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74753ec3-f7de-41b5-90fe-d77a48989052 | As shown in the figure, in △ABC, ∠A=40°, BC=3. Arcs are drawn with B and C as centers and BC as the radius on the right side of BC, intersecting at point D, and intersecting the extensions of AB and AC at points E and F, respectively. The sum of the lengths of arcs $\overset\frown{DE}$ and $\overset\frown{DF}$ is. | $\frac{5\pi }{3}$ | math | |
cadf4246-7790-49cb-b5e5-f99b44e67c85 | As shown in the figure, this is a snake model in computer science. The 4th number from left to right in the 20th row is. | 194 | math | |
c8c02455-9bb3-42ad-a37e-2830135167ce | As shown in the figure, a square piece of paper with a side length of $$\quantity{6}{cm}$$ is cut along the dotted lines and then precisely forms a hexagonal prism with a regular hexagon as its base. What is the lateral surface area of this hexagonal prism in ___$$\unit{cm^{2}}$$? | $$36-12\sqrt{3}$$ | math | |
22fa98c7-e398-41b5-87da-9dcfcea5cc7c | The positions of real numbers $$a$$ and $$b$$ on the number line are shown in the figure. Simplify $$|a+b|+|b-a|=$$ ___. | $$-2a$$ | math | |
1cc07f7d-fef3-4737-b8ed-f5c9c0e7c497 | A school has two math interest classes, $$A$$ and $$B$$. Class $$A$$ has 40 students, and class $$B$$ has 50 students. In a recent exam, the average score of class $$A$$ is 90 points, and the average score of class $$B$$ is 81 points. What is the average score of the school's math interest classes? | $$85$$ | math | |
105122f7-8104-4f98-a275-ed366bae87cd | As shown in the figure, a particle moves in the first quadrant and on the positive half-axes of the $$x$$ and $$y$$ axes. In the first second, it moves from the origin to $$\left ( 0,1\right ) $$. Then, it moves back and forth in directions parallel to the $$x$$ and $$y$$ axes as shown in the figure, i.e., $$\left (0,0 \right ) \rightarrow \left (0,1 \right ) \rightarrow \left ( 1,1\right ) \rightarrow \left ( 1,0\right ) \rightarrow \left ( 2,0\right )\rightarrow \cdots $$. It moves one unit length per second. Therefore, at $$2016$$ seconds, the position of the particle is ___. | $$\left (8,44 \right ) $$ | math | |
cb777310-fb3a-4d39-addd-ad3bb04288fb | As shown in the figure, $$D$$ is a point on the side $$BC$$ of $$\triangle ABC$$, and $$\angle 1=\angle 2$$, $$\angle 3=\angle 4$$, $$\angle BAC=63^{\circ}$$, then $$\angle DAC=$$ ___. | $$24$$ | math | |
b7d25a03-b055-473f-929d-60890f37caea | As shown in the figure, $$AA_{1}$$ and $$BB_{1}$$ intersect at point $$O$$, $$AB∥A_{1}B_{1}$$, and $$AB=\dfrac{1}{2}A_{1}B_{1}$$. If the diameter of the circumcircle of $$\triangle AOB$$ is $$1$$, then the diameter of the circumcircle of $$\triangle A_{1}OB_{1}$$ is ___. | $$2$$ | math | |
f99d14a8-1b4e-4e3c-b831-9e6ccecdffe2 | As shown in the figure, there are 5 sets of data (x, y). After removing one set of data, the linear correlation of the remaining 4 sets of data can be maximized. Which set of data should be removed? | $$D(3,10)$$ | math | |
538a04da-d419-42f6-9ea0-dbaa07aab062 | The table below records the results of a player's free throws. As the number of attempts increases, the frequency of successful shots gradually stabilizes to ___. (Round to $$0.1$$) | $$0.5$$ | math | |
faa18c3b-75ca-4db5-bbc0-62157cd5da3a | A region monitors the speed of vehicles on a certain road section, and 40 vehicles are selected for speed analysis, resulting in the frequency distribution histogram of speeds shown in the figure. According to the graph, the number of vehicles traveling at less than 70 km/h is: | 16 | math | |
67941629-6fab-442e-80d9-65acf5b813c7 | In the figure, in parallelogram $ABCD$, the angle bisector $AE$ of $\angle DAB$ intersects $CD$ at $E$, $AB=5$, $BC=3$. Find the length of $EC$. | 2 | math | |
