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cc7f65d5-bdd5-43ff-942b-c4e25a2804dd | A cube loses one vertex after a 'corner' is removed. This geometric shape is ___ (fill in the number). | 3 | math | |
6fd886c5-8a54-48bd-a6c0-b267e05e5062 | In a certain public welfare activity, Xiao Ming conducted a statistical analysis of the donation situation of his classmates and drew an incomplete statistical chart as shown in the figure. Among them, the number of people who donated 100 yuan accounts for 25% of the total number of people in the class. What is the median of the donations this time? | 20 | math | |
d14ce504-7fa7-4ba7-829a-f7a4a97b3319 | As shown in the figure, a car departs from station $$A$$, passes through station $$B$$ to station $$C$$, and then returns. On the way out, it stops at station $$B$$, but does not stop on the return trip. The speed of the car on the way out is $$40$$ kilometers per hour. What is the average speed of the car for the round trip in kilometers per hour? | 45 | math | |
82055694-be2b-4de9-8895-d72edf48de61 | Execute the program flowchart shown in the figure. If the input value of $$n$$ is $$3$$, then the output value of $$s$$ is ______. | $$4$$ | math | |
d2570007-6964-4b20-bd93-ddd8dcf9c8da | As shown in the figure, $$BC$$ is the diameter of $$ \odot O$$, and $$BC=6$$. The extension of $$CB$$ intersects the tangent line at point $$D$$ on $$ \odot O$$ at point $$A$$. If $$AD=4$$, then $$AB=$$ ___. | $$2$$ | math | |
545fd10a-3b77-4fbe-8c62-fdef037fed77 | As shown in the figure, in ΔABC, the angle bisectors of ∠B and ∠C intersect at point O. A line DE parallel to BC is drawn through point O, intersecting sides AB and AC at points D and E, respectively. If AB = 5 and AC = 4, then the perimeter of ΔADE is ______. | 9 | math | |
8e64c030-29b0-4ded-b904-707791935593 | As shown in the figure, given lines $$l_{1}$$, $$l_{2}$$, $$l_{3}$$, and $$l_{1} \parallel l_{2} \parallel l_{3}$$, line $$AC$$ intersects $$l_{1}$$, $$l_{2}$$, $$l_{3}$$ at points $$A$$, $$B$$, $$C$$ respectively, and line $$FD$$ intersects $$l_{1}$$, $$l_{2}$$, $$l_{3}$$ at points $$F$$, $$E$$, $$D$$ respectively. Given $$AB:BC=3:2$$ and $$DF=20$$, then $$DE=$$ ___. | $$8$$ | math | |
31c0fc56-97b4-4596-a408-2f3a0192efae | Plant 5 different types of crops in the 5 experimental fields shown in the figure (all 5 types of crops must be selected), with each field planting only 1 type of crop. How many different planting methods are there? | 120 | math | |
dd561fd1-acad-41ea-8228-5d22774399d7 | As shown in the figure, $$AB\parallel CD$$, $$PB$$ bisects $$\angle ABC$$, $$PC$$ bisects $$\angle DCB$$. Then $$\angle P=$$ ___ degrees. | $$90$$ | math | |
3e6760d9-fc51-4906-97b1-95fa95bf4a78 | Given the following program: When $$x=3$$, $$y=4$$, $$z=5$$, the output result is ___. | $$48$$ | math | |
f33246e3-819a-47b7-8159-6b6281d1e425 | As shown in the figure, the line $$y = - \dfrac{4}{3}x + 4$$ intersects the $$x$$-axis and $$y$$-axis at points $$A$$ and $$B$$, respectively. After rotating $$\triangle AOB$$ 90° clockwise around point $$A$$, we get $$\triangle A O'B'$$. What are the coordinates of point $$B'$$? | $$(7,3)$$ | math | |
b6df39cf-e850-4b01-8e19-b8fb18b0b349 | The figure shows the unfolded net of a cubic box, with different numbers and letters on its surfaces. When folded along the dotted lines to form a cube, the sum of the numbers on opposite faces is equal. Then, $$x+y=$$ ______ . | 11 | math | |
9f32c32b-ae44-4ea2-b1b9-55df20703eab | As shown in the figure, the perimeter of $$\square ABCD$$ is $$16cm$$. $$AC$$ and $$BD$$ intersect at point $$O$$, and $$OE \bot AC$$ intersects $$AD$$ at $$E$$. Then the perimeter of $$\triangle DCE$$ is ___. | $$8cm$$ | math | |
