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c3d7a84d-3e63-436e-868d-7bfbd2deea93 | Given the function $$f(x)=2 \sin \dfrac{ \pi x}{4}$$, if $$a$$, $$b$$, and $$c$$ are three different numbers from the set $$A=\left \lbrace f(0),f(1),f\left(\dfrac{4}{3}\right),f(2)\right \rbrace $$, execute the algorithm flowchart as shown, then the probability that the output $$a=f\left(\dfrac{4}{3}\right)$$ is ___. | $$\dfrac{1}{4}$$ |
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7fa5007a-0d7a-44cd-8146-e3e3d2aed3a8 | As shown in the figure, if there are m line segments and n rays in the figure, then $m+n=$. | 26 |
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b1a1df8d-eac3-48b2-b453-0ee7540c85e4 | Regarding the volume of a solid of revolution, there is the following Guldin's theorem: "The volume of the solid of revolution obtained by revolving a region D in a plane around a straight line (each point of region D is on the same side of the line, including the line) for one full rotation is equal to the product of the area of D and the path passed by the centroid (also known as the center of mass) of D." Using this theorem, we can find the volume of the spatial figure formed by rotating the semicircular disk $\left\{ \begin{array}{*{35}{l}} {{x}^{2}}+{{y}^{2}}\le 1 x\le 0 \end{array} \right.$ around the line $x=\frac{2}{3\pi }$ for one full rotation: | 2π |
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1e0b0f03-af16-4c69-9710-d48a29ce2e66 | The shelf life of a certain food item $$t$$ (unit: hours) is related to its storage temperature $$x$$ (unit: $$ \unit{\degreeCelsius} $$) by the function $$t= \begin{cases} 64, x \leqslant 0, 2^{kx+6}, x > 0, \end{cases}$$ and the food's shelf life at $$4 \unit{\degreeCelsius} $$ is $$16$$ hours. It is known that on a certain day, Person A bought the food item at 10 a.m. and left it outside, with the outside temperature changing as shown in the graph that day. Given the following four conclusions: 1. The shelf life of the food item at $$6 \unit{\degreeCelsius} $$ is $$8$$ hours; 2. When $$x \in [-6,6]$$, the shelf life $$t$$ of the food item gradually decreases as $$x$$ increases; 3. By 1 p.m. on that day, the food item purchased by Person A is still within its shelf life; 4. By 2 p.m. on that day, the food item purchased by Person A has exceeded its shelf life. The sequence numbers of all correct conclusions are ___. | 1.4. |
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1bee1f1d-9a44-4997-89ef-b54b5df241e0 | As shown in the figure, in the right triangle △ABC, ∠ACB=90°, AC=6, BC=8. From the right angle vertex C, draw CA$_{1}$ perpendicular to AB, with the foot of the perpendicular being A$_{1}$. Then, from A$_{1}$, draw A$_{1}$C$_{1}$ perpendicular to BC, with the foot of the perpendicular being C$_{1}$. From C$_{1}$, draw C$_{1}$A$_{2}$ perpendicular to AB, with the foot of the perpendicular being A$_{2}$. Then, from A$_{2}$, draw A$_{2}$C$_{2}$ perpendicular to BC, with the foot of the perpendicular being C$_{2}$, and so on, constructing in this way continuously, forming a series of line segments CA$_{1}$, A$_{1}$C$_{1}$, C$_{1}$A$_{2}$, A$_{2}$C$_{2}$, …, A$_{n}$C$_{n}$. Thus, A$_{1}$C$_{1}$= , A$_{n}$C$_{n}$= . | $6\times {{\left( \frac{4}{5} \right)}^{2}}$
$6\times {{\left( \frac{4}{5} \right)}^{2n}}$ |
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5faf2886-3ed3-477a-be3d-106f3c100b59 | As shown in the figure, the side length of rhombus $$ABCD$$ is $$2$$, and the height $$AE$$ bisects $$BC$$. Find the area of the rhombus ___. | $$2\sqrt{3}$$ |
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eb6589e5-0471-4204-8bc8-10ef7c2df26c | Learning has become a modern trend. A certain city department has compiled statistics on the occupational distribution of library visitors over the past 6 months. Here are two incomplete statistical charts.
(1) During the statistical period, a total of ___ 10,000 people visited the library, with the percentage of businesspeople being ___\%.
