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 $x$ is a multiple of $46200$, determine the greatest common divisor of $f(x) = (3x + 5)(5x + 3)(11x + 6)(x + 11)$ and $x$."
}
] | math | {
"style": "rule",
"ground_truth": "990"
} | {
"index": 15000,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$ is a multiple of $54896$, find the greatest common divisor of $f(x)=(5x+4)(9x+7)(11x+3)(x+12)$ and $x$."
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"index": 15001,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$ is a positive integer less than 100, how many solutions does the congruence $x + 13 \\equiv 55 \\pmod{34}$ have?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 15002,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$ is a root of the equation $x^{2}+x-6=0$, simplify $\\frac{x-1}{\\frac{2}{{x-1}}-1}$ and find its value."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"index": 15003,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$ is real and $x^3+\\frac{1}{x^3}=52$, find $x+\\frac{1}{x}$."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 15004,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$ is the geometric mean of $4$ and $16$, find the value of $x$."
}
] | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"index": 15005,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$ is the median of the data set $1$, $2$, $3$, $x$, $5$, $6$, $7$, and the average of the data set $1$, $2$, $x^{2}$, $-y$ is $1$, find the minimum value of $y- \\frac {1}{x}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {23}{3}"
} | {
"index": 15006,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$ varies directly as the square of $y$, and $y$ varies directly as the cube root of $z$, determine the power $n$ such that $x$ varies as $x^n$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 15007,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$, $y$, $z \\in \\mathbb{R}$, if $-1$, $x$, $y$, $z$, $-3$ form a geometric sequence, calculate the value of $xyz$."
}
] | math | {
"style": "rule",
"ground_truth": "-3\\sqrt{3}"
} | {
"index": 15008,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$, $y$, $z$ are positive real numbers, find the maximum value of $\\dfrac{xy+yz}{x^{2}+y^{2}+z^{2}}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{2}}{2}"
} | {
"index": 15009,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x$, $y$, and $z$ are all non-negative numbers and $x + y + z = 2$, find the minimum value of $\\frac{1}{3}x^{3} + y^{2} + z$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{12}"
} | {
"index": 15010,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x+\\sin y=2008$ and $x+2008 \\cos y=2007$, where $0 \\leq y \\leq \\frac{\\pi}{2}$, find the value of $x+y$."
}
] | math | {
"style": "rule",
"ground_truth": "2007 + \\frac\\pi 2"
} | {
"index": 15011,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x+\\sin y=2008$ and $x+2008 \\cos y=2007$, where $0 \\leq y \\leq \\pi / 2$, find the value of $x+y$."
}
] | math | {
"style": "rule",
"ground_truth": "2007+\\frac{\\pi}{2}"
} | {
"index": 15012,
"solution": "Subtracting the two equations gives $\\sin y-2008 \\cos y=1$. But since $0 \\leq y \\leq \\pi / 2$, the maximum of $\\sin y$ is 1 and the minimum of $\\cos y$ is 0 , so we must have $\\sin y=1$, so $y=\\pi / 2$ and $x+y=2007+\\frac{\\pi}{2}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x+y = 10$ and $2x+y = 13$, evaluate $x^2-y^2$."