be91e477-7892-497e-a48f-40fa2223ee9c | The age distribution of 200 employees in a certain unit is shown in the figure. Now, 40 employees are to be selected as a sample using systematic sampling. All employees are randomly numbered from 1 to 200 and divided into 40 groups in numerical order (1 to 5, 6 to 10, ... 196 to 200). If the number drawn from the 5th group is 22, then the number that should be drawn from the 8th group is ___. If stratified sampling is used, the number of people to be drawn from the age group under 40 should be ___. | 37 20 | math | |
6b5ac825-b818-4c9a-a42a-647e5d8e4a0d | As shown in the figure, there are different paths connecting locations $$A$$, $$B$$, $$C$$, and $$D$$, and each segment can only be traversed once. The number of different ways to travel from location $$A$$ to location $$C$$ is ___. | $$7$$ | math | |
8a8516ea-3ae0-450d-8c10-d86248be7ab6 | In the rectangle $$ABCD$$, $$AB=3$$, $$BC=4$$. If the rectangle is folded along the diagonal $$BD$$, then the area of the shaded part in the figure is ___. | $$\dfrac{75}{16}$$ | math | |
3d28d623-3631-4f6c-8797-6d4467eb213f | In early winter 2014, a small convenience store wanted to understand the relationship between the sales volume of hot tea $$y$$ (cups) and the temperature $$x (^{\circ}C)$$, and randomly recorded the number of cups of hot tea sold and the temperature on 4 days, creating the following table: From the data in the table, the linear regression equation $$\hat{y}=\hat{b}x+\hat{a}$$ was calculated, with $$\hat{b}\approx-2$$. Predict the sales volume of hot tea when the temperature is $$-5^{\circ}C$$. (Given the regression coefficient $$\hat b=\dfrac{\sum\limits_{i=1}^{n}{x_iy_i}-n\overline{x}\overline{y}}{\sum\limits_{i=1}^{n}x_i^{2}-n\overline{x}^{2}}$$, $$\hat{a}=\overline{y}-\hat b\overline{x}$$) | $$70$$ | math | |
fd9a6da9-d5cc-43d3-bcca-fda9de56675e | The table below shows the frequency distribution of a sample data with a capacity of $$10$$. If the midpoint of each group is used to approximately calculate the mean of this set of data $$\overline{x}$$, then the value of $$\overline{x}$$ is ___. | $$\number{19.7}$$ | math | |
dfe70af0-0f8e-4109-9ccf-e0d280e9940b | A middle school has launched a 'One Hour of Sunshine Sports' activity. Based on the school's actual situation, as shown in the figure, it has decided to offer four sports activities: 'A: Kick Shuttlecock, B: Basketball, C: Jump Rope, D: Table Tennis' (each student must choose one). To understand which activity the students like best, a random sample of students was surveyed, and the survey results were plotted in the statistical chart shown in the figure. Among the students who participated in the survey, the number of students who like the Jump Rope activity is ___. | 40 | math | |
de6e074c-bd47-4a96-a686-3ded2a6ed91e | As shown in the figure, the line $$l_{1}$$: $$y=k_{1}x+4$$ intersects with the line $$l_{2}$$: $$y=k_{2}x-5$$ at point $$A$$. They intersect the $$y$$-axis at points $$B$$ and $$C$$ respectively. Points $$E$$ and $$F$$ are the midpoints of segments $$AB$$ and $$AC$$ respectively. The length of segment $$EF$$ is ___ | $$\dfrac{9}{2}$$ | math | |
229fab7e-5765-45d2-a219-f388d065dca1 | As shown in the figure, in rhombus $$ABCD$$, $$\angle B=60^{\circ}$$, and $$AB=4$$. What is the perimeter of the square $$ACEF$$ with side $$AC$$? | $$16$$ | math | |
bcd463d7-1a5e-4582-81fb-272d0399236f | In the Tang Dynasty of China, the poet Wang Wei wrote in his poem: 'The bright moon shines among the pines, the clear spring flows over the stones.' Here, the bright moon and the clear spring are natural scenes, which remain unchanged. The adjective 'bright' corresponds to 'clear,' the noun 'moon' corresponds to 'spring,' the parts of speech remain unchanged, and the same applies to the other nouns. The unchanging properties in changes are widely present in both literature and mathematics. For example, we can use a geometric function board software to draw the parabola $$C$$: $$x^{2}=y$$ (as shown in the figure). Draw a line $$l$$ through the focus $$F$$ intersecting $$C$$ at points $$A$$ and $$B$$. Draw tangents to $$C$$ at points $$A$$ and $$B$$, and let these tangents intersect at point $$P$$. Draw a vertical line from point $$P$$ to the $$x$$-axis, intersecting $$C$$ at point $$N$$. Moving point $$B$$ along $$C$$, it is found that $$\dfrac{|NP|}{|NF|}$$ is a constant value. What is this constant value? | $$1$$ | math | |