(2) If the number of readers visiting the library in May is \number{28000}, estimate the number of visits by working professionals. | (1) 16 12.5
(2) \number{10500} |
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d5c75bfb-3ddb-4d8c-ad68-c1fe0c12898c | As shown in the figure, in $\Delta ABC$, points $D$ and $E$ are on $AB$ and $BC$, respectively. If $AD:DB=CE:EB=2:3$, then ${{S}_{\vartriangle DBE}}:{{S}_{\vartriangle ADC}}=$ ? | $9:10$ |
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7d42d181-db25-47fc-b5f8-22243983ff35 | In a certain region, in order to understand the average daily sleep time (in hours) of elderly people aged 70-80, a random selection of 50 elderly people was surveyed. The table below shows the frequency distribution of the daily sleep time for these 50 elderly people. Based on the statistical data mentioned above, a part of the calculation is shown in the flowchart below, then the output value of S is ___. | $$6.42$$ |
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dd348c10-d186-4797-95e5-59a8db5b32eb | Given that $AB = 4$, points $M, N$ are any two points on the semicircle with diameter $AB$, and $MN = 2$, $\overrightarrow{AM} \cdot \overrightarrow{BN} = 1$, find $\overrightarrow{AB} \cdot \overrightarrow{NM}$. | $-6$ |
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8cf2ca99-2554-40d8-b7c1-823ca20d635e | As shown in the figure, in the rectangular coordinate plane xOy, the graph of the function y$_{1}$=$\frac{k}{x}$ intersects with the line y$_{2}$=x+1 at point A (1,a). Then: (1) The value of k is; (2) When x satisfies , y$_{1}$ > y$_{2}$. | 2;
x<−√2 or 0<x<1. |
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697da351-4d06-48a7-a946-e22aa28e91b6 | In triangle DABC, AB = AC = 6, DE is the perpendicular bisector of AC and intersects at E. If the perimeter of quadrilateral DEBC is 10, then the perimeter of triangle DABC is what? | 16 |
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f2cb7a73-7ea8-4314-8962-a6db019ed2fa | As shown in the figure, in $$\triangle ABC$$, $$AB=a$$, $$AC=b$$, the perpendicular bisector $$DE$$ of side $$BC$$ intersects $$BC$$ and $$AB$$ at points $$D$$ and $$E$$, respectively. Then the perimeter of $$\triangle AEC$$ is equal to ___. | $$a+b$$ |
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f382592d-73b0-4c07-ac18-4f0d1a095d72 | As shown in the figure, D is the midpoint of line segment AB, and E is the midpoint of line segment BC. If AC=10, what is the length of line segment DE? | 5 |
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74e89e1e-50f1-4001-9797-0014177ff5b3 | In the Southern Song Dynasty of China, mathematician Yang Hui used triangles to explain the square law of binomial expansion, naming it the "Yang Hui Triangle." This triangle provided the coefficient rules for expanding $$(a+b)^{n}(n=1,2,3,4,\cdots )$$ in order of decreasing powers of $$a$$: . According to the above rules, write out the coefficient of the term containing $$x^{\number{2015}}$$ in the expansion of $$\left (x-\dfrac{x}{2} \right )^{\number{2017}}$$. | $$-\number{4034}$$ |
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03340094-ea00-4861-80fe-02e1ac24a86e | As shown in the figure, lines $$a$$ and $$b$$ intersect at point $$O$$. Given $$\angle 1=50{{}^\circ}$$, find $$\angle 2=$$______ degrees. | $$50$$ |
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1f512ec4-6aeb-495c-86e4-0533dc9ab9be | Ships $$A$$ and $$B$$ set sail from locations Jia and Yi, which are $$145 \ \unit{km}$$ apart in the east-west direction. $$A$$ travels westward from Jia. $$B$$ travels southward from Yi. The speed of $$A$$ is $$40 \ \unit{km / h}$$, and the speed of $$B$$ is $$16 \ \unit{km / h}$$. After ___ hours, the distance between $$A$$ and $$B$$ is the shortest. | $$\dfrac{25}{8}$$ |
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4d70d263-b230-49ba-a826-f298e39dfd5c | As shown in the figure, please complete a condition: ___, so that $$ \triangle ABC\sim \triangle AED$$. | $$ \angle B= \angle AED$$ or $$ \angle C= \angle ADE$$ or $$\dfrac{AB}{AE}=\dfrac{AC}{AD}$$ |
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e33f2145-e202-4bb8-97ca-2ce58cbe69c7 | Execute the program flowchart as shown. If the input value of $$n$$ is $$4$$, then the output value of $$s$$ is ___. | $$7$$ |
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aa3cc382-8e3b-4763-ad49-83a7faa244a9 | As shown in the figure, this is a part of the map of Taizhou City. A right-angle coordinate system is established with the positive directions in the east and north as the $$x$$-axis and $$y$$-axis, respectively. It is stipulated that one unit length represents $$1km$$. Jia and Yi describe the location of district $$A$$ in the bridge area as follows; then the coordinates of Jiangjiao District $$B$$ are ___. | $$(10,8\sqrt{3})$$ |
