data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given $\\overrightarrow{a}=(2,3)$ and $\\overrightarrow{b}=(-4,7)$, if $\\overrightarrow{a}+ \\overrightarrow{c}=0$, then the projection of $\\overrightarrow{c}$ in the direction of $\\overrightarrow{b}$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "- \\dfrac { \\sqrt {65}}{5}"
} | {
"index": 12600,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\overrightarrow{a}=(2\\sin x,1)$ and $\\overrightarrow{b}=(2\\cos (x-\\frac{\\pi }{3}),\\sqrt{3})$, let $f(x)=\\overrightarrow{a}\\bullet \\overrightarrow{b}-2\\sqrt{3}$.\n\n(I) Find the smallest positive period and the zeros of $f(x)$;\n\n(II) Find the maximum and mini... | math | {
"style": "rule",
"ground_truth": "-\\sqrt{2}"
} | {
"index": 12601,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\overrightarrow{a}=(\\sin \\pi x,1)$, $\\overrightarrow{b}=( \\sqrt {3},\\cos \\pi x)$, and $f(x)= \\overrightarrow{a}\\cdot \\overrightarrow{b}$:\n(I) If $x\\in[0,2]$, find the interval(s) where $f(x)= \\overrightarrow{a}\\cdot \\overrightarrow{b}$ is monotonically inc... | math | {
"style": "rule",
"ground_truth": "-\\frac{16\\sqrt{481}}{481}"
} | {
"index": 12602,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\overrightarrow{a}=(\\tan (\\theta+ \\frac {\\pi}{12}),1)$ and $\\overrightarrow{b}=(1,-2)$, where $\\overrightarrow{a} \\perp \\overrightarrow{b}$, find the value of $\\tan (2\\theta+ \\frac {5\\pi}{12})$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{1}{7}"
} | {
"index": 12603,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\overrightarrow{m}=(2\\sqrt{3},1)$, $\\overrightarrow{n}=(\\cos^2 \\frac{A}{2},\\sin A)$, where $A$, $B$, and $C$ are the interior angles of $\\triangle ABC$; \n$(1)$ When $A= \\frac{\\pi}{2}$, find the value of $|\\overrightarrow{n}|$; \n$(2)$ If $C= \\frac{2\\pi}{3}... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 12604,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\overrightarrow{m}=(\\sin \\omega x,-1)$, $\\overrightarrow{n}=(1,- \\sqrt {3}\\cos \\omega x)$ where $x\\in\\mathbb{R}$, $\\omega > 0$, and $f(x)= \\overrightarrow{m}\\cdot \\overrightarrow{n}$, and the distance between a certain highest point and its adjacent lowest p... | math | {
"style": "rule",
"ground_truth": "\\dfrac {48}{25}"
} | {
"index": 12605,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin (30^{\\circ}+\\alpha)= \\frac {3}{5}$, and $60^{\\circ} < \\alpha < 150^{\\circ}$, solve for the value of $\\cos \\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3-4\\sqrt{3}}{10}"
} | {
"index": 12606,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin (\\frac{\\pi }{3}-\\theta )=\\frac{3}{4}$, find $\\cos (\\frac{\\pi }{3}+2\\theta )$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"index": 12607,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin (x-\\frac{5π}{12})=\\frac{1}{3}$, find $\\cos (\\frac{2021π}{6}-2x)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{9}"
} | {
"index": 12608,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin (α- \\frac {π}{6})= \\frac {2}{3}$, $α∈(π, \\frac {3π}{2})$, $\\cos ( \\frac {π}{3}+β)= \\frac {5}{13}$, $β∈(0,π)$, find the value of $\\cos (β-α)$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{10+12 \\sqrt{5}}{39}"
} | {
"index": 12609,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin (α- \\frac{π}{3})= \\frac{3}{5} $, where $a∈(\\frac{π}{4}, \\frac{π}{2})$, find $\\tan α=\\_\\_\\_\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{48+25 \\sqrt{3}}{11}"
} | {
"index": 12610,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin 2\\alpha = 3\\sin 2\\beta$, calculate the value of $\\frac {\\tan(\\alpha-\\beta)}{\\tan(\\alpha +\\beta )}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 12611,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin 2α - 2 = 2\\cos 2α$, find the value of $\\sin^{2}α + \\sin 2α$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{5}"
} | {
"index": 12612,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin \\alpha + 2\\cos \\alpha = \\frac{\\sqrt{10}}{2}$, find the value of $\\tan 2\\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{3}{4}"
} | {
