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340fb804-8d8b-4040-a7af-ca856adaeceb | As shown in the figure, given the linear functions y = kx - b and y = $\frac{1}{3}$x intersect at point A(a, 1), then the solution for the equation $(k-\frac{1}{3})x = b$ is x =. | 3 | math | |
29c1d3e1-35d7-413f-9f83-b81ecf7ecb9d | A person is riding a bike on a road parallel to the tram tracks. He notices that a tram passes him from behind every 6 minutes and a tram passes him from the opposite direction every 2 minutes. Assuming the speeds of the trams and the person are constant (denoted as u$_{1}$ and u$_{2}$, respectively), please refer to the following diagram to find how often (in minutes, denoted as t) trams leave the station. | 3 | math | |
394169f4-8f13-44ef-8e47-0a4e2defaee3 | There are several (more than 20) pieces of a naturally grown traditional Chinese medicine. From these, 20 pieces are randomly selected, and their weights are measured precisely in grams. It is specified that each piece of the medicine weighing no less than 15 grams is considered a high-quality product. The program flowchart shown below is used to count the number of high-quality products among the 20 samples, where $m$ represents the weight of each piece of medicine. The integers that should be filled in at points 1. and 2. in the chart are respectively. | 14, 19 | math | |
8b5b85e8-94db-4fbe-ba26-2ae449355978 | As shown in the figure, the side length of rhombus ABCD is 3 cm, E is the midpoint of AB, and DE is perpendicular to AB. What is the area of rhombus ABCD in cm²? | $\frac{9\sqrt{3}}{2}$ | math | |
45cf16fd-440e-47ed-a113-56de4e029ece | As shown in the figure, in △ABC, ∠C=90°, AC=3, AB=5, point D is a point on side BC. If △ACD is folded along AD, point C exactly lands on point E on AB, then BD=． | 2.5 | math | |
d6cc217c-163f-4546-b5e7-f140ca78898c | May 10th is Mother's Day. On Mother's Day, Xiaoming gave his mother 5 Carnations and 5 gift boxes. According to the information in the image, how much did Xiaoming spend in total? | 440 | math | |
e78191c2-89e7-47b4-9de0-066c68f9b2f0 | Execute the program flowchart shown in the figure. If the input value of $n$ is $4$, then the output value of $s$ is. | $7$ | math | |
d5d28b6b-433d-473d-94ef-32f98003be91 | A workshop divides 10 workers into two groups, Group A and Group B, to process a certain type of part. The number of qualified parts processed by each worker in a unit of time is shown in the stem-and-leaf plot. It is known that the average number of qualified parts processed by the workers in both groups in a unit of time is 20, then m+n = | 11 | math | |
ffec0638-7dd9-45e8-94f4-de97ed9ed17f | Given: As shown in the figure, isosceles right triangles are constructed outward on the three sides of the right triangle ABC. If the hypotenuse AB = 3, then the area of the shaded part in the figure is. | 4.5 | math | |
134f6361-adfd-476c-ba22-f85fa5b53e7a | As shown in the figure, two bamboo poles AB and AD are leaning against the edge CE. It is measured that ∠ABC = α, ∠ADC = β. The ratio of the lengths of bamboo poles AB and AD is. | $\frac{\sin \beta }{\sin \alpha }$ | math | |
aded07b5-ce35-41a6-ac7d-436ffdee5d62 | In a physical education class, the PE teacher tested the standing long jump for 40 students in Grade 9, Class 1. The scores and the corresponding number of students are shown in the figure. What is the average score of this test? | $\frac{35}{4}$ | math | |
76be66a2-1284-4f34-9da3-35c7ced55b4a | In the figure, in △ABC, DE∥BC, $\frac{DE}{BC}=\frac{2}{3}$, the area of △ADE is 8, then the area of △ABC is | 18 | math | |
cb6f75d0-f38e-4904-9a57-7d3b17c3e6b1 | As shown in the figure, in △ABC, BP and CP are the angle bisectors of ∠ABC and ∠ACB, respectively, and PD ∥ AB, PE ∥ AC. The perimeter of △PDE is 6 cm. Then BC = ______ cm. | 6 | math | |
c0857def-714a-4466-8831-13e008ebacda | Given the following program, the output result is $$S=$$ ___. | $$56$$ | math | |