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3edddd17-79b3-4b73-abba-8c9ce1b8fe36 | As shown in the figure, the acute angle between line $OD$ and the $x$-axis is ${{30}^{\circ }}$. The length of $O{{A}_{1}}$ is $1$. Triangles \(\Delta {{A}_{1}}{{A}_{2}}{{B}_{1}}, \Delta {{A}_{2}}{{A}_{3}}{{B}_{2}}, \Delta {{A}_{3}}{{A}_{4}}{{B}_{3}},..., \Delta {{A}_{n}}{{A}_{n+1}}{{B}_{n}}\) are all equilateral triangles. Points \({{A}_{1}}, {{A}_{2}}, {{A}_{3}},..., {{A}_{n+1}}\) are ordered on the positive x-axis, and points \({{B}_{1}}, {{B}_{2}}, {{B}_{3}},..., {{B}_{n}}\) are sequentially arranged on line $OD$. Then, the coordinates of point ${{B}_{n}}$ are. | $\left( 3\times {{2}^{n-2}},\sqrt{3}\times {{2}^{n-2}} \right)$ |
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26f21f70-8abd-4b5e-852b-1edea6adf5cb | As shown in the figure, given the cube $ABCD-{{A}_{1}}{{B}_{1}}{{C}_{1}}{{D}_{1}}$, if 3 edges are randomly selected from a total of 12, what is the probability that these three edges are pairwise opposite edges? (Express the result as the simplest fraction) | $\frac{2}{55}$ |
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eef6e70d-1356-4a8c-b43b-f2fd6870ab1c | As shown in the figure, points A, B, and C are on the same straight line. Triangles △ABD and △BCE are both equilateral triangles. AE and CD intersect BD and BE at points F and G, respectively. Connect FG. The following conclusions are drawn: 1. AE = CD 2. ∠BFG = 60°; 3. EF = CG; 4. AD ⊥ CD 5. FG ∥ AC Of these, the correct conclusions are. (Fill in the numbers) | 1.2.3.5. |
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ac45587e-2d60-4206-8925-d17549a96121 | In rectangle $$ABCD$$, point $$E$$ is the midpoint of side $$BC$$. The bisector of $$\angle AEC$$ intersects side $$AD$$ at point $$F$$. If the vector magnitude $$\left \lvert \overrightarrow{AB}\right \rvert =3$$, $$\left \lvert \overrightarrow{AD}\right \rvert =8$$, then $$\left \lvert \overrightarrow{FD}\right \rvert =$$___. | $$3$$ |
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2538a15c-a6c1-4993-a49f-d0a7bcf5aab8 | As shown in the figure, in $$\triangle ABC$$, $$D$$ and $$E$$ are points on sides $$AB$$ and $$AC$$, respectively, and $$DE \parallel BC$$. If the ratio of the perimeter of $$\triangle ADE$$ to $$\triangle ABC$$ is $$2:3$$, and $$AD=4$$, then $$DB=$$___. | $$2$$ |
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4fa673e4-363a-41e9-bc9e-cdcfcda27eee | Given the function $f(x)=A\cdot sin(\omega x+\phi )$ with $(A > 0,\omega > 0,\left| \phi \right| < \frac{\pi}{2})$, a portion of its graph is shown in the figure, find $f(0)=$. | 1 |
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a1b15cf1-996d-4562-a247-a1b8083fcde2 | As shown in the figure, suppose the length of the bamboo pole (dashed-line part) is 8m. Then the maximum area of the rectangle ABCD that can be enclosed is m. | 16 |
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a78f0da7-3bac-4e01-b461-52a093ee8e0d | As shown in the figure, the regular hexagon $$ABCDEF$$ is inscribed in circle $$\odot O$$, and the radius of $$\odot O$$ is $$1$$. Find the length of the arc $$\overset{\frown} {AB}$$. | $$\dfrac{ \pi }{3}$$ |
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7e6d11a9-4ae1-4604-949f-ea1c5db4223b | As shown in the figure, in rectangle $ABCD$, $AB=12$, $BC=5$, and the hyperbola $M$ with foci at points $A$ and $B$ is given by: $\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1$, which passes through points $C$ and $D$. The standard equation of the hyperbola $M$ is. | $\frac{{{x}^{2}}}{16}-\frac{{{y}^{2}}}{20}=1$ |
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be69244c-4a07-40f4-81db-8d4cc0161079 | As shown in the diagram, O is a point inside triangle ABC, and OA = OB = OC. If angle OBA = 20°, angle OCB = 30°, then angle OAC =.?[](612abefc6245de0597011f8ba870b62b4ac90b42c255194a2c9f1303e9a1c665) | 40° |
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5e27fcb2-0f5c-41d4-9f3e-2fc2ee291c73 | As shown in the figure, $$AB$$ is the diameter of the semicircle $$O$$. Point $$C$$ is on the extended line of $$AB$$, and $$CD$$ is tangent to the semicircle $$O$$ at point $$D$$. Also, $$AB=2CD=4$$, then the area of the shaded part in the figure is ___. | $$\dfrac{\pi }{2}$$ |
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a8a9656a-9873-4a36-b886-285e635363bc | As shown in the figure, point $E$ and $F$ are on side $AB$ of $\vartriangle ABC$. If $AB = 8$, $\angle ACB = \frac{\pi }{2}$, $\angle ABC = \angle ECF = \frac{\pi }{6}$, then the maximum area of $\vartriangle ECF$ is: | $4\sqrt{3}$ |
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4a5b8ce6-e4af-4225-9a00-5dbd34b4a130 | As shown in the figure, it is known that AB=AC. The perpendicular bisector MN of AB intersects AB at point M and AC at point D. If ∠A=36°, then ∠DBC= ? | 36° |
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59865982-df06-46d2-bd8f-b3cb88c7c2fb | As shown in the figure, given that point $$A$$ is a moving point on the graph of the inverse function $$y=-\dfrac{2}{x}$$, and connecting $$OA$$. If the line segment $$OA$$ is rotated $$90^{ \circ }$$ clockwise around point $$O$$ to obtain line segment $$OB$$, then the functional expression of the graph on which point $$B$$ lies is ___. | $$y=\dfrac{2}{x}$$ |
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1adf5b9a-578e-4e1f-a2a3-d7824afe137a | As shown in the figure, the volume of the rectangular prism $ABCD-{A}_{1}{B}_{1}{C}_{1}{D}_{1}$ is 120, and E is the midpoint of $C{C}_{1}$. Find the volume of the tetrahedron E-BCD. | 10. |
