data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\overrightarrow{OA}=(-2,4)$, $\\overrightarrow{OB}=(-a,2)$, $\\overrightarrow{OC}=(b,0)$, $a > 0$, $b > 0$, and $O$ is the coordinate origin. If points $A$, $B$, and $C$ are collinear, find the minimum value of $\\frac{1}{a}+\\frac{1}{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3 + 2\\sqrt{2}}{2}"
} | {
"index": 14700,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\overrightarrow{OA}=(1,0)$, $\\overrightarrow{OB}=(1,1)$, and $(x,y)=λ \\overrightarrow{OA}+μ \\overrightarrow{OB}$, if $0\\leqslant λ\\leqslant 1\\leqslant μ\\leqslant 2$, then the maximum value of $z= \\frac {x}{m}+ \\frac{y}{n}(m > 0,n > 0)$ is $2$. Find the min... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}+ \\sqrt{6}"
} | {
"index": 14701,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\overrightarrow{a}$ and $\\overrightarrow{b}$ are both unit vectors, and $(2\\overrightarrow{a}+ \\overrightarrow{b})\\cdot( \\overrightarrow{a}-2\\overrightarrow{b})=- \\frac{3\\sqrt{3}}{2}$, determine the angle between vectors $\\overrightarrow{a}$ and $\\overrig... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"index": 14702,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\overrightarrow{a}$ and $\\overrightarrow{b}$ are both unit vectors, and their angle is $120^{\\circ}$, calculate the magnitude of the vector $|\\overrightarrow{a}-2\\overrightarrow{b}|$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{7}"
} | {
"index": 14703,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\overrightarrow{a}$ and $\\overrightarrow{b}$ are both unit vectors, if $|\\overrightarrow{a}-2\\overrightarrow{b}|=\\sqrt{3}$, then the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is ____."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}\\pi"
} | {
"index": 14704,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\overrightarrow{a}$ and $\\overrightarrow{b}$ are non-zero vectors and $(\\overrightarrow{a} - 6\\overrightarrow{b}) \\perp \\overrightarrow{a}$, $(2\\overrightarrow{a} - 3\\overrightarrow{b}) \\perp \\overrightarrow{b}$, find the angle between $\\overrightarrow{a}... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"index": 14705,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\overrightarrow{a}$ and $\\overrightarrow{b}$ are non-zero vectors, $\\overrightarrow{m} = \\overrightarrow{a} + t \\overrightarrow{b} (t \\in \\mathbb{R})$, $|\\overrightarrow{a}| = 1$, $|\\overrightarrow{b}| = 2$, the minimum value of $|\\overrightarrow{m}|$ is o... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"index": 14706,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\overrightarrow{a}$ and $\\overrightarrow{b}$ are unit vectors and satisfy $\\overrightarrow{a} \\cdot \\overrightarrow{b} = 0$, find the maximum value of $(\\overrightarrow{a} + \\overrightarrow{b} + \\overrightarrow{c}) \\cdot (\\overrightarrow{a} + \\overrightar... | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{5}"
} | {
