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|---|---|---|---|---|
55455215-db86-40f2-89f5-4d7501252464
|
<image>The flowchart of a program is shown in the figure. If the input value of $x$ is $2$, then the output value of $y$ is.
|
$\sqrt{2}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/55455215-db86-40f2-89f5-4d7501252464.png"
}
] |
aa142044-e96b-49ae-b508-4411c7ad7b3c
|
<image>As shown in the figure, $\vartriangle ABC$ is an equilateral triangle. $P$ is a point on the angle bisector $BD$ of $\angle ABC$. $PE \perp AB$ at point $E$, and the perpendicular bisector of line segment $BP$ intersects $BC$ at point $F$, with the foot of the perpendicular being point $Q$. If $BF=2$, then the length of $PE$ is.
|
$\sqrt{3}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/aa142044-e96b-49ae-b508-4411c7ad7b3c.png"
}
] |
2eb4bc9c-895e-489f-9ae8-ce6c765f0460
|
<image>To measure the height of an unclimbable tree on the school's flat ground, Xiao Wen conducted the following exploration: According to the law of reflection in physics, using a mirror and a measuring tape, he designed the measurement plan as shown in the figure: placing a small mirror at a suitable position, he could just see the top of the tree in the mirror. At this moment, the horizontal distance between Xiao Wen and the mirror was 2.0 meters, and the horizontal distance between the base of the tree and the mirror was 8.0 meters. If Xiao Wen's eyes were 1.6 meters above the ground, then the height of the tree is approximately meters. (Note: The angle of reflection equals the angle of incidence)
|
6.4
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/2eb4bc9c-895e-489f-9ae8-ce6c765f0460.png"
}
] |
8ed45e8c-11b1-477a-b096-cd408c066d56
|
<image>As shown in the figure, two squares with side lengths of $\sqrt{3}$ are cut along their diagonals, and the resulting four triangles are rearranged to form a larger square. What is the side length of this larger square?
|
$\sqrt{6}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/8ed45e8c-11b1-477a-b096-cd408c066d56.png"
}
] |
9cd68124-6d1a-4acb-a23b-575fc45f8215
|
<image>As shown in the figure, given: $DE//BC$, $\angle A=54{}^\circ $, $\angle C=60{}^\circ $, then $\angle 1=$°.
|
$66{}^\circ $
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/9cd68124-6d1a-4acb-a23b-575fc45f8215.png"
}
] |
8e325469-9153-4b0c-a1fd-6b27a025e14d
|
<image>As shown in the figure, the robot Liangliang moves from point A to point B along a unit grid, moving only one unit length at a time. How many of the shortest paths can Liangliang take from A to B?
|
$80$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/8e325469-9153-4b0c-a1fd-6b27a025e14d.png"
}
] |
743758a9-8664-472b-9abd-fd9e020bccd3
|
<image>In the three scales below, '△' and '□' represent two objects of different masses. The mass of the weight on the right pan of the third scale is:
|
10
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/743758a9-8664-472b-9abd-fd9e020bccd3.png"
}
] |
02e459b2-46a5-4802-a926-a5ca7f04a41b
|
<image>As shown in the figure, point P(12, 5) lies on the graph of an inverse proportion function, and PH is perpendicular to the x-axis at point H. Then the value of tan∠POH is.
|
$\frac{5}{12}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/02e459b2-46a5-4802-a926-a5ca7f04a41b.png"
}
] |
91d1c13e-55c7-4648-b174-d59b229044a5
|
<image>As shown in the figure, in the Cartesian coordinate system $$xOy$$, $$A(1,1)$$, $$B(2,2)$$, the hyperbola $$y=\dfrac{k}{x}$$ intersects the line segment $$AB$$. The range of values for $$k$$ is ___.
|
$$1 \leqslant k \leqslant 4$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/91d1c13e-55c7-4648-b174-d59b229044a5.png"
}
] |
50f14522-a669-40b9-b070-a6cd8cf5577d
|
<image>As shown in the figure, the first figure has 1 black ball; the second figure is a shape made of 3 balls of the same size, with the 2 balls on the bottom layer being black and the rest white; the third figure is a shape made of 6 balls of the same size, with the 3 balls on the bottom layer being black and the rest white; ..., then the probability of randomly picking a black ball from the nth figure is ___. (n is a positive integer).
|
$$\dfrac{2}{n + 1}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/50f14522-a669-40b9-b070-a6cd8cf5577d.png"
}
] |
eaf2d81d-eb26-4fcb-93dd-3b5d1e0644e3
|
<image>As shown in the figure, a boat departs from point $$A$$ and sails in a direction $$60^{\circ}$$ north of east (i.e., $$\angle 1$$) to point $$B$$, then returns in a direction $$40^{\circ}$$ south of west (i.e., $$\angle 2$$). Then, $$\angle ABC=$$ ___.
|
$$20^{\circ}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/eaf2d81d-eb26-4fcb-93dd-3b5d1e0644e3.png"
}
] |
8488cdb7-7e0b-482c-a288-8a7f07f0fa95
|
<image>As shown in the figure, $$AB$$ is the diameter of $$ \odot O$$, $$CB$$ is tangent to $$ \odot O$$ at $$B$$, and $$CD$$ is tangent to $$ \odot O$$ at $$D$$, intersecting the extension of $$BA$$ at $$E$$. If $$EA=1$$ and $$ED=2$$, then the length of $$BC$$ is ___.
|
$$3$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/8488cdb7-7e0b-482c-a288-8a7f07f0fa95.png"
}
] |
b1a4271a-9064-40e9-804d-671434cc1434
|
<image>As shown in the figure, $$\triangle ABC$$ is a right-angled triangle paper piece, where $$\angle C=90^{\circ}$$, the two perpendicular sides are $$AC=6 \ { \mathrm {cm}}$$ and $$BC=8\ { \mathrm {cm}}$$. First, fold $$\triangle ABC$$ so that point $$B$$ coincides with point $$A$$, and the fold line is $$EF$$. Then, $$\tan \angle CAE=$$ ___.