fc04a9bf-34da-4d14-9520-7160c52b41af | Integers are arranged in a regular pattern as shown in the figure. The last number in the 2nd row is $-4$, the last number in the 3rd row is 9, the last number in the 4th row is $-16$, and so on. What is the 21st number in the 21st row? | 421 | math | |
a51ea971-c732-4238-b67c-5f291c3e949d | In the figure, in $\vartriangle ABC$, $DE\text{//}BC$, AB=5, AC=3. If BD=AE, then the length of AD is. | $\frac{25}{8}$ | math | |
32107ceb-c99e-429a-90cf-faa99470cc18 | As shown in the figure, point E is on the extension of side BC of ΔABC. CD bisects ∠ACE. If ∠A = 70° and ∠DCA = 65°, then the measure of ∠B is. | 60° | math | |
62387b2d-b8cf-4378-9ef2-71c6ce2ed004 | As shown in the flowchart, if the input value of $x$ is $-5.5$, then the output result $c=$. | 1 | math | |
31aab6ac-0fa0-4cd6-8751-552901d96699 | Given that the distance between places A and B is 10 kilometers, at 9:00 AM, person 甲 starts riding an electric bike from A to B, and at 9:10 AM, person 乙 starts driving from B to A. The relationship between the distance y (kilometers) from A and the time x (minutes) used by 甲 is shown in the figure below, then the time when 乙 arrives at A is. | 9:20 | math | |
d1bf96ea-c327-43e3-9b55-6c9508cd7676 | As shown in the figure, in the Cartesian coordinate system, a perpendicular line is drawn from point $$A_{1}\left(0,-\dfrac{1}{3}\right)$$ to the $$y$$-axis, intersecting the line $$y=-x$$ at point $$B_{1}$$. Then, a perpendicular line is drawn from point $$B_{1}$$ to the line $$y=-x$$, intersecting the $$y$$-axis at point $$A_{2}$$. Next, a perpendicular line is drawn from point $$A_{2}$$ to the $$y$$-axis, intersecting the line $$y=-x$$ at point $$B_{2}$$, and so on. What are the coordinates of $$B_{6}$$? | $$\left(\dfrac{32}{3},-\dfrac{32}{3}\right)$$ | math | |
8c349d34-1a8c-483e-8cf3-759298e831fb | The graph of the quadratic function $$y=ax^{2}+bx+c$$ is shown in the figure, with its axis of symmetry being the line $$x=1$$. If one of its intersections with the $$x$$-axis is $$A\left ( 3,0\right ) $$, then from the graph, the solution set of the inequality $$ax^{2}+bx+c < 0$$ is ___. | $$-1 < x < 3$$ | math | |
7f2f01a4-7f62-4f0f-940c-72c411d1ae10 | In the figure, in $$\triangle ABC$$, $$AB=BC$$, $$∠ABC=110^{\circ}$$, the perpendicular bisector $$DE$$ of $$AB$$ intersects $$AC$$ at point $$D$$, and $$BD$$ is connected, then $$∠ABD=$$ ___ degrees. | $$35$$ | math | |
4f8e96e0-7d71-4a95-a959-e15a91119992 | Given that the side length of rhombus ${{A}_{1}}{{B}_{1}}{{C}_{1}}{{D}_{1}}$ is $2$, $\angle {{A}_{1}}{{B}_{1}}{{C}_{1}}$=120°, and the diagonals ${{A}_{1}}{{C}_{1}},{{B}_{1}}{{D}_{1}}$ intersect at point $O$. Using point $O$ as the origin, and the lines $O{{A}_{1}},O{{B}_{1}}$ as the $x$-axis and $y$-axis respectively, a Cartesian coordinate system is established as shown in the figure. Rhombus ${{B}_{2}}{{C}_{1}}{{D}_{2}}{{A}_{1}}$ is constructed with ${{A}_{1}}{{C}_{1}}$ as the diagonal, similar to rhombus ${{A}_{1}}{{B}_{1}}{{C}_{1}}{{D}_{1}}$. Then, rhombus ${{A}_{2}}{{B}_{2}}{{C}_{2}}{{D}_{2}}$ is constructed with ${{B}_{2}}{{D}_{2}}$ as the diagonal, similar to rhombus ${{B}_{2}}{{C}_{1}}{{D}_{2}}{{A}_{1}}$. Next, rhombus ${{B}_{3}}{{C}_{2}}{{D}_{3}}{{A}_{2}}$ is constructed with ${{A}_{2}}{{C}_{2}}$ as the diagonal, similar to rhombus ${{A}_{2}}{{B}_{2}}{{C}_{2}}{{D}_{2}}$, and so on. Let the area of rhombus ${{A}_{1}}{{B}_{1}}{{C}_{1}}{{D}_{1}}$ be ${{S}_{1}}$, the area of rhombus ${{A}_{2}}{{B}_{2}}{{C}_{2}}{{D}_{2}}$ be ${{S}_{2}}$, ..., and the area of rhombus ${{A}_{n}}{{B}_{n}}{{C}_{n}}{{D}_{n}}$ be ${{S}_{n}}$. Then, ${{S}_{n}}=$. | $2\sqrt{3}\times {{9}^{n-1}}$ | math | |
e15190d0-d585-44e0-833e-f762fe5f50aa | As shown in the figure, there is a right-angled triangular piece of paper, with the two perpendicular sides AC=6cm and BC=8cm. Now, the right-angled side is folded along the line AD, so that it falls on the hypotenuse AB and coincides with AE. What is the length of CD? | 3cm | math | |