"index": 14707,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\overrightarrow{a}$ and $\\overrightarrow{b}$ are unit vectors, and $(2\\overrightarrow{a}+ \\overrightarrow{b})\\cdot (\\overrightarrow{a}-2\\overrightarrow{b})=- \\frac {3 \\sqrt {3}}{2}$, calculate the angle between vectors $\\overrightarrow{a}$ and $\\overright... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"index": 14708,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\overrightarrow{a}$ and $\\overrightarrow{b}$ are unit vectors, and the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is $90^{\\circ}$, if vector $\\overrightarrow{c}$ satisfies $|\\overrightarrow{c}- \\overrightarrow{a} \\overrightarrow{b}|=2$, cal... | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{2}"
} | {
"index": 14709,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\overrightarrow{a}$ and $\\overrightarrow{b}$ are unit vectors, if $|\\overrightarrow{a}-2\\overrightarrow{b}|=\\sqrt{3}$, then the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is ____."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}\\pi"
} | {
"index": 14710,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\overrightarrow{a}=(2,3)$, $\\overrightarrow{b}=(-4,7)$, and $\\overrightarrow{a}+\\overrightarrow{c}=\\overrightarrow{0}$, find the projection of $\\overrightarrow{c}$ on the direction of $\\overrightarrow{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{65}}{5}"
} | {
"index": 14711,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\overrightarrow{e}$ is a unit vector, $|\\overrightarrow{a}|=4$, and the angle between $\\overrightarrow{a}$ and $\\overrightarrow{e}$ is $\\frac{2}{3}π$, find the projection of $\\overrightarrow{a}$ onto $\\overrightarrow{e}$."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 14712,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\overset{→}{a\\_n}=\\left(\\cos \\frac{nπ}{6},\\sin \\frac{nπ}{6}\\right)$, $n∈ℕ^∗$, $\\overset{→}{b}=\\left( \\frac{1}{2}, \\frac{\\sqrt{3}}{2}\\right)$, calculate the value of $y={\\left| \\overset{→}{{a\\_1}}+ \\overset{→}{b}\\right|}^{2}+{\\left| \\overset{→}{{... | math | {
"style": "rule",
"ground_truth": "4029"
} | {
"index": 14713,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\overset{⇀}{m}=(2,1)$, $\\overset{⇀}{n}=(\\sin θ,\\cos θ)$, where $θ∈(0, \\dfrac{π}{2})$ is the inclination angle of line $l$ passing through point $A(1,4)$, if $\\overset{⇀}{m}· \\overset{⇀}{n}$ is at its maximum when line $l$ is tangent to the circle $(x+1)^{2}+(... | math | {
"style": "rule",
"ground_truth": "\\dfrac{2 \\sqrt{5}}{5}"
} | {
"index": 14714,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sec x + \\tan x = \\frac{4}{3},$ enter all possible values of $\\sin x.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{25}"
} | {
"index": 14715,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sec x - \\tan x = \\frac{5}{4},$ find all possible values of $\\sin x.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 14716,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin 2α - 2 = 2 \\cos 2α$, find the value of ${\\sin}^{2}α + \\sin 2α$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{5}"
} | {
"index": 14717,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin 2α = - \\frac{24}{25}$, and $α ∈ (\\frac{3π}{4},π)$, find the value of $\\sin α$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 14718,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin A+\\sin B=1$ and $\\cos A+\\cos B= \\frac{3}{2}$, what is the value of $\\cos(A-B)$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{8}"
} | {
"index": 14719,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin A+\\sin B=1$ and $\\cos A+\\cos B=3 / 2$, what is the value of $\\cos (A-B)$?"
