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cf9f294f-ce0d-4a81-b79f-5582af77952e | As shown in the figure, $\text{DE}//\text{BC}$. If $\text{AD}=4$, $\text{DB}=6$, and $\text{BC}=12$, then the length of $\text{DE}$ is. | $\frac{24}{5}$ | math | |
eb8c6110-8ed7-4c39-8ec5-13ce15432e06 | As shown in the figure, quadrilateral $ABCD$ is a square, $AB=2$, point $O$ is the midpoint of diagonal $AC$. The right triangle $OEF$ is rotated around point $O$, where $\angle EOF=90{}^\circ$, the two legs $OE$ and $OF$ intersect sides $BC$ and $CD$ at points $P$ and $Q$ respectively, and line segment $PQ$ is drawn. During the rotation, the minimum value of $PQ$ is. | $\sqrt{2}$ | math | |
f4a0d42f-94e8-4fb9-8c92-8366bc54a648 | As shown in the figure, to make a small conical funnel with a slant height of 8 cm and a base circumference of 12π cm, if there is no loss, the area of the required cardboard is. | 48π | math | |
09269de0-513e-4ce5-bf92-d280b3aef05b | In △ABC, AD is the altitude, and AE is the angle bisector. Given ∠B=28° and ∠C=60°, find ∠DAE=°. | 16° | math | |
c4106016-ad24-4ad7-b3ec-f87c0101de7a | In an isosceles right triangle $ABC$ where $AB=AC$, points $D$ and $E$ are on $BC$ and $AB$ respectively, and it is given that $AE=BE=1$, $CD=3BD$. Then $\overrightarrow{AD} \cdot \overrightarrow{CE}=$. | $\frac{1}{2}$ | math | |
bb5b335a-4031-46cc-9d45-6d7fcba9a589 | The city's statistics bureau surveyed 10,000 residents regarding their monthly income and drew a frequency distribution histogram based on the data collected (each group includes the left endpoint but not the right endpoint, such as the first group representing [1000, 1500)). Try to find the median of the sample data based on the frequency distribution histogram. | $2400$ | math | |
0aecb48a-e301-431c-aa06-0d8e8a096244 | To understand whether the interest in playing basketball is related to gender among students in a class, a survey was conducted among 50 students, and the following 2×2 contingency table was obtained: Then, under the premise of not exceeding a certain probability of error, it can be concluded that the interest in playing basketball is related to gender (expressed as a percentage). Note: ${{\chi }^{2}}=\frac{n{{(ad-bc)}^{2}}}{(a+b)(c+d)(a+c)(b+d)}$ P(${{\chi }^{2}}\ge k$) 0.10 0.05 0.025 0.010 0.005 0.001 k 2.706 3.841 5.024 6.635 7.879 10.828 | 0.5% | math | |
26a483ea-3c7b-4ef2-a44f-505df87b3237 | As shown in the figure, in the regular triangular prism ABC-A$_{1}$B$_{1}$C$_{1}$, AB=1, AA$_{1}$=2, then the cosine value of the dihedral angle C$_{1}$-AB-C is. | $\frac{\sqrt{57}}{19}$ | math | |
d7eb0f1f-4490-4f8e-8fe9-03e55e685010 | As shown in the figure, to measure the distance between the two mountain peaks D and C, an aircraft measures from two points A and B along a horizontal direction. At point A, the angle of depression to point D is 75°, and to point C is 30°; at point B, the angle of depression to point D is 45°, and to point C is 60°, and $AB = \sqrt{6} \text{km}$. What is the distance between C and D in kilometers? | $\sqrt{10}$ | math | |
db9365d5-24e3-4ccf-8b1f-0f676f5c217d | As shown in the figure, AD is the median of $\vartriangle \text{ABC}$. It is known that the perimeter of $\vartriangle \text{ABD}$ is 25 cm, and AB is 6 cm longer than AC. What is the perimeter of $\vartriangle \text{ACD}$ in cm? | 19 | math | |
a351cf20-5195-4628-a86a-f1d2436eb2a1 | As shown in the figure, the diagonals AC and BD of rhombus ABCD intersect at point O, with AC = 6 and BD = 8. The area of rhombus ABCD is. | 24 | math | |
d07da592-20d5-4b77-a828-60f6ed8504a3 | In the figure, in the Cartesian coordinate system, the line $y=-\frac{1}{2}x+4$ intersects the $x$-axis at point $A$ and the $y$-axis at point $B$. On the $x$-axis, take point $A_1$ such that $OA_1 = \frac{1}{2}OB$, and connect $A_1B$. Draw $A_1B_1 \perp x$-axis, intersecting the line $AB$ at point $B_1$. Draw $B_1A_2 \parallel BA_1$, intersecting the $x$-axis at point $A_2$. Draw $A_2B_2 \perp x$-axis, intersecting the line $AB$ at point $B_2$. Draw $B_2A_3 \parallel BA_1$, intersecting the $x$-axis at point $A_3$, and so on. What is the $y$-coordinate of point $B_{10}$? | $\frac{3^{10}}{2^{18}}$ | math | |
