id stringlengths 4 8 | image_url stringlengths 77 81 | query stringlengths 7 1.32k | answer stringlengths 1 148 | choice stringlengths 4 597 | question_type stringclasses 2 values |
|---|---|---|---|---|---|
func426 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/426.png | As shown in the figure, in a rectangular coordinate system, a moving point starts from the origin O and moves upward, to the right, and downward continuously. After moving one unit length each time, the points obtained are A_2(1, 1), A_3(1, 0), A_4(2, 0), ..., then the coordinates of point A_{100} are ____________. | (50, 0) | NULL | free_form |
func427 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/427.png | As shown in the figure, for the inverse proportional function y = k/x (k ≠ 0), there is a point P on the curve. A perpendicular line to the y-axis is drawn from point P and intersects the curve at point Q. Connect OP. If the area of △OPQ is 3, then k = ______. | −6 | NULL | free_form |
func428 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/428.png | In the plane rectangular coordinate system (O, y), the graphs of the linear functions y = kx and y = mx + n are as shown. What is the solution set for the linear inequality kx ≤ mx + n with respect to x? | x ≥ -1 | NULL | free_form |
func43 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/43.png | As shown in the figure, P(12,a) lies on the hyperbolic function y=60/x. PH is perpendicular to the x-axis at point H. Find the value of sin∠POH ________. | 5/13 | NULL | free_form |
func431 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/431.png | As shown in the figure, the line y = -1/2x + 2 intersects the x-axis and y-axis at points A and B respectively. Point C is on segment OA, and ∠ABC = 45°. What are the coordinates of point C ______? | (-3/2, 0) | NULL | free_form |
func432 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/432.png | As shown in the figure, in a rectangular coordinate plane, A(1, 0), B(0, 1), a circle is drawn with point A as the center and AB as the radius, intersecting the positive x-axis at point C. The real number represented by point C is ______. | √2 + 1 | NULL | free_form |
func433 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/433.png | In a plane rectangular coordinate system, A(2,0), B(0,4), a line l through point B is parallel to the y-axis, and point P(a,4) is any point on this line. An isosceles right triangle △APQ is formed on the right side of AP with ∠APQ=90°.
(1) When a=0, the coordinates of circle O are ________.
(2) When point P moves along the line, point O also moves. The minimum value of |O| is ________. | (4,6), 5√2 | NULL | free_form |
func434 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/434.png | As shown in the figure, vertex A of rectangle ABCD lies on the graph of the inverse proportional function y = -k/x (x < 0), vertex C lies on the graph of the inverse proportional function y = k/x (x > 0), AB ⊥ x-axis, and the intersection point of the diagonals of rectangle ABCD is the origin O. If the area of △ABCD = 5, then k = ________. | 1/2 | NULL | free_form |
func435 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/435.png | As shown in the figure, in the parallelogram ABCD, AD = 10, AB = 14, and the coordinates of point A are (−6, 0). What are the coordinates of point C? | (14, 8) | NULL | free_form |
func436 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/436.png | As shown in the figure, points A and D are on the graph of the inverse proportional function y = k/x (k < 0). CD is perpendicular to the y-axis, with C as the foot of the perpendicular. AB is perpendicular to CD, with B as the foot of the perpendicular. If the area of quadrilateral OABD is 8 and BD = 2CD, then the value of k is ________. | -4 | NULL | free_form |
func437 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/437.png | The working conditions of three couriers on a certain day are shown in the diagram, where the horizontal and vertical coordinates of points A1, A2, A3, B1, B2, and B3 represent the time spent and the number of parcels delivered in the morning by couriers A, B, and C, respectively. The horizontal and vertical coordinates of points A1, A2, and A3 represent the time spent and the number of parcels delivered in the afternoon by couriers A, B, and C, respectively. The following three conclusions are made: ① The courier who spent the shortest time delivering parcels in the morning is A; ② The courier who delivered the most parcels in the afternoon is C; ③ The courier who delivered the most total parcels throughout the day is B. Among the above conclusions, the serial numbers of all correct conclusions are ____________. | ①③ | NULL | free_form |
func438 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/438.png | As shown in the figure, point A is on the graph of the inverse proportional function y = k/x (k < 0, x < 0). AB is perpendicular to the x-axis at point B, point C is on the positive half of the y-axis, connect BC, and AD is parallel to BC intersecting the x-axis at point D. If the area of quadrilateral ABCD is 0.5, what is the value of k? | -0.5 | NULL | free_form |