a14bbb0f-dbbf-43bd-b946-f485034ce5e1 | As shown in the figure, the line y=1 passes through point A and intersects the y-axis at point B, with OA=2. There is a point P on line AB. If triangle POA is an isosceles triangle with OA as a leg, then the coordinates of point P are. | $(-\sqrt{3},1);(-2+\sqrt{3},1);(2+\sqrt{3},1)$ | math | |
8d461976-01a8-4375-86c2-bd8ba6581fcb | In rectangle $\text{ABCD}$, $\text{M}$ is the midpoint of side $\text{AD}$, $\text{P}$ is a point on side $\text{BC}$, $\text{PE}\bot \text{MC}$, $\text{PF}\bot \text{MB}$. When $\text{AB}$ and $\text{BC}$ satisfy certain conditions, quadrilateral $\text{PEMF}$ is a rectangle. | $\text{AB}=\frac{1}{2}\text{BC}$ | math | |
cee55e9b-6f1a-4eb7-8e8a-4d7fc594ca8a | As shown in the figure, in $\Delta ABC$, $BF$ bisects $\angle ABC$, $AG \perp BF$, with the foot of the perpendicular at point $D$, intersecting $BC$ at point $G$. $E$ is the midpoint of $AC$, and $DE$ is connected. Given that $DE = 2.5cm$ and $AB = 4cm$, the length of $BC$ is $cm$. | 6.5 | math | |
a1b67524-cb35-4654-831c-7587e305d837 | In the figure, in rectangle ABCD, DE is perpendicular to AC at point F and intersects side BC at point E. Given AB = 6, AD = 8, find the length of CE. | 4.5 | math | |
88b856b9-eef3-4bc0-af20-c5a577039c90 | As shown in the figure, line segment AD intersects with BC at point O, AB∥CD. If AB:CD = 2:3, and the area of △ABO is 2, then the area of △CDO is | 4.5 | math | |
15677376-ad9d-4161-81fa-0807ac81978c | Execute the algorithm flowchart shown in the figure. If the input value of $x$ is 2, then the output value of $n$ is. | 2 | math | |
c03f9481-4e14-48cf-a0b7-b300fe93c51a | The graph of the linear function $y=ax+b$ is shown in the figure. The solution set of the inequality $ax+b > 2$ is. | $x > 2$ | math | |
6208a1e0-1ef9-428c-bedf-577711c05c01 | As shown in the figure, it is known that the graphs of the functions y=kx+b and y=$\frac{1}{2}$x-2 intersect at point P. According to the graph, the solution to the inequality system kx+b<$\frac{1}{2}$x-2<0 is. | 2＜x＜4 | math | |
b8d49160-1526-4948-a6da-c54ce4882fc6 | As shown in the figure, A and B are two grid points on a grid composed of small squares with side length 1. Place point C arbitrarily on any grid point (excluding points A and B). The probability that a triangle can be drawn with points A, B, and C as vertices is. | $\frac{31}{34}$ | math | |
3597395d-1ca9-46ae-ab2a-70fc2b8f0564 | As shown in the figure, in the right trapezoid $$ABCD$$, $$DC \parallel AB$$, $$CB \perp AB$$, $$AB=AD=a$$, $$CD=\dfrac{a}{2}$$, points $$E$$ and $$F$$ are the midpoints of segments $$AB$$ and $$AD$$, respectively. Then $$EF=$$ ___. | $$\dfrac{a}{2}$$ | math | |
7d90d0b8-8473-482c-bbd3-bc7707f84b50 | A middle school randomly surveyed 50 students to understand their physical exercise time at school during the week. The results are shown in the following table: What is the average physical exercise time for these 50 students at school this week? ___ hours. | 6.4 | math | |
ee149518-edcc-47a7-817b-66192241f058 | As shown in the figure, given that line $$AB \parallel CD$$, $$\angle C=125^{\circ}$$, and $$\angle A=45^{\circ}$$, then $$\angle E=$$ ___. | $$80^{\circ}$$ | math | |
9ca92fcf-d93e-4a34-84fd-dc3176f8e7fc | Referring to the flowchart shown in the figure, if $$a=5^{0.6}$$, $$b=0.6^{5}$$, $$c=\log_{0.5}5$$, then the number output is among $$a$$, $$b$$, and $$c$$, which is ___. | $$a$$ | math | |
ecdd1c47-24ce-4d55-b95c-ae86d10fc1cf | As shown in the figure, the area of the large square is $$13$$, and four congruent right triangles form a smaller square. The length of the shorter leg of the right triangle is $$2$$. If a dart is thrown at the large square, what is the probability that the dart lands inside the smaller square? | $$\dfrac{1}{13}$$ | math | |
