id
stringlengths 4
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| image_url
stringlengths 77
81
| query
stringlengths 7
1.32k
| answer
stringlengths 1
148
| choice
stringlengths 4
597
| question_type
stringclasses 2
values |
|---|---|---|---|---|---|
func227
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/227.png
|
A and B are 60 km apart. Two individuals, A and B, start cycling simultaneously from points A and B, respectively, towards each other at a constant speed. B rests for 0.5 hours during the journey before continuing at the original speed. The relationship between the distance (km) from point A and the time (hours) is shown in the graph. After how many hours of departure will the two meet?
|
2.1
|
NULL
|
free_form
|
func229
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/229.png
|
The graph of the linear function y = kx + b and y = -2x + 4 is shown in the figure. What is the solution set for the inequality kx + b ≥ -2x + 4?
|
x ≥ 1/2
|
NULL
|
free_form
|
func230
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/230.png
|
In the same coordinate system, as shown in the figure, the graphs of the linear functions y = k1x, y = k2x, y = k3x, y = k4x are represented by l1, l2, l3, and l4 respectively. The relationship of the magnitudes of k1, k2, k3, and k4 is __________________.
|
k3 > k4 > k1 > k2
|
NULL
|
free_form
|
func231
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/231.png
|
As shown in the figure, in the plane Cartesian coordinate system, E is the midpoint of BC. Given A(0,4), B(-2,0), C(8,0), and D(4,4), point P is a moving point on line segment BC. When the length of BP is ______, the quadrilateral with vertices at P, A, D, and E is a parallelogram.
|
1 or 9
|
NULL
|
free_form
|
func232
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/232.png
|
As shown in the figure, given the inverse proportional function y = k / x (k is a constant, k ≠ 0) passes through point A. Draw AB perpendicular to the x-axis at point A, with the foot of the perpendicular at B. If the area of △AOB is 4, then k =
|
-8
|
NULL
|
free_form
|
func233
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/233.png
|
As shown in the figure, P is a point on the graph of the parabola y = x^2 − 3x − 4 in the fourth quadrant. Draw perpendicular lines from point P to the x-axis and y-axis, with foot points A and B, respectively. What is the maximum perimeter of the quadrilateral OAPB? ________.
|
16
|
NULL
|
free_form
|
func234
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/234.png
|
It is known that the inverse proportional function y = -6 / x (x < 0), y = 2 / x (x > 0) has its graph shown in the figure. From any point P on the positive half of the x-axis, draw a line parallel to the x-axis, intersecting the graphs of these two functions at points M and N respectively. If point A is any point on the x-axis, and lines M A and N A are connected, then S_AMN equals ________.
|
4
|
NULL
|
free_form
|
func235
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/235.png
|
As shown in the figure, in the plane Cartesian coordinate system, the coordinates of points A and B are (-2, 0) and (0, 3), respectively. Taking point A as the center of the circle and AB as the radius, draw an arc intersecting the positive x-axis at point C. What are the coordinates of point C?
|
(√13 − 2, 0)
|
NULL
|
free_form
|
func236
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/236.png
|
As shown in the figure, a rectangular vegetable garden ABCD is to be enclosed. One side AD is a wall, and the length of AD cannot exceed 26 meters. The other three sides AB, BC, and CD are surrounded by a fence, with the total length of these three sides being 40 meters. The following conclusions are given: ① The length of AB can be 6 meters; ② The length of AB has two different values that meet the condition where the area of the garden ABCD is 192㎡; ③ The maximum area of the garden ABCD is 200㎡. Among these, the correct conclusions are ________.
|
③②
|
NULL
|
free_form
|
func237
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/237.png
|
As shown in the figure, in the rectangle OABC and the square CDEF, point A lies on the positive half of the y-axis, points C and F are both on the positive half of the x-axis, and point D is on side BC, where BC = 2CD, and AB = 1. If points B and E lie on the same rectangular hyperbola, then the expression for this rectangular hyperbola is __________.
|
y = 2/x
|
NULL
|
free_form
|
func238
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/238.png
|
As shown in the figure, in a Cartesian coordinate system, the rectangle OABC has two sides OA and OC lying on the x-axis and y-axis, respectively, and OA = 4, OC = 3. If rectangle OABC is rotated counterclockwise around point O such that point A lands exactly on point A1 on the BC side, what are the coordinates of point A1?