8c0e8a26-9be9-4c80-a601-a802e0f6ac60 | As shown in the figure, segment $$AB$$ is the diameter of circle $$\odot{O}$$, point $$C$$ is on the circle, and $$\angle{AOC}=80^{\circ}$$. Point $$P$$ is a moving point on the extension of segment $$AB$$. Connect $$PC$$. Then the measure of $$\angle{APC}$$ is ___ degrees (write one possible value). | $$\number{30}$$ | math | |
7244d32c-50b7-4745-8fd6-9173c38e4c30 | As shown in the figure, a ship is sailing from west to east at sea. At point A, an island M is observed to be at a bearing of north-east α. After sailing m nautical miles, at point B, the island is observed to be at a bearing of north-east β. It is known that there are reefs within n nautical miles around the island (including the boundary). If the ship continues to sail eastward, under what condition of α and β will the ship not be in danger of hitting the reefs? | $$m\cos \alpha \cos \beta >n\sin \left ( \alpha-\beta \right ) $$ | math | |
d21e9301-a9b3-449f-a695-b1e920358aaf | The growth height of a certain plant is observed at different temperatures. The observation results are as follows: . The slope of the regression line of y with respect to x is ___. | 927 | math | |
b34085b6-5742-4abd-97a9-5e279bcf4027 | Given the planar region $$\triangle ABC$$ (including the boundaries) as shown in the figure, if the objective function $$z=ax+y(a > 0)$$ has infinitely many optimal solutions that maximize it, then the value of the real number $$a$$ is ___. | $$\dfrac{3}{5}$$ | math | |
6afb6113-a1da-46b2-bdeb-fad5dc37312f | As shown in the figure, in $$\triangle ABC$$, $$D$$ is the midpoint of $$AB$$, and $$F$$ lies on line segment $$CD$$. Let $$\overrightarrow{AB}=\boldsymbol{a}$$, $$\overrightarrow{AC}=\boldsymbol{b}$$, and $$\overrightarrow{AF}=x\boldsymbol{a}+y\boldsymbol{b}$$. Then the minimum value of $$\dfrac{1}{x}+\dfrac{2}{y}$$ is ___ | $$8$$ | math | |
f1da12a4-b698-407d-a629-5d4ce5454ff2 | In △ABC, ∠BAC = 90°, AB = AC = 5. The triangle is folded so that point B lands on point D on AC, with EF as the crease. If BE = 3, then the value of sin∠CFD is. | $\frac{2}{3}$ | math | |
ad5f0ec2-714e-4679-9f5d-efcf8592a727 | A school randomly selected some students from the second year of high school and divided their module test scores into 6 groups: $[40,50)$, $[50,60)$, $[60,70)$, $[70,80)$, $[80,90)$, $[90,100]$. The frequency distribution histogram is shown in the following figure. It is known that there are 600 students in the second year of high school. According to this, estimate the number of students whose module test scores are no less than 60 points. | 480 | math | |
6978c9c4-4ba0-41ba-a5c6-a5525b8a4870 | As shown in the figure, the curve C: y = 2^x (0 ≤ x ≤ 2) has two endpoints M and N, and NA is perpendicular to the x-axis at point A. Divide the line segment OA into n equal parts, and construct rectangles with each part as one side, such that the side parallel to the x-axis has one endpoint on curve C and the other endpoint below curve C. Let the sum of the areas of these n rectangles be S_n. Then, \(\underset{n\to \infty }{\mathop{\text{lim}}}\,\left[ \left( 2n-3 \right)\left( \sqrt[n]{4}-1 \right){{S}_{n}} \right]\) =. | 12 | math | |
a216feec-c961-4467-b63b-6cbd0992ee0a | As shown in the figure on the right, in the triangular prism $ABC-{{A}_{1}}{{B}_{1}}{{C}_{1}}$, E and F are the midpoints of AB and AC, respectively. The plane $EF{{C}_{1}}{{B}_{1}}$ divides the triangular prism into two parts with volumes ${{V}_{1}}$ and ${{V}_{2}}$. Then ${{V}_{1}}$ : ${{V}_{2}}$ =. | $\frac{7}{5}$ | math | |
94444278-ba0b-400c-b314-c0b7cf44ba55 | As shown in the figure, in the right triangle $\text{Rt}\vartriangle ABC$, $\angle ACB=90{}^\circ$, $\angle A=15{}^\circ$, the perpendicular bisector of $AB$ intersects $AC$ at point $D$ and $AB$ at point $E$, and $BD$ is connected. If $AD=12$, then the length of $BC$ is. | 6 | math | |
