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4d613249-84c9-401f-bd27-ac6869e87a8b | Given the function $f\left( x \right)=2\text{sin}\left( \omega x+\varphi \right)$ ($\omega > 0$, $\left| \varphi \right| < \frac{\pi }{2}$), the graph is shown as in the figure. Among them, $A\left( \frac{5\pi }{6},-2 \right)$ and $B\left( \frac{19\pi }{12},0 \right)$, then the function $f\left( x \right)=$ . | $2\sin (2x-\frac{\pi }{6})$ |
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145c10ec-1d80-4c8a-847e-cc6bc8fc6efc | As shown in the figure, the ellipse $$C$$: $${x^2\over a^2}+{y^2\over4}=1$$ (where $$a>2$$) and the circle $$O$$: $$x^2+y^2=a^2+4$$. The left and right foci of the ellipse $$C$$ are $$F_1$$ and $$F_2$$, respectively. A line $$l$$ passes through a point $$P$$ on the ellipse and the origin $$O$$, intersecting the circle $$O$$ at points $$M$$ and $$N$$. If $$|PF_1|\cdot|PF_2|=6$$, then the value of $$|PM|\cdot|PN|$$ is ______. | $$6$$ |
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515c1262-9e5d-4411-8b80-de2cd465e11a | $$6$$ balls labeled with different numbers are placed in a column-shaped paper box with lids at both ends, as shown in the figure. If a ball is randomly drawn from one end for $$6$$ times and arranged in sequence, the number of different arrangements is ___ (fill in the blank with a number). | $$32$$ |
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256989c1-5212-40b6-adde-6fa2d2cc452c | As shown in the figure, in triangle ABC, ∠C = 90°, point D is on BC, DE ⊥ AB at E, and AE = EB, DE = DC, then the measure of ∠B is. | 30° |
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31c2d142-fc32-4431-8442-9d39e439ef1a | A TV transmission tower in a city is built on a small hill outside the city. As shown in the figure, the height of the hill $$BC$$ is about $$30$$ meters. There is a point $$A$$ on the ground, and the distance between points $$A$$ and $$C$$ is approximately $$50$$ meters. From point $$A$$, the angle of elevation to the TV transmission tower $$ ( \angle CAD) $$ is approximately $$45^{ \circ }$$. Then the height of the transmission tower $$CD$$ is ___ meters. | $$250$$ |
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a7b2f611-b7f3-4b34-8eea-5766a42f828b | In order to understand the growth of a certain economic forest, a random sample of 60 trees was taken. The circumference at the base of these trees (in cm) is distributed within the interval [80, 130], as shown in the frequency distribution histogram. How many of the 60 sampled trees have a base circumference of less than 100 cm? | 24 |
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24ce7a8e-2004-4377-8595-babd66223d8d | 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 surfaces of the cylinder and the generatrix. Let the volume of the cylinder $$O_{1}O_{2}$$ be $$V_{1}$$, and the volume of the sphere $$O$$ be $$V_{2}$$. Then the value of $$\dfrac{V_{1}}{V_{2}}$$ is ___. | $$\dfrac{3}{2}$$ |
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6ebe2caa-0fb1-4b3b-9453-eef91f5cc6b7 | As shown in the figure, AB is the diameter of circle O, and the chord CD is parallel to AB. If angle ABC = 65°, then angle ACD = ______. | 25°. |
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49692190-8264-4d31-8ce6-4497fb12373d | Given the probability distribution of the random variable $$X$$ is and $$E\left ( X\right ) =1.1$$, find $$D\left ( X\right ) =$$___. | $$0.49$$ |
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a6948233-6f51-485f-b4dd-3a5d3121268a | The following pseudocode represents a value-finding problem of a piecewise function, as illustrated by the graph. The condition that should be filled in the '△' is ___. | $$x \geqslant 2$$(or $$x > 2$$) |
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42aaccf7-211d-4666-a867-a14d609b6935 | As shown in the figure, point $$P$$ is any point on the graph of the inverse proportional function $$y={6\over x}$$. Draw vertical lines through point $$P$$ to each of the coordinate axes, forming the rectangle $$OAPB$$ with the axes. Point $$D$$ is any point inside rectangle $$OAPB$$. Connect $$DA$$, $$DB$$, $$DP$$, and $$DO$$. Then, the area of the shaded part in the figure is ______. | $$3$$ |
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bfa06e5c-2506-42e1-87e8-9db66b4bce81 | A sample of 150 students from a certain school was taken. After measuring the students' heights, the histogram of height distribution is shown in the figure. It is known that the school has 1,500 students. Estimate the number of students whose height is between 160 cm and 165 cm in the school, which is approximately ______ people. | From the problem, it is known that there are 30 people with heights between 160-165 cm in the sample of 150 students. Thus, the frequency is 30/150 = 0.2. Therefore, among the 1,500 students, the number of students with heights between 160 cm and 165 cm is approximately 1,500 * 0.2 = 300 people; Thus, it is 300. |
