vertex B of rectangle OABC are (12,5). The line y = (1/4)x + b exactly divides the rectangle OABC into two equal areas. Therefore, b = . | 1 | 9 | false |
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math_4317 | <image>
In the figure, the perpendicular bisector of side $AB$ intersects $BC$ at point $D$ and $AB$ at point $E$. If $AE=3$, and the perimeter of $\Delta ADC$ is 9, then the perimeter of $\Delta ABC$ is . | 15 | 15 | false |
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math_2257 | <image>
As shown in the figure, the graph of the function $$f(x)$$ is the line segment $$ABC$$, where the coordinates of $$A$$, $$B$$, and $$C$$ are ($$0$$, $$4$$), ($$2$$, $$0$$), and ($$6$$, $$4$$), respectively. Then $$f(f(0))=$$ ___. | 2 | 3 | false |
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math_3717 | <image>
As shown in the figure, the area of ABCD is 12, points E and F are on AC, and AE = EF = FC, then the area of △BEF is. | 2 | 6 | false |
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math_6414 | <image>
As shown in the figure, points B, C, and D are on the same straight line, CE is parallel to AB, and ∠ACB = 90°. If ∠A = 60°, then ∠ECD = ___ degrees? | 30 | 6 | false |
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math_689 | <image>
As shown in the figure, given $AC \parallel DE$, $\angle B = 24^{\circ}$, $\angle D = 58^{\circ}$, then $\angle C =$ ___ degrees. | 34 | 0 | false |
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math_811 | <image>
Arrange 'goldfish' using matchsticks as shown in the figure: . According to the pattern above, the number of matchsticks needed for the $$n\left(n \in \mathbf{N}^{*}\right)$$th 'goldfish' is ___. | 6n+2 | 0 | false |
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math_4537 | <image>
The class president of a ninth-grade class at a school compiled the number of extracurricular books read by all classmates from January to August last year (unit: books) and created the line graph shown below. The median of this set of data is ______. | 58 | 16 | false |
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math_5930 | <image>
Execute the flowchart shown in the figure, then the output value of k is. | 5 | 13 | false |
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math_1363 | <image>
As shown in the figure, with point 0 as the center of similarity, the pentagon ABCDE is enlarged to obtain the pentagon A′B′C′D′E′. Given that OA = 10 cm and OA′ = 20 cm, the ratio of the perimeter of pentagon ABCDE to the perimeter of pentagon A′B′C′D′E′ is ______. | 1:2 | 13 | false |
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math_3289 | <image>
As shown in the figure, BM is the median of △ABC. If AB = 5 cm and BC = 3 cm, then the difference in the perimeters of △ABM and △BCM is ______ cm. | 2 | 16 | false |
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math_6138 | <image>
As shown in the figure, a curved pipe, after two bends, remains parallel. If $\angle C = 59^\circ$, then $\angle B =$ ___ degrees. | 121 | 1 | false |
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math_7752 | <image>
As shown in the figure, points $$A$$, $$C$$, $$E$$, $$B$$, $$D$$ are on a straight line, $$AB = CD$$, point $$E$$ is the midpoint of $$CB$$, if $$AE = 10$$, $$CB = 4$$, then the length of $$BD$$ is ___. | 8 | 13 | false |
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math_6704 | <image>
As shown in the figure, in $\Delta ABC$, $AB=AC$, and the measure of $\angle A$ is ${{40}^{\circ }}$. Point $D$ is a point on side $BC$. On $AB$, take $BE=CD$, and on $AC$, take $CF=BD$. Connect $DE$ and $DF$. What is the measure of $\angle EDF$ in degrees? | 70 | 7 | false |
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math_3279 | <image>
As shown in the figure, in $\vartriangle ABC$, $\overrightarrow{AD}=\frac{1}{2}\overrightarrow{AB}$, $\overrightarrow{AE}=\frac{1}{3}\overrightarrow{AC}$, $CD$ and $BE$ intersect at point $P$, $AB=2$, $AC=4$, and $\overrightarrow{AP}\cdot \overrightarrow{BC}=2$. Find the value of $\overrightarrow{AB}\cdot \overrightarrow{AC}$. | 2 | 0 | false |
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math_3587 | <image>
In the 'Creating a Beautiful Campus, Striving to be Civilized Students' model school evaluation activity, the scores given by 10 judges to a certain school are shown in the table below. What is the average score given by these 10 judges? | 89 | 16 | false |
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math_7958 | <image>