|
(√7, 3)
|
NULL
|
free_form
|
func239
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/239.png
|
As shown in the figure, the equation of line AB is y1 = k1x + b1, and the equation of line AC is y2 = k2x + b2. They intersect the x-axis at points B and C, respectively, and the x-coordinates of points B, A, and C are -2, -1, and 2. The range of x values that satisfy y1 > y2 > 0 is ______.
|
-1 < x < 2
|
NULL
|
free_form
|
func24
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/24.png
|
The graph of the quadratic function y = ax^2 + bx + c is shown. When the function value y < 0, what is the range of the independent variable x?
|
-1 < x < 3
|
NULL
|
free_form
|
func240
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/240.png
|
Given the quadratic function y = -x^2 + x + 6 and the linear function y = -2x + m, the graph of the quadratic function above the x-axis is reflected downward across the x-axis, while the rest of the graph remains unchanged, resulting in a new function (as shown in the diagram). When the line y = -2x + m intersects this new graph at 4 points, the range of m is ______.
|
-25/4 < m < -4
|
NULL
|
free_form
|
func241
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/241.png
|
The graph of the quadratic function y = x² - x - 2 is shown in the figure. Then the range of x when the function value y ≥ 0 is __.
|
x ≤ -1 or x ≥ 2
|
NULL
|
free_form
|
func242
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/242.png
|
As shown in the figure, given points M(1,3) and N(3,3), one branch of the hyperbolic function y=k/x (k≠0) intersects the line segment MN. How many integer values of k satisfy this condition?
|
7
|
NULL
|
free_form
|
func243
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/243.png
|
As shown in the figure, A and B are points on the hyperbolic function y = k / x (x > 0). Connect OA and OB. Through point A, draw AC perpendicular to the x-axis at point C, intersecting OB at point D. If AD : DC = 2 : 3, the area of △AOD is 2, and the coordinates of point B are (m, 2), then the value of m is ______.
|
5
|
NULL
|
free_form
|
func244
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/244.png
|
As shown in the figure, the line y = kx + b passes through the point (-1, 2). The solution set of the inequality (k + 2)x + b > 0 with respect to x is ________.
|
x > -1
|
NULL
|
free_form
|
func245
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/245.png
|
The quadratic function y = ax^2 + bx has a graph as shown. If the quadratic equation ax^2 + bx = m in terms of x has real roots, the value of m could be _________ (provide one possible value).
|
-2 (the answer is not unique)
|
NULL
|
free_form
|
func246
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/246.png
|
Ice and snow sports are becoming increasingly popular. In a freestyle skiing big air training session, an athlete slides off a jump 2m high, and their trajectory follows a parabolic path. Let their horizontal distance from the edge of the platform be x m, and their vertical height from the horizontal plane at the base of the jump be y m. The functional relationship between y and x is given by y = -1/24x^2 + 2x + 2 (0 ≤ x ≤ 16). When their horizontal distance from the edge of the platform is ______ m, the vertical height reaches its maximum value.
|
6
|
NULL
|
free_form
|
func247
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/247.png
|
As shown in the figure, given that the line y₁ = x + m and y₂ = kx - 1 intersect at point P(-3, 2), what is the solution set for the inequality x + m < kx - 1 with respect to x?
|
x < -3
|
NULL
|
free_form
|
func248
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/248.png
|
It is known that the graph of the linear function y = kx + b is shown in the figure. When y > -4, the range of x is __________.
|
x > 0
|
NULL
|
free_form
|
func249
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/249.png
|
As shown in the figure, point P(3a, a) is an intersection point of the hyperbolic function graph y = k/x (k > 0) with O. The area of the shaded region in the figure is 10π. Then, k = ______.
|
12
|
NULL
|
free_form
|
func25
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/25.png
|
As shown in the figure, point A lies on the graph of the inverse proportional function y = k/x in the first quadrant, point B is on the positive half-axis of the x-axis, OA = AB, and the area of △AOB is 3. What is the value of k?
