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math_4905 | <image>
As shown in the figure, the sides OC and CD of squares ABOC and EFCD lie on the x-axis, and point F is on side AC. The graph of the inverse proportion function $y=\frac{k}{x}$ passes through points A and E, and ${S}_{\triangle OAE}=3$. Find $k$. | 6 | 6 |
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math_6682 | <image>
In the rhombus ABCD, E and F are the midpoints of AD and BD, respectively. If EF = 2, then the side length of rhombus ABCD is. | 4 | 16 |
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math_221 | <image>
In the parallelogram $ABCD$, $DE$ bisects $\angle ADC$ and intersects $BC$ at point $E$, $AF \perp DE$, with the foot of the perpendicular at $F$. If $\angle DAF = 40{}^\circ$, then $\angle B$ equals ___ degrees. | 100 | 12 |
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math_4706 | <image>
As shown in the figure, $$PA$$ and $$PB$$ are tangents to circle $$\odot O$$ at points $$A$$ and $$B$$, respectively. A tangent through point $$C$$ intersects $$PA$$ and $$PB$$ at points $$D$$ and $$E$$. Given that $$PA=8cm$$, the perimeter of $$\triangle PDE$$ is ___ $$cm$$. | 16 | 13 |
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math_259 | <image>
In the figure, in parallelogram $$ABCD$$, diagonals $$AC$$ and $$BD$$ intersect at point $$O$$. Point $$E$$ is the midpoint of $$AB$$, and $$OE = \quantity{5}{cm}$$. What is the length of $$AD$$ in $$\unit{cm}$$? | 10 | 16 |
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math_6462 | <image>
A workshop needs to determine the time required for processing parts to set labor hour quotas. For this purpose, 5 trials were conducted, and the data collected is as follows: From the data in the table, the linear regression equation is $$\bar{y}=0.65x+\bar{a}$$, and according to the regression equation, the predicted time for processing 70 parts is ______ minutes. | 102 | 11 |
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math_1831 | <image>
It is known that a population consists of 20 individuals numbered $$01$$, $$02$$, $$\cdots$$, $$19$$, $$20$$. Using the random number table below, select $$5$$ individuals. The selection method is to start from the 5th column of the 1st row of the random number table and select two digits at a time from left to right. The number of the 4th selected individual is ___. | 07 | 14 |
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math_5755 | <image>
As shown in the flowchart, it can determine whether any input integer x is odd or even. What is the condition in the decision box? | m=0? | 3 |
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math_966 | <image>
As shown in the figure, AB is the diameter of circle O and AB = 10, ∠B = 30°, then the length of chord AC is. | 5 | 8 |
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math_4165 | <image>
As shown in the figure, a cylindrical drink bottle made of a certain material has a volume of $$250mL$$. When the radius of its base $$r=$$___$$cm$$, the material used is minimized. (Express the answer in terms of $$\pi$$) | 5\pi ^{-\dfrac{1}{3}} | 14 |
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math_4868 | <image>
As shown in the figure, AB is the diameter of circle O, point C is a point on line segment AB, and ∠BAC = 30°. Point B is the midpoint of arc CD. What is the measure of ∠ABD in degrees? | 60 | 3 |
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math_6041 | <image>
As shown in the figure, the length of line segment AB is 10 cm. Point D is a moving point on AB, not coinciding with point A. An equilateral triangle ACD is constructed with AD as one side. A perpendicular DP is drawn from point D to CD. A rectangle CDGH is constructed with a moving point G (not coinciding with point D) on DP. The diagonals of the rectangle intersect at point O. Connecting OA and OB, the minimum value of segment OB is . | 5 | 12 |
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math_2143 | <image>
As shown in the figure, the shape is composed of fan-shaped sections of the same size. Xiao Jun is going to randomly paint each fan section with one of the colors: red, yellow, or blue. The probability that the topmost fan section is painted red is. | \frac{1}{3} | 16 |
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math_13 | <image>
As shown in the figure, it is known that the graphs of the functions $$y=ax+2$$ and $$y=bx-3$$ intersect at point $$A\left(2,-1\right)$$. Based on the graph, the solution set for the inequality $$ax>bx-5$$ is ___. | x < 2 | 16 |