56d229d3-301b-42ca-8149-0066acc3beb2 | As shown in the figure, the vertex O of quadrilateral OABC is the origin of the coordinate system. With O as the center of similarity, quadrilateral OA1B1C1 is similar to quadrilateral OABC. If A(6,0) corresponds to A1(4,0), and the area of quadrilateral OABC is 27, then the area of quadrilateral OA1B1C1 is. | 12 | math | |
b63b2f2b-16df-4268-b2a9-99507b117e12 | A rectangular piece of paper ABCD is folded in the manner shown in the figure to form rhombus AECF. If AB=6, then the length of BC is . | $2\sqrt{3}$ | math | |
756d8f90-faea-47a2-ac4e-2f3aae1ac477 | As shown in the figure, O is the origin of the coordinate system, and the coordinates of vertex A of rhombus OABC are (-3, -4). Vertex C is on the negative half of the x-axis. The graph of the function y = $\frac{k}{x}$ (x < 0) passes through the center E of rhombus OABC. Find the value of k. | 8 | math | |
d83704f5-4cc1-42ea-b5e3-886670f23582 | As shown in the figure, the line y = ax intersects the hyperbola y = $\frac{\text{k}}{\text{x}}$ (x > 0) at point A(1, 2). The solution set of the inequality ax > $\frac{\text{k}}{\text{x}}$ is. | x > 1 | math | |
9697d24d-5d6b-4297-9671-a415a28695a8 | As shown in the figure, the graph of the parabola y=x^2+bx+b^2-9. What is the value of b? | -3 | math | |
8b8888eb-68c1-4499-a0b2-505735615ccd | The graph of the function $y=ax^2+bx+c(0 \leq x \leq 3)$ is shown in the figure. What is the minimum value of the function? | -1 | math | |
d0512657-aa28-427c-938f-75abf03ab2b8 | If Xiao Qiang randomly throws a dart at the square wooden board shown in the figure, what is the probability that the dart lands in the shaded area? | $\frac{1}{9}$ | math | |
63c5d293-e853-40c3-b099-94a25c3bf3c1 | As shown in the figure, construct a regular quadrilateral ABCD inscribed in a circle ⊙O with a radius of 2, then construct the incircle of the regular quadrilateral ABCD to get the second circle, then construct a regular quadrilateral A$_{1}$B$_{1}$C$_{1}$D$_{1}$ inscribed in the second circle, and then construct the incircle of the regular quadrilateral A$_{1}$B$_{1}$C$_{1}$D$_{1}$ to get the third circle, and so on. What is the radius of the sixth circle? | $\frac{\sqrt{2}}{4}$ | math | |
e55caca2-1a3a-469b-bbda-703b96a1127d | The students of a class participated in an environmental knowledge competition, and it is known that the competition scores are all integers. After organizing the scores of the participating students into 6 groups, a frequency distribution histogram of the competition scores was drawn (as shown in the figure). According to the information in the figure, the percentage of students who scored higher than 60 points out of the total number of participants in the class is. | 80% | math | |
653aef1c-d1b8-431a-a97f-1cef972a62be | As shown in the figure, in a square grid composed of 25 small squares with side lengths of 1, the intersection points of the grid lines are called grid points. Given that A and B are two grid points, if C is also a grid point and makes $\vartriangle ABC$ an isosceles triangle, then the number of grid points C that satisfy this condition is m, where $m=$. | 10 | math | |
3cf07a41-5aaf-44d0-b3fc-bb89ccb46b37 | As shown in the figure, ⊙O is the circumcircle of △ABC, ∠AOB = 70°, then ∠C is ____ degrees. | 35º | math | |
d1f26c41-a0a5-4a3b-9189-46bef24bf452 | The basic principle of the election for the 12th National People's Congress representatives is: equal election between urban and rural areas, achieving equality among people, regions, and ethnic groups. According to the Xinhua News Agency on February 28, the representatives of the National People's Congress from the 5 autonomous regions of ethnic minorities are shown in the table below. The median of the number of representatives from these five regions is ______. | 58 | math | |