|
3
|
NULL
|
free_form
|
func250
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/250.png
|
As shown in the figure, let points A and B lie on the hyperbolic function y = k/x (x > 0). From point A, draw AC perpendicular to the x-axis at point C, and from point B, draw BD perpendicular to the y-axis at point D. Draw BE perpendicular to the x-axis at point E, and connect AD. Given that AC = 1, BE = 1, and the area of rectangle OEBD is 5, find tan ∠ADC = ______.
|
1/4
|
NULL
|
free_form
|
func251
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/251.png
|
As shown in the figure, in the Cartesian coordinate plane, O is the origin, A(4, 4.8), B(10, 0). Connect AO and AB. Points C and D are respectively on line segments AB and OB. The triangle △CDB is folded along the line containing CD such that the corresponding point D' of D falls on line segment AO, and BD' = OB'. What are the coordinates of point D ________?
|
(40/13, 30/13)
|
NULL
|
free_form
|
func252
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/252.png
|
As shown in the figure, the hypotenuse AB of the right triangle △AOB intersects the x-axis at point B, and AC = BC. The graph of the inverse proportional function y = k / (x < 0) passes through point A. If B(3, 1) and the area of △AOC is 7.5, then the value of k is ______.
|
-27
|
NULL
|
free_form
|
func253
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/253.png
|
As shown in the figure, the equation y = k/x represents the hyperbolic curve above the x-axis. The area of parallelogram OABC is 5. If point A lies on the x-axis, point B lies on the curve y = k1/x, and point C lies on the curve y = k2/x, what is the value of k2 - k1?
|
-5/2
|
NULL
|
free_form
|
func255
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/255.png
|
As shown in the figure, in the rectangular coordinate system, it is known that line l1 and line l2 intersect at point C(0,1), and the angle between them is 90°. Line l1 intersects the negative x-axis at point A, and line l2 intersects the positive x-axis at point B(2,0). What are the coordinates of point A?
|
(-1/2, 0)
|
NULL
|
free_form
|
func256
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/256.png
|
As shown in the figure, in the plane Cartesian coordinate system, the straight line AB intersects the hyperbolic function y = k/x (x > 0) at points A and B, intersects the x-axis at point C(3, 0), and intersects the y-axis at point D(0, 2). If AD = AB = BC, then k = ______.
|
4/3
|
NULL
|
free_form
|
func257
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/257.png
|
As shown in the figure, point A is on the negative half of the x-axis, point C is on the positive half of the y-axis, and the quadrilateral OABC is a rectangle. Point D is the midpoint of AB. The graph of the inverse proportion function y = -k/x (x < 0) passes through point D and intersects BC at point E. Connect DE. If S_ABDE = 1, what is the value of k?
|
-4
|
NULL
|
free_form
|
func258
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/258.png
|
As shown in the diagram, a moving point P starts from point B and travels along the edges of the figure (adjacent edges are perpendicular) at a speed of 2 cm per second, following the path B→C→D→E→F→A. The corresponding area (cm²) of △ABP and the graph of the motion time t (s) of point P are shown in the second diagram. Given that AB = 8 cm and the area S = 18 cm², t = _______ seconds.
|
3 or 14
|
NULL
|
free_form
|
func259
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/259.png
|
As shown in the figure, A(2√2, 0), AB = 3√2. Taking point A as the center of the circle and the length of AB as the radius, an arc intersects the negative x-axis at point C. The coordinates of point C are ______.
|
(-√2, 0)
|
NULL
|
free_form
|
func26
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/26.png
|
As shown in the graph of the quadratic function y = ax² + bx + c, the following information is given: ①c > 0; ②a < 0; ③a − b + c > 0; ④b² > 4ac; ⑤2a − 2b − 2h. Which of these are correct (fill in the sequence numbers).
|
③④⑤
|
NULL
|
free_form
|
func260
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/260.png
|
As shown in the figure, the graph of the direct proportion function y = k₁x intersects with the graph of the inverse proportion function y = k₂/x at point A. If the coordinates of point A are (1, 2), then the solution set for the inequality k₁x > k₂/x with respect to x is ______.
|
−1 < x < 0 or x > 1
|
NULL
|
free_form
|
func261
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/261.png
|
As shown in the figure, in the isosceles triangle △ABO, the side OA lies on the negative half of the x-axis, point B is in the second quadrant, and the area of △ABO is 6. The hyperbolic function y=k/x passes through point B. What is the value of k?