c426dd60-e310-40ee-81ca-348fa76857ef | To understand the situation of teachers using multimedia for teaching at a school, a simple random sampling method was used to select 20 teachers from 200 teaching staff. The number of times they used multimedia for teaching last semester was surveyed and is represented in the stem-and-leaf plot shown in the figure. Based on this, it can be estimated that the number of teachers out of 200 who used multimedia for teaching between 15 and 30 times last semester is ___. | $$100$$ | math | |
eb9c7348-8989-4dee-9da3-bfad25d8ba62 | As shown in the figure, the radius of $$\odot O$$ is $$OC =\quantity{5}{cm}$$, the line $$l \perp OC $$, with the foot of the perpendicular at $$H$$, and $$l$$ intersects $$\odot O$$ at points $$A$$ and $$B$$, where $$AB=\quantity{8}{cm}$$. How many $$\unit{cm}$$ must $$l$$ be moved downward along the line of $$OC$$ to be tangent to $$\odot O$$? | $$2$$ | math | |
6be44eb0-15d5-4a16-bff5-e6c430b10653 | As shown in the figure, take two points $$C$$ and $$D$$ on the ray $$AB$$. How many rays are there in the figure? | $$4$$ | math | |
9edbb849-2808-48f0-973a-af66c411c187 | As shown in the figure, in rectangle $$ABCD$$, the length of $$AB$$ is $$a$$, and the length of $$BC$$ is $$b$$, where $$\dfrac{2}{3}b < a$$. The rectangular paper is folded in the following sequence, then the length of $$C'D'$$ is ___ (expressed in terms of $$a$$ and $$b$$). | $$3a-2b$$ | math | |
94f1ca94-35fb-40e3-84cb-7d468b5bbafa | In the $$xOy$$ plane, consider the closed figure $$D$$ formed by the two semicircular arcs $$(x-1)^{2}+y^{2}=1(x \geqslant 1)$$ and $$(x-3)^{2}+y^{2}=1(x \geqslant 3)$$, and the two lines $$y=1$$ and $$y=-1$$, as shown in the shaded area in the figure. Let the solid of revolution formed by rotating $$D$$ around the $$y$$-axis be denoted as $$\Omega$$. If a horizontal cross-section of $$\Omega$$ at $$(0,y)(|y| \leqslant 1)$$ has an area of $$4 \pi \sqrt{1-y^{2}}+8 \pi$$, use Zu Geng's principle, a horizontally placed cylinder, and a rectangular prism to determine the volume of $$\Omega$$ as ___. | $$2 \pi ^{2}+16 \pi$$ | math | |
74719e94-99a5-46ac-8465-a282b144db80 | We can measure the distance to the Moon using the following method: as shown in the figure, during a full moon, place a 5-fen coin (with a diameter of about 2.4 cm) at a distance of about 2.6 meters from the eye (point O) at point AB, which exactly covers the Moon. Given that the diameter of the Moon is about 3500 kilometers, the distance between the Moon and the Earth is approximately ___ kilometers. | $$3.8\times 10^{5}$$ | math | |
7cc7a71d-d0d8-4511-9526-aea55f17aee7 | In the parallelogram $$ABCD$$, point $$E$$ is the midpoint of side $$AD$$. $$EC$$ intersects diagonal $$BD$$ at point $$F$$. Then, the ratio $$EF:FC$$ equals ___. | $$1:2$$ | math | |
41d6f508-327f-4a49-9f5c-205b76cfdd80 | As shown in the figure, in $$\triangle ABC$$, $$DE$$ is the perpendicular bisector of $$BC$$, with the foot of the perpendicular at $$E$$, and it intersects $$AC$$ at point $$D$$. If $$AB=6$$ and $$AC=9$$, then the perimeter of $$\triangle ABD$$ is ___. | $$15$$ | math | |
aa5e4ece-ca75-40e3-8f3f-15e94c4d8583 | To investigate the basic mathematical problem-solving abilities of students in a class, a random sample of students was taken to examine their scores on a mathematics test with a full score of 100 points. The score ranges were [45,55), [55,65), [65,75), [75,85), [85,95). The frequency distribution histogram is shown in the figure below. What is the average score of these students? | 64 | math | |
22b48c01-3a24-4b9d-88c1-5edffe73ff30 | A marksman's score in rings, denoted as $$\xi$$, has the following distribution: Given that the expectation of $$\xi$$, $$E\xi=8.9$$, what is the value of $$y$$? ______. | 0.4 | math | |
