id
stringlengths 4
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stringlengths 77
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| query
stringlengths 7
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| answer
stringlengths 1
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| choice
stringlengths 4
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| question_type
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|---|---|---|---|---|---|
func520
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/520.png
|
As shown in the figure, point A lies on the graph of the inverse proportion function y=k1/x (x>0, k1>0), and point B lies on the graph of the inverse proportion function y=k2/x (x>0, k2>0). Segment AB is parallel to the x-axis, CD is perpendicular to the x-axis at point D, and intersects AB at point E. When the difference between the areas of triangle △ABC and triangle △DBC is 6, and k1=3k2, the value of k2 is _______. The area of quadrilateral ABDC is _______ (expressed in an algebraic expression with k2).
|
-18, 5k2+90/6
|
NULL
|
free_form
|
func521
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/521.png
|
As shown in the figure, in a square grid, if point A(-1,1) and point B(0,-1), what are the coordinates of point C ______.
|
(1, -2)
|
NULL
|
free_form
|
func523
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/523.png
|
As shown in the figure, in the parallelogram ABCD, the coordinates of vertices A, B, and D are A(3,0), B(-2,0), and D(0,4), respectively. What are the coordinates of point C?
|
(-5,4)
|
NULL
|
free_form
|
func524
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/524.png
|
As shown in the figure, in the rectangular coordinate system xOy, the graph of the function y = 9/x (x > 0) passes through two points A(x1, y1) and B(x2, y2). If the area of △ABO is 9/2, then the value of x1 + 1/x1 + x2 + 1/x2 is ________.
|
25/2.
|
NULL
|
free_form
|
func525
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/525.png
|
As shown in the figure, point P is a point on the function graph y=k/x (k≠0). A perpendicular is drawn from point P to the x-axis at point A. If point B is the midpoint of OA and the area of △PAB is 2, then k = ________.
|
8
|
NULL
|
free_form
|
func526
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/526.png
|
In the activity "Functions in Life," a study group designed a problem scenario: Xiao Ming runs from home to the stadium, exercises there for a while, then walks to the stationery store to buy a compass, and finally strolls back home. The relationship between Xiao Ming's distance from home (y/km) and the time he spent (x/min) is shown in the diagram. When Xiao Ming is 2 km away from home, the time he has been away from home is ______ minutes.
|
12 minutes
|
NULL
|
free_form
|
func527
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/527.png
|
As shown in the figure, in the Cartesian coordinate system, the graph of the inverse proportional function y = k/x (k is a constant, k ≠ 0, x < 0) intersects the sides OA and AB of the right triangle △AOB at points C and D, respectively. ∠ABO = 90°, point C is the midpoint of OA, and point D lies on the x-axis. Given that tan∠A = 1/2 and OB = 2, what is the value of k?
|
-2
|
NULL
|
free_form
|
func528
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/528.png
|
A school designs the main gate of a newly constructed library in the shape of a parabolic arch. The design requires that the product of the span and the arch's height of the designed parabola equals 48㎡. The design also needs to consider factors like aesthetics, harmony, and smoothness. The design department proposed a scheme based on this requirement, and the shape of the parabolic arch is represented in a plane Cartesian coordinate system, as shown in the figure. Here, point N lies on the x-axis, PE⊥ON, and OE=EN. If the span of the parabolic arch ON = 12m and the arch height PE = 4m, then the expression for the parabolic function is ______.
|
y = -1/9 x² + 4/3 x
|
NULL
|
free_form
|
func529
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/529.png
|
7. Car A and Car B depart from point M simultaneously to head to point N. As shown in the graph, the polyline O-A-B-C reflects the relationship between distance traveled and time for both cars. It is known that Car A stopped for 36 minutes midway before continuing to point N, and both cars arrived at N simultaneously. Based on the information, the following statements are proposed: (1) The speed of Car B is 70 km/h; (2) The speed of Car A after resuming its journey is 100 km/h; (3) The two cars will not meet before arriving at N; (4) When Car A resumes its journey, the two cars are 60 km apart. Which statements are correct? ________.
|
(2)(3)(4)
|
NULL
|
free_form
|
func53
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/53.png
|
As shown, given the inverse proportional function y = k - 2 / x, whose graph passes through vertex A of rectangle ABOC with an area of 8, what is the value of k?
|
-6
|
NULL
|
free_form
|
func530
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/530.png
|
As shown in the figure, in a Cartesian coordinate system, a moving point moves one unit length in the direction indicated by the arrow each time, obtaining points P1(0, 1), P2(1, 1), P3(1, 0), P4(1, -1), P5(2, -1), ..., what is the coordinate of P2023?
|
(674, 1)
|
NULL
|
free_form
|
func531
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/531.png
|
Given that the graph of the linear function y = kx + b intersects the y-axis at point A(0,2), the x-axis at point B(4,0), and the graph of the direct proportion function y = mx at point C(2,a). What is the solution of the system of equations (mx - y = 0, kx - y = -b)? ________.
|
x = 2, y = 1
|
NULL
|
free_form
|
func532
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/532.png
|
As shown in the figure, in the plane Cartesian coordinate system xOy, the line y = 2x intersects y^2 = -x + a at point P(1, 2). Then the solution set of the inequality 2x ≥ -x + a is ______.