b1e4e85f-48dd-44af-beff-dea52761c510 | Let the probability distribution of a random variable be as shown in the table below, and given that $$E(\xi) = 1.6$$, then $$a - b =$$ ___. | $$-0.2$$ | math | |
e24f1f1f-63e9-4e39-bef5-8efbe5d8bcef | A daily commodity is classified into five quality levels according to industry standards, with the quality coefficients $$X$$ being $$1$$, $$2$$, $$3$$, $$4$$, and $$5$$ respectively. Now, $$200$$ items are randomly selected from a batch of this commodity, and the frequency $$f$$ of their quality coefficients is recorded in the following distribution table: Among the $$200$$ selected items, the number of items with a quality coefficient $$X=1$$ is ___. | $$20$$ | math | |
bfc5d0eb-c650-49cd-8588-79914b0738ac | As shown in the figure, in a semicircle with a radius of $$1$$, a square $$ABCD$$ with a side length of $$\dfrac{1}{2}$$ is placed. If a point is randomly thrown into the semicircle, the probability that the point lands inside the square is ___. | $$\dfrac{1}{2\pi }$$ | math | |
ce2a955a-bef0-4479-b3f2-ca3ee478040b | A junior high school's ninth grade has the following table on the health bulletin board. What is the average score of Class (2) for the week? | $$9.7$$ | math | |
be863258-1d8d-4dc9-8d7f-8c06908c8084 | As shown in the figure, the pattern is arranged according to a certain rule. Following this rule, in the 1st to the 2012th pattern, the total number of '♣' is ___. | 503 | math | |
131acc2f-0983-46eb-9ff3-2ac7b8ceae58 | In the flowchart shown, if the input $$x \in [-1,4]$$, then the probability that the output $$y \in (0,1]$$ is ___. | $$\dfrac{1}{5}$$ | math | |
6dcb075c-f270-4ef4-bcc8-b2ef2f41b512 | As shown in the figure, lines $$m \parallel n$$, and $$\triangle ABC$$ is an isosceles right triangle with $$\angle BAC = 90^{\circ}$$, then $$\angle 1 =$$ ___ degrees. | $$45$$ | math | |
129c0f1b-27e7-4db0-b11f-e393d50d1b25 | As shown in the figure, there is a movable frame in the shape of a parallelogram, with the diagonals being two rubber bands. If the shape of the frame is changed, then the angle $$∠\alpha$$ also changes, and the lengths of the two diagonals also vary. When $$∠\alpha$$ is ___, the lengths of the two diagonals are equal. | $$90^{\circ}$$ | math | |
b522a987-e20d-4665-b909-fe9ca009be53 | The figure below is a flowchart for calculating the value of $$1^{2}+2^{2}+3^{2}+\cdots +100^{2}$$, then the positive integer $$n=$$ ___. | $$100$$ | math | |
a11b7916-733f-4eda-8e39-c2323af72f79 | Given that the front view and side view of a certain geometric body are as shown in the figure, among the following 5 figures: the number of figures that can be used as the top view of the geometric body is ___ | $$4$$ | math | |
9726e46b-3ae7-462a-ba23-79ecc1476172 | As shown in the figure, line $$AB$$ and line $$OC$$ intersect at point $$O$$, $$∠AOC=100^{\circ}$$, then $$∠1=$$ ___ degrees. | $$80$$ | math | |
087378ac-83cf-4999-8596-d0c85b40b2ec | As shown in the figure, in the Cartesian coordinate system, circle $$⊙A$$ passes through point $$O$$, and intersects the $$x$$-axis and $$y$$-axis at points $$B$$ and $$C$$ respectively. Given that $$B\left ( 12,0\right ) $$ and $$C\left ( 0,5\right )$$, the radius of $$⊙A$$ is ___. | $$\dfrac{13}{2}$$ | math | |
64723254-89e9-4c00-b03b-85a4b12b1165 | As shown in the figure, line $$AB \parallel CD$$, line $$EF$$ intersects $$AB$$ and $$CD$$ at points $$E$$ and $$F$$ respectively, $$FP \perp EF$$ at point $$F$$, and intersects the bisector of $$\angle BEF$$ at point $$P$$. If $$\angle 1=20^{ \circ }$$, then the measure of $$\angle P$$ is ___. | $$55^{\circ}$$ | math | |
