instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
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In the tetrahedron \(ABCD\), \(AB = 2\), \(AC = 3\), \(AD = 4\), \(BC = \sqrt{13}\), \(BD = 2\sqrt{5}\), and \(CD = 5\). Calculate the volume of the tetrahedron \(ABCD\). Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the complex number expression \(\frac{5i}{1-2i}\). Express your answer as a simplified complex number in the form a + bi. | -2 + i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Find the symmetry center of the function $y=3\tan\left(\frac{1}{2}x+\frac{\pi}{3}\right)$. Express your answer as an ordered pair (x, y). | \left(-\frac{2\pi}{3},0\right) | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
The function f(x) = a^x (a > 0 and a!= 1) is a decreasing function on R. Find the range of a. Express your answer as an interval in standard interval notation. | (0, 1) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ satisfies $z\left(1+i\right)=1-i$ (where $i$ is the imaginary unit), find the magnitude $|z|$ of the complex number. Express your answer as a numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given \( S = (x-1)^{4} + 4(x-1)^{3} + 6(x-1)^{2} + 4(x-1) + 1 \), simplify S to its most basic form. Express your answer as a simplified algebraic expression. | x^4 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
If \( x = -2 \) and \( y = -1 \), evaluate the expression \( (3x + 2y) - (3x - 2y) \). Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number $z$ satisfies the equation $(1+i)z=3+i$, find the modulus of the complex number $z$. Express your answer as a numerical value in the form $\sqrt{x}$, where $x$ is a whole number. | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the unit vectors $\overset{→}{a}$ and $\overset{→}{b}$ satisfy $\left|\overset{→}{a}-\overset{→}{b}
ight|=\sqrt{3}\left|\overset{→}{a}+\overset{→}{b}
ight|$, find the angle between $\overset{→}{a}$ and $\overset{→}{b}$ in radians. | \dfrac{2\pi}{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sin\left(\alpha + \frac{\pi}{2}\right) = \frac{1}{3}$, and $\alpha\in \left(0, \frac{\pi}{2}\right)$, find the value of $\tan\alpha$. Express your answer as a simplified radical. | 2\sqrt{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Find the monotonically decreasing interval of the function $f(x)=3+x\ln x$. Express your answer as an interval in standard notation. | (0, \frac {1}{e}) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of abc^2 in the expansion of (a + 2b - 3c)^4 using the multinomial theorem. Express your answer as a single numerical value. | 216 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
If $a$ is a real number, and $(1+ai)(a-i)=2$, then find the value of $a$. Express your answer as a single real number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of terms in the expansion of the binomial $(a + b)^{2n}$. Express your answer as a single integer. | 2n + 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Find the symmetric axis of the quadratic function y = x^2 + 8x + 9. Express your answer as an equation of a line in the form x = a. | x = -4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Consider a function \( y = f(x) \) defined on \(\mathbf{R}\) with the following properties:
1. For any \( x \in \mathbf{R} \), \( f\left(x^{3}\right) = f^{3}(x) \).
2. For any \( x_{1}, x_{2} \in \mathbf{R} \) with \( x_{1} \neq x_{2} \), \( f\left(x_{1}\right) \neq f\left(x_{2}\right) \).