92b584d3-0bdb-4dab-9dd5-925af3ead768 | As shown in the figure, the 'Zhao Shuang Xian Tu' is composed of four congruent right-angled triangles and a square, forming a larger square. Let the longer leg of the right-angled triangle be $a$, and the shorter leg be $b$. If $ab=6$, and the area of the larger square is 25, then the side length of the smaller square is. | $\sqrt{13}$ | math | |
7ee1bb42-33b5-40c0-b31e-85c1f0cc1e15 | Given the function $f(x) = A\sin(\omega x + \varphi)$ (where $A > 0$, $\omega > 0$, and $|\varphi| < \frac{\pi}{2}$), part of its graph is shown below. Find the smallest positive value of $m$ that satisfies $f(x + m) - f(m - x) = 0$. | $\frac{\pi}{12}$ | math | |
1e4f1837-e25e-42f9-b3ec-66671790db2a | As shown in the figure, inside the acute angle ∠AOB, drawing 1 ray results in 3 acute angles; drawing 2 different rays results in 6 acute angles; drawing 3 different rays results in 10 acute angles; … Following this pattern, drawing 6 different rays results in how many acute angles? | 28 | math | |
22b4fed9-5f53-469f-ae2b-0183238e12da | Given the set $$M=\{1,2,3,4\}$$, let $$f(x)$$ and $$g(x)$$ be functions from set $$M$$ to set $$M$$, with the following correspondence rules: Then $$f(g(1))=$$ ___. | $$1$$ | math | |
6067a41a-5cae-4805-b4d2-152070ff8090 | A teacher conducted a survey to understand the time students spend learning online during weekends. The teacher randomly surveyed 10 students from the class, and the statistical data is shown in the table: The average time these 10 students spent learning online during the weekend is ______ hours. | 2.5 | math | |
cdca90e0-c383-474a-b4c1-14f431685dcc | As shown in the figure, in $\vartriangle ABC$, points $D$, $E$, and $F$ are on sides $AB$, $AC$, and $BC$, respectively. $DE\,//\,BC$, $EF\,//\,AB$, and $AD:AB=3:8$. Therefore, ${{S}_{\vartriangle ADE}}:{{S}_{\vartriangle EFC}}=$. | $9:25$ | math | |
20336063-bf3c-497f-b22a-8ec311b64b52 | As shown in the figure, $\angle AOB$ is a right angle, $OB$ bisects $\angle COD$, and $\angle COD = 40^\circ$. Therefore, $\angle AOD =$. | $110^\circ$ | math | |
d9f003bf-0787-4713-86ad-c9f28973f251 | As shown in the figure, BD bisects $\angle ABC$, and $\angle DBE={{90}^{{}^\circ }}$. If $\angle ABC={{40}^{{}^\circ }}$, then $\angle ABE=$. | 70° | math | |
3f49e55a-4bfd-411e-b32d-a30e2ae53694 | As shown in the figure, it is a certain calculation program of a computer. If the initial input is x = -2, then the final output result is. | -10 | math | |
08023938-8392-4e66-b0ed-3c47fec4316f | In ancient China, there was Qin Jiushao's algorithm for calculating the value of polynomials. The following is a flowchart implementing this algorithm. Execute the flowchart. If the input is $x=2, n=2$, and the values of $a$ are sequentially 2, 2, 5, then the output $s=$ | 17 | math | |
5b30ec80-0270-453b-a420-eb901e8825eb | As shown in the figure, in the quadrilateral pyramid $P-ABCD$, all four lateral faces are isosceles triangles with a vertex angle of $15^\circ$, and the length of each lateral edge is $a$. Points $E$, $F$, and $G$ are on $PB$, $PC$, and $PD$ respectively. The minimum perimeter of quadrilateral $AEFG$ is: | $a$ | math | |
007b75e9-91ba-4be5-b0b5-4405685b016a | As shown in the figure, the coordinates of the two points on line segment $AB$ are $A\left( 2.5,5 \right)$ and $B\left( 5,0 \right)$. With the origin as the center of similarity, line segment $AB$ is reduced to line segment $CD$. If the coordinates of point $D$ are $\left( 2,0 \right)$, then the coordinates of point $C$ are. | $\left( 1,2 \right)$ | math | |
584b278a-6b74-4d4f-aed6-0191739bcff1 | The surface development of a cube is shown in the figure. Each face of the cube is filled with a number, and the numbers on opposite faces are reciprocals of each other. Then the value of ${(yz)}^{x}$ is. | $-\frac{1}{8}$ | math | |
1e1e685f-7ba5-453d-9ab2-e28ece2ec0c7 | As shown in the figure, it is known that △ABC ∼ △DBE, AB = 6, DB = 8, then $\frac{{{S}_{\vartriangle ABC}}}{{{S}_{\vartriangle DBE}}}$ =. | $\frac{9}{16}$ | math | |