As shown in the figure, the regular hexagon $$ABCDEF$$ is inscribed in $$\odot{}O$$, with the radius of $$\odot{}O$$ being $$1$$. The length of the arc $$\overset{\frown} {AB}$$ is ___. | \dfrac{\pi }{3} | 14 | false |
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math_2670 | <image>
A certain unit conducts a professional skills test for 2000 employees, with test scores ranging from [50, 100]. The scores are grouped into [50, 60), [60, 70), [70, 80), [80, 90), and [90, 100], and the frequency distribution histogram of the employees' test scores is shown in the figure. If a stratified sampling method is used to select 50 people's test scores for detailed analysis, how many people with test scores of 80 or above (including 80) should be selected? | 15 | 11 | false |
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math_6379 | <image>
As shown in the figure, the chord $$AB=8\rm{cm}$$ of circle $$\odot O$$, and the distance from the center $$O$$ to the chord $$AB$$ is $$3\rm{cm}$$. What is the diameter of $$\odot O$$? ______ $$\rm{cm}$$ | 10 | 16 | false |
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math_7349 | <image>
Run the following pseudocode, the result is ___. | 17 | 12 | false |
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math_441 | <image>
In a senior high school, Class A and Class B of the third grade each selected 7 students to participate in a high school mathematics competition. The stem-and-leaf plot of their scores is shown below, where the median score of Class A is 81, and the average score of Class B is 86. What is the value of $x+y$? | 5 | 1 | false |
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math_3947 | <image>
Read the following flowchart and run the corresponding program. The output value of $$S$$ is ___. | 4 | 1 | false |
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math_6573 | <image>
As shown in the figure, in $\Delta ABC$, $DE$ is the perpendicular bisector of $BC$, with the foot of the perpendicular at $E$, and intersects $AC$ at point $D$. If $AB=6, AC=9$, then the perimeter of $\Delta ABD$ is. | 15 | 16 | false |
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math_1914 | <image>
As shown in the figure, $$⊙O$$ is the circumcircle of $$\triangle ABC$$, the radius of $$⊙O$$ is $$R=2$$, and $$\sin B=\dfrac{3}{4}$$, then the length of the chord $$AC$$ is ___. | 3 | 14 | false |
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math_5320 | <image>
The graph of the function $$y=f(x)$$ is shown below. Which of the following statements is correct? 1. $$f'(3) > 0$$; 2. $$f'(3) < 0$$; 3. $$f'(3)=0$$; 4. The sign of $$f'(3)$$ is uncertain. | 2 | 7 | false |
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math_5438 | <image>
Given the flowchart of a program as shown in the figure, if $$a=0.6^{2}$$, $$b=3^{0.5}$$, $$c= \log \nolimits _{0.5}5$$, then the number output by the program is ___. | \sqrt{3} | 0 | false |
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math_5680 | <image>
As shown in the figure, $$AB$$ is a chord of $$\odot O$$, with a length of $$8$$. $$P$$ is a moving point on $$\odot O$$ (not coinciding with $$A$$ or $$B$$). $$OC \perp AP$$ at point $$C$$, and $$OD \perp PB$$ at point $$D$$. The length of $$CD$$ is ______. | 4 | 16 | false |
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math_2944 | <image>
As shown in the figure, $$\triangle ABC$$ is an isosceles right triangle, $$\angle BAC = 90^{\circ}$$, $$AB = AC = 1$$. When $$\triangle ABC$$ is folded along the altitude $$AD$$ on the hypotenuse $$BC$$, the plane $$ABD$$ is perpendicular to the plane $$ACD$$. Then, $$BC =$$ ___. | 1 | 0 | false |
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math_2624 | <image>
As shown in the figure, points $$A$$, $$B$$, and $$C$$ are on circle $$O$$, and $$AB=4$$, $$\angle ACB=45^{\circ}$$, then the area of circle $$O$$ is ___. | 8\pi | 8 | false |
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math_852 | <image>
If the graph of the linear function $$y=kx+b(b \neq 0)$$ is as shown in the figure, then the solution set of the inequality $$kx+b > 0$$ is ___. | x < 2 | 16 | false |
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math_381 | <image>
As shown in the figure, during a surveying activity, a student stands at point $$A$$ to observe markers placed at points $$B$$ and $$C$$. The data shows that point $$B$$ is located 20 meters away from point $$A$$ in the direction of 75° north of east, and point $$C$$ is located 20 meters away from point $$A$$ in the direction of 15° south of west. What is the distance between points $$B$$ and $$C$$ in meters? | 20 \sqrt{2} | 0 | false |