|
-6
|
NULL
|
free_form
|
func262
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/262.png
|
In a Cartesian coordinate system, a moving point follows the direction as shown in the diagram: (0, 0) → (1, 1) → (2, 2) → (2, 3) → (3, 3) → (4, 4) → ..., and so on. According to this pattern, with (0, 0) being the 1st point, find the coordinates of the 78th point.
|
(520, 521)
|
NULL
|
free_form
|
func263
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/263.png
|
As shown in the figure, the graph of the direct proportional function y = -3x intersects with the linear function y = kx + b (k > 0) at the point (m, -6). The solution set of the inequality kx + b < -3x with respect to x is ________.
|
x < 2
|
NULL
|
free_form
|
func264
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/264.png
|
As shown in the figure, the graph of the linear function y = kx + b intersects the x-axis and y-axis at points A(2, 0) and B(0, -1.5), respectively. What is the range of the independent variable x when y < 0?
|
x < 2
|
NULL
|
free_form
|
func265
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/265.png
|
A straightedge with a 45° angle and a triangle board are placed as shown in the diagram on the Cartesian coordinate plane, where A(-2, 0) and B(0, 1). What is the equation of the line BC?
|
y = -1/3x + 1
|
NULL
|
free_form
|
func266
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/266.png
|
As shown in the figure, the graph of the linear function y = kx + b passes through the points (-1,0) and (0,-2). The solution set for the inequality kx + b < 0 is ______.
|
x < -1
|
NULL
|
free_form
|
func267
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/267.png
|
As shown in the figure, given the line l: y = √3x, passing through point M(2, 0), draw a perpendicular to the axis intersecting line l at point N. From point N, draw a perpendicular to line l intersecting the x-axis at point M1; from point M1, draw a perpendicular to line l intersecting line l at point N1; from point N1, draw a perpendicular to line l intersecting the x-axis at point M2...; continuing this process, the coordinates of point Mn are ______.
|
(2^4n, 0)
|
NULL
|
free_form
|
func268
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/268.png
|
A and B both ride bicycles at a constant speed, departing from location A towards location B. The relationship between their distance from A, v (in km), and the riding time, t (in hours), is shown in the graph. When t = 2, how far apart are A and B (in km)?
|
5
|
NULL
|
free_form
|
func269
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/269.png
|
As shown in the diagram, a rectangular vegetable garden ABCD is enclosed using a 60-meter-long fence, with one side adjacent to a wall (the wall's length is unlimited). Let the length of the fence AB parallel to the wall be x meters, and the area of the vegetable garden be y m². Write the relationship between y and x ______.
|
y = -1/2x² + 30x
|
NULL
|
free_form
|
func27
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/27.png
|
As shown in the figure, the vertex of the parabola y = x² - 4x + 2 is A, and it intersects the y-axis at point B. What is the equation of line AB? ______
|
y = -2x + 2
|
NULL
|
free_form
|
func270
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/270.png
|
As shown in the figure, the straight line l parallel to the x-axis intersects the graphs of the functions y=6/x (x>0) and y=2/x (x>0) at points A and B, respectively. Connecting AO and BO, the area of △ABO is _______.
|
2
|
NULL
|
free_form
|
func271
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/271.png
|
The water dispenser in the classroom automatically heats when powered on. During heating, the water temperature rises by 10°C per minute, stopping at 100°C. Afterward, the temperature begins to drop, and at this stage, the water temperature y (in °C) is inversely proportional to time t (in minutes). When the water temperature drops to 30°C, the dispenser restarts the heating process. The relationship between water temperature y (in °C) and time t (in minutes) is shown in the diagram. During the process of heating from 30°C to 100°C and then cooling back to 30°C, the time during which the water temperature remains no less than 50°C is ______min.
|
12
|
NULL
|
free_form
|
func272
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/272.png
|
As shown in the figure, in the rectangular coordinate system xOy, there are three points A(0, 1), B(4, 1), C(5, 6). What is sin∠BAC? ______.
|
√2/2
|
NULL
|
free_form
|
func273
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/273.png
|
As shown in the figure, in the Cartesian coordinate plane xOy, the quadrilateral ABCD is a rhombus, ∠ABC=120°, and the coordinates of point B are (0, -3). What are the coordinates of point A?______.