}
] | math | {
"style": "rule",
"ground_truth": "5/8"
} | {
"index": 14720,
"solution": "Squaring both equations and add them together, one obtains $1+9 / 4=2+2(\\cos (A) \\cos (B)+\\sin (A) \\sin (B))=2+2 \\cos (A-B)$. Thus $\\cos A-B=5 / 8$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin \\alpha $ and $\\cos \\alpha $ are the two roots of the quadratic equation $2x^{2}-x-m=0$ with respect to $x$, find $\\sin \\alpha +\\cos \\alpha =$______ and $m=\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 14721,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin \\alpha + \\cos \\alpha = -\\frac{3}{\\sqrt{5}}$, and $|\\sin \\alpha| > |\\cos \\alpha|$, find the value of $\\tan \\frac{\\alpha}{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{5} + 1}{2}"
} | {
"index": 14722,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin \\alpha + \\cos \\alpha = -\\frac{\\sqrt{5}}{2}$ and $\\frac{5\\pi}{4} < \\alpha < \\frac{3\\pi}{2}$, find the value of $\\cos \\alpha - \\sin \\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"index": 14723,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin \\alpha + \\cos \\alpha = \\frac{\\sqrt{2}}{3}$, and $0 < \\alpha < \\pi$, find $\\tan\\left(\\alpha - \\frac{\\pi}{4}\\right) = \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_.$"
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 14724,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin \\alpha - 2\\cos \\alpha = \\frac{\\sqrt{10}}{2}$, find $\\tan 2\\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 14725,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin \\alpha - \\cos \\alpha = \\frac{1}{5}$, and $0 \\leqslant \\alpha \\leqslant \\pi$, find the value of $\\sin (2\\alpha - \\frac{\\pi}{4})$ = $\\_\\_\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{31\\sqrt{2}}{50}"
} | {
"index": 14726,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin \\alpha = 2 \\cos \\alpha$, find the value of $\\cos ( \\frac {2015\\pi}{2}-2\\alpha)$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac {4}{5}"
} | {
"index": 14727,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin \\alpha = 3 \\sin \\left(\\alpha + \\frac{\\pi}{6}\\right)$, find the value of $\\tan \\left(\\alpha + \\frac{\\pi}{12}\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{3} - 4"
} | {
"index": 14728,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin \\alpha = \\frac{2\\sqrt{2}}{3}$, $\\cos(\\alpha + \\beta) = -\\frac{1}{3}$, and both $\\alpha$ and $\\beta$ are within the interval $(0, \\frac{\\pi}{2})$, find the value of $\\sin(\\alpha - \\beta)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10\\sqrt{2}}{27}"
} | {
"index": 14729,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin \\alpha = \\frac{4}{5}$ and $\\alpha$ is an angle in the second quadrant, find the value of $\\cot (\\frac{\\pi}{4} - \\frac{\\alpha}{2})$."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 14730,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin \\alpha = \\frac{\\sqrt{5}}{5}$ and $\\sin (\\alpha - \\beta) = -\\frac{\\sqrt{10}}{10}$, where both $\\alpha$ and $\\beta$ are acute angles, find the value of $\\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 14731,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin \\alpha \\cos \\alpha = \\frac{1}{8}$, and $\\alpha$ is an angle in the third quadrant. Find $\\frac{1 - \\cos^2 \\alpha}{\\cos(\\frac{3\\pi}{2} - \\alpha) + \\cos \\alpha} + \\frac{\\sin(\\alpha - \\frac{7\\pi}{2}) + \\sin(2017\\pi - \\alpha)}{\\tan^2 \\alpha... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"index": 14732,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin \\alpha$ is a root of the equation $5x^2 - 7x - 6 = 0$, find the value of $\\frac{\\sin \\left(\\alpha - \\frac{\\pi}{2}\\right)\\sin^2\\left(-\\alpha + \\pi\\right)\\sin \\left( \\frac{3\\pi}{2}-\\alpha\\right)\\tan \\left(\\alpha - \\pi\\right)}{\\sin \\left... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 14733,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin \\alpha$ is a root of the equation $5x^{2}-7x-6=0$, calculate the value of $\\frac {\\sin (-\\alpha- \\frac {3}{2}\\pi)\\cdot \\sin ( \\frac {3}{2}\\pi-\\alpha)\\cdot \\tan ^{2}(2\\pi-\\alpha)}{\\cos ( \\frac {\\pi}{2}-\\alpha)\\cdot \\cos ( \\frac {\\pi}{2}+\... | math | {