Find the value of \( f(0) + ... | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Let $A=\{x|y=\log _{2}(1-x)\}$, $B=\{x|x\leqslant -1\}$. Find the intersection of set $A$ and the complement of set $B$ in the real numbers, denoted as $A\cap (\complement _{R}B)$. Express your answer as an interval. | (-1, 1) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given a circle with a radius of 1 cm, find the arc length corresponding to a central angle of 150°. Express your answer in centimeters (cm) as a fraction or decimal with π. | \frac{5\pi}{6} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Let $a, b, c, d$ be real numbers. Suppose that all the roots of $z^4+az^3+bz^2+cz+d=0$ are complex numbers lying on a circle in the complex plane centered at $0+0i$ and having radius $1$. Find the sum of the reciprocals of the roots. Express your answer in terms of the coefficients of the polynomial. | -a | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given two vectors $\vec{a}=(1,2)$ and $\vec{b}=(x, 1)$, find the value of $x$ such that $(\vec{a}+2\vec{b})$ is parallel to $(2\vec{a}-2\vec{b})$. Express your answer as a single value. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures 8 inches high and 10 inches wide. Calculate the area of the border, in square inches. | 88 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
6 students are lined up in a row, and among them, students A and B must stand together. How many different arrangements are there? Express your answer as a whole number. | 240 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, each of \( \triangle QPT, \triangle QTS \) and \( \triangle QSR \) is an isosceles, right-angled triangle, with \( \angle QPT = \angle QTS = \angle QSR = 90^{\circ} \). The combined area of the three triangles is 56. If \( QP = PT = k \), find the value of \( k \). Express your answer as a single numeri... | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Distribute 2 identical novels and 2 identical picture albums among 3 students, with each student receiving at least 1 book. Find the total number of different distribution methods. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the inverse proportion function y = (k-1)/x, and two points A(x1, y1) and B(x2, y2) on the graph, where 0 < x1 < x2 and y1 < y2, what is the range of values for k that satisfies these conditions? Express your answer as an inequality in terms of k. | k < 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A high school has 1200 students in grade 10, 900 students in grade 11, and 1500 students in grade 12. A sample of 720 students is to be selected for a study using stratified sampling based on grade levels. Calculate the number of students that should be selected from grade 12. Express your answer as a whole number. | 300 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=\begin{cases}2^{x},x\leqslant 0 \\\sin x,x > 0\end{cases}$, find the value of $f(f( \frac{7π}{6}))$. Express your answer as a single value, simplified if possible. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x^8$ in the expansion of $(x^{2}-1)^{2}(x^{3}+ \frac {1}{x})^{4}$. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the set M = {1, 2,..., 10}, how many subsets A of M have a sum of elements equal to 8? Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Four people, including A and B, participate in a 4x100m relay race. Given that A does not run the first leg, find the probability that B does not run the second leg. Express your answer as a simplified fraction. | \dfrac{7}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of n in the curve y = x^n where the derivative at x = 2 is 12. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given $\sin \left(\alpha +\frac{\pi }{6}\right)=\frac{1}{3}$, calculate the value of $\cos \left(2\alpha -\frac{2\pi }{3}\right)$. Express your answer as a fraction or decimal. | -\frac{7}{9} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A can do a piece of work in 9 days. B is 50% more efficient than A. Calculate the number of days it takes B to do the same piece of work. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ with the general term formula $a_n = 2n - 5$, calculate the sum of the absolute values of the first 10 terms, $|a_1| + |a_2| + \ldots + |a_{10}|$. Express your answer as a single integer. | 68 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
In triangle \( \triangle ABC \), the heights corresponding to sides \(a, b,\) and \(c\) are \( h_a = 6, h_b = 4, \) and \( h_c = 3 \), respectively. Calculate the ratio of the sides \(a: b: c \). Express your answer as a ratio of integers. | 2 : 3 : 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
The mean (average) of 6, 9, and 18 is equal to the mean (average) of 12 and y. What is the value of y that makes the two means equal? Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola y = 2(x-3)^2 + 2, find its equation after shifting it 3 units to the left and then 2 units down. Express your answer as a simplified algebraic equation. | y = 2x^2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the sum of the arithmetic sequence 2+5+8+…+29, where the common difference is 3. Express your answer as a single integer. | 155 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Two men and two women are lined up in a row. Find the total number of ways they can be arranged alternately by gender. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