e1e18ec4-c087-48a3-8f25-4514aecbb73d | Uncle Li used a 60-meter-long fence to enclose a right trapezoid small garden against a wall (as shown in the figure). The area of this garden is ______ square meters. | 400 | math | |
2951450d-ee4f-4727-9d62-017893d1e013 | As shown in Figure 1, it is a right-angled triangular paper piece, $$ \angle A=30^{ \circ }$$, $$BC=\quantity{4}{cm}$$, it is folded so that point $$C$$ lands on point $$C'$$ on the hypotenuse, with the fold line being $$BD$$, as shown in Figure 2. Then, Figure 2 is folded along $$DE$$, so that point $$A$$ lands on point $$A'$$ on the extension of $$DC'$$, as shown in Figure 3. The length of the fold line $$DE$$ is ___. | $$\dfrac{8}{3} \ \unit{cm}$$ | math | |
c3819de4-274c-4be9-a6d3-7d1225838d77 | Given the development of a cone as shown in the figure, where the central angle of the sector is $$120^{\circ}$$, and the radius of the base circle is $$1$$, what is the volume of the cone? | $$\dfrac{2\sqrt{2} \pi }{3}$$ | math | |
cdfce964-0ad0-40ea-ae87-184637d9ff3e | In a marathon race, the stem-and-leaf plot of the results (in minutes) of 35 athletes is shown in the figure. If the athletes are ranked from best to worst as numbers 1 to 35, and a systematic sampling method is used to select 7 people, then the number of athletes whose results fall within the interval [139, 151] is ___. | 4 | math | |
30ee48f8-f947-4d9b-a5cb-d2c709be9451 | As shown in the figure, fold one side of the rectangle $$ABCD$$, $$AD$$, so that point $$D$$ lands on point $$F$$ on side $$BC$$. It is known that the crease $$AE=5\sqrt{5}\ \unit{cm}$$, and $$\tan \angle EFC=\dfrac{3}{4}$$, then the perimeter of rectangle $$ABCD$$ is ___. | $$\quantity{36}{cm}$$ | math | |
c5ffeeeb-133d-4d5c-a30e-38114cb7fe78 | As shown in the figure, it is known that the graphs of the functions $$y=ax+2$$ and $$y=bx-3$$ intersect at point $$A\left(2,-1\right)$$. Based on the graph, the solution set for the inequality $$ax>bx-5$$ is ___. | $$x < 2$$ | math | |
3c77a3d4-66c7-4228-93e4-6a4f71835753 | As shown in the figure, △ABC is inscribed in ⊙O, ∠BAC = 120°, AB = AC = 4, and BD is the diameter of ⊙O. Then BD = ______. | 8 | math | |
ba6dcd36-93b1-4771-bf92-2f7a793630de | As shown in the figure, $$\triangle ABD$$ and $$\triangle ACE$$ are both equilateral triangles. The basis for determining that $$\triangle ADC \cong \triangle ABE$$ is ___. | $$SAS$$ | math | |
867fd117-056f-41ca-803f-7927a1f2e237 | As shown in the figure, in the 'Pascal's Triangle', the numbers above the diagonal line $$AB$$, indicated by the arrows, form a zigzag sequence: $$1$$, $$2$$, $$3$$, $$3$$, $$6$$, $$4$$, $$10$$, $$\cdots$$, let the sum of the first $$n$$ terms of this sequence be $$S_{n}$$, then $$S_{16}=$$ ___. | $$164$$ | math | |
10d523e6-12ee-4fc5-8145-69b8e8fe56c9 | Execute the program flowchart shown. If the input value of $$x$$ is $$1$$, then the output value of $$n$$ is ___. | $$3$$ | math | |
31eb6997-de90-4869-8b28-90b5304c45be | As shown in the figure, the diagonals AC and BD of quadrilateral ABCD intersect at point O, and BD bisects AC. If BD = 8, AC = 6, and ∠BOC = 120°, then the area of quadrilateral ABCD is ___. (Leave the answer in radical form) | 12√3 | math | |
9b71b5bd-53c9-4787-bdab-2e506cf5c8a9 | As shown in the figure, in rhombus $$ABCD$$, $$DE \bot AB$$ at $$E$$, $$DE=\quantity{6}{cm}$$, and $$\sin A=\dfrac{3}{5}$$, then the area of rhombus $$ABCD$$ is ___ $$\unit{cm^{2}}$$. | $$60$$ | math | |
23d9570e-9c14-4675-8cab-cc88e8730f91 | As shown in the figure, point O is the center of the regular pentagon ABCDE. Then the measure of ∠BAO is ___. | 54° | math | |
e910ad3b-0064-4792-bf86-c9780353938e | In the figure, in circle $$\odot O$$, chord $$AD \parallel BC$$, $$DA=DC$$, and $$\angle AOC=160^{\circ}$$, then $$\angle BCO$$ is equal to ___ degrees. | $$30$$ | math | |