"style": "rule",
"ground_truth": "\\frac{25}{16}"
} | {
"index": 14734,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin \\alpha$ is a root of the equation $5x^{2}-7x-6=0$, find:\n$(1)$ The value of $\\frac {\\cos (2\\pi-\\alpha)\\cos (\\pi+\\alpha)\\tan ^{2}(2\\pi-\\alpha)}{\\cos ( \\frac {\\pi}{2}+\\alpha)\\sin (2\\pi-\\alpha)\\cot ^{2}(\\pi-\\alpha)}$.\n$(2)$ In $\\triangle A... | math | {
"style": "rule",
"ground_truth": "-2- \\sqrt {3}"
} | {
"index": 14735,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin \\theta$ and $\\cos \\theta$ are the two roots of the equation $4x^{2}-4mx+2m-1=0$, and $\\frac {3\\pi}{2} < \\theta < 2\\pi$, find the angle $\\theta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {5\\pi}{3}"
} | {
"index": 14736,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin ^{10} x+\\cos ^{10} x=\\frac{11}{36}$, find the value of $\\sin ^{14} x+\\cos ^{14} x$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{41}{216}"
} | {
"index": 14737,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin x + \\cos x = \\frac{1}{2}$, where $x \\in [0, \\pi]$, find the value of $\\sin x - \\cos x$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}}{2}"
} | {
"index": 14738,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin x - \\cos x < 0$, determine the range of the function $y = \\frac{\\sin x}{|\\sin x|} + \\frac{\\cos x}{|\\cos x|} + \\frac{\\tan x}{|\\tan x|}$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 14739,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin x \\cdot \\cos x = -\\frac{1}{4}$ and $\\frac{3\\pi}{4} < x < \\pi$, find the value of $\\sin x + \\cos x$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{2}}{2}"
} | {
"index": 14740,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin x-2\\cos x= \\sqrt {5}$, find the value of $\\tan x$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\dfrac{1}{2}"
} | {
"index": 14741,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin x= \\frac {3}{5}$, and $x\\in( \\frac {\\pi}{2},\\pi)$, find the values of $\\cos 2x$ and $\\tan (x+ \\frac {\\pi}{4})$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {1}{7}"
} | {
"index": 14742,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin(\\alpha + \\frac{\\pi}{5}) = \\frac{1}{3}$ and $\\alpha$ is an obtuse angle, find the value of $\\cos(\\alpha + \\frac{9\\pi}{20})$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{2} + 4}{6}"
} | {
"index": 14743,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin(\\theta + 3\\pi) = -\\frac{2}{3}$, find the value of $\\frac{\\tan(-5\\pi - \\theta) \\cdot \\cos(\\theta - 2\\pi) \\cdot \\sin(-3\\pi - \\theta)}{\\tan(\\frac{7\\pi}{2} + \\theta) \\cdot \\sin(-4\\pi + \\theta) \\cdot \\cot(-\\theta - \\frac{\\pi}{2})} + 2 \\... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 14744,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin(a + \\frac{\\pi}{4}) = \\sqrt{2}(\\sin \\alpha + 2\\cos \\alpha)$, determine the value of $\\sin 2\\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{5}"
} | {
"index": 14745,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin(x+\\pi) + \\cos(x-\\pi) = \\frac{1}{2}, x \\in (0, \\pi)$.\n(1) Find the value of $\\sin x \\cos x$;\n(2) Find the value of $\\sin x - \\cos x$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}}{2}"
} | {
"index": 14746,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin(α + \\frac{π}{6}) - \\cos α = \\frac{1}{3}$, find the value of $2 \\sin α \\cos(α + \\frac{π}{6})$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{18}"