|
x > 1
|
NULL
|
free_form
|
func533
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/533.png
|
When the water dispenser is plugged in, it will automatically heat the water. During heating, the water temperature rises by 10°C per minute and stops heating when it reaches 100°C. After that, the water temperature starts to drop. At this point, the water temperature v(°C) and time t(min) follow an inverse proportional function. When the water temperature drops to 30°C, the water dispenser repeats the process and starts heating again. The relationship between water temperature v(°C) and time t(min) during heating is shown in the figure. In the process where the water temperature rises from 30°C to 100°C and then drops back to 30°C, the duration for which the water temperature is not below 50°C is ______min.
|
12
|
NULL
|
free_form
|
func534
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/534.png
|
As shown in the figure, in the plane Cartesian coordinate system, the graph of the quadratic function y = ax^2 + bx + c intersects the x-axis at one point (-3, 0), and its axis of symmetry is the line x = -1. Then, the coordinate of the other intersection point of the graph with the x-axis is ________.
|
(1, 0)
|
NULL
|
free_form
|
func535
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/535.png
|
Class 9-2 plans to plant vegetables in the labor practice area. The class monitor bought an 8-meter-long fence, intending to enclose a garden with one side against a wall (the wall is long enough). To maximize the area of the garden, the students proposed three solutions: a rectangle, an isosceles triangle (with the base against the wall), and a semicircle. The best solution is _________.
|
Option 3
|
NULL
|
free_form
|
func536
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/536.png
|
As shown in the figure, the line v₁ = kx + b (k ≠ 0) intersects with v₂ = −x at point P(a, 1). Then the solution set of the inequality kx + b > −x regarding x is ________.
|
x > −1
|
NULL
|
free_form
|
func537
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/537.png
|
As shown in the figure, OA = AB, ∠BAO = 90°, OB = 2, and the parabola passes through points O, A, and B. Find the equation of the parabola __________________.
|
y = x² + 2x
|
NULL
|
free_form
|
func539
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/539.png
|
6. The partial graph of the quadratic function y = ax² + bx + c is shown in the figure. It intersects the y-axis at (0, -1) and its axis of symmetry is the line x = 1. The following conclusions are given: ① a > 0; ② ac > 0; ③ a > 1; ④ a + b + c = 0; ⑤ On the graph of this quadratic function, for any real number k, k² - 1 > 2; ⑥ The sum of all roots of the equation |ax² + bx + c| = 1 is 2. Which of the conclusions are correct?
|
②④⑥
|
NULL
|
free_form
|
func54
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/54.png
|
As shown in the figure, points A and B are on the x-axis, AB = 4, and the coordinates of point C are (0, -8). The graph of a quadratic function y = ax^2 + bx + c passes through points A and B, and the vertex is point D. If the quadrilateral ABCD is a parallelogram, and the graph of the quadratic function shifts downward to just pass through point C, then the equation of the shifted function is ________.
|
y = 2x^2 - 16x - 8
|
NULL
|
free_form
|
func540
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/540.png
|
As shown in the figure, in a rectangular coordinate plane, the coordinates of point P are (-4, 3). What is the length of OP?
|
5
|
NULL
|
free_form
|
func541
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/541.png
|
As shown in the figure, it is known that the side length of the equilateral triangle △ABC is 2. O is the origin of the coordinate system, point A lies on the x-axis, and point B is in the second quadrant. △ABC rolls along the x-axis without sliding in the positive direction. After the first roll, it becomes △A1B1C1, and so on. After 2024 rolls, the y-coordinate of the corresponding point M2024 to the midpoint M1 of △A is ______.
|
0
|
NULL
|
free_form
|
func542
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/542.png
|
As shown in the figure, the line y = -x + b intersects the line y = kx + 6 at point P(2,4). Then the solution set of the inequality kx + 6 < x + b with respect to x is ______.
|
x > 2
|
NULL
|
free_form
|
func543
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/543.png
|
As shown in the figure, this is a spinning wheel made during a math activity class. The surface of the wheel is divided into four equal fan-shaped regions, which are labeled with the numbers -3, -2, -1, and 1. If the wheel is spun twice, and the numbers in the regions where the pointer stops are recorded as m and n respectively (if the pointer lands exactly on a boundary line, the wheel needs to be spun again), then what is the probability that the line y = mx + n does not pass through the fourth quadrant?
|
1/16
|
NULL
|
free_form
|
func544
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/544.png
|
As shown in the figure, in the rectangle ABCD, a moving point P starts from point A and moves at a constant speed along the path A→B→C→D→A until it stops at point A. Let the path traveled by point P be x, and the area of △PCD be y. If the relationship between y and x is as shown in the figure, what is the area of rectangle ABCD?
|
21
|
NULL
|
free_form
|
func545
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/545.png
|
1. A circular fountain is to be built with a vertical pipe installed at the center of the pool. A spray head is attached to the top of the pipe, such that the water forms a parabolic trajectory reaching its highest point 1m horizontally from the center of the pool, with a height of 3m. The water falls at a point 3m away from the center of the pool. What should be the height of the pipe in meters?
|
9/4
|
NULL
|
free_form
|
func546
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/546.png
|
As shown in the figure, △OAB1, △B1A1B2, △B2A2B3, etc., are all equilateral triangles with a side length of 2. Points B1, B2, B3…, all lie on the straight line y = √3/3·x. What are the coordinates of point A2024?