6461cfb7-ea0e-4ecd-987a-7fbd00f9c115 | During the large class break activities, students actively participate in physical exercises. Xiao Hong randomly selected a portion of students from the entire school to test their 'one-minute jump rope' performance and drew the partial frequency distribution histogram (divided into six groups from left to right, each group includes the minimum value but not the maximum value) and the pie chart as shown in the figure. If the number of 'one-minute jump rope' skips is no less than 130, the performance is considered excellent. The entire school has 1200 students. Based on the information provided in the figures, estimate the number of students in the school with excellent 'one-minute jump rope' performance. | 480 | math | |
b07eff77-7d96-4863-b847-5939761a6b02 | As shown in the figure, lines $$AB$$ and $$CD$$ intersect line $$EF$$ at points $$E$$ and $$F$$, respectively, and $$∠1=105^{\circ}$$. When $$∠2=$$___, $$AB∥CD$$. | $$75^{\circ}$$ | math | |
7d6bc469-7b1f-4c90-9d28-8ff93232761f | Use 6 different colors to color the 'smiley face' in the figure, with the requirement that the 'eyes' (i.e., the regions marked as $$A$$ and $$B$$ in the figure) use the same color. How many different ways are there to color it? (Answer with a number) | $$216$$ | math | |
14f00ad8-85f0-47ce-93d6-5bd38e4384ea | As shown in the figure, the following geometric bodies are formed by arranging small cubes with an edge length of $$1$$ on the ground according to a certain pattern. If the exposed surfaces are all painted (the bottom surface is not painted), then in the $$n$$-th geometric body, the number of small cubes with only two faces painted is ___ . | $$(8n-4)$$ | math | |
9dffc4a7-fe81-4382-8534-4120b93be070 | The figure below shows the graphs of the functions: $$y=x^{2}-1$$, $$y=x^{2}+6x+8$$, $$y=x^{2}-6x+8$$, $$y=x^{2}-12x+35$$ in the same Cartesian coordinate system. The most likely graph for $$y=x^{2}-6x+8$$ is ___. | 3 | math | |
1ca36e5d-3747-4441-84ec-bd645c4dc760 | As shown in the figure, the lines $$AD∥BE$$, $$AC$$ and $$BC$$ bisect $$∠BAD$$ and $$∠ABE$$ respectively, and $$∠CAD=55^{\circ}$$. Then $$∠CBE=$$ ___. | $$35^{\circ}$$ | math | |
c42d89fc-592b-49d7-818c-50b894100e1f | As shown in the figure, $$A$$, $$B$$, and $$C$$ are three points on a number line (unit length is $$1$$), and the numbers they correspond to are all integers. If the number corresponding to point $$B$$ is $$7$$ more than twice the number corresponding to point $$A$$, then the number corresponding to point $$C$$ is ___. | $$3$$ | math | |
367d9b97-a7a2-4305-a8c9-d1a9b9eda6e8 | As shown in the figure, in rhombus $$ABCD$$, the diagonals $$AC$$ and $$BD$$ intersect at point $$O$$, and $$AC=8$$, $$BD=6$$. Then the height $$DH$$ of rhombus $$ABCD$$ is ___. | $$4.8$$ | math | |
bbbc2f1a-3fc1-4f24-97a5-4f6bc47ad889 | The top view of a geometric solid is a rectangle as shown in the figure, the front view (or main view) is an isosceles triangle with a base length of 8 and a height of 5, and the side view (or left view) is an isosceles triangle with a base length of 6 and a height of 5. The volume of the geometric solid is ______. | $$80$$ | math | |
0dc0a891-8a05-4dd5-a861-9f75edb642f2 | The chart below shows the scores of Xiao Hua's five math tests. The average score of Xiao Hua's five tests is ______. | 92 | math | |
bc6e7cdd-ea13-4c3d-9efb-db4b9c1b2f2d | A student walks to school, and in 10 minutes, he covers a part of the total distance (as shown in the figure). Estimating that he cannot arrive on time by walking, he switches to a taxi to reach the school. The relationship between his journey and time is shown in the following graph (the total distance is '1'). How many minutes earlier does he arrive at the school compared to if he had walked the entire way? | 24 | math | |
f2f3a696-df4c-4518-9e5b-15c0ad17ba58 | As shown in the figure, it is known that $$\angle 1 = 34\degree$$, then $$\angle 2 =$$______． | 56° | math | |
da168b9a-05ce-4e40-afca-dd801bb3390a | Two right-angled triangular rulers are placed with their right-angle vertices coinciding as shown in the figure. If $$∠AOD=110^{\circ}$$, then $$∠BOC=$$ ___. | $$70^{\circ}$$ | math | |