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05585e43-e494-4b22-88ac-dd02fa79d456 | The numbers 1, 3, 6, 10, 15, 21, … are called triangular numbers because these numbers of points can be arranged to form an equilateral triangle (as shown in the figure below). Find the eighth triangular number. | 36 |
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47faf686-f39a-4505-b70c-253dba23c07b | As shown in the figure, given the ellipse $$\dfrac{x^{2}}{9}+\dfrac{y^{2}}{4}=1$$, $$A$$ is a point where the ellipse intersects the $$y$$-axis. $$\triangle ABC$$ is an equilateral triangle inscribed in the ellipse. Find the side length of $$\triangle ABC$$, which is ___. | $$\dfrac{72\sqrt{3}}{31}$$ |
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401996b1-7526-4395-a7c2-e68d512f2e1c | As shown in the figure, points $$E$$ and $$F$$ are the midpoints of sides $$AD$$ and $$BC$$ of the parallelogram $$ABCD$$, respectively. If quadrilateral $$AEFB$$ is similar to quadrilateral $$ABCD$$, and $$AB=4$$, then the length of $$AD$$ is ___. | $$4\sqrt{2}$$ |
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14bdacc0-4606-49c6-8bae-b59848610306 | As shown in the figure, if vertex $$B$$ of the square $$OABC$$ and vertex $$E$$ of the square $$ADEF$$ are both on the graph of the function $$y=\dfrac{1}{x}(x > 0)$$, then the coordinates of point $$E$$ are (___,___). | $$\dfrac{\sqrt{5}+1}{2}$$ $$\dfrac{\sqrt{5}-1}{2}$$ |
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1a54735a-b79c-4d36-ac42-021b1ebd7bce | Regarding the inequality in x: -2x + a ≥ 3, as shown in the figure, the value of a is: | 1 |
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571ba4e5-eede-4bfc-8b90-389a2118f803 | As shown in the figure, in $$ \triangle ABC$$, $$ \angle ACB=90^{ \circ }$$, point $$F$$ is on the extension of side $$AC$$, and $$FD \perp $$$$AB$$, with the foot of the perpendicular being point $$D$$. If $$AD=6$$, $$AB=10$$, $$ED=2$$, then $$FD=$$___. | $$12$$ |
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2c5200f8-355b-49ad-b460-b7bf0eff662a | As shown in the figure, in $$\triangle ABC$$, $$D$$ is the midpoint of side $$AC$$. Let $$BD=\overrightarrow{a}$$, $$BC=\overrightarrow{b}$$, find how $$\overrightarrow{CA}$$ can be expressed using $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$. | $$2\overrightarrow{a}-2\overrightarrow{b}$$ |
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56bc2dfb-f489-41da-8784-5e20d01860f0 | Given a rod $AB$, and an isosceles triangle $A'ED$ with base altitude $A'B'$ and a square $GHFK$ as shadows at the same moment as shown in the figure, find the similar triangles among them. | $\vartriangle A'EB'\backsim \vartriangle A'DB'$, $\vartriangle B'C'E\backsim \vartriangle B'C'D$, $\vartriangle ABC\backsim \vartriangle A'B'C'\backsim \vartriangle GHM\backsim \vartriangle KFN$ |
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70120019-be0e-4025-82b8-b25789c07e02 | As shown in the figure, a mathematics study group needs to measure the height of a building $CD$ on the ground (the building $CD$ is perpendicular to the ground). The measurement plan is to first select two points $A$ and $B$ on the ground, with a distance of $100$ meters between them. Then measure $\angle DAB={{60}^{\circ }}$ at point $A$, $\angle DBA={{75}^{\circ }}$, and $\angle DBC={{30}^{\circ }}$ at point $B$. What is the height of the building $CD$ in meters? | $25\sqrt{6}$ |
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51af8702-6c1f-40b8-8c7b-08593c5a61fd | Given the function $$f(x)=A \sin ( \omega x+ \varphi )$$$$\left(A > 0, \omega > 0,| \varphi | < \dfrac{ \pi }{2}\right)$$, a part of its graph is shown as in the figure. Then the expression for the function is ___. | $$f(x)=2 \sin \left(2x+\dfrac{ \pi }{6}\right)$$ |
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fabde2a5-e12b-4460-821f-72e535964851 | After learning how to solve a linear inequality in one variable, the teacher assigned the following exercise: Solve the inequality $\frac{15-3x}{2}$≥$7-x$, and represent its solution set on the number line. Below is Xiaoming's process: Question: Please indicate from which step Xiaoming made a mistake and explain the reasoning. | Error in step 5, reasoning based on the property of inequality 3 |
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ea3b5ca4-311c-4ca4-a115-5d375cb8ef5f | In the circuit diagram shown, if any switch among them is closed, the probability of the bulb emitting light is . | $\frac{1}{3}$ |
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d6ceee09-b9e6-4970-9590-6a188454e76a | As shown in the figure, △ABC ≅ △DCB, ∠DBC = 40°, then ∠AOB = °. | 80° |
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a768b222-d794-4ae1-94b9-48d601035633 | As shown in Figure 1, it is a diagram of a small bed that can be folded and supported on the ground after unfolding. At this time, points A, B, and C are on the same straight line, and \angle ACD = 90^{\circ}. Figure 2 is a diagram of the folding of the small bed support leg CD. During the folding process, \triangle ACD transforms into a quadrilateral ABC'D', and finally folds into a line segment BD''.