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math_5724 | <image>
Read the program and answer the question. The correct judgment about the two programs (Program A and Program B) and their output results is ___. 1. Programs are different, results are different; 2. Programs are different, results are the same; 3. Programs are the same, results are different; 4. Programs are the same, results are the same. | 2 | 11 |
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math_7377 | <image>
Execute the given pseudocode, the output result is ___. | 5 | 16 |
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math_4627 | <image>
As shown in the figure, in $$\triangle ABC$$, $$\angle C=90^{\circ}$$, point $$D$$ is on $$AC$$. When $$\triangle BCD$$ is folded along $$BD$$, point $$C$$ lands on the hypotenuse $$AB$$. If $$AC=12\ \unit{cm}$$, $$DC=5\ \unit{cm}$$, then $$\sin A=$$ ___. | \dfrac{5}{7} | 0 |
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math_2690 | <image>
As shown in the figure, the midpoint of the line segment $$AB$$, which is $$12\ \unit{cm}$$ long, is $$M$$, and $$MC:CB = 1:2$$. What is the length of line segment $$AC$$ in $$\unit{cm}$$? | 8 | 15 |
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math_3052 | <image>
As shown in the figure, △ABC is rotated 20° clockwise around point A to get △AB'C'. If B'C' passes through point C, then the measure of ∠C' is ___ degrees? | 80 | 1 |
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math_1658 | <image>
As shown in the figure, the right triangle ABC is rotated 34° counterclockwise around point A to get the right triangle AB′C′, with point C′ exactly landing on the hypotenuse AB. Connecting BB′, what is the measure of ∠BB′C′ in degrees? | 17 | 2 |
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math_2930 | <image>
Xiao Ming randomly checked the air quality data for several days from the city's environmental monitoring network as a sample for statistics. He then drew the bar chart and pie chart shown in the figure. According to the information provided in the figure, what is the measure of the central angle of the sector representing excellent air quality in the pie chart, in degrees? | 108 | 0 |
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math_2015 | <image>
As shown in the figure, an open umbrella can be approximately regarded as a cone. The diameter of the circle formed by the ends of the ribs (the frame under the fabric that supports it) $$AC$$ is $$12$$ decimeters, and the rib $$AB$$ is $$9$$ decimeters long. Therefore, the minimum amount of silk fabric needed to make such an umbrella is ___ square decimeters. | \quantity{54}{\pi } | 16 |
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math_7150 | <image>
During the Dragon Boat Festival, Mr. Wang's family went on a self-driving trip to a place 170km away from home. The graph below shows the function of the distance y (km) from home and the driving time x (h). When they were 20km away from their destination, the total driving time was. | 2.25h | 0 |
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math_2075 | <image>
As shown in the figure, trees are planted on a slope with a gradient of $$1:3$$. The horizontal distance (the horizontal distance between adjacent trees) is $$6$$ meters. Therefore, the sloping distance $$AB$$ between adjacent trees on the slope equals ___ meters (the result should be in radical form)? | 2 \sqrt{10} | 16 |
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math_3672 | <image>
In triangle ABC, OB and OC are the angle bisectors of ∠ABC and ∠ACB, respectively, and ∠A = 70°. What is ∠BOC in degrees? | 125 | 14 |
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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 |
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math_2062 | <image>
As shown in the figure, ${{P}_{1}}$ is a semicircular cardboard with a radius of 2. A semicircle with a radius of 1 is cut from the lower left end of ${{P}_{1}}$ to get the shape ${{P}_{2}}$. A semicircle with a radius of $\frac{1}{2}$ is cut from the lower left end of ${{P}_{2}}$ to get the shape ${{P}_{3}}$, and so on. Each time, a smaller semicircle (with a diameter equal to the radius of the previous semicircle cut out) is cut to obtain the shapes ${{P}_{1}},{{P}_{2}},{{P}_{3}},{{P}_{4}},\cdots ,{{P}_{n}},\cdots$. Let the area of the cardboard ${{P}_{n}}$ be ${{S}_{n}}$. Find: $\underset{n\to \infty }{\mathop{\lim }}\,{{S}_{n}}=$ | \frac{4}{3}\pi | 4 |