f4387a69-ca69-46b1-8aa8-81875b61cc6b | As shown in the figure, a square ABCD with side length 4 is folded along the crease EF, such that point B lands on the midpoint G of side AD. The area of quadrilateral BCEF is ___. | 6 | math | |
6ce81a0e-4e76-45b8-94f1-aee2cd6b6c48 | As shown in the figure, a residential area has a rectangular open space that is 30m long and 24m wide. It is planned to construct two identical rectangular green spaces within it, with their total area being 480m². There are pedestrian walkways of equal width between the two green spaces and around them. What is the width of the pedestrian walkway in meters? | 2 | math | |
ed44f172-0285-4b16-837e-ad6528d52533 | As shown in the figure, line $$AD \parallel BE$$, $$AC$$ and $$BC$$ bisect $$\angle BAD$$ and $$\angle ABE$$ respectively, and $$\angle CAD = 55^{°}$$. What is $$\angle CBE =$$ ___. | $$35^{\circ}$$ | math | |
1d4b895c-a57c-4475-82b7-d10e597e2ef9 | As shown in the figure, draw the perpendicular bisector of line segment $$AB$$, intersecting $$AB$$ at point $$O$$. On this perpendicular bisector, intercept $$OC = OA$$. With point $$A$$ as the center and the length of $$CA$$ as the radius, draw an arc intersecting $$AB$$ at point $$P$$. The ratio of $$AP$$ to $$AB$$ is ___. | $$\dfrac{\sqrt{2}}{2}$$ | math | |
84c34316-9b90-430c-985e-a64c2247ec34 | As shown in the figure, a rectangle is cut into four pieces along the dotted lines (where $$x>y$$). These four pieces can be exactly assembled into a square. If $$y=2$$, then the value of $$x$$ is ___. | $$1+\sqrt{5}$$ | math | |
74d0eab7-a761-488e-bf98-9693e5b8fc2f | In the semicircle shown in the figure, $$AD$$ is the diameter, and $$AD=3$$, $$AC=2$$. The value of $$\sin B$$ is ___. | $$\dfrac{2}{3}$$ | math | |
7accc25d-fbc2-4e2b-b386-fbefac985d2b | As shown in the figure, in rectangle $$ABCD$$, $$M$$ and $$N$$ are the midpoints of sides $$AD$$ and $$BC$$, respectively, and $$E$$ and $$F$$ are the midpoints of segments $$BM$$ and $$CM$$, respectively. If $$AB=8$$ and $$AD=12$$, then the perimeter of quadrilateral $$ENFM$$ is ______. | $$20$$ | math | |
718b11ad-6569-4765-b3e4-4ce81c3fa842 | In the figure, ▱ABCD has EF passing through the intersection point O of the diagonals. If AB = 4 cm, AD = 3 cm, and OF = 1 cm, then the perimeter of quadrilateral BCFE is ______. | 9cm | math | |
dce43017-d982-44d3-b4cb-a927aead82a0 | According to the 'Thresholds and Testing for Blood and Breath Alcohol Content for Vehicle Drivers' released by the General Administration of Quality Supervision, Inspection and Quarantine, the thresholds for blood alcohol content for drivers after drinking or being drunk are shown in Table 1. The traffic law enforcement department of a certain region has compiled the law enforcement record data for May, as shown in Table 2. The estimated probability of driving after drinking in this region in May is ___. | $$0.09$$ | math | |
f2e43b91-a789-4569-8692-46ff42152ac8 | From a physical fitness test involving 400 participants, the scores of 50 randomly selected individuals are summarized in the following table. The estimated variance of the scores for the 400 participants is ___. | 1.16 | math | |
45b103c6-0dfa-48cd-b776-e0d67cbb53ed | The swimming ring we use when swimming can be considered as formed by rotating the following shape ___ (fill in the number) around its axis of symmetry. | 3 | math | |
0b0608ec-e8de-45dc-9404-be5c5e9b3ba1 | As shown in the figure, in the Cartesian coordinate system, △P$_{1}$OA$_{1}$, △P$_{2}$A$_{1}$A$_{2}$, △P$_{3}$A$_{2}$A$_{3}$, ... are all isosceles right triangles, with their right-angle vertices P$_{1}$(3,3), P$_{2}$, P$_{3}$, ... all lying on the line y = -$\frac{1}{3}$x + 4. Let the areas of △P$_{1}$OA$_{1}$, △P$_{2}$A$_{1}$A$_{2}$, △P$_{3}$A$_{2}$A$_{3}$, ... be S$_{1}$, S$_{2}$, S$_{3}$, ..., respectively. According to the pattern reflected in the figure, S$_{2018}$ =. | $\frac{9}{{{4}^{2017}}}$ | math | |