"index": 12613,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin \\alpha + \\cos \\alpha = - \\frac{\\sqrt{5}}{2}$, and $\\frac{5\\pi}{4} < \\alpha < \\frac{3\\pi}{2}$, calculate the value of $\\cos \\alpha - \\sin \\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"index": 12614,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin \\alpha + \\cos \\alpha = \\frac{1}{5}$, and $- \\frac{\\pi}{2} \\leqslant \\alpha \\leqslant \\frac{\\pi}{2}$, find the value of $\\tan \\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{3}{4}"
} | {
"index": 12615,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin \\alpha + \\cos \\alpha = \\frac{\\sqrt{2}}{3}$ ($\\frac{\\pi}{2} < \\alpha < \\pi$), find the values of the following expressions: \n$(1) \\sin \\alpha - \\cos \\alpha$; \n$(2) \\sin^2\\left(\\frac{\\pi}{2} - \\alpha\\right) - \\cos^2\\left(\\frac{\\pi}{2} + \\a... | math | {
"style": "rule",
"ground_truth": "-\\frac{4\\sqrt{2}}{9}"
} | {
"index": 12616,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin \\alpha + \\cos \\beta = \\frac{1}{3}$ and $\\sin \\beta - \\cos \\alpha = \\frac{1}{2}$, find $\\sin (\\alpha-\\beta)=$ ______."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{59}{72}"
} | {
"index": 12617,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin \\alpha - \\cos \\alpha = \\frac{\\sqrt{10}}{5}$, $\\alpha \\in (\\pi, 2\\pi)$,\n\n$(1)$ Find the value of $\\sin \\alpha + \\cos \\alpha$;\n\n$(2)$ Find the value of $\\tan \\alpha - \\frac{1}{\\tan \\alpha}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{8}{3}"
} | {
"index": 12618,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin \\alpha = \\frac{3}{5}$ and $\\cos (\\alpha - \\beta) = \\frac{12}{13}$, where $0 < \\alpha < \\beta < \\frac{\\pi}{2}$, determine the value of $\\sin \\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{56}{65}"
} | {
"index": 12619,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin \\left( \\frac{\\pi}{4}-x\\right)= \\frac{1}{5} $, and $-\\pi < x < - \\frac{\\pi}{2}$. Find the values of the following expressions:\n\n$(1)\\sin \\left( \\frac{5\\pi}{4}-x\\right)$; \n\n$(2)\\cos \\left( \\frac{3\\pi}{4}+x\\right)$; \n\n$(3)\\sin \\left( \\frac{... | math | {
"style": "rule",
"ground_truth": "-\\frac{2 \\sqrt{6}}{5}"
} | {
"index": 12620,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin \\left(\\alpha+ \\frac {\\pi}{3}\\right)=- \\frac {4}{5}$, and $- \\frac {\\pi}{2} < \\alpha < 0$, find $\\cos \\alpha=$ ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {3-4 \\sqrt {3}}{10}"
} | {
"index": 12621,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin \\left(\\frac{3\\pi }{2}+\\theta \\right)=\\frac{1}{4}$,\n\nfind the value of $\\frac{\\cos (\\pi +\\theta )}{\\cos \\theta [\\cos (\\pi +\\theta )-1]}+\\frac{\\cos (\\theta -2\\pi )}{\\cos (\\theta +2\\pi )\\cos (\\theta +\\pi )+\\cos (-\\theta )}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{32}{15}"
} | {
"index": 12622,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin \\left(\\theta -\\frac{\\pi }{3}\\right)=\\frac{3}{5}$, then $\\sin \\left(2\\theta -\\frac{\\pi }{6}\\right)=$_______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{25}"
} | {
"index": 12623,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin \\left(x+ \\frac {\\pi}{3}\\right)= \\frac {1}{3}$, then the value of $\\sin \\left( \\frac {5\\pi}{3}-x\\right)-\\cos \\left(2x- \\frac {\\pi}{3}\\right)$ is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {4}{9}"
} | {
"index": 12624,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin \\theta + \\cos \\theta = \\frac{1}{5}$, with $\\theta \\in (0,\\pi)$.\n\n$(1)$ Find the value of $\\tan \\theta$; \n$(2)$ Find the value of $\\frac{1-2\\sin \\theta \\cos \\theta}{\\cos^2 \\theta - \\sin^2 \\theta}$."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"index": 12625,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin a= \\frac{ \\sqrt{5}}{5}$, $a\\in\\left( \\frac{\\pi}{2},\\pi\\right)$, find:\n\n$(1)$ The value of $\\sin 2a$;\n\n$(2)$ The value of $\\tan \\left( \\frac{\\pi}{3}+a\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "5 \\sqrt{3}-8"
} | {