(1) The mathematical principle applied in the design of this small bed is ___.
(2) If AB:BC=1:4, then the value of \tan \angle CAD is ___. | (1) The stability of triangles and the instability of quadrilaterals (2) \dfrac{8}{15} |
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1027b44e-b10a-40d0-966d-3a314059c0fa | As shown in the figure, represent the three numbers $\sqrt{2}$, $\sqrt{5}$, and $\sqrt{18}$ on the number line. Then, the numbers included in the solution set represented in the figure are: | $\sqrt{5}$ |
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aceaab49-05a6-41b8-b6f2-2892fab551dc | As shown in the figure, an ant crawls horizontally to the right from point $$A$$ along the number line for $$2$$ units to reach point $$B$$. The point $$A$$ represents $$-\sqrt{2}$$, and let the number represented by point $$B$$ be $$m$$. Then the value of $$\left \lvert m-1\right \rvert $$ is ___. | $$\sqrt{2}-1$$ |
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fdafacb5-18c7-4670-b40c-b6d4d2a7ee47 | As shown in the figure, in order to measure the height of the school flagpole $$AB$$, the math activity group uses a bamboo pole $$CD$$ with a length of $$\quantity{2}{m}$$ as a measuring tool. Move the bamboo pole so that the shadow of the top of the bamboo pole coincides with the shadow of the top of the flagpole at point $$O$$ on the ground. Measure the distances as $$OD=4\ \unit{m}$$ and $$BD=14\ \unit{m}$$. What is the height of the flagpole $$AB$$ in meters? | $$9$$ |
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9cc84d97-4b66-4826-a4f3-e3ade4981546 | As shown in the figure, $BD=CF$, $FD\bot BC$ at point $D$, $DE\bot AB$ at point $E$, $BE=CD$. If $\angle AFD$$=140{}^\circ $, then $\angle EDF$=. | $50{}^\circ $. |
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aaa35bec-6c84-48b6-9520-84798d35f7f1 | In a factory, the data on the production and cost of a certain product is as follows: From the data in the table, we obtain the linear regression equation $$\widehat{y}=\widehat{b}x+\widehat{a}$$ where $$\widehat{b}=1.1$$. Predict the cost when the production is $$9$$ thousand units, which is approximately ___ ten thousand yuan. | $$14.5$$ |
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c59c6388-c13d-4605-bd79-a0ddefceb4c7 | As shown in the figure, it is known that the distance from island $$A$$ to coastal highway $$BC$$ is $$AB = \quantity{50}{km}$$, and the distance between $$B$$ and $$C$$ is $$\quantity{100}{km}$$. From $$A$$ to $$C$$, first take a boat with a speed of $$\quantity{25}{km/h}$$, then take a car with a speed of $$\quantity{50}{km/h}$$. Select the landing point to be ___ $$\unit{km}$$ from point $$B$$ such that the total travel time is minimized. | $$\dfrac{50\sqrt{3}}{3}$$ |
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fceb014a-0917-49a8-965a-9e4a883e8947 | The Hundred Sons Backtracking Diagram is a square number table formed by arranging numbers 1, 2, 3, \cdots , 100 without repetition. It is a numeric representation of the simplified history of Macau, for example: the central four positions '19\ 99\ 12\ 20' indicate the Macau handover date, and the middle two positions in the last line '23\ 50' indicate the area of Macau, \cdots \cdots , at the same time it is also a 10-level phantom square, the sum of 10 numbers in each row, the sum of 10 numbers in each column, the sum of 10 numbers in each diagonal line are all equal. Then this sum is ___. | 505 |
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0b4aa39f-c176-4fb2-a9ca-aa9bcb801286 | As shown in the figure, it is a schematic diagram of a mechanical component with an outer rim in the shape of a rectangle. According to the dimensions marked in the figure (unit: $mm$), the distance between the centers of the two circular holes $A$ and $B$ is $mm$. | 100 |
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240e879a-2a76-465f-bd8f-fcdc6c894092 | Just before the sports meeting, the school bought $$4$$ soccer balls and $$1$$ basketball for the storage room, costing a total of $$162$$ yuan. It is known that each soccer ball is $$2$$ yuan cheaper than each basketball. How much does each soccer ball cost______ yuan. | 32 |
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6c844ca4-51c2-4f4d-a12a-19adc2190c60 | As shown in the flowchart, if the result of the program execution is $S=1320$, then what should be filled in the decision box. | K<10? |
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8bf9076e-42fe-465b-b436-53fb3c2a1715 | As shown in the figure, in $▱\text{ABCD}$, $\text{AC}$ bisects $\angle \text{DAB}$, and $\text{AB}=7$. Find the perimeter of $\text{ABCD}$. | $28$ |
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e14c06e1-df83-496f-a029-73de48e8722b | The figure shows a waterwheel with a radius of $$r$$ being $$\quantity{3}{m}$$. The center of the cross-sectional circle of the waterwheel, point $$O$$, is $$\quantity{2}{m}$$ above the water surface. It is known that starting from point $$A$$, the waterwheel completes a rotation in $$\quantity{15}{s}$$. The distance of point $$P$$ on the waterwheel to the water surface, $$y$$ (unit: $$\unit{m}$$), satisfies the functional relationship with time $$x$$ (unit: $$\unit{s}$$): $$y=A \sin ( \omega x+ \varphi )+2$$. Then $$\omega =$$___, $$A=$$___. | $$\dfrac{2 \pi }{15}$$ $$3$$ |