}
] | math | {
"style": "rule",
"ground_truth": "-40"
} | {
"index": 15013,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x+y=12$, $xy=9$, and $x < y$, find the value of $\\frac {x^{ \\frac {1}{2}}-y^{ \\frac {1}{2}}}{x^{ \\frac {1}{2}}+y^{ \\frac {1}{2}}}=$ ___."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac { \\sqrt {3}}{3}"
} | {
"index": 15014,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x, y > 0$ and $\\frac{1}{x} + \\frac{1}{y} = 2$, find the minimum value of $x + 2y$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3 + 2\\sqrt{2}}{2}"
} | {
"index": 15015,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x, y,$ and $z$ are real numbers that satisfy: \\begin{align*} x &= \\sqrt{y^2-\\frac{1}{16}}+\\sqrt{z^2-\\frac{1}{16}}, \\\\ y &= \\sqrt{z^2-\\frac{1}{25}}+\\sqrt{x^2-\\frac{1}{25}}, \\\\ z &= \\sqrt{x^2 - \\frac 1{36}}+\\sqrt{y^2-\\frac 1{36}}, \\end{align*} and th... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 15016,
"solution": "Note that none of $x,y,z$ can be zero.\n Each of the equations is in the form \\[a=\\sqrt{b^2-d^2}+\\sqrt{c^2-d^2}\\]\n Isolate a radical and square the equation to get \\[b^2-d^2=a^2-2a\\sqrt{c^2-d^2}+c^2-d^2\\]\n Now cancel, and again isolate the radical, and square the equation to ... |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x<1$ and \\[(\\log_{10} x)^2 - \\log_{10}(x^2) = 48,\\]compute the value of \\[(\\log_{10}x)^3 - \\log_{10}(x^3).\\]"
}
] | math | {
"style": "rule",
"ground_truth": "-198"
} | {
"index": 15017,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x=1$ is a root of the quadratic equation $\\left(m-2\\right)x^{2}+4x-m^{2}=0$, calculate the value(s) of $m$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 15018,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x\\_1$ is a root of the equation $x \\ln x = 2006$ and $x\\_2$ is a root of the equation $xe^{x} = 2006$, find the value of $x\\_1 \\cdot x\\_2$."
}
] | math | {
"style": "rule",
"ground_truth": "2006"
} | {
"index": 15019,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x^2 + y^2 = 14x + 6y + 6,$ find the largest possible value of $3x + 4y.$"
}
] | math | {
"style": "rule",
"ground_truth": "73"
} | {
"index": 15020,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x^2+x-6$ is a factor of the polynomial $2x^4+x^3-ax^2+bx+a+b-1$, find the value of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 15021,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x^{2}+y^{2}=1$, determine the maximum and minimum values of $x+y$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{2}"
} | {
"index": 15022,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x_{0}$ is a zero of the function $f(x)=2a\\sqrt{x}+b-{e}^{\\frac{x}{2}}$, and $x_{0}\\in [\\frac{1}{4}$,$e]$, find the minimum value of $a^{2}+b^{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{{e}^{\\frac{3}{4}}}{4}"
} | {
"index": 15023,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x_{1}$ and $x_{2}$ are two real roots of the one-variable quadratic equation $x^{2}-6x+k=0$, and (choose one of the conditions $A$ or $B$ to answer the following questions).<br/>$A$: $x_1^2x_2^2-x_1-x_2=115$;<br/>$B$: $x_1^2+x_2^2-6x_1-6x_2+k^2+2k-121=0$.<br/>$(1)$ ... | math | {
"style": "rule",
"ground_truth": "-11"
} | {
"index": 15024,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $x_{1}$ is a root of the one-variable quadratic equation about $x$, $\\frac{1}{2}m{x^2}+\\sqrt{2}x+{m^2}=0$, and ${x_1}=\\sqrt{a+2}-\\sqrt{8-a}+\\sqrt{-{a^2}}$ (where $a$ is a real number), find the values of $m$ and the other root of the equation."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 15025,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $y < 1$ and \\[(\\log_{10} y)^2 - \\log_{10}(y^3) = 75,\\] compute the value of \\[(\\log_{10}y)^3 - \\log_{10}(y^4).\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2808 - 336\\sqrt{309}}{8} - 6 + 2\\sqrt{309}"
} | {
"index": 15026,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $y = f(x) + x^2$ is an odd function, and $f(1) = 1$, determine the value of $g(x) = f(x) + 2$ when $x = -1$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 15027,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $y = f(x) + x^2$ is an odd function, and $f(1) = 1$, if $g(x) = f(x) + 2$, then $g(-1) = \\_\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 15028,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $y$ is a multiple of $3456$, what is the greatest common divisor of $g(y) = (5y+4)(9y+1)(12y+6)(3y+9)$ and $y$?"
}
] | math | {
"style": "rule",
"ground_truth": "216"
} | {
"index": 15029,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $y$ is a multiple of $42522$, what is the greatest common divisor of $g(y)=(3y+4)(8y+3)(14y+9)(y+17)$ and $y$?"
}
] | math | {
"style": "rule",
"ground_truth": "102"
} | {
"index": 15030,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $y$ is a multiple of $45678$, what is the greatest common divisor of $g(y)=(3y+4)(8y+3)(14y+9)(y+14)$ and $y$?"