690a625d-de56-4d04-abf9-f2efa3074306 | As shown in the figure, a small ball bounces from P to Q, then reflects to R, and from R reflects to S, and finally from S reflects back to the original point P. The angles of incidence and reflection are equal (for example, ∠PQA = ∠RQB, etc.). Given AB = 8, BC = 15, and DP = 3, the length of the path taken by the ball is. | 34 | math | |
d42ec5a7-403e-4f2d-b0c3-4475296d2591 | As shown in the figure, line AE∥BD, point C is on BD. If AE＝4, BD＝8, and the area of △ABD is 16, then the area of △ACE is. | 8 | math | |
0aed6867-2fca-4e62-bf87-d37bb1897168 | As shown in the figure, in $\vartriangle ABC$, $AC=6$cm, $AB=8$cm, $BC=10$cm, $DE$ is the perpendicular bisector of side $AB$. What is the perimeter of $\vartriangle ADC$ in cm? | 16 | math | |
21af45a8-7b11-45e3-95ea-7fc66bb22e6d | As shown in the figure, in △OAB, ∠AOB = 72°, the angle bisector of ∠OAB intersects the line containing the bisector of the exterior angle ∠ABN of △OBA at point D. Find the measure of ∠ADB. | 36° | math | |
5f1831c1-4121-4e20-a28a-9151794e9975 | As shown in the figure, it is a flowchart of an algorithm, then the value of $$n$$ output is ___. | $$5$$ | math | |
5f4238f4-dc9b-4e7c-98ea-1daa2afcbf0e | As shown in the figure, in $$\triangle ABC$$, $$AM:AB=1:3$$, $$AN:AC=1:4$$, $$BN$$ intersects $$CM$$ at point $$E$$, $$\overrightarrow{AB}=\boldsymbol{a}$$, $$\overrightarrow{AC}=\boldsymbol{b}$$, then $$\overrightarrow{AE}=$$ ___ (express the answer in terms of $$\boldsymbol{a}$$ and $$\boldsymbol{b}$$). | $$\dfrac{3}{11}\boldsymbol{a}+\dfrac{2}{11}\boldsymbol{b}$$ | math | |
318577cf-7c23-4d18-baac-4d13c77e485c | As shown in the figure, $$\odot O$$ is the circumcircle of $$\triangle ABC$$, $$\angle AOB=70^{\circ}$$, and $$AB=AC$$. Then $$\angle ABC=$$ ___. | $$35^{\circ}$$ | math | |
7dbc5d1f-51e0-4f64-9c86-c4f4551f0f33 | In a robot soccer match, Robot No.1 of Team A starts from point A and moves in a straight line at a constant speed to point B. Upon reaching point B, it discovers that the soccer ball at point D is rolling towards point A at a speed twice its own, as shown in the figure. Given that AB = 4√2 dm, AD = 17 dm, and ∠BAC = 45°, if the time required for the robot to rotate on the spot is negligible, then the robot can intercept the ball at point C, which is ___ dm away from point A. | 7 | math | |
55623439-6c8c-4189-8695-954dcfbf7bf6 | As shown in the figure, in $$\triangle ABC$$, $$M$$ and $$N$$ are the midpoints of $$AB$$ and $$BC$$, respectively. $$AN$$ and $$CM$$ intersect at point $$O$$. The ratio of the area of $$\triangle MON$$ to the area of $$\triangle AOC$$ is ___. | $$1:4$$ | math | |
96b26637-adeb-4f74-b821-c92c09576931 | As shown in the figure, the surface of a cube consists of 6 identical squares. Let the side length of the square be a, then the surface area of the cube is $6a^2$, and the volume is $a^3$. If the surface area of cube A is 4 times the surface area of cube B, then the volume of cube A is how many times the volume of cube B? | 8 | math | |
1b25f4ee-6ff5-43d1-8abe-6d8e28b71540 | In the figure, in △ABC, AB=AC, and the perpendicular bisector DE of AB intersects AC and AB at points D and E, respectively. Given AB=8, and the perimeter of △CBD is 15, then BC=． | 7 | math | |
bdb212a5-c7d6-4d1b-8af7-e0842e4772fd | As shown in the chart, the recorded data of output $x$ (tons) and the corresponding production energy consumption $y$ (tons of standard coal) for the production of product A. According to the data provided in the table, the linear regression equation for $y$ with respect to $x$ is $\hat{y}=0.7x+0.35$. What is the value of $m$ in the table? | 3 | math | |
0a4c143e-c312-455e-8e45-d1a7e9584022 | The position of the real number $a$ on the number line is shown in the figure below, then $|a-3|=$. | 3-a | math | |