"index": 12626,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin x = \\frac{3}{5}$, with $x \\in \\left( \\frac{\\pi}{2}, \\pi \\right)$, find the values of $\\cos 2x$ and $\\tan\\left( x + \\frac{\\pi}{4} \\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{7}"
} | {
"index": 12627,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin x_{1}=\\sin x_{2}=\\frac{1}{3}$ and $0 \\lt x_{1} \\lt x_{2} \\lt 2\\pi$, find $\\cos |\\overrightarrow{a}|$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{9}"
} | {
"index": 12628,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin(3\\pi + \\alpha) = -\\frac{1}{2}$, find the value of $\\cos\\left(\\frac{7\\pi}{2} - \\alpha\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 12629,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin(\\alpha - \\beta) = \\frac{1}{3}$ and $\\cos \\alpha \\sin \\beta = \\frac{1}{6}$, calculate the value of $\\cos(2\\alpha + 2\\beta)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"index": 12630,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin\\alpha + \\cos\\alpha = \\frac{\\sqrt{2}}{3}$, where $\\alpha \\in (0, \\pi)$, calculate the value of $\\sin\\left(\\alpha + \\frac{\\pi}{12}\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{2} + \\sqrt{3}}{6}"
} | {
"index": 12631,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin\\alpha= \\frac {2 \\sqrt {2}}{3}$, $\\cos(\\alpha+\\beta)=- \\frac {1}{3}$, and $\\alpha, \\beta\\in(0, \\frac {\\pi}{2})$, determine the value of $\\sin(\\alpha-\\beta)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {10 \\sqrt {2}}{27}"
} | {
"index": 12632,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin\\alpha= \\frac{1}{2}+\\cos\\alpha$, and $\\alpha\\in(0, \\frac{\\pi}{2})$, then $\\sin2\\alpha= \\_\\_\\_\\_\\_\\_$, $\\cos2\\alpha= \\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{7}}{4}"
} | {
"index": 12633,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin\\left( \\frac{\\pi}{3} + a \\right) = \\frac{5}{13}$, and $a \\in \\left( \\frac{\\pi}{6}, \\frac{2\\pi}{3} \\right)$, find the value of $\\sin\\left( \\frac{\\pi}{12} + a \\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17\\sqrt{2}}{26}"
} | {
"index": 12634,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin\\left(\\frac{\\pi}{3} - \\alpha\\right) = \\frac{1}{3}$, calculate $\\sin\\left(\\frac{\\pi}{6} - 2\\alpha\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{7}{9}"
} | {
"index": 12635,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin\\left(\\frac{\\pi}{3}+\\frac{\\alpha}{6}\\right)=-\\frac{3}{5}$, $\\cos\\left(\\frac{\\pi}{12}-\\frac{\\beta}{2}\\right)=-\\frac{12}{13}$, $-5\\pi < \\alpha < -2\\pi$, $-\\frac{11\\pi}{6} < \\beta < \\frac{\\pi}{6}$,\n\nFind the value of $\\sin \\left(\\frac{\\alph... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{65}"
} | {
"index": 12636,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin\\left(\\theta - \\frac{\\pi}{6}\\right) = \\frac{1}{4}$ with $\\theta \\in \\left( \\frac{\\pi}{6}, \\frac{2\\pi}{3}\\right)$, calculate the value of $\\cos\\left(\\frac{3\\pi}{2} + \\theta\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{15} + \\sqrt{3}}{8}"
} | {
"index": 12637,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin\\theta + \\cos\\theta = \\frac{1}{5}$, with $\\theta \\in (0,\\pi)$,\n1. Find the value of $\\tan\\theta$;\n2. Find the value of $\\frac{1+\\sin 2\\theta + \\cos 2\\theta}{1+\\sin 2\\theta - \\cos 2\\theta}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{4}"
} | {
"index": 12638,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin\\theta + \\cos\\theta = \\frac{3}{4}$, where $\\theta$ is an angle of a triangle, find the value of $\\sin\\theta - \\cos\\theta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{23}}{4}"
} | {
"index": 12639,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sin\\theta= \\frac {m-3}{m+5}$ and $\\cos\\theta= \\frac {4-2m}{m+5}$ ($\\frac {\\pi}{2} < \\theta < \\pi$), calculate $\\tan\\theta$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac {5}{12}"
} | {
"index": 12640,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sqrt[3]{2.37} \\approx 1.333$ and $\\sqrt[3]{23.7} \\approx 2.872$, determine the approximate value of $\\sqrt[3]{2370}$."