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be899930-b063-4c1e-9c19-c5557bd22424 | In the diagram shown (three regions $$A$$, $$B$$, $$C$$), a bean is randomly dropped. The probability that the bean falls in the ______ region is the greatest (fill in $$A$$, $$B$$, or $$C$$). | $$A$$ |
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9e4fcaed-7e98-4b37-be5f-93c6bffe7ec3 | Given that △ABC ∼ △A$_{1}$B$_{1}$C$_{1}$ and the similarity ratio is, the area of △ABC is 8, then the area of △A$_{1}$B$_{1}$C$_{1}$ is______. | 18. |
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65fb26bc-a83d-4add-b2b0-c93d2b912c6a | As shown in the figure, line $$a$$ passes through vertex $$A$$ of the square $$ABCD$$. Lines are drawn through vertices $$B$$ and $$D$$ respectively, such that $$DE\bot a$$ at point $$E$$, and $$BF\bot a$$ at point $$F$$. If $$DE=4$$, $$BF=3$$, then the length of $$EF$$ is ___. | $$7$$ |
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25fc4549-dbf8-40a9-ab7b-6c06be2d30f8 | As shown in the figure, it is known that the graphs of functions $$y=2x+b$$ and $$y=ax-3$$ intersect at point $$P\left(-2,-5\right)$$. According to the graph, the solution to the equation $$2x+b=ax-3$$ is ___. | $$-2$$ |
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bfd1f080-a740-4338-a361-3361a3738e19 | Given the ellipse $$C$$: $$\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1(a > b > 0)$$ with eccentricity $$\dfrac{\sqrt{2}}{2}$$, $$F_{1}$$ and $$F_{2}$$ are its left and right foci, respectively. $$A$$ and $$B$$ are the right vertex and the top vertex of the ellipse, respectively. $$PF_{1}$$ is perpendicular to the x-axis and intersects the ellipse at point $$P$$ (as shown in the figure). If the straight line $$PF_{2}$$ intersects the ellipse $$C$$ at another point $$Q$$, and the area of quadrilateral $$OAQB$$ is $$\dfrac{16}{5}$$, then the standard equation of the ellipse $$C$$ is ___. | $$\dfrac{x^{2}}{8}+\dfrac{y^{2}}{4}=1$$ |
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e4ec7b0c-9f16-47ac-90b5-6170a89920b5 | (1) Given the function $$f\left ( x\right )=x $$, the graph is shown as in the figure, fill in the blank: 1. The graph trends ___ from left to right (fill in "increasing" or "decreasing"). 2. On the domain of the function $$f\left ( x\right ) $$, as $$x$$ increases, the value of $$f\left ( x\right ) $$ gradually ___ (fill in "increases" or "decreases"), the function $$f\left ( x\right ) $$ is a ___ function on its domain (fill in "increasing" or "decreasing"). (2) Given the function $$f\left ( x\right )=-2x+1$$, the graph is shown as in the figure, fill in the blank: 1. The graph trends ___ from left to right (fill in "increasing" or "decreasing"). 2. On the domain of the function $$f\left ( x\right ) $$, as $$x$$ increases, the value of $$f\left ( x\right ) $$ gradually ___ (fill in "increases" or "decreases"), the function $$f\left ( x\right ) $$ is a ___ function on its domain (fill in "increasing" or "decreasing"). | (1)1. increasing 2. increases increasing (2)1. decreasing 2. decreases decreasing |
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3a0fef71-9621-40d5-9e10-86039a491851 | The purchase price of a certain brand of bicycle is $400, and the listed selling price is $500. The shop is preparing to offer a discount on the sale, but it must ensure that the profit margin is not less than $5%$. What is the maximum discount that can be offered? | 8.4 |
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ff5f68c7-31a4-4b72-8d1d-03d1394fcdb4 | The graph of a linear function y = kx + b is shown in the figure. When x ______, y < 4. | >-2 |
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dff9a53c-ee62-4ddf-9fec-9848adf5d156 | The three views of a certain quadrilateral pyramid are shown in the diagram, then the volume of this quadrilateral pyramid is___. | $$3$$ |
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22f96c8c-2673-4939-8602-24d154e166af | The two real numbers $a, b$ correspond to points on the number line as shown in the figure. Determine whether $3-\left| a \right|$ is greater than or less than $3-\left| b \right|$ (fill in “>” or “<”). | $ < $ |
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39a4a149-bc16-4bdd-99b8-7f8a81427e84 | As shown in the figure, ∠ACD is an exterior angle of triangle ABC. If ∠ACD=125°, ∠A=75°, then find ∠B=°. | 50 |
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39c83a46-8a84-42cb-9dda-25e879c63349 | As shown in the figure, in \( \triangle ABC \), \( \angle CAB = 70^{\circ} \). In the same plane, rotate \( \triangle ABC \) around point \( A \) so that point \( B \) falls on point \( B' \) and point \( C \) falls on point \( C' \). If \( C{C}' \parallel AB \), then \( \angle BAB' = □ \). | 40° |
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9cccb4c9-8b3e-41a1-9e26-a8ed3b5598df | As shown in the figure, the first (1), (2), (3), (4)... have 1, 5, 11, 19... 'small squares' respectively. How many 'small squares' are there in the 10th (10) figure? | 109 |
