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
value |
|---|---|---|---|---|---|
If the square of the complex number $(2+ai)^2$ (where $a \in \mathbb{R}$) is a real number (where $i$ is the imaginary unit), then find the value of the real number $a$. Express your answer as a single real number. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle \(O: x^2 + y^2 = 9\), and a line \(l\) intersects the circle \(O\) at points \(M\) and \(N\), with \(|MN| = 4\), calculate the dot product of vectors \(\overrightarrow{MN}\) and \(\overrightarrow{MO}\). Express your answer as a single numerical value. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Peter's family ordered a 12-slice pizza for dinner. Peter ate one slice and shared another slice equally with his brother Paul. What fraction of the pizza did Peter eat? Express your answer as a simplified fraction. | \frac{1}{8} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
A set of five different positive integers has a mean (average) of 20 and a median of 18. What is the greatest possible integer in the set? Express your answer as a single integer. | 60 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
A chipmunk and a squirrel are hiding acorns in holes. The chipmunk hides 3 acorns in each hole it digs, while the squirrel hides 4 acorns in each hole it digs. If they both hide the same total number of acorns, and the squirrel needs 4 fewer holes than the chipmunk, calculate the total number of acorns hidden by the ch... | 48 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \begin{cases} \log_{2}(5-x), & x\leq 1 \\ f(x-1)+1, & x>1 \end{cases}$, calculate the value of $f(2016)$. Express your answer as a single integer. | 2017 | big_math | [
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Algebra -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given α ∈ (π, 3π/2), and tan α = 4/3, find the value of sin α. Express your answer as a fraction, including the negative sign if applicable. | -\frac{4}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
A figure consists of a grid arrangement of squares, with 5 groups of squares, and each group contains 5 squares. How many squares are in the figure? Express your answer as a whole number. | 25 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given a cone with a base radius of 1 and a slant height of 2√7, find the volume of the cone. Express your answer in terms of π. | \sqrt{3}\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression (-1)^3 + (-1)^2 + (-1). Express your answer as an integer in the form \boxed{ }. | -1 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Observe the following equations: $m+n=1$, $m^{2}+n^{2}=3$, $m^{3}+n^{3}=4$, $m^{4}+n^{4}=7$, $m^{5}+n^{5}=11$. A pattern is observed from the third term onwards, where the number on the right side is the sum of the right side numbers of the previous two terms. Calculate $m^{9}+n^{9}$ based on this pattern. Express your... | 76 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $A=\{x|0 \lt x \lt 2\}$ and $U=\{x|-2 \lt x \lt 2\}$, find the complement of $A$ in $U$, denoted as $\complement _{U}A$, and express your answer in interval notation. | (-2,0] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the constant term in the expansion of $(x^{2}+1)(\\frac{1}{\\sqrt{x}}-2)^{5}$. Express your answer as a single integer. | -42 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a^{2}$, $b^{2}$, $c^{2}$ form an arithmetic sequence, and the area of $\triangle ABC$ is $\frac{{{b^2}}}{3}$, then find the value of $\tan B$. Express your answer as a numerical value. | \frac{4}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given \(x > 0\), \(y > 0\), and \(x+2y+2xy=8\), find the minimum value of \(x+2y\). Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^2(a * 2^x - 2^-x) is an odd function, find the value of a that satisfies this condition. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $U={0,1,2,4}$, and the complement of set $A$ in $U$ is ${1,2}$, determine the set $A$. Express your answer as a set of elements enclosed in curly brackets, with elements separated by commas. | {0,4} | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A quadrilateral is bounded by the lines with equations \( x=0 \), \( x=4 \), \( y=x-2 \), and \( y=x+3 \). Find the area of this quadrilateral. Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
A three-digit positive integer "$a_1a_2a_3$" is called a "convex number" if $a_1 < a_2$ and $a_2 > a_3$. Calculate the total number of convex numbers. Express your answer as a single integer. | 240 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the tens digit of $2015^{2016} - 2017$. Express your answer as a single digit. | 0 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Let the function $f$ be defined on the set of ordered pairs of positive integers, and satisfy:
1. $f(x, x) = x$;
2. $f(x, y) = f(y, x)$;
3. $(x+y)f(x, y) = yf(x, x+y)$;