|
$$\dfrac{7}{24 }$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/b1a4271a-9064-40e9-804d-671434cc1434.png"
}
] |
885521fe-b868-4dc1-8ba5-72419fb28429
|
<image>As shown in the figure, given that $$O$$ is the origin of the coordinate system, point $$A(3,0)$$, $$B(4,4)$$, and $$C(2,1)$$, then the coordinates of the intersection point $$P$$ of $$AC$$ and $$OB$$ are ___.
|
$$\left(\dfrac{3}{2},\dfrac{3}{2}\right)$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/885521fe-b868-4dc1-8ba5-72419fb28429.png"
}
] |
f07c9722-6d7b-4d64-9008-998d4cce2614
|
<image>A company has 50,000 yuan to invest in a development project. If successful, it can earn a 12% profit after one year; if it fails, it will lose 50% of the total funds after one year. The table below shows the results of 200 similar projects implemented in the past: Therefore, the estimated expected profit the company can gain after one year is ___.
|
4760
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/f07c9722-6d7b-4d64-9008-998d4cce2614.png"
}
] |
e8967b7c-1c47-4838-b676-5a430dd11352
|
<image>As shown in the figure, points $$E$$ and $$F$$ are on the sides $$AB$$ and $$BC$$ of square $$ABCD$$, respectively, and $$EF$$ intersects the diagonal $$BD$$ at point $$G$$. If $$BE=5$$ and $$BF=3$$, then the ratio $$FG:EF$$ is ___.
|
$$\dfrac{3}{8}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/e8967b7c-1c47-4838-b676-5a430dd11352.png"
}
] |
1243a383-231e-4f35-9673-fff7d70ce2dc
|
<image>As shown in the figure, in $$\triangle ABC$$, $$\angle A = m^{\circ}$$, the angle bisectors of $$\angle ABC$$ and $$\angle ACD$$ intersect at point $$A_{1}$$, yielding $$A_{1}$$; the angle bisectors of $$\angle A_{1}BC$$ and $$\angle A_{1}CD$$ intersect at point $$A_{2}$$, yielding $$A_{2}$$, and so on, until the angle bisectors of $$\angle A_{2015}BC$$ and $$\angle A_{2015}CD$$ intersect at point $$A_{2016}$$, then $$\angle A_{2016} =$$ ___ degrees.
|
$$\left ( \dfrac{m}{2^{2016}}\right ) $$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/1243a383-231e-4f35-9673-fff7d70ce2dc.png"
}
] |
d724d73c-81c8-458a-98ec-b3b1dceec96d
|
<image>A factory, in order to reasonably price a newly developed product, conducted a trial sale at a pre-determined price, obtaining the following data: From the data in the table, the linear regression equation is $$\hat{y}=-4x+a$$, then $$a=$$ ___.
|
$$106$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/d724d73c-81c8-458a-98ec-b3b1dceec96d.png"
}
] |
5faebd1a-298e-43c9-8e85-dc1ee60a3bcd
|
<image>As shown in the figure, the side length of the square is 2 cm. The difference in area between regions I and II, divided by the arc, is ___ square centimeters.
|
$$\number{0.14}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/5faebd1a-298e-43c9-8e85-dc1ee60a3bcd.png"
}
] |
c67e1cff-58d6-448f-a3db-bf31639abfa1
|
<image>As shown in the figure, lines $$a$$ and $$b$$ intersect, $$\angle 2 = 3\angle 1$$, then $$\angle 3 =$$ ___.
|
$$45^{\circ}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/c67e1cff-58d6-448f-a3db-bf31639abfa1.png"
}
] |
776bc316-4f85-4d96-a2ab-2fb4a919b61c
|
<image>As shown in the figure, Hao Shuai glued a large and a small cube together to form a new 3D shape. The surface area of this 3D shape is ______ square centimeters. (Unit: centimeters)
|
232
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/776bc316-4f85-4d96-a2ab-2fb4a919b61c.png"
}
] |
0573674e-5974-4950-80f4-32fa1bae23d8
|
<image>To understand the basic situation of the monthly average electricity consumption of residents in a certain area, the electricity consumption data of several households in the region were extracted, resulting in the frequency distribution histogram shown in the figure. If there are 150 households with a monthly average electricity consumption in the range $$\left[ {110,120} \right)$$, then the number of households with a monthly average electricity consumption in the range $$\left[ {120,140} \right)$$ is ______.
|
$$300$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/0573674e-5974-4950-80f4-32fa1bae23d8.png"
}
] |
26e9e28f-6fbc-4a01-9f99-d42a89f52622
|
<image>As shown in the figure, in rectangle $$ABCD$$, diagonals $$AC$$ and $$BD$$ intersect at point $$O$$. Points $$E$$ and $$F$$ are the midpoints of $$AO$$ and $$AD$$, respectively. If $$AB=6cm$$ and $$BC=8cm$$, then the perimeter of $$\triangle AEF$$ is ______ $$cm$$.
|
9
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/26e9e28f-6fbc-4a01-9f99-d42a89f52622.png"
}
] |
a935aab7-1ec0-4ee6-9da3-1d3e76ecfc26
|
<image>As shown in the figure, the line $$y=kx+b$$ ($$k\neq 0$$) intersects the $$x$$-axis at the point $$(-4,0)$$, then the solution to the equation $$kx+b=0$$ with respect to $$x$$ is $$x=$$______．
|
-4
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/a935aab7-1ec0-4ee6-9da3-1d3e76ecfc26.png"
}
] |
ec23cdbe-9240-41e6-bfb4-433506605c0e
|
<image>The figure is a three-view diagram of a solid shape composed of two rectangular prisms. According to the dimensions marked in the figure (unit: $$mm$$), calculate the surface area of this solid shape, which is ______ $$mm^2$$.