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7858c5df-6b28-4937-89cb-5d6aae737f29 | As shown in the figure, the sector is the unfolded side view of a conical frustum. If $$\angle AOB=120^\circ$$ and the length of arc $$AB$$ is $$12\mathrm \pi\ {\mathrm{cm}}$$, then the lateral surface area of the conical frustum is ___$${\mathrm{cm}}^2$$. | $$108\mathrm \pi$$ |
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1e631009-82bc-476a-bd11-7c9d3271a258 | Arrange the positive integers in the order shown in the figure. If an ordered real number pair $$\left ( n,m\right ) $$ represents the m-th number from left to right in the n-th row, for example, $$\left ( 4,3\right ) $$ represents the number $$9$$, then the real number represented by $$\left ( 7,2\right ) $$ is___. | $$23$$ |
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6499816c-44ec-490c-9dad-a6ebfb166a9b | As shown in the figure, in $\Delta ABC$, $\angle C=90{}^\circ$, $AC=BC$, $BD$ bisects $\angle ABC$ and intersects $AC$ at point $D$. $DE\perp AB$ at point $E$. If $AB=10cm$, find the perimeter of $\Delta ADE$ in cm. | 10. |
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c45377fc-b14d-4320-82d6-67d2bf2eacf6 | As shown in the figure, in $\vartriangle ABC$, $AB=AC$, $\angle A={{40}^{{}^\circ }}$. The perpendicular bisector of AB intersects AB at point D and intersects AC at point E. Connect BE, then the measure of $\angle CBE$ is ( ) | 30${}^\circ $ |
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a52a5115-7d8f-4a81-a853-7de0601ed24f | As shown in the figure, points $$A$$, $$B$$, and $$C$$ are on the same straight line, $$\angle A=\angle C=90^{\circ}$$, $$AB=CD$$, please add an appropriate condition: ___, so that "$$HL$$" can be used to determine that $$Rt\triangle EAB\cong Rt\triangle BCD$$. | $$EB=BD$$ |
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e5bf0b2c-45f5-49b0-9326-6a15a00c4d31 | As shown in the diagram, in $\Delta ABC$, $\angle A=60{}^\circ $, $\angle B=50{}^\circ $, $D$ and $E$ are two points on $AB$ and $AC$, respectively. Connect $DE$ and extend it to intersect the extension line of $BC$ at point $F$. At this time, $\angle F=35{}^\circ $, then the measure of $\angle 1$ is ? | 145° |
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f7c72041-7036-4dc4-92bb-46b771e39d3c | As shown in the figure, $$OA$$ and $$OC$$ are radii of circle $$\odot O$$, point $$B$$ is on $$\odot O$$, connect $$AB$$, $$BC$$. If $$\angle ABC=40^{\circ}$$, then $$\angle AOC=$$ ___ degrees. | $$80$$ |
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4a24b13e-1e23-4de7-9b0a-9845e70672cc | As shown in the figure, the right-angled isosceles triangle represents the orthographic projection of a horizontal planar figure. The area of this planar figure is ___. | $$2\sqrt{2}$$ |
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eadd15da-0e4c-4223-8047-10455ab295a1 | As shown in the figure, $$AA_{1}$$ and $$BB_{1}$$ intersect at point $$O$$, and $$AB\parallel A_{1}B_{1}$$, also $$AB=\dfrac{1}{2}A_{1}B_{1}$$. If the diameter of the circumcircle of $$\triangle AOB$$ is $$1$$, then the diameter of the circumcircle of $$\triangle A_{1}OB_{1}$$ is___. | $$2$$ |
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dfcd55b9-f9dd-4f72-b44f-e8ac4793d9b9 | From one corner of a cube with a side length of $$2$$, a smaller cube with a side length of $$1$$ is removed. The resulting piece is as shown in the figure, and the surface area of this piece is ___. | $$\number{24} $$ |
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c1fbda08-e3d9-44a1-8f32-1c61d9ab9bf5 | As shown in the figure, $$\triangle ABC$$ and $$\triangle DOE$$ are similar figures. $$A\left ( 0,3\right ) $$, $$B\left ( -2,0\right ) $$, $$C\left ( 1,0\right )$$, $$E\left ( 6,0\right ) $$, then $$D$$(___), the coordinates of the center of similarity $$M$$ of $$\triangle ABC$$ and $$\triangle DOE$$ are(___). | $$4$$,$$6$$ $$-4$$,$$0$$ |
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a6b4bc99-a543-41f3-a52d-ac2ecd318946 | As shown in the figure, it is the frequency distribution bar chart of 60 students in a class participating in the 2011 New Curriculum Island Mathematics Academic Level Test after tidying up their scores (the scores are all integers). According to the figure, the number of students in the class who passed is ______. | 45 |
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70830e7e-20a3-4c95-b6c3-6253d42dfb45 | As shown in the figure, when a student who is 1.6 m tall stands upright at the tip of the shadow of the flagpole, it is measured that the lengths of the shadows of the student and the flagpole are 1.2 m and 9 m, respectively. Find the height of the flagpole, which is ______. | 12 meters. |
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4dfb00e6-e658-4485-8044-92c4ee4db654 | As shown in the figure, it is known that $$AB \parallel CD$$, $$CE$$ bisects $$\angle ACD$$, and intersects $$AB$$ at $$E$$, $$\angle A=118^{ \circ }$$. Then $$\angle AEC=$$___. | $$31$$ |