c57a1e4a-7b37-42da-b6b4-55f2d1e791b4 | If the probability distribution table of the random variable $$\eta$$ is as follows: then when $$P( \eta < x)=0.8$$, the range of the real number $$x$$ is ___. | $$\left ( 1,2\right \rbrack $$ | math | |
fe1267b5-0830-45f1-8931-f7ec5105cd20 | In the right-angled triangle ABC, ∠C = 90°, ∠A = 30°, and AB = 8 cm. Therefore, BC = ______ cm. | 4 | math | |
dfc6f1f9-8862-4fc5-9486-fcaa87d63806 | Translating the small rhombus $$\lozenge$$ can create a beautiful 'Chinese Knot' pattern. Among the following four patterns, they are similar 'Chinese Knot' patterns formed by translating $$\lozenge$$. According to the pattern in the figures, the number of small rhombuses in the 20th pattern is ______. | $$800$$ | math | |
8a3d7f64-1c08-4fde-b905-324c5caf290c | In the parallelogram $$ABCD$$, point $$E$$ is on side $$BC$$, and $$DE$$ intersects diagonal $$AC$$ at $$F$$. If $$CE=2BE$$, and the area of $$\triangle ABC$$ is $$15$$, then the area of $$\triangle FEC$$ is ______. | 4 | math | |
8b932a5f-7a05-4848-befc-22d994a96f69 | As shown in the figure, the lines $$l_{1}$$, $$l_{2}$$, and $$l_{3}$$ represent three intersecting highways. A cargo transfer station is to be built such that it is equidistant from the three highways. How many possible locations are there? | $$4$$ | math | |
75904c30-7651-4576-b05d-3440a26b174b | As shown in the figure, all triangles are right-angled triangles, and all quadrilaterals are squares. Given that $$S_1=9$$, $$S_2=16$$, and $$S_3=144$$, then $$S_4=$$______． | 169 | math | |
36448863-4eb4-4a05-bd95-b31c5aa03503 | As shown in the figure, the outer contour of a park is quadrilateral $$ABCD$$, which is divided into four parts by the diagonals $$AC$$ and $$BD$$. The area of $$\triangle BOC$$ is $$2$$ square kilometers, the area of $$\triangle AOB$$ is $$1$$ square kilometer, and the area of $$\triangle COD$$ is $$3$$ square kilometers. The land area of the park is $$6.92$$ square kilometers. Therefore, the area of the artificial lake is ______ square kilometers. | 0.58 | math | |
c04c703c-ddda-407f-8e24-1643ada24adf | As shown in the figure, the three views of a geometric solid are two rectangles and one sector. The surface area of this geometric solid is ___. | $$12+15\pi $$ | math | |
16c85c36-218f-403f-bc5a-63de08eb8623 | As shown in the figure, it is given that $$\angle 1 = \angle 2$$, if one more condition is added to make the conclusion "$$AB \cdot DE = AD \cdot BC$$" true, then this condition can be ___. | $$\angle B = \angle D$$ | math | |
8370d586-7cc8-443a-8df1-36006173001d | As shown in the figure, given that $$ABCD$$ is a square, $$P$$ is a point outside the plane of $$ABCD$$, and the projection of $$P$$ onto the plane $$ABCD$$ is exactly the center $$O$$ of the square. $$Q$$ is the midpoint of $$CD$$. If $$\overrightarrow{PA}=x\overrightarrow{PO}+y\overrightarrow{PQ}+\overrightarrow{PD}$$, then $$x+y=$$ ___. | $$0$$ | math | |
0cc4b7e5-fcf0-4205-b77d-fbd7c86e338b | As shown in the figure, points $$A$$ and $$C$$ are on the graph of the inverse proportion function $$y=\dfrac{a}{x}$$ ($$a>0$$), and points $$B$$ and $$D$$ are on the graph of the inverse proportion function $$y=\dfrac{b}{x}$$ ($$b<0$$). $$AB\parallel CD\parallel x$$-axis, and $$AB$$ and $$CD$$ are on opposite sides of the $$x$$-axis. Given that $$AB=3$$, $$CD=2$$, and the distance between $$AB$$ and $$CD$$ is $$5$$, the value of $$a-b$$ is ___. | $$6$$ | math | |
a736fa8e-edcf-491a-8ab3-237e8da13cbe | The students of a school come from three regions: A, B, and C, with a ratio of $2:3:7$. The pie chart shown below represents this distribution, where the central angle corresponding to region A is °. | 60 | math | |
6b6f4d5e-b503-40ad-b0f5-7cfd5399d3ee | As shown in the figure, in the right triangle $ABC$, $\angle ACB = 90^\circ$, $D$ is the midpoint of $AB$, and a perpendicular line from point $D$ to $AB$ intersects $AC$ at point $E$. Given $AC = 8$ and $\cos A = \frac{4}{5}$, find $DE$. | $\frac{15}{4}$ | math | |