func439 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/439.png | As shown in the figure, the coordinates of point A are (0,6). The triangle △AOB is translated rightward along the x-axis to form △A'OB'. If the corresponding point A' lies on the line y=2x−1, then the distance between point B and its corresponding point B' is ________. | 7/2 | NULL | free_form |
func44 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/44.png | As shown in the figure, the parabola y = 15/32 * (x − 6)^2 − 15 intersects the y-axis at point A and the x-axis at points B and C. The point A’s symmetric point about the axis of symmetry of the parabola is point D, which is on the y-axis. Point F is on the circle with center C and radius 1.5. What are the coordinates of point D ______, and what is the minimal value of DE + EF ______? | (12, 15), 23.5 | NULL | free_form |
func440 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/440.png | As shown in the figure, a beam of light starts from point A(-4,10), passes through point B(0,2) on the y-axis, and is reflected to pass through point C(m,n). What is the value of 2m−n? ________ | -2 | NULL | free_form |
func441 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/441.png | As shown in the figure, in the plane rectangular coordinate system, based on this pattern of exploration, point A₁(1, 2), A₂(2, 0), A₃(3, -2), A₄(4, 0)... Based on this pattern, determine the coordinates of point A₂₀₂₄ ______. | (2024, 0) | NULL | free_form |
func442 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/442.png | As shown in the figure, rectangle OABC is in the rectangular coordinate system, where A(4,0), C(0,3), point E is the midpoint of BC. A moving point P starts from point O and moves at a speed of 1 cm per second along O→A→B→E, eventually reaching point E. If the time for point P's movement is in seconds, then when the area of △OPE equals 5 cm², the coordinates of point P are __________. | (10/3, 0) or (4, 1) | NULL | free_form |
func443 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/443.png | As shown in the figure, line BC passes through the origin O, point A lies on the x-axis, AD⊥BC at D. If B(m, 3), C(n, −5), A(4, 0), then AD·BC = ______. | 32 | NULL | free_form |
func444 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/444.png | 9. Car A and car B travel from point A to point B, with a total distance of 800 km. The function graphs of their traveled distances with respect to time are shown in the figure. The following statements are given: ① Car B departs 2 hours earlier than car A; ② Car A catches up with car B in 5 hours, at which point car B has traveled 300 km; ③ The speed of car B is less than the speed of car A; ④ Car A uses 3 hours less to complete the entire trip. Which of the above statements are correct? ______. | ③④ | NULL | free_form |
func446 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/446.png | As shown in the figure, the sides of the square OBCD with side length 3 coincide with the coordinate axes. What are the coordinates of point C? | (-3,3) | NULL | free_form |
func447 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/447.png | As shown in the figure, in the rectangular coordinate system, A(0, 4), B(−4, 5), C(−5, 0), D(2, 0), what is the area of quadrilateral ABCD? | 24.5 | NULL | free_form |
func448 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/448.png | As shown in the figure, a parabolic curve: y = −x(x−3)(0≤x≤3), denoted as C₁, intersects the x-axis at points 0 and 4. Rotating C₁ 180° around 1 produces C₂, which intersects the x-axis at A₂. Rotating C₂ 180° around A₂ produces C₃, which intersects the x-axis at A₃. This process continues until Cₙ is obtained. If point P(m, n) lies on the 11th parabolic curve, then the value of mn is ______. | 31 or 32 | NULL | free_form |
func449 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/449.png | As shown in the figure, it is known that points P(−1,3) and Q(−3,2) are located on opposite sides of the curve L: y=−kx (x<0). Write an integer value of k that satisfies the condition: _________________ . | −4 (not unique) | NULL | free_form |
func45 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/45.png | In ancient times, Emperor Qianlong once passed by Lugou Bridge in autumn and composed a poem: "Half an axis retains the reflection of three autumns, a streak divides the waves brighter than a mirror," and inscribed the phrase "Lugou Morning Moon" on a stele at the bridgehead. The main arch of Lugou Bridge can be approximated as a parabola. If the span OA of the arch over the water surface is about 22 meters and a plane rectangular coordinate system is built as shown in the diagram, the equation of the parabola representing the main bridge arch is y = -13/121(x-11)^2 + k. Then the distance between the highest point P of the main arch and its reflection P' in the water is ___ meters. | 26 | NULL | free_form |
