prompt list | ground_truth string |
|---|---|
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 20.2 km/hr |
[
{
"content": "Let \\(T\\) be a topological space with Borel \\(\\sigma\\)-algebra \\(B(T)\\), generated by the open sets of \\(T\\). If \\(S \\in B(T)\\), consider the set \\(C = \\{A \\subset S : A \\in B(T)\\}\\). This set forms a \\(\\sigma\\)-algebra on \\(S\\). Now, generate the Borel \\(\\sigma\\)-algebra... | Yes |
[
{
"content": "Suppose that the graph of \\[2x^2 + y^2 + 8x - 10y + c = 0\\]consists of a single point. (In this case, we call the graph a degenerate ellipse.) Find $c.$",
"role": "user"
}
] | 33 |
[
{
"content": "On a plane, points are colored according to the following rules:\n1. Choose any positive integer \\( m \\), and let \\( K_{1}, K_{2}, \\cdots, K_{m} \\) be circles with different non-zero radii such that \\( K_{i} \\subset K_{j} \\) or \\( K_{j} \\subset K_{i} \\) for \\( i \\neq j \\).\n2. Points... | 2019 |
[
{
"content": "Using the integral test, find an upper bound for the error in approximating the series \\( S = \\sum_{n=3}^\\infty \\frac{1}{n(\\ln n)^4} \\) by the partial sum \\( S_{30} = \\sum_{n=3}^{30} \\frac{1}{n(\\ln n)^4} \\). Provide your answer accurate to three significant digits.",
"role": "user"
... | 0.00847 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 723.08 N |
[
{
"content": "Find the largest integer $m$ such that \\( \\sum_{n=0}^m 9(n+1)10^n \\le 10^{100000} \\).",
"role": "user"
}
] | 99994 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Covariance is 304.92 and Correlation is 0.1732 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 18 h |
[
{
"content": "Count the number of 10-digit numbers using the digits from \\( \\{1, 2, 3, 4\\} \\) such that no two adjacent digits differ by 1.",
"role": "user"
}
] | 21892 |
[
{
"content": "Jack rewrites the quadratic $9x^2 - 30x - 42$ in the form of $(ax + b)^2 + c,$ where $a,$ $b,$ and $c$ are all integers. What is $ab$?",
"role": "user"
}
] | -15 |
[
{
"content": "Given that \\( \\tan^2\\alpha \\tan^2\\beta +\\tan^2\\beta \\tan^2\\gamma + \\tan^2\\gamma \\tan^2\\alpha + 2\\tan^2\\alpha \\tan^2\\beta \\tan^2\\gamma =1 \\), find the value of \\( \\sin^2\\alpha + \\sin^2\\beta +\\sin^2\\gamma \\).",
"role": "user"
}
] | 1 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 6 degrees |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | -1.4 m |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1157.4 J/K |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.5 m/s |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 29.90 hours |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 7.638e-7 mol |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.73 m/s |
[
{
"content": "Let \\( f(x) \\) be defined as follows:\n\\[\nf(x) = \\begin{cases} e^{-1/x^2}, & x \\neq 0 \\\\ 0, & x = 0 \\end{cases}\n\\]\nCalculate \\( f'''(0) \\).",
"role": "user"
}
] | 0 |
[
{
"content": "In an equilateral triangle \\(ABC\\), points \\(M\\) and \\(N\\) are on side \\(AB\\), point \\(P\\) is on side \\(BC\\), and point \\(Q\\) is on side \\(CA\\) such that \\(MA + AQ = NB + BP = AB\\). Determine the angle formed by the lines \\(MP\\) and \\(NQ\\).",
"role": "user"
}
] | 60^\circ |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Social Security Tax Payable, SUTA Tax Payable, FUTA Tax Payable, and Medicare Tax Payable |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.44 V |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 25 years |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 6.880 ft/s |
[
{
"content": "Consider a continuous function $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ such that $f(f(x)) = x$ for all $x \\in \\mathbb{R}$. Does this function necessarily have a fixed point?",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the nested radical expression: \n\n$$A = \\sqrt{1 + 2\\sqrt{1 + 3\\sqrt{1 + 4\\sqrt{1 + 5\\sqrt{1 + \\cdots}}}}}$$\n\nDetermine the value of $A$.",