Calculate the value of $f(12, 16) + f(16, 12)$. Express your answer as a single integer. | 96 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic function y = ax^2 + bx + c, where coefficients a, b, and c are restricted to the set {0, 1, 2}, and a cannot be 0, calculate the total number of different quadratic functions that can be formed under these conditions. | 18 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given a point $P(-4,3)$ on the terminal side of angle $α$, find the value of $\cos α$. Express your answer as a fraction. | -\frac{4}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Scientist Konstantin Tsiolkovsky proposed in 1903 that the maximum velocity v of a single-stage rocket in an ideal situation without considering air resistance and Earth's gravity satisfies the formula: v = v0 * ln((m1 + m2) / m1), where m1 and m2 are the structural mass of the rocket and the mass of the propellant, re... | 3.6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = (a+1)x^2 + (a-2)x + a^2 - a - 2, find the value of a that makes this function even. Provide your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Find the conjugate of the complex number $\dfrac{-5}{i-2}$. Express your answer as a complex number in the form a + bi. | 2-i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, $AB = AC = 3.6$, point $D$ lies on $AB$ with $AD = 1.2$, and point $E$ lies on the extension of $AC$. The area of $\triangle AED$ is equal to the area of $\triangle ABC$. Find the length of $AE$. Express your answer as a single number with one decimal place. | 10.8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given a three-digit number $m$ and a two-digit number $n$, form a five-digit number by placing $m$ in front of $n$. Express the resulting five-digit number as an algebraic expression in terms of $m$ and $n$. Provide your answer in the form $\boxed{expression}$. | 100m+n | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given sin(θ) = 1/3 and θ ∈ (π/2, π), find tan(θ). Express your answer as a simplified fraction or exact decimal, including the negative sign if applicable. | -\frac{\sqrt{2}}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the integer part of 5 - sqrt(3) is x and the decimal part is y, calculate the value of 2x^3 - (y^3 + 1/y^3). Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\sqrt {a-1}+b^{2}-4b+4=0$, find the value of $ab$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $\frac{\cos 85^{\circ} + \sin 25^{\circ} \cos 30^{\circ}}{\cos 25^{\circ}}$. Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of integer solutions x that satisfy the equation (x^2 - x - 1)^(x+2) = 1. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
For a geometric sequence $\{a_n\}$, the sum of its first $n$ terms is $S_n$. Given $S_{10} = 10$ and $S_{20} = 30$, find the value of $S_{30}$. Express your answer as a single number. | 70 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $A=\{y\in \mathbb{N}|y=-x^{2}+4x,x\in \mathbb{Z}\}$ and $B=\{x|\ln x > 1\}$, find the intersection of $A$ and the complement of $B$ in $\mathbb{R}$, denoted as $A\cap (\complement_{\mathbb{R}}B)$. Express your answer as a set of numbers using standard set notation. | \{0\} | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given two lines $l_{1}$: $(3+m)x-4y=5-3m$ and $l_{2}$: $2x-y=8$ are parallel, find the value of the real number $m$. Express your answer as a single real number. | 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $C$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ passing through the point $(\sqrt{2}, 2\sqrt{2})$, and the line $l$ passing through the point $(0, -2)$ is parallel to one of the asymptotes of the hyperbola $C$. The distance between the line $l$ and this asymptote is $\frac{2}{3}$... | 2 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given sets A and B defined by the inequalities:
A = {x | x^2 - 3x - 4 <= 0, x in Z}
B = {x | 2x^2 - x - 6 > 0, x in Z}
Find the number of proper subsets of A cap B. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given that circle M passes through points A(-1,-4) and B(6,3), and the center of the circle lies on the line x-y-4=0, find the standard equation of circle C in the format (x-a)^2 + (y-b)^2 = r^2. | (x-3)^2 + (y+1)^2 = 25 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
If the decimal representation of $2^n$ starts with 7, what is the first digit in the decimal representation of $5^n$? Express your answer as a single digit. | 1 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
The figure shows the map of an (imaginary) country consisting of five states. The goal is to color this map with green, blue, and yellow in such a way that no two neighboring states share the same color. Calculate the exact number of different ways the map can be painted. | 6 | big_math | [