|
(-3√3, 0)
|
NULL
|
free_form
|
func274
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/274.png
|
As shown in the figure, it is a part of the graph of the inverse proportional function y = k + 3 / x (k is a constant and k ≠ -3, x < 0). What could be the value of k? (Write only one)
|
-5 (not unique)
|
NULL
|
free_form
|
func275
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/275.png
|
As shown in the figure, it is known that the graphs of a directly proportional function and a linear function y2=1/2x+b intersect at point P. Below are four statements: ①a<0; ②b<0; ③When x>0, y1>y2; ④When x=-2, y1=y2. Which of these statements are correct? (Only fill in the serial numbers)
|
①④
|
NULL
|
free_form
|
func276
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/276.png
|
Given the function y=|x²-4| with a graph roughly as shown in the figure, for the equation |x²-4|=m (where m is a real number), if the equation has exactly 3 distinct real roots, what is the value of m?
|
4
|
NULL
|
free_form
|
func277
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/277.png
|
As shown in Figure ①, in the right triangle △ABC, ∠B = 90°, a moving point P starts at point A and moves along the broken line A—B—C at a speed of 2 units per second (stopping at point C). The function graph of the length of BP changing with the time t (in seconds) is shown in Figure ②. What is the length of BC?
|
8
|
NULL
|
free_form
|
func278
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/278.png
|
As shown in the figure, the graphs of the linear function y = kx + b and the reciprocal function y = m/x intersect at points A(1,6) and B(n,-2). Then the solution of the equation kx + b = m/x is ____________.
|
1 or -3
|
NULL
|
free_form
|
func279
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/279.png
|
Jialu conducts an electric experiment with a sliding rheostat in a dynamic circuit. The power supply voltage remains constant, and the relationship between current I (A) and resistance R (Ω) is shown in the diagram.
(1) The power supply voltage is ______ V;
(2) The sliding rheostat is labeled as '20Ω 2A,' where '20Ω' indicates that the maximum resistance of the rheostat in the circuit is 20Ω, and '2A' indicates that the maximum allowable current through the rheostat is 2A. The minimum resistance of the sliding rheostat connected to the circuit is ______ Ω.
|
3; 1.5
|
NULL
|
free_form
|
func28
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/28.png
|
As shown in the figure, a radar detector measures the results. If the position of target A in the figure is recorded as (2, 90°), the position of target B is (3, 30°), and there is a target C at position (4, x°), where the distance between target C and target B is 5, then the value of x is______.
|
120 or 300
|
NULL
|
free_form
|
func280
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/280.png
|
As shown in the figure, in the rectangular coordinate plane, the quadrilateral OABC is a rhombus. If the coordinates of point A are (8, 15), what is the perimeter of the rhombus__________?
|
68
|
NULL
|
free_form
|
func281
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/281.png
|
As shown in the figure, D is a point on the graph of the inverse proportional function y=1/k (k>0) in the first quadrant. Connect O and extend to point B, such that BD=OD. From point D, draw a perpendicular BA to point A, which intersects the graph of the inverse proportional function y=1/k (k>0) at point C.
(1) If ∠AOB=60° and OD=2, then k=____.
(2) BC÷AC=____.
|
√3
|
NULL
|
free_form
|
func283
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/283.png
|
Place 5 identical squares in a Cartesian coordinate plane. If the coordinates of vertices M and N are (-3, 2) and (9, 2) respectively, then the coordinates of vertex A are ________.
|
(13, -6)
|
NULL
|
free_form
|
func284
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/284.png
|
As shown in the figure, in a rectangular coordinate system, point A lies on the graph of the inverse proportional function y = k/x (x > 0), B and C are two points on the positive half of the x-axis, AB = AC, OB = BC. If the area of △ABC is 4, the value of k is ______.
|
12
|
NULL
|
free_form
|
func285
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/285.png
|
As shown in the figure, in the plane Cartesian coordinate system, the coordinates of points A, B, and C are (1,1), (4,1), and (2,3) respectively. If the line y = x + b intersects ΔABC, the range of b is ______.
|
−3 ≤ b ≤ 1
|
NULL
|
free_form
|
func286
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/286.png
|
As shown in the figure, a chain consists of several uniform iron rings fully stretched to the maximum extent. Each iron ring is 4 cm long and 0.5 cm thick, and the rings are in a state of maximum stretch. Let the length of the chain with n rings be y cm. Write the relationship between y and n: ________.