|
$$200$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/ec23cdbe-9240-41e6-bfb4-433506605c0e.png"
}
] |
02b95386-34d8-43e3-ac8d-c88de143ac61
|
<image>As shown in the figure, $$\triangle ABC$$ is a right-angled triangular piece of colored paper, where $$AC=\quantity{15}{cm}$$ and $$BC=\quantity{20}{cm}$$. If the altitude $$CD$$ on the hypotenuse is divided into $$n$$ equal parts, and then $$(n-1)$$ rectangular strips of equal width are cut out. Then the sum of the areas of these $$(n-1)$$ strips is ___ $$\unit{cm^{2}}$$.
|
$$\dfrac{150(n-1)^{2}}{n}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/02b95386-34d8-43e3-ac8d-c88de143ac61.png"
}
] |
9206b7dd-fe9e-44d1-84c0-a3c757115e80
|
<image>Given the circle $$O$$: $$x^{2} + y^{2} = r^{2} (r > 0)$$ and a point $$A(0, -r)$$ on the circle, a line $$l$$ passing through point $$A$$ intersects the circle at another point $$B$$ and the $$x$$-axis at point $$C$$. If $$OC = BC$$, then the slope of line $$l$$ is ___.
|
$$ \pm \sqrt{3}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/9206b7dd-fe9e-44d1-84c0-a3c757115e80.png"
}
] |
344e3e53-4e14-476e-839d-c9cd08846835
|
<image>As shown in the figure, in the Cartesian coordinate system $$xOy$$, points $$A$$ and $$B$$ are both on the unit circle. It is known that point $$A$$ is in the first quadrant with an x-coordinate of $$\dfrac{\sqrt{3}}{3}$$, and point $$B$$ is in the second quadrant. If $$\triangle AOB$$ is an equilateral triangle, then the coordinates of point $$B$$ are ___.
|
$$\left(\dfrac{\sqrt{3}-3\sqrt{2}}{6},\dfrac{3+\sqrt{6}}{6}\right)$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/344e3e53-4e14-476e-839d-c9cd08846835.png"
}
] |
f8326283-70b7-47f8-8e4f-005e35680830
|
<image>A school selected the scores of 60 students (from 40 to 100 points) who participated in a computer test, divided them into six segments, and plotted them as a frequency distribution histogram (where the segment 70~80 is unclear due to some reason). If 60 points and above (including 60 points) is considered passing, estimate the passing rate of this test based on the information in the graph.
|
75%
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/f8326283-70b7-47f8-8e4f-005e35680830.png"
}
] |
ef2760e7-c8de-4b21-bfb3-364642a14c81
|
<image>As shown in the figure, in the rectangular coordinate system $$xOy$$, it is known that $$A$$, $$B_{1}$$, and $$B_{2}$$ are the right, lower, and upper vertices of the ellipse $$C$$: $$\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1(a > b > 0)$$, respectively, and $$F$$ is the right focus of the ellipse $$C$$. If $$B_{2}F \perp AB_{1}$$, then the eccentricity of the ellipse $$C$$ is ___.
|
$$\dfrac{\sqrt{5}-1}{2}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/ef2760e7-c8de-4b21-bfb3-364642a14c81.png"
}
] |
c50861b0-65be-474c-8c62-8912f65898b0
|
<image>Given: As shown in the figure, $$\triangle OAD \cong \triangle OBC$$, and $$\angle O = 72^{\circ}$$, $$\angle D = 21^{\circ}$$, then $$\angle DEC =$$ ___.
|
$$114^{\circ}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/c50861b0-65be-474c-8c62-8912f65898b0.png"
}
] |
dfe4f5d4-c982-480f-a410-43d7f1ee5926
|
<image>As shown in the figure, $$AC$$ is the wiper on the car's windshield. If $$AO=\quantity{45}{cm}$$ and $$CO=\quantity{5}{cm}$$, when $$AC$$ rotates 90 degrees clockwise around point $$O$$, the area swept by the wiper $$AC$$ is ___ $$\unit{cm^{2}}$$ (the result should be in terms of $$π$$).
|
$$500π$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/dfe4f5d4-c982-480f-a410-43d7f1ee5926.png"
}
] |
e1141687-4b60-4549-a3fd-3ad76a5f798f
|
<image>In the triangular prism $$ABC-A_{1}B_{1}C_{1}$$, if $$\overrightarrow{CA}=\boldsymbol{a}$$, $$\overrightarrow{CB}=\boldsymbol{b}$$, $$\overrightarrow{CC_{1}}=\boldsymbol{c}$$, and $$E$$ is the midpoint of $$A_{1}B$$, then $$\overrightarrow{CE}=$$ ___. (Express in terms of $$\boldsymbol{a}$$, $$\boldsymbol{b}$$, and $$\boldsymbol{c}$$)
|
$$\dfrac{1}{2}(\boldsymbol{a}+\boldsymbol{b}+\boldsymbol{c})$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/e1141687-4b60-4549-a3fd-3ad76a5f798f.png"
}
] |
e989f36f-a8db-4497-8ff1-e2852cc670c1
|
<image>Execute the algorithm flowchart shown in the figure, then the value of the output $$S$$ is ___.
|
$$-5$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/e989f36f-a8db-4497-8ff1-e2852cc670c1.png"
}
] |
d61c4330-f74b-4b23-ba17-43872f9a9ca8
|
<image>As shown in the figure, the real numbers corresponding to points $$A$$ and $$B$$ on the number line are $$1$$ and $$\sqrt{2}$$, respectively. Point $$C$$ is the symmetric point of point $$B$$ with respect to point $$A$$. The real number represented by point $$C$$ is ___.
|
$$2-\sqrt{2}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/d61c4330-f74b-4b23-ba17-43872f9a9ca8.png"
}
] |
9777f2ea-c183-4d5d-b0b7-9ae0b54a9854
|
<image>In an experiment, four sets of values for $$\left (x, y \right ) $$ were measured as . According to the table, the regression equation is $$\hat{y}=-5x+\hat{a}$$, and using this model, the predicted value of $$y$$ when $$x$$ is $$20$$ is ___.