"Mathematics -> Discrete Mathematics -> Graph Theory -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In a 3x3 grid of squares, where each square can be colored black or white, what is the largest number of squares that can be colored black such that the design created has rotational symmetry of order 2, but no lines of symmetry? Provide your answer as a single integer. | 5 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, the endpoints of line segment AB are A(-4, -1) and B(1, 1). After translating line segment AB, we obtain line segment A'B'. If the coordinates of point A' are (-2, 2), then find the coordinates of point B'. Express your answer as an ordered pair (x, y). | (3, 4) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given that \\(\\frac{x-y}{z-y}=-10\\), calculate the value of \\(\\frac{x-z}{y-z}\\). Express your answer as a single number. | 11 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality m ≤ 8x - 1 ≤ n, where the length of the interval of solutions is 25, find the value of n - m. Express your answer as a single integer. | 200 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Let A, B, and C be three distinct points on line l, and let O be a point not on line l. If $p\vec{OA} + q\vec{OB} + r\vec{OC} = \vec{0}$ ($p, q, r \in \mathbb{R}$), find the value of $p+q+r$. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given $$\frac {x}{1+i}=1-yi$$, where $x$, $y$ are real numbers, and $i$ is the imaginary unit, find the conjugate of $x+yi$. Express your answer as a complex number in the form $a + bi$, where $a$ and $b$ are real numbers. | 2-i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Represent 505,000 in scientific notation as $a \\times 10^b$. Provide your answer in the format $\boxed{a \\times 10^b}$. | 5.05 \\times 10^5 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the differential equation f'(x) = (3/x)f(x) for all x in R+, and f(2^2016) is not equal to 0, calculate the ratio f(2^2017)/f(2^2016). Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Calculus -> Differential Equations -> Ordinary Differential Equations (ODEs)"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given a normal distribution N(0, $\frac{4}{9}$), calculate the probability of data falling outside the interval (-2, 2). Express your answer as a decimal value rounded to 4 decimal places, using the boxed notation. | 0.0026 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Let $z=\frac{{3+i}}{{1+i}}$. Calculate the coordinates of the point corresponding to the complex number $z$ in the complex plane. Express your answer as an ordered pair $(x, y)$. | (2, -1) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
The digit "日" can be modified by removing sticks to create different numerical representations. By analyzing the stick counts for each digit from 0 to 9, determine the number of distinct representations that can be created. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A set of data $2$, $1$, $4$, $x$, $6$ has an average of $4$. Calculate the value of $x$ that makes this average true. Express your answer as a single number. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
The surface area of the Earth is approximately 780,000,000 km^2. Represent this value using scientific notation in the form a×10^n, where 1≤|a|＜10, and include the unit (km^2). | 7.8×10^8 km^2 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 58000000000 in scientific notation. The answer should be in the form a × 10^b, where a is a number between 1 and 10, and b is an integer. | 5.8 \times 10^{10} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Randomly select two numbers, denoted as $a$ and $b$, within the interval $[-\pi, \pi]$. Calculate the probability that the function $f(x) = x^2 + 2ax - b^2 + \pi$ has a root. Express your answer as a fraction. | \frac {3}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the expression |x-1| + |x-2| + |x-3|. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Suppose $S = \{1, 2, 3, x\}$ is a set with four distinct real numbers for which the difference between the largest and smallest values of $S$ is equal to the sum of elements of $S.$ Calculate the value of $x$. Express your answer as a fraction or decimal, including a negative sign if applicable. | -\frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (1, k)$ and $\overrightarrow{b} = (2, 2)$, and that $\overrightarrow{a} + \overrightarrow{b}$ is collinear with $\overrightarrow{a}$, find the value of $k$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality x^2 - 5x + 6 < 0, find the range of values for the function P = x^2 + 5x + 6. Express your answer as an interval. | (20, 30) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given that point P is in the plane of triangle ABC, and the vector equation $\overrightarrow{PB} + \overrightarrow{PC} + 4\overrightarrow{PA} = \overrightarrow{0}$ holds, find the probability that a randomly chosen point inside triangle ABC falls inside triangle PBC. Express your answer as a fraction or decimal value b... | \dfrac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Find the absolute value of -2035. Express your answer as a positive integer. | 2035 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given an increasing geometric sequence ${a_n}$, where $a_1+a_4=9$ and $a_2a_3=8$, find the common ratio $q$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