To construct a conical funnel with a slant height of 30 cm, find the height that maximizes its volume. Express your answer in centimeters (cm), including the square root symbol (√) if necessary. | 10\sqrt{3}\text{ cm} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Person A took away 70% of the whole cookie, and then Person B took one-third of what was left. What percentage of the whole cookie remains? Express your answer as a percentage, including the percentage sign (%). | 20% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given two points in space $P(-1, 2, -3)$ and $Q(3, -2, -1)$, calculate the distance between points $P$ and $Q$ using the formula:
$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$
Express your answer as a single number, possibly in simplest radical form if not a whole number. | 6 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $\sqrt[3]{\sqrt{9}}$. Express your answer as a simplified radical expression. | \sqrt[3]{3} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \(A C = C B = 10 \, \text{m}\), where \(A C\) and \(C B\) are each the diameter of the small equal semi-circles. The diameter of the larger semi-circle is \(A B\). Calculate the difference in lengths between the semi-circular arc from \(A\) to \(B\) and the combined semi-circular arcs from \(A\) to \(C\... | 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the triangle shown is isosceles with \( AB = AC \). The measure of angle \( BAC \) is \( 40^\circ \). Points \( B \), \( C \), and \( D \) lie on a straight line, and the measure of angle \( DBC \) is \( 2x \). Find the value of \( x \). Express your answer as a single integer. | 55 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {an} where all terms are positive, and the relation 3a2 + 2a3 = a4, find the value of S3/a2. Express your answer as a fraction or decimal. | 13/3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given that {a_n} is a geometric sequence, a_4 + a_7 = 2, and a_2 * a_9 = -8, find the value of a_1 + a_10. Express your answer as a single integer. | -7 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow {a}=(1, -1)$, $\overrightarrow {b}=(1, 2)$, vector $\overrightarrow {c}$ satisfies $(\overrightarrow {c}+\overrightarrow {b}) \perp \overrightarrow {a}$ and $(\overrightarrow {c}-\overrightarrow {a}) \parallel \overrightarrow {b}$. Determine $\overrightarrow {c}$. Express your answer as a... | (2, 1) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given that point P(m,n) is located in the second quadrant, determine the quadrant in which point P'(-m^2, -n) lies. Express your answer as the name of the quadrant (e.g., first, second, third, fourth). | third | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola $y^{2}=x$ has its focus as one of the foci of the ellipse $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{3}=1$, find the eccentricity of the ellipse. Express your answer as a fraction. | \frac{1}{7} | big_math | [
"Mathematics -> Geometry -> Differential Geometry -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
The Rhind Papyrus is one of the oldest mathematical texts in the world. It contains the following problem: Divide 120 loaves of bread into 5 portions such that the number of loaves in each portion forms an arithmetic sequence, and the sum of the three larger portions is exactly 7 times the sum of the two smaller portio... | 2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Twelve girls met in a cafe. On average, they ate \(1 \frac{1}{2}\) cupcakes each, although no cupcakes were actually divided. None of them ate more than two cupcakes and two of them ate no cupcakes at all. Determine the number of girls who ate two cupcakes. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
The population of a town increased by 1200 people, then this new population decreased by 11%. Now the population of the town is 32 people less than it was before the 1200 increase. What was the original population of the town? Express your answer as a single integer. | 10000 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation z(2-i) = 11+7i, solve for z. Express your answer in the form a+bi. | 3+5i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(m,1)$ and $\overrightarrow{b}=(2,n)$, if $|\overrightarrow{a}|=2$ and $\overrightarrow{a}\perp \overrightarrow{b}$, calculate the product $mn$. Express your answer as a single numerical value. | -6 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the range of values for $x$ that makes the expression $\frac{1}{\sqrt{x-2}}$ meaningful within the real number range. Express your answer as an inequality in the format $x$ [relation] [value]. | $x > 2$ | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the expression \(1+2+3-4+5+6+7-8+9+10+11-12\). Provide your answer as a single integer. | 30 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given two vectors $\overrightarrow{a}=(2,m,3)$ and $\overrightarrow{b}=(4,-1,n)$, and $\overrightarrow{a} \parallel \overrightarrow{b}$, find the value of $mn$. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \( \triangle PQR \) is an equilateral triangle. If \( PQ = 4x \) and \( PR = x + 12 \), what is the value of \( x \)? Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\cos (\frac{\pi}{2}+\phi)=\frac{2}{3}$ and $|\phi| < \frac{\pi}{2}$, calculate the value of $\tan \phi$. Express your answer as a simplified fraction, including any necessary square roots. | -\frac{2\sqrt{5}}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