|
(2024√3, 2026)
|
NULL
|
free_form
|
func547
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/547.png
|
As shown in the figure, the ray ○ represents the function graph of the profit difference (total ticket income minus operating cost) versus the passenger volume x for a certain bus route. Currently, the route is operating at a loss. To improve the situation without increasing ticket prices, the bus company decides to lower operating costs through optimized management. The graph of the adjusted relationship between x is represented by ray △. The intersection coordinates of the two rays with the x-axis are (1.5, 0) and (0.6, 0), respectively. When the passenger volume reaches 1 million people, the profit difference after optimization increases by ________ million yuan compared to before.
|
2/3
|
NULL
|
free_form
|
func548
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/548.png
|
As shown in the figure, in the rectangular coordinate system, point A(0,2), B(2,5), C(4,2), and D(7,0) are given. Connect AB and CD, and rotate line segment AB around a certain point by a certain angle so that it coincides with line segment CD (point A coincides with point C, and point B coincides with point D). What are the coordinates of the center of rotation? ______.
|
(2,0)
|
NULL
|
free_form
|
func549
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/549.png
|
As shown in the figure, vertex A of the square ABCD lies on the x-axis, E is the midpoint of AD, and the hyperbolic function y = -k / (x + k) (k ≠ 0) passes through vertex B of the square. If OA = 1 and tan∠OAE = 2, then k = ________.
|
10
|
NULL
|
free_form
|
func55
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/55.png
|
As shown in the diagram, on a grid with a unit length of 1, A1, A2, A3, A4, A5, A6, ..., all are isosceles right triangles with their hypotenuses lying on the x-axis. The hypotenuse lengths are 2, 4, 6, ... respectively. If the coordinates of the vertices of A1, A2, A3, A4 are A1(2, 0), A2(5, 1), A3(10, 2), A4(17, 3), then according to the pattern shown in the diagram, the x-coordinate of A2022 is _______.
|
1
|
NULL
|
free_form
|
func551
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/551.png
|
Which of the following scenarios can represent the functional relationship between y and x as shown in the given graph? ( )
|
C
|
['The area of a circle and its radius x', 'The perimeter of a square and its side length x', 'The height of a thrown object and the time t', 'The distance a child rides from school and the speed x']
|
multi_choice
|
func552
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/552.png
|
As shown in the figure, P(12, a) is on the graph of the inverse proportion function y = 60/x. PH ⊥ x-axis at point H, then sin∠POH = ( )
|
B. 5/13
|
['5/12', '5/13', '12/5', '12/13']
|
multi_choice
|
func553
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/553.png
|
As shown in the figure, if the parabola y = -x^2 + 4x - 2 is shifted upward by m units (m > 0), and there is only one intersection with the x-axis in the range -1 < x < 4, what is the range of values for m?
|
B
|
['2 ≤ x ≤ 7', '2 ≤ x < 7', '2 < x < 7', '2 ≤ x ≤ 7']
|
multi_choice
|
func554
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/554.png
|
As shown in the figure, the vertices of the equilateral triangle ABC are all located on the coordinate axes, with A(-2, 0). Draw a perpendicular BD from point B to AB, and the perpendicular BD intersects the x-axis at point D. What are the coordinates of point D?
|
C
|
['(4, 0)', '(4√3, 0)', '(6, 0)', '(8, 0)']
|
multi_choice
|
func555
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/555.png
|
A certain balloon is filled with a fixed mass of gas. When the temperature remains constant, the gas pressure P (kPa) inside the balloon is an inverse proportional function of the gas volume V (m³). Its graph is shown in the figure. When the gas volume is 2 m³, the gas pressure is ( ).
|
D
|
['150kPa', '120kPa', '96kPa', '48kPa']
|
multi_choice
|
func556
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/556.png
|
As shown in the figure, the graph of the quadratic function y = ax^2 + bx + c intersects the x-axis at points A(-1, 0) and B. The axis of symmetry is the line x = 1. Which of the following statements is correct?
A. a > 0
B. The coordinates of point B are (4, 0)
C. 4a + 2b + c > 0
D. When x -> -∞, the value of y increases as the value of x increases.
|
C
|
['a > 0', 'The coordinates of point B are (4, 0)', 'When \\(x > 1\\), the value of \\(y\\) increases as \\(x\\) increases', '4a + 2b + c > 0']
|
multi_choice
|
func557
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/557.png
|
As shown in the figure, the coordinates of the intersection of the linear functions y=x+2 and y=-3x+b are (a,3). Determine the solution of the linear equation system {x-y=-2, 3x+y=b} and the value of b. Then, what is the solution for the equation -3x+b=3 in terms of x?
|
A. x=1
|
['x = 1', 'x = 2', 'x = 3', 'x = 4']
|
multi_choice
|
func558
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/558.png
|
As shown in the figure, in the Cartesian coordinate plane, the isosceles right triangle OAA1 has its right angle side OA lying on the x-axis, and the coordinates of point A1 are (1,1). Using point A1 as the vertex of the right angle and OA1 as one of the right-angle sides, another isosceles right triangle OA1A2 is constructed. Then, using point A2 as the vertex of the right angle and OA2 as one of the right-angle sides, another isosceles right triangle OA2A3 is constructed, and so on. According to this pattern, what are the coordinates of point A2024 ( )?
|
D
|
['(2^1011, -2^1011)', '(2^1011, 0)', '(2^1012, -2^1012)', '(2^1012, 0)']
|
multi_choice
|
func559
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/559.png
|
It is known that the quadratic function y = ax^2 + bx + c (a, b, and c are constants, a ≠ 0) has a graph as shown in the figure. The following conclusions are given: ① a > 0; ② b² - 4ac < 0; ③ 2a + b = 0; ④ -a - b + c < 0. The number of correct conclusions is ( ).