8dd77802-5e7d-4bdd-a3ad-2865a52510d4 | The right table gives a 'triangular number array', where each column forms an arithmetic sequence, and from the third row onwards, each row forms a geometric sequence. Let the number in the $i$th row and $j$th column be ${{a}_{ij}}(i\ge j,i,j\in {{N}^{*}})$, then ${{a}_{mn}}=$$(m\ge 3)$. | $\frac{m}{{{2}^{n+1}}}$ | math | |
52691b9d-f705-403c-9b93-584e88ec072b | In the figure, in rectangle $ABCD$, $AB=1$, $BC=\sqrt{3}$. A circle is drawn with $B$ as the center and $BD$ as the radius, intersecting the extension of $BC$ at point $M$. Another circle is drawn with $D$ as the center and $CD$ as the radius, intersecting $AD$ at point $N$. The area of the shaded region in the figure is. | $\frac{7\pi }{12}-\frac{\sqrt{3}}{2}$ | math | |
751d4fa6-4dfd-4b49-a9af-8091001b7d93 | In the Pascal's Triangle, replace each number with a fraction to form a fraction triangle known as the Leibniz Triangle, as shown in the figure. If an ordered pair of real numbers (m, n) represents the nth number from the left in the mth row, such as (4, 2) representing the fraction $\frac{1}{12}$, then the fraction represented by (8, 2) is. | $\frac{1}{56}.$ | math | |
b40dbb22-d872-4ce6-a2c7-a3bec3493a86 | The relationship between the quantity of bananas sold (in kilograms) and the selling price (in yuan) at a fruit shop is shown in the following table: If the quantity of bananas sold is represented by x (kilograms) and the selling price by y (yuan), then the relationship between y and x is; | y=3x | math | |
c8ecb5ab-17f7-4726-b12e-16f0ca0a059c | As shown in the figure, in rectangle ABCD, DE is perpendicular to AC at E, and ∠ADE:∠EDC = 3:2. What is the measure of ∠BDE? | 18° | math | |
736e103b-f7eb-45d1-abad-e87bdceb5c9a | Cities A and B are 600 kilometers apart. Two cars, Car A and Car B, start from City A and drive towards City B at the same time. Car A, after reaching City B, immediately returns and meets Car B on its way back. The graph below shows the distance $y$ (km) from City A and the driving time $x$ (h) for both cars. When they have been driving for 7 hours, the two cars meet. What is the speed of Car B? | 75 km/h | math | |
951aef9f-fc04-4abc-a0bb-03f5fb9bddac | As shown in the figure, person 甲 starts from point $A$ and walks in a direction $70{}^\circ $ north of east to point $B$, while person 乙 starts from point $A$ and walks in a direction $15{}^\circ $ south of west to point $C$. The degree measure of $\angle BAC$ is. | $125{}^\circ $ | math | |
486d489d-d67e-4fcf-945d-398d4ec4d8e1 | As shown in the figure, several squares with patterns and without patterns, all of the same size, are combined to form a regular pattern. It is known that the side length of each small square is $0.3m$. Please express the relationship between the number of squares with patterns $n$ and the length of the pattern ${{L}_{n}}$ using an algebraic expression. | ${{L}_{n}}=0.3(2n+1)$ | math | |
600e60fe-03e5-4ed2-9ee1-31b78b6bccc2 | As shown in the figure, in rectangle $ABCD$, $AB=10$, $AD=6$, and $E$ is a point on $BC$. When $\Delta CDE$ is folded along $DE$, point $C$ lands on point $F$ on $AB$. Find the length of $CE$. | $\frac{10}{3}$ | math | |
20c655f6-649a-40c1-9b4a-ceb33fb65e57 | As shown in the figure, there are four points $$A$$, $$B$$, $$C$$, and $$D$$ on the number line. The point representing the opposite number of $$2$$ is point ___. | $$A$$ | math | |
10cbfb04-2b11-4fc1-b8c1-1402996ece1b | As shown in the figure, in $$\triangle ABC$$, $$AB=2$$, $$BC=1.5$$, $$\angle ABC=120^{\circ}$$, if $$\triangle ABC$$ is rotated around the line $$BC$$ for one full revolution, then the volume of the resulting solid of revolution is ___. | $$\dfrac{3 \pi }{2}$$ | math | |
47115a04-0840-4af8-8e8d-ab0fb238f0be | Let the universal set $U=\mathbb{R}$, $A=\{x\in \mathbb{N} | 1 \le x \le 10\}$, $B=\{x\in \mathbb{R} | x^2 + x - 6 = 0\}$. Then the set represented by the shaded area in the following figure is: | $\{2\}$ | math | |
57abbf22-8a73-4f90-90fe-851bc392d3b8 | As shown in the figure, the lengths of the two diagonals of rhombus $$ABCD$$ are $$AC=8$$ and $$BD=6$$, respectively. Therefore, the area of rhombus $$ABCD$$ is ___. | $$24$$ | math | |