ddf5bc77-ba8c-4c7b-b9a4-76e552dd181a | There are 3 squares placed as shown in the figure, the areas of the shaded parts are denoted as $S_1$ and $S_2$ respectively, then $S_1:S_2=$. | 4:9 | math | |
040dc4cc-93bf-4557-9bd1-bea08e8c5990 | The figure shows a square and its inscribed circle. If a grain of rice is randomly thrown into the square, the probability that it lands inside the circle is. | $\frac{\pi }{4}$ | math | |
8b62200b-fab5-459a-88c3-d602557d4678 | In the right triangle ABC, ∠BAC = 90°, AB = AC. Perpendiculars BD and CE are drawn from points B and C to a line passing through point A, with feet of the perpendiculars at D and E, respectively. If BD = 3 and CE = 2, then DE =. | 5 | math | |
935bee9f-a448-492f-9919-577b8d16866b | As shown in the figure, in the Cartesian coordinate system, the line y=x+b intersects the x-axis at point A and the y-axis at point B. A circular arc is drawn with point A as the center and the length of segment AB as the radius, intersecting the positive half of the x-axis at point C. If AC=$\sqrt{2}$, then the value of b is. | 1 | math | |
1c26d650-575f-4383-bd78-af5db7364700 | As shown in the figure, in rectangle ABCD, $AB=3$, $AD=2$. With point A as the center and the length of AD as the radius, an arc is drawn intersecting AB at point E. The area of the shaded part in the figure is (the result should be expressed in terms of $\pi$). | $6-\pi$ | math | |
3e372c4c-0e31-4c8a-9ef8-6f84e40548e3 | This summer vacation, Xiaoming's family plans to drive from City A to City G. He has planned a route time map, where the numbers on the arrows represent the required time (unit: hours). What is the shortest time needed to travel from City A to City G in hours? | 10 | math | |
09f868fa-aa49-4ccb-a77e-8aea6f9872c0 | An equilateral triangle, a right triangle, and an isosceles triangle are placed as shown in the figure. Given that the base angle of the isosceles triangle ∠3 = 72°, then ∠1 + ∠2 = | 138° | math | |
8b5c6b6c-b335-4313-8b0e-cc73bba823b2 | Squares A$_{1}$B$_{1}$C$_{1}$O, A$_{2}$B$_{2}$C$_{2}$C$_{1}$, A$_{3}$B$_{3}$C$_{3}$C$_{2}$, … are placed as shown in the figure. Points A$_{1}$, A$_{2}$, A$_{3}$, … and points C$_{1}$, C$_{2}$, C$_{3}$, … lie on the line y = x + 1 and the x-axis, respectively. What is the y-coordinate of point B$_{2018}$? | ${{2}^{2017}}$ | math | |
f8fe2b82-0c17-441e-bb3d-1f20acfad5d1 | In a skewed triangular prism ABC—A$_{1}$B$_{1}$C$_{1}$ with a volume of 9, S is a point on C$_{1}$C. The volume of S—ABC is 2. What is the volume of the tetrahedron S—A$_{1}$B$_{1}$C$_{1}$? | $1$ | math | |
a1820433-865a-4260-b98d-7bc37b13b956 | As shown in the figure, the side length of square $$ABCD$$ is $$1$$, and $$M$$ and $$N$$ are points on $$BC$$ and $$CD$$, respectively. If $$\triangle AMN$$ is an equilateral triangle, then the side length of $$\triangle AMN$$ is ___. | $$\sqrt{6}-\sqrt{2}$$ | math | |
5f7a4506-8f4e-4372-a656-bb7ebf037f58 | As shown in the figure, river banks AD and BC are parallel, and bridge AB is perpendicular to both banks. From point C, the angle between the ends of the bridge A and B is ∠BCA = 60°, and the length BC is measured to be 7m. Then the length of the bridge AB is ___ m. | 7\sqrt{3} | math | |
f383c147-7858-4dde-8e93-0109b4a263a9 | As shown in the figure, to measure the height of a building, from the top of the building $$A$$, the angle of depression to a point $$B$$ on the ground is $$30^{\circ}$$. It is known that the horizontal distance $$BC$$ from this point on the ground to the building is $$\quantity{30}{m}$$. Therefore, the height of the building $$AC$$ is ___ $$\unit{m}$$ (express the result in radical form). | $$10\sqrt{3}$$ | math | |
bba22c44-c302-44c7-99f5-f57b8e485070 | As shown in the figure, $$AB$$ and $$CD$$ are two chords of a circle $$O$$ with radius $$a$$. They intersect at the midpoint $$P$$ of $$AB$$, with $$CP=\dfrac{9}{8}a$$ and $$\angle AOP=60^{\circ}$$. Then, $$PD=$$ ___. | $$\dfrac{2}{3}a$$ | math | |