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math_7093 | <image>
Draw a square, then connect the midpoints of each side to form a second square, and continue this process to draw a total of 3 squares, as shown in the figure. If a point is randomly thrown into the figure, what is the probability that the point lands inside the third square? | \frac{1}{4} | 16 |
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math_7833 | <image>
As shown in the figure, the three vertices of the equilateral triangle $ABC$ are on the surface of sphere $O$. The distance from the center of the sphere $O$ to the plane $ABC$ is 1, and $AB=3$. What is the surface area of sphere $O$? | 16\pi | 12 |
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math_1973 | <image>
As shown in the figure, line $$AB$$ and line $$CD$$ intersect at point $$O$$, $$\angle AOC=50\degree$$, $$OE$$ bisects $$\angle BOD$$, then $$\angle BOE=$$______ degrees. | 25 | 14 |
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math_4896 | <image>
As shown in the figure, points $$B$$, $$C$$, $$E$$, $$F$$ lie on the same straight line, $$AB \parallel DC$$, $$DE \parallel GF$$, $$\angle B = \angle F = 72^{\circ}$$, then $$\angle D =$$ ___ degrees. | 36 | 14 |
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math_556 | <image>
Happy with Statistics. The number of triangles in the figure is ______. | 6 | 0 |
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math_3572 | <image>
As shown in the figure, in $\Delta ABC$, $\Delta ABC$ is translated along the ray $BC$ such that point $B$ moves to point $C$, resulting in $\Delta DCF$. Connect $AF$. If the area of $\Delta ABC$ is 4, then the area of $\Delta ACF$ is. | 4 | 7 |
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math_2873 | <image>
As shown in the figure, $$AB$$ is tangent to $$\odot O$$ at point $$B$$, $$OA=2$$, and $$\angle OAB=30^{\circ}$$, with chord $$BC \parallel OA$$. The length of the minor arc $$BC$$ is ___. | \dfrac{\pi}{3} | 0 |
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math_4690 | <image>
The partial graph of the function $f(x)=A\sin (\omega x+\varphi )$ $(A > 0, 0 \le \varphi < 2\pi )$ is shown in the figure. What is the value of $f(2019)$? | -1 | 2 |
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math_1882 | <image>
As shown in the figure, in parallelogram $$ABCD$$, $$AC$$ and $$BD$$ intersect at point $$O$$, and $$E$$ is the midpoint of segment $$AO$$. If $$\overrightarrow{BE}=\lambda \overrightarrow{BA}+\mu \overrightarrow{BD}$$ ($$ \lambda $$, $$ \mu \in \mathbf{R}$$), then $$\lambda + \mu =$$___. | \dfrac{3}{4} | 3 |
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math_5017 | <image>
As shown in the figure, DE∥BC, DE:BC = 3:4, then AE:CE =. | 3 | 9 |
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math_8086 | <image>
As shown in the figure, in circle $\odot O$, $\angle AOB = 150{}^\circ$, $E$ is a point on the major arc $AB$, and $C$ and $D$ are two different points on the minor arc $AB$ (not coinciding with points $A$ and $B$). What is the degree measure of $\angle C + \angle D$? | 105 | 1 |
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math_7465 | <image>
The running result of the following program is ___. | \dfrac{19}{3} | 10 |
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math_7089 | <image>
As shown in the figure, ∠BOD=140°, then the degree measure of ∠BCD is ___ degrees. | 110 | 16 |
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math_5767 | <image>
The figure shows a certain operation program. According to the instructions of this program, if the initial input value of x is 1, the output value is 4, which is recorded as the first operation; if the output value of the first operation is input again, the output value is 2, which is recorded as the second operation; …, and so on in a loop, then the output value of the 2020th operation is. | 4 | 15 |
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math_2278 | <image>
In the figure, in △ABC, AB＝BC＝12cm. D, E, and F are the midpoints of BC, AC, and AB, respectively. What is the perimeter of quadrilateral BDEF in cm? | 24 | 15 |
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math_785 | <image>
As shown in the figure, AC is the angle bisector of $\angle BAE$, $\angle B=40{}^\circ $, $\angle E=70{}^\circ $, then $\angle ACE=$°. | 75 | 14 |
