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values | query stringlengths 19 440 | pos listlengths 1 1 | neg listlengths 15 15 | train_group_size int64 16 16 | batch_size int64 8 8 |
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Given a Mathematics problem, retrieve relevant passages that help solve the problem. | What subject-area collaboration maintains the definition used for the real number in the context of signal analysis? | [
"A sound signal is given by the equation f(t) = sin(2πt) + sin(4πt) + sin(6πt) where t is time in seconds. Use Fourier transform to find the frequencies, amplitudes, and phases of the individual sinusoidal components in the signal.\nTo find the frequencies, amplitudes, and phases of the individual sinusoidal compon... | [
"Here is a numerical problem for a math student to solve related to the subtopic of Applying the Fourier transform to signal processing:\n\nA continuous-time signal in the time domain is defined as x(t) = 2cos(2π(10t + 3)) + 4cos(2π(25t - 1)).\n\na) What is the period of the signal?\nb) Compute the Fourier transfor... | 16 | 8 |
Given a Mathematics problem, retrieve relevant passages that help solve the problem. | What other types of mathematical transforms, such as the integral transform, are used in signal analysis? | [
"A sound signal is given by the equation f(t) = sin(2πt) + sin(4πt) + sin(6πt) where t is time in seconds. Use Fourier transform to find the frequencies, amplitudes, and phases of the individual sinusoidal components in the signal.\nTo find the frequencies, amplitudes, and phases of the individual sinusoidal compon... | [
"What is the Fourier transform of a periodic signal defined as:\n\nf(t) = 3sin(2πt) + 2sin(4πt) + sin(6πt) \n\nover the interval [-π, π]?\nThe Fourier transform of a periodic signal can be found using the Fourier series representation. For a given function f(t) with period T, the Fourier series representation is gi... | 16 | 8 |
Given a Mathematics problem, retrieve relevant passages that help solve the problem. | What are the frequencies, amplitudes, and phases of the individual sinusoidal components in a signal that uses the definition of a real number in its analysis? | [
"A sound signal is given by the equation f(t) = sin(2πt) + sin(4πt) + sin(6πt) where t is time in seconds. Use Fourier transform to find the frequencies, amplitudes, and phases of the individual sinusoidal components in the signal.\nTo find the frequencies, amplitudes, and phases of the individual sinusoidal compon... | [
"Suppose a signal is defined as f(t) = sin(2πt) + sin(4πt) + sin(6πt), where t is time in seconds. Use the Fourier transform to compute the frequency spectrum of the signal and determine the frequencies and amplitudes of the individual sinusoids that make up the signal.\nTo find the frequency spectrum of the signal... | 16 | 8 |
Given a Mathematics problem, retrieve relevant passages that help solve the problem. | What is the surface area of a shape that includes a straight line segment from the base to the apex? | [
"A right circular cone has a radius of 6cm and a slant height of 10cm. Determine the surface area of the cone.\nTo find the surface area of a right circular cone, we need to calculate the area of the base and the lateral surface area, and then add them together.\n\nThe base of the cone is a circle with radius r = 6... | [
"The solid shown was formed by cutting a right circular cylinder in half. If the base has a radius of 6 cm and the height is 10 cm, what is the total surface area, in terms of $\\pi$, of the solid? [asy]\nimport three;\nsize(100);\ndefaultpen(linewidth(.7pt));\ncurrentprojection=orthographic(4,2.5,3);\n\npath3 p=ar... | 16 | 8 |
Given a Mathematics problem, retrieve relevant passages that help solve the problem. | Find the point on a line segment that is twice as far from one endpoint as it is from the other in an orthogonal coordinate system, given two endpoints with coordinates (5,-2) and (-1,1.5). | [
"On the xy co-ordinate plane, point C is (5,-2) and point D is (-1,1.5). The point on line segment CD that is twice as far from C as from D is:\nAnswer Choices: (A) (1,-1) (B) (1,1) (C) (2,0.25) (D) (3,0.5) (E) (3,1) \nLet's think about the multi-choice question step by step.\nWe want the point on the line... | [
"On the xy co-ordinate plane, point C is (5,-2) and point D is (-2,2.5). The point on line segment CD that is twice as far from C as from D is:\nAnswer Choices: (A) (1,-1) (B) (1,1) (C) (2,0.25) (D) (3,-0.5) (E) (3,1) \nLet's think about the multi-choice question.\nWe want the point on the line that is twi... | 16 | 8 |
Given a Mathematics problem, retrieve relevant passages that help solve the problem. | What are the coordinates of a point on a line segment in a rectilinear coordinate system that is twice as far from one endpoint as it is from the other, given that the endpoints are (5,-2) and (-1,1.5)? | [