439d0a63-b142-4e03-969e-c41db6a34049 | As shown in the figure, point $$A$$ initially is located at the origin on the number line. Now, point $$A$$ is moved as follows: the first time, it moves 1 unit to the right to point $$B$$; the second time, it moves 3 units to the left to point $$C$$; the third time, it moves 6 units to the right to point $$D$$; the fourth time, it moves 9 units to the left to point $$E$$; and so on. How many times at least must the point be moved so that its distance from the origin is no less than 41? | $$41$$ | math | |
1233666f-ce24-4a20-ac6d-52b9a184eb52 | Execute the program flowchart as shown in the figure, then the output of $$n$$ is ___. | $$13$$ | math | |
20424627-5a71-40dd-ad5a-5f1b277b83a0 | On a plane, if we use a straight line to cut one corner of a square, we cut off a right-angled triangle. As shown in Figure 1, by labeling the side lengths, we get the Pythagorean theorem: $$c^{2}=a^{2}+b^{2}$$. Imagine replacing the square with a cube and changing the cutting line to a cutting plane as shown in Figure 2. In this case, a tetrahedron $$O-LMN$$ with three edges mutually perpendicular is cut from the cube. If $$S_{1}$$, $$S_{2}$$, and $$S_{3}$$ represent the areas of the three side faces, and $$S_{4}$$ represents the area of the cutting plane, through analogy, the conclusion is ___. | $$S_{4}^2=S_{1}^2+S_{2}^2+S_{3}^2$$ | math | |
3c57d216-9145-4968-92fa-3ab48f002329 | The three views of a geometric solid are shown in the figure (unit: m). What is the volume of the geometric solid? | $(6 + \pi) m^3$ | math | |
69e1a3b7-b389-4e20-b790-747139befefe | As shown in the figure, use 4 different colors to color the rectangles A, B, C, and D, with the requirement that adjacent rectangles must be colored differently. How many different ways are there to color them? | 72 | math | |
0d4a88ea-46eb-4b36-a562-af07b89d25ca | Arrange the terms of the sequence $\left\{ {{2}^{n}} \right\}$ in order as shown in the figure. What is the 15th number in the 11th row? | ${{2}^{115}}$ | math | |
a161c439-d028-4440-850c-87213aecc04b | A survey on monthly pocket money was conducted among 100 randomly selected students from a high school. It was found that the monthly pocket money amounts ranged from 50 to 350 yuan. The frequency distribution histogram is shown in the figure. Among these students, the number of students whose monthly pocket money falls within the interval $\left[ 100,250 \right)$ is. | 70 | math | |
977ea5eb-cda9-4da1-a399-3cc14abc81df | As shown in the figure, there is a sphere O inside the cylinder O$_{1}$ O$_{2}$. The sphere is tangent to the top and bottom bases and the generatrix of the cylinder. Let the volume of cylinder O$_{1}$ O$_{2}$ be V$_{1}$, and the volume of sphere O be V$_{2}$. The value of $\frac{V_{1}}{V_{2}}$ is | $\frac{3}{2}$ | math | |
c754b98f-bc6b-4ea9-8d65-8c0ca5b92591 | As shown in the figure, an equilateral triangle CDE is constructed outside the square ABCD, and line segments AE and BE are connected. What is the measure of ∠AEB in degrees? | 30 | math | |
45a952e0-276a-4509-a2e7-424b828d8c82 | In the right triangle $\text{Rt}\vartriangle ABC$, $\angle C=90{}^\circ$, $\angle A=30{}^\circ$, and $BC+AB=6\text{cm}$. Find the length of AB in cm. | 4 | math | |
8b5da1f2-86b2-4c61-8e81-a2669b2bdca0 | As shown in the figure, AB = AC = AD. If ∠BAC = 28° and AD ∥ BC, then ∠D = . | 38° | math | |
dbff000c-4f79-4f4c-b53f-f6c2a2a90a9b | Given the functions $y=f(x)$ and $y=g(x)$ are defined on $[-3,3]$ as even and odd functions, respectively, and their graphs on $[0,3]$ are shown in the figure, then the solution set of the inequality $\frac{f(x)}{g(x)} \ge 0$ on $[-3,3]$ is. | $(-3,-2] \cup (-1,0) \cup (1,2]$ | math | |
6814d392-1dcd-40ea-acde-921718619fcf | In the figure, in quadrilateral ABCD, ∠B = 75°, BC = 4, the circle ⊙O with AD as its diameter intersects CD at point E. Connect OE. Then the area of the shaded region is. | $\frac{\pi }{3}$ | math | |