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math_6954 | <image>
Observe the number of small squares in each of the following figures. According to this pattern, the number of small squares in the 11th figure is ______. | 66 | 15 |
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math_4608 | <image>
In the right-angled triangle paper piece $$ABC$$, $$\angle C=90^{\circ}$$, $$\angle B=30^{\circ}$$, point $$D$$ (not coinciding with $$B$$ or $$C$$) is any point on $$BC$$. The triangle paper piece is folded in the following manner. If the length of $$EF$$ is $$a$$, then the perimeter of $$\triangle DEF$$ is ___ (express your answer in terms of $$a$$). | 3a | 5 |
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math_3852 | <image>
In the figure, in $\Delta ABC$, the angle bisectors $BE$ and $CD$ of $\angle ABC$ and $\angle ACB$ intersect at point $F$, and $\angle A=60{}^\circ$. What is the measure of $\angle BFC$ in degrees? | 120 | 15 |
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math_6260 | <image>
As shown in the figure, the first figure contains $$1$$ triangle, the second figure contains a total of $$5$$ triangles, and the third figure contains a total of $$9$$ triangles. Following this pattern, the sixth figure will contain a total of ___ triangles. | 21 | 16 |
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math_1899 | <image>
As shown in the figure, a rectangle is formed by 4 small squares, each with a side length of 2 cm. The area of the shaded part in the figure is cm². | 8 | 1 |
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math_7611 | <image>
Execute the program flowchart as shown in the figure, the output is $z=$. | 18 | 16 |
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math_2011 | <image>
Given the flowchart of a program as shown in the figure, if the values of $$x$$ input are $$0$$, $$1$$, and $$2$$, the corresponding outputs of $$y$$ are $$a$$, $$b$$, and $$c$$, respectively. Then, $$a+b+c=$$______. | 6 | 11 |
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math_6253 | <image>
The stem-and-leaf plot below shows the scores given by seven judges to two contestants, A and B, in a youth singer competition (where $$m$$ is a digit from $$0$$ to $$9$$). After removing the highest and lowest scores, the average scores of contestants A and B are $$a_{1}$$ and $$a_{2}$$, respectively. Which of the following is true? 1. $$a_{1}>a_{2}$$; 2. $$a_{2}>a_{1}$$; 3. $$a_{1}=a_{2}$$; 4. The size of $$a_{1}$$ and $$a_{2}$$ depends on the value of $$m$$. | 2 | 4 |
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math_2429 | <image>
The pseudocode of an algorithm is shown in the figure. If the input value of $x$ is $32$, then the output value of $y$ is. | 5 | 16 |
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math_4957 | <image>
Student A and Student B are playing a spinning wheel game. The figure shows two identical spinning wheels, each divided into four equal areas, marked with the numbers '1', '2', '3', and '4'. The pointers are fixed, and both wheels are spun simultaneously, allowing them to stop freely. If the product of the numbers pointed to by the two pointers is odd, Student A wins; if the product is even, Student B wins; if a pointer lands on the boundary line of a sector, they spin again. In this game, the probability of Student B winning is. | \frac{3}{4} | 12 |
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math_1495 | <image>
A middle school has a total of 120 teachers in the junior high school department and 180 teachers in the senior high school department, with the gender ratio as shown in the figure. It is known that, using stratified sampling, the union representatives include 6 female teachers from the senior high school department. What is the total number of male teachers in the union representatives? | 12 | 0 |
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math_5984 | <image>
As shown in the figure, the lines y = x + b and y = kx + 2 intersect the x-axis at points A (﹣2, 0) and B (3, 0), respectively. The solution set for $\left\{ \begin{array}{*{35}{l}} x + b > 0 \\ kx + 2 > 0 \end{array} \right.$ is | ﹣2 < x < 3 | 16 |
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math_867 | <image>
As shown in the figure, in parallelogram ABCD, point P is the midpoint of AB, PQ is parallel to AC and intersects BC at point Q. Connect AQ and CP. How many triangles in the figure have the same area as triangle APC? | 3 | 7 |