44276b2c-df5e-4c59-a61b-0fb62132752c | Below is a simple numerical operation program. When the input value of $x$ is 4, the output value is. | 2 | math | |
045d621e-a814-4806-8de0-b5ba10f6d18b | As shown in the figure, in the complex plane, the complex number corresponding to point A is z, then the complex number z = ___. | 2-i | math | |
f4a47afd-9713-460c-bef8-f3e4de60dcd0 | As shown in the figure, $$AD$$ is the median of $$\triangle ABC$$, $$AB > AC$$, $$AB = 8\unit{cm}$$, and the difference in the perimeters of $$\triangle ABD$$ and $$\triangle ACD$$ is $$2\unit{cm}$$, then $$AC = $$___$$\unit{cm}$. | $$6$$ | math | |
c4f3ce75-621a-486c-b94b-1e88684be70d | As shown in the figure, a line parallel to $$BC$$, $$DE$$, divides $$\triangle ABC$$ into two parts of equal area. What is the value of $$\dfrac{AD}{AB}$$? | $$\dfrac{\sqrt{2}}{2}$$ | math | |
567bb63d-fd8e-4ae0-bee3-ae9b06a5926d | A company has three factories producing the same type of electronic product. The production distribution of the three factories is shown in the figure. Now, using stratified sampling, 100 items are to be drawn from the products of the three factories for a lifespan test. The number of items that should be drawn from the first factory is ___. The test results show that the average lifespan of the products from the first, second, and third factories are 1020 hours, 980 hours, and 1030 hours, respectively. Estimate the average lifespan of the products produced by this company to be ___ hours. | 50 1015 | math | |
1e3d56aa-130b-4ebe-91c8-c6070529c4bd | The flowchart of a program is shown in the figure. After the program runs, the output value of $$x$$ is $$31$$. Then, $$a$$ equals ______. | $$3$$ | math | |
3c966eb6-2736-44d9-be16-e97e44049a2e | A clothing mall conducted a study to understand the relationship between the monthly sales volume of sweaters $$y$$ (pieces) and the average monthly temperature $$x$$ ($$\degree{\rm C}$$). They randomly collected data on the monthly sales volume and the average temperature for four months, as shown in the table below: From the data in the table, the linear regression equation $${\mathstrut{\hat y}}={\mathstrut{\hat b}}x+{\mathstrut{\hat a}}$$ was calculated, where $${\mathstrut{\hat b}}\approx-2$$. The meteorological department predicts that the average temperature for the next month will be about $$6\degree{\rm C}$$. Based on this statistical data, the estimated sales volume of sweaters for the next month is ______ pieces. | $$46$$ | math | |
610fa059-ba68-4a94-8000-42e4fae60d90 | As shown in the figure, if the front view of a cone is an isosceles right triangle with a hypotenuse of length $$4$$, then the lateral surface area of the cone is ___. | $$4\sqrt{2} \pi$$ | math | |
03d4ef55-5bc6-4240-8bae-468166d7925a | In an opaque bag, there are $$n$$ balls that are identical except for their color, including $$5$$ black balls. A ball is randomly drawn from the bag, its color is noted, which is called one ball-drawing trial, and then it is put back into the bag. The bag is mixed again, and another ball is drawn. Below is a table of the number of ball-drawing trials and the number of times a black ball was drawn, simulated using a computer: According to the table, the estimated value of $$n$$ is ___. | $$10$$ | math | |
0ff2d417-783f-424d-a108-60e027d9bd9e | As shown in the figure, in the right triangle $$\text{Rt}\triangle ABC$$, $$∠C=90^{°}$$, $$BC=3$$, $$AC=6$$, point $$D$$ is a moving point on the side $$AC$$, and point $$D$$ moves from point $$C$$ to point $$A$$. If $$CD=x$$ and the area of $$\triangle ABD$$ is $$y$$, then the relationship between $$y$$ and $$x$$ is ___. (No need to specify the range of $$x$$) | $$y=9 -\dfrac{3}{2}x$$ | math | |
cb6bc8c7-b39e-45ef-915b-e89a322ac547 | As shown in the figure, $$\triangle ABC \cong \triangle DEF$$, $$AD = 10\ \unit{cm}$$, $$BE = 6\ \unit{cm}$$, then the length of $$AE$$ is ___. | $$\quantity{2}{cm}$$ | math | |
