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a4daee92-00d3-473e-97cd-d5d994f12299 | As shown in the figure, one side of $n$ consecutive squares with side length $1$ is on the same straight line. Points $M_{1}$, $M_{2}$, $M_{3}$, $\cdots$, $M_{n}$ are the midpoints of sides $B_{1}B_{2}$, $B_{2}B_{3}$, $B_{3}B_{4}$, $\cdots$, $B_{n}B_{n+1}$, respectively. The area of $\triangle B_{1}C_{1}M_{1}$ is $S_{1}$, the area of $\triangle B_{2}C_{2}M_{2}$ is $S_{2}$, $\cdots$, and the area of $\triangle B_{n}C_{n}M_{n}$ is $S_{n}$. Find $S_{n}$ expressed as a formula containing $n$. | $\frac{1}{4\left( 2\text{n}-1 \right)}$ |
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21c7c3f1-11b0-405f-818a-23f6f432f6f8 | As shown in the figure, overlay two right triangles such that the right-angle vertex coincides at point $$O$$. Then $$ \angle AOB + \angle DOC = $$___ degrees. | $$180$$ |
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098f9b44-56fb-4869-b22c-c3bc977364a1 | As shown in Figure 1, in rectangle $ABCD$, $AB=2BC$, $E$ and $F$ are the midpoints of $AB$ and $CD$ respectively. Now, fold this rectangle along $EF$ to form a right dihedral angle $A-EF-C$ (as shown in Figure 2). Then in Figure 2, find the angle formed by line $AF$ and the plane $EBCF$. | $\frac{\pi}{4}$ |
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319852ae-0d64-4298-9afa-c6d905d05396 | As shown in the figure, it is known that the equilateral triangle ABC has side length 1. The three medians of triangle ABC form triangle A$_{1}$B$_{1}$C$_{1}$, and the three medians of triangle A$_{1}$B$_{1}$C$_{1}$ form triangle A$_{2}$B$_{2}$C$_{2}$. Continuing this process results in the perimeter of triangle A$_{5}$B$_{5}$C$_{5}$ being . | $\frac{3}{32}$ |
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2f25ad37-ea13-4d3b-bbf4-16da2ecf55b4 | As shown in the figure, △A'B'C' is obtained by a homothetic transformation of △ABC with respect to point O. If the area ratio of △A'B'C' to △ABC is 4:9, then OB':OB is . | $2:3$ |
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8933599a-ae09-4cc4-b6c5-e4a3c209de59 | As shown in the figure, in rhombus ABCD, the diagonals AC and BD intersect at point O. AC = 8, BD = 6, and OE is perpendicular to AD at point E, intersecting BC at point F. What is the length of EF? | $\frac{\text{24}}{\text{5}}$ |
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6f3a99cf-99e9-4e08-bfd6-b0a991153f50 | In order to develop more houses suitable for consumer demand in the market, thereby leading to rational development and rational consumption. A real estate marketing planning company conducted a survey of the needs of 2000 customers and used specialized software for statistical analysis. They created a statistical distribution chart of consumers' demand for area needs as shown in the figure (where demand rate = number of customers in need / total number of surveyed customers). Please observe and calculate the number of customers whose demand for the area is 100~140 (inclusive of 140, not inclusive of 100) m². | 1234 |
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2ffd163d-fb85-4cb2-9e50-246f9e4ff809 | As shown in the figure, given that $$O$$ is a point on the side $$CD$$ of rectangle $$ABCD$$, and rotating this rectangle around the line $$CD$$ as the axis results in a solid of volume $$1$$, where the volume of the cylinder with $$OA$$ as the generatrix is $$\dfrac{1}{4}$$, then the volume of the cylinder with $$OB$$ as the generatrix is___. | $$\dfrac{1}{12}$$ |
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2769b0c0-bbe9-489f-8af2-f94dab2bf03b | Given that point $$P$$ is on the graph of a linear function $$y = kx + b$$ ($$k$$, $$b$$ are constants, and $$k < 0$$, $$b > 0$$), point $$P$$ moves left by $$1$$ unit and then up by $$2$$ units to reach point $$Q$$, which is also on the graph of the function $$y = kx + b$$. (1) The value of $$k$$ is ___. (2) In the diagram, the graph of this linear function intersects the $$x$$-axis and $$y$$-axis at points $$A$$ and $$B$$, respectively, and intersects the graph of the reciprocal function $$y = \dfrac{-4}{x}$$ at points $$C$$ and $$D$$ (where point $$C$$ is in the second quadrant). A perpendicular line from point $$C$$ to the $$x$$-axis intersects at point $$E$$. Let $$S_{\number{1}}$$ denote the area of quadrilateral $$CEOB$$ and $$S_{\number{2}}$$ denote the area of $$\triangle OAB$$. If $$\dfrac{S_{\number{1}}}{S_{\number{2}}} = \dfrac{7}{9}$$, then the value of $$b$$ is ___. | (1)$$-2$$ (2)$$3\sqrt{\number{2}}$$ |
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f2a333c2-5ba9-475b-8990-3eb6dfe0fc9c | As shown in the figure, in the inscribed pentagon $$ABCDE$$ of circle $$\odot O$$, $$\angle CAD=35^{\circ}$$, then $$\angle B+\angle E=$$___$$^{\circ}$$. | $$215$$ |
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fe91fc15-7e62-435f-aa8e-38e7c6f1cdb1 | As shown in the figure, on the graph of the inverse proportional function y = $\frac{2}{x}$ (x > 0), there are points P$_{1}$, P$_{2}$, P$_{3}$, P$_{4}$, whose horizontal coordinates are 1, 2, 3, 4 in sequence. Vertical lines passing through these points, parallel to the x-axis and y-axis, enclose shaded regions in the figure. The areas of these shaded regions from left to right are S$_{1}$, S$_{2}$, S$_{3}$ respectively. Then S$_{1}$ + S$_{2}$ + S$_{3}$ =. | $\frac{3}{2}$ |