"role": "user"
}
] | 3 |
[
{
"content": "Let $A$ be a real symmetric $n \\times n$ matrix. If $A^4 = I_n$, does this imply that $A = I_n$? Provide a justification for your answer.",
"role": "user"
}
] | No |
[
{
"content": "The circle \\( C: x^{2}+(y-2)^{2}=16 \\) is symmetric about the line \\( a x + b y - 12 = 0 \\). A moving point \\( S \\) lies on the line \\( y + b = 0 \\). From point \\( S \\), two tangents \\( S A \\) and \\( S B \\) are drawn to the circle \\( C \\), touching the circle at points \\( A \\) an... | (0, 0) |
[
{
"content": "Is it possible to construct an order on \\(\\Bbb R^2\\) that turns it into a set with the least-upper-bound property, while ensuring the order is non-degenerate (i.e., it can distinguish distinct elements)?",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $500 billion |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 716.5kg |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 65.62 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $150 |
[
{
"content": "Let \\( U \\subseteq \\mathbb{R}^2 \\) be an open, bounded, connected subset, and let \\( f: U \\to U \\) be a smooth map such that \\( Jf = 1 \\) (i.e., \\( f \\) is area-preserving). Suppose that \\( h = f \\circ f \\) is affine. Must \\( f \\) be affine?",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit: $$\\lim_{r \\to \\infty} \\frac{\\int_0^{1} x^{r-1}\\sin x\\,dx}{\\int_0^{1} x^r\\sin{x}\\,dx}$$",
"role": "user"
}
] | 1 |
[
{
"content": "How many positive and negative integers is $12$ a multiple of?",
"role": "user"
}
] | 12 |
[
{
"content": "Find all real numbers \\( x \\) such that \\( \\arcsin(2x^2-1) - 2\\arcsin x = -\\frac{\\pi}{2} \\).",
"role": "user"
}
] | [0, 1] |
[
{
"content": "In a quadrilateral pyramid \\(SABCD\\), the base is a trapezoid \\(ABCD\\) with the bases \\(AD\\) and \\(BC\\) in the ratio 2:1. A plane passes through point \\(D\\) and the midpoints of edges \\(SA\\) and \\(SB\\). Determine the ratio in which this plane divides the edge \\(SC\\).",
"role": ... | 2:1 |
[
{
"content": "Given two mappings $f$ and $g$ of a set $S$ into itself, where $f^2 = g^2 = I$ and $f \\circ g = g \\circ f$, determine if it is always true that $(f \\circ g)^4 = I$. Provide a proof or counterexample.",
"role": "user"
}
] | (f \circ g)^4 = I |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $\rm Mn^{+2}(N^{+5}O^{-2}_3)_2\rightarrow H^{+1}Mn^{+7}O^{-2}_4+5e^-$ |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 5.95 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 48.0% |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 16.76 psi |
[
{
"content": "Solve for \\( x \\) in the equation: \n\\[ 3^{\\frac{x-1}{2}} - 2^{\\frac{x+1}{3}} = 2^{\\frac{x-2}{3}} + 3^{\\frac{x-3}{2}} \\]",
"role": "user"
}
] | 5 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 20.9% |
[
{
"content": "A system consists of 7 components with independent, identically distributed lifetimes. The system will operate until any one of the components fails. If the lifetime \\( X \\) of each component has the probability density function:\n\\[\nf(x) =\n\\begin{cases}\n\\frac{3}{x^4}, & \\text{for } x > ... | \dfrac{21}{20} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 16Ga + 3S_8 -> 8Ga_2S_3 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | CH_3OCH_3 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.00523M |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 8,182,769 foot pounds |
[
{
"content": "Find the minimum value of the expression \\((s+5-3|\\cos t|)^{2}+(s-2|\\sin t|)^{2}\\) as $s$ and $t$ range over all real numbers.",
"role": "user"
}
] | 2 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 300 g |
[
{
"content": "Evaluate the integral: $$\\int_{0}^{\\infty} \\frac{\\sin^4x}{x^4}dx$$",
