messages listlengths 2 2 | deepmath_final_answer stringlengths 0 305 | solution_column stringclasses 1
value | solution_clipped bool 2
classes |
|---|---|---|---|
[
{
"role": "user",
"content": "Evaluate the limit: \\[ \\lim_{x \\to \\infty} \\sqrt{x} \\left( \\sqrt[3]{x+1} - \\sqrt[3]{x-1} \\right) \\]"
},
{
"role": "assistant",
"content": "Okay, so I have this limit to evaluate: the limit as x approaches infinity of the square root of x times the differen... | 0 | r1_solution_1 | true |
[
{
"role": "user",
"content": "Find the auxiliary equation for the ordinary differential equation with constant coefficients: \\((x^2D^2 + xD + 1)y = \\sin(2\\log x)\\sin(\\log x)\\)."
},
{
"role": "assistant",
"content": "Okay, so I need to find the auxiliary equation for the given ordinary diff... | m^2 + 1 = 0 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\left(\\dfrac{1}{\\tan^2 x}-\\dfrac{1}{x^2} \\right) \\]"
},
{
"role": "assistant",
"content": "Okay, so I need to find the limit as x approaches 0 of (1/tan²x - 1/x²). Hmm, let me start by understanding what's going on here... | -\dfrac{2}{3} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Determine the minimum sample size required such that the probability of at least two people being a match at all six genetic markers exceeds 0.001, given that the probability of a match at each individual marker is 1/9."
},
{
"role": "assistant",
"content": "Okay, s... | 34 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Find the limit: \\[ \\lim_{x \\to \\infty} (x!)^{1/x} \\]"
},
{
"role": "assistant",
"content": "Okay, so I need to find the limit as x approaches infinity of (x!)^(1/x). Hmm, factorials can be tricky because they grow really fast, but maybe I can use some approxima... | \infty | r1_solution_1 | false |
[
{
"role": "user",
"content": "Find the length of the polar curve given by \\( r = \\sqrt{1 + \\cos(2\\theta)} \\) for \\( 0 \\leq \\theta \\leq \\frac{\\pi\\sqrt{2}}{4} \\)."
},
{
"role": "assistant",
"content": "Okay, so I need to find the length of the polar curve given by \\( r = \\sqrt{1 + \... | \dfrac{\pi}{2} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Let $A$ be a proper infinite subset of a set $X$. If $x$ and $y$ are two distinct elements of $X$ that are not in $A$, and we define $B = \\{x, y\\} \\cup A$, what is the cardinality of $B$ in terms of the cardinality of $A$?"
},
{
"role": "assistant",
"content": "O... | |A| | r1_solution_1 | false |
[
{
"role": "user",
"content": "Consider a regular pentagon circumscribed in a circle. Connect each vertex of this pentagon to every other non-adjacent vertex with a straight line segment to form a pentagram, which contains a smaller pentagon. What is the ratio of the area of the original (large) pentagon to ... | \phi^4 | r1_solution_1 | true |
[
{
"role": "user",
"content": "Find the longest geometric progression with a common ratio greater than 1 that can be formed from the set \\( \\{100, 101, 102, \\ldots, 1000\\} \\). What is the length of this progression?"
},
{
"role": "assistant",
"content": "Okay, so I need to find the longest g... | 6 | r1_solution_1 | true |
[
{
"role": "user",
"content": "Find the expectation \\( \\mathbb{E}[X_t] \\) where \\( X_t = \\sin(B_t) \\) and \\( B_t \\) is a standard Brownian motion for \\( t \\geq 0 \\)."
},
{
"role": "assistant",
"content": "Okay, so I need to find the expectation of X_t, which is defined as sin(B_t), whe... | 0 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Determine the values of \\( k \\) for which the equation \\( x^3 - 3x^2 + 6x + k = 0 \\) has three real roots."
},
{
"role": "assistant",
"content": "Okay, so I need to figure out for which values of \\( k \\) the equation \\( x^3 - 3x^2 + 6x + k = 0 \\) has three r... | \emptyset | r1_solution_1 | false |
[
{
"role": "user",
"content": "Verify that the distance from the function \\( f = x \\) to the set \\( Y = \\{ f \\in C[0,1] : \\int_0^1 f = 0 \\} \\) is 0.5, given that \\( Y \\) is a closed subset of \\( X = \\{ f \\in C[0,1] : f(0) = 0 \\} \\)."
},
{
"role": "assistant",
"content": "Okay, so I... | 0.5 | r1_solution_1 | true |
[
{
"role": "user",
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\frac{\\sum_{k=1}^n k^p}{n^{p+1}} \\]"
},
{
"role": "assistant",
"content": "Okay, so I need to evaluate the limit: \\[ \\lim_{n \\to \\infty} \\frac{\\sum_{k=1}^n k^p}{n^{p+1}} \\] Hmm, let's see. I remember that when... | \dfrac{1}{p+1} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Provide an example of a field that properly contains the field of complex numbers \\( \\mathbb{C} \\)."
},
{
"role": "assistant",
"content": "Okay, so the problem is asking for an example of a field that properly contains the complex numbers, ℂ. Hmm, first I need to... | \mathbb{C}(t) | r1_solution_1 | false |
End of preview. Expand in Data Studio
README.md exists but content is empty.
- Downloads last month
- 38