}
] | math | {
"style": "rule",
"ground_truth": "13.33"
} | {
"index": 12641,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sqrt{15129}=123$ and $\\sqrt{x}=0.123$, calculate the value of $x$."
}
] | math | {
"style": "rule",
"ground_truth": "0.015129"
} | {
"index": 12642,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sqrt{18} < \\sqrt{n^2}$ and $\\sqrt{n^2} < \\sqrt{200}$, determine the number of integers n."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 12643,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sqrt{2 + \\frac{2}{3}} = 2\\sqrt{\\frac{2}{3}}, \\sqrt{3 + \\frac{3}{8}} = 3\\sqrt{\\frac{3}{8}}, \\sqrt{4 + \\frac{4}{15}} = 4\\sqrt{\\frac{4}{15}}\\ldots$, if $\\sqrt{6 + \\frac{a}{b}} = 6\\sqrt{\\frac{a}{b}}$ (where $a,b$ are real numbers), please deduce $a = \\_\\_... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 12644,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sqrt{2.1}=1.449$ and $\\sqrt{21}=4.573$, find the value of $\\sqrt{21000}$."
}
] | math | {
"style": "rule",
"ground_truth": "144.9"
} | {
"index": 12645,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sqrt{20} \\approx 4.472, \\sqrt{2} \\approx 1.414$, find $-\\sqrt{0.2} \\approx$____."
}
] | math | {
"style": "rule",
"ground_truth": "-0.4472"
} | {
"index": 12646,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\sqrt{99225}=315$, $\\sqrt{x}=3.15$, then $x=(\\ )$."
}
] | math | {
"style": "rule",
"ground_truth": "9.9225"
} | {
"index": 12647,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\tan ( \\frac {π}{4}+x)=- \\frac {1}{2}$, find the value of $\\tan 2x$ (Part 1) and simplify the trigonometric expression $\\sqrt { \\frac {1+\\sin x}{1-\\sin x}}+ \\sqrt { \\frac {1-\\sin x}{1+\\sin x}}$ if $x$ is an angle in the second quadrant. Then, find its value (... | math | {
"style": "rule",
"ground_truth": "2\\sqrt {10}"
} | {
"index": 12648,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\tan (\\alpha +\\beta )=7$ and $\\tan (\\alpha -\\beta )=1$, find the value of $\\tan 2\\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\dfrac{4}{3}"
} | {
"index": 12649,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\tan (\\alpha+ \\frac {π}{3})=2$, find the value of $\\frac {\\sin (\\alpha+ \\frac {4π}{3})+\\cos ( \\frac {2π}{3}-\\alpha)}{\\cos ( \\frac {π}{6}-\\alpha )-\\sin (\\alpha + \\frac {5π}{6})}$."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 12650,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\tan (\\alpha-\\beta)= \\frac {1}{2}$, $\\tan \\beta=- \\frac {1}{7}$, and $\\alpha$, $\\beta\\in(0,\\pi)$, find the value of $2\\alpha-\\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac {3\\pi}{4}"
} | {
"index": 12651,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\tan (\\theta-\\pi)=2$, then $\\sin ^{2}\\theta+\\sin \\theta\\cos \\theta-2\\cos ^{2}\\theta=$ \\_\\_\\_\\_\\_\\_ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {4}{5}"
} | {
"index": 12652,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\tan 2\\alpha= \\frac {3}{4}$, $\\alpha\\in(- \\frac {\\pi}{2}, \\frac {\\pi}{2})$, $f(x)=\\sin (x+\\alpha)+\\sin (\\alpha-x)-2\\sin \\alpha$, and for any $x\\in\\mathbb{R}$, it always holds that $f(x)\\geqslant 0$, find the value of $\\sin (\\alpha- \\frac {\\pi}{4})$.... | math | {
"style": "rule",
"ground_truth": "- \\frac {2 \\sqrt {5}}{5}"
} | {
"index": 12653,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\tan \\alpha = -\\frac{1}{2}$, find the value of $\\frac{1+2\\sin \\alpha \\cos \\alpha}{\\sin^2 \\alpha - \\cos^2 \\alpha}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"index": 12654,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\tan \\alpha = 3$, evaluate the expression $2\\sin ^{2}\\alpha + 4\\sin \\alpha \\cos \\alpha - 9\\cos ^{2}\\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{21}{10}"
} | {
"index": 12655,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\tan \\alpha = \\frac{1}{3}$, calculate the value of $\\frac{\\cos^2 \\alpha - 2\\sin^2 \\alpha}{\\cos^2 \\alpha}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{9}"