func450 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/450.png | As shown in the figure, in the plane Cartesian coordinate system xOy, the vertex C of the rhombus OABC is on the positive half of the x-axis. If the coordinates of point A are (3, 4), then the coordinates of point B are __________. | (8, 4) | NULL | free_form |
func451 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/451.png | As shown in the figure, the straight line l passes through point P and intersects the graphs of the inverse proportional functions y₁ = k₁ / x (x > 0) and y₂ = k₂ / x (x > 0) at points A and B, respectively. Connecting OA and OB, it is known that the area of △AOB is 6. Then, k₁ - k₂ = _______. | 12 | NULL | free_form |
func452 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/452.png | As shown in the figure, points A and B are respectively on the graphs of the inverse proportional functions y=2k/(x+1) and y=k/x. Vertical lines are drawn from points A and B to the x-axis and y-axis, forming a shaded region with an area of 5. What is the value of k? ______. | 4 | NULL | free_form |
func453 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/453.png | After studying the biology lesson 'Blood', the biology interest group found that doctors usually advise 'do not take another dose of the medicine within 4 hours', which is related to the concentration of the drug in the blood. After class, they looked up information and found the following: the blood drug concentration of a certain medicine for adults increases gradually after administration, reaches a maximum, and then begins to gradually decline. During the decline, the blood drug concentration y (mg/ml) is an inverse proportion function of the medication time x (h). Point A(4, 10) is on the graph of this inverse proportional function. When the blood drug concentration is 3 mg/ml, the drug is almost ineffective. Therefore, it is more reasonable for an adult to take this medicine ________ hours later. | 5 | NULL | free_form |
func454 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/454.png | On the grid paper shown in the figure, establish an appropriate plane Cartesian coordinate system such that the coordinates of point A are (-2, 0) and the coordinates of point B are (0, 1). Then the coordinates of point C are __________. | (-1, 2) | NULL | free_form |
func455 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/455.png | As shown in the figure, a moving point P moves in the direction indicated by the arrows. The first time, it moves from the origin to the point (2,2). The second time, it moves to the point (4,0). The third time, it moves to the point (6,4), and so on, following this pattern. What will be the coordinates of the point after the 2024th move? | (4048, 0) | NULL | free_form |
func456 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/456.png | As shown in the figure, in the rectangular coordinate system xOy, the coordinates of points A and B are (-2, 0) and (0, 3) respectively. Draw an arc with A as the center and AB as the radius, intersecting the positive x-axis at point C. What is the x-coordinate of point C? | -2 + √13 | NULL | free_form |
func457 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/457.png | As shown in the diagram, in a Cartesian coordinate system, a moving point starts from the origin O and moves continuously in directions of upward, rightward, downward, and rightward. Each movement covers a unit distance, and the following points are obtained: A₁(0,1), A₂(1,1), A₃(1,0), A₄(2,0), ... What are the coordinates of the point A₁₀₁₂₄? | (1012, 0) | NULL | free_form |
func458 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/458.png | When Xiaoliang was conducting an experiment in the laboratory, he couldn't find a balance to measure the mass of the required chemical. Instead, he used the principle of the lever to create a balance to measure the mass of the chemical (Principle of the lever: Force × Force Arm = Resistance × Resistance Arm). As shown in Figure 1, when a weight of 60 grams is placed on the left tray of the balance, and a weight of 20 grams is placed on the right tray, the balance is leveled. As shown in Figure 2, after placing the chemical to be measured on the right tray, a 15-gram weight is placed on the left tray to balance the balance again. What is the mass of the chemical in grams? | 5 | NULL | free_form |
func459 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/459.png | As shown in the figure, the graphs of the directly proportional function y = ax (where a is a constant and a ≠ 0) and the inversely proportional function y = b/x (where b is a constant and b ≠ 0) intersect at points A and B. Draw a line through point B parallel to the x-axis (denoted as BC) and connect AC, ensuring AC is parallel to the y-axis. If the area of triangle △ABC is 12, the value of b is _______. | 6 | NULL | free_form |