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math_7917 | <image>
Execute the program shown in the figure, after inputting a=3, b=-1, n=4, the output result is _____. | 4 | 8 |
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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 |
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math_3728 | <image>
In the square ABCD, O is the intersection point of the diagonals AC and BD. Draw OE⊥OF, where OE and OF intersect AB and BC at points E and F respectively. Given AE=3 and FC=2, what is the length of EF? | \sqrt{13} | 11 |
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math_1761 | <image>
As shown in the figure, in the Cartesian coordinate system, the coordinates of point A are (0,3). After triangle OAB is translated to the right along the x-axis, it becomes triangle O'A'B'. The corresponding point of point A lies on the line y = \dfrac{3}{4}x. What is the distance between point B and its corresponding point B'? | 4 | 10 |
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math_4600 | <image>
As shown in the figure, in rhombus $$ABCD$$, $$\angle A=60^{\circ}$$, and $$BD=7$$. What is the perimeter of rhombus $$ABCD$$? | 28 | 3 |
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math_1625 | <image>
As shown in the figure, in the tetrahedron $A-BCD$, $E$, $F$, and $G$ are the midpoints of $AB$, $BC$, and $AD$ respectively, and $\angle GEF=135{}^\circ$. What is the degree measure of the angle formed by $BD$ and $AC$? | 45 | 9 |
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math_4526 | <image>
As shown in the figure, the line $y=\frac{1}{3}x+2$ intersects the $x$-axis at point $A$ and the $y$-axis at point $B$. Point $D$ is on the positive half of the $x$-axis, with $OD=OA$. A perpendicular line from point $D$ to the $x$-axis intersects line $AB$ at point $C$. If the graph of the inverse proportion function $y=\frac{k}{x} (k\ne 0)$ passes through point $C$, then the value of $k$ is. | 24 | 16 |
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math_1378 | <image>
In the ancient Chinese mathematical text 'Nine Chapters on the Mathematical Art', there is a problem about 'carrying gold through checkpoints'. The gist of the problem is: 'In ancient times, one had to pay taxes when passing through checkpoints. One day, a person carried a certain amount of money to pass through checkpoints, paying \(\frac{1}{2}\) of the money at the first checkpoint, \(\frac{1}{3}\) of the remaining money at the second checkpoint, \(\frac{1}{4}\) of the remaining money at the third checkpoint, and so on...' Based on the mathematical idea contained in this story, a program flowchart is designed as shown in the figure. Running this program, the output value of n is: | 6 | 3 |
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math_1869 | <image>
Given the sequence $$\left \lbrace a_{n}\right \rbrace $$ where $$a_{1}=2$$, $$a_{n}=a_{n-1}+2\left (n \geqslant 2\right )$$, and the sum of the first $$n$$ terms is $$S\left (n\right )$$. Let $$g\left (n\right )=\dfrac{1}{S(n)}$$. If the flowchart shown is executed, and the output result is $$S=\dfrac{2016}{2017}$$, then the condition that should be filled in the blank in the decision box is ___. | 2016 | 0 |
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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 |
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math_439 | <image>
As shown in the figure, the side length of rhombus $$ABCD$$ is $$1$$, line $$l$$ passes through point $$C$$, intersects the extension of $$AB$$ at $$M$$, and intersects the extension of $$AD$$ at $$N$$. Then $$\dfrac{1}{AM} + \dfrac{1}{AN} =$$ ___. | 1 | 5 |
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math_7182 | <image>
As shown in the figure, lines AB and CD intersect at point O, ∠COE is a right angle, OF bisects ∠AOE, and ∠COF = 24°. What is the measure of ∠BOD in degrees? | 42 | 0 |
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math_3170 | <image>
As shown in the figure, $\angle ACB=\angle BDC={{90}^{\circ }}$, $BD=4$, $BC=5$. Then, $AC=$ when $\vartriangle ACB\backsim \vartriangle BDC$. | \frac{20}{3} | 10 |
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math_1375 | <image>