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0f6438dc-75c7-4867-bce3-922bda57fd83 | As shown in the figure: (1) The base of this prism is a ______ shape. (2) This prism has ______ lateral faces, and the shape of the lateral faces is ______. (3) The number of lateral faces equals the number of edges on the base ______. (4) This prism has ______ lateral edges, with a total of ______ edges. (5) If CC' = 3cm, then BB' is ______ cm. | (1)3;(2)3, quadrilateral;(3)equal;(4)3, 9;(5)3. |
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57ac1aef-1135-4f8a-852c-ed4ccd9d223a | Below is a statistical table of a certain mall's business revenue for each quarter in the year 2002. 1. Please calculate the average monthly business revenue for the year 2002, in ten thousand yuan. 2. Predict the average monthly business revenue for the year 2003, assuming a 5% increase; the average monthly business revenue for 2003 can reach ______ ten thousand yuan. | 254.4
267.12 |
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7a205c55-6e2f-421f-a750-15ef86c95f97 | As shown in the figure, in triangle ABC, D, E, and F are the midpoints of sides BC, AD, and CE respectively, and the area of triangle ABC is 16 cm². Find the area of triangle BEF. | 4 cm² |
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0e2dfcd4-032a-476d-9123-642790add88c | As shown in the figure, in $\vartriangle ABC$, the perpendicular bisector of AB intersects A at point D and intersects BC at point E. If $BC=6$, $AC=5$, then the perimeter of $\vartriangle ACE$ is . | 11. |
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f12cc35c-e207-48a1-a268-281cd28b84b3 | As shown in the figure, the horizontal surface has a gray sector $$OAB$$ with an area of $$120\pi \text{c}{{\text{m}}^{2}}$$, where the length of $$OA$$ is $$12\text{cm}$$, and $$OA$$ is perpendicular to the ground. If the sector rolls to the right without sliding until point $$A$$ touches the ground again, the path length moved by point $$O$$ is ______ . | 2π |
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9021adc4-cecf-4428-8c7d-342f6a157f7a | As shown in the figure, the little monkey first welds an iron wire into a rectangular frame and then pastes paper boards on the six sides to create a paper box. So, how many centimeters of iron wire and how many square centimeters of paper board are needed? (Ignore any wastage) | 96
376 |
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7325cf9e-a46d-4090-815d-ed66b3451a97 | As shown in the figure, in the quadrilateral $$\square ABCD$$, $$\overrightarrow{AB}=\overrightarrow{a}$$, $$\overrightarrow{AD}=\overrightarrow{b}$$, $$\overrightarrow{AN}=3\overrightarrow{NC}$$, and $$M$$ is the midpoint of $$BC$$. Find $$\overrightarrow{MN}=$$___ (expressed in terms of $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$). | $$\dfrac{1}{4}\left ( \overrightarrow{b}-\overrightarrow{a}\right ) $$ |
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69fe755c-968b-4c81-bc12-095f27444d39 | As shown in the figure, in $\vartriangle ABC$, $D$ is the midpoint of $BC$, and $E,F$ are two trisection points on $AD$. If $\overrightarrow{BE}\cdot \overrightarrow{CE}=\frac{7}{8}$ and $BC=\frac{\sqrt{26}}{2}$, then $\overrightarrow{BF}\cdot \overrightarrow{CF}=$. | -1 |
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2a232391-5ca6-40c1-958b-80d8abd30529 | The cross-section of a certain tunnel is designed as a half-ellipse shape with the maximum height $h$ of 6 meters (as shown in the diagram). The road is designated as a dual carriageway with a total width of $8\sqrt{7}$ meters. If the height restriction for vehicles passing through is not to exceed 4.5 meters, then the minimum design width $d$ of the tunnel should be at least meters. | 32 |
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811d50e4-acf7-499f-858f-785528658293 | As shown in the figure, △ABC is an equilateral triangle, AE=CD, AD and BE intersect at point P, BQ⊥AD at Q. If PQ=4 and PE=1, find the length of AD. | 9. |
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2070bfa3-06bf-4c61-b0a3-66b1cd3cb55e | As shown in the figure, in triangle ABC, C$_{1}$ and C$_{2}$ are trisection points on side AB, A$_{1}$, A$_{2}$, and A$_{3}$ are quartiles on side BC. Line AA$_{1}$ intersects CC$_{1}$ at point B$_{1}$, line CC$_{2}$ intersects C$_{1}$A$_{2}$ at point B$_{2}$. Denote the areas of triangles AC$_{1}$B$_{1}$, C$_{1}$C$_{2}$B$_{2}$, and C$_{2}$BA$_{3}$ as S$_{1}$, S$_{2}$, and S$_{3}$, respectively. If S$_{1}$+S$_{3}$=6, find S$_{2}$. | $\frac{8}{3}$ |
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5320de33-2495-4d81-8e92-35e1654e35c7 | As shown in the figure, in the grid, each side of the small square is 1. The points $A, B, C$ are on the grid points, then the tangent value of $\angle ABC$ is | $\frac{1}{2}$ |