}
] | math | {
"style": "rule",
"ground_truth": "1512"
} | {
"index": 15031,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $y=\\left(m-2\\right)x+(m^{2}-4)$ is a direct proportion function, find the possible values of $m$."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 15032,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $y=f(x)$ is an odd function, if $f(x)=g(x)+x^{2}$ and $g(1)=1$, then $g(-1)=$ _____ ."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 15033,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $y=f\\left(x\\right)+x^{2}$ is an odd function, and $f\\left(1\\right)=1$, if $g\\left(x\\right)=f\\left(x\\right)+2$, then $g\\left(-1\\right)=$____."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 15034,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $z$ is a complex number such that $z+\\frac 1z=2\\cos 3^\\circ$, find $z^{2000}+\\frac 1{z^{2000}}$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 15035,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $z$ is a complex number such that $z+\\frac 1z=2\\cos 3^\\circ$, find the least integer that is greater than $z^{2000}+\\frac 1{z^{2000}}$."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 15036,
"solution": "Let $z=re^{i\\theta}$. Notice that we have $2\\cos(3^{\\circ})=e^{i\\frac{\\pi}{60}}+e^{-i\\frac{\\pi}{60}}=re^{i\\theta}+\\frac{1}{r}e^{-i\\theta}.$\n$r$ must be $1$ (or else if you take the magnitude would not be the same). Therefore, $z=e^{i\\frac{\\pi}{\\theta}}$ and plugging into... |
DeepScaleR | [
{
"role": "user",
"content": "Given that $z$ is a complex number such that $z+\\frac{1}{z}=2\\cos 5^\\circ$, find $z^{1500}+\\frac{1}{z^{1500}}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{3}"
} | {
"index": 15037,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that ${(1-2x)^{2016}}=a_{0}+a_{1}(x-2)+a_{2}(x-2)^{2}+\\cdots+a_{2015}(x-2)^{2015}+a_{2016}(x-2)^{2016}$ $(x\\in\\mathbb{R})$, find the value of $a_{1}-2a_{2}+3a_{3}-4a_{4}+\\cdots+2015a_{2015}-2016a_{2016}$."
}
] | math | {
"style": "rule",
"ground_truth": "4032"
} | {
"index": 15038,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that ${{a_n}}$ is an arithmetic sequence with a non-zero common difference, and ${{a_9}}$, ${{a_1}}$, ${{a_5}}$ form a geometric sequence, and ${{a_1} + 3{{a_5}} + {{a_9}} = 20}$, determine the value of ${{a_{13}}}$."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 15039,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $| \\overrightarrow{a}|=1$, $| \\overrightarrow{b}|= \\sqrt {2}$, and $\\overrightarrow{a} \\perp ( \\overrightarrow{a}- \\overrightarrow{b})$, find the angle between vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {\\pi}{4}"
} | {
"index": 15040,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $| \\overrightarrow{a}|=12$, $| \\overrightarrow{b}|=9$, and $\\overrightarrow{a} \\cdot \\overrightarrow{b}=-54 \\sqrt {2}$, find the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi}{4}"
} | {
"index": 15041,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $| \\overrightarrow{a}|=2$, $\\overrightarrow{e}$ is a unit vector, and the angle between $\\overrightarrow{a}$ and $\\overrightarrow{e}$ is $\\dfrac {\\pi}{3}$, find the projection of $\\overrightarrow{a}+ \\overrightarrow{e}$ on $\\overrightarrow{a}- \\overrightarr... | math | {
"style": "rule",
"ground_truth": "\\sqrt {3}"
} | {
"index": 15042,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $| \\overrightarrow{a}|=3 \\sqrt {2}$, $| \\overrightarrow{b}|=4$, $\\overrightarrow{m}= \\overrightarrow{a}+ \\overrightarrow{b}$, $\\overrightarrow{n}= \\overrightarrow{a}+λ \\overrightarrow{b}$, $ < \\overrightarrow{a}, \\overrightarrow{b} > =135^{\\circ}$, if $\\... | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"index": 15043,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $| \\overrightarrow{a}|=5$, $| \\overrightarrow{b}|=3$, and $\\overrightarrow{a} \\cdot \\overrightarrow{b}=-12$, find the projection of vector $\\overrightarrow{a}$ on vector $\\overrightarrow{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 