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144cddb5-d448-4551-a09d-4abfb2e2e52d | As shown in the figure, the foci of the ellipse $$\dfrac{x^{2}}{9}+\dfrac{y^{2}}{4}=1$$ are $$F_{1}$$ and $$F_{2}$$. When $$\angle F_{1}PF_{2}$$ is an obtuse angle, the range of values for the x-coordinate of point $$P$$ is ___. | $$-\dfrac{3}{\sqrt{5}} < x_{P} < \dfrac{3}{\sqrt{5}}$$ or $$ -\dfrac{3\sqrt{5}}{5} < x < \dfrac{3\sqrt{5}}{5}$$ |
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e6459a05-daf7-4361-8afa-10064957f368 | As shown in the figure, in the rectangle $$ABCD$$, it is known that $$AB=3$$, $$AD=2$$, and $$\overrightarrow{BE}=\overrightarrow{EC}$$, $$\overrightarrow{DF}=\dfrac{1}{2}\overrightarrow{FC}$$. Then $$\overrightarrow{AE}\cdot \overrightarrow{AF}=$$___. | $$5$$ |
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22d970e5-127f-4c47-a916-7bd3e1b36139 | As shown in the figure, in the planar Cartesian coordinate system, the edge $\text{OB}$ of the trapezoid $\text{AOBC}$ is on the positive half of the $\text{x}$-axis, $\text{AC}\,//\,\text{OB}$, $\text{BC}\bot \text{OB}$. A branch of the hyperbola $\text{y}=\frac{\text{k}}{\text{x}}$ passing through point $\text{A}$ intersects the diagonal line of the trapezoid $\text{OC}$ at point $\text{D}$ in the first quadrant, and intersects edge $\text{BC}$ at point $\text{E}$. If $\frac{\text{OD}}{\text{OC}}=\frac{1}{2}$ and ${{\text{S}}_{\vartriangle \text{OAC}}}=2$, then the value of $\text{k}$ is. | $\frac{4}{3}.$ |
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67ae4958-43f0-444a-90e3-0d88ea23fa37 | As shown in the figure, it is known that point A is any point on the graph of the inverse proportional function y=$\frac{2}{x}$. A line AB is drawn through point A perpendicular to the y-axis at point B, then the area of triangle AOB is what? | 1 |
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8b43b0e0-cd20-4353-9876-3044cba67a63 | A set of 60 items are numbered consecutively from 00, 01, ..., 59. You need to draw a sample of size 6 from these items. Please start reading from the 10th column of the random number table below moving to the right until enough samples are drawn. The numbers drawn for the sample are ___. | 01
47
20
28
17
02 |
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bbb36598-dcc9-4362-bb4f-ac65863dfe42 | As shown in the figure, it is known that ${{\text{l}}_{1}}\parallel {{\text{l}}_{2}}\parallel {{\text{l}}_{3}}$, if AB : $\text{BC}=2$ : 3, $\text{DE}=4$, then the length of EF is . | 6 |
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08c30564-77ee-4e3b-b546-2f229f6457bc | Read the program shown in the figure. When inputting separately x=2, x=1, x=0, the output values of y are ___, ___, ___. | 1 1 -1 |
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f063dd19-71b1-4f35-a4c2-f59a6676f50f | As shown in the figure, fold the square ABCD along BE so that point A falls on the point A' on diagonal BD, then connect A'C. What is the angle ∠BA'C=? | 67.5° |
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dc78d40b-3650-47d2-a102-f9411d235751 | As shown in the figure, the image displays the variation of the water level height $$y ( \unit{m} ) $$ of a certain bay relative to the average sea level over a period of $$24\ \unit{h}$$ in a day. Therefore, the functional relationship between the water level height $$y$$ and the time $$x$$ starting from midnight is ___. | $$y = - 6 \sin \dfrac{ \pi }{6}x$$ (not unique) |
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1aa55fce-e042-46b0-92e0-9334d3132a2f | As shown in the figure, in the cube $$ABCD-A_{1}B_{1}C_{1}D_{1}$$, point $$P$$ moves along the diagonal line $$AC$$, and the following four propositions are given: (1) $$D_{1}P \parallel$$ plane $$A_{1}BC_{1}$$; (2) $$D_{1}P \perp BD$$; (3) plane $$PDB_{1} \perp$$ plane $$A_{1}BC_{1}$$; (4) the volume of the tetrahedron $$A_{1}-BPC_{1}$$ remains constant. The sequence numbers of all correct propositions are ___. | (1)(3)(4) |
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f4b34456-eb03-47af-91a7-ef840e3d5595 | A production line produces a type of resistor. Every fixed period, 4 resistors are randomly selected to measure their resistance values (unit: kΩ). A total of 10 batches were measured. The data is as follows (sorted from smallest to largest): The average resistance value is \(\overline{x}=81.795\), standard deviation \(s=1.136\). Then the frequencies of these 40 resistors' resistance values falling within \((\overline{x}-s,\overline{x}+s)\) and \((\overline{x}-2s,\overline{x}+2s)\) are ___, ___. | $$0.65$$ $$0.95$$ |
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ed42c652-b117-4961-8e95-9ed1808104e9 | As shown in the figure, it is known that $$\parallelogram ABCD$$ has vertices $$A$$ and $$C$$ located on the lines $$x=2$$ and $$x=5$$, respectively. $$O$$ is the origin of the coordinate system. Find the minimum length of the diagonal $$OB$$. | $$7$$ |