0988cba8-e436-4cb4-9789-7f1c8482d8a7 | Two masts on a ship, each 7.5m high, are 15m apart. A 30m long rope is tied to the tops of the masts and is stretched as shown in the figure. Assuming the rope lies in the plane containing the two masts, the distance from the point P where the rope touches the deck to mast AB is ___ m. | 4.75 | math | |
61824613-d34b-499c-89e4-63400179e5ed | As shown in the figure, in $$\triangle ABC$$, point $$D$$ is on side $$BC$$, $$AD \bot AC$$, $$\sin\angle BAC = \dfrac{2\sqrt{2}}{3}$$, $$AB = 3\sqrt{2}$$, $$AD = 3$$, then the length of $$BD$$ is ___. | $$\sqrt{3}$$ | math | |
729b2a9c-e708-47d1-82a1-41be61d53e27 | As shown in the figure, in the rhombus $$ABCD$$, $$AB=10$$, $$AC=12$$, then its area is ___. | $$96$$ | math | |
96ef0f0f-963c-485c-b645-c2155cfd1319 | As shown in the figure, this is a cross-section of a water pipe with a diameter of $$2m$$ laid horizontally. The width of the water surface is $$1.6m$$. The deepest point of the water in the pipe at this time is ___$$m$$. | $$0.4$$ | math | |
428b3aff-1aaf-4f6a-94ad-e1fde3b89682 | The English oral test scores (unit: points) of 10 students in groups $$A$$ and $$B$$ are as follows: Group $$A$$: $$82$$, $$84$$, $$85$$, $$89$$, $$79$$, $$80$$, $$91$$, $$89$$, $$79$$, $$74$$; Group $$B$$: $$76$$, $$90$$, $$84$$, $$86$$, $$81$$, $$87$$, $$86$$, $$82$$, $$85$$, $$83$$. The scores of the two groups are represented in a stem-and-leaf plot (as shown in the figure). The scores of group ___ are more consistent. | $$B$$ | math | |
8a5f5d30-0069-4b93-bd5d-752c9e9c2c16 | As shown in the figure, given that $$O$$ is the origin, point $$A\left (3,0\right )$$, $$B\left (4,4\right )$$, and $$C\left (2,1\right )$$, the coordinates of the intersection point $$P$$ of $$AC$$ and $$OB$$ are ___. | $$\left (\dfrac{3}{2},\dfrac{3}{2} \right )$$ | math | |
cbf67581-a84e-4488-bb83-9af322c2e554 | According to the program shown in the figure, the output value of $$k$$ is ___. | $$3$$ | math | |
5d710ccf-bc87-443d-af72-c3dcfd246b95 | Draw a square, then connect the midpoints of each side of this square to form a second square, and so on. In this way, a total of 3 squares are drawn, 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? | $$\dfrac{1}{4}$$ | math | |
28d2dcaa-de76-476b-96a5-72682dac6b4b | As shown in the figure, given points $$A(a,0,0)$$, $$B(0,b,0)$$, $$C(0,0,c)$$, then a normal vector of plane $$ABC$$ is ___. | $$(bc,ac,ab)$$ | math | |
01fb4f15-6f38-4b4e-8186-7b0d778706a2 | As shown in the figure, in the tetrahedron $$C-ABD$$, $$E$$ and $$F$$ are the midpoints of $$AC$$ and $$BD$$ respectively. If $$CD=2AB=4$$ and $$EF \bot AB$$, then the angle formed by $$EF$$ and $$CD$$ is ___. | $$30^{\circ}$$ | math | |
b0702489-0f31-47f8-afe9-5a415d47db64 | The figure below shows a simple numerical operation program. When the input value of $$x$$ is $$-1$$, the output value is ___. | $$-2$$ | math | |
2211f64e-0747-41b4-9ebe-a756414c224c | A factory conducts a sampling inspection of a batch of products. Based on the net weight (unit: grams) data of the inspected products, a frequency distribution histogram is drawn as shown in the figure. It is known that the range of the product's net weight is $$[96,106]$$, and the number of products with a net weight in the range $$[96,100)$$ is $$24$$. The number of products with a net weight in the range $$[98,104)$$ in the sample is ______. | $$60$$ | math | |
abbd794a-88ff-417b-bf60-e4066864cd15 | In the right triangle $$ABC$$, $$\angle B=90^{\circ}$$, the perpendicular bisector of $$AC$$, $$DE$$, intersects $$AC$$ at point $$D$$ and $$BC$$ at point $$E$$. If $$\angle BAE=10^{\circ}$$, then the measure of $$\angle C$$ is ___. | $$40^{\circ}$$ | math | |
c1854fd1-2739-4a9f-9d51-0d1a46ba5847 | To welcome the school arts festival, a class in the seventh grade is holding a class song collection activity, with submissions accepted from Monday to Friday. The class committee grouped the number of works submitted by the students by day and created a frequency distribution histogram as shown in the figure. It is known that the ratio of the heights of the rectangles from left to right is $$2:3:4:6:1$$, and the frequency of the second group is $$9$$. How many works were submitted by the whole class? | $$48$$ | math | |