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4c68b841-e266-4e11-91af-fdf846426efa | As shown in the figure, points A and B are separated by a pond. A point C is chosen outside of AB, and AC and BC are connected. Find the midpoints M and N of AC and BC, respectively. If it is measured that MN = 20m, then the distance between points A and B is ______ m. | 40m. |
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35ceb1f5-5984-4389-af3d-982e22342506 | An algorithm's flowchart is shown in the figure, and the output value of $a$ is. | 9 |
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60f0db97-b53e-4806-a75c-a4a8c97d31b7 | As shown in the figure, points $$A$$ and $$C$$ are on the graph of the function $$y=\dfrac{\sqrt{3}}{x}(x < 0)$$, while points $$B$$ and $$D$$ are on the x-axis. Triangles $$\triangle OAB$$ and $$\triangle BCD$$ are both equilateral triangles. What are the coordinates of point $$C$$? | $$(-1-\sqrt{2},\sqrt{3}-\sqrt{6})$$ |
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3dce00d4-b7c5-4dd8-9e11-230c7d5dad7a | As shown in the figure, in $$\triangle ABC$$, $$\angle C=90^{\circ}$$, $$\angle A=30^{\circ}$$, $$BD$$ is the angle bisector of $$\angle ABC$$. If $$AB=6$$, then the distance from point $$D$$ to $$AB$$ is ___. | $$\sqrt{3}$$ |
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29486612-6016-40c8-9f7d-2ab1108e8820 | As shown in the figure, a cylinder with a volume of 48π and a generatrix of length 3 is intersected by a plane that is not parallel to the base, forming an ellipse. The length of the minor axis of this ellipse is ______. | Let the radius of the base of the cylinder be r, then from the problem statement, the length of the minor axis of the ellipse is 2r. Using the volume formula 48π = πr$^{2}$×3, we find r = 4. Therefore, the length of the minor axis is 8, so it is 8. |
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2584bd67-de95-4098-a1eb-25108fbcc5a4 | As shown in the figure, the quadrilateral DEFG is an inscribed rectangle within the triangle ABC, where points D and G lie on sides AB and AC respectively, and points E and F lie on side BC. Given that DG = 2DE, AH is the height of triangle ABC, BC = 20, and AH = 15, what is the perimeter of the rectangle DEFG? | 36 |
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e57ecf77-6fd5-434b-bd5f-2405e3e94bdd | The diagram below is a certain calculation program on a calculator. If the initial input is $x = -2$, the final output result is. | $-17$ |
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fd23563f-f268-49cd-9f3f-f692e2d21861 | As shown in the figure, point C is a point on the line segment AB, CB = a, points D and E are the midpoints of AC and AB, respectively. Find the length of the line segment DE (expressed in terms of a). | $\frac{a}{2}$ |
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cc071576-d49f-46fc-9ff9-0ded68a15f49 | As shown in the figure, in the Cartesian coordinate plane, the coordinates of point E are ______ . | (1, 2) |
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9227a402-550e-48c0-81e0-366a4e7aed1f | As shown in the right figure, the number of triangles is ____/____ (fill in the fraction) of the total number of figures, and the colored triangles are ____/____ (fill in the fraction) of the number of triangles. | 9
19
5
9 |
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d64a2751-ca51-4fb0-bb79-c64c12adfedd | A student looked at the following statistical diagram and said: "This diagram shows that the permanent population of City A increased significantly from 2015 to 2016." Do you think this student's statement is reasonable? ___ (Fill in "reasonable" or "unreasonable"), and your reason is ___. | Unreasonable. The reason is not unique, such as the increase of 24,000 compared to 21,705,000 does not manifest a "significant" increase. |
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9cc362cd-b70d-4b8d-a483-d56177dbdc23 | As shown in the figure, in the parallelogram $$ABCD$$, $$AC$$ and $$BD$$ intersect at point $$O$$. $$E$$ is the midpoint of segment $$AO$$. If $$\overrightarrow{BE}= \lambda\ \overrightarrow{BA}+ \mu\ \overrightarrow{BD}( \lambda , \mu \in \mathbf{R})$$, then $$\lambda + \mu=$$___. | $$\dfrac{3}{4}$$ |
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d9f02337-c845-47ae-8290-705d5cfa0e24 | As shown in the figure, $$AD \parallel BC$$, $$AB \parallel DC$$, $$AC$$ intersects with $$BD$$ at point $$O$$, and line $$EF$$ passes through point $$O$$, intersecting $$AD$$ and $$BC$$ at points $$E$$ and $$F$$, respectively. How many pairs of congruent triangles are there in the figure? | $$6$$ |
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bf0e60b8-9cd7-431a-b141-5f6f0d462e54 | As shown in diagrams (A) and (B), there are two squares with equal side lengths. In diagram A, an arc is drawn inside the square with the side as the radius, connecting the diagonal lines. In diagram B, semicircles are drawn on each side of the square. The areas of the shaded parts are noted as $$S_A$$ and $$S_B$$, respectively. Then the relation between $$S_A$$ and $$S_B$$ is: $$S_A$$ ______ $$S_B$$. (Fill in with “$$>$$”, “$$=$$”, or “$$<$$”.) | = |