func46 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/46.png | As shown in the figure, the sides OA and OC of the square OABC lie on the x-axis and y-axis respectively. Point D(5, 3) is located on side AB. When △CDB is rotated 90° around point C, the coordinates of the corresponding point D′ after rotation are ______. | (2, 10) or (-2, 0) | NULL | free_form |
func460 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/460.png | As shown in the figure, the point O(0, 0) is first translated 1 unit to the right, then 1 unit upward, resulting in point A1. Then, point A1 is translated 1 unit upward and 2 units to the right, resulting in point A2. Next, point A2 is translated 2 units upward and 4 units to the right, resulting in point A3. Lastly, point A3 is translated 4 units upward and 8 units to the right, resulting in point A4. Following this pattern of translation, the coordinates of point A2024 are _______. | (2^2024−1, 2^2023) | NULL | free_form |
func461 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/461.png | As shown in the figure, the graph of the inverse proportional function y = k/x (x > 0) passes through point A(2,1). If x > 2, what is the range of values for y________. | 0 < y < 1 | NULL | free_form |
func462 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/462.png | As shown in the figure, the graph of the linear function y = -x + b intersects with y = x + 2 at the point M(2,4). Then the solution set of the linear inequality -x + b > x + 2 with respect to x is ____________. | x < 2 | NULL | free_form |
func463 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/463.png | As shown in the figure, the line l₁: y = 2x intersects the line l₂: y = kx + 4 at point P. The solution set of the inequality 2x < kx + 4 is ________. | x < 1. | NULL | free_form |
func464 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/464.png | As shown in the figure, in a rectangular coordinate plane, the side AB of the rectangle ABCD lies on the positive x-axis. The coordinates of points D and B are (4, 3) and (10, 0), respectively. On the graph of the proportional function y = kx passing through point D, there is a point P such that P is the midpoint of D. When the graph of y = kx is shifted downward along the y-axis, forming the graph of y = kx + b, if point P lies within the closed interior region of rectangle ABCD, what is the value range of b? ______. | -6 < b < -3 | NULL | free_form |
func465 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/465.png | As shown in the figure, given points A(1,4) and B(5,4), point P is an integer point on line segment AB (not coinciding with A or B, and both x and y coordinates are integers). If the hyperbola y=k/x (x>0) passes through point P, provide one possible value of k: ______. | 8, 12, or 16 (any one of them is acceptable). | NULL | free_form |
func466 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/466.png | As shown in the figure, point A is on the hyperbola y = 3/x. Through point A, draw AB // x-axis, meeting the axis at point C and intersecting the hyperbola y = k/x (x < 0) at point B. If BC = 2AC, then the value of k is __. | -6 | NULL | free_form |
func467 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/467.png | As shown in the figure, the line y = kx + b intersects with the line y = kx + 6 at point P(3, 5). The solution set of the inequality kx + 6 > x + b is ______. | x < 3 | NULL | free_form |
func468 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/468.png | As shown in the figure, point P starts from vertex D of rectangle ABCD and moves along the path D→C→B→A at a speed of 1 cm per second. The graph of the function between the motion time x and the area y of △DAP is shown in Figure 2. What is the area of rectangle ABCD? | 12 | NULL | free_form |
func469 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/469.png | As shown in the figure, rectangle OABC and square ADEF have their vertices A and D on the positive x-axis, vertex C on the positive y-axis, vertex F on edge AB, and vertices B and E on the graph of the inverse proportional function y = k/(x > 0). If the coordinates of point B are (8, 6), what are the coordinates of point E? | (12, 4) | NULL | free_form |
func47 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/47.png | As shown in the figure, △ABC is an equilateral triangle with a side length of 4. D is a point on AC. A perpendicular line is drawn from point D to BC, intersecting BC at E. Using x to represent the length of segment BE, it is clear that the area y of Rt△CDE is a quadratic function of BE. The vertex form of this function is __________. | y = √3 / 2 * (x - 4)^2 (2 < x < 4) | NULL | free_form |
func470 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/470.png | As shown in the figure, points B, B₁, B₂, ... lie on the x-axis, and point A lies on the y-axis. AC is perpendicular to the y-axis, BC is perpendicular to the x-axis, with the intersection being point C. Line l₁ passes through the origin O and point C. Point A₁ is the midpoint of BC, B₁B₂ = 1/2OB. A₁C₁ is perpendicular to the y-axis, and B₁C₁ is perpendicular to the x-axis. Line l₂ passes through point O and point C₁. Point A₂ is the midpoint of B₁C₁, B₂B₃ = 1/2B₁B₂. A₂C₂ is perpendicular to the y-axis, and B₂C₂ is perpendicular to the x-axis. Line l₃ passes through point O and point C₂, and so on. If point C(4,8), what is the equation of line l₄? | y = 2/3^(1/x) | NULL | free_form |