} | {
"index": 12656,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\tan \\alpha =2$, find the values of the following expressions: \n$(1) \\frac{\\sin \\alpha - 3\\cos \\alpha}{\\sin \\alpha + \\cos \\alpha}$ \n$(2) 2\\sin ^{2} \\alpha - \\sin \\alpha \\cos \\alpha + \\cos ^{2} \\alpha$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{5}"
} | {
"index": 12657,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\tan \\alpha=4$, calculate the value of $\\frac{1+\\cos 2\\alpha+8\\sin^2\\alpha}{\\sin 2\\alpha}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{65}{4}"
} | {
"index": 12658,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\tan \\left(α- \\frac {π}{4}\\right)=2$, find the value of $\\sin \\left(2α- \\frac {π}{4}\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {\\sqrt {2}}{10}"
} | {
"index": 12659,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\tan \\theta = 5,$ find\n\\[\\frac{1 - \\cos \\theta}{\\sin \\theta} - \\frac{\\sin \\theta}{1 + \\cos \\theta}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 12660,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\tan \\theta =2$, find $\\cos 2\\theta =$____ and $\\tan (\\theta -\\frac{π}{4})=$____."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 12661,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\tan(\\theta-\\pi)=2$, find the value of $\\sin^2\\theta+\\sin\\theta\\cos\\theta-2\\cos^2\\theta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"index": 12662,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\tan\\alpha= \\dfrac {1}{3}$, find the values of the following expressions: \n1. $\\dfrac {\\sin \\alpha+\\cos\\alpha}{5\\cos\\alpha-\\sin\\alpha}$\n2. $\\dfrac {1}{2\\sin\\alpha\\cdot \\cos\\alpha+\\cos ^{2}\\alpha}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {2}{3}"
} | {
"index": 12663,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\tan\\alpha= \\frac {1}{2}$ and $\\tan(\\alpha-\\beta)=- \\frac {2}{5}$, calculate the value of $\\tan(2\\alpha-\\beta)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{12}"
} | {
"index": 12664,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\tan\\left( \\frac{\\pi}{4} + \\alpha \\right) = \\frac{1}{7}$, with $\\alpha \\in \\left( \\frac{\\pi}{2}, \\pi \\right)$, find the value of $\\tan\\alpha$ and $\\cos\\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{5}"
} | {
"index": 12665,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\theta \\in (0, \\frac{\\pi}{2})$, and $\\sin\\theta = \\frac{4}{5}$, find the value of $\\cos\\theta$ and $\\sin(\\theta + \\frac{\\pi}{3})$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4 + 3\\sqrt{3}}{10}"
} | {
"index": 12666,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\theta∈(0,\\frac{π}{2}$, $\\sin \\theta - \\cos \\theta = \\frac{\\sqrt{5}}{5}$, find the value of $\\tan 2\\theta$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{3}"
} | {
"index": 12667,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\triangle ABC$ and a point $P$ on one of its sides, call line $\\ell$ the $\\textit{splitting line}$ of $\\triangle ABC$ through $P$ if $\\ell$ passes through $P$ and divides $\\triangle ABC$ into two polygons of equal perimeter. Let $\\triangle ABC$ be a triangle where... | math | {
"style": "rule",
"ground_truth": "459"
} | {
"index": 12668,
"solution": "We wish to solve the Diophantine equation $a^2+ab+b^2=3^2 \\cdot 73^2$. It can be shown that $3|a$ and $3|b$, so we make the substitution $a=3x$ and $b=3y$ to obtain $x^2+xy+y^2=73^2$ as our new equation to solve for.\n Notice that $r^2+r+1=(r-\\omega)(r-{\\omega}^2)$, where $\\omega=... |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\triangle ABC$ is an oblique triangle, with the lengths of the sides opposite to angles $A$, $B$, and $C$ being $a$, $b$, and $c$, respectively. If $c\\sin A= \\sqrt {3}a\\cos C$.\n\n(Ⅰ) Find angle $C$; \n(Ⅱ) If $c= \\sqrt {21}$, and $\\sin C+\\sin (B-A)=5\\sin 2A$, fi... | math | {