|
y = 3n + 1
|
NULL
|
free_form
|
func287
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/287.png
|
As shown in the figure, given A1(1,0), A2(1,-1), A3(-1,-1), A4(-1,1), A5(2,0), ..., find the coordinates of point A2024.
|
(-506,506)
|
NULL
|
free_form
|
func288
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/288.png
|
As shown in the figure, in the coordinate plane oxy, the line y = kx + b intersects with y = mx + n at point M(2,4). Which of the following conclusions are correct? (Fill in the number). ① The solution to the system of equations y = kx + b and y = mx + n with respect to y is x = 2; ② The solution set of the two inequalities kx + b < mx + n and kx + b > mx + n with respect to x is x < 2; ③ k + b > 0; ④ m > 0.
|
③④
|
NULL
|
free_form
|
func29
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/29.png
|
As shown in the figure, a square grid is placed in a certain plane Cartesian coordinate system. The coordinates of point B are (-1, -2). What are the coordinates of point C?
|
(2, 1)
|
NULL
|
free_form
|
func290
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/290.png
|
As shown in the figure, in a Cartesian coordinate plane, circle ⊙A is tangent to the x-axis at point B. Draw the diameter BC. The graph of the function y = 20/x (x > 0) passes through point C. Point D is any point on the y-axis. Then the area of △ACD is ________.
|
5
|
NULL
|
free_form
|
func291
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/291.png
|
As shown in the figure, the line y = -2x + 2 intersects with the line y = kx + b (where k, b are constants and k ≠ 0) at point A(m, 4). Then the solution set of the inequality -2x + 2 < kx + b is ______.
|
x > -1
|
NULL
|
free_form
|
func292
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/292.png
|
In the rectangular coordinate system xOy, point A(-2,3), point B(-1,0), point D(2,3), and point C lies on the x-axis. If CD = AB, what are the coordinates of point C?
|
(1,0) or (3,0)
|
NULL
|
free_form
|
func293
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/293.png
|
As shown in the figure, the coordinates of three vertices of □ABCD are A(4, -2), B(5, 2), C(2, 2). What are the coordinates of the fourth vertex D?
|
(1, -2)
|
NULL
|
free_form
|
func294
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/294.png
|
As shown in the figure, the 'heart' pattern is composed of part of the graph of the function y = −x^2 + 10 and its symmetric graph about the line y = −x. Point A is any point on the 'heart' pattern above the line y = −x, and point B is its symmetric point. When AB = √2, the coordinates of point B are __________.
|
(-2, 2) or (1, 5)
|
NULL
|
free_form
|
func295
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/295.png
|
As shown in the figure, given that the straight line y = kx (k ≠ 0) intersects the hyperbola y = 2/x at two points A(m, 1) and B, the solution set of the inequality y > kx is ____________.
|
x < -2 or 0 < x < 2
|
NULL
|
free_form
|
func296
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/296.png
|
In the coordinate grid shown in the figure, write an integer coordinate where its distances to the two coordinate axes are equal: ________.
|
(1,1) (the answer is not unique)
|
NULL
|
free_form
|
func297
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/297.png
|
As shown in the figure, in △OAB, the coordinates of vertices A and B are (2, 4) and (6, 0), respectively. When △OAB is translated horizontally along the x-axis to form △CDE:
(1) If CB = 3, then the length of OE is ________;
(2) Connect AD. Under the condition in (1), the area of quadrilateral OADE is ________.
|
(1) 21/2
(2) 30
|
NULL
|
free_form
|
func299
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/299.png
|
As shown in the figure, the graphs of the functions y = kx + b and y = mx intersect at point P. Then the solution to the two-variable linear equation system { y = kx + b, y = mx } concerning x and y is ________.
|
{ x = 2, y = 1 }
|
NULL
|
free_form
|
func3
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/3.png
|
As shown in the figure, the parabola y = x² is translated to obtain parabola m, which passes through point A(-2, 0) and the origin. Its vertex is P, and its axis of symmetry intersects the parabola y = -x² at point O. What is the area of the shaded region in the figure?