func471 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/471.png | As shown in the figure, it is known that the area of the square ABCD is 9. Two vertices B and D are points on the graph of the inverse proportional function y = k/x (k > 0, x > 0). If the coordinates of point D are (m, n), then the value of m − n is ________. | -3 | NULL | free_form |
func472 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/472.png | As shown in the figure, point A is on the graph of the inverse proportional function y = k/x (x > 0). A perpendicular line is drawn from point A to the y-axis, intersecting the y-axis at point B. If point C lies on the x-axis and the area of △ABC is 1, then the value of k is ______. | 2 | NULL | free_form |
func473 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/473.png | As shown in the figure, the graph of the linear function y = kx + b passes through points A and B. The solution to the equation kx + b = 0 with respect to x is _______. | x = 2 | NULL | free_form |
func474 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/474.png | In a rectangular coordinate system, point A1 starts from the origin and moves along the direction shown in the figure. The coordinates of its positions are sequentially: A2 (1, 0), A3 (1, 1), A4 (−1, 1), A5 (−1, −1), A6 (2, −1), A7 (2, 2), ..., then the coordinates of A2024 are ______. | (−506, 506) | NULL | free_form |
func475 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/475.png | In the Cartesian coordinate system, a right-angled triangle is placed as shown in the figure, where ∠ACB = 90°, ∠A = 30°, B(0, 1), C(√3, 0). What are the coordinates of point A? | (2√3, 3) | NULL | free_form |
func476 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/476.png | A phone is being charged using a charger, and its screen displays the current battery level as 20% (as shown in Figure 1). Through testing, it is observed that when using a fast charger and a regular charger to charge the phone, the battery level y (unit: %) as a function of charging time x (unit: hours) corresponds to line segments AB and AC in Figure 2, respectively. First, the phone is charged with the regular charger for a hours, and then switched to the fast charger to fully charge the phone. If the total charging time is 4 hours, find the value of a. | 3 | NULL | free_form |
func477 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/477.png | As shown in the diagram, in a Cartesian coordinate plane, point O(0, 0) and point A(2, 4) are given. Using point O as the center and the length of OA as the radius, an arc is drawn intersecting the positive x-axis at point B. What are the coordinates of point B? | (2√5, 0) | NULL | free_form |
func478 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/478.png | 4. As shown in the figure, in the plane rectangular coordinate system, given point A(3,0), B(-1,0), C(0,2), if quadrilateral ABCD is a parallelogram, the coordinates of point D are ________. | (4,2) | NULL | free_form |
func479 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/479.png | As shown in the figure, the quadrilateral OABC is a parallelogram, where O(0,0), A(5,0), and C(1,4). If the line y = kx − 3 bisects the area of the parallelogram OABC, then k = _____. | 5/3 | NULL | free_form |
func48 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/48.png | As shown in the figure, point A is on the graph of the inverse proportional function y = k / x (k < 0, x < 0). Point B is on the x-axis such that AB ⊥ x-axis, point C is on the positive semi-axis of the y-axis, line BC is connected, and line AD || BC intersects the x-axis at point D. If the area of quadrilateral ABCD, S_ABCD, is 6, what is the value of k? | -6 | NULL | free_form |
func481 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/481.png | As shown in the figure, given that the graphs of the functions y1 = x + b and y2 = ax + 3 intersect at point P, the solution set for the inequality ax + 3 < x + b is ________. | x > 1 | NULL | free_form |
func482 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/482.png | As shown in the figure, in the plane rectangular coordinate system, given the parallelogram ABOC with A(6,0), C(1,3), the line y=kx−1 intersects BC and OA at M and N, respectively, and divides the area of ▱ABOC into two equal parts. What is the value of k? | 5/7 | NULL | free_form |
func483 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/483.png | A fast car departs from Location A heading towards Location B. After reaching Location B, it immediately returns to Location A along the same route. A short time after the fast car's departure, a slow car departs from Location A heading towards Location B. Midway, the slow car stops for 1/4 hour before continuing at the same speed towards Location B. The distances of the two cars from Location A, represented as v1 (in km), are plotted against the driving time of the slow car, as shown in the figure. What is the speed of the slow car in ______ km/h? | 80 | NULL | free_form |