} | {
"index": 14747,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin(α + \\sqrt{2}\\cos(α) = \\sqrt{3})$, find the value of $\\tan(α)$.\n\nA) $\\frac{\\sqrt{2}}{2}$\nB) $\\sqrt{2}$\nC) $-\\frac{\\sqrt{2}}{2}$\nD) $-\\sqrt{2}$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 14748,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin2α + \\sinα = 0, α ∈ (\\frac{π}{2}, π)$, find the value of $\\tan(α + \\frac{π}{4})$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3} - 2"
} | {
"index": 14749,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin\\alpha + \\cos\\alpha = \\frac{1}{5}$, and $0 \\leq \\alpha < \\pi$, find the value of $\\tan\\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac {4}{3}"
} | {
"index": 14750,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin\\alpha + \\cos\\alpha = \\frac{\\sqrt{2}}{3}$ and $0 < \\alpha < \\pi$, find the value of $\\tan(\\alpha - \\frac{\\pi}{4})$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 14751,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin\\alpha + \\sin\\beta = \\frac{1}{3}$, find the maximum and minimum values of $y = \\sin\\beta - \\cos^2\\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{11}{12}"
} | {
"index": 14752,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin\\alpha = \\frac{1}{2} + \\cos\\alpha$, and $\\alpha \\in (0, \\frac{\\pi}{2})$, find the value of $\\frac{\\cos 2\\alpha}{\\sin(\\alpha - \\frac{\\pi}{4})}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{14}}{2}"
} | {
"index": 14753,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin\\alpha = \\frac{3}{5}$, and $\\alpha \\in \\left(\\frac{\\pi}{2}, \\pi \\right)$. \n(1) Find the value of $\\tan\\left(\\alpha+\\frac{\\pi}{4}\\right)$; \n(2) If $\\beta \\in (0, \\frac{\\pi}{2})$, and $\\cos(\\alpha-\\beta) = \\frac{1}{3}$, find the value of ... | math | {
"style": "rule",
"ground_truth": "\\frac{6\\sqrt{2} - 4}{15}"
} | {
"index": 14754,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin\\alpha=\\frac{\\sqrt{5}}{5}$ and $\\sin\\beta=\\frac{\\sqrt{10}}{10}$, where both $\\alpha$ and $\\beta$ are acute angles, find the value of $\\alpha+\\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 14755,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sin{\\alpha} = -\\frac{3}{5}$, $\\sin{\\beta} = \\frac{12}{13}$, and $\\alpha \\in (\\pi, \\frac{3\\pi}{2})$, $\\beta \\in (\\frac{\\pi}{2}, \\pi)$, find the values of $\\sin({\\alpha - \\beta})$, $\\cos{2\\alpha}$, and $\\tan{\\frac{\\beta}{2}}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 14756,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sqrt {3} \\overrightarrow{a}+ \\overrightarrow{b}+2 \\overrightarrow{c}= \\overrightarrow{0}$, and $| \\overrightarrow{a}|=| \\overrightarrow{b}|=| \\overrightarrow{c}|=1$, find the value of $\\overrightarrow{a}\\cdot ( \\overrightarrow{b}+ \\overrightarrow{c})$."... | math | {
"style": "rule",
"ground_truth": "-\\dfrac{\\sqrt{3}}{2}"
} | {
"index": 14757,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sqrt {3}\\sin x+\\cos x= \\frac {2}{3}$, find the value of $\\tan (x+ \\frac {7\\pi}{6})$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{4}"
} | {
"index": 14758,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sqrt{51.11}\\approx 7.149$ and $\\sqrt{511.1}\\approx 22.608$, determine the value of $\\sqrt{511100}$."
}
] | math | {
"style": "rule",
"ground_truth": "714.9"
} | {
"index": 14759,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sqrt{x}+\\frac{1}{\\sqrt{x}}=3$, determine the value of $\\frac{x}{x^{2}+2018 x+1}$."
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{1}{2025}$"
} | {
"index": 14760,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\square + q = 74$ and $\\square + 2q^2 = 180$, what is the value of $\\square$?"