"role": "user"
}
] | \dfrac{\pi}{3} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.37 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | H = (1+m/M)*h |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 96,779.27 |
[
{
"content": "Consider a power series \\( \\sum_{n \\ge 0} a_n z^n \\) where all coefficients \\( a_i \\) are non-zero complex numbers, and the radius of convergence is finite and non-zero, specifically \\( R = 2 \\). Is it necessary that the limit \\( \\lim_{n \\rightarrow \\infty} \\frac{|a_{n+1}|}{|a_n|} \\)... | No |
[
{
"content": "Consider the vector field \\( \\mathbf{F} = \\langle 0, 0, x + z \\rangle \\) and the surface \\( S \\), which is the portion of the plane \\( x + y + 2z = 4 \\) lying within the first octant. Assuming the outward unit normal vector \\( \\mathbf{n} \\) to \\( S \\) has a positive third component, ... | 16 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | m*sqrt(v{eq}_1{/eq}^2 + v{eq}_2{/eq}^2) |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 7.0 |
[
{
"content": "Determine whether there exists an infinite non-nilpotent group $G$ such that every proper nontrivial subgroup $H$ satisfies $N_G(H) = H$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.115e-3 N |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 44.5 g/mol |
[
{
"content": "Calculate the expectation \\( \\mathbb{E}(Y) \\) where \\( Y = X^6 \\) and \\( X \\sim \\mathcal{N}(0,1) \\). Use the Law of the Unconscious Statistician (LOTUS) and integration by parts to evaluate the integral:\n\\[ \\mathbb{E}(Y) = \\int_{-\\infty}^{\\infty} x^6 \\cdot \\frac{1}{\\sqrt{2\\pi}} ... | 15 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 8 Newtons |
[
{
"content": "Given the system of equations:\n\\[ x^2 - y^2 - u^3 + v^2 + 4 = 0 \\]\n\\[ 2xy + y^2 - 2u^2 + 3v^4 + 8 = 0 \\]\nfind \\( \\frac{\\partial u}{\\partial x} \\) at the point \\((x,y) = (2,-1)\\).",
"role": "user"
}
] | \dfrac{13}{32} |
[
{
"content": "Evaluate the limit \\( \\lim_{n \\to \\infty} n^k \\left(1 - \\frac{1}{\\sqrt{n}}\\right)^n \\) for any fixed positive integer \\( k \\).",
"role": "user"
}
] | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Net Sales 100%
less: COGS 60%
Gross Profit 40%
less: Operating Expenses 32%
NET INCOME 8% |
[
{
"content": "Find all triples \\((m, p, q)\\) where \\(m\\) is a positive integer and \\(p, q\\) are primes such that \\(2^m p^2 + 1 = q^5\\).",
"role": "user"
}
] | (1, 11, 3) |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 348 N |
[
{
"content": "Let $a \\in \\mathbb{R}$ and consider the vector space $E$ of $C^{\\infty}$ functions $f : \\mathbb{R} \\to \\mathbb{R}$. Define the endomorphism $u$ on $E$ by $u(f) = f' - af$. Determine the dimension of $\\ker u^n$ and provide a basis for $\\ker u^n$, where $u^n$ is the composition of $u$ with i... | n |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 52 |
[
{
"content": "Determine the maximum value of the function \\( N(a - b) \\) over all \\( a, b \\in U(V) \\), where \\( N: \\text{End}_\\mathbb{C} \\to \\mathbb{R}_{\\ge 0} \\) is defined as \\( N(a) := \\max_{\\{v \\in V\\, :\\, |v| = 1\\}} |av| \\). What is the diameter of the group \\( U(V) \\)?",
"role": ... | 2 |
[
{
"content": "Find the minimum value of the expression \\(x^2+y^2+z^2+\\dfrac{1}{x^2}+\\dfrac{1}{y^2}+\\dfrac{1}{z^2}\\) for non-zero real numbers \\(x, y, z\\) such that \\(x+y+z=6\\).",
"role": "user"
}
] | \dfrac{51}{4} |
[
{
"content": "The distance between two cities on a map is 15 inches. If the scale is 0.25 inches = 3 miles, how many miles apart are the actual cities?",
"role": "user"
}
] | 180\text{ miles} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 500 |
[
{
"content": "Consider polynomials $f, g \\in P^5(-1,1)$ satisfying $||f|| = \\sqrt{6}$, $||f+g||= 4$, and $||f-g||=2$, where the norm is the $L^2$ norm. What is $||g||$?",