061a5b7c-94a6-4b4e-8d92-460cd78bb113 | Let the height of a frustum of a cone be $$3$$, and its axial section is shown in the figure. The angle between the generatrix $$AA_{1}$$ and the diameter $$AB$$ of the bottom circle is $$60^{\circ}$$, and one of the diagonals in the axial section is perpendicular to the side. What is the volume of the frustum? | $$21 \pi$$ | math | |
3ed81d55-d087-4ad2-a3c3-b3df5d8f9929 | The three views of a triangular pyramid are shown in the figure. Its front view, side view, and top view are all right-angled triangles, with areas of $$1$$, $$2$$, and $$4$$, respectively. The volume of this geometric body is ___. | $$\dfrac{4}{3}$$ | math | |
918df50e-e3ef-45f6-b34f-91c480e0c465 | The figure shows the three views of a geometric solid. The volume of this solid is ___. (The result should be in terms of $$π$$) | $$250\pi$$ | math | |
5330815c-bf04-48ce-9dd1-029134b9f68b | As shown in the figure, in the circular spinner, the central angles of the three sectors $$A$$, $$B$$, and $$C$$ are $$150^{\circ}$$, $$120^{\circ}$$, and $$90^{\circ}$$, respectively. After spinning the spinner, the probability of the pointer stopping at any position is the same (if the pointer stops on the boundary line, the spinner is spun again). What is the probability that the pointer stops in region $$B$$ after one spin? | $$\dfrac{1}{3}$$ | math | |
4e347e89-aa66-409b-8d33-795fb0df4b9f | As shown in the figure, the distances from two lighthouses $$A$$ and $$B$$ to the marine observation station $$C$$ are both $$a\,km$$. Lighthouse $$A$$ is located at a bearing of $$20^{\circ}$$ north of east from station $$C$$, and lighthouse $$B$$ is located at a bearing of $$40^{\circ}$$ south of east from station $$C$$. The distance between lighthouses $$A$$ and $$B$$ is ___ $$km$$. | $$\sqrt{3}a$$ | math | |
cbbae228-5b58-4cdc-9a1b-4d1659dda48f | As shown in the figure, it is known that all the edges of the triangular prism $$ABC{-}A_{1}B_{1}C_{1}$$ are $$1$$, and $$AA_{1} \perp $$ the base $$ABC$$. What is the volume of the tetrahedron $$B_{1}-ABC_{1}$$? | $$\dfrac{\sqrt{3}}{12}$$ | math | |
52538d7a-f24f-4b3a-84ac-f6f038f46b4d | The stem-and-leaf plot of a class's Chinese scores is shown in the figure. The excellence rate (90 points and above) is ___, and the lowest score is ___. | 4% 51 | math | |
270c879c-f33f-4e75-9c88-ced5ca17b992 | As shown in the figure, $$AB$$ is the diameter of $$\odot O$$, $$C$$ is a point on $$\odot O$$, $$CD \perp AB$$, with the foot of the perpendicular at $$D$$, $$F$$ is the midpoint of the arc $$\overset{\frown} {AC}$$, $$OF$$ intersects $$AC$$ at point $$E$$, $$AC=8$$, $$EF=2$$. The length of $$AD$$ is ___. | $$\dfrac{32}{5}$$ | math | |
132ac66c-6ab4-48ba-b24e-8b24419b1164 | As shown in the figure, a beam of parallel sunlight shines on a regular pentagon, then $$ \angle 1=$$___$$^{\circ}$$. | $$30$$ | math | |
986dede4-6ead-45ec-827c-f3feafea1a54 | Use small cubes to build a geometric shape such that the views from the right side and the top are the two shapes shown below. The minimum number of small cubes needed to build such a geometric shape is $$x$$, and the maximum number of small cubes needed is $$y$$. Then $$x+y=$$ ______. | 12 | math | |
2d0dc165-a5d1-4f84-ad45-a6d1b15c03e5 | The distribution of the random variable $$\xi$$ is as follows: The variance of the random variable $$\xi$$, $$D(\xi)$$, is ______. | $$1$$ | math | |
c72aba52-2348-441f-b200-b956dc0c49ec | As shown in the figure, in the equilateral triangle $$ABC$$, $$D$$ and $$E$$ are on $$AC$$ and $$AB$$ respectively, and $$\dfrac{AD}{AC}=\dfrac{1}{3}$$, $$AE=BE$$. Then, $$ \triangle AED\sim $$___. | $$\triangle CBD$$ | math | |
76dfa7cd-e19f-41f9-b7f9-e484798796fb | As shown in the figure, in the right trapezoid $$ABCD$$, $$AB \parallel CD$$, $$\angle DAB=90^{\circ}$$, $$AD=AB=4$$, $$CD=1$$. A moving point $$P$$ is on the side $$BC$$, and satisfies $$\overrightarrow{AP}=m\overrightarrow{AB}+n\overrightarrow{AD}$$ ($$m$$, $$n$$ are both positive real numbers). Then the minimum value of $$\dfrac{1}{m}+\dfrac{1}{n}$$ is ___. | $$\dfrac{7+4\sqrt{3}}{4}$$ | math | |