func484 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/484.png | As shown in Figure 1, in the rectangle MNPQ, a moving point R starts from point N and moves along the direction N → P → Q → M until it stops at point M. Let the distance traveled by point R be x and the area of triangle MNR be y. If the graph of y as a function of x is shown in Figure 2, what is the maximum area of triangle MNR? | 12 | NULL | free_form |
func485 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/485.png | In the rectangular coordinate system xOy, rectangle OABC is placed as shown in the figure. A moving point P starts from (0,3) and moves in the indicated direction. Each time it hits the rectangle's edge, it bounces off, with the reflection angle equal to the incidence angle. When point P hits the rectangle's edge for the 5th time, the coordinates of point P are ___________; when point P hits the rectangle's edge for the 2024th time, the coordinates of point P are ___________. | (1,4); (7,4) | NULL | free_form |
func486 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/486.png | As shown in the figure, the endpoints A and B of the line segment AB passing through the origin are respectively on the graphs of the inverse proportional function y = k / x (x > 0) and y = k / x (x < 0). A perpendicular line is drawn from point A to the x-axis, with the foot of the perpendicular being C. If the area of △BOC is 3, then k=__________. | 18 | NULL | free_form |
func487 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/487.png | Given the quadratic function y = -x² + 2x + m as shown in the graph, when y < 0, the range of values for x is ____________. | x < -1 or x > 3 | NULL | free_form |
func488 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/488.png | As shown in the figure, the vertex A of a square ABCD with a side length of 4 is located on the positive x-axis. The reciprocal function y=k/x (k≠0) passes through the origin O and the midpoint E of side BC in the first quadrant. What is the value of k? ________. | 16 | NULL | free_form |
func489 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/489.png | As shown in the figure, the graph of the linear function y = x + b intersects the graph of the reciprocal function y = k / x at points A(2, 3) and B(-3, m). What is the solution set of the inequality x + b > k / x? | x > 2 or x < -3 | NULL | free_form |
func49 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/49.png | As shown in the figure, the graph of the direct proportion function y = kx (k ≠ 0) intersects with the graph of the inverse proportion function y = 4/x at points A and B. Through point A, draw AC perpendicular to the y-axis at point C, and connect BC. Then the area of △BOC is ______. | 2 | NULL | free_form |
func490 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/490.png | As shown in the figure, in the Cartesian coordinate plane, the line y=2x+1 is translated to obtain the line y=kx+b. If point A(1, 1) lies on the line y=kx+b, then the value of k+b is ________. | 1 | NULL | free_form |
func491 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/491.png | As shown in the figure, in a rectangular coordinate plane, the coordinates of points A and B are (1, 4) and (m, 4) respectively. If the line y=3x intersects line segment AB, then the value of m can be ______ (provide one possible value). | 2 (not unique) | NULL | free_form |
func492 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/492.png | As shown in the figure, line AB: y = -1/2x + 2 intersects line AC at point A on the y-axis, and they intersect the x-axis at points B and C, respectively. If AO/BO = OC/OA, what are the coordinates of point C? | (-1, 0) | NULL | free_form |
func493 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/493.png | As shown in the figure, in the rectangular coordinate system, the area of square ABCD is 9. The graph of the inverse proportional function y = k/x (k≠0, x＞0) passes through points A and C. If the coordinates of point C are (m, n), then the value of m−n is ______. | 3 | NULL | free_form |
func494 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/494.png | As shown in the figure, the vertices A, B, C of the parallelogram ▭ABCD have coordinates (0,2), (-1,-1), and (2,-1), respectively. What are the coordinates of vertex D? ________. | (3,2) | NULL | free_form |
func495 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/495.png | As shown in the figure, it is known that point P lies on the line y = 3/2x + 3, and the coordinates of point P are (a, a + 5). Point P is moved downward by a units and then moved left by b units to obtain point P', which also lies on the same linear function. What is the value of b? ________ | 8/3 | NULL | free_form |