78120392-3822-42be-bc18-bab4cc36fcd4 | As shown in the figure, a straight line intersects the positive halves of the two coordinate axes at points A and B, respectively. P is any point on the line segment AB (excluding the endpoints). Perpendicular lines are drawn from P to the coordinate axes, forming a rectangle whose perimeter is 10. What is the function expression of the line? | $y=-x+5$ | math | |
d083ee86-32dd-4a0b-b364-1f1ad2551f60 | Execute the program flowchart shown below. If the input value of x is 6, then the output value of x is. | $0$ | math | |
cab0e1a9-e93c-41f6-925d-8ed0684809ee | 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? | $5cm^2$ | math | |
bbd03531-f3da-4fe3-af27-8eef5558ca23 | As shown in the figure, quadrilateral $ABCD$ is inscribed in circle ⊙O, and point E is on the extension of the diameter CD, with AB∥CD. If ∠C=70°, then the measure of ∠ADE is. | 110° | math | |
6128bd83-99c4-4be2-8f7b-8293aecfdbf8 | In the figure, AD is the median of △ABC, point E is the midpoint of AD, point F is the midpoint of BE, S$_{△ABC}$ = 41, then S$_{△BFC}$ =. | $\frac{41}{4}$ | math | |
147c1e49-857c-4ac1-a97f-cb39c40acd6b | In the figure, in △ABC, D and E are the midpoints of BC and AC, respectively. BF bisects ∠ABC and intersects DE at point F. If AB = 12 and BC = 9, then the length of EF is. | 1.5 | math | |
bacbac19-2a4a-46aa-af13-4450b337aabf | Given, as shown in the figure, the side length of square $ABCD$ is 4, points $E$ and $F$ are on sides $AB$ and $BC$ respectively, and $AE=3$, $BF=2$. Line $AF$ intersects $DE$ at point $G$ and the extension of $DC$ at point $H$. Find the area of quadrilateral $DGFC$. | $\frac{84}{11}$ | math | |
16f166fc-efd3-447f-a302-2b14148353d6 | As shown in the figure, the side length of each small square in the grid is 1, and the thick lines represent the three views of a certain geometric solid. What is the surface area of the circumscribed sphere of this solid? | $12\pi $ | math | |
e00b6623-23d6-4273-9fdf-daefa3c403d5 | As shown in the figure, the area of $$\triangle ABC$$ is $$\quantity{12}{cm^{2}}$$. Points $$D$$ and $$E$$ are the midpoints of sides $$AB$$ and $$AC$$, respectively. What is the area of trapezoid $$DBCE$$ in $$\unit{cm^{2}}$$? | $$9$$ | math | |
47c8d860-cc05-4b06-9a38-827b4c4a6ee1 | Among the following functions, their graphs all intersect the $$x$$-axis. Which of them cannot use the bisection method to find the function's zeros? (Fill in the sequence number). | (1)(3) | math | |
cf98076b-e780-4fca-899a-12199f62d88d | As shown in the figure, $$AB \bot CD$$ at point $$B$$, $$BE$$ is the bisector of $$∠ABD$$. The measure of $$∠CBE$$ is ___. | $$135^{\circ}$$ | math | |
e8947e4a-801e-48b1-9e99-5977e3e71085 | The figure below shows a geometric solid obtained by cutting off two corners of a cube, where $$M$$ and $$N$$ are the midpoints of their respective edges. The front view of this solid is ___ (fill in the number). | 2 | math | |
57b404c1-1f34-42f1-a95b-eaf770b23835 | As shown in the figure, point $$P$$ is a point on the unit circle. It starts from its initial position $$P_{0}$$ and moves counterclockwise along the unit circle by an angle $$\alpha \left(0 < \alpha < \dfrac{ \pi }{2}\right)$$ to reach point $$P_{1}$$, and then continues to move counterclockwise along the unit circle by $$\dfrac{ \pi }{3}$$ to reach point $$P_{2}$$. If the x-coordinate of point $$P_{2}$$ is $$-\dfrac{4}{5}$$, then the value of $$\cos \alpha$$ is ___. | $$\dfrac{3\sqrt{3}-4}{10}$$ | math | |
76c4a8fa-ec97-4ac1-9cfa-f87c3e140637 | The three views of a geometric solid are shown in the figure. What is its lateral surface area? | $$16$$ | math | |
352d8d24-9964-408b-95dc-39176ebbbac3 | As shown in the figure, the front view and left view of a spatial geometric body are both equilateral triangles with a side length of $$1$$, and the top view is a circle. Therefore, the lateral surface area of this geometric body is ___. | $$\dfrac{ \pi }{2}$$ | math | |
45e30759-9b7a-4c78-a9e0-337182db9d39 | The figure shows the intuitive diagram of $$\triangle AOB$$ drawn using the oblique projection method, $$\triangle A^{'}O^{'}B^{'}$$. What is the area of $$\triangle AOB$$? | $$16$$ | math | |