6c3d50cb-b5cb-419f-b840-503957daadfe | As shown in the figure, point E is on the extension of side BC of square ABCD, and CE = AC. Connect AE. Then, ∠ACE = ______ degrees. | 135 | math | |
b491d833-5a5b-4249-95b0-93023ada5bfb | Draw a perpendicular from the pole to the line $$l$$, and the polar coordinates of the foot of the perpendicular are $$\left(3,\dfrac{2 \pi }{3}\right)$$. Then the polar equation of $$l$$ is ___. | $$ \rho \cos \left(\dfrac{2 \pi }{3}- \theta \right)=3$$ | math | |
b705a75a-5890-430b-bbf0-3268515d2fb4 | As shown in the figure, the diagonals of $$\square ABCD$$ intersect at point $$O$$. Please add one condition: ___ (fill in only one), to make $$ABCD$$ a rectangle. | $$AC=BD$$ | math | |
4e18a787-d35c-44a2-9e82-1f9b1665204c | As shown in the figure, in the right triangle $$ABC$$, $$D$$ is the midpoint of the hypotenuse $$AB$$, $$AC=6cm$$, $$BC=8cm$$, $$EC\bot$$ plane $$ABC$$, $$EC=12cm$$. Then, $$ED=$$ ___ $$cm$$. | $$13$$ | math | |
4dc640d5-6c87-48e8-9bad-9d88261a4853 | As shown in the figure, a flea starts from point M, first climbs up 2 units, then moves left 3 units to reach point P, and then jumps to the point P1, which is the reflection of point P over the x-axis. What are the coordinates of point P1? | (-3, -3) | math | |
9e5510c7-c00e-409e-be3a-7b08eaed8109 | Given that the side length of square $ABCD$ is 6, points $E$ and $F$ are on $AD$ and $DC$ respectively, with $AE=DF=2$. $BE$ intersects $AF$ at point $G$, and point $H$ is the midpoint of $BF$. Connecting $GH$, the length of $GH$ is. | $\sqrt{13}$ | math | |
15ce77fc-9e16-41d1-b0b6-85e22942d847 | The positions of the rational numbers $a, b, c$ on the number line are shown in the figure: Then the simplified result of the algebraic expression $\left| a+c \right|-2\left| a-b \right|+\left| b-c \right|$ is. | $a-3b$ | math | |
ad7df31a-7525-467a-9180-52123ad4acfb | As shown in the figure, above the x-axis, a line parallel to the x-axis intersects the graphs of the inverse proportion functions y = $\frac{k_1}{x}$ and y = $\frac{k_2}{x}$ at points A and B, respectively. Connecting OA and OB, if the area of △AOB is 6, then $k_1 - k_2$ = . | -12 | math | |
03be736f-0715-450f-98d0-7035995a6229 | As shown in the figure, ABCD—A$_{1}$B$_{1}$C$_{1}$D$_{1}$ is a cube with edge length a. The angle formed by AB$_{1}$ and the plane D$_{1}$B$_{1}$BD is ＝． | ${{30}^{\circ }}$ | math | |
99018e15-fca6-4ee9-b689-61940b105128 | In the right triangle ABC, ∠BAC = 90°, AB = 3, AC = 4, and AD is perpendicular to BC at point D. Find the length of AD. | $\frac{12}{5}$ | math | |
bb66eda6-c9f6-4917-9ae2-347b50ec0b83 | As shown in the figure, the radius of the sector is 3, and the central angle θ is 120°. Using this sector to form the lateral surface of a cone, the radius of the base of the cone is. | 1 | math | |
3dcade39-0749-4369-9efb-7d20c0218545 | As shown in the figure, the side length of square ABCD is 4, the diagonals AC and BD intersect at point O, and a semicircle is drawn with BC as the diameter. The area of the shaded region in the figure is. | 12 - 2π | math | |
769bccba-997f-4ab6-9dd1-b6605ad98ea2 | A component is composed of four electronic elements connected in the manner shown in the diagram. The component operates normally if element 1 or element 2 is functioning, and element 3 or element 4 is functioning. The lifespans (in hours) of the four electronic elements all follow a normal distribution $N(1000,{{50}^{2}})$, and the functioning of each element is independent of the others. What is the probability that the component's lifespan exceeds 1000 hours? | $\frac{9}{16}$ | math | |