}
] | math | {
"style": "rule",
"ground_truth": "66"
} | {
"index": 14761,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sum_{k=1}^{35}\\sin 5k=\\tan \\frac mn,$ where angles are measured in degrees, and $m$ and $n$ are relatively prime positive integers that satisfy $\\frac mn<90,$ find $m+n.$"
}
] | math | {
"style": "rule",
"ground_truth": "177"
} | {
"index": 14762,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sum_{k=1}^{35}\\sin 5k=\\tan \\frac mn,$ where angles are measured in degrees, and $m_{}$ and $n_{}$ are relatively prime positive integers that satisfy $\\frac mn<90,$ find $m+n.$"
}
] | math | {
"style": "rule",
"ground_truth": "177"
} | {
"index": 14763,
"solution": "We note that $\\sin x = \\mbox{Im } e^{ix}$. We thus have that \\begin{align*} \\sum_{k = 1}^{35} \\sin 5k &= \\sum_{k = 1}^{35} \\mbox{Im } e^{5ki}\\\\ &= \\mbox{Im } \\sum_{k = 1}^{35} e^{5ki}\\\\ &= \\mbox{Im } \\frac{e^{5i}(1 - e^{180i})}{1 - e^{5i}}\\\\ &= \\mbox{Im } \\frac{2\\c... |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sum_{k=1}^{36}\\sin 4k=\\tan \\frac{p}{q},$ where angles are measured in degrees, and $p$ and $q$ are relatively prime positive integers that satisfy $\\frac{p}{q}<90,$ find $p+q.$"
}
] | math | {
"style": "rule",
"ground_truth": "73"
} | {
"index": 14764,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\sum_{k=1}^{40}\\sin 4k=\\tan \\frac{p}{q},$ where angles are measured in degrees, and $p$ and $q$ are relatively prime positive integers that satisfy $\\frac{p}{q}<90,$ find $p+q.$"
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"index": 14765,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\tan (3 \\alpha-2 \\beta)=\\frac{1}{2}$ and $\\tan (5 \\alpha-4 \\beta)=\\frac{1}{4}$, find $\\tan \\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{16}"
} | {
"index": 14766,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\tan \\alpha +\\tan \\beta -\\tan \\alpha \\tan \\beta +1=0$, and $\\alpha ,\\beta \\in \\left(\\frac{\\pi }{2},\\pi \\right)$, calculate $\\alpha +\\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7\\pi}{4}"
} | {
"index": 14767,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\tan \\alpha = -2$, find the value of the following expressions:\n\n$(1) \\frac{\\sin \\alpha - 3 \\cos \\alpha}{\\sin \\alpha + \\cos \\alpha}$\n\n$(2) \\frac{1}{\\sin \\alpha \\cdot \\cos \\alpha}$"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{2}"
} | {
"index": 14768,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\tan \\alpha = -\\frac{1}{3}$ and $\\cos \\beta = \\frac{\\sqrt{5}}{5}$, with $\\alpha, \\beta \\in (0, \\pi)$, find:\n1. The value of $\\tan(\\alpha + \\beta)$;\n2. The maximum value of the function $f(x) = \\sqrt{2} \\sin(x - \\alpha) + \\cos(x + \\beta)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 14769,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\tan \\alpha = 2$, find the value of $\\frac{2 \\sin \\alpha - \\cos \\alpha}{\\sin \\alpha + 2 \\cos \\alpha}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 14770,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\tan \\alpha = 2$, where $\\alpha$ is an angle in the first quadrant, find the value of $\\sin 2\\alpha + \\cos \\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{4 + \\sqrt{5}}{5}"
} | {
"index": 14771,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\tan \\alpha = \\frac{1}{3}$ and $\\tan (\\alpha + \\beta) = \\frac{1}{2}$, find the value of $\\tan \\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{7}"
} | {
"index": 14772,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\tan \\alpha =2$, find the value of $\\frac{4\\sin^{3} \\alpha -2\\cos \\alpha }{5\\cos \\alpha +3\\sin \\alpha }$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"index": 14773,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\tan \\alpha$ and $\\frac{1}{\\tan \\alpha}$ are the two real roots of the equation $x^2 - kx + k^2 - 3 = 0$, and $3\\pi < \\alpha < \\frac{7}{2}\\pi$, find $\\cos \\alpha + \\sin \\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{2}"