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076ee053-835f-4e99-ba08-aa11c5c924df | Execute the flowchart as shown, and the output value of $$S$$ is ___. | $$8$$ |
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40e71ae9-3604-47ea-99d1-24394522c0b4 | As shown in the figure, $PA$ and $PB$ are tangent to the circle $\odot O$ at points $A$ and $B$, respectively. The secant line $EF$ of the circle $\odot O$ intersects $PA$ and $PB$ at points $E$ and $F$, respectively, and the intersection point $C$ is on the arc $\widehat{AB}$. If the perimeter of $\Delta PEF$ is $8cm$, then the length of $PA$ is $cm$. | 4 |
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2b7eaf10-76bb-4a64-b432-dcdee7ada150 | As shown in the figure, it is known that the line segment $$AB$$ is within the plane $$\alpha$$, the line segment $$AC\bot \alpha$$, the line segment $$BD\bot AB$$, and the line segment $$DD'\bot \alpha$$ at point $$D'$$. If $$\angle DBD'=30^{\circ}$$, $$AB=a$$, $$AC=BD=b$$, then the length of $$CD$$ is ___. | $$\sqrt{a^{2}+3b^{2}}$$ |
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4576e566-6192-4600-aa68-d02b1a694aec | As shown in the figure, in the rectangle ABCD, AC is a diagonal, and ∠ACD = 60°, AB = 2, then the area of rectangle ABCD is equal to . | 4$\sqrt{3}$ |
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d9504f64-2f96-460e-9faf-d56e61da619a | As shown in the figure, in the right triangular prism $ABC-{{A}_{1}}{{B}_{1}}{{C}_{1}}$, $AB=1$, $BC=2$, $AC=\sqrt{5}$, and $A{{A}_{1}}=3$. Let $M$ be a moving point on the line segment $B{{B}_{1}}$. Find the area of triangle $\Delta AM{{C}_{1}}$ when $AM+M{{C}_{1}}$ is minimized. | $\sqrt{3}$ |
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12ecc06d-a8ed-424f-9d37-7cbf90c62da0 | As shown in the figure, in the spatial Cartesian coordinate system $$C-xyz$$, $$AB$$ is the diameter of circle $$O$$, $$AC=BC=2\sqrt{2}$$, $$DC\parallel EB$$, $$DC=EB$$, $$tan\angle EAB=\dfrac{1}{4}$$, then the cosine value of the dihedral angle of $$D-AE-B$$ is ___. | $$-\dfrac{\sqrt{2}}{6}$$ |
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8026853b-4289-4891-87a6-badd7d19dee4 | As shown in the figure, in $$\triangle ABC$$, $$\angle 1=\angle 2$$, $$BE=CD$$, $$AB=5$$, $$AE=2$$, then $$CE=$$___. | $$3$$ |
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c1a361da-167b-4a58-a563-97f4e659b9cf | According to the "Individual Income Tax Law of the People's Republic of China": Starting from September 1, 2011, the threshold for individual income tax was increased from 2,000 yuan to 3,500 yuan, meaning that the portion of monthly income exceeding 2,000 yuan originally subject to tax now requires tax payment only for the portion exceeding 3,500 yuan. If the tax rates for portions that exceed the threshold remain the same before and after the tax law amendment, the tax is calculated according to the following bracket: In May 2011, an employee paid 295 yuan in individual income tax. Assuming the income remains unchanged, how much individual income tax does the employee need to pay in October 2011? | 145 |
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eca7f575-8869-4b82-9fe7-7b78df51d821 | As shown in the figure, it is known that $\triangle \text{ABC}$ and $\triangle \text{ADE}$ are both equilateral triangles. Point $\text{D}$ is on the edge $\text{BC}$, and $\text{DE}$ intersects $\text{AC}$ at point $\text{F}$. If $\text{AB}=9$ and $\text{BD}=3$, then the length of $\text{CF}$ is . | $2$ |
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aa39bab1-b35b-487d-bd82-d1b471ab1c4d | Read the following material: In the math class, the teacher presents the following problem: Xiao Yun's method is as follows: The teacher says: 'Xiao Yun's method is correct.' Please complete the diagram according to Xiao Yun's method. | A quadrilateral with all four sides equal is a rhombus; opposite sides of a rhombus are parallel (not unique). |
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29a0522f-1d5b-4d34-9afe-fa1d372a9ff0 | According to the following figures and corresponding number of points, write a general formula for the sequence of numbers formed by the number of points ${{a}_{n}}=$. | $5n-4$ |
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d9fa7e98-e28a-4b49-ab9d-891a0ceff633 | As shown in the figure, let $\overrightarrow{AB}=\overrightarrow{a}$, $\overrightarrow{AC}=\overrightarrow{b}$, $\overrightarrow{AD}=\overrightarrow{c}$ be three non-zero vectors on a plane that are pairwise non-parallel. $x \in \mathbf{R}$, there are the following problems:
1. The equation $\overrightarrow{a}{{x}^{2}}+\overrightarrow{b}x+\overrightarrow{c}=0$ with respect to $x$ may have two different real solutions;
2. The equation $\overrightarrow{a}{{x}^{2}}+\overrightarrow{b}x+\overrightarrow{c}=0$ with respect to $x$ certainly has no real solutions;
3. The real solutions for the equation $\overrightarrow{a}{{x}^{2}}+\overrightarrow{b}x=0$ with respect to $x$ are $x=0$ or $x=-\frac{\overrightarrow{b}}{\overrightarrow{a}}$;