A line $l$ passing through point P(1,1) divides the circular region {(x,y)|x^2+y^2≤4} into two parts with areas S1 and S2. Find the equation of line $l$ in the form ax + by + c = 0 when $|S1-S2|$ is at its maximum. | x+y-2=0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x+1) = x^2$, find the value of $f(2)$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given an ellipse x^2/a^2 + y^2/b^2 = 1 (a > b > 0) and a hyperbola x^2/m^2 - y^2/n^2 = 1 (m > 0, n > 0) sharing the same foci (-c, 0) and (c, 0), where c is the geometric mean of a and m, and n^2 is the arithmetic mean of 2m^2 and c^2, find the eccentricity of the ellipse. Express your answer as a numerical value. | \dfrac{1}{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Let $f(x) = ax^3 + bx - 5$, and $f(-7) = 7$. Calculate the value of $f(7)$. Express your answer as a single integer in the format $\boxed{[answer]}$. | -17 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given the matrix pattern that involves multiplying the diagonals and summing the results, as illustrated by the example:
$$
\begin{array}{c|c}
3 & 4 \\
\hline
5 & 6 \\
\end{array}
\rightarrow 3 \times 6 + 4 \times 5 = 38
$$
apply this pattern to find the missing number x in the following matrix, which is evaluated as 1... | 5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
In a frequency distribution histogram of a sample, there are eight small rectangles. The area of the last rectangle is equal to a quarter (1/4) of the sum of the areas of the other seven rectangles. Given that the sample size is 200, calculate the frequency of the eighth group. Express your answer as a whole number. | 40 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $\dfrac {x^{2}}{9}- \dfrac {y^{2}}{m}=1$ has asymptotes with the equation $y=± \dfrac { \sqrt {5}}{3}x$, find the distance from the focus $F$ of the hyperbola to the asymptote $l$. Express your answer as a single number. | \sqrt{5} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the velocity of a freely falling object as V = gt, calculate the distance the object travels from t = 0 to t = t_0 using a definite integral. Express your answer as a mathematical expression involving all necessary variables and constants. | \frac{1}{2}gt_0^2 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
In the complex plane, the point corresponding to the complex number $z$ is $z(-1, 2)$. Find the conjugate of the complex number $z$, denoted as $\overline{z}$. Express your answer in the form $a + bi$, where $a$ and $b$ are real numbers. | -1-2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
The chord length of the line $ax+y-5=0$ intersecting the circle $C: x^2+y^2-4x-2y+1=0$ is 4. Calculate the value of a. Express your answer as an integer. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
The domain of the function $f(x)$ is $[0,8]$. Find the domain of the function $\frac{f(2x)}{x-4}$. Express your answer in interval notation. | [0,4) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Express $214.7$ billion in scientific notation. Provide your answer in the form $\boxed{a \times 10^b}$, where $a$ is the coefficient and $b$ is the exponent. | 2.147 \times 10^{11} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
A company has 450 employees, consisting of 45 senior employees, 135 mid-level employees, and 270 junior employees. A sample of 30 people is to be drawn using stratified sampling. Calculate the number of people to be sampled from each job title, assuming the sampling ratio is the same for all job titles. Express your an... | 3, 9, 18 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x-1)=x^{2}+6x$, determine the expression for $f(x)$. Express your answer as a function of $x$. | x^{2}+8x+7 | big_math | [
"Mathematics -> Algebra -> Functions -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Find the constant term in the binomial expansion of ${\\left(\\frac{2}{x}-x\\right)}^{6}$. Express your answer as a single integer. | -160 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