|
1
|
NULL
|
free_form
|
func30
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/30.png
|
As shown in the figure, it is known that the graphs of the linear functions y = ax - 3 and y = bx + 12 intersect at point P(5, 7). Then the solution set for the inequality ax - 3 < bx + 12 is _______.
|
x < 5
|
NULL
|
free_form
|
func300
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/300.png
|
As shown in the diagram, in a plane Cartesian coordinate system, a moving point starts from the origin O and moves continuously in the upward and rightward directions, moving one unit at a time. Points A1(0, 1), A2(1, 1), A3(1, 0), A4(2, 0) ... are obtained as a result. What are the coordinates of point A42001 (n is a natural number)?
|
(1000, 1)
|
NULL
|
free_form
|
func301
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/301.png
|
To reasonably schedule commute and school time, a school investigation team surveyed the time spent commuting to and from school in one day among grade 9 students. In this survey, 20 students from grade 9 were randomly selected. The time spent commuting to school was used as the horizontal coordinate and the time spent commuting from school was used as the vertical coordinate to establish a plane Cartesian coordinate system. Based on the survey results, corresponding points were plotted, as shown in the figure: Given that the grade 9 cohort of the school has 400 students in total, estimate the number of grade 9 students whose commute to school does not exceed 15 minutes. ______ students.
|
280
|
NULL
|
free_form
|
func302
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/302.png
|
As shown in the figure, the graph of the linear function y = kx + b passes through the point (1, -3). When y < -3, the range of values for x is ______.
|
x > 1
|
NULL
|
free_form
|
func303
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/303.png
|
Under the condition of constant temperature, the pressure P (kPa) produced by a gas is inversely proportional to the volume V (mL) of the gas. The graph shows the function of the pressure P of a certain gas after compression with respect to the volume V of the gas. If the pressure decreases from 100 kPa to 80 kPa, then the corresponding increase in the gas volume is ______ mL.
|
15
|
NULL
|
free_form
|
func305
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/305.png
|
As shown in the figure, the functions y = -2x and y = kx + 4 intersect at point A(m, 3). Then the solution set of the inequality kx + 4 + 2x ≤ 0 is ________.
|
x ≤ -1.5
|
NULL
|
free_form
|
func306
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/306.png
|
As shown in Figure 1, in rectangle ABCD, moving point P starts at point B and moves along BC, CD, and DA until it stops at point A. Let the path of point P be x, and the area of △ABP be y. If the function graph of y with respect to x is as shown in Figure 2, the length of the diagonal of the rectangle is ________.
|
√41
|
NULL
|
free_form
|
func307
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/307.png
|
As shown in the figure, the linear function y = kx + b passes through points A(5, 0) and B(0, -2). Then, the solution set of the inequality kx + b < 0 with respect to x is ________.
|
x < 5.
|
NULL
|
free_form
|
func308
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/308.png
|
As shown in the figure, the straight lines y = kx + b and y = mx + n intersect the x-axis at points A(-1/2, 0) and B(2, 0), respectively. Then the solution set of the inequality system {kx + b < 0, mx + n < 0} is ______.
|
-1/2 < x < 2
|
NULL
|
free_form
|
func309
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/309.png
|
As shown in the figure, in the 2D Cartesian coordinate system, the quadrilateral AOCD is a parallelogram. The coordinates of A are (1,2), O is (0,0), and C is (3,0). What are the coordinates of point D?
|
(4,2)
|
NULL
|
free_form
|
func31
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/31.png
|
The hyperbola y=4/x intersects the straight line y=kx at points A and B. Passing through point B, draw BC perpendicular to the x-axis at point C, and connect AC. Then, the area of △AOC is ________.
|
2
|
NULL
|
free_form
|
func310
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/310.png
|
As shown in the figure, in a Cartesian coordinate system, the line y=4/3x+4 intersects the axes at points A and B, respectively. From point O, draw OC perpendicular to AB, intersecting AB at point C. What is the length of OC? ______.
|
12/5
|
NULL
|
free_form
|
func311
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/311.png
|
As shown in the figure, the intersection of the two lines l1: y = 2x + b and l2: y = x - 2 is (5, 3). Then, the solution set of the inequality 2x + b > x - 2 with respect to x is ________.
|
x > 5.