4. The equation $\overrightarrow{a}{{x}^{2}}+\overrightarrow{b}x=0$ with respect to $x$ has no non-zero real solutions.
The true statements are: . | 2.4. |
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7f0ae199-ac67-4003-a07f-752d5fe692df | The age distribution of the members of a men's soccer team at a certain school is shown in the figure. Based on the information in the figure, the median age of these team members is ______ years. | $$15$$ |
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5f1a3725-5f4c-4d0f-bba0-3b5657b16b29 | As shown in the figure, it is known that $$AB$$ and $$AC$$ are the chord and secant of circle $$\odot O$$, respectively. $$A$$ is the point of tangency, and $$AD$$ is the angle bisector of $$\angle BAC$$, intersecting $$\odot O$$ at $$D$$. The extension line of $$BD$$ intersects $$AC$$ at $$C$$. Given $$AC=6$$, $$AD=5$$, find $$CD=$$___. | $$4$$ |
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4e94d916-4c1d-4650-8674-59216d84d939 | As shown in the figure, $$M$$ is the midpoint of $$CD$$, and $$EM\perp CD$$. If $$CD=4$$ and $$EM=8$$, then the radius of the circle on which $$\overset{\frown} {CED}$$ lies is ___. | $$\dfrac{17}{4}$$ |
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d7aae7e6-33f0-4a0a-9051-9bf562e6d506 | As shown in Figure 1, in the square $$ABCD$$, point $$P$$ moves along side $$DA$$ from point $$D$$ towards $$A$$ at a speed of $$\quantity{1}{cm/s}$$; simultaneously, point $$Q$$ moves along sides $$AB$$ and $$BC$$ from point $$A$$ towards point $$C$$ at a speed of $$\quantity{2}{cm/s}$$. When point $$P$$ reaches point $$A$$, both $$P$$ and $$Q$$ stop moving. Let point $$P$$ start at $$x\ \unit{s}$$, and $$y$$ be the area of $$\triangle PAQ$$ in $$\unit{cm^{2}}$$; the function graph of $$y$$ against $$x$$ is shown in Figure 2. The function equation corresponding to the line segment $$EF$$ is ___. | $$y=-3x+18$$ |
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6c06b409-9c09-475d-a296-bf0087e81622 | As shown in the figure, in $\triangle ABC$, $BD\bot AD$, $\angle A=15{}^\circ$, $AC=BC=6$, find the length of $CD$. | $3\sqrt{3}$ |
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6cb5524d-aed1-4ad9-8288-e755ffea5f05 | As shown in the figure, it is known that AB=AC, AD=AE, ∠BAC=∠DAE, BD=7cm, then CE=______cm. | ∵∠BAC=∠DAE ∴∠BAD=∠CAE ∵AB=AC, AD=AE ∴△ABD≌△ACE (SAS) ∴BD=CE=7cm. Therefore, fill in 7. |
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29696170-3192-47b8-b762-7828c8874d60 | Given that the slant height of a cone with apex point $$P$$ is $$3$$, and the base radius is $$1$$. An ant departs from a point $$A$$ on the circumference of the base, travels around the surface of the cone, and returns to point $$A$$. Find the length of the shortest path the ant can take. | $$3\sqrt{3}$$ |
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2eebe8c2-1d53-409a-8804-301a22062eef | The distance between ports A and B is 900 kilometers. A ship departs from port A heading towards port B. After sailing for 10 hours, the distance traveled is 80 kilometers less than the remaining distance. What is the average speed of this ship in kilometers per hour? | 41 |
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a5a09087-65ea-41f1-ba5c-1d38c4d231fb | As shown in the figure, an airplane detects a target C in mid-air at point A, with the flying altitude AC = 1200m. The angle of depression to a horizontal directional station B from the airplane is α = 16°31′. Then the distance between the airplane at point A and the directional station at point B is (round the result to the nearest whole number) (referencing trigonometric values: sin16°31′ = 0.28, cos16°31′ = 0.95, tan16°31′ = 0.30) | 4286m |
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2cc9e067-3d6c-4284-9bf7-39876fe19ba2 | According to the pseudocode given in the figure, when the input $a$ is 3, the final output value of $S$ is . | 21 |
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6899b357-2e15-463a-8a61-b73b051ab600 | A polyhedron is composed of many small cubes, and its front view and left view are as shown in the figure. To form this polyhedron, at least ___ small cubes are needed, and at most ___ small cubes are needed. | $$7$$ $$11$$ |
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c14ce521-9039-4d6b-905a-89a3bbec29c0 | As shown in the figure, in triangle ABC, ∠C = 90°, AD is the angle bisector of ∠BAC, DE ⊥ AB at E. If CB = 6, then what is DE + DB = ? | 6 |
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57beeb87-f010-4841-82dc-4a806f4e5b72 | Jia and Yi set off from location $$A$$ to location $$B$$ at the same time. Jia is riding a bicycle, while Yi is walking. After arriving at location $$B$$, Jia takes half an hour to complete the task and then returns at the original speed. The distance $$y$$ (unit: kilometers) between Jia and Yi and the time $$t$$ (unit: hours) are related as shown in the figure. What is the value of $$a$$ in the figure? | $$\dfrac{25}{14}$$ |