In a school with 1000 second-year high school students, 200 of them are arts students. A stratified sampling method by subject (arts and sciences) is employed to draw a sample of 100 students. Find the number of arts students in the sample. Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Let the sum of the coefficients of the expansion of the binomial $$(3 \sqrt[3]{x} + \frac{1}{x})^{n}$$ be $P$, and the sum of all the binomial coefficients be $S$. If $P + S = 272$, then find the value of $n$. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the circle and the square have the same center O and equal areas. The circle has a radius of 1 and intersects one side of the square at P and Q. Find the length of PQ. Express your answer as a mathematical expression, including the square root symbol if necessary. | \sqrt{4 - \pi} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(m,-1)$, and knowing that $\overrightarrow{a}⊥\overrightarrow{b}$, calculate the magnitude $|\overrightarrow{b}|$. Express your answer as a single value, including any necessary square roots. | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola $C: x^{2}=2py$ ($p > 0$) with focus $F$ and directrix $l$, and point $P(x_{0}, 1)$ ($x_{0} > 0$) lies on the parabola $C$. A perpendicular line is drawn from point $P$ to line $l$ with the foot of the perpendicular being $Q$. If $|PO|=|PQ|$ (where $O$ is the origin), calculate the value of $x_{0}$. E... | 2\sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $m+2n-1=0$, find the fixed point (x, y) that the line $mx+3y+n=0$ passes through. Express your answer as an ordered pair (x, y) with exact fractions. | \left( \frac {1}{2},- \frac {1}{6}\right) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
The sum of the first 6 terms of an arithmetic sequence \\(\\{a\_{n}\\}\\) is 30, and the sum of the first 12 terms is 100. Find the sum of the first 18 terms. Express your answer as a single numerical value. | 210 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, O is the center of each circle. The circumferences of the circles are 24π and 14π. B is a point on the outer circle and OB intersects the inner circle at A. Find the length of segment AB. | 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the derivative of the function y = (2x+1)^2 at x = 1. | 12 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
For a certain type of corn seed, the probability of each seed germinating is 90%. If 5 seeds are sown, find the probability that exactly two seeds do not germinate. Express your answer as a decimal approximation rounded to two decimal places. | 0.07 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
A journalist wants to take a photo of 3 volunteers and the 2 elderly people they are helping, arranging them in a row. The requirement is that the 2 elderly people must stand next to each other but not at the ends of the row. Find the total number of different arrangements. Express your answer as a single numerical val... | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\log_{5}45=a$, find the value of $\log_{5}3$. Express your answer in terms of $a$. | \dfrac {a-1}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
If $\overrightarrow{a}=(\frac{3}{2}, \sin\alpha)$ and $\overrightarrow{b}=(\cos\alpha, \frac{1}{3})$, and $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the acute angle $\alpha$ in degrees. | 45 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
The length of a circular arc is equal to the side length of the inscribed square of its circle. Find the radian measure of its central angle. Express your answer in radians. | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the surface area of a cone is $12\pi \text{cm}^2$, and its lateral surface unfolds into a semicircle, find the radius of the base of the cone. Express your answer in centimeters (cm). | 2 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given $\lim_{Δx→0}\frac{{f(4+Δx)-f(4-Δx)}}{{Δx}}=-10$, find the derivative f'(4). Express your answer as a single number. | -5 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = e^x*cos(x), find the value of f'(π/2). Express your answer as a single mathematical expression. | = -e^(π/2) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Let $M = \{(x, y) \,|\, |\tan(\pi y)| + \sin^2(\pi x) = 0\}$, and $N = \{(x, y) \,|\, x^2 + y^2 < 1\}$. Calculate the number of elements in $M \cap N$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n} with the general term formula a_n = \begin{cases}-n,\;n\leqslant 4 \\\sqrt {n^{2}-4n}-n,\;n > 4\end{cases}(n\in\mathbb{N}^*), find the limit of a_n as n approaches infinity. Express your answer as a single numerical value. | -2 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Find the last two digits of the sum \(0! + 5! + 10! + 15! + \cdots + 100!\). Express your answer as a two-digit number, including leading zeros if necessary. | 21 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
A survey of the hair colour of 600 people found that 30% of them have blonde hair. Calculate the number of people with blonde hair. Express your answer as a whole number. | 180 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} =(2,3)$ and $\overrightarrow{b} =(-1,2)$, if $m \overrightarrow{a}+4 \overrightarrow{b}$ is collinear with $\overrightarrow{a}-2 \overrightarrow{b}$, determine the value of $m$. Express your answer as a single numerical value. | -2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
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