ad1a602e-610f-4a1c-9d76-6a2e59778869 | Execute the pseudocode shown in the figure. When the input values of $$a$$ and $$b$$ are $$1$$ and $$3$$ respectively, the final output value of $$a$$ is ___. | $$5$$ | math | |
dfbd83f7-feca-4f49-a7b7-5a03d2692c76 | As shown in the figure, during a snow disaster, a large tree broke at a height of $$3\unit{m}$$ from the ground, and the top of the tree fell $$4 \unit{m}$$ away from the base of the tree. What was the height of the tree before it broke? ___$$\unit{m}$$ | $$\quantity{8}{}$$ | math | |
d52603c1-0a70-4c29-b654-5829deb90998 | At the top platform of Banzhang Mountain Forest Park in Zhuhai City (also known as the Macao Return Park), there is a Hundred Sons Return Stele. The Hundred Sons Return Stele is a brief history of Macao over a hundred years, recording major historical events in recent years and relevant historical, geographical, and cultural data, such as the central four numbers read as $$\number{1999}\cdot 12\cdot 20$$ indicating the day of Macao's return, and the number $$23\cdot 50$$ below the center indicating that Macao's area is approximately $$23.50$$ square kilometers. The Hundred Sons Return Stele is actually a ten-order magic square, filled with $$100$$ integers from $$1$$ to $$100$$, where the sum of the numbers in each row, each column, and each diagonal are all equal. What is the sum of the numbers on the diagonal (from the top left to the bottom right) in the following figure 2? | $$505$$ | math | |
5e70d43e-0d6f-4bc0-bc7e-c9e38c945514 | The pseudocode of a certain algorithm is as follows. The statement $$k \leftarrow k+1$$ is executed ___ times. | $$0$$ | math | |
10bb0429-8747-4fe9-98f1-59587b83d051 | The graph of the function $$f(x)=ax^{3}+bx^{2}+cx+d$$ is shown in the figure. $$f'(x)$$ is the derivative of the function $$f(x)$$. Then the solution set of the inequality $$xf'(x) < 0$$ is ___. | $$(- \infty ,-\sqrt{3})\cup (0,\sqrt{3})$$ | math | |
61df7e0a-c00b-46ba-83a0-31b772fa23a9 | As shown in the figure, there are ______ triangles. | 9 | math | |
7f4cc7c7-b281-47b3-b9d6-38a9f4b8a0ab | Given points $$P$$, $$Q$$, $$R$$, and $$S$$ are the midpoints of four edges of a cube, then the figure in which line $$PQ$$ and line $$RS$$ are skew lines is ___. | 3 | math | |
1e85b1b6-aeb9-4686-944d-f916e0c7426b | As shown in the figure, $$AB$$ is the diameter of a semicircle, and $$AB=4$$. The semicircle is rotated clockwise around point $$B$$ by $$45^{\circ}$$, and point $$A$$ rotates to position $$A'$$. The area of the shaded part in the figure is ___. | $$2\pi$$ | math | |
8ce1851b-17f9-434f-b822-077540c628eb | As shown in the figure, in $$\triangle ABC$$, $$AB=\quantity{5}{cm}$$, $$BC=\quantity{12}{cm}$$, $$AC=\quantity{13}{cm}$$, then the length of the median $$BD$$ on side $$AC$$ is ___ $$\unit{cm}$$. | $$\dfrac{13}{2}$$ | math | |
bc0a7d43-1b46-451f-970e-86af17f9fdb3 | As shown in the figure, in the rectangular prism $$ABCD-A_{1}B_{1}C_{1}D_{1}$$, $$AB=AD=\quantity{3}{cm}$$, $$AA_{1}=\quantity{2}{cm}$$, then the volume of the quadrilateral pyramid $$A-BB_{1}D_{1}D$$ is ___ $$\unit{cm^{3}}$$. | $$6$$ | math | |
8932e168-80b8-4cdb-9197-975969321edc | In May 2016, relevant departments conducted a survey on the travel methods of some citizens planning to visit Shanghai Disneyland. Figures 1 and 2 are two incomplete statistical charts based on the collected data. According to the information provided in the charts, the number of people who chose to travel by bus is ___. | 6000 | math | |
ef3e3a28-e1fc-4a81-9e84-140ce688ee0a | To solve the electricity supply problem for four villages, the government has invested in setting up power transmission lines between an existing power plant and the four villages. It is known that the distances between the four villages and the power plant are as shown in the figure (distance unit: $$km$$). The shortest total length of the power transmission lines that can deliver electricity to these four villages should be ___. | $$20.5km$$ | math | |