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627eed50-e477-4db9-955d-a18d074ce525 | Choose any $$4$$ vertices on a cube, they may form $$4$$ vertices of the following polyhedra, these polyhedra are ___. (Write the serial numbers of all correct conclusions)1. Rectangle; 2. Quadrilateral that is not a rectangle; 3. A tetrahedron with three faces as isosceles right triangles and one face as an equilateral triangle; 4. A tetrahedron with each face as an equilateral triangle; 5. A tetrahedron with each face as a right triangle. | 1.3.4.5. |
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5d0f9964-50ab-4194-92f4-3c2986a71c0a | The school decided to make uniforms for seventh-grade students, with four sizes available: small, medium, large, and extra-large. A random survey of the heights of 100 students was conducted, yielding the following frequency distribution table for height: It is known that there are 800 seventh-grade students in the school, so how many medium-sized uniforms should be ordered? | 360 sets |
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51355fdf-ef99-450a-9625-2f024e12a84f | The image illustrates the forest area statistics of four countries—China, the United States, Japan, and Australia—in 1996. (Unit: km) (1) The tree height in the image represents ______. (2) The forest area of the United States is approximately twice the forest area of ______. | (1) The forest area of a country; (2) Since the forest area of the United States is 295,000 and the forest area of Australia is 146,000, 295,000 ÷ 146,000 ≈ 2, so the forest area of the United States is approximately twice the forest area of Australia. |
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fd530e4d-e003-4945-aa10-d67957d18f40 | In the figure below, points $$P$$, $$Q$$, $$R$$, $$S$$ are the midpoints of the four edges of a cube, respectively, then $$PQ$$ and $$RS$$ are skew lines ___. (Fill in the serial number). | 3. |
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0668823c-43ab-4ab1-9445-aecb4281abb2 | Please complete the following questions (1) The population of a certain city's urban area is $$a$$ ten thousand, and the green area is $$b$$ ten thousand $$\unit{m^{2}}$$, then the average green area per person is ___ $$\unit{m^{2}}$$; (2) In a certain city, the average annual income per person was $$n$$ yuan 5 years ago. It is estimated that this year's average annual income per person is 2 times more than 5 years ago plus 500 yuan. Then this year's average annual income per person will be ___ yuan; (3) As shown in the figure, the volume of this cuboid is ___, and the surface area is ___. | (1)$$\dfrac{b}{a}$$ (2)$$(2n+500)$$ (3)$$abc$$ $$2ab + 2bc + 2ac$$ |
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e0ca7beb-27e0-4add-816f-a6bb3ed5d7d5 | As shown in the figure, point $$A$$ is a point on the graph of the inverse proportional function $$y=\dfrac{k}{x}$$. Line segment $$AB$$ is perpendicular to the y-axis at point $$B$$. Point $$P$$ is a point on the x-axis. The area of $$\triangle APB$$ is $$5$$. Then the value of $$k$$ is ___. | $$10$$ |
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ca95cd8c-7099-4847-9e52-cc479aded9ae | As shown in the figure, the graphs of three proportional functions correspond to expressions as follows: 1. \(y=ax\), 2. \(y=bx\), 3. \(y=cx\). Arrange \(a\), \(b\), \(c\) from smallest to largest and connect them with the "\(<\)" symbol as ___. | \(a < c < b\) |
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638389aa-c3b2-4c92-af9c-8c9695cf9c77 | Among the following objects: which are the same object ______ (fill in the serial number of the same shapes) | (1)(3). |
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3e84886d-af60-41b4-874e-4b766a472209 | As shown in the figure, in the equilateral triangle $ABC$, $D$ is the midpoint of $AB$, $DE\bot AC$ at point $E$, $EF\bot BC$ at point $F$, and it is known that $AB=8$, then the length of $BF$ is . | $5$ |
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266b993a-d931-4af9-b270-ce8c12d539ee | As shown in the figure, $$AD \parallel EG \parallel FH \parallel BC$$, $$E$$ and $$F$$ divide $$AB$$ into three equal parts, $$G$$ and $$H$$ are on $$DC$$, with $$AD=4$$ and $$BC=13$$. Find $$EG=$$___, $$FH=$$___. | $$7$$ $$10$$ |
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2298f587-bdd1-4e48-853d-c478f366c0f1 | As shown in the figure, the area of triangle ABC is 1. The median line AD$_{1}$ divides triangle ABD$_{1}$ such that the area is S$_{1}$. The median line AD$_{2}$ along triangle AD$_{1}$C divides triangle AD$_{1}$D$_{2}$ such that the area is S$_{2}$. According to the aforementioned method, the areas of the triangles taken subsequently are S$_{3}$, S$_{4}$, ... S$_{n}$. What is the sum of the areas of the triangles taken? | $\frac{{{2}^{n}}-1}{{{2}^{n}}}$ |