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f844b6dd-7b5a-4309-9bc0-8df306644273 | As shown in the figure, in the equilateral triangle $$ABC$$, $$BE$$ and $$CD$$ are the altitudes from the sides $$AC$$ and $$AB$$ respectively, and they intersect at point $$O$$. Then the measure of $$\angle BOC$$ is ___. | $$120^{\circ}$$ |
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38094ae0-d7ad-4028-874c-923d6dbaf288 | As shown in the figure, $$O$$ is the midpoint of side $$AC$$ of the equilateral triangle $$ABC$$, and is also the midpoint of side $$A_{1}C_{1}$$ of the equilateral triangle $$A_{1}B_{1}C_{1}$$. Then, $$AA_{1}:BB_{1}=$$___. | $$1:\sqrt{3}$$ |
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af75896e-8502-4853-8d55-a7f5aa180a73 | As shown in the figure, it is the net of a cube-shaped paper box. If the numbers on the opposite faces are opposite numbers, then the value of x-y in the figure is | 5 |
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26f7f1e3-65de-434a-a852-b6be7963036f | As shown in the figure, cut a cross-shaped paperboard composed of 5 smaller squares with side length 1, so that the cut pieces can just fit into a larger square $ABCD$. What is the side length of the larger square that is formed? | $\sqrt{5}$ |
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4d5f0c49-540e-4fc8-9299-83d0e405e0d5 | As shown in the figure, in $$\triangle ABC$$, $$AB=1.8$$, $$BC=3.9$$, $$\angle B=60^\circ$$, rotate $$\triangle ABC$$ counterclockwise around point $$A$$ by a certain angle to obtain $$\triangle ADE$$. When the corresponding point $$D$$ of point $$B$$ happens to fall on the side $$BC$$, the length of $$CD$$ is ______. | 2.1 |
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4fc483b2-3bcd-45ef-9d50-e19d4d95ecf8 | Given: As shown in the figure, in $$\triangle ABC$$, points $$A_{1}$$, $$B_{1}$$, $$C_{1}$$ are the midpoints of $$BC$$, $$AC$$, $$AB$$ respectively. Points $$A_{2}$$, $$B_{2}$$, $$C_{2}$$ are the midpoints of $$B_{1}C_{1}$$, $$A_{1}C_{1}$$, $$A_{1}B_{1}$$ respectively, and so on. If the perimeter of $$\triangle ABC$$ is $$1$$, then the perimeter of $$\triangle A_{n}B_{n}C_{n}$$ is___. | $$\dfrac{1}{2^{n}}$$ |
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0c417702-ef29-4413-8464-71669498ccef | As shown in the figure, there are two concentric circles. The radius of the larger circle is $$5~cm$$, and the radius of the smaller circle is $$3~cm$$. If the chord $$AB$$ of the larger circle intersects the smaller circle, then the range of values for the chord $$AB$$ is ___. | $$8~cm< AB\leqslant 10~cm$$ |
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de7b1a49-aac0-4772-aa78-6578fce4cf6e | In the cube $$ABCD-A_{1}B_{1}C_{1}D_{1}$$, points $$E$$ and $$F$$ are the midpoints of $$BB_{1}$$ and $$D_{1}B_{1}$$, respectively. The edge length of the cube is $$1$$. Establish a spatial rectangular coordinate system as shown in the figure. Then the coordinates of point $$E$$ are ___, and the coordinates of point $$F$$ are ___. | $$(1,1,\dfrac{1}{2})$$ $$(\dfrac{1}{2},\dfrac{1}{2},1)$$ |
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aa617366-93b1-4623-866e-56a18823378a | As shown in the figure, there is a relationship between the taxi fare and the distance traveled in a certain city. If Xiaoming's uncle took a taxi to Xiaoming's home and spent 22 yuan, then Xiaoming's uncle traveled a distance of a kilometer. | 13 |
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669b0d11-6210-435c-9261-eddb64a6dcf8 | Given two vectors $$\overrightarrow{OA}$$ and $$\overrightarrow{OB}$$ with length $$1$$, and the angle between them is $$120^{\circ}$$, as shown in the figure, point $$C$$ moves on the arc $$\overrightarrow{AB}$$ with center $$O$$. If $$\overrightarrow{OC}=x\overrightarrow{OA}+y\overrightarrow{OB}$$, where $$x$$, $$y \in \mathbf{R}$$, then the maximum value of $$x+y$$ is ___. | $$2$$ |
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5d590fa6-4dd6-4499-8b98-f6fe07375e10 | As shown in the figure, in triangle ABC, AB = AC. With point C as the center of a circle and CB as the radius, an arc is drawn intersecting the extension of line AC at point D. Connect BD. If ∠A = 36°, then what is the measure of ∠CDB in degrees? | 36. |
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ebfbabfd-ff51-4cc2-a1ae-d495559b91a9 | As shown in the figure, AB is the diameter of circle O, D is a point on the extended line of AB, DC intersects circle O at C, ∠DAC=30°, OD=10, then the radius of circle O is ___; the length of AC is ___. | 5
5√3 |
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07aaa55f-ebe3-4f58-837e-9a6f75a6ea92 | It is known that the slope of line l is the slope of the line 2x - 3y + 12 = 0, and the y-intercept of l is twice the y-intercept of the line 2x - 3y + 12 = 0. For line l, the equation is | $x-3y+24=0$ |
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acfa8c03-6dcf-42ee-a30b-09f5cd22688d | As shown in the figure, $$O$$ is a point on the line $$AB$$. Draw a ray $$OC$$ through $$O$$ such that $$\angle AOC=3\angle BOC$$. Then $$\angle BOC=$$___$$^{\circ}$$. | $$45$$ |