} | {
"index": 14774,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\tan \\alpha$ and $\\frac{1}{\\tan \\alpha}$ are the two real roots of the equation $x^2 - kx + k^2 - 3 = 0$, and $\\pi < \\alpha < \\frac{3}{2}\\pi$.\n\n$(1)$ Find the value of $\\cos \\alpha + \\sin \\alpha$.\n\n$(2)$ Find the value of $\\frac{\\sin (\\alpha-3\\p... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 14775,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\tan \\alpha=-2$, the focus of the parabola $y^{2}=2px (p > 0)$ is $F(-\\sin \\alpha\\cos \\alpha,0)$, and line $l$ passes through point $F$ and intersects the parabola at points $A$ and $B$ with $|AB|=4$, find the distance from the midpoint of segment $AB$ to the ... | math | {
"style": "rule",
"ground_truth": "\\frac{21}{10}"
} | {
"index": 14776,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\tan \\beta= \\frac{4}{3}$, $\\sin (\\alpha+\\beta)= \\frac{5}{13}$, and both $\\alpha$ and $\\beta$ are within $(0, \\pi)$, find the value of $\\sin \\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{63}{65}"
} | {
"index": 14777,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\tan x = -\\frac{12}{5}$ and $x \\in (\\frac{\\pi}{2}, \\pi)$, find the value of $\\cos(-x + \\frac{3\\pi}{2})$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{12}{13}"
} | {
"index": 14778,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\tan α$ and $\\tan β$ are the roots of the equation $x^{2}+3 \\sqrt {3}x+4=0$, and $\\(- \\frac {π}{2} < α < \\frac {π}{2}\\)$, $\\(- \\frac {π}{2} < β < \\frac {π}{2}\\)$, find $α+β$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac {2\\pi}{3}"
} | {
"index": 14779,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\tan(\\alpha+ \\frac {\\pi}{4})= \\frac {3}{4}$, calculate the value of $\\cos ^{2}(\\frac {\\pi}{4}-\\alpha)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{25}"
} | {
"index": 14780,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\tan\\alpha=3$, calculate the following:\n(1) $\\frac{\\sin\\alpha+\\cos\\alpha}{2\\sin\\alpha-\\cos\\alpha}$\n(2) $\\sin^2\\alpha+\\sin\\alpha\\cos\\alpha+3\\cos^2\\alpha$"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 14781,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\tan\\left(\\alpha + \\frac{\\pi}{3}\\right)=2$, find the value of $\\frac{\\sin\\left(\\alpha + \\frac{4\\pi}{3}\\right) + \\cos\\left(\\frac{2\\pi}{3} - \\alpha\\right)}{\\cos\\left(\\frac{\\pi}{6} - \\alpha\\right) - \\sin\\left(\\alpha + \\frac{5\\pi}{6}\\right... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 14782,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\tan{\\alpha} = 3$, calculate:\n\n1. $$\\frac{4\\sin{\\alpha} - \\cos{\\alpha}}{3\\sin{\\alpha} + 5\\cos{\\alpha}}$$\n2. $$\\sin{\\alpha} \\cdot \\cos{\\alpha}$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"index": 14783,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\textstyle\\binom{2k}k$ results in a number that ends in two zeros, find the smallest positive integer $k$."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 14784,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\text{1 mile} = \\text{8 furlongs}$ and $\\text{1 furlong} = \\text{40 rods}$, the number of rods in one mile is"
}
] | math | {
"style": "rule",
"ground_truth": "320"
} | {
"index": 14785,
"solution": "To find the number of rods in one mile, we use the given conversion factors between miles, furlongs, and rods.\n\n1. **Convert miles to furlongs:**\n Given that 1 mile equals 8 furlongs, we can write:\n \\[\n 1\\ \\text{mile} = 8\\ \\text{furlongs}\n \\]\n\n2. **Convert furlon... |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\theta$ and $\\phi$ are acute angles such that $\\tan \\theta = \\frac{1}{7}$ and $\\sin \\phi = \\frac{1}{\\sqrt{10}},$ find $\\theta + 2 \\phi,$ measured in radians."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 14786,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\theta$ is an angle in the fourth quadrant, and $\\sin\\theta + \\cos\\theta = \\frac{1}{5}$, find: \n(1) $\\sin\\theta - \\cos\\theta$; \n(2) $\\tan\\theta$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{4}"