func496 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/496.png | As shown in the figure, △OAC and △BAD are both isosceles right triangles, ∠LACO=∠LADB=90°, and the graph of the inverse proportional function y=10/x in the first quadrant passes through point B. Then the difference in the areas of △OAC and △BAD, SOAC - SBAD, is __________. | 5 | NULL | free_form |
func497 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/497.png | As shown in the figure, points A and B lie on the graphs of the direct proportion function y = x and the inverse proportion function y = 3 / x (x > 0), respectively. Point C is on the x-axis, AC ⊥ OC, and BD ⊥ AC at point D. If AB ⊥ OA, then the area difference between △AOC and △ABD is ______. | 3/2 | NULL | free_form |
func498 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/498.png | 2. The quadratic function y = ax² + bx is shown in the figure, then the linear function y = x + b must not pass through which quadrant? | Fourth | NULL | free_form |
func499 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/499.png | As shown in the figure, the hyperbolas y=2/x and y=4/x are in the first quadrant. The line BC is parallel to the x-axis and intersects the two curves at points B and C, respectively. Point A lies on the y-axis. What is the area of △ABC? ________. | 1 | NULL | free_form |
func5 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/5.png | As shown in the figure, the straight line y = ax + c intersects the straight line y = mx + n at point A(-4, 0). What is the solution set of the inequality ax + c > mx + n with respect to x? | x > -4 | NULL | free_form |
func50 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/50.png | As shown in the figure, the diagonals of rectangle OABC intersect at point D. The graph of the inverse proportional function y = k/x passes through point D and intersects BC at point E. Given that OA = 8 and OC = 6, find the ratio CE:BE. | 1:3 | NULL | free_form |
func500 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/500.png | The combination of numbers and shapes is a common method for solving mathematical problems. As shown in the figure, the graph of a linear function y = kx + b (where k and b are constants, and k < 0) and the line y = 1/3 both pass through point A(3, 1). When kx + b >= 1/3, the range of x is ________. | 0 ≤ x ≤ 3 | NULL | free_form |
func502 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/502.png | As shown in the figure: Given that the coordinates of point A are (-2√3, 2), and the diagonals of rhombus ABCD intersect at the origin O, the coordinates of point C are ________. | (2√3, -2) | NULL | free_form |
func503 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/503.png | As shown in the figure, in the plane rectangular coordinate system, O is the origin of the coordinates. In quadrilateral ABCD, the coordinates of points A and C are (-1, -1) and (3, 2) respectively, point B is in the fourth quadrant, and point D is on the y-axis. What are the coordinates of point B? ________. | (2, -1) | NULL | free_form |
func504 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/504.png | As shown in the figure, the straight line y = x + m intersects the straight line y = -1/2x - n at point A. Then the solution set of the inequality x + m ≥ -1/2x - n regarding x is ____________. | x ≥ -4 | NULL | free_form |
func505 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/505.png | As shown in the figure, the line l1: y = x + 1 intersects the line l2: y = ax + c at the point P(1, b). Then the solution to the system of equations {y = x + 1, y = ax + c} with respect to x and y is __________; | {x = 1, y = 2} | NULL | free_form |
func506 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/506.png | In response to the initiative 'Lucid waters and lush mountains are invaluable assets' and to build ecological civilization, a factory began limiting production and working on pollution control modifications starting from January 2019. Its monthly profit y (in ten thousand yuan) and month x are depicted in the graph. Before the pollution control was completed, it formed a part of the inverse proportional function graph. After the pollution control was completed, it formed a part of the linear function graph. Which of the following conclusions are correct? (Fill in the numbers only). | ②④ | NULL | free_form |
func507 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/507.png | As shown in the figure, in a rectangular coordinate plane, the side length of the equilateral triangle AOB is 12, side OB lies on the x-axis, and point A is in the first quadrant. If the graph of the inverse proportional function y = k/x passes through the midpoint C of OA in the first quadrant, then k = ______. | 9√3 | NULL | free_form |
func508 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/508.png | In physics, power represents the rate of doing work. The ratio of work done to the time taken is called power. That is, when the work done is constant, power P(w) is an inverse proportional function of the time t(s) taken to do the work. The graph is shown as follows. When t = 50, P = _____? | 1200 | NULL | free_form |