15044,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $| \\overrightarrow{a}|=5$, $| \\overrightarrow{b}|=4$, and $\\overrightarrow{a} \\cdot \\overrightarrow{b}=-10$, find the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$ (denoted as $\\langle \\overrightarrow{a}, \\overrightarrow{b} \\rangle$)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"index": 15045,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $| \\overrightarrow{a}|=6$, $| \\overrightarrow{b}|=3$, and $\\overrightarrow{a} \\cdot \\overrightarrow{b}=-12$, find the projection of vector $\\overrightarrow{a}$ onto vector $\\overrightarrow{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 15046,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $| \\overrightarrow{a}|=| \\overrightarrow{b}|=| \\overrightarrow{c}|=1$, and $ \\overrightarrow{a}+ \\overrightarrow{b}+ \\sqrt {3} \\overrightarrow{c}=0$, find the value of $ \\overrightarrow{a} \\overrightarrow{b}+ \\overrightarrow{b} \\overrightarrow{c}+ \\overri... | math | {
"style": "rule",
"ground_truth": "\\dfrac {1}{2}- \\sqrt {3}"
} | {
"index": 15047,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $| \\vec{e} | = 1$ and it satisfies $| \\vec{a} + \\vec{e} | = | \\vec{a} - 2\\vec{e} |$, find the projection of vector $\\vec{a}$ in the direction of $\\vec{e}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 15048,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $|$$\\overrightarrow {a}$$ $|=1$, $\\overrightarrow {b}$ $=$ ($ $\\frac { \\sqrt {3}}{3} $, $ \\frac { \\sqrt {3}}{3}$), and $|$ $\\overrightarrow {a}$ $+3 \\overrightarrow {b}$ $|=2$, find the projection of $\\overrightarrow {b}$ in the direction of $\\overrig... | math | {
"style": "rule",
"ground_truth": "- \\frac {1}{2}"
} | {
"index": 15049,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $|\\cos\\theta|= \\frac {1}{5}$ and $\\frac {5\\pi}{2}<\\theta<3\\pi$, find the value of $\\sin \\frac {\\theta}{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{15}}{5}"
} | {
"index": 15050,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=2$, and $(\\overrightarrow{a}+\\overrightarrow{b})\\perp\\overrightarrow{a}$, determine the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"index": 15051,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=\\sqrt{2}$, and $\\overrightarrow{a} \\bot (\\overrightarrow{a}-\\overrightarrow{b})$, find the projection of vector $\\overrightarrow{a}$ in the direction of vector $\\overrightarrow{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 15052,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $|\\overrightarrow{a}|=4, |\\overrightarrow{b}|=8$, and the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is $\\frac{2\\pi}{3}$.\n\n(I) Find $|\\overrightarrow{a}+\\overrightarrow{b}|$;\n\n(II) Find the value of $k$ such that $(\\overrightarrow{a}+2\\... | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"index": 15053,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $|\\overrightarrow{a}|=5$, $|\\overrightarrow{b}|=3$, and $\\overrightarrow{a} \\cdot \\overrightarrow{b} = -12$, find the projection of vector $\\overrightarrow{a}$ on vector $\\overrightarrow{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 15054,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $|\\overrightarrow{a}|=5$, $|\\overrightarrow{b}|=3$, and $\\overrightarrow{a}\\cdot\\overrightarrow{b}=-12$, find the projection of vector $\\overrightarrow{a}$ onto vector $\\overrightarrow{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 15055,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $|\\vec{a}| = 1$ and $\\vec{b} = (1, \\sqrt{3})$, and that $(\\vec{b} - \\vec{a}) \\perp \\vec{a}$, find the angle between vector $\\vec{a}$ and vector $\\vec{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"index": 15056,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $|\\vec{a}| = 2$, $|\\vec{b}| = 1$, and $(2\\vec{a} - 3\\vec{b}) \\cdot (2\\vec{a} + \\vec{b}) = 9$.\n\n(I) Find the angle $\\theta$ between vectors $\\vec{a}$ and $\\vec{b}$;\n\n(II) Find $|\\vec{a} + \\vec{b}|$ and the projection of vector $\\vec{a}$ in the directi... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{7}}{7}"