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46407eff-1975-45fb-8447-df846a5d7ed3 | As shown in the figure, the central angle of the sector is $$90^{\circ}$$. The chord $$AB$$ divides the sector into two parts. If these two parts, respectively, rotate one full turn around the line on which $$AO$$ lies, let the volumes of the resulting solids of revolution be $$V_1$$ and $$V_2$$. Then the ratio of $$V_1$$ to $$V_2$$ is ___. | $$1:1$$ |
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fd330aff-57b3-4ee8-b103-ea45a2da91f0 | As shown in the figure, in $$\angle ABC$$, $$AB=AC$$, $$\angle A=80^{\circ}$$, $$E$$, $$F$$, $$P$$ are points on $$AB$$, $$AC$$, $$BC$$ respectively and $$BE=BP$$, $$CP=CF$$, then $$\angle EPF=$$___$$^{\circ}$$. | $$50$$ |
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cc329968-fa5b-4757-9a00-7b891f2ce1c7 | As shown in the diagram, there are structural formulas and molecular formulas for three types of compounds. According to their pattern, please write the molecular formula for the 2013th compound___. | $$\ce{C_{\number{2013}}H_{\number{4028}}}$$. |
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a2120f48-93ff-4072-b895-ff6fb5261b55 | In the right triangle ABC, with ∠C = 90°, AC = 3, and ∠B = 37°, what is the length of BC? (Note: tan ∠B = 0.75, sin ∠B = 0.6, cos ∠B = 0.8) | 4 |
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06ce1684-9738-4c7b-afc1-f5164843b9ee | As shown in the figure, in the parallelogram $$ABCD$$, $$AB=10$$, $$BC=6$$, $$E$$ and $$F$$ are the midpoints of $$AD$$ and $$DC$$ respectively. If $$EF=7$$, then the perimeter of the quadrilateral $$EACF$$ is ___. | $$\number{29}$$ |
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688d31b9-55a8-447c-82ff-63358c160d3d | Write down the answers for each of the following problems on the lines provided. (No need to show calculations or reasoning processes) (1) As shown in the figure, fold the rectangle $$ABCD$$ along the line $$AE$$ so that vertex $$D$$ falls exactly on point $$F$$ on edge $$BC$$. Given that $$CE=\quantity{3}{cm}$$, and $$AB=8\ \unit{cm}$$, find the area of the shaded region in the figure. The area of the shaded region is ___. (2) Given: The lengths of the diagonals of a rhombus are $$\quantity{6}{cm}$$ and $$\quantity{8}{cm}$$, respectively. Find the area of the rhombus. The area of this rhombus is ___ $$\unit{cm^{2}}$$. (3) Given that the side length of rhombus $$ABCD$$ is $$4\sqrt{3}$$, and one of its interior angles is $$60^{ \circ }$$, find the length of the longer diagonal of the rhombus. The length of the longer diagonal is ___. (4) Given: As shown in the figure, in rhombus $$ABCD$$, $$\angle A=60^{ \circ }$$, $$AB=6\ \unit{cm}$$, $$E$$ and $$F$$ are on $$AB$$ and $$BC$$, respectively, and $$\triangle DEF$$ is an equilateral triangle, $$AE=2\ \unit{cm}$$, find the area of $$\triangle CDF$$. The area of $$\triangle CDF$$ is ___. | (1)$$30\ \unit{cm^{2}}$$
(2)$$\quantity{24}{cm^{2}}$$
(3)$$12$$
(4)$$6\sqrt{3}\ \unit{cm^{2}}$$ |
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7d33110d-a37c-4ecd-93c7-114c8c5f4d5e | From a collection of 800 products, 60 are selected for quality inspection using the random number table method. Firstly, the 800 products are numbered from 001, 002, \cdots, 800. If the reading begins from the number 8 in the 8th row and 8th column of the random number table, what are the serial numbers of the first four products selected? (Below is an excerpt of numbers from the 7th to the 9th row of the random number table) | 169 555 671 105 |
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8c33dcc7-c87c-4ab3-b8b5-d1adbeaae8d8 | Below is a statistical chart of sales of summer and winter clothing from a certain clothing store from September to December. Please analyze based on the statistical chart. Here, represents ______, and represents ______. (Fill in 'summer clothing' or 'winter clothing') | summer clothing
winter clothing |
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3faea494-5c94-4cbb-9d15-267febefe4ce | As shown in the figure, in a rectangular piece of paper $$ABCD$$, $$AB=4$$, $$BC=8$$, points $$E$$ and $$F$$ are on $$AD$$ and $$BC$$ respectively. Fold the paper $$ABCD$$ along the straight line $$EF$$ so that point $$C$$ falls on point $$H$$ on $$AD$$, and point $$D$$ falls on point $$G$$. There are the following four conclusions: 1. Quadrilateral $$CFHE$$ is a rhombus; 2. $$CE$$ bisects $$ \angle DCH$$; 3. The range of values for line segment $$BF$$ is $$3 \leqslant BF \leqslant 4$$; 4. When point $$H$$ coincides with point $$A$$, $$EF=2\sqrt{5}$$. Among these, the correct conclusions are ___. (Fill in the sequence numbers) | 1.3.4. |
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4423ae55-1b1d-45df-a392-987104ba04b2 | As shown in the figure, points $$A$$, $$B$$, $$C$$, and $$D$$ lie on circle $$\odot O$$, with $$OC\perp AB$$ and $$\angle AOC=40^{\circ}$$. What is the measure of $$\angle BDC$$? | $$20^{\circ}$$ |
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28c456ce-6968-461c-a56d-d954534f639a | In the expansion of , the coefficient of the term is (answer in numerical form). | 35 |