The graph of the function y = a^(x-1) + 1 (0 < a ≠ 1) must pass through a specific point. Find this point. Express your answer as an ordered pair (x, y). | (1,2) | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
A certain unit encourages its employees to save water by implementing the following policy: Each employee is charged a yuan per cubic meter for the first 10 cubic meters of water used each month. For any usage above 10 cubic meters, the charge is doubled for the excess amount. If an employee paid 16a yuan for water in ... | 13 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
In the figure, ∆ABC is a right-angled triangle with ∠ACB = 90°, AC = AE, and BC = BF. Find the measure of ∠ECF in degrees. | 45 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Let \( F = \lg \frac{1+x}{1-x} \). If each \( x \) in the expression is replaced by \( \frac{3x + x^3}{1 + 3x^2} \), simplify the new function \( G \) and express it in terms of \( F \). Use the format \( G = [expression] \). | G = 3F | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Among line segments, isosceles trapezoids, parallelograms, rectangles, rhombuses, squares, and equilateral triangles, how many figures are both centrally symmetric and axially symmetric? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
If the sequence {a_n} is defined by a_1 = 2 and a_{n+1} = a_n + 2n for n >= 1, calculate the value of a_{100}. Express your answer as a single integer. | 9902 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression \( 5 + 4 - 3 + 2 - 1 \) and provide the exact result as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Peyton puts 30 liters of oil and 15 liters of vinegar into a large empty can. He then adds 15 liters of oil to create a new mixture. What percentage of the new mixture is oil? Express your answer as a percentage. | 75 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given that line $l$ has a slope angle of $45°$ and passes through the point $(0,1)$, find the equation of line $l$ in the standard form (Ax + By + C = 0). | x - y + 1 = 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $a=(1,-2)$ and $b=(m,4)$, where $a$ is parallel to $b$, calculate the result of $2a-b$. Express your answer as a vector in the form $(x, y)$. | (4,-8) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
In a certain month in Shangri-La, Yunnan, the average highest temperature is $9^{\circ}\mathrm{C}$, and the average lowest temperature is $-5^{\circ}\mathrm{C}$. Calculate the difference between the average highest temperature and the average lowest temperature. Express your answer in degrees Celsius ($^{\circ}\mathrm{... | 14^{\circ}\mathrm{C} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given a finite sequence \(P = \left(p_1, p_2, \cdots, p_n\right)\), the Cesaro sum is defined as \(\frac{1}{n}(S_1 + S_2 + \cdots + S_n)\), where \(S_k = p_1 + p_2 + \cdots + p_k\) for \(1 \leq k \leq n\). If a sequence \(\left(p_1, p_2, \cdots, p_{99}\right)\) of 99 terms has a Cesaro sum of 1000, then find the Cesaro... | 991 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the numbers 1.000, 0.101, 0.011, and 0.001. Express your answer as a decimal number with all significant digits. | 1.113 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Decimals"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given a function f(x) defined on R with its derivative denoted as f''(x), and f(x) + f''(x) > 1, let a = f(2) - 1 and b = e^(f(3) - 1). Analyze the relationship between a and b. Express your answer as a relationship (e.g., <, >, =) between a and b. | a < b | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ with the general term formula $a_n = 2n - 49$ ($n \in \mathbb{N}$), find the value of $n$ when the sum of the first $n$ terms $S_n$ reaches its minimum. Express your answer as a whole number. | 24 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