|
C. 3
|
['1', '2', '3', '4']
|
multi_choice
|
func56
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/56.png
|
The driving speed of vehicles on a certain elevated bridge, y (km/h), is related to the number of vehicles per 100 meters on the elevated bridge, x (x is a positive integer), as shown in the figure. When x >= 10, y is inversely proportional to x. When the driving speed of vehicles drops below 21 km/h, traffic congestion occurs. To avoid traffic congestion, the number of vehicles per 100 meters on the elevated bridge should satisfy the range of ______.
|
0 < x <= 50
|
NULL
|
free_form
|
func560
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/560.png
|
Given the quadratic function y = ax^2 + bx + c (a ≠ 0), the graph is shown below. In which quadrant is point A(a, b, c) located?
A. First quadrant B. Second quadrant C. Third quadrant D. Fourth quadrant
|
B
|
['First quadrant', 'Second quadrant', 'Third quadrant', 'Fourth quadrant']
|
multi_choice
|
func561
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/561.png
|
As shown in the figure, two individuals, A and B, are riding a bicycle and a motorcycle respectively, both departing from the same location and traveling along the same route toward a destination 120 km away, as depicted in the figure. The functions l1 and l2 represent the relationships between the distance s (km) from the starting point and the travel time t (h) for A and B, respectively. How long after B's departure do they meet?
|
C. 2/3 hour
|
['7/3 hours', '5/3 hours', '2/3 hours', '1.5 hours']
|
multi_choice
|
func562
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/562.png
|
As shown in the diagram, in the same plane rectangular coordinate system, the linear functions y = k₁x + b₁ and y = k₂x + b₂ (where k₁/k₂ ≠ 0) are represented by lines l₁ and l₂, respectively. Then, the graph of the linear function y = k₁x - b₁ + b₂ passes through ( )
A. The first and third quadrants
B. The first, second, and fourth quadrants
C. The second, third, and fourth quadrants
D. The second, third, and fourth quadrants
|
C
|
['One, Three quadrants', 'One, Four quadrants', 'Two, Three quadrants', 'Two, Four quadrants']
|
multi_choice
|
func563
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/563.png
|
As shown in the figure, the parabola y = ax² intersects with the straight line y = bx at point A, where the x-coordinate of point A is 2. What is the solution set of the inequality ax² + bx > 0?
|
A. -2 < x < 0
|
['-2 < x < 0', '0 < x < 2', 'x > 2', 'x < -2']
|
multi_choice
|
func564
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/564.png
|
As shown in the figure, a parabola intersects the x-axis at points A and B. Its vertex P moves along a fold line. If the coordinates of points C, D, and E are (-1, 4), (3, 4), and (3, 1) respectively, and the maximum value of the x-coordinate of point A is 2, what is the minimum value of the x-coordinate of point B?
|
A
|
['1', '2', '3', '4']
|
multi_choice
|
func566
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/566.png
|
In the rectangular coordinate system, the graph of the linear function y = kx + b is shown in the figure. What are the ranges of k and b?
|
C
|
['k > 0, b > 0', 'k > 0, b < 0', 'k < 0, b > 0', 'k < 0, b < 0']
|
multi_choice
|
func568
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/568.png
|
It is known that points A and C are on the graph of the inverse proportional function y1 = k1 / x (k1 ≠ 0), and points B and D are on the graph of the inverse proportional function y2 = k2 / x (k2 ≠ 0). Given AB = 3, CD = 6, AB ∥ x-axis, CD ∥ x-axis, and the combined vertical distances of AB and CD from the x-axis is 9, then k2 - k1 = ( ).
|
C
|
['9', '16', '18', '20']
|
multi_choice
|
func569
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/569.png
|
As shown in the figure, the area of rhombus OABC is 8. Point B is on the y-axis, and point C is on the graph of the inverse proportional function y = (m/x). What is the value of m?
|
B
|
['-2', '-4', '-6', '-8']
|
multi_choice
|
func57
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/57.png
|
As shown in the figure, the square OABC has its vertices O and B on the x-axis and y-axis respectively. Point A(0,4), and point D(4,3) is located on the side BC. Triangle △ACD is rotated 90° clockwise about point A, resulting in △AOD'. Line AM bisects ∠DAD' and intersects OB at point M. What are the coordinates of point M?
|
(2.4, 0)
|
NULL
|
free_form
|
func570
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/570.png
|
As shown in the figure, the graph of the direct proportional function y = kx (k ≠ 0) intersects with the graph of the inverse proportional function y2 = k2/x (k2 > 0) at points A and B. Given that the x-coordinate of point B is 3, when 0 < y2 < y1, what is the range of values for x?
|
B
|
['-3 < x < 0', 'x < -3', 'x > 3', '-3 < x < 0 or x > 3']
|
multi_choice
|
func571
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/571.png
|
As shown in the figure, the parabola y = ax² + bx + c intersects the x-axis at points A and B, and intersects the y-axis at point C(0, -1). Point A lies between (-4, 0) and (-3, 0) (exclusive). The vertex of the parabola is D, and the minimum value of y is -2. The following statements are given: ① If a > 0, there exists a point (-5/2, -2) that lies on the parabola; ② If points M(-5/2, -2) and N(-5/2, 3) lie on the parabola, then v1 > v2; ③ a > -3; ④ If a = -1, then △ABD forms an equilateral triangle. Which of the statements is correct?