857d1256-e2ba-4ee8-97b6-3a998ee5967c | As shown in the figure, there are nine cards printed with different car logos (except for the patterns, all other aspects of the cards are the same). Now, the patterned side is placed face down, and one card is drawn at random. The probability of drawing a card with a centrally symmetric car logo is. | $\frac{1}{9}$ | math | |
1bfaa1ff-5d4d-4e8d-b9d7-1c807cd56ee8 | Given the line y = kx + b (k > 0, b > 0) intersects the x-axis and y-axis at points A and B, respectively, and intersects the hyperbola y = 16/x (x > 0) at point C in the first quadrant. If BC = 2AB, then the area of triangle AOB is: | 4/3 | math | |
1d726f8d-61f8-46a7-b031-fcca4d3dd041 | As shown in the figure, in the right triangle ABC, ∠C=90°, AB=5, AC=4. Semicircles are drawn with the three sides of the right triangle ABC as diameters. The area of the shaded region is. | 6 | math | |
5c51a86d-84d5-4d4e-b97e-93ef2c177e39 | As shown in the figure, the feasible region of the objective function $u=ax-y$ is the quadrilateral $OACB$ (including the boundaries). If point $C\left( \frac{2}{3},\frac{4}{5} \right)$ is the unique optimal point of this objective function, then the range of values for $a$ is. | $\left( -\frac{12}{5},-\frac{3}{10} \right)$ | math | |
07257871-fcb6-411e-b1a0-b62de9e5f703 | In the figure, in $\vartriangle ABC$, $AC=AD=BD$, and $\angle DAC=80{}^\circ$. What is the measure of $\angle B$? | $25{}^\circ$ | math | |
93de362d-665f-4914-9882-b2d9ae3094fc | As shown in the figure, all quadrilaterals are squares, and all triangles are right triangles. If the sum of the areas of squares A, B, and C is 9, then the side length of square D is. | 3 | math | |
6eb025d9-5a53-4ba1-831a-e5781c7cef80 | Given the function $y=A\sin (\omega x+\varphi )(A > 0,\omega > 0,|\varphi | < \pi )$ with the graph shown below, the formula for this function is: | $y=2\sin \left( 2x+\frac{\pi }{6} \right)$ | math | |
716bce19-1769-4f2b-8b9d-2d4c716cae2d | As shown in the figure (units: cm), express the area of the shaded part in the triangular ruler in cm$^{2}$. | $\frac{1}{2}ab-\pi {{r}^{2}}$ | math | |
51666a19-35ce-417b-ac84-dea97c37a367 | The results of a germination test for a certain type of rapeseed under the same conditions are shown in the table below: The probability of this type of rapeseed germinating is ___ (result rounded to $$0.01$$). | $$0.95$$ | math | |
f5252fd8-f715-4942-8dae-4f52ff4d1c18 | As shown in the figure, in the rectangular coordinate system $$xOy$$, the vertices of triangle $$ABC$$ are $$A(0,a)$$, $$B(b,0)$$, and $$C(c,0)$$, and point $$P(0,p)$$ is a point on the line segment $$AO$$ (excluding the endpoints). Here, $$a$$, $$b$$, $$c$$, and $$p$$ are non-zero real numbers. The lines $$BP$$ and $$CP$$ intersect the sides $$AC$$ and $$AB$$ at points $$E$$ and $$F$$, respectively. A student has correctly derived the equation of line $$OE$$ as: $$\left ( \dfrac{1}{c}-\dfrac{1}{b}\right )x-\left ( \dfrac{1}{p}-\dfrac{1}{a}\right )y=0$$. Please complete the equation of line $$OF$$: (___)$$x-$$$$\left(\dfrac{1}{p}-\dfrac{1}{a}\right)y=0$$. | $$\dfrac{1}{b}-\dfrac{1}{c}$$ | math | |
f2504ae3-e320-47a7-a84c-bdea65915e48 | The curve corresponding to the equation $$y=-\sqrt{4-x^{2}}$$ is ___ (fill in the number). | 1 | math | |
922ddf47-d6fc-4b21-addf-49dec05ac965 | Following the operation steps shown in the figure, if the input value of $$x$$ is $$1$$, then the output value is ___. | $$11$$ | math | |
48bd00c6-116a-464f-a33d-184748046f38 | Point $$P(3a,a)$$ is an intersection point of the graph of the inverse proportion function $$y=\dfrac{k}{x}(k>0)$$ and circle $$⊙O$$. The area of the shaded part in the figure is $$10π$$. What is the formula for the inverse proportion function? | $$y=\dfrac{12}{x}$$ | math | |
fa57dd98-7b79-4914-a18c-1da1739da302 | Given that the solution to the equation ax+by=4 is , please write an equation containing a and b ______. | 2a+b=4 | math | |