"role": "user"
}
] | 2 |
[
{
"content": "Determine the cardinality of the set of $\\mathbb{Q}$-morphisms from the algebraic closure of $\\mathbb{Q}$ to $\\mathbb{C}$. Provide your answer as a cardinal number.",
"role": "user"
}
] | 2^{\aleph_0} |
[
{
"content": "Convert $1230_{4}$ to base 10.",
"role": "user"
}
] | 108 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 7.2 |
[
{
"content": "Determine the values of \\( z \\) for which the series \\( \\sum (in)^2 z^n \\) converges.",
"role": "user"
}
] | |z| < 1 |
[
{
"content": "Find all integer solutions to the equation \\( y^3 - 1 = x^4 + x^2 \\).",
"role": "user"
}
] | (0, 1) |
[
{
"content": "Compute the Lebesgue outer measure of the set \\((0,1) \\cap \\mathbb{Q}^c\\), where \\(\\mathbb{Q}\\) is the set of rational numbers.",
"role": "user"
}
] | 1 |
[
{
"content": "Solve the exponential equation: \\[ 7 \\cdot 3^{x+1} - 5^{x+2} = 3^{x+4} - 5^{x+3} \\]",
"role": "user"
}
] | -1 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | CH_3OH + NaCl |
[
{
"content": "Determine the order of the group $G = \\langle a, b \\mid a^3 = b^2 = (ab)^3 = e \\rangle$. Provide your answer as a single integer.",
"role": "user"
}
] | 12 |
[
{
"content": "Determine if the function \\( f(x, y, z) = (x + y + z)\\sin\\left(\\frac{1}{x}\\right)\\sin\\left(\\frac{1}{y}\\right) \\) has a limit at the point \\((0,0,0)\\) in \\(\\mathbb{R}^3\\). If a limit exists, find its value.",
"role": "user"
}
] | 0 |
[
{
"content": "Find the exact length of the curve defined by the equation \\(x=\\frac{y^4}{8}+\\frac{1}{4y^2}\\) for \\(1 \\le y \\le 2\\) using the arc length formula \\(L = \\int_{a}^{b} \\sqrt{1+\\left(\\frac{dx}{dy}\\right)^2} \\, dy.\\)",
"role": "user"
}
] | \dfrac{33}{16} |
[
{
"content": "Express the output $Y(s)$ in terms of the input $X(s)$ for a control system with a nested feedback loop, where the plant transfer function is given by \\( F(s)=\\frac{R^3+R^2sL}{s^2R^2L-R^3s} \\). The input $X(s)$ is subtracted from the output $Y(s)$ to form $X_1(s) = X(s) - Y(s)$, which is then m... | Y(s) = X(s) |
[
{
"content": "Given a complex matrix \\(\\mathbf{X} \\in \\mathbb{C}^{m \\times 2}\\) with full column rank, determine the matrix \\(\\mathbf{M}\\) that completes the following valid equality:\n\n\\[ \\mathbf{I}_{m} - \\mathbf{X} \\left( \\mathbf{X}^{*} \\mathbf{X} \\right)^{-1} \\mathbf{X}^{*} = \\mathbf{M} \\... | D |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Service revenue
902,105
Interest income
1,500
Income and Expense Summary
903,605
To close the revenue accounts.
Income and Expense Summary
816,215
Rent expense 63,500
Wages expense
539,260
Supplies expense
42,520
Utilities expense
8,595
Depreciation expense
155,500
Interest expense
6,840
To close the expense accounts.... |
[
{
"content": "Let $A$ be any $n \\times n$ diagonalizable matrix. Is $p(A)$ diagonalizable for any polynomial $p(x)$? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 53515855.58 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.3 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\frac{n(n^{1/n}-1)}{\\ln (n+2)! - \\ln n!}. \\]",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Given the set \\( P = \\{a_1, a_2, \\cdots, a_n, \\cdots\\} \\), where \\( a_1 = 1 \\) and \\( a_n = a_{n-1} + 4(n-1) \\) for \\( n \\in \\mathbb{N}^*, n \\neq 1 \\), and the set \\( Q = \\{ n \\mid (1+i)^{2n} = 2^n i, n \\in \\mathbb{N}^* \\} \\), determine the relationship between sets \\( P \\)... | A |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $16.40 |
[
{
"content": "If a function $f(x)$ is differentiable at $x = a$, is it necessarily differentiable in an open interval including $x = a$?",
"role": "user"
}
] | No |
[
{
"content": "Evaluate the integral: $$\\int x^{x^2+1}(2\\ln x+1)dx.$$",
"role": "user"
}
] | x^{x^2} |
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