|
C
|
['1', '2', '3', '4']
|
multi_choice
|
func572
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/572.png
|
10. As shown in the figure, in the rectangular coordinate system, point O has coordinates (0,2). Using OA as a side, construct an equilateral triangle △OAA1 on the right. From point A1, draw a perpendicular line to the x-axis, with the foot of the perpendicular being O1. Using O1A1 as a side, construct an equilateral triangle △O1A1A2 on the right. Then, from point A2, draw a perpendicular line to the x-axis, with the foot of the perpendicular being O2. Continuing this process, construct equilateral triangles △O2A2A3, and so on. Following this pattern, what is the y-coordinate of point A2024?
|
A
|
['(\\frac{1}{2}, 1)^{2023}', '(\\frac{1}{2}, 1)^{2024}', '(\\frac{\\sqrt{3}}{2}, 1)^{2023}', '(\\frac{\\sqrt{3}}{2}, 1)^{2024}']
|
multi_choice
|
func573
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/573.png
|
8. Eight squares with side length 1 are placed as shown in the plane rectangular coordinate system. The line y=kx divides these eight squares into two equal areas. What is the value of k?
|
A
|
['1', '10/9', '3/4', '4/3']
|
multi_choice
|
func574
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/574.png
|
As shown in the figure, OB intersects the x-axis at point A, and OB:OA=5:3. If the area of △ABC is 8, ∠C=90°, and AC is parallel to the y-axis, what is the value of k ( )?
|
B. 36
|
['18', '36', '12', '200/9']
|
multi_choice
|
func575
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/575.png
|
As shown in the figure, it depicts the flight formation of a bomber squadron. It is known that the coordinates of bombers B and C are (-2, -3) and (2, -1), respectively. What are the coordinates of bomber A?
|
A. (-2, 1)
|
['(-2, 1)', '(2, 1)', '(-2, 3)', '(-3, 2)']
|
multi_choice
|
func577
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/577.png
|
In the Cartesian coordinate system, a snail starts from the origin and moves continuously along the path shown in the figure, with each move being 1 unit in length. What are the coordinates of point A20?
|
A
|
['(10, 0)', '(10, -1)', '(9, 0)', '(9, -1)']
|
multi_choice
|
func578
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/578.png
|
As shown in the figure, in a Cartesian coordinate system, there are four points A, B, C, and D. The graph of the linear function y = kx + 1 (0 < k < 1) may pass through ( ).
|
A
|
['Point A', 'Point B', 'Point C', 'Point D']
|
multi_choice
|
func579
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/579.png
|
A farmer wants to use a fence to enclose a rectangular chicken farm, as shown in the figure, with one side of the chicken farm against the wall. If the total length of the fence is 20m, let the side of the rectangle against the wall be xm, and the area be ym². When x changes within a certain range, y changes accordingly. The functional relationship between y and x is ( ).
|
D
|
['y = 20x', 'y = 20 - 2x', 'y = 20/x', 'y = x(20 - 2x)']
|
multi_choice
|
func58
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/58.png
|
As shown in the diagram, through point P(4,5), draw PC perpendicular to the x-axis at point C and PD perpendicular to the y-axis at point D. PC and PD intersect the hyperbolic function y = 8/x (x > 0) at points A and B, respectively. What is the area of quadrilateral BOAP?
|
12
|
NULL
|
free_form
|
func580
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/580.png
|
As shown in the figure, in the Cartesian coordinate system xOy, the radius of the circle ρ is 2, and the coordinates of point ρ are (-5, 0). If ⊙ρ is translated to the right along the x-axis such that ⊙ρ becomes tangent to the y-axis, how far does ⊙ρ translate to the right?
|
D
|
['1', '5', '3', '1 or 5']
|
multi_choice
|
func581
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/581.png
|
As shown in the figure, the vertex of the parabola y=ax^2+bx+c has coordinates (-1/2, m), and one intersection of the parabola with the x-axis lies between 0 and 1. Among the following conclusions: ①a>0; ②h+c>0; ③If the graph passes through the points (-2, y1) and (-2, y2), then y1≠y2; ④The quadratic equation in one unknown, ax^2+hx+c+3=0, has no real roots, then m<3. How many of these conclusions are correct? ( )
|
C
|
['①②', '②③', '①③④', '①②④']
|
multi_choice
|
func582
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/582.png
|
Given a function that satisfies the following table (x is the independent variable):
| x | 3 | 4 | 5 | 9 |
|----|----|----|----|----|
| y | -9 | -7 | -5 | -3 |
What is the expression of this function? ( )
A. y = -9/x
B. y = -9/x
C. y = x/9
D. y = -x/9
|
B
|
['y = -9/x', 'y = -9x', 'y = x/9', 'y = -x/9']
|
multi_choice
|
func583
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/583.png
|
As shown in the figure, in △AOB, S△AOB = 4, AB // y-axis, point A is on the graph of the inverse proportional function y = 2 / x. If point B is on the graph of the inverse proportional function y = k / x, then the value of k is ( ).