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340c462b-9efd-4dd4-a2a2-6003e01cde4d | As shown in the figure, in the parallelogram $ABCD$, point $E$ is the midpoint of side $CD$. Line segments $AE$ and $BD$ intersect at point $F$. If $\overrightarrow{BC}=\overrightarrow{a}, \overrightarrow{BA}=\overrightarrow{b}$, express $\overrightarrow{DF}$ using $\overrightarrow{a}$ and $\overrightarrow{b}$. | $- \frac{1}{3} \overrightarrow{a} - \frac{1}{3} \overrightarrow{b}$ |
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43013755-4a48-43f9-8071-fd19d8450c3b | The three-view diagram of a rectangular prism is shown in the figure. If the top view is a square, then the volume of this rectangular prism is ___. | $$36$$ |
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95480254-b34e-4bac-8adc-c3ffce2a1a7e | By randomly surveying 110 different college students about their interest in a certain sport, the following contingency table was obtained: Use the formula $$\chi ^{2}=\dfrac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)}$$ to calculate $$\chi ^{2}=\dfrac{110\times (40\times 30-20\times 20)^{2}}{60\times 50\times 60\times 50}\approx 7.8$$. Refer to the attached table: , the correct conclusion sequence number obtained is ___. 1. It is concluded with more than 99% confidence that "interest in this sport is related to gender"; 2. It is concluded with more than 99% confidence that "interest in this sport is not related to gender"; 3. Under the premise that the probability of error does not exceed 0.1%, it is concluded that "interest in this sport is related to gender"; 4. Under the premise that the probability of error does not exceed 0.1%, it is concluded that "interest in this sport is not related to gender". | 1. |
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ec666e6f-098a-411b-b976-bbdd57cc9b90 | As shown in the figure, the square $$ABCD$$ is divided into two smaller squares and two rectangles. If the areas of the two smaller squares are $$4$$ and $$5$$ respectively, then the area of the square $$ABCD$$ is ___. | $$9+4\sqrt{5}$$ |
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d3579257-bddd-4f8f-8d47-76fe5dad3ce7 | According to the image, the time is ______ hour(s) and ______ minute(s), and the angle formed by the hour hand and the minute hand is ______. After 40 minutes, the time will be ______ hour(s) and ______ minute(s). | 7
25
acute
8
5 |
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1433ffdc-59df-4fbc-a203-279b631af5fc | As shown in the figure, along the path shaped as the letter "Tian" (田), walk from point $$A$$ to point $$N$$, and only walk towards the right or downward, choosing a path randomly. What is the probability of passing through point $$C$$? | $$\dfrac{2}{3}$$ |
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0e6ed77d-ee80-4735-aed6-2d2f6dfe9345 | A certain market conducted a 6-day trial sale for a certain product (costing $$5$$ yuan/piece), and obtained the following data: . It was found that the sales volume $$y$$ (pieces) has a linear relationship with the unit price $$x$$ (yuan) and the regression linear equation $$\hat {y}=\hat {b}x+\hat {a}$$ (where $$\hat {b}=-20$$ and $$\hat {a}=\overline{y}-\hat b\overline{x}$$). Therefore, in order to achieve maximum profit in the future, the unit price of the product should be ___ yuan. | $$\dfrac{35}{4}$$(or fill in $$8.75$$) |
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9d74c609-a85d-49e0-8a08-d18aefc2131d | The axial cross-section dimensions of a cylindrical container are shown in the figure. Inside the container is a solid sphere with its diameter exactly equal to the height of the cylinder. Now, water is used to fill the container. Then, the sphere is removed (assuming the density of the sphere is greater than that of water and no water is lost during the process), and the height of the water surface in the container after the sphere is removed is cm. | $\frac{25}{3}$ |
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c71af849-2b5e-4022-bbf6-f05860e12aa8 | Given the probability distribution of a discrete random variable $$X$$ as shown in the table below, if $$E\left (X \right ) =0$$ and $$D\left (X \right ) =1$$, find $$a=$$___, $$b=$$___. | $$\dfrac{5}{12}$$ $$\dfrac{1}{4}$$ |
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5f97ae08-550e-4c7a-8f74-1b30255bcad7 | According to the pseudo code shown in the diagram, the output value of $$S$$ is ___. | $$20$$ |
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63d7cd25-66f2-465d-b481-63497b164b52 | Given the figure, in triangle ABC, E is on BC, D is on BA, and through E, EF is perpendicular to AB at F. ∠BAC=90°, ∠1=∠ACD, AE=CD, EF= $\frac{4}{3}$, find the length of AD. | $\frac{4}{3}$ |
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997d907f-d4fa-4247-9e9b-f89815421029 | As shown in the figure, there is an object AB of length 21 cm located 24 cm in front of a small hole O. The image A'B' formed by object AB passing through the small hole O falls exactly on a screen 16 cm behind the small hole. What is the length of the image A'B' in cm? | 14 |