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6ee7e677-c66c-4152-b204-20a6d5fa6786 | The three-view diagram of a polyhedron composed of identical square blocks is shown on the right. The number of square blocks in this polyhedron is ______. | 5 |
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116f88ea-3078-4e0b-97c9-48abecaf4709 | Please complete the following problems: (1) If the random variable $$X$$ has a distribution given by $$P(X=k)=\dfrac{C}{k(k+1)}(1 \leqslant k \leqslant 10,k \in \mathbf{N})$$, where $$C$$ is a constant, find the value of $$P\left(\dfrac{1}{2} < X < \dfrac{5}{2}\right)$$. (2) Given that the discrete random variable $$X$$ has the following distribution: determine the value of $$P(X=10)$$. | (1)$$\dfrac{11}{15}$$ (2)$$\dfrac{1}{3^{9}}$$ |
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494d3907-2929-47d4-99a7-f9bcbccec531 | To monitor the water quality of a certain river, 6 water quality tests were conducted, and a line graph of the ammonia nitrogen content was plotted as shown in the figure. If the average ammonia nitrogen content of these 6 water quality tests is \( \quantity{1.5}{mg/L} \), then the ammonia nitrogen content obtained in the 3rd test is ___ \( \unit{mg/L} \). | $$1$$ |
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2d78e09e-052f-4194-bdc4-8c029e07d9d0 | In the 2D rectangular coordinate system $$xOy$$, for any point $$P(x,y)$$ not on the coordinate axes, we define the point $$P'\left(\dfrac{1}{x},\dfrac{1}{y}\right)$$ as the "inverse point" of point $$P$$. There are two points $$A$$ and $$B$$ on the line $$y=-x+1$$, and their inverse points $$A'$$ and $$B'$$ are on the graph of the inverse proportional function $$y=\dfrac{k}{x}$$. If $$AB=2\sqrt{2}$$, then $$k=$$___. | $$-\dfrac{4}{3}$$ |
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cbfe3ec1-c1d6-45ce-9eb9-f7b5ee3ede72 | As shown in the flowchart below, based on the value of the variable $$x$$ finally output, the last digit of $$s$$ is ______. | $$4$$ |
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3b0d1eb6-d196-43cd-9423-be2ffad1cb78 | As shown in the figure, the straight line y = -\frac{4}{3}x + 4 intersects the x-axis and y-axis at points A and B, respectively. Rotate triangle \triangle AOB 90° clockwise around point A to obtain \triangle AO'B'. Then the coordinates of point B' are (?). | (7, 3) |
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aabd649c-819a-42ec-90c3-f889a2e3ee9c | As shown in the figure, after rotating $Rt\Delta ABC$ counterclockwise around point $C$ by $40{}^\circ$, you get $\Delta EDC$. At this time, point $D$ is on $AB$. What is the measure of $\angle BAC$? | $20{}^\circ $ |
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038ceea7-7bd2-415a-b628-d3eef0f6d616 | As shown in the flowchart, the output result is $$y= $$___. | $$4$$ |
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a97e70ce-3544-434c-9b02-28d5f986d8cf | As shown in the figure, in $$\triangle ABC$$, $$AB=6$$, $$DE \parallel AC$$. Rotate $$\triangle BDE$$ around point $$B$$ clockwise to obtain $$\triangle BD'E'$$, and the corresponding point $$D'$$ of point $$D$$ falls on the side $$BC$$. Given $$BE'=5$$ and $$D'C=4$$, the length of $$BC$$ is ___. | $$2+\sqrt{34}$$ |
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a6a89991-dcbb-4ba5-9887-927c2d6bac35 | As shown in the figure, the length of line segment AB is 9 meters. Shift it 5 meters to the left to get line segment $A'B'$, where point $A$ corresponds to point $A'$ and point $B$ corresponds to point $B'$. Then the length of line segment AB’= meters. | 4. |
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1ee54914-206f-4e4d-b127-0deb31f13dfe | The figure contains ______ line segments and ______ rays. | Line segments are: PA, PB, PC, AB, AC, BC, a total of 6; Rays are: AB, BC, BA, CA, as well as rays extending to the left starting from point C and rays extending to the right starting from point A, a total of 6. |
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1f74b035-9438-4d50-a92d-d050ec3f7f18 | Read the program and complete the following questions: The program is as shown: (1) If the statement y=x+1 is not executed, then the range of input x is _____; (2) If the execution result is 3, then the assignment statement executed is _____, and the value of input x is _____. | (1)x<1 (2)y=x+1 2 |
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eb6b2c99-1f82-4629-b940-8118b92815af | As shown in the figure, in $\vartriangle ABC$, when $C=\frac{\pi }{3}$ and $BC=4$, point $D$ is on side $AC$, $AD=DB$, $DE\bot AB$, and $E$ is the foot of the perpendicular. If $DE=2\sqrt{2}$, then $\text{cos} A=$ | $\frac{\sqrt{6}}{4}$ |
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70f29199-d8d3-4ce5-874f-33572c5e8001 | Person A and Person B are traveling towards each other on a straight path. A is riding a bicycle from point A to point B, while B is driving a car from point B to point A. They are traveling at different constant speeds. It is known that A departs 6 minutes before B. Throughout the entire process, the relationship between the distance y (in kilometers) between A and B and the time x (in minutes) from A's departure is shown in the diagram. When B reaches the endpoint A, A still needs ___ minutes to reach the endpoint B. | 18 |