1a5ef5aa-5f8a-42b7-97f2-760c01a23b0a | In the stem-and-leaf plot shown, the sum of the medians of Group A and Group B is | 64 | math | |
fd2a02c7-a718-4094-bf13-53696a21998d | As shown in the figure, $$\triangle ABC$$ is an equilateral triangle, and $$BM=CN$$. $$AM$$ and $$BN$$ intersect at point $$P$$. The measure of $$\angle APN$$ is ______ degrees. | 60 | math | |
403e26ad-d25d-420c-a48b-602fa1f00779 | As shown in the figure, extend line segment $$AB$$ to $$C$$, such that $$BC=4$$. If $$AB=8$$, then the length of line segment $$AC$$ is ___ times the length of $$BC$$. | $$3$$ | math | |
5f137a38-61d1-4843-b54b-298d16173882 | The graph of the function $$y=f(x)=a^{x}-\dfrac{1}{a}$$ (where $$a>0$$ and $$a \neq 1$$) could be ___. | 4 | math | |
df698860-1445-4fb5-bb12-89c595e5cfa7 | As shown in the figure, given the hyperbola $$\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1(a>0,b>0)$$ with the left focus at $$F_1$$, the left and right vertices at $$A$$ and $$B$$ respectively, and point $$M$$ on the hyperbola and above the $$x$$-axis, $$MF_1⊥x$$-axis. The lines $$MA$$ and $$MB$$ intersect the $$y$$-axis at points $$P$$ and $$Q$$ respectively. If $$|OP|=\sqrt{2}|OQ|$$, then the eccentricity of the hyperbola $$e=$$ ___. | $$3+2\sqrt{2}$$ | math | |
a65d00e4-aef0-44f1-bbc0-6488bcb017d2 | The figure below is a flowchart of an algorithm. Answer the following question: When the input value is $$3$$, the output result is ___. | $$8$$ | math | |
44cd4307-b724-402e-a471-4b0f5787a7f5 | As shown in the figure, regular octagons with side lengths of $$1$$, $$2$$, and $$3$$ are stacked together, with the distance between adjacent beads on the same side being $$1$$. If regular octagons with side lengths of $$4$$, $$5$$, $$6$$, $$\cdots$$, $$10$$ are placed in the same manner, then the total number of beads in the tessellation of these $$10$$ regular octagons is ___. | $$341$$ | math | |
6df9187b-c38e-44ce-b414-cf60f19772c5 | As shown in the figure, in $$\triangle ABC$$, it is given that $$\overrightarrow{AN}=\dfrac{1}{2}\overrightarrow{AC}$$, and $$P$$ is a point on $$BN$$. If $$\overrightarrow{AP}=m\overrightarrow{AB}+\dfrac{1}{4}\overrightarrow{AC}$$, then the value of the real number $$m$$ is ___. | $$\dfrac{1}{2}$$ | math | |
b904c107-0276-4fe6-8700-451c8d00a19a | As shown in the figure, $$\triangle ABC$$ is an equilateral triangle with a side length of $$2\sqrt{3}$$, and $$P$$ is any point on the circle centered at $$C$$ with a radius of $$1$$. Then, $$\left (\overrightarrow{AP}\cdot \overrightarrow{BP}\right )_{ \min }=$$ ___. | $$1$$ | math | |
f9eb7ea9-debb-4031-a02a-dce5ac5c0b36 | The graph shown is of the inverse proportion function $y=\frac{k}{x}$ in the second quadrant. If the area of rectangle OABC in the graph is 2, then k=． | -2 | math | |
ec216eff-611e-4d88-bbbb-09a0f869750c | As shown in the figure, the light bulb $P$ is directly above the horizontal rod $AB$. The shadow of $AB$ under the light is $CD$, and $AB \parallel CD$. Given that $AB = 1.5m$, $CD = 4.5m$, and the distance from point $P$ to $CD$ is $2.7m$, what is the distance between $AB$ and $CD$ in meters? | $1.8$ | math | |
3b62e521-79c4-4c66-b9d6-6864d014e96b | As shown in the figure, in rectangle ABCD, DC = 6 cm. There is a point E on DC. Folding triangle ADE along line AE makes point D fall exactly on side BC, and this point is labeled as F. If the area of triangle ABF is 24, then the length of CE is cm². | $\frac{8}{3}$ | math | |
9c5cb87d-67f7-43e6-b143-6c52fd2effb2 | As shown in the figure, a right-angled triangular board ABC with a ${30}^\circ$ angle is rotated around point A, so that points B, A, and C′ lie on the same straight line. The angle of rotation of the triangular board ABC is _____ degrees. | 150° | math |
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