c2dce387-741b-43e1-bf03-38c3e801d55c | Given the function $$f(x)=\begin{cases} 4x, &0< x \leqslant 5, \\ 20, &5 < x \leqslant 9, \\ 56-4x, &9< x \leqslant 14, \end{cases}$$ find $$f(a) (a \in (0,14])$$ in the algorithm, a selection structure is needed, where the shape of the decision box is ___ (fill in the number). | 4 | math | |
0ab786dd-aba0-405d-b06a-8fc3b74cc7c4 | On the 90th anniversary of the founding of the Communist Party of China, the County Education Bureau held a 'Red Songs Resound in China' activity. A 'Red Songs' singing competition was held in June. The organizing committee stipulates: any participant's score x satisfies: 60 ≤ x < 100. After the competition, the scores of all participants were organized as shown in the following table: According to the information provided in the table, n = ___. | 0.25 | math | |
5c3a8d6c-e254-419c-9ae6-0fb537a26e0c | The perspective view of a geometric solid is shown in the figure. Among the following four top views, which one is correct? | 2 | math | |
6c70e63b-43ef-4133-9431-581c7bf1111f | As shown in the figure, $$PD$$ is perpendicular to the plane of the square $$ABCD$$. Given that $$AB=2$$, $$E$$ is the midpoint of $$PB$$, and $$\cos〈\overrightarrow{DP},\overrightarrow{AE}〉=\dfrac{\sqrt{3}}{3}$$, if a spatial rectangular coordinate system is established with the lines containing $$DA$$, $$DC$$, and $$DP$$ as the $$x$$-axis, $$y$$-axis, and $$z$$-axis respectively, then the coordinates of point $$E$$ are ___. | $$(1,1,1)$$ | math | |
8e3a55e0-4a4b-476b-8daa-22fd1c57d408 | As shown in the figure, in $\Delta ABC$, $\angle B=40{}^\circ $, $\angle C=28{}^\circ $, point $D$ is on the extension of $BA$. What is the measure of $\angle CAD$? | 68° | math | |
8d2e9b25-4383-4a8c-8051-229f672be130 | If the front view of a triangular prism with a regular triangular base is shown in the figure below, then its surface area is. | $6+2\sqrt{3}$ | math | |
f7afdc83-fc56-46cf-8612-dc2747f29cf0 | The three views of a geometric body are shown in the figure, and all three triangles are right-angled triangles. What is the maximum value of xy? | 64 | math | |
c53d2068-0a9a-42c6-8e32-037bfabe27f7 | In the parallelogram $ABCD$, $\overrightarrow{AB}=\overrightarrow{a}$, $\overrightarrow{AD}=\overrightarrow{b}$, point $O$ is the intersection of diagonals $AC$ and $BD$, and point $E$ is on side $CD$ such that $DE=2EC$. Then $\overrightarrow{OE}=$ (express using $\overrightarrow{a}$ and $\overrightarrow{b}$). | $\frac{1}{2}\overrightarrow{b}+\frac{1}{6}\overrightarrow{a}$ | math | |
64b0f1e1-e3c4-469e-8f65-1d1c99233378 | To promote green travel in the city, the Little Blue Bike company plans to build a shared bike parking point E along the AB section of the Shiwu Port riverbank. There are two squares C and D nearby (as shown in the figure), with CA perpendicular to AB at A, and DB perpendicular to AB at B. AB = 4 km, CA = 2 km, and DB = 1 km. At what distance from point A should the parking point E be built so that its distance to both squares is equal? | $\frac{13}{8}$ | math | |
90d0b5e7-718e-4813-989c-5c410419baaf | As shown in the figure, in parallelogram $ABCD$, $AB=2AD=2$, $\angle BAD=60{}^\circ $, and $E$ is the midpoint of $DC$. The cosine of the angle between vector $\overrightarrow{AC}$ and $\overrightarrow{EB}$ is equal to. | $\frac{\sqrt{7}}{14}$ | math | |
57877655-362d-4ebf-a684-434b8f5202d6 | As shown in the figure, there is an isosceles triangle paper with a base angle of $35^\circ$. Now, it is cut along a line perpendicular to the base through a point on the base, dividing it into a triangle and a quadrilateral. What is the degree measure of the largest angle in the quadrilateral? | 125 | math |
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