In the Cartesian coordinate system, the vertices of the square $ABCD$ have coordinates $A$(1, 1), $B$(1, -1), $C$(-1, -1), $D$(-1, 1), and there is a point $P$(0, 2) on the $y$-axis. Construct the point $P_1$ symmetric to point $P$ with respect to point $A$, the point $P_2$ symmetric to point $P_1$ with respect to point $B$, the point $P_3$ symmetric to point $P_2$ with respect to point $C$, the point $P_4$ symmetric to point $P_3$ with respect to point $D$, the point $P_5$ symmetric to point $P_4$ with respect to point $A$, and the point $P_6$ symmetric to point $P_5$ with respect to point $B$, and so on. What are the coordinates of $P_{2020}$? | (0, 2) | 9 |
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math_6348 | <image>
As shown in the figure, in rhombus $$ABCD$$, $$AC$$ and $$BD$$ intersect at point $$O$$, and $$E$$ is the midpoint of $$AB$$. If $$OE=3$$, then the perimeter of rhombus $$ABCD$$ is ___. | 24 | 13 |
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math_6686 | <image>
The figure shows a numerical converter. If the input value of x is 3, then the output value is. | -4 | 0 |
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math_5098 | <image>
The Forbidden City in Beijing was the imperial palace of the Ming and Qing dynasties in China, also known as the 'Purple Forbidden City.' It is located at the center of Beijing's central axis. The Forbidden City is 750 meters wide from east to west and 960 meters long from north to south, covering an area of 720,000 square meters. 300 students from Ape School are going to visit the Forbidden City. There are two types of buses available for rent: large buses and minibuses. Please first write down the most cost-effective rental plan, and then calculate the total cost of ______ yuan. | 6100 | 0 |
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math_5956 | <image>
As shown in the figure, given ∠O=30°, CD is the perpendicular bisector of OA, then the measure of ∠ACB is ______ degrees. | 60 | 0 |
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math_4269 | <image>
The annual salary details of all employees in a certain advertising company are shown in the table: The median annual salary of all employees in the company is ___ million yuan. | 8 | 0 |
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math_1384 | <image>
In the figure, in △ABC, ∠ACB = 90°, squares are constructed outward on sides AC and AB with areas S$_{1}$ and S$_{2}$ respectively. If S$_{1}$ = 2 and S$_{2}$ = 5, then BC = ___. | \sqrt{3} | 13 |
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math_4560 | <image>
The oblique axonometric drawing of a certain plane figure is shown in the figure, with side AB parallel to the y-axis, and BC, AD parallel to the x-axis. It is known that the area of quadrilateral ABCD is $2\sqrt{2}\text{cm}^2$. What is the area of the original plane figure in cm^2? | 8 | 0 |
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math_5369 | <image>
Execute the program flowchart shown below. If the input $a=5$, then the output $n=$. | 3 | 16 |
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math_7661 | <image>
If you want to output the larger value between a and b, then the condition box in the structure chart shown in the figure should be filled with. | b \ge a | 1 |
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math_805 | <image>
As shown in the figure, $$OA=1$$. A point $$B$$ is randomly chosen on the semicircular arc with center $$O$$ and radius $$OA$$. The probability that the area of $$\triangle AOB$$ is less than $$\dfrac{1}{4}$$ is ___. | \dfrac{1}{3} | 5 |
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math_5092 | <image>
Using the data from the following profit table to make a decision, the scheme that should be chosen is ___. | A_{3} | 16 |
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math_2768 | <image>
In the figure, in $\vartriangle ABC$, point $D$ is on line segment $BC$, $AB=AD=DC$, $\angle B=70{}^\circ $, then $\angle C$=${}^\circ $. | 35 | 2 |
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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 |
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math_2840 | <image>
Given f(x) = , the domain of definition is R, then the range of values for k is ______. | [0, 16) | 1 |
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math_7005 | <image>
In the figure, in $\Delta ABC$, $\overrightarrow{AD}=\frac{1}{3}\overrightarrow{AB}$, and point E is the midpoint of $CD$. Given $\overrightarrow{CA}=\overrightarrow{a}$ and $\overrightarrow{CB}=\overrightarrow{b}$, then $\overrightarrow{AE}=$ (express using $\overrightarrow{a}$ and $\overrightarrow{b}$). | \frac{1}{6}\overrightarrow{b}-\frac{2}{3}\overrightarrow{a} | 15 |