func509 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/509.png | Four identical rectangular paper pieces are arranged in the Cartesian coordinate system, forming the pattern shown in the diagram. If point A is (1.5, 3.5), then the coordinates of point B are ________. | (-3.5, 1) | NULL | free_form |
func51 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/51.png | As shown in the figure, the straight line y = x + 4 intersects the hyperbolic function y = -3/x (x < 0) at point C, and the line segment BC is bisected by the y-axis. What is the value of k? ________. | 5 | NULL | free_form |
func510 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/510.png | 1. As shown in the figure, △OABC has the side OA lying on the x-axis, vertex C on the graph of an inverse proportional function, BC intersects the y-axis at point D, and D is the midpoint of BC. If the area of △OABC is 10, then the expression for this inverse proportional function is ___________. | y = 5/x | NULL | free_form |
func511 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/511.png | As shown in the figure, in the Cartesian coordinate system, the vertices A, B, and C of rectangle ABCD have coordinates (0, 3), (-1/2, 3/2), and (4, 0), respectively. Given BD ∥ x-axis, the coordinates of point D are ___. | (4.5, 1.5) | NULL | free_form |
func512 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/512.png | The position of the line y = kx + b and y = mx in the Cartesian coordinate plane is shown in the figure. What is the solution set of the inequality −1 < kx + b < mx? | 0 < x < 2 | NULL | free_form |
func513 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/513.png | Quadrilaterals have instability. As shown in the figure, in the plane rectangular coordinate system, the side AB of square ABCD with side length 2 has its midpoint O at the origin. Points A and B are fixed, and point D is located on the positive half of the y-axis at point D. Then the coordinates of the corresponding point C' are ________. | (2, √3) | NULL | free_form |
func514 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/514.png | As shown in the figure, the outer diameter of a kind of circular ring is 8 cm, and the ring width is 1 cm. If x such rings are straightened together and tightened, their length is y cm. What is the relationship between y and x? | y = 6x + 2 (x ≥ 1 and x is an integer) | NULL | free_form |
func515 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/515.png | As shown in the figure, given point A(2,1) and B(2,2), if the hyperbolic function y = k/x (x > 0) intersects line segment AB, then the value of k could be _________. (Provide one possible value.) | 3 (the answer is not unique) | NULL | free_form |
func516 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/516.png | As shown in the figure, in the Cartesian coordinate system, point P lies on the hyperbolic curve y = k / x (x > 0, y > 0), and its y-coordinate is 4. Through point P, draw PQ intersecting the x-axis at point Q. Rotate segment QP clockwise by 60° about the origin O to generate segment QM. If point M lies on the hyperbolic curve after the rotation, what is the value of k? ________. | 8√3 | NULL | free_form |
func517 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/517.png | As shown in the figure, fishing boat A and port B are 19 nautical miles apart. We use the ordered pair (39° south of west, 19 nautical miles) to describe the position of fishing boat A relative to port B. Then the position of port B relative to fishing boat A can be described as _____. | (39° north of east, 19 nautical miles) | NULL | free_form |
func518 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/518.png | In the plane rectangular coordinate system, a robot starts from the origin O and moves one unit length each time in the directions: right, up, left, and down. The movement follows a path as shown in the figure. For the 1st move, it reaches point A1(1,0); for the 2nd move, it reaches point A2(1,1); for the 3rd move, it reaches point A3(2,1); for the 4th move, it reaches point A4(2,0). What are the coordinates of point A2024 after the 2024th move? ________. | (1012,0) | NULL | free_form |
func519 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/519.png | A new courier store has been added to the community, and the number of parcels collected in the first three days is shown in the figure. Assuming that the average daily growth rate of parcels collected by the courier store is x, based on the information in the figure, the equation satisfied by x is ________. | 200(1+x)^2=242 | NULL | free_form |
func52 | /home/yz979/scratch/chengye/math/data/original_set/funcset/images/images/52.png | As shown in the figure, given that the area of rectangle OABC is 100, the diagonal OB intersects one branch of the inverse proportional function graph at point D. If OB:BD = 3:2, then the expression for the inverse proportional function is ________. | y = 4/x | NULL | free_form |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.