|
C. -6
|
['6', '±6', '-6', '-3']
|
multi_choice
|
func584
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/584.png
|
As shown in Figure 1, this is a parabolic arch bridge in a park. According to Figure 2, a plane rectangular coordinate system is established, and the function y = -1/25x^2 is obtained. At normal water level, the width of the water surface AB is 30 meters. When the water level rises by 7 meters, the width of the water surface CD is ( ).
|
D. 10√2 meters
|
['5 meters', '5√2 meters', '10 meters', '10√2 meters']
|
multi_choice
|
func585
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/585.png
|
9. A beam of light starts from point A(4,8), passes through point C on the x-axis, and then reflects to pass through point B(4,0). What is the total path length of the light from point A to point B? ( )
|
B
|
['13', '15', '16', '17']
|
multi_choice
|
func586
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/586.png
|
As shown in the figure, the point covered by the small hand is 3 units away from the x-axis and 2 units away from the y-axis. What are the coordinates of this point ( )?
|
A
|
['(-2, -3)', '(3, -2)', '(2, -3)', '(-2, 3)']
|
multi_choice
|
func588
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/588.png
|
As shown in the figure, a car passes a certain section of road at a constant speed. The time required t (h) and the driving speed v (km/h) are represented by a segment of a hyperbolic curve. If the driving speed on this road must not exceed 80 km/h, the shortest time required for the car to pass this section of road is ( )
A. 40 minutes B. 45 minutes C. 55 minutes D. 60 minutes
|
B
|
['40 minutes', '45 minutes', '55 minutes', '60 minutes']
|
multi_choice
|
func59
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/59.png
|
As shown in the figure, the vertices of △ABC are A(2,3), B(-2,0), and C(0,-1). Point D lies on the coordinate axis. If the quadrilateral formed by A, B, C, and D is a parallelogram, then the coordinates of point D are ______.
|
(0,4)
|
NULL
|
free_form
|
func590
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/590.png
|
A courier vehicle departs from the company, reaches 4 stations, unloads parcels at each station, and immediately returns to the company along the same route. The vehicle's speed is constant, and the time taken to unload parcels at each station is the same. The relationship between the distance from the company to the stations and the travel time (partial data) is shown in the diagram.
[Question] The time the courier vehicle spends unloading parcels at each station is ( ).
|
B. 5 minutes
|
['4 minutes', '5 minutes', '6 minutes', '7 minutes']
|
multi_choice
|
func592
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/592.png
|
As shown in the figure, the graphs of the functions y=2x and y=ax+6 intersect at point P(m, 4). Then, the solution set of the inequality ax+6 > 2x is ( ).
|
B. x < 2
|
['x > 2', 'x < 2', 'x > 3', 'x < 3']
|
multi_choice
|
func593
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/593.png
|
As shown in the figure, the straight line y = kx + b passes through points A(-1, m) and B(-2, 0), and the straight line y = 2x passes through point A. Then the solution set of the inequality system {2x < kx + b, kx + b < 0} is ( ).
|
B. -2 < x < -1
|
['x < -2', '-2 < x < -1', '-2 < x < 0', '-1 < x < 0']
|
multi_choice
|
func595
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/595.png
|
As shown in the figure, the graph of the linear function y = ax + b and the hyperbolic function y = k/x (k > 0) intersect at points A(1,2) and B(m,-1). Then the solution set of ax + b > k/x is ( ).
|
C
|
['\\(-2 < x < 0\\) or \\(0 < x < 1\\)', '\\(-1 < x < 0\\) or \\(0 < x < 2\\)', '\\(-2 < x < 0\\) or \\(x > 1\\)', '\\(-1 < x < 0\\) or \\(x > 2\\)']
|
multi_choice
|
func596
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/596.png
|
After dinner, Xiao Ming and his parents took a walk to a newsstand 500 meters away from home, stayed for a moment, and then returned. The following situation describes the change in distance over time during their walk. Which of the following scenarios is possible?
|
A
|
['They walked to the newspaper stand at a constant speed, returned home at a faster speed, and walked home quickly.', 'They walked to the newspaper stand at a constant speed, returned home at a slower speed, and walked home slowly.', 'They walked to the newspaper stand at a constant speed, returned home at a speed that gradually increased, and walked home faster and faster.', 'They walked to the newspaper stand at a speed that gradually increased, returned home at a speed that gradually decreased, and walked home slower and slower.']
|
multi_choice
|
func597
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/597.png
|
As shown in the figure, the linear function y = -2/3x - 4 and the direct variation function y = kx intersect at point A in the third quadrant and at point B on the y-axis. Additionally, AO = AB. What is the equation of the direct variation function? ( ).
|
B
|
['y = \\frac{3}{4}x', 'y = \\frac{2}{3}x', 'y = \\frac{4}{3}x', 'y = \\frac{5}{6}x']
|
multi_choice
|
func598
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/598.png
|
As shown in the figure, a linear function y = kx + b intersects y = -2x + 1 at point P(1,3). Which of the following statements is incorrect ( ).
|
D
|
['k > 0', 'b > 0', 'The solution to the equation kx + b = -1 is x = -1', 'The solution set of the inequality kx + b < -2x + 1 is x < 3']
|
multi_choice
|
func599
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/599.png
|
As shown in the figure, in the rectangular coordinate plane, the vertex coordinates of △ABC are A(1, 1), B(3, 1), and C(1, 2). When the line y = x + b intersects △ABC, what is the range of values for b?