"style": "rule",
"ground_truth": "\\frac {5 \\sqrt {3}}{4}"
} | {
"index": 12669,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\triangle ABC$ with $AC=1$, $\\angle ABC= \\frac{2\\pi}{3}$, $\\angle BAC=x$, let $f(x)= \\overrightarrow{AB} \\cdot \\overrightarrow{BC}$.\n\n$(1)$ Find the analytical expression of $f(x)$ and indicate its domain;\n\n$(2)$ Let $g(x)=6mf(x)+1$ $(m < 0)$, if the range of... | math | {
"style": "rule",
"ground_truth": "- \\frac{5}{2}"
} | {
"index": 12670,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\triangle ABC$ with its three interior angles $A$, $B$, and $C$, and $2\\sin^{2}(B+C)= \\sqrt{3}\\sin 2A$.\n\n(Ⅰ) Find the degree of $A$; \n(Ⅱ) If $BC=7$ and $AC=5$, find the area $S$ of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "10\\sqrt{3}"
} | {
"index": 12671,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\triangle ABC$ with the sides opposite to angles $A$, $B$, $C$ being $a$, $b$, $c$ respectively, and it satisfies $\\frac {\\sin (2A+B)}{\\sin A}=2+2\\cos (A+B)$. \n(I) Find the value of $\\frac {b}{a}$; \n(II) If $a=1$ and $c= \\sqrt {7}$, find the area of $\\triangl... | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {3}}{2}"
} | {
"index": 12672,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\triangle ABC$, where $AB=6$, $BC=8$, $AC=10$, and $D$ is on $\\overline{AC}$ with $BD=6$, find the ratio of $AD:DC$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{18}{7}"
} | {
"index": 12673,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\triangle ABC$, where the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it satisfies $a\\cos 2C+2c\\cos A\\cos C+a+b=0$.\n\n$(1)$ Find the size of angle $C$;\n\n$(2)$ If $b=4\\sin B$, find the maximum value of the area $S$ of $\\tria... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 12674,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\triangle PQR$ with $\\overline{RS}$ bisecting $\\angle R$, $PQ$ extended to $D$ and $\\angle n$ a right angle, then:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}(\\angle p + \\angle q)"
} | {
"index": 12675,
"solution": "1. **Identify the angles in $\\triangle PQR$ and the extended line:**\n - Let $\\angle PRS = \\theta$ since $\\overline{RS}$ bisects $\\angle R$.\n - $\\angle PQR = p$ and $\\angle QRP = q$.\n - $\\angle PRD = m$ since $\\angle n$ is a right angle, and $\\angle PRD$ is the exter... |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\vec{a}=(\\cos \\alpha, \\sin \\alpha), \\vec{b}=(\\cos \\beta, \\sin \\beta)$, and $|\\vec{a}-\\vec{b}|=\\frac{2 \\sqrt{5}}{5}$. If $0 < \\alpha < \\frac{\\pi}{2}$, $-\\frac{\\pi}{2} < \\beta < 0$, and $\\sin \\beta=-\\frac{5}{13}$, then $\\sin \\alpha=$"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{33}{65}$"
} | {
"index": 12676,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\{1,a, \\frac{b}{a}\\}=\\{0,a^2,a+b\\}$, calculate the value of $a^{2005}+b^{2005}$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 12677,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\{a_{n}\\}$ is a geometric sequence, $a_{2}a_{4}a_{5}=a_{3}a_{6}$, $a_{9}a_{10}=-8$, then $a_{7}=\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 12678,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\{a_{n}\\}\\left(n\\in N*\\right)$ is an arithmetic sequence with a common difference of $-2$, and $a_{6}$ is the geometric mean of $a_{2}$ and $a_{8}$. Let $S_{n}$ be the sum of the first $n$ terms of $\\{a_{n}\\}$. Find the value of $S_{10}$."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 12679,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $\\|\\mathbf{v}\\| = 4,$ find $\\|-3 \\mathbf{v}\\|.$"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 12680,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a = 1 + 2\\binom{20}{1} + 2^2\\binom{20}{2} + \\ldots + 2^{20}\\binom{20}{20}$, and $a \\equiv b \\pmod{10}$, determine the possible value(s) for $b$."