} | {
"index": 14787,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\theta$ is an angle in the second quadrant and $\\tan(\\theta + \\frac{\\pi}{4}) = \\frac{1}{2}$, find the value of $\\sin\\theta + \\cos\\theta$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{10}}{5}"
} | {
"index": 14788,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\theta$ is an angle in the second quadrant, and $\\tan 2\\theta = -2\\sqrt{2}$.\n(1) Find the value of $\\tan \\theta$.\n(2) Calculate the value of $\\frac {2\\cos^{2} \\frac {\\theta}{2}-\\sin\\theta-\\tan \\frac {5\\pi}{4}}{\\sqrt {2}\\sin(\\theta + \\frac {\\pi}... | math | {
"style": "rule",
"ground_truth": "3 + 2\\sqrt{2}"
} | {
"index": 14789,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\theta$ is an angle in the second quadrant, and $\\tan(\\theta + \\frac{\\pi}{4}) = \\frac{1}{2}$, calculate the value of $\\sin\\theta + \\cos\\theta$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{10}}{5}"
} | {
"index": 14790,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\theta$ is an angle in the third quadrant, and $sin^{4}\\theta+cos^{4}\\theta= \\frac {5}{9}$, then $sin2\\theta= \\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {2 \\sqrt {2}}{3}"
} | {
"index": 14791,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\theta$ is an angle in the third quadrant, if $\\sin^{4}{\\theta} + \\cos^{4}{\\theta} = \\frac{5}{9}$, find the value of $\\sin{2\\theta}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{2}}{3}"
} | {
"index": 14792,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\theta=\\arctan \\frac{5}{12}$, find the principal value of the argument of the complex number $z=\\frac{\\cos 2 \\theta+i \\sin 2 \\theta}{239+i}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 14793,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\triangle ABC$ is an acute triangle, vector $\\overrightarrow{m} = (\\cos(A + \\frac{\\pi}{3}), \\sin(A + \\frac{\\pi}{3}))$, $\\overrightarrow{n} = (\\cos B, \\sin B)$, and $\\overrightarrow{m} \\perp \\overrightarrow{n}$.\n\n(Ⅰ) Find the value of $A-B$; \n(Ⅱ) If... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3} + 3"
} | {
"index": 14794,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\triangle ABC$ is an acute triangle, vector $\\overrightarrow{m}=(\\cos (A+ \\frac{\\pi}{3}),\\sin (A+ \\frac{\\pi}{3}))$, $\\overrightarrow{n}=(\\cos B,\\sin B)$, and $\\overrightarrow{m} \\perp \\overrightarrow{n}$.\n(I) Find the value of $A-B$;\n(II) If $\\cos B... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}+3"
} | {
"index": 14795,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\triangle ABC$ is an equilateral triangle with side length $s$, determine the value of $s$ when $AP = 2$, $BP = 2\\sqrt{3}$, and $CP = 4$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{14}"
} | {
"index": 14796,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\triangle ABC$ is an isosceles right triangle with one leg length of $1$, determine the volume of the resulting geometric solid when $\\triangle ABC$ is rotated around one of its sides"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}\\pi}{6}"
} | {
"index": 14797,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\triangle+q=59$ and $(\\triangle+q)+q=106$, what is the value of $\\triangle$?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 14798,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $\\underbrace{9999\\cdots 99}_{80\\text{ nines}}$ is multiplied by $\\underbrace{7777\\cdots 77}_{80\\text{ sevens}}$, calculate the sum of the digits in the resulting product."
}
] | math | {
"style": "rule",
"ground_truth": "720"
} | {
"index": 14799,
"solution": ""
} |
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