|
$$26.5$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/9777f2ea-c183-4d5d-b0b7-9ae0b54a9854.png"
}
] |
332eb846-4b1e-4bc7-b896-c32ed683fcb7
|
<image>As shown in the figure, this is the front view and top view of a rectangular prism. From the given data (unit: $$\unit{cm}$$), the volume of the rectangular prism can be determined as ___ $$ \unit{cm^3}$$.
|
$$18$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/332eb846-4b1e-4bc7-b896-c32ed683fcb7.png"
}
] |
7ddacc97-6bba-4ee7-8594-3b200e1d719e
|
<image>As shown in the figure, the perimeter of rectangle $$ABCD$$ is $$12cm$$. $$E$$ is the midpoint of $$BC$$, and $$AE \bot ED$$ at point $$E$$. Then, $$AB=$$______$$cm$$.
|
$$2$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/7ddacc97-6bba-4ee7-8594-3b200e1d719e.png"
}
] |
da0cbcd6-a444-4187-b29e-bba0563ce86b
|
<image>A school holds an extracurricular comprehensive knowledge competition, and randomly selects the scores of 400 students. All scores are between 50 and 100 points. The scores are divided into 5 groups as follows: Group 1, scores greater than or equal to 50 points and less than 60 points; Group 2, scores greater than or equal to 60 points and less than 70 points; ... Group 5, scores greater than or equal to 90 points and less than or equal to 100 points. Based on this, a frequency distribution histogram is drawn as shown in the figure. How many of the 400 students have excellent scores?
|
$$100$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/da0cbcd6-a444-4187-b29e-bba0563ce86b.png"
}
] |
3523062f-53af-4e1b-b873-b5e418c8735d
|
<image>As shown in the figure, a particle moves in the first quadrant. In the first second, it moves from the origin to $(0,1)$, and then it continues to move back and forth in directions parallel to the $x$-axis and $y$-axis (i.e., $(0,0)\to(0,1)\to(1,1)\to(1,0)\to(2,0)\to\cdots$), moving one unit length per second. What is the position of the particle at the 2006th second?
|
$(18, 44)$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/3523062f-53af-4e1b-b873-b5e418c8735d.png"
}
] |
4195459e-7e25-4cde-a4f3-d2f405f681af
|
<image>As shown in the figure, in the right triangle $$\triangle ABC$$, $$\angle ACB=90^{\circ}$$, $$AC=8$$, $$BC=6$$, and $$CD \perp AB$$, with the foot of the perpendicular at $$D$$. Then, $$\tan \angle BCD =$$ ___.
|
$$\dfrac{3}{4}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/4195459e-7e25-4cde-a4f3-d2f405f681af.png"
}
] |
03f09aea-b7c6-490e-9d33-7db68ed95a44
|
<image>As shown in the figure, the entire circle represents the total number of students in a class participating in extracurricular activities. The number of students jumping rope accounts for $$30\%$$, the central angle of the sector representing students playing shuttlecock is $$60^{\circ}$$, and the ratio of the number of students playing shuttlecock to those playing basketball is $$1:2$$. What percentage of the total number of students participate in 'other' activities?
|
$$20$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/03f09aea-b7c6-490e-9d33-7db68ed95a44.png"
}
] |
ae6e851b-ff67-4e57-b56b-3469c615735d
|
<image>As shown in the figure, a square with side length $$3$$ is cut from a square paper with side length $$(a+3)$$. The remaining part is then cut and rearranged along the dotted lines into a rectangle (without overlapping or gaps) as shown in the figure. The length of the other side of the formed rectangle is ___.
|
$$a+6$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/ae6e851b-ff67-4e57-b56b-3469c615735d.png"
}
] |
8c925712-a34a-447d-b0b5-0b747414cba5
|
<image>Given the values of $$x$$ and $$y$$ as shown in the table: From the scatter plot, $$y$$ is linearly related to $$x$$, and the regression equation is $$\widehat{y}=0.95x+\widehat{a}$$, then $$\widehat{a}=$$____.
|
$$2.6$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/8c925712-a34a-447d-b0b5-0b747414cba5.png"
}
] |
643ad4a2-497b-4614-91ad-7b423f9b7264
|
<image>As shown in the figure, in a grid composed of small squares with side lengths of $$1$$, a circle $$\odot O$$ with radius $$1$$ is centered at a grid point. The tangent value of $$\angle AED$$ is ___.
|
$$\dfrac{1}{2}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/643ad4a2-497b-4614-91ad-7b423f9b7264.png"
}
] |
a3f5931e-f597-4ec0-abdc-8a9932a2f3f0
|
<image>In the square shown in the figure, if $$\number{10000}$$ points are randomly selected, the estimated number of points that fall into the shaded area (where the boundary curve $$C$$ is part of the density curve of the normal distribution $$N(-1,1)$$) is ___.
|
$$\number{1359}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/a3f5931e-f597-4ec0-abdc-8a9932a2f3f0.png"
}
] |
cc073eb3-fccf-44d9-83bf-6d64d3c54b37
|
<image>Given that the length of line segment $$AB$$ is $$2$$, a square $$ACDB$$ is constructed below $$AB$$. Take a point $$E$$ on $$AB$$, and construct a square $$AENM$$ above $$AB$$. Draw $$EF \bot CD$$, with the foot of the perpendicular at point $$F$$, as shown in the figure. If the area of square $$AENM$$ is equal to the area of quadrilateral $$EFDB$$, then the length of $$AE$$ is ___.