} | {
"index": 15057,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $|\\vec{a}|=1$, $|\\vec{b}|=2$, and $(\\vec{a}+\\vec{b})\\cdot \\vec{b}=3$, find the angle between $\\vec{b}$ and $\\vec{a}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"index": 15058,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $|\\vec{a}|=1$, $|\\vec{b}|=\\sqrt{2}$, $\\vec{c}=\\vec{a}+\\vec{b}$, and $\\vec{c} \\perp \\vec{a}$, the angle between vectors $\\vec{a}$ and $\\vec{b}$ is $\\_\\_\\_\\_\\_\\_\\_\\_\\_.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi}{4}"
} | {
"index": 15059,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $|\\vec{a}|=4$, and $\\vec{e}$ is a unit vector. When the angle between $\\vec{a}$ and $\\vec{e}$ is $\\frac{2\\pi}{3}$, the projection of $\\vec{a} + \\vec{e}$ on $\\vec{a} - \\vec{e}$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{21}}{7}"
} | {
"index": 15060,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $|\\vec{a}|=|\\vec{b}|=1$ and $\\vec{a} \\perp \\vec{b}$, find the projection of $2\\vec{a}+\\vec{b}$ in the direction of $\\vec{a}+\\vec{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{3\\sqrt{2}}{2}"
} | {
"index": 15061,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $|x|=3$, $y^{2}=4$, and $x < y$, find the value of $x+y$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 15062,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $||\\overrightarrow{OA}||=||\\overrightarrow{OB}||=2$, point $C$ is on line segment $AB$, and the minimum value of $||\\overrightarrow{OC}||$ is $1$, find the minimum value of $||\\overrightarrow{OA}-t\\overrightarrow{OB}||$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 15063,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α \\in ( \\frac{5}{4}π, \\frac{3}{2}π)$, and it satisfies $\\tan α + \\frac{1}{\\tan α} = 8$, find the value of $\\sin α \\cos α$ and $\\sin α - \\cos α$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{\\sqrt{3}}{2}"
} | {
"index": 15064,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α \\in (0, \\frac{π}{2})$ and $\\sin (\\frac{π}{4} - α) = \\frac{\\sqrt{10}}{10}$,\n(1) find the value of $\\tan 2α$;\n(2) find the value of $\\frac{\\sin (α + \\frac{π}{4})}{\\sin 2α + \\cos 2α + 1}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{8}"
} | {
"index": 15065,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α \\in (0, \\frac{π}{2})$ and $\\sin α - \\cos α = \\frac{1}{2}$, find the value of $\\frac{\\cos 2α}{\\sin (α + \\frac{π}{4})}$.\n\nA) $-\\frac{\\sqrt{2}}{2}$\nB) $\\frac{\\sqrt{2}}{2}$\nC) $-1$\nD) $1$"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{2}}{2}"
} | {
"index": 15066,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α \\in (0, \\frac{π}{2})$, and $\\sin (\\frac{π}{6} - α) = -\\frac{1}{3}$, find the value of $\\cos α$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{6} - 1}{6}"
} | {
"index": 15067,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α \\in (0, \\frac{π}{3})$ satisfies the equation $\\sqrt{6} \\sin α + \\sqrt{2} \\cos α = \\sqrt{3}$, find the values of:\n1. $\\cos (α + \\frac{π}{6})$\n2. $\\cos (2α + \\frac{π}{12})$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{30} + \\sqrt{2}}{8}"
} | {
"index": 15068,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α \\in (0,π)$, and $\\sin α= \\frac {3}{5}$, find the value of $\\tan (α- \\frac {π}{4})$."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"index": 15069,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α \\in (0,π)$, if $\\sin α + \\cos α = \\frac{\\sqrt{3}}{3}$, find the value of $\\cos^2 α - \\sin^2 α$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{3}"
} | {
"index": 15070,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α \\in \\left( \\frac{π}{2}, π \\right)$ and $3\\cos 2α = \\sin \\left( \\frac{π}{4} - α \\right)$, find the value of $\\sin 2α$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{17}{18}"