|
B. -2 ≤ b ≤ 1
|
['-1 ≤ b ≤ 1', '-2 ≤ b ≤ 1', '-1 ≤ b ≤ 1/2', '-2 ≤ b ≤ 1/2']
|
multi_choice
|
func6
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/6.png
|
As shown in the figure, Xiao Ming plays golf, and the ball's flight path follows a parabolic trajectory. Ignoring air resistance, the relationship between the ball's flight height h (meters) and flight time t (seconds) satisfies the function h = 20t - 5t². Then the time required for the ball to reach its highest point from being hit is ______ seconds.
|
2
|
NULL
|
free_form
|
func60
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/60.png
|
As shown in the figure, a small rectangular wooden board is cut from a larger rectangular wooden board with a length of 5 dm and a width of 3 dm. The function relationship between the area S (dm²) of the remaining part of the wooden board and x (dm) is ___________, and the value range of the variable x is ___________.
|
S = -x² - 2x + 15; 0 <= x <= 1
|
NULL
|
free_form
|
func600
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/600.png
|
Regarding the graph of the linear function y = kx + b as shown in the figure below, which of the following statements is incorrect? ( )
|
D
|
['k < 0', 'Passes through the point (0, 1)', 'y decreases as x increases', 'When x > 0, y < 0']
|
multi_choice
|
func601
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/601.png
|
As shown in the figure, the graphs of the functions y = kx + b (k ≠ 0) and y = mx + n (m ≠ 0) are displayed. What are the coordinates of the point corresponding to the solution of the system of equations y = kx + b and y = mx + n, with respect to its symmetric point about the y-axis?
|
B
|
['(-2, 3)', '(2, 3)', '(2, -3)', '(-2, -3)']
|
multi_choice
|
func602
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/602.png
|
As shown in the figure, the graph of the linear function y = x + m intersects the x-axis at the point (-3, 0). What is the solution to the inequality x + m > 0?
|
A
|
['x > -3', 'x < -3', 'x > 3', 'x < 3']
|
multi_choice
|
func603
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/603.png
|
In the plane Cartesian coordinate system, △ABC shown in Figure 1 undergoes axisymmetric transformations sequentially as shown in Figure 2. If the coordinates of point A4 are (x, y), then after the 2023rd transformation, the coordinates of point A2023 are _______.
|
B
|
['(x, y)', '(-x, y)', '(x, -y)', '(-x, -y)']
|
multi_choice
|
func604
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/604.png
|
As shown in the figure, in the Cartesian coordinate system, the coordinates of point A are (0,3). △OAB is translated rightward along the x-axis to form △O′A′B′. If the corresponding point A′ lies on the line y=3/4x, then the distance between point B and its corresponding point B′ is ( ).
|
D
|
['4/3', '4√3', '3', '4']
|
multi_choice
|
func605
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/605.png
|
As shown in the figure, the quadratic function y = ax^2 + bx + c (a ≠ 0, a, b, and c are constants, and x is a part of the graph of the function) has an axis of symmetry at the line x = −1, passes through the point (1, 0), and the x-axis intercept is between the points (0, −2) and (0, −3). Among the given options, the correct one is ( ).
|
D
|
['\\(b^2 < 4ac\\)', '\\(2ab = 0\\)', '\\(a - 3b + c > 0\\)', '\\(\\frac{4}{3} < b < 2\\)']
|
multi_choice
|
func606
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/606.png
|
The graph of the linear function y = kx + b (where k and b are constants, and k ≠ 0) is shown in the figure. Which of the following could be the point (b, h)?
|
C
|
['(-1, -1)', '(-1, 0)', '(1, -1)', '(1, 1)']
|
multi_choice
|
func607
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/607.png
|
A bullet train departs from location A to location B, and a regular train departs from location B to location A. Both trains travel at constant speeds and leave simultaneously. Let the time taken by the regular train be x (hours), and the distance between the two trains be y (kilometers). If the graph shows the relationship between y and x, which of the following conclusions is incorrect?
A. The distance between location A and location B is 1000 kilometers
B. Point D indicates that the two trains meet 3 hours after departure
C. The regular train's speed is 100 km/h
D. The bullet train takes 4 hours to travel from A to B
|
C
|
['The distance between points A and B is 1000 thousand meters', 'The actual meaning of point B is that two hours have passed since departure', 'The speed of the regular train is 100 thousand meters per hour', 'The time taken by the train to travel from point A to point B is 4 hours']
|
multi_choice
|
func608
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/608.png
|
As shown in the figure, if the graph of the linear function y = kx + b (k≠0) intersects the x-axis at point (3, 0), then the solution set of the inequality k(x+5) + b > 0 with respect to x is ( ).
|
B. x > -2
|
['x < -2', 'x > -2', 'x < 8', 'x > 8']
|
multi_choice
|
func609
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/609.png
|
Given the inverse proportional function y = k/x (k ≠ 0), a point P lies on the graph. From point P, draw PM ⊥ x-axis intersecting at point M, and connect OP. The area of △PMO is 3, then k = （ ）.
|
C. -6
|
['-3', '3', '-6', '6']
|
multi_choice
|
func61
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/61.png
|
As shown in the figure, in the right triangle △ABC, ∠B=90°, AB=6cm, BC=8cm. A moving point P starts from point A and moves towards point B along AB at a constant speed of 1cm/s, while a moving point Q starts from point B and moves towards point C along BC at a constant speed of 2cm/s. When one point reaches its destination, the other point also stops moving. If moving points P and Q start simultaneously from points A and B, __________, the area of △PBQ is maximized, and the maximum area is __________ cm².