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987b56d9-1684-496b-a474-96453e124711 | As shown in the diagram, Team A and Team B both consist of 7 members who participated in the 'Safety Knowledge Contest'. The mode of Team B's scores is $\text{m}+81$. If one member is chosen from each team, what is the probability that the scores of the two selected members are the same? | $\frac{4}{49}$. |
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34a1463e-3813-4dd8-a46f-57b23dcf83f7 | A problem concerning 'the simultaneous growth of pine and bamboo' is mentioned in the book 'Mathematical Enlightenment' by the Chinese mathematician Zhu Shijie of the Yuan Dynasty: The pine tree grows to five chi, and the bamboo grows to two chi. The pine tree grows by half its height each day, and the bamboo grows by its full height each day. On which day will the pine and bamboo be of equal height? The problem is to find the day when the height of the pine tree and the bamboo are the same, given that the initial height of the pine tree is 5 chi and that of the bamboo is 2 chi, with the growth rates mentioned above. A flowchart is based on this idea: if the input $a=5$, $b=2$, then the result of the output for $n$ is | 4 |
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6e09cf03-73c4-457f-8749-aed4f33d4bd6 | As shown in the figure, in a two-dimensional Cartesian coordinate system, there are several points with integer x-coordinates, arranged in the order shown by the arrow “$$\rightarrow$$”, such as $$(1,0)$$, $$(2,0)$$, $$(2,1)$$, $$(1,1)$$, $$(1,2)$$, $$(2,2)$$$$\cdots$$ According to this pattern, the x-coordinate of the 60th point is ______. | $$5$$ |
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6dc08e8c-1278-4339-9752-5e989bc04d91 | As shown in the figure, it is known that $$\triangle ABE\cong \triangle ACF$$, $$\angle E= \angle F=90^{ \circ }$$, $$\angle CMD=70^{ \circ }$$, then $$\angle 2=$$___. | $$20^{\circ}$$ |
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c29a713c-f525-4b98-91b2-0e213ed8ddf4 | As shown in the figure, in a plane rectangular coordinate system, there are two points $A\left( 6,0 \right)$ and $B(6,3)$. Using the origin $O$ as the center of similarity, with a ratio of $\frac{1}{2}$, line segment $AB$ is contracted to line segment $CD$, where point $C$ corresponds to point $A$ and point $D$ corresponds to point $B$. Also, $CD$ is on the right side of the y-axis. Then the coordinates of point $D$ are: | $\left( 3,\frac{3}{2} \right)$ |
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a7558bef-30c1-4e7e-a3f2-220002296d68 | As shown in the figure, in the rhombus ABCD with a side length of 6, ∠DAB=60°, taking point D as the center of the circle, the height DF of the rhombus as the radius is used to draw an arc, intersecting AD at point E and intersecting CD at point G. The area of the shaded part in the figure is | $18\sqrt{3}-9\pi $ |
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f0ce3448-fd38-4981-97e3-84b397dc42eb | $$AE$$ is the angle bisector of $$\triangle ABC$$, $$AD\perp BC$$ at point $$D$$. If $$\angle BAC=130^{\circ}$$, $$\angle C=30^{\circ}$$, then the measure of $$\angle DAE$$ is ___. | $$5^{\circ}$$ |
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b88918f3-34b2-4362-8833-200dd4d52e44 | As shown in the figure, it is known that $$OE$$ bisects $$\angle AOB$$, $$OD$$ bisects $$\angle BOC$$, $$\angle AOB$$ is a right angle, and $$\angle EOD=70^{\circ}$$. Find $$\angle BOC=$$___. | $$50^{\circ}$$ |
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a8efd500-c0df-478c-9fad-ca69c6c51d2b | As shown in the figure, if point $$P$$ is on the graph of the inverse proportional function $$y=-\dfrac{3}{x}(x < 0)$$, draw $$PM \perp x$$-axis at point $$M$$, and $$PN \perp y$$-axis at point $$N$$, then the area of rectangle $$PMON$$ is ___. | $$3$$ |
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8f813c2e-ed3f-4aed-ba92-643ed35dc74d | As shown in the figure, this flowchart is used to calculate the value of $$\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{6}+\cdots+\dfrac{1}{20}$$. The condition that should be filled in the decision box is ___. | $$i\leqslant 10?$$ |
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a5d8a44f-d900-4972-90d3-e76718f15ab2 | A university conducted a statistical analysis of the independent admission test scores of 1000 students and obtained a frequency distribution histogram (as shown in the figure). The number of students among the 1000 who scored no less than 70 in this independent admission test is ______. | 600 |
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837381f0-fa4c-421c-ab3e-1a5ff4921cab | As shown in the figure, $$\odot O$$ is the incircle of $$\triangle ABC$$. If $$\angle ABC=70^{\circ}$$ and $$\angle ACB=40^{\circ}$$, then $$\angle BOC=$$ ___ $$^{\circ}$$. | $$125$$ |
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