} | {
"index": 15071,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α$ and $β ∈ ( \\frac{π}{2},π)$, and $sinα + cosα = a$, $cos(β - α) = \\frac{3}{5}$.\n(1) If $a = \\frac{1}{3}$, find the value of $sinαcosα + tanα - \\frac{1}{3cosα}$;\n(2) If $a = \\frac{7}{13}$, find the value of $sinβ$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{16}{65}"
} | {
"index": 15072,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α$ and $β$ are acute angles, and $\\cos(α+β)=\\frac{3}{5}$, $\\sin α=\\frac{5}{13}$, find the value of $\\cos β$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{56}{65}"
} | {
"index": 15073,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α$ and $β$ are acute angles, and it is given that $\\sin \\alpha =\\frac{3}{5}$ and $\\cos (\\alpha +\\beta )=\\frac{5}{13}$, calculate the value of $\\cos \\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{56}{65}"
} | {
"index": 15074,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α$ and $β$ are both acute angles, $cosα= \\frac {3}{5}$, and $cos(α+β)=- \\frac {5}{13}$, find the value of $sinβ$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {56}{65}"
} | {
"index": 15075,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α$ and $β$ are both acute angles, and $\\cos (α+β)=\\dfrac{\\sin α}{\\sin β}$, the maximum value of $\\tan α$ is \\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{ \\sqrt{2}}{4}"
} | {
"index": 15076,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α$ and $β$ belong to the interval $(0, \\frac{π}{2})$, and they satisfy the equation $\\tan(α + β) = 9\\tan(β)$, find the maximum value of $\\tan(α)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 15077,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α$ is an acute angle and $\\sin α= \\frac {3}{5}$, find the value of $\\cos α$ and $\\cos (α+ \\frac {π}{6})$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {4\\sqrt {3}-3}{10}"
} | {
"index": 15078,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α$ is an angle in the first quadrant, it satisfies $\\sin α - \\cos α = \\frac{\\sqrt{10}}{5}$. Find $\\cos 2α$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{4}{5}"
} | {
"index": 15079,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α$ is an angle in the fourth quadrant and $\\cos α= \\frac {3}{5}$.\n(1) Find the value of $\\tan α$;\n(2) Find the value of $\\frac {\\sin ( \\frac {3}{2}π-α)+2\\cos (α+ \\frac {π}{2})}{\\sin (α-π)-3\\cos (2π-α)}$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 15080,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α$ is an angle in the second quadrant and $\\cos (α+π)= \\frac {3}{13}$.\n(1) Find the value of $\\tan α$;\n(2) Find the value of $\\sin (α- \\frac {π}{2}) \\cdot \\sin (-α-π)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{12\\sqrt{10}}{169}"
} | {
"index": 15081,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α$ is an angle in the second quadrant and $\\sin α= \\frac {3}{5}$, find $\\sin 2α$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{24}{25}"
} | {
"index": 15082,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α$ is an angle in the second quadrant, let point $P(x, \\sqrt {5})$ be a point on the terminal side of $α$, and $\\cos α= \\frac { \\sqrt {2}}{4}x$. Find the value of $4\\cos (α+ \\frac {π}{2})-3\\tan α$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt {15}- \\sqrt {10}"
} | {
"index": 15083,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α$ is an angle in the third quadrant and $\\cos 2α=-\\frac{3}{5}$, find $\\tan (\\frac{π}{4}+2α)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{7}"
} | {
"index": 15084,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α$ is an angle in the third quadrant and $\\cos(α+π)=\\frac{4}{5}$, find the value of $\\tan 2α$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{24}{7}"
} | {