|
$$\sqrt[]{5}-1$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/cc073eb3-fccf-44d9-83bf-6d64d3c54b37.png"
}
] |
4a02891e-218d-42fe-acfb-bb482711c26e
|
<image>As shown in the figure, in the Cartesian coordinate system $$xOy$$, $$N$$ is a moving point on the circle $$A$$: $$(x+1)^{2}+y^{2}=16$$. Point $$B(1,0)$$, and the perpendicular bisector of $$BN$$, denoted as $$MP$$, intersects the radius $$AN$$ at point $$P$$. The equation of the trajectory of the moving point $$P$$ is ___.
|
$$\dfrac{x^{2}}{4}+\dfrac{y^{2}}{3}=1$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/4a02891e-218d-42fe-acfb-bb482711c26e.png"
}
] |
44d2a690-9ea5-43b5-9db5-f4cc678492f4
|
<image>As shown in the figure, in a $$3\times 3$$ grid, points $$A$$, $$B$$, $$C$$, $$D$$, $$E$$, and $$F$$ are located at the grid points. If one point is randomly selected from points $$C$$, $$D$$, $$E$$, and $$F$$ to form a triangle with points $$A$$ and $$B$$, what is the probability that the triangle formed is an isosceles triangle?
|
$$\dfrac{3}{4}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/44d2a690-9ea5-43b5-9db5-f4cc678492f4.png"
}
] |
63f89360-d3b0-4b7b-b75b-f3b2b4693179
|
<image>A small ball rolls freely on the floor shown in the figure and randomly stops on one of the square tiles. What is the probability that the ball will eventually stop on a black area?
|
$$\dfrac{2}{9}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/63f89360-d3b0-4b7b-b75b-f3b2b4693179.png"
}
] |
ac1f8d2a-51be-4553-b661-64b80b8998e6
|
<image>As shown in the figure, a little sheep is tied at vertex A of rectangle ABCD. There is a basket of green grass at each of B, C, and D. Given that AB = 5 and BC = 12, to ensure that the sheep can reach at least one basket of grass and cannot reach at least one basket, the range of the length l of the rope is ___.
|
5 < l < 13
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/ac1f8d2a-51be-4553-b661-64b80b8998e6.png"
}
] |
6426f05d-cc58-4a04-9bf7-c4b21aa2ea50
|
<image>As shown in the figure, $$AB\parallel CD\parallel EF$$. If $$\angle A=30^{\circ}$$ and $$\angle AFC=15^{\circ}$$, then $$\angle C=$$ ___.
|
$$15^{\circ}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/6426f05d-cc58-4a04-9bf7-c4b21aa2ea50.png"
}
] |
e080045c-6372-425c-9a8f-3b6258c0decd
|
<image>As shown in the figure, fold one side $$AD$$ of the rectangle $$ABCD$$ so that point $$D$$ falls on point $$F$$ on side $$BC$$. It is known that the crease $$AE=5\sqrt{5}\ \unit{cm}$$, and $$\tan \angle EFC=\dfrac{3}{4}$$. What is the perimeter of rectangle $$ABCD$$?
|
$$36$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/e080045c-6372-425c-9a8f-3b6258c0decd.png"
}
] |
11c129f8-5123-46c4-933e-3a7cdd408ea6
|
<image>As shown in the figure, the line $$y=ax$$ intersects the hyperbola $$y={k\over x}$$ $$(x>0)$$ at point $$A(1,2)$$. The solution set of the inequality $$ax>{k\over x}$$ is ______.
|
$$x>1$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/11c129f8-5123-46c4-933e-3a7cdd408ea6.png"
}
] |
fbe44225-26a0-431a-9dee-9839a5ff37fe
|
<image>As shown in the figure, the pattern is made using matchsticks. The number of matchsticks needed for the $$n$$th pattern is ______.
|
$$2n+1$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/fbe44225-26a0-431a-9dee-9839a5ff37fe.png"
}
] |
c66c1618-7803-43cd-8383-ca9d3d0919d4
|
<image>Execute the program flowchart shown in the figure. If the output $$S=7$$, then the value of the input $$k(k\in{\bf{N^*}})$$ is ______.
|
$$3$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/c66c1618-7803-43cd-8383-ca9d3d0919d4.png"
}
] |
0431acb7-051a-417e-9548-da64417d6efd
|
<image>As shown in the figure, $$AM$$ is the diameter of $$\odot O$$, the line $$BC$$ passes through point $$M$$, and $$AB=AC$$, $$\angle BAM= \angle CAM$$. The line segments $$AB$$ and $$AC$$ intersect $$\odot O$$ at points $$D$$ and $$E$$ respectively, and $$\angle BMD=40^{ \circ }$$. Then $$\angle EOM=$$ ___.
|
$$80^{ \circ }$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/0431acb7-051a-417e-9548-da64417d6efd.png"
}
] |
53038256-8757-4825-b8d9-48b2fcd28d72
|
<image>As shown in the figure, $$ABCD$$ is a rectangular area, where $$AB=18$$ meters and $$AD=11$$ meters. The width of the paths from entrances $$A$$ and $$B$$ is $$1$$ meter, and the width of the path where the two paths meet is $$2$$ meters. The rest of the area is planted with grass. What is the area of the grass in square meters?
|
$$160$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/53038256-8757-4825-b8d9-48b2fcd28d72.png"
}
] |
693f1f37-0777-4a9b-8e2a-f9e9aa7ed60f
|
<image>As shown in the figure, quadrilateral $$ABCD$$ is an inscribed quadrilateral of $$ \odot O$$. Given that $$ \angle C= \angle D$$, the positional relationship between $$AB$$ and $$CD$$ is ___.
|
$$AB \parallel CD$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/693f1f37-0777-4a9b-8e2a-f9e9aa7ed60f.png"
}
] |
78037754-394c-4fcb-9a96-917548c64544
|
<image>As shown in the figure, in rhombus $$ABCD$$, if $$AC=6$$ and $$BD=8$$, then the area of rhombus $$ABCD$$ is ___.