PQRST is a regular pentagon. The point U lies on ST such that ∠QPU is a right angle. Find the ratio of the interior angles in triangle PUT. Express your answer as a ratio of integers. | 1:3:6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the parabola $x^{2}=16y$, find the coordinates of its focus. Express your answer as a coordinate pair. | (0,4) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Among four non-coplanar points, how many planes can be formed by any three of these points? Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of real roots of the quadratic function f(x) = x^2 + bx - 3, where b is a real number. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given $f(x)=2x+3$ and $g(x)=4x-5$, determine the expression for $h(x)$ such that $f(h(x))=g(x)$. Express your answer as a function of x. | h(x)=2x-4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given $(1-2x)^{2016} = a_0 + a_1(x-2) + a_2(x-2)^2 + \ldots + a_{2015}(x-2)^{2015} + a_{2016}(x-2)^{2016}$ ($x \in \mathbb{R}$), find the value of $a_1 - 2a_2 + 3a_3 - 4a_4 + \ldots + 2015a_{2015} - 2016a_{2016}$ after differentiating the expansion with respect to $x$ and setting $x=1$. Express your answer as a single ... | 4032 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given two sets M = {z | z = t/(1+t) + i*(1+t)/t, t ∈ R, t ≠ -1, t ≠ 0} and N = {z | z = sqrt(2)*[cos(arcsin t) + i*cos(arccos t)], t ∈ R, |t| ≤ 1}, determine the number of elements in the intersection M ∩ N. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(2,3)$ and $\overrightarrow{b}=(1,2)$, if $\overrightarrow{a}+2\overrightarrow{b}$ is parallel to $m\overrightarrow{a}+\overrightarrow{b}$, find the value of $m$. Express your answer as a single number. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola equation $y^{2}=-8x$, find the coordinates of the focus. Express your answer as a coordinate pair (x, y). | (-2,0) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation \(x^2 + px + 1 = 0\) with \(p > 0\), find the value of \(p\) when the difference between the roots is 1. Express your answer as a single value, including the square root symbol if necessary. | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence \\(\{a_n\}\) with the general term \\(a_n = (-1)^n(3n-2)\\), find the sum of the first 10 terms of the sequence. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the plane passing through point A(1,2,3) has a normal vector (1,-1,1), calculate the distance from point P(-2,3,1) to this plane. Express your answer as a numerical value or a radical expression if necessary. | 2\sqrt{3} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\tan \theta = 2$, simplify the expression $\frac {2\cos\theta}{\sin( \frac {\pi}{2}+\theta )+\sin(\pi +\theta )}$. Express your answer as a single numerical value. | -2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
The heights of two trees are in the ratio 3:4. The top of one tree is 16 feet higher than the top of the other tree. Find the height of the taller tree. Express your answer in feet as a whole number. | 64 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the units digit of 3^2011 by observing and summarizing the patterns of the units digits in the powers of 3. Express your answer as a single digit. | 7 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Let the function $f(x)=xe^x$. Find the value of x that corresponds to a local extremum of $f(x)$. Clearly state whether this point is a local minimum or maximum. | -1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$, where $a_1=1$, and $a_na_{n-1}=a_{n-1}+(-1)^n$ $(n\geqslant 2,n\in\mathbb{N}^*)$, calculate the ratio $\dfrac{a_3}{a_4}$. Express your answer as a simplified fraction. | \dfrac{1}{6} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ satisfying $z \cdot \overline{z} + z + \overline{z} = 17$, calculate the minimum value of $|z + 2 - i|$. Express your answer as a numerical value. | 2 \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=\frac{1}{3}{{x}^{3}}+a{{x}^{2}}+{{b}^{2}}x+1$, where $a$ is a number chosen from $1$, $2$, $3$, and $b$ is a number chosen from $0$, $1$, $2$, calculate the probability that this function has two extreme points. Express your answer as a simplified fraction. | \frac{2}{3} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\sqrt{(-3)^2}$. Provide your answer as a simplified numerical value. | 3 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
A square ABCD with side length $a$ is folded along the diagonal AC, making triangle ADC fold upwards. If $\angle DAB = 60^\circ$, then calculate the dihedral angle formed by D—AC—B. Express your answer in degrees. | 90^\circ | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z = \frac{2\mathrm{i}}{1-\mathrm{i}}$, find the value of $\overline{z} \cdot z$, where $\overline{z}$ is the conjugate of $z$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 1/3*x^3 - x for x >= 0, and a point (m, n) on its graph, find the minimum value of n - m + 2*sqrt(2). Express your answer as a single number in the form \boxed{[answer]}. | \frac{2\sqrt{2}}{3} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given \( k \in \mathbf{Z}_{+} \). Express the fraction \(\frac{7}{51}\) in base \(k\) as \(\overline{0.23}_{k} = (0.2323)_{k}\). Find the value of \(k\) that satisfies this condition. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x)$ and $g(x)$ are both odd functions, and $F(x) = f(x) + g(x) + 2$ has a maximum value of 8 on $(0, +\infty)$, determine the minimum value of $F(x)$ on $(-\infty, 0)$. Express your answer as a single numerical value. | -4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
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