query stringlengths 34 640 | positive_passages listlengths 1 1 | negative_passages listlengths 20 30 | score float64 65.2 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": 1794,
"text": "The Fourier transform can help in determining the characteristics of the individual sinusoidal components in the given signal by converting the signal into the frequency domain. To do this, we can apply the Fourier transform to the signal, which will reveal the amplitudes of the s... | 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": 4284,
"text": "A triangle has a leg of length 5 and a hypotenuse of length 13. Find the area of the triangle using the Pythagorean theorem.",
"title": ""
},
{
"doc_id": 401,
"text": "A right-angled triangle has one leg of length 6 cm and the hypotenuse of length 10 cm. Find the l... | 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": 1048,
"text": "A network with 12 nodes, where each node is connected to exactly 4 other nodes, requires the minimum number of colors such that no two adjacent nodes share the same color. Since the network forms a perfect square grid, only 2 colors are needed to ensure no two adjacent nodes are t... | 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": 1032,
"text": "A student needs to calculate the total number of edges in a graph G where each vertex is connected to exactly 4 other vertices. If G has 12 vertices, how many edges are there in G? This problem involves understanding the degree of vertices and the properties of graphs, which is re... | 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": 44796,
"text": "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 ang... | 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": 682,
"text": "A sinusoidal function has an amplitude of 2, a period of 6, and a frequency of 2. Write the equation for the function and find its phase angle in degrees.",
"title": ""
},
{
"doc_id": 683,
"text": "A sinusoidal function has an amplitude of 2, a period of 6, and a ph... | 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": 5745,
"text": "If the period of the sinusoidal function were to double, how would it affect the amplitude A in the equation, and what would be the new value of A in degrees? 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 ... | 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":682,"text":"Given a sinusoidal function with an amplitude of 4, a period of 10, and no ve(...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":5830,"text":"If the period of the sinusoidal function were changed to 8 seconds, how woul(...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":1789,"text":"A surface S is a torus with a major radius of 5 and a minor radius of 2. How(...TRUNCATED) | 73.75 |
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