9c541544-dd6c-47ae-a95c-04b8352ceecc | As shown in the figure, point D is inside triangle ABC, AC = 4, AD = BC = 3, and angles ACB and ADB are supplementary. The maximum value of the sum of the areas of triangles ACD and BCD is. | $\frac{9}{2}$ | math | |
424791d5-87df-4ff7-aef2-d24b867276b8 | It is known that the horizontally placed △ABC is obtained by the 'oblique cavalier projection' method as shown in the figure, where $B'O'=C'O'=2$, $A'O'=\sqrt{3}$. What is the area of the original △ABC? | $4\sqrt{3}$ | math | |
2f3145ea-c815-4ca1-991e-757dec905ea6 | As shown in the figure, the following is the unfolded diagram of a rectangular prism (unit: $$cm$$). The area of the largest face among the six faces of this rectangular prism is ______ $$cm^2$$. | 27 | math | |
f79425fd-5531-49f0-84b9-a3fb32258eef | Execute the following program flowchart. If the input value of $$\epsilon$$ is $$0.25$$, then the output value of $$n$$ is ______. | $$3$$ | math | |
27cfd1bc-3339-408a-b813-a9bb6356cb03 | As shown in the figure, in the right triangle $$ABC$$, $$\angle C=90^{°}$$, $$\angle A=60^{°}$$, $$AB=\quantity{2}{cm}$$, the triangle $$ABC$$ is rotated around point $$B$$ to the position of $$\triangle A_{1}BC_{1}$$, and points $$A$$, $$B$$, and $$C_{1}$$ are collinear. The length of the path that point $$A$$ travels is ___. | $$\dfrac{5\pi }{3}\unit{cm}$$ | math | |
640700ab-dd3d-4f3d-b669-3a143175b12c | Place a right-angled triangular board $$OAB$$ with a $$30^{\circ}$$ angle in a Cartesian coordinate system as shown in the figure, with $$OB$$ on the $$x$$-axis. If $$OA=2$$, and the triangular board is rotated $$75^{\circ}$$ clockwise around the origin $$O$$, then the coordinates of the corresponding point $$A'$$ of point $$A$$ are ___. | $$\left ( \sqrt{2},-\sqrt{2}\right ) $$ | math | |
6839123d-9695-4b8f-92a9-d7b84bb49dd3 | As shown in the figure, given the spatial quadrilateral $$OABC$$, $$OB=OC$$, and $$∠AOB=∠AOC=\dfrac {π}{3}$$. Then the value of $$cos<\overrightarrow {OA},\overrightarrow {BC}>$$ is ___. | $$0$$ | math | |
26e69a1d-7058-4a55-b1a2-ee945dfdba44 | The three views of a geometric solid are shown in the figure. The surface area of the solid is ___. | $$92$$ | math | |
166ab2dd-dc63-4c91-91ed-9613c4c2d32d | As shown in the figure, $$AB$$ and $$CD$$ intersect at point $$O$$. $$\triangle AOC \backsim \triangle BOD$$, $$OC:OD=1:2$$, and $$AC=5$$. What is the length of $$BD$$? | $$10$$ | math | |
9b3a7d29-ef56-4e2d-b96e-53a69a791a6e | As shown in the figure, when $$\triangle AOB$$ is rotated $$45^{\circ}$$ counterclockwise around point $$O$$, it becomes $$\triangle COD$$. If $$\angle AOB=15^{\circ}$$, then the measure of $$\angle AOD$$ is ___. | $$60^{\circ}$$ | math | |
dbc701d5-f72f-423d-a0ff-d1ef60046cfb | In the $$8 \times 8$$ square grid paper shown in the figure, a needle is randomly thrown onto the paper. The probability of hitting the shaded area is ___. | $$\dfrac{1}{8}$$ | math | |
35a4fc6c-8338-4c27-adb6-7552a65da3e8 | As shown in the figure, in the right triangle $$ABC$$, $$∠ACB=90^{\circ}$$, $$∠B=15^{\circ}$$, the perpendicular bisector of $$AB$$ intersects $$AB$$ at point $$E$$ and $$BC$$ at point $$D$$. Given that $$BD=13cm$$, then $$AC=$$ ___. | $$6.5cm$$ | math | |
b4608027-b396-4bf6-9a71-5364f9c0742e | As shown in the figure, there is a point A on the graph of the inverse proportion function y = $\frac{k}{x}$ (k ≠ 0). A perpendicular line AP is drawn from A to the x-axis at point P. If the area of triangle AOP, S$_{△}$$_{AOP}$, is 1, then k = . | 2 | math | |
60750105-1e9f-44f2-92ed-43a8ab94f591 | The German mathematician Leibniz discovered a unit fraction triangle (a unit fraction refers to a fraction with a numerator of $1$ and a denominator that is a positive integer), known as the Leibniz triangle. According to the pattern in the first $6$ rows, the third number from the left in the $7$th row is. | $\frac{1}{105}$ | math | |