|
$$24$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/78037754-394c-4fcb-9a96-917548c64544.png"
}
] |
90883f65-1ac8-4854-9a66-65b805446748
|
<image>As shown in the figure, $$A$$ and $$B$$ are the centers of two quarter circles. The difference in the areas of the two shaded regions is ___ square centimeters.
|
$$\number{1.42}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/90883f65-1ac8-4854-9a66-65b805446748.png"
}
] |
dcc6308d-fb60-4569-a950-509e6be8ec2c
|
<image>As shown in the figure, in the tetrahedron $$A-BCD$$, $$AB=1$$, $$AD=2\sqrt{3}$$, $$BC=3$$, $$CD=2$$, $$ \angle ABC= \angle DCB=\dfrac{ \pi }{2}$$, then the size of the dihedral angle $$A-BC-D$$ is ___.
|
$$60^{ \circ }$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/dcc6308d-fb60-4569-a950-509e6be8ec2c.png"
}
] |
33e54747-755b-492f-ae30-e7cae6971e2e
|
<image>The figure shows 4 patterns created by Xiaoqiang using copper coins. According to the pattern of arrangement, try to guess how many copper coins are needed for the nth pattern.
|
$$\dfrac{1}{2}n(n+1)$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/33e54747-755b-492f-ae30-e7cae6971e2e.png"
}
] |
33c0656c-1c96-4e9b-b51f-b65e1ea78b81
|
<image>In the figure, in parallelogram ABCD, diagonals AC and BD intersect at point O, and point E is the midpoint of CD. What is the ratio of the area of △ODE to the area of △AOB?
|
1:2
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/33c0656c-1c96-4e9b-b51f-b65e1ea78b81.png"
}
] |
4e5a9c8f-a2e2-43a0-8985-ca4e314efe9f
|
<image>(2016 Zhuzhou City, Hunan Province) Given that A, B, C, and D are points on the coordinate axes in a plane coordinate system, and △AOB ≌ △COD. Let the equation of line AB be y$_{1}$=k$_{1}$x+b$_{1}$, and the equation of line CD be y$_{2}$=k$_{2}$x+b$_{2}$. Then k$_{1}$k$_{2}$=．
|
1
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/4e5a9c8f-a2e2-43a0-8985-ca4e314efe9f.png"
}
] |
92b59a3b-8358-45f4-89ef-08089b94d8e0
|
<image>As shown in the figure, given that a//b, and ∠2 is twice the size of ∠1, then the degree measure of ∠2 is °;
|
120
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/92b59a3b-8358-45f4-89ef-08089b94d8e0.png"
}
] |
040e7649-e389-4938-8766-d535886df95f
|
<image>In the figure, in $\Delta ABC$, the three interior angles are A, B, and C, $AB=3$, $AC=6$, and the length of the angle bisector AD is 2. What is the measure of $\angle A$?
|
$\frac{2\pi }{3}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/040e7649-e389-4938-8766-d535886df95f.png"
}
] |
bbd6dd5b-cb72-48c4-b761-7d1f18ea571d
|
<image>As shown in the figure, the line $l$ is the tangent line to the curve $y=f(x)$ at $x=3$, then ${f}'(3)=$.
|
$\frac{1}{2}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/bbd6dd5b-cb72-48c4-b761-7d1f18ea571d.png"
}
] |
1ae98211-7d4a-4b7a-afa6-0ac386c6093f
|
<image>In the figure, in $\Delta ABC$, the angle bisectors of $\angle B$ and $\angle C$ intersect at point $O$. A line $MN$ is drawn through point $O$ parallel to $BC$, intersecting $AB$ and $AC$ at points $M$ and $N$ respectively. If $AB=8$ and $AC=10$, then the perimeter of $\Delta AMN$ is.
|
18
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/1ae98211-7d4a-4b7a-afa6-0ac386c6093f.png"
}
] |
3584109e-e01b-4be7-bd59-a6100d2fa475
|
<image>According to the pseudocode shown in the figure, when the input value of $a$ is 4, the output value of $S$ is.
|
28
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/3584109e-e01b-4be7-bd59-a6100d2fa475.png"
}
] |
bb286753-7f01-4db8-928b-52e6f4ca150d
|
<image>As shown in the figure, O is a point on the line AB, ∠COB = 29°30′, then ∠1 = .
|
150.5°
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/bb286753-7f01-4db8-928b-52e6f4ca150d.png"
}
] |
6992959f-43ac-402a-8fbd-7e2cc6c6e454
|
<image>A container with an inlet pipe and an outlet pipe starts to only fill with water for the first 4 minutes. In the following 8 minutes, it both fills and drains water. The inflow and outflow rates per minute are constant. The relationship between the water volume y (unit: L) in the container and the time x (unit: min) is shown in the figure: What is the water volume in the container at 8 minutes?
|
25
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/6992959f-43ac-402a-8fbd-7e2cc6c6e454.png"
}
] |
90395465-0b61-4806-9b8d-67cd3e6cb586
|
<image>As shown in the figure, the square $ABCD$ is rotated $90^\circ$ clockwise around point $O$ to obtain the square $A_1B_1C_1D_1$, where $AB=4$, $BO=3$, and $AO=5$. What is the area swept by $\Delta ABO$ to $\Delta A_1B_1O$? (Express the result in terms of $\pi$)
|
$\frac{25\pi }{4}+6$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/90395465-0b61-4806-9b8d-67cd3e6cb586.png"
}
] |
20599b8b-ea51-4d3f-b325-2b23fb189fee
|
<image>As shown in the figure, the diameter AB of a semicircle is 2, O is the center, and C is any point on the semicircle different from A and B. If P is a moving point on the radius OC, then the minimum value of (${PA}^⇀$ + ${PB}^⇀$) · ${PC}^⇀$ is.
|
$-\frac{1}{2}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/20599b8b-ea51-4d3f-b325-2b23fb189fee.png"
}
] |
c4e33cd7-ee0d-4ae8-9009-3dcae768a0e5
|
<image>As shown in the figure, in rhombus OABC, the coordinates of point A are (2, 1), and the x-coordinate of point B is 3. What are the coordinates of point C?