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423e85ef-9913-404e-9636-7d1cc2d6d600 | As shown in the figure, the circumcircle of triangle ABC has a radius of 2, and ∠C = 30°, then the area of the sector AOB is. | $\frac{2}{3}$π. |
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2c764e01-979c-460c-8dc5-86845c98b477 | As shown in the figure, extend line segment AB to C such that BC = 2AB, extend line segment BA to D such that AD = 3AB. Point E is the midpoint of line segment DB, and point F is the midpoint of line segment AC. If EF = 10cm, then the length of AB is cm. | 4 |
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d5b508f8-3a7e-4b6e-ba2d-ea9b2d318a36 | As shown in the figure, in a square with side length $$a$$, a smaller square with side length $$b$$ is cut off from one corner ($$a > b$$). The remaining part is assembled into a trapezoid. Calculate the area of the shaded portions of these two figures separately; they verify the formula ___. | $$a^2-b^2=(a+b)(a-b)$$ |
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e59a766d-204e-4519-8866-11fb99e41463 | As shown in the figure, to make the surface unfolding of the figure fold along the dotted line into a cube, the sum of the two numbers on opposite faces is equal, $$a+b-c=$$___. | $$6$$ |
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b7e73c32-47ff-4006-b5b8-86852be5f87d | In the Yellow race, the proportion of people with each blood type is as shown in the table below: It is known that people with the same blood type can donate blood to each other, people with blood type $$\text{O}$$ can donate blood to individuals of any blood type, and anyone's blood can be donated to someone with blood type $$\text{AB}$$. People with different blood types cannot donate blood to each other. Xiaoming has blood type $$\text{A}$$. If Xiaoming becomes ill and requires a blood transfusion, the probability that a randomly selected person can donate blood to Xiaoming is ___. | $$\number{0.63}$$ |
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33c3fe9c-33ee-45c7-90a9-aed97692da8c | Given the function $f(x)=\text{A}\sin (\omega x+\varphi )$ (A>0, $\omega $>0, 0<$\varphi $<$\pi $), part of its graph on R is shown in the figure. Then the value of $f(36)$ is . | $-\frac{3\sqrt{2}}{2}$ |
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eaa9e3dc-5609-4596-8265-306ad4502640 | As shown in the figure, in quadrilateral ABCD, AB = 3, BC = 5, draw an arc with center B and any length as the radius, intersecting BA and BC at points P and Q, respectively. Then, with P and Q as centers, draw arcs with radii greater than $\frac{1}{2}$PQ, intersecting inside $\angle ABC$ at point M. Extend BM to intersect AD at point E. What is the length of DE? | 2. |
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a9cf3652-1efa-4531-af70-7281ede01c27 | As shown in the figure, in the parallelogram $ABCD$, $AB=8$, the bisector of $\angle BAD$ intersects the extended line of $BC$ at point $E$, and intersects $DC$ at point $F$. Additionally, point $F$ is the midpoint of side $DC$, and $DG \bot AE$, perpendicular from $G$. If $DG=3$, then the length of $AE$ is | $4\sqrt{7}$ |
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b05be671-79d6-48bc-acd7-e6aa64102410 | (1)$$30+$$______$$=$$______.(2)$$36-$$______$$=$$______. | 6
36
6
30 |
|
4831aba7-692c-4bbd-8079-b85ac74d2f8b | As shown in the figure, among the angles $$\angle 1$$, $$\angle 2$$, $$\angle 3$$, $$\angle 4$$, $$\angle 5$$, and $$\angle B$$ in the figure, corresponding angles are ___, interior alternate angles are ___, and co-interior angles are___. | $$\angle 1$$ and $$\angle B$$, $$\angle B$$ and $$\angle 4$$ $$\angle 3$$ and $$\angle 4$$, $$\angle 2$$ and $$\angle 5$$ $$\angle 3$$ and $$\angle 5$$, $$\angle 2$$ and $$\angle 4$$, $$\angle 3$$ and $$\angle B$$, $$\angle B$$ and $$\angle 5$$ |
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e43b80a2-870a-40b2-a912-d9fdb8c2a676 | As shown in the figure, in the right triangle $\text{Rt}\vartriangle ABC$, $\angle A=90{}^\circ$, $AB=3$, $AC=4$. Now fold $\vartriangle ABC$ along $BD$, making point $A$ fall exactly on $BC$. Then, what is $CD$? | $\frac{5}{2}$ |
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07dfcf6c-d2ac-43c1-a927-eb9fb99cbbde | As shown in the figure, in the square $ABCD$, the diagonals $AC$ and $BD$ intersect at point $O$. $E$ is a point on $BC$ with $CE=5$, and $F$ is the midpoint of $DE$. If the perimeter of $\Delta CEF$ is 18, then the length of $OF$ is . | $\frac{7}{2}$ |
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acdf781a-829b-4f4a-bf68-2cbc1e044d69 | As shown in the figure, three congruent triangles $\vartriangle ABF$, $\vartriangle BCD$, and $\vartriangle CAE$ are arranged to form an equilateral triangle $ABC$, and $\vartriangle DEF$ is an equilateral triangle, with $EF=2AE$. Let $\angle ACE=\theta$, then $\sin 2\theta =$ | $\frac{7\sqrt{3}}{26}$ |