"On the xy co-ordinate plane, point C is (5,-2) and point D is (-1,1.5). The point on line segment CD that is twice as far from C as from D is:\nAnswer Choices: (A) (1,-1) (B) (1,1) (C) (2,0.25) (D) (3,0.5) (E) (3,1) \nLet's think about the multi-choice question step by step.\nWe want the point on the line... | [
"On the xy co-ordinate plane, point C is (5,-2) and point D is (-1,3.5). The point on line segment CD that is twice as far from C as from D is:\nAnswer Choices: (A) (1,-1) (B) (1,1) (C) (2,0.25) (D) (3,0.5) (E) (3,1) \nLet's think about the multi-choice question.\nWe want the point on the line that is twic... | 16 | 8 |
Given a Mathematics problem, retrieve relevant passages that help solve the problem. | Find the point on a line segment that is twice as far from one endpoint as it is from the other in a rectilinear coordinate system, given the endpoints are (5,-2) and (-1,1.5). | [
"On the xy co-ordinate plane, point C is (5,-2) and point D is (-1,1.5). The point on line segment CD that is twice as far from C as from D is:\nAnswer Choices: (A) (1,-1) (B) (1,1) (C) (2,0.25) (D) (3,0.5) (E) (3,1) \nLet's think about the multi-choice question step by step.\nWe want the point on the line... | [
"On the xy co-ordinate plane, point C is (5,-2) and point D is (-1,3.5). The point on line segment CD that is twice as far from C as from D is:\nAnswer Choices: (A) (1,-1) (B) (1,1) (C) (2,0.25) (D) (3,0.5) (E) (3,1) \nLet's think about the multi-choice question.\nWe want the point on the line that is twic... | 16 | 8 |
Given a Mathematics problem, retrieve relevant passages that help solve the problem. | Find the point on a line segment that is twice as far from one endpoint as it is from the other, using the definition from 'Perpendicular Distance from Straight Line in Plane to Point/General Form/Proof 1'. | [
"On the xy co-ordinate plane, point C is (5,-2) and point D is (-1,1.5). The point on line segment CD that is twice as far from C as from D is:\nAnswer Choices: (A) (1,-1) (B) (1,1) (C) (2,0.25) (D) (3,0.5) (E) (3,1) \nLet's think about the multi-choice question step by step.\nWe want the point on the line... | [
"On the xy co-ordinate plane, point C is (5,-2) and point D is (-1,2). The point on line segment CD that is twice as far from C as from D is:\nAnswer Choices: (A) (1,-1) (B) (1,1) (C) (2,0.25) (D) (3,0.5) (E) (3,1) \nLet's think about the multi-choice question step by step.\nWe want the point on the line t... | 16 | 8 |
Given a Mathematics problem, retrieve relevant passages that help solve the problem. | How can you use a continuous mathematical theory to find the length of the missing leg in a right triangle with one leg of length 5 and a hypotenuse of length 13? | [
"A triangle has a leg of length 5 and a hypotenuse of length 13. Find the length of the other leg using trigonometry.\nWe can use the Pythagorean theorem to find the length of the other leg. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite... | [
"A right triangle has a hypotenuse of length 13 and one leg of length 5. Find the length of the other leg.\nWe can use the Pythagorean theorem to solve this problem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the leng... | 16 | 8 |
Given a Mathematics problem, retrieve relevant passages that help solve the problem. | How can you find the length of one leg of a right triangle when given the length of another leg and the hypotenuse using a theorem related to Pythagoras's Theorem? | [
"A triangle has a leg of length 5 and a hypotenuse of length 13. Find the length of the other leg using trigonometry.\nWe can use the Pythagorean theorem to find the length of the other leg. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite... | [
"A right triangle has a hypotenuse of length 10 cm and one leg of length 6 cm. Find the length of the other leg using the Pythagorean theorem.\nThe Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares o... | 16 | 8 |
Given a Mathematics problem, retrieve relevant passages that help solve the problem. | What is the theory of right triangles that involves the calculation of side lengths? | [
"A triangle has a leg of length 5 and a hypotenuse of length 13. Find the length of the other leg using trigonometry.\nWe can use the Pythagorean theorem to find the length of the other leg. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite... | [
"A right triangle has a base of 6 cm and a height of 8 cm. What is the length of the hypotenuse of the triangle according to the Pythagorean theorem?\nThe Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the othe... | 16 | 8 |
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