query stringlengths 34 640 | positive_passages listlengths 1 1 | negative_passages listlengths 1 20 | score float64 50.8 98.5 |
|---|---|---|---|
Query: How would the Fourier transform help in determining the characteristics of the individual sinusoidal components in the given signal? | [
{
"doc_id": 0,
"text": "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 t... | [
{
"doc_id": 44096,
"text": "A function is given by f(t) = sin(2πt) + cos(4πt). Use Fourier transform to find the frequency and amplitude of the two sinusoidal components in f(t).\nTo find the frequency and amplitude of the two sinusoidal components in f(t) using the Fourier transform, we first need to expre... | 78 |
Query: If the hypotenuse of a right-angled triangle is 13 and one of the legs is 5, how does this affect the length of the other leg? | [
{
"doc_id": 0,
"text": "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 th... | [
{
"doc_id": 2412,
"text": "A right triangle has one leg measuring 9 cm and the hypotenuse measuring 15 cm. What is the length of the other leg of the triangle?\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 ... | 77.15 |
Query: What is the minimum number of colors needed to color a network with 12 nodes where each node is connected to exactly 4 other nodes, ensuring no two connected nodes share the same color? | [
{
"doc_id": 0,
"text": "A student needs to determine the chromatic number of a graph G, where G has 12 vertices and the degree of each vertex is 4. What is the minimum number of colors needed to color the vertices of G such that no two adjacent vertices have the same color?\nTo determine the chromatic numbe... | [
{
"doc_id": 4803,
"text": "What is the chromatic number of a graph G, given that G has 8 vertices and the degree of each vertex is 3?\nThe chromatic number of a graph is the smallest number of colors needed to color the vertices of the graph such that no two adjacent vertices share the same color.\n\nIn thi... | 83.9 |
Query: If each vertex in the graph G is connected to exactly 4 other vertices, what is the minimum number of colors needed to color the vertices such that no two adjacent vertices share the same color? | [
{
"doc_id": 1,
"text": "A student needs to determine the chromatic number of a graph G, where G has 12 vertices and the degree of each vertex is 4. What is the minimum number of colors needed to color the vertices of G such that no two adjacent vertices have the same color?\nTo determine the chromatic numbe... | [
{
"doc_id": 4803,
"text": "What is the chromatic number of a graph G, given that G has 8 vertices and the degree of each vertex is 3?\nThe chromatic number of a graph is the smallest number of colors needed to color the vertices of the graph such that no two adjacent vertices share the same color.\n\nIn thi... | 77.15 |
Query: A pendulum's motion can be modeled by a sinusoidal function. If a pendulum has an amplitude of 2 units, a period of 6 seconds, and starts its swing at a phase shift of π/3 radians, what is the equation of its motion in the form y = A * sin(B(x - C)) + D, and what is the phase angle in degrees when it starts its ... | [
{
"doc_id": 0,
"text": "A sinusoidal function has an amplitude of 2, a period of 6, and a phase shift of π/3. Write the equation for the function and find its phase angle in degrees.\nTo write the equation for the sinusoidal function, we need to determine the values of A, B, C, and D in the general equation... | [
{
"doc_id": 1006,
"text": "A sine function has an amplitude of 5 and a period of π/3. Find its equation.\nTo find the equation of the sine function with an amplitude of 5 and a period of π/3, we can use the general form of a sine function:\n\ny = A * sin(B(x - C)) + D\n\nwhere A is the amplitude, B determin... | 84.75 |
Query: Given a sinusoidal function with a specific amplitude and period, how would you determine the frequency and phase angle in degrees, and what steps are involved in converting the phase shift from radians to degrees? | [
{
"doc_id": 1,
"text": "A sinusoidal function has an amplitude of 2, a period of 6, and a phase shift of π/3. Write the equation for the function and find its phase angle in degrees.\nTo write the equation for the sinusoidal function, we need to determine the values of A, B, C, and D in the general equation... | [
{
"doc_id": 1006,
"text": "A sine function has an amplitude of 5 and a period of π/3. Find its equation.\nTo find the equation of the sine function with an amplitude of 5 and a period of π/3, we can use the general form of a sine function:\n\ny = A * sin(B(x - C)) + D\n\nwhere A is the amplitude, B determin... | 78 |
Query: If the period of the sinusoidal function were to double, how would it affect the frequency B in the equation, and what would be the new value of B in degrees? | [
{
"doc_id": 2,
"text": "A sinusoidal function has an amplitude of 2, a period of 6, and a phase shift of π/3. Write the equation for the function and find its phase angle in degrees.\nTo write the equation for the sinusoidal function, we need to determine the values of A, B, C, and D in the general equation... | [
{
"doc_id": 1006,
"text": "A sine function has an amplitude of 5 and a period of π/3. Find its equation.\nTo find the equation of the sine function with an amplitude of 5 and a period of π/3, we can use the general form of a sine function:\n\ny = A * sin(B(x - C)) + D\n\nwhere A is the amplitude, B determin... | 83.5 |
"Query: Given a sinusoidal function with an amplitude of 4, a period of 10, and no vertical shift, w(...TRUNCATED) | [{"doc_id":3,"text":"A sinusoidal function has an amplitude of 2, a period of 6, and a phase shift o(...TRUNCATED) | [{"doc_id":1006,"text":"A sine function has an amplitude of 5 and a period of π/3. Find its equatio(...TRUNCATED) | 73.75 |
"Query: If the period of the sinusoidal function were changed to 8 seconds, how would the frequency (...TRUNCATED) | [{"doc_id":4,"text":"A sinusoidal function has an amplitude of 2, a period of 6, and a phase shift o(...TRUNCATED) | [{"doc_id":1006,"text":"A sine function has an amplitude of 5 and a period of π/3. Find its equatio(...TRUNCATED) | 77.55 |
"Query: How does the number of holes in the surface S affect the total Gaussian curvature according (...TRUNCATED) | [{"doc_id":0,"text":"A surface S is a portion of a sphere with radius r=5 centered at the origin. Fi(...TRUNCATED) | [{"doc_id":42544,"text":"Consider a surface given by the equation x^2 + y^2 + z^2 = 1. Find the Gaus(...TRUNCATED) | 73.75 |
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