|
3 9
|
NULL
|
free_form
|
func610
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/610.png
|
As shown in the figure, the graph of the linear function y₁ = x + b intersects with the graph of the linear function y₂ = kx + 4 at point P(1, 3). Then the solution set of the inequality b < kx + 4 with respect to x is ( ).
|
D. x < 1
|
['x > 2', 'x > 0', 'x > 1', 'x < 1']
|
multi_choice
|
func611
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/611.png
|
Given the quadratic function y = -x^2 + hx + c whose graph is as shown, and the following conclusions: ① The graph of the function intersects the positive y-axis; ② When x > 0, y decreases as x increases, then the origin of the coordinate system might be ( ).
|
C
|
['Point A', 'Point B', 'Point C', 'Point D']
|
multi_choice
|
func612
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/612.png
|
Driving under the influence (DUI) is defined as having a blood alcohol content of 200 micrograms or more per milliliter. A research institute experimentally determined the relationship between the blood alcohol concentration y (micrograms/milliliter) and drinking time x (hours) after consuming a certain brand of 38-degree liquor, as shown in the graph (when 4 ≤ x ≤ 10, y and x are inversely proportional). Which of the following statements is incorrect?
A. Within the first 4 hours of drinking, the longer the drinking time, the higher the blood alcohol concentration.
B. When x = 8, the blood alcohol concentration is 320.
C. When x = 6, the blood alcohol concentration is 100.
D. For drinking times between 4 hours and 10 hours, the duration of blood alcohol concentration above the legal limit is less than 1 hour.
|
D
|
['If the driver drinks within 4 hours, the longer the drinking time, the higher the blood alcohol concentration.', 'When x = 8 hours, the blood alcohol concentration is 320 micrograms per liter.', 'When x = 9 hours, the blood alcohol concentration is 1600/9 micrograms per liter.', 'The blood alcohol concentration will remain above 200 micrograms per liter for 7 hours after drinking.']
|
multi_choice
|
func613
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/613.png
|
Given the line v1 = kx + b (k > 0) intersects the x-axis at the point (−3, 0), and the line v2 = mx + b (m < 0) intersects the x-axis at the point (4, 0), determine the solution set for the inequality kx + b > mx + n. ( )
|
D. x > 4
|
['-3 < x < 4', 'x > -3', 'x < 4', 'x > 4']
|
multi_choice
|
func614
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/614.png
|
As shown in the figure, it is a leaf specimen placed in a Cartesian coordinate system. The coordinates of the leaf tip points A and B are (-3, 3) and (-1, 0), respectively. What are the coordinates of the leaf base point C?
|
B
|
['(2, 0)', '(2, 1)', '(1, 0)', '(1, -1)']
|
multi_choice
|
func615
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/615.png
|
The graph of a quadratic function is given (0 ≤ x ≤ 4). For this function within the given range of the variable, the following statement is correct ( ).
|
A
|
['There is a maximum value of 2 and a minimum value of -2.5', 'There is a maximum value of 2 and a minimum value of 1.5', 'There is a maximum value of 1.5 and a minimum value of -2.5', 'There is a maximum value but no minimum value']
|
multi_choice
|
func616
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/616.png
|
In an exploratory lesson, the teacher posed a question: 'Using the graphs of a quadratic function and a linear function, find the approximate roots of the quadratic equation 2x² = x + 2.' Xiao Hua used a computer to draw the graphs shown below. By observation, it can be seen that the two approximate roots of the equation, x1 and x2, satisfy -1 < x1 < 0 and 1 < x2 < 2. The mathematical idea reflected in Xiao Hua's method is ( ).
|
C
|
['Rationalization', 'Classification discussion', 'Graphical combination', 'From special to general']
|
multi_choice
|
func617
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/617.png
|
As shown in the figure, in the same plane rectangular coordinate system, the graph of a linear function y₁ = kx + b (k and b are constants, and k ≠ 0) intersects with the graph of an inverse proportional function y₂ = C / x (C is a constant, and C ≠ 0) at points A(-3, -2) and B(2, m). Then the solution set of the inequality y₁ > y₂ is ( )
A. -3 < x < 2
B. x < -3 or x > 2
C. -3 < x < 0 or x > 2
D. 0 < x < 2
|
C
|
['-3 < x < 2', 'x < -3 or x > 2', '-3 < x < 0 or x > 2', '0 < x < 2']
|
multi_choice
|
func618
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/618.png
|
The chart shows the functional relationship between the fare y (in yuan) of a ride-hailing platform and the distance traveled x (in kilometers). Based on the information in the chart, when Xiao Ming takes a ride-hailing car from home to the airport and pays a total fare of 64 yuan, if the car's speed is consistently maintained at 60 kilometers/hour without considering other factors (like traffic lights, traffic jams, etc.), the time needed to travel from home to the airport is ( ).
A. 10 minutes B. 15 minutes C. 18 minutes D. 20 minutes
|
D
|
['10 minutes', '15 minutes', '18 minutes', '20 minutes']
|
multi_choice
|
func619
|
/home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/619.png
|
Given the functions y1 = 3x and y2 = kx + b intersecting at point A(1, 3), determine the solution set for the inequality kx + b < 3x with respect to x ( ).
|
D. x > 1
|
['x < 3', 'x > 3', 'x < 1', 'x > 1']
|
multi_choice
|
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