|
(1, 2)
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/c4e33cd7-ee0d-4ae8-9009-3dcae768a0e5.png"
}
] |
9fd5b013-5e1e-4216-926e-e67361196913
|
<image>As shown in the figure, in the right triangle ABC, ∠BCA = 90°, ∠ABC = 60°, and AC = $2\sqrt{3}$. The triangle ABC is rotated counterclockwise around point B, so that point C rotates to point C' on the extension of AB. The area of the shaded region swept by side AC is =. (The result should be expressed in terms of π)
|
4π
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/9fd5b013-5e1e-4216-926e-e67361196913.png"
}
] |
55cccbfe-d0b1-4a87-bbe7-3c97cabed67f
|
<image>Given that AB//CD//EF, the quantitative relationship between $x$, $y$, and $z$ is:
|
$x-y+z=180{}^\circ $
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/55cccbfe-d0b1-4a87-bbe7-3c97cabed67f.png"
}
] |
82fe4784-1f8a-4ffe-a117-68041d82875c
|
<image>The graph of the function $f(x)=\sqrt{3}\sin (\omega x+\varphi )\left( \varphi > 0,\frac{\pi }{2} < \varphi < \pi \right)$ is shown in the figure. What is the smallest positive period of this function?
|
$8$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/82fe4784-1f8a-4ffe-a117-68041d82875c.png"
}
] |
e5685642-ee01-40c4-a1de-55cfb758906b
|
<image>As shown in the figure, a rectangular paper ABCD is folded along CE, and point B lands exactly on the edge AD, let this point be F. After unfolding, if AB:BC = 4:5, then ∠CFD ≈ (accurate to 0.01°)
|
53.13°
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/e5685642-ee01-40c4-a1de-55cfb758906b.png"
}
] |
d5067e6b-2106-486c-8eb7-002ba3ec47b9
|
<image>In the 'Nine Chapters on the Mathematical Art', a right triangular prism is called a 'qian du' (堑堵), and a pyramid with a rectangular base and one edge perpendicular to the base is called a 'yang ma' (阳马). Given that the three views of a geometric solid formed by a 'qian du' and a 'yang ma' are shown in the figure, and the volume of the solid is $\frac{5}{6}\sqrt{3}$, then $x$ =.
|
$\sqrt{3}$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/d5067e6b-2106-486c-8eb7-002ba3ec47b9.png"
}
] |
7bc8fbbf-e941-4ad2-ad18-76387f98b020
|
<image>As shown in the figure, fold one side AD of the rectangular paper ABCD so that point D lands on point F on side BC. Given that AB = 8 cm and BC = 10 cm, find the perimeter of triangle ADE.
|
$\left( 15\text{+}5\sqrt{5} \right)cm$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/7bc8fbbf-e941-4ad2-ad18-76387f98b020.png"
}
] |
c1de477c-a337-47ed-8a92-27cb0d7140ba
|
<image>As shown in the figure, in $$\triangle ABC$$, $$\angle ACB=90{{}^\circ}$$, $$AB=8cm$$, and $$D$$ is the midpoint of $$AB$$. Now, $$\triangle BCD$$ is translated 1cm in the direction of $$BA$$ to obtain $$\triangle EFG$$, and $$FG$$ intersects $$AC$$ at $$H$$. What is the length of $$GH$$ in cm?
|
$$3$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/c1de477c-a337-47ed-8a92-27cb0d7140ba.png"
}
] |
cb52efce-5c6f-4662-8071-ac72c83e3879
|
<image>The figure shows the sample frequency distribution histogram based on the average temperature (unit: $$^{\circ}C$$) data of some cities in June of a certain year. The range of average temperatures is $$[20.5,26.5]$$, and the sample data is grouped as $$[20.5,21.5)$$, $$[21.5,22.5)$$, $$[22.5,23.5)$$, $$[23.5,24.5)$$, $$[24.5,25.5)$$, $$[25.5,26.5]$$. It is known that the number of cities with an average temperature below $$22.5^{\circ}C$$ is $$11$$. The number of cities with an average temperature of no less than $$25.5^{\circ}C$$ in the sample is ___.
|
$$9$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/cb52efce-5c6f-4662-8071-ac72c83e3879.png"
}
] |
8073bfd2-0869-471c-8e93-4f84038164f8
|
<image>The pseudocode of a certain algorithm is shown in the figure. The result output by the algorithm is ___.
|
$$6$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/8073bfd2-0869-471c-8e93-4f84038164f8.png"
}
] |
ec3c26ff-6d3a-4892-a1dc-948c168530b1
|
<image>The distribution of the random variable $$\xi$$ is given by . Find the value of $$E(5 \xi +4)$$.
|
$$13$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/ec3c26ff-6d3a-4892-a1dc-948c168530b1.png"
}
] |
79f38a41-2902-4715-a476-ad2dd5cf2c09
|
<image>In the square $$ABCD$$, if $$AF=BE$$, then the degree measure of $$\angle AOD$$ is ___.
|
$$90^{\circ}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/79f38a41-2902-4715-a476-ad2dd5cf2c09.png"
}
] |
4dcfc5ff-2579-4e6a-921b-269b7c75205c
|
<image>Each small square in the grid has a side length of $$1$$. The vertices of $$\triangle ABC$$ are all at the intersections of the grid. Then $$\sin A=$$ ___.
|
$$\dfrac{3}{5}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/4dcfc5ff-2579-4e6a-921b-269b7c75205c.png"
}
] |
c0ccf2b0-378e-48f9-a0c1-55198b710eef
|
<image>As shown in the figure, the set of angles whose terminal sides fall within the shaded area (including the boundaries) is ___.