|
NULL
|
free_form
|
func312
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/312.png
|
As shown in the figure, point A is on the graph of the inverse proportional function y = k * (x > 0). Through point A, draw AB perpendicular to the y-axis. Point C is on the positive half of the y-axis. If the area of △ABC is 3, then the value of k is ________.
|
-6
|
NULL
|
free_form
|
func313
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/313.png
|
As shown in the figure, it is known that the graph of the function y = ax + b intersects the graph of the function y = kx - 3 at point P(4, -6). Then the solution set of the inequality ax + b ≤ kx - 3 is __________.
|
x ≤ 4
|
NULL
|
free_form
|
func314
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/314.png
|
As shown in the figure, P is a point on the graph of the inverse proportional function y = k/x (k ≠ 0, x > 0). A perpendicular is drawn from point P to the x-axis at point A, and point B is the reflection of point A about the x-axis. Connect PB. If the area of △PAB is 16, the value of k is ________.
|
16
|
NULL
|
free_form
|
func315
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/315.png
|
As shown in the figure, it is known that parallelogram OABC is translated to the right to obtain parallelogram OA′B′C′. Point A′ happens to be the midpoint of AB, where point A has coordinates (√3, 2) and point B has coordinates (3√3, 2). What are the coordinates of point A′? _______
|
(2√3, 2)
|
NULL
|
free_form
|
func316
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/316.png
|
As shown in the figure, in the rectangular coordinate system, the coordinates of point A are (-1, 0), point B is on the positive half-axis of the y-axis. Taking point B as the center and the length of BA as the radius, an arc intersects the positive half-axis of the x-axis at point C. What are the coordinates of point C? ________.
|
(1,0)
|
NULL
|
free_form
|
func317
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/317.png
|
As shown in the figure, during an activity, Class A located at point A intends to travel 5 km to point B to meet with Class B. Describe the position of Class B relative to Class A in terms of direction and distance.
|
5 km in the direction of 40° west of south.
|
NULL
|
free_form
|
func318
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/318.png
|
As shown in the figure, in a rectangular coordinate system, quadrilateral ABCD is a square with A(1, 1), B(-1, 1), C(-1, -1), and D(1, -1). The curve A4A1, A1A2, A2A3... is called the 'involute of the square,' where the centers of arcs A4A1, A1A2, A2A3... are points B, C, D, and A in sequence, repeating cyclically. What are the coordinates of point A2024?
|
(4049, -1)
|
NULL
|
free_form
|
func32
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/32.png
|
As shown in the figure, in the Cartesian coordinate system, point A lies on the graph of the function y = -(k/x) (x < 0). A perpendicular is drawn from point A to the x-axis, intersecting the y-axis at point B. Point C lies on the x-axis. If the value of k is -8, then the area of △ABC is ________.
|
4
|
NULL
|
free_form
|
func322
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/322.png
|
As shown in the figure, O is the origin of the coordinates. In the parallelogram OABC, the coordinates of vertex A are (-3, 4), vertex C lies on the negative half of the x-axis, and the graph of the inverse proportional function y = k/x passes through vertex B in the second quadrant. If the area of parallelogram OABC is 16, then the value of k is ________.
|
-28
|
NULL
|
free_form
|
func323
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/323.png
|
As shown in the figure, the height AD of triangle ABC is 4, BC = 8, point E is on the side BC, and AE is connected. If the length of BE is x, and the area of triangle ACE is y, then the relationship between y and x is __________________.
|
y = 16 - 2x
|
NULL
|
free_form
|
func324
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/324.png
|
In the figure, the graphs of the inverse proportional function y = k / x and the linear function y = ax + b are shown. The coordinates of the intersection points A(2, 3) and B(m, −2) are given. Directly write the inequality solution for ax + b > k / x: __________.
|
x > 2 or −3 < x < 0
|
NULL
|
free_form
|
func325
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/325.png
|
In the rectangular coordinate plane xOy, the graphs of the linear functions y1 = kx + b and y2 = x + m are shown as in the figure. If the x-coordinate of their intersection is 2, which of the following statements are correct? ① The acute angle between the line y2 = x + m and the x-axis is 45°; ② k + b > 0; ③ The solution set of the inequality kx + b < x + m with respect to x is x < 2; ④ mk > 0.
|
①②④
|
NULL
|
free_form
|
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