}
] | math | {
"style": "rule",
"ground_truth": "2011"
} | {
"index": 12681,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a > 0$, $b > 0$, and it satisfies the equation $3a + b = a^2 + ab$. Find the minimum value of $2a + b$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2} + 3"
} | {
"index": 12682,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a > 1$, $b > 1$, and $$\\frac {1}{a-1} + \\frac {1}{b-1} = 1$$, find the minimum value of $a + 4b$."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 12683,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a \\gt 0$, $b \\gt 0$, and $a+2b=1$, find the minimum value of $\\frac{{b}^{2}+a+1}{ab}$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{10} + 6"
} | {
"index": 12684,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a \\gt 0$, $b \\gt 0$, if ${a}^{2}+{b}^{2}-\\sqrt{3}ab=1$, determine the maximum value of $\\sqrt{3}{a}^{2}-ab$."
}
] | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{3}"
} | {
"index": 12685,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a \\gt 0$, $b\\in R$, if the inequality $\\left(ax-2\\right)(-x^{2}-bx+4)\\leqslant 0$ holds for all $x \\gt 0$, then the minimum value of $b+\\frac{3}{a}$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 12686,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a \\in \\mathbb{R}$, the function $f(x) = ax^3 - 3x^2$, and $x = 2$ is an extreme point of the function $y = f(x)$.\n\n1. Find the value of $a$.\n2. Find the extreme values of the function $f(x)$ in the interval $[-1, 5]$."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 12687,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a \\in \\{0, 1, 2\\}$ and $b \\in \\{-1, 1, 3, 5\\}$, find the probability that the function $f(x) = ax^2 - 2bx$ is an increasing function on the interval $(1, +\\infty)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"index": 12688,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a \\in \\{0,1,2\\}, b \\in \\{-1,1,3,5\\}$, determine the probability that the function $f(x)=ax^{2}-2bx$ is increasing in the interval $(1,+\\infty)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{5}{12}"
} | {
"index": 12689,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a \\neq 0,$ solve for $x$ in\n\\[\\begin{vmatrix} x + a & x & x \\\\ x & x + a & x \\\\ x & x & x + a \\end{vmatrix} = 0.\\]Give your answer in terms of $a.$"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{a}{3}"
} | {
"index": 12690,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a \\star b = a^2 + 2ab + b^2$, what is the value of $a \\star b$ when $a = 4$ and $b = 6?$"
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 12691,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a$ and $b$ are positive numbers such that $a^b = b^a$ and $b = 4a$, solve for the value of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[3]{4}"
} | {
"index": 12692,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a$ and $b$ are the roots of the equation $x^2-2cx-5d = 0$, and $c$ and $d$ are the roots of the equation $x^2-2ax-5b=0$, where $a,b,c,d$ are distinct real numbers, find $a+b+c+d$."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 12693,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a$, $b$, $c > 0$ and $$a(a+b+c)+bc=4-2 \\sqrt {3}$$, calculate the minimum value of $2a+b+c$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}-2"
} | {
"index": 12694,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a$, $b$, $c \\in \\mathbb{R}$, and $2a+2b+c=8$, find the minimum value of $(a-1)^2+(b+2)^2+(c-3)^2$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{49}{9}"
} | {
"index": 12695,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a$, $b$, $c \\in \\{1, 2, 3, 4, 5, 6\\}$, if the lengths $a$, $b$, and $c$ can form an isosceles (including equilateral) triangle, then there are \\_\\_\\_\\_\\_\\_ such triangles."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 12696,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a$, $b$, $c$, $d$, $e$ are 5 elements taken from the set $\\{1, 2, 3, 4, 5\\}$ without repetition, calculate the probability that $abc+de$ is an odd number."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"index": 12697,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a$, $b$, and $c$ are the three sides of $\\triangle ABC$, and $3a^2+3b^2-3c^2+2ab=0$, then $\\tan C= \\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "-2 \\sqrt {2}"
} | {
"index": 12698,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $a+b+c=0$ and $a^2+b^2+c^2=1$, find the values of $ab+bc+ca$ and $a^4+b^4+c^4$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 12699,
"solution": ""
} |
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