|
$$\{\alpha |k\cdot 180^{ \circ }+45^{ \circ } \leqslant \alpha \leqslant k\cdot 180^{ \circ }+90^{ \circ },k \in \mathbf{Z}\}$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/c0ccf2b0-378e-48f9-a0c1-55198b710eef.png"
}
] |
0d0a076f-d398-4ccb-9edb-0f70e96cbe87
|
<image>As shown in the figure, $$AB \perp BC$$ at $$B$$, $$AB \perp AD$$ at $$A$$. Given $$AB=12$$, $$AD=5$$, $$BC=10$$. Point $$E$$ is the midpoint of $$CD$$. The length of $$AE$$ is ___.
|
$$6.5$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/0d0a076f-d398-4ccb-9edb-0f70e96cbe87.png"
}
] |
b5318866-f2f4-47ca-9eb8-b3eb55d0fa12
|
<image>If the input is $$8$$, then the result of the following program execution is ___.
|
$$0.7$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/b5318866-f2f4-47ca-9eb8-b3eb55d0fa12.png"
}
] |
6a10095f-f39d-4f47-b3ef-272ee034b0a7
|
<image>As shown in the figure, a circular spinner is divided into five equal扇形 sectors, each labeled with the numbers $$1$$, $$2$$, $$3$$, $$4$$, and $$5$$. The position of the spinner's pointer is fixed, and the spinner is allowed to stop freely after being spun. After one spin, when the spinner stops, the probability of the pointer landing on a sector labeled with an even number is denoted as $$P$$(even), and the probability of the pointer landing on a sector labeled with an odd number is denoted as $$P$$(odd). If the pointer lands on a line, it is counted as the sector to its right. Then, $$P$$(even)___$$P$$(odd) (fill in with "$$>$$", "$$<$$", or "$$=$$").
|
$$<$$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/6a10095f-f39d-4f47-b3ef-272ee034b0a7.png"
}
] |
54a5afcf-43a7-4c9b-b580-f4e3f0ae54bf
|
<image>As shown in the figure, $\vartriangle APT$ is symmetric to $\vartriangle CPT$ with respect to the line $PT$, and $AT=PT$. Extend $AT$ to intersect $PC$ at point $F$. When $\angle A=$, $\vartriangle TFC$ is an isosceles triangle.
|
36°
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/54a5afcf-43a7-4c9b-b580-f4e3f0ae54bf.png"
}
] |
e8cdbb99-875f-40ed-aa42-f98794bc28f5
|
<image>According to the algorithm flowchart shown, the output value of $S$ is.
|
9
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/e8cdbb99-875f-40ed-aa42-f98794bc28f5.png"
}
] |
efbd48c0-a8e4-4b52-bad9-2a700b340324
|
<image>A questionnaire survey was conducted by randomly selecting 100 citizens in the age groups [10, 20), [20, 30), …, [50, 60). The frequency distribution histogram of the sample is shown in the figure. If 12 people are randomly selected using stratified sampling from those aged 40 and over, then the number of people selected from the [50, 60) age group is.
|
3
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/efbd48c0-a8e4-4b52-bad9-2a700b340324.png"
}
] |
e5282d37-26ee-495f-a6ae-e42a61772d11
|
<image>The 'Nine Chapters on the Mathematical Art' is an ancient Chinese mathematical treatise with a wealth of content. One of the problems in the book is: 'There is a pile of rice placed against the corner of a wall, with a base circumference of 8 feet and a height of 5 feet. Question: What is the volume of the pile and how much rice is in it?' The problem means: 'Rice is piled up against the corner of a room (as shown in the figure, the rice pile is a quarter of a cone), with the arc length at the base being 8 feet and the height of the pile being 5 feet. What is the volume of the rice pile and how much rice is in it?' Given that 1 hu of rice has a volume of approximately 1.62 cubic feet and the value of pi is approximately 3, how many hu of rice are there (round the result to the nearest whole number)?
|
22
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/e5282d37-26ee-495f-a6ae-e42a61772d11.png"
}
] |
85a0e667-80c1-453c-8e73-285db8d58fe1
|
<image>According to the pseudocode shown in the figure, if the output value $y$ is 3, then the input value $x$ is.
|
$-2$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/85a0e667-80c1-453c-8e73-285db8d58fe1.png"
}
] |
5116e5c9-555a-4af3-a16a-0f28b8da9af2
|
<image>As shown in the figure, in quadrilateral $ABCD$, $AD=BC$ and $AD//BC$, $AB=8$, $AD=5$, $AE$ bisects $\angle DAB$ and intersects the extension of $BC$ at point $F$. Then $CF=$.
|
3
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/5116e5c9-555a-4af3-a16a-0f28b8da9af2.png"
}
] |
5543e8da-8d56-430e-b246-a1c495193d51
|
<image>In the generalized third-order magic square shown in the figure, three numbers are given. Try to find the values of x and y.
|
-1, 2
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/5543e8da-8d56-430e-b246-a1c495193d51.png"
}
] |
0e94fb30-c606-4811-b286-0a138e3493cd
|
<image>As shown in the figure, in $\Delta ABC$, $AC=6$, $BC=10$, $\tan C=\frac{3}{4}$, point $D$ is a moving point on side $AC$ (not coinciding with point $C$), and $DE \bot BC$ with the foot of the perpendicular at $E$. Point $F$ is the midpoint of $BD$, and $EF$ is connected. Let $CD=x$, and the area of $\Delta DEF$ be $S$. The function relationship between $S$ and $x$ is:
|
$y=-\frac{3}{25}{{x}^{2}}+\frac{3}{2}x$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/0e94fb30-c606-4811-b286-0a138e3493cd.png"
}
] |
76d1d325-0dca-4481-8365-ed83b5b74378
|
<image>Given the graph of the cubic function $y=f(x)$ as shown, the expression for the function $f(x)$ is.
|
$f(x)=2x^3-3x^2+2$
|
math
|
[
{
"path": "/home/xywang96/Training_datas/MMK12/76d1d325-0dca-4481-8365-ed83b5b74378.png"
}
] |
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