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math_6346 | <image>
As shown in the figure, point $$A$$ is on the graph of an inverse proportion function. A perpendicular line $$AB$$ is drawn from point $$A$$ to the y-axis at point $$B$$. Point $$P$$ is on the x-axis, and the area of $$\triangle ABP$$ is $$2$$. The expression for this inverse proportion function is ___. | y=\dfrac{4}{x} | 13 | false |
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math_4650 | <image>
As shown in the figure, the interior angles of hexagon $ABCDEF$ are all equal, and $AD//BC$. What is the measure of $\angle DAB$ in degrees? | 60 | 14 | false |
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math_7824 | <image>
As shown in the figure, OB⊥OA, line CD passes through point O, and ∠AOC＝25°, then ∠BOC＝ ______ degrees, ∠BOD＝ ______ degrees. | 65 | 0 | false |
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math_7050 | <image>
According to the pseudocode shown in the figure, when the value of $$a$$ is $$3$$, the final output value of $$S$$ is ______. | 21 | 16 | false |
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math_6324 | <image>
As shown in the figure, $M$ is a fixed point on the circumference of a circle with radius $R$. A point $N$ is chosen at random on the circumference, and $MN$ is connected. What is the probability that the length of the chord $MN$ does not exceed $\sqrt{3}R$? | \frac{2}{3} | 13 | false |
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math_4305 | <image>
As shown in the figure, in square $ABCD$, $AB=2$. Connect $AC$, and draw an arc with $C$ as the center and $AC$ as the radius, intersecting the extension of $BC$ at point $E$. What is the length of the arc $\overset\frown{AE}$? | \frac{3 \sqrt{2}}{2} \pi | 0 | false |
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math_3492 | <image>
As shown in the figure, in quadrilateral $$ABCD$$, $$DB=DC$$, $$\angle C=70^{\circ}$$, and $$AE\bot BD$$ at $$E$$. What is the measure of $$\angle DAE$$ in degrees? | 20 | 3 | false |
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math_4572 | <image>
Quadrilateral $ABCD$ is similar to quadrilateral ${A}'{B}'{C}'{D}'$, with $O$ as the center of similarity. If $OA:O{A}'=2:3$, then ${{S}_{ABCD}}:{{S}_{{A}'{B}'{C}'{D}'}}=$. | 4:9 | 16 | false |
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math_6962 | <image>
In the figure, in $Rt\Delta ABC$, $\angle ACB=90°$, $AC=4$, $BC=3$, and $CD\bot AB$. Then $\tan \angle BCD=$. | \frac{3}{4} | 3 | false |
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math_6987 | <image>
The figure below shows the frequency distribution histogram of the speeds of cars passing through a certain speed monitoring point on National Highway 204. Among the 300 cars passing through this monitoring point, approximately how many cars have speeds in the range [60, 80)? | 150 | 4 | false |
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math_5631 | <image>
The circular fountain in Minhang Sports Park has water jets as shown in Figure 1. If the curve $$APB$$ represents the water flow with the farthest landing point $$B$$ from point $$O$$ (as shown in Figure 2), the height $$y$$ (meters) of the water droplets as a function of the horizontal distance $$x$$ (meters) is given by $$y=-x^{2}+4x+\dfrac{9}{4}$$. What is the minimum radius of the circular fountain in meters to ensure that the water does not fall outside the fountain? | \dfrac{9}{2} | 2 | false |
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math_4965 | <image>
As shown in the figure, line $$PA$$ is tangent to circle $$\odot O$$ at point $$A$$, $$OP=2\sqrt{3}$$, $$AP=3$$, and chord $$AB \perp OP$$ at point $$C$$. Then, $$AC=$$ ___. | \dfrac{3}{2} | 5 | false |
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math_3107 | <image>
From a pile of apples, 20 apples are randomly selected, and their mass (in grams) data distribution is as follows: Then, the number of apples with a mass of at least 120 grams in the pile is approximately ___% of the total number of apples. | 70 | 13 | false |
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math_5822 | <image>
As shown in the figure, $$\angle B = \angle D$$, $$AE \perp BC$$, $$\angle ACD = 90^{\circ}$$, and $$AB = 6$$, $$AC = 4$$, $$AD = 12$$. Then $$AE =$$ ___. | 2 | 2 | false |
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