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math_7523 | <image>
The execution result of the following pseudocode is ___. | 65 | 16 |
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math_6337 | <image>
A textile factory, to understand the quality of cotton, randomly tested the length of 100 cotton fibers (the length of cotton fibers is an important indicator of cotton quality). The obtained data are all within the interval [5, 40], and the frequency distribution histogram is shown in the figure. How many of the 100 tested cotton fibers have a length less than 20 mm? | 30 | 0 |
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math_7263 | <image>
As shown in the figure, $$P$$ is a point outside the circle $$\odot O$$. $$PA$$ and $$PB$$ are tangents to $$\odot O$$ at $$A$$ and $$B$$ respectively. $$C$$ is any point on the arc $$\overset{\frown} {AB}$$. A tangent to $$\odot O$$ through $$C$$ intersects $$PA$$ and $$PB$$ at $$D$$ and $$E$$ respectively. If the perimeter of $$\triangle PDE$$ is $$12$$, then the length of $$PA$$ is ___. | 6 | 16 |
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math_1338 | <image>
Execute the program flowchart on the right. If the input value of $$x$$ is $$1$$, then the output value of $$y$$ is ______. | 13 | 16 |
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math_348 | <image>
As shown in the figure, it is a flowchart of an algorithm, then the output value of $k$ is. | 5 | 15 |
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math_862 | <image>
As shown in the flowchart, what is the running result of the algorithm? | 6 \sqrt{6} | 16 |
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math_1271 | <image>
As shown in the figure, the radius of sector $OAB$ is $6cm$. $AC$ is tangent to arc $AB$ at point $A$ and intersects the extension of $OB$ at point $C$. If $AC=4cm$ and the length of arc $AB$ is $3cm$, then the area of the shaded part is $c{{m}^{2}}$. | 3 | 9 |
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math_1680 | <image>
As shown in the figure, there are several cards of two types of squares, A and C, and several cards of type B rectangles. Using these, one can form some new rectangles. If one wants to form a rectangle with a length of (3a+2b) and a width of (a+b), then the number of type B rectangular cards needed is. | 5 | 15 |
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math_4515 | <image>
As shown in the figure, in $$\triangle ABC$$, $$∠C=90^{\circ}$$, $$BE$$ bisects $$∠ABC$$, and $$ED⊥AB$$ at point $$D$$. If $$AC=3$$, then $$AE+DE=$$ ___. | 3 | 16 |
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math_4986 | <image>
As shown in the figure, the lengths of the three sides AB, BC, and CA of △ABC are 4, 5, and 6, respectively. The three angle bisectors intersect at point O, and S$_{△}$$_{CAO}$ = 9. Find S$_{△}$$_{ABO}$. | 6 | 9 |
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math_1250 | <image>
In the figure, point O is a point inside △ABC, and the distances from point O to the three sides AB, BC, and CA are equal. Connecting OB and OC, if ∠A = 78°, then the degree measure of ∠BOC is ___ degrees. | 129 | 14 |
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math_5226 | <image>
In a 'Poetry in Every Class' competition, the line graph below shows the scores of 10 students from a class. If one student is randomly selected from these 10 students, the score with the highest probability of being selected is ___. | 90 | 16 |
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math_3258 | <image>
In the circle ⊙$O$, $\overset\frown{AB}=\overset\frown{AC}$, and $\angle C={{75}^{\circ }}$. Then $\angle A=$°. | 30 | 11 |
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math_694 | <image>
The age and number of members of a school's 'Reading and Writing' club are shown in the table below. What is the median age of the club members? | 14 | 16 |
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math_3972 | <image>
A certain region has a total of 100,000 households, with the ratio of urban to rural households being 4:6. According to stratified sampling, the ownership of refrigerators was surveyed in 1,000 households in the region, with the results shown in the table below. Therefore, the estimated number of rural households without refrigerators in the region is ___. | 16,000 | 2 |
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