"index": 15085,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α$ is an angle in the third quadrant, $f(α)= \\frac {\\sin (π-α)\\cdot \\cos (2π-α)\\cdot \\tan (-α-π)}{\\tan (-α )\\cdot \\sin (-π -α)}$.\n(1) Simplify $f(α)$;\n(2) If $\\cos (α- \\frac {3}{2}π)= \\frac {1}{5}$, find the value of $f(α)$;\n(3) If $α=-1860^{\\circ}$,... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 15086,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α$ is an internal angle of a triangle, and $\\sin α + \\cos α = \\frac{1}{5}$. \n$(1)$ Find the value of $\\tan α$; \n$(2)$ Find the value of $\\frac{\\sin(\\frac{3π}{2}+α)\\sin(\\frac{π}{2}-α)\\tan^{3}(π-α)}{\\cos(\\frac{π}{2}+α)\\cos(\\frac{3π}{2}-α)}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{3}"
} | {
"index": 15087,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α$ is an obtuse angle, if $\\sin (α- \\frac{3π}{4})= \\frac{3}{5}$, find $\\cos (α+ \\frac{π}{4})=\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_.$"
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{4}{5}"
} | {
"index": 15088,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α+β= \\frac {π}{3}$ and $tanα+tanβ=2$, find the value of $cos(α-β)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {3}-1}{2}"
} | {
"index": 15089,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α, β ∈ (0, \\frac{π}{2})$, and $\\frac{\\sin β}{\\sin α} = \\cos(α + β)$, \n(1) If $α = \\frac{π}{6}$, then $\\tan β =$ _______; \n(2) The maximum value of $\\tan β$ is _______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{4}"
} | {
"index": 15090,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α,β$ satisfy $\\frac{\\sin α}{\\sin (α +2β)}=\\frac{2018}{2019}$, find the value of $\\frac{\\tan (α +β)}{\\tan β}$."
}
] | math | {
"style": "rule",
"ground_truth": "4037"
} | {
"index": 15091,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α,β∈(0,π), \\text{and } cosα= \\frac {3}{5}, cosβ=- \\frac {12}{13}$.\n(1) Find the value of $cos2α$,\n(2) Find the value of $sin(2α-β)$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{253}{325}"
} | {
"index": 15092,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α\\in\\mathbb{R}$ and $\\sin α + 2\\cos α = \\frac{\\sqrt{10}}{2}$, find the value of $\\tan α$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"index": 15093,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α∈( \\dfrac {π}{2},π)$, and $\\sin \\dfrac {α}{2}+\\cos \\dfrac {α}{2}= \\dfrac {2 \\sqrt {3}}{3}$.\n(1) Find the values of $\\sin α$ and $\\cos α$;\n(2) If $\\sin (α+β)=- \\dfrac {3}{5},β∈(0, \\dfrac {π}{2})$, find the value of $\\sin β$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {6 \\sqrt {2}+4}{15}"
} | {
"index": 15094,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α∈(0, \\dfrac {π}{2})$ and $β∈(0, \\dfrac {π}{2})$, with $cosα= \\dfrac {1}{7}$ and $cos(α+β)=- \\dfrac {11}{14}$, find the value of $sinβ$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{3}}{2}"
} | {
"index": 15095,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α∈(0, \\dfrac{π}{2})$ and $β∈(\\dfrac{π}{2},π)$, with $\\cos β=-\\dfrac{1}{3}$ and $\\sin (α+β)=\\dfrac{7}{9}$.\n\n(1) Find the value of $\\sin α$;\n\n(2) Find the value of $\\sin (2α+β)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{10\\sqrt{2}}{27}"
} | {
"index": 15096,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α∈(0,π)$, and $\\sin α + \\cos α = \\frac{\\sqrt{2}}{2}$, find the value of $\\sin α - \\cos α$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"index": 15097,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α∈[\\dfrac{π}{2}, \\dfrac{3π}{2}]$, $β∈[-\\dfrac{π}{2}, 0]$, and the equations $(α-\\dfrac{π}{2})^{3}-\\sin α-2=0$ and $8β^{3}+2\\cos^{2}β+1=0$ hold, find the value of $\\sin(\\dfrac{α}{2}+β)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{2}}{2}"
} | {
"index": 15098,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given that $α∈\\left( \\frac{π}{2},π\\right) $, and $\\sin \\left(π-α\\right)+\\cos \\left(2π+α\\right)= \\frac{ \\sqrt{2}}{3} $. Find the values of:\n\n$(1)\\sin {α} -\\cos {α} .$\n\n$(2)\\tan {α} $."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{9+4 \\sqrt{2}}{7}"
} | {
"index": 15099,
"solution": ""
} |
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