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85e9c0d1-e9e1-4181-9b8c-31ab855e84dd | In order to study the relationship between color blindness and gender, a survey of 1000 people was conducted. The results of the survey are shown in the table below: Based on the data above, can it be seen whether color blindness and gender are independent of each other? ___ (fill in "Yes" or "No"). | No |
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a248d4ed-5a39-4229-a8dc-ba36df4d1869 | As shown in the figure, the slope of a hill is i=1: $\sqrt{3}$, Xiaoming starts from the foot of the mountain at point A, and walks 200 meters up the hill to reach point B. How many meters does Xiaoming ascend? | 100 |
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a8909f48-5406-47da-811a-0d13b6ed7fe9 | As shown in the figure, AB ∥ CD, and AD and BC intersect at point E. Draw EF through point E parallel to CD, intersecting BD at point F. Given that AB : CD = 2 : 3, find the ratio $\frac{EF}{AB}$. | $\frac{3}{5}$ |
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c182e9af-b244-4009-a71f-e9d1b466740e | A batch of products' lengths (unit: millimeter) were sampled for inspection. The figure below shows the frequency distribution histogram of the inspection results. According to the standard, products with lengths in the range $$[20,25)$$ are classified as first-grade, those in the ranges $$[15,20)$$ and $$[25,30)$$ are classified as second-grade, and those in the ranges $$[10,15)$$ and $$[30,35)$$ are classified as third-grade. Using frequency to estimate probability, if a product is randomly selected from this batch, the probability that it is second-grade is ___. | $$0.45$$ |
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8c065490-19f0-440c-928c-8930032670d2 | As shown in the figure, it is known that the graphs of the functions $$y=ax+b$$ and $$y=kx$$ intersect at point $$P$$. According to the graph, the solution for the system of linear equations in two variables $$\left\lbrace\begin{align}&y=ax+b, \cr& y=kx\end{align}\right .$$ concerning $$x$$ and $$y$$ is ___. | $$x=-4$$,$$y=-2$$ |
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69136059-f5d8-4401-97f5-3ef2bf79f35b | Given that the base of a quadrilateral pyramid is a parallelogram, and its three-view diagram is as shown (unit: $$m$$), find the volume of this quadrilateral pyramid in $$m^{3}$$! | $$2$$ |
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b179c9f6-ef03-4186-a8b0-6fb6fd192f48 | As shown in the figure, in parallelogram ABCD, BC=24, E, F are trisection points of BD, then BM=___, DN=___. | 12
6 |
|
6a768263-2b0a-431c-b36d-955e69ab54cc | On Little Monkey's birthday, his mother bought him a box of strawberries, containing $$36$$ pieces. Little Monkey let his mother eat first. After she ate $$\frac29$$ of the box of strawberries, Little Monkey ate $$\frac17$$ of the remaining strawberries. How many strawberries did the mother and Little Monkey eat, respectively: Mother ate ______ strawberries, Little Monkey ate ______ strawberries. | 8
4 |
|
0ec3ecc6-54a0-49e0-978f-8069c7f80fc8 | As shown in the figure, it is known that a linear function y$_{1}$ = ax + b and an inverse proportional function ${{y}_{2}}=\frac{k}{x}$ intersect at two points A (−2, m) and B (1, n). When y$_{1}$ > y$_{2}$, the range for variable x is . | x < −2 or 0 < x < 1. |
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ee4969cf-6b54-4bd1-a986-b0d1b7456016 | As shown in the figure, ΔABC is an isosceles right triangle with ∠ACB=90°, BC=AC. After rotating ΔABC around point A counterclockwise by 45°, we get ΔAB'C'. If AB=2, then the area of the shadowed part swept by line segment BC during the above rotation is (result in terms of π) | $\frac{\pi }{4}$ |
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dfe75b51-66a4-4270-b420-0433662ba70f | For a sample data set with a size of $$20$$, the grouped frequency distribution is shown in the following table: What is the frequency of the sample data falling in the interval $$[10,40)$$? | $$\number{0.45}$$ |
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37e6e357-871e-432b-a362-c9c45af81bb4 | As shown in the figure, divide a circular turntable into four sector areas A, B, C, and D in the ratio 1:2:3:4. After the turntable spins freely and stops, what is the probability that the pointer lands in area C? | $\frac{3}{10}$ |
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cafe008f-8a59-4b87-9d09-08651ddaa4ea | The diagram is a flowchart of an algorithm, what is the output value of $S$? | 22 |
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44b121e2-84c3-43f4-9917-b5a532510467 | In the rectangle $ABCD$, it is known that $AB=2$, $AD=4$, and $E$, $F$ are on the line segments $AD$, $BC$ respectively, such that $AE=1$ and $BF=3$. As shown in the figure, the quadrilateral $AEFB$ is folded along $EF$ to form ${A}'EF{B}'$. During the folding process, the maximum value of the tangent of the dihedral angle ${B}'-CD-E$ is ?. | $\frac{3\sqrt{2}}{5}$ |
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