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1afbad79-4b9b-43f9-9fb3-e4241aa9c1be | In a competition, 9 out of 11 players from a certain basketball team participate in the game. The scoring stem-and-leaf plot is shown in the image. From the players who scored more than 10 points, randomly select 2 players. The probability that the sum of the points scored by these 2 players exceeds 35 points is: | $\frac{3}{10}$ |
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8934eb6c-08e9-4072-9277-f25ca70c8d34 | As shown in the figure, suppose $$C_{1}$$, $$C_{2}$$, $$\cdots $$, $$C_{n}$$, $$\cdots $$ are a series of circles on the coordinate plane. Their centers are all on the positive half of the x-axis and each circle is tangent to the line $$y=\dfrac{\sqrt{3}}{3}x$$. For each positive integer $$n$$, circle $$C_{n}$$ is externally tangent to circle $$C_{n+1}$$. Let $$r_{n}$$ represent the radius of $$C_{n}$$. It is known that $$\left \lbrace r_{n}\right \rbrace $$ is an increasing sequence, and $$r_{1}=1$$. Find the general formula for the sequence $$\{r_{n}\}$$, which is $$r_{n}=$$___. | $$3^{n-1}$$ |
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7af36674-b8c1-4a4d-a09c-4e5b62875bab | As shown in the figure, $\Delta ABC$ is translated along $BC$ to $\Delta DEF$, where $AB=10$, $DO=4$, and the translation distance is 6. Find the area of the shaded region. | 48 |
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ed5a5b00-4d5d-484a-bbd0-770e2601e1aa | As shown in the figure, to ensure $$AB \parallel CD$$, only one additional condition needs to be added. This condition could be ___. (Fill in one condition you think is correct) | $$ \angle 2 = \angle 4$$ or $$ \angle B + \angle BCD = 180^{\circ}$$ or $$ \angle BAD + \angle D = 180^{\circ}$$ (choose one) |
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59adda4d-6087-42ae-8417-dd3d888e9406 | The orthographic view of a certain geometric figure is shown in the diagram, then the area of the side view of this geometric figure is ___. | $$5a^{2}$$ |
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6a3a5f10-4d5d-4472-a306-16d126d2639f | As shown in the figure, the diagonal center of square ABCD is at the coordinate origin, AB is parallel to the x-axis, AD and BC intersect the x-axis at E and F, respectively. If square ABCD has two vertices on the hyperbola y=\frac{a+2}{x}, with the real number a satisfying a^3 imes a = 1, then the area of the quadrilateral DEBF is. | 6 or 2 or 10 |
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0d835eb2-943d-4d6d-8eb4-ea7198005ff6 | As shown in the figure, in $$Rt\Delta ABC$$, $$E$$ is the midpoint of the hypotenuse $$AB$$. If $$AB=10$$, then $$CE=$$___. | $$5$$ |
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39f7d654-8647-49c8-ad55-b6257cbcee0d | As shown in the figure, from the right-angled side AC and the hypotenuse AB of $$\text{Rt}\triangle ABC$$, equilateral triangles $$ACD$$ and $$ABE$$ are constructed externally. Given $$EF \perp AB$$ and with F as the foot of the perpendicular, connect DF. When $$\dfrac{AC}{AB}=$$ ___, the quadrilateral $$ADFE$$ is a parallelogram. | $$\dfrac{\sqrt{3}}{2}$$ |
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ce6c0eee-6c38-4809-9091-8a240c1c09e9 | As shown in the figure, in the five regions, the central region $E$ is a painting. Now, it is required to color the remaining four regions. There are four colors available for selection. Each region should be painted with only one color, and neighboring regions should not have the same color. The number of different coloring methods is
. | 84 |
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d79b80d9-744a-4e5d-9b3e-9a852ccb7980 | It is known that in the rectangle $$ABCD$$, the coordinates of points $$A$$ and $$C$$ are $$A\left( 3,8 \right)$$ and $$C\left( 7,6 \right)$$, respectively. Then the coordinates of points $$B$$ and $$D$$ are: point $$B$$______, point $$D$$______. | (3,6)
(7,8) |
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2132dea2-0449-42fc-81af-04cafa9f5277 | Execute the program flowchart as shown in the figure, with the output result being $$a$$. If the coefficient of the $$x^{3}$$ term in the expansion of the binomial $$\left(\sqrt{m}x^{2}+\dfrac{1}{\sqrt{x}}\right)^{4}$$ is $$\dfrac{a}{2}$$, then the constant $$m=$$___. | $$\dfrac{1}{4}$$ |
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3b3916f0-c90c-4ff3-9362-3824ec1a9790 | In the right triangle $$\mathrm{Rt}\triangle ABC$$ as shown, $$\angle C=90^{\circ}$$, $$BC=3$$, point $$O$$ is on $$AB$$, $$OB=2$$, a circle with $$OB$$ as the radius $$\odot O$$ is tangent to $$AC$$ at point $$D$$, and intersects $$BC$$ at point $$F$$, where $$OE\perp BC$$, then the length of arc $$BF$$ is ___. | $$2$$ |