9d8bea15-45d0-4f30-b3ef-b8d9c0924f25 | The pattern shown in the figure consists of three blades, which can coincide with itself after rotating 120° around point O. If the area of each blade is 4 cm² and ∠AOB = 120°, then the area of the shaded part in the figure is. | 4 cm² | math | |
0d99048d-c249-417d-a44e-c111fb26322c | In the figure, in $\Delta ABC$, $E$ and $D$ are the midpoints of $AB$ and $CE$ respectively, and $S_{\Delta ABC} = 24$. Then $S_{\Delta EDB} =$. | 6 | math | |
397330e9-6aa5-4646-8ef7-3f4032345b7c | As shown in the figure, $A,B$ are two points on the sea level, 800m apart. At point $A$, the angle of elevation to the mountain top $C$ is 45°, and $\angle BAD=120°$. Also, at point $B$, $\angle ABD=45°$, where point $D$ is the projection of point $C$ on the sea level. The height of the mountain $CD$ is. | $800(\sqrt{3}+1)\text{m}$ | math | |
4980fe8f-457c-4a6e-b3d4-6f24267952cf | As shown in the figure, a car is driving straight on a city street. At a certain moment, it is exactly 30m in front of the speed detection device A on the opposite side of the road at point C. After 2 seconds, the distance between the car and the speed detection device is measured to be 50m. What is the speed of the car in m/s? | 20 | math | |
6a271400-e17d-4091-9d7e-a864875d8041 | As shown in the figure, after a geometric body is cut from a cube with edge length $a$, the volume of the remaining part is. | $\frac{5}{6}{{a}^{3}}$ | math | |
0d934e7d-33ac-49c8-bb59-10130ce63bc0 | As shown in the figure, in the Cartesian coordinate system, the line $y=ax$ intersects the hyperbola $y=\frac{k}{x} (k > 0)$ at points $A$ and $B$. A perpendicular line is drawn from point $A$ to the x-axis, intersecting at point $C$. Given that the area of triangle $BOC$ is 3, find the value of $k$. | 6 | math | |
e1329385-0afb-4cc4-abaa-3d61be020f5d | There is a numerical converter, the principle of which is shown in the following diagram: When the input x=16, the output y equals. | $\sqrt{2}$ | math | |
17105f43-f65d-4ad4-9cfe-11f02b42ec8a | As shown in the figure, in the rectangle OABC, $OA=3$, $OC=4$. Then the coordinates of point B are. | $\left( -4,3 \right)$ | math | |
0cb4d450-3411-4d8e-b767-0fef47433cc0 | A supermarket is holding a grand New Year's promotion. Customers who spend over 100 yuan can participate in a lottery activity. The rules are as follows: The customer places a ball of appropriate size at the top of the container shown in the figure. The ball will freely fall and encounter black obstacles 3 times, finally landing in either bag A or bag B, and the customer will receive the prize in the corresponding bag. It is known that each time the ball encounters a black obstacle, the probability of falling to the left or right is $\frac{1}{2}$. If on the day of the event, Xiao Ming spends 108 yuan in the supermarket, according to the rules of the activity, he can participate in the lottery once, then the probability of Xiao Ming winning the prize in bag A is. | $\frac{3}{4}$ | math | |
1d1e1fad-aa75-4f43-bc10-d2af00a49cfd | There is a straight running track between points A and B. Person A and Person B start from A and B respectively at the same time, running towards each other at a constant speed, and Person B's speed is 80% of Person A's speed. When A and B reach B and A respectively, they immediately turn back and run, with A maintaining the same speed and B increasing his speed by 25% (still at a constant speed). The relationship between the distance y (meters) between A and B and the running time x (minutes) is shown in the figure. What is the distance from B when they meet for the second time? | 1687.5 | math | |
943ece05-b6d4-4da9-bcba-894d604b58b5 | As shown in the figure, both Figure 1 and Figure 2 are composed of 8 identical small rectangles, and the area of the small square (shaded part) in Figure 2 is 1cm². Then the perimeter of the small rectangle is equal to. | 16 | math |
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