4c85ae65-0dd0-4939-9a3d-4380bc162536 | In a certain military exercise, the Red side, in order to accurately assess the battlefield situation, measured the Blue side's two elite units at points A and B from two military bases C and D, which are $$\dfrac{\sqrt{3}a}{2}$$ apart. It is given that $$\angle ADB=30^{ \circ }$$, $$\angle BDC=30^{ \circ }$$, $$\angle DCA=60^{ \circ }$$, and $$\angle ACB=45^{ \circ }$$. As shown in the figure, the distance between the two elite units of the Blue side is ___. | $$\dfrac{\sqrt{6}}{4}a$$ | math | |
045ead1c-16f7-4a28-96a3-0641a23201f1 | As shown in Figure -1-17, Rt△AOB ≌ Rt△CDA, and A(-1, 0), B(0, 2). What are the coordinates of point C? ______ . | (-3, 1) | math | |
72b71522-153c-4319-ae99-d8db55087fc8 | As shown in the figure, in rectangle $$ABCD$$, $$AB=6$$, $$BC=8$$, connect $$AC$$. The incircles of $$\triangle ABC$$ and $$\triangle ADC$$ are $$\odot O_{1}$$ and $$\odot O_{2}$$, respectively, and the points of tangency with $$AC$$ are $$E$$ and $$F$$, respectively. The length of $$EF$$ is ___. | $$2$$ | math | |
07fd1500-5ba3-45fa-b666-774e41fe0606 | The price control department of a city surveyed the sales volume and price of a certain product in 5 malls on a certain day, obtaining a set of data on the price $$x$$ (unit: yuan) and sales volume $$y$$ (unit: pieces) as shown in the following table: It is known that there is a good linear correlation between sales volume $$y$$ and price $$x$$, and its linear regression equation is $$\hat{y}=-3.2x+a$$. Then, $$a=$$ ___. | $$40$$ | math | |
0b5c300f-2c4b-4a4b-8160-1d34a61aa49f | As shown in the figure, $$BC$$ is the diameter of the semicircle $$\odot O$$, $$EF \perp BC$$ at point $$F$$, and $$\dfrac{BF}{FC}=5$$. Given that point $$A$$ is on the extension of $$CE$$, $$AB$$ intersects the semicircle at point $$D$$, and $$AB=8$$, $$AE=2$$, then $$AD=$$ ___. | $$\dfrac{\sqrt{3}+1}{2}$$ | math | |
57c2da3b-8a7b-4503-a185-7d7de2d164c7 | As shown in the figure, points A, B, C, and D lie on the same circle, BC = CD, and AC intersects BD at point E. If AC = 8, CD = 4, and segments BE and ED are positive integers, then BD = ___. | 7 | math | |
61cfed06-903b-4531-b194-dff9a47fe8cd | As shown in the figure, the graph of a linear function intersects the $$x$$-axis and $$y$$-axis at points $$A$$ and $$B$$, respectively. When $$\triangle AOB$$ is folded along the line $$AB$$, it forms $$\triangle ACB$$. If $$C\left(\dfrac{3}{2},\dfrac{\sqrt{3}}{2}\right)$$, then the equation of the linear function is ___ . | $$y=-\sqrt{3}x+\sqrt{3}$$ | math | |
76856daa-0869-49e6-91b7-63eff060f25a | A school has introduced a 'Reading Reward Program' to encourage students to read more outside of class. After the program was announced, opinions were randomly solicited from 100 students, and the number of students who 'approved', 'disapproved', or 'abstained' was statistically analyzed, resulting in the pie chart shown below. In this survey of 100 students, the number of students who approved of the program is ______. | 70 | math | |
0f3ea9b4-edf6-4612-ae36-e3cef8a27bf4 | The Engel's Coefficient $$y(\%)$$ of a certain region and the year $$x$$ are statistically recorded in the following table: From the scatter plot, it can be seen that $$y$$ and $$x$$ are linearly related, and the regression line equation is $$\hat{y}=\hat{b}x+4055.25$$. According to this model, the predicted Engel's Coefficient $$(\%)$$ for the year $$2013$$ is ______. | $$29.25$$ | math | |
69e8c665-39cb-4dfd-b9a1-d47c3bb71d86 | The toy airplane was discounted by ______ yuan during the special offer period. | 17 | math | |
5dc9188f-af5f-409f-898d-c280506d835b | As shown in the figure, the perimeter of rectangle $$ABCD$$ is $$16$$. Four squares are constructed outward from the four sides of the rectangle, and the sum of the areas of these four squares is $$68$$. What is the area of rectangle $$ABCD$$? | $$15$$ | math |
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