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e102b7f4-4b54-425c-9475-426afbd44654 | As shown in the figure, $AB=AC, BD=CD, AD=AE, \angle BAD={{30}^{{}^\circ }}$, find $\angle EDC=$. | $15{}^\circ $ |
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2bc1ad60-acc0-4317-96b5-e8fff61ded0e | Find the area of the shaded region in the figure below (unit: $${\rm m}$$). ______ square meters. | 24 |
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afe11f3d-0b7b-4c30-9b55-ed682a217529 | As shown in the figure, the area of rectangle $$ABCD$$ is $$15$$, and the length of side $$AB$$ is greater than the length of side $$AD$$ by $$2$$. Then the length of $$AD$$ is ___. | $$3$$ |
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dd6bccf5-7342-43e4-b1c2-4b28d3c461c0 | As shown in the figure, it is a schematic diagram of five tourist attractions in a certain city's urban area (the side length of each small square in the figure is $$1$$ unit length). Please take a certain attraction as the origin to establish a plane Cartesian coordinate system. (1) Use coordinates to represent the locations of the following attractions: with ___ as the origin, Animal Park ___, Martyr's Cemetery ___, Waterside Park ___; (2) Calculate the area of the triangle formed by Riverside Park, Martyr's Cemetery, and Beads Garden as the vertices, which is ___. | (1) Beads Garden $$\left ( -1,-3\right ) $$ $$\left (2,5\right ) $$ $$\left ( 3,2\right ) $$ (2)$$10$$ |
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47e4a457-1bfe-4a5f-842e-d7c95e84a24f | (1) As shown in the figure, it is a figure formed by piecing together a ______, a ______, a ______, and a ______. (2) The number of ______ and ______ in the figure is equal, and the number of ______ is the least. | rectangle
square
circle
triangle
rectangle
circle
triangle |
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2ad0fd6e-0956-4c51-aaac-a0072b23f1a3 | As shown in the figure, $BD$ bisects $\angle ABC$, $AB=4$, $BC=6$. When $BD=$, $\Delta ABD\backsim \Delta DBC$. | $2\sqrt{6}$ |
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9e30584e-61ca-4963-8a18-6d2c2d51614d | Given the flowchart shown in the figure, the value of $$s$$ is ___. | $$\number{7500}$$ |
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ba6c551a-a1c0-42fa-bf60-4d54311343f6 | The graph of the function $$f(x)$$ is shown in the figure. The range of $$f(x)$$ is ___. | $$[-4,3]$$ |
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b97655c6-09ac-4c5f-a5ca-3aa8e30761e2 | In order to understand the situation of seventh-grade students participating in social practice activities during the school term, a random survey was conducted on the number of days for which 200 seventh-grade students in a certain city participated in social practice activities. An incomplete bar chart was created based on the survey results. Therefore, the average number of days that the surveyed 200 students participated in social practice activities is ___. | 5.3 |
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8b2dee2d-a147-45c8-ad65-f56aa77a5ff2 | As shown in the figure, the side AB of rectangle ABCD is parallel to the y-axis, and the coordinates of vertex A are (1, m), C (3, m+6). Then the inverse proportional function equation that goes through points B and D is ? | $y=\frac{9}{x}$ |
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828ca344-7e28-4654-85bc-696ad08cc202 | As shown in the figure, in a plane Cartesian coordinate system, it is known that A(1,0), D(3,0), △ABC is similar to △DEF, and the origin O is the center of similarity. If the area of △ABC is 5, then the area of △DEF is . | 45 |
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6a9b6748-418b-4874-912e-6f1cb590d994 | As shown in the figure, the sequences $$\left \lbrace A_{n} \right \rbrace$$ and $$\left \lbrace B_{n} \right \rbrace$$ are on two sides of a certain vertex angle, and $$|A_{n}A_{n+1}|=|A_{n+1}A_{n+2}|$$, $$A_{n} \neq A_{n+2}$$, $$n \in \mathbf{N}^{*}$$, $$|B_{n}B_{n+1}|=|B_{n+1}B_{n+2}|$$, $$B_{n} \neq B_{n+2}$$, $$n \in \mathbf{N}^{*}$$ ($$P \neq Q$$ means the points $$P$$ and $$Q$$ do not coincide). If $$d_{n}=|A_{n}B_{n}|$$, and $$S_{n}$$ is the area of $$\triangle A_{n}B_{n}B_{n+1}$$, then the sequence $$\left \lbrace S_{n} \right \rbrace$$ is an ___ sequence (fill in either "arithmetic" or "geometric"). | arithmetic |
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52c66a62-9ca1-4e5b-93ea-38cd7f9390e9 | Please complete the following problems. (1) Draw a ray $$OC$$ inside $$\angle AOB$$. In figure 1, there are ___ different angles. (2) Draw two rays $$OC$$ and $$OD$$ inside $$\angle AOB$$. In figure 2, there are ___ different angles. (3) Draw three rays $$OC$$, $$OD$$, and $$OE$$ inside $$\angle AOB$$. In figure 3, there are ___ different angles. (4) Draw $$n$$ rays $$OC$$, $$OD$$, $$OE$$, $$\cdots$$ inside $$\angle AOB$$. In the figure, there are ___ different angles. (5) Draw 100 rays $$OC$$, $$OD$$, $$OE$$, $$\cdots$$ inside $$\angle AOB$$. In the figure, there are ___ different angles. | (1)$$3$$ (2)$$6$$ (3)$$10$$ (4)$$\dfrac{1}{2}(n+2)(n+1)$$ (5)$$5151$$ |
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