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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|---|---|---|---|---|---|
Given two non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, and that the magnitude of $\overrightarrow{a} + \overrightarrow{b}$ equals the magnitude of $\overrightarrow{a} - \overrightarrow{b}$, i.e., $|\overrightarrow{a} + \overrightarrow{b}| = |\overrightarrow{a} - \overrightarrow{b}|$, find the angle b... | 90 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other",
"Mathematics -> Linear Algebra -> Vectors"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $F_1$ and $F_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, respectively. A circle with the coordinate origin $O$ as the center and $OF_1$ as the radius intersects the hyperbola at point $P$ in the first quadrant. When the area of $\triangle PF_1F_2$ i... | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of the $x^3$ term in the expansion of the binomial $(2x- \frac {1}{x})^{5}$. Express your answer as a single integer, including the negative sign if applicable. | -80 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Binomial Expansion"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $3(\overrightarrow{a} + \overrightarrow{x}) = \overrightarrow{x}$, solve for $\overrightarrow{x}$ in terms of $\overrightarrow{a}$. Express your answer as a vector. | - \frac{3}{2} \overrightarrow{a} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Three equilateral triangles with sides of length 1 are shown shaded in a larger equilateral triangle. The total shaded area is half the area of the larger triangle. What is the side length of the larger equilateral triangle? Express your answer as a single value, including the square root symbol if necessary. | \sqrt{6} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Convert the binary number 10011_2 to its decimal equivalent. Express your answer as a single integer. | 19 | big_math | [
"Mathematics -> Other -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
A school cafeteria surveys students' preferences for three types of zongzi to determine the final procurement plan. What statistical measure should the cafeteria use to identify the most preferred type of zongzi among the students? Provide your answer as the name of the statistical measure. | Mode | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
When five students are lining up to take a photo, two teachers happen to pass by. The students strongly request the teachers to join the photo as a memento. If the order of the five students is fixed, then find the total number of ways for the two teachers to stand in line with the students for the photo. Express your ... | 42 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\frac{2}{3}$, $\frac{4}{5}$, $\frac{6}{7}$, $\frac{8}{9}$,..., derive its general term formula and use it to calculate the 10th term of the sequence. Express your answer as a fraction in simplest form. | \frac{20}{21} | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Express 361000000 in scientific notation in the form $a \times 10^b$, where a is between 1 and 10 (not including 10). | 3.61 \times 10^8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z = 1 - \frac{1}{i}$, where $i$ is the imaginary unit, calculate the modulus of $z$. Express your answer using the absolute value sign or modulus notation. | |z| = \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The average energy consumption per seat of a maglev train is only one-third of that of an airplane and 70% of that of a car. What is the average energy consumption per seat of a car as a fraction of that of an airplane? Express your answer as a simplified fraction. | \frac{10}{21} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, if (a_5, a_7) are the two roots of the equation x^2 - 2x - 6 = 0, then calculate the sum of the first 11 terms of {a_n}. Express your answer as a single numerical value. | 11 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Observe the sequence of expressions:
$$a+b=1, a^2+b^2=3, a^3+b^3=4, a^4+b^4=7, a^5+b^5=11,\ldots$$
Using the pattern of the sequence, calculate the value of $a^8+b^8$. Express your answer as a single integer. | 47 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \begin{cases} (3-a)x-3, & x\leqslant 7 \\ a^{x-6}, & x > 7 \end{cases}$, if the sequence $\{a_n\}$ satisfies $a_n=f(n)$ ($n\in\mathbb{N}^*$) and $\{a_n\}$ is an increasing sequence, then find the range of the real number $a$. Express your answer as an interval. | (2,3) | big_math | [
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Randomly draw two balls from seven balls marked with numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$. What is the probability that the sum of the numbers on the two balls is divisible by $3$? Express your answer as a simplified fraction. | \dfrac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
A hockey team has 6 more red helmets than blue helmets. The ratio of red helmets to blue helmets is 5:3. Calculate the total number of red helmets and blue helmets. | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
The displacement S (in meters) of an object varies with time t (in seconds) according to the equation S(t) = 3t - t^2. Find the instantaneous velocity of the object at t = 2 seconds. Express your answer in meters per second (m/s). | -1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of distinct pairs (x, y) of real numbers that satisfy both of the following equations:
\[x = x^2 + y^2\]
\[y = 2xy\]
Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality $$\frac {ax}{x-1} < 1$$, its solution set is $\{x|x < 1, \text{ or } x > 3\}$. Calculate the value of $a$ that satisfies this condition. Express your answer as a single fraction. | \frac {2}{3} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Solve for \(x\) in the equation \(\sqrt{5+11-7}=\sqrt{5+11}-\sqrt{x}\). Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
For $x \in (0, +\infty)$, the power function $y = (m^2 - m - 1)x^{-m+1}$ is a decreasing function. Find the real number $m$ that satisfies this condition. | 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n} which satisfies 5^{a_{n+1}} = 25 * 5^{a_n}, and a_2 + a_4 + a_6 = 9, find the value of log_{1/3}(a_5 + a_7 + a_9). Express your answer as a single number. | -3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions",
"Mathematics -> Algebra -> Sequences and Series"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Among the following real numbers, count the number of irrational numbers:
$3.1415926$, $\sqrt[3]{3}$, $\frac{10}{3}$, $\sqrt{16}$, $\sqrt{12}$, $\frac{5}{6}$
Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
In a certain information transmission process, a permutation of four digits (with repetition allowed) is used to represent a message, with different permutations representing different messages. If only the digits 0 and 1 are used, calculate the number of messages that have at most two digits in common with the message... | 11 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Five years ago today, Sally was 7 years old. How old will Sally be in two more years? Express your answer as a whole number. | 14 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
A student is practicing shooting. The probabilities of hitting at three different positions, A, B, and C, are $\frac{1}{3}$, $\frac{1}{2}$, and $p$, respectively. If the student shoots once at each position, and the probability of hitting exactly two shots is $\frac{7}{18}$, then what is the value of $p$? Express your ... | \frac{2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, right triangles PSQ and RSQ are shown. Given QR = 25, SR = 20, and PS = 8, find the perimeter of triangle PQR. Express your answer as a whole number. | 70 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola y^2 = 8(x + 2), a line with an inclination angle of 60 degrees is drawn through the focus F. If this line intersects the parabola at points A and B, and the perpendicular bisector of chord AB intersects the x-axis at point P, calculate the length of segment PF. Express your answer as a single number. | \frac{16}{3} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given a regular polygon where at most 5 diagonals can be drawn from a vertex, find the measure of one of its interior angles. Express your answer in degrees. | 135 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
In a bag, there are 5 white balls and 5 red balls of the same size. If 3 balls are randomly selected from the bag and it is known that there is at least one white ball among the 3 balls, then find the probability of exactly 2 red balls being selected. Express your answer as a simplified fraction. | \frac{5}{11} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Combinations"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given the line \(l_{1}\): \(4x-3y+6=0\) and the line \(l_{2}\): \(x=-1\), find the minimum sum of distances from a moving point \(P\) on the parabola \(y^{2}=4x\) to the lines \(l_{1}\) and \(l_{2}\). Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
All the diagonals of a convex heptagon divide it into several non-overlapping smaller convex polygons. What is the maximum possible number of sides of these smaller polygons? Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $|\vec{a}|=4, |\vec{b}|=5$, and $\vec{a} \perp \vec{b}$, calculate the dot product $\vec{a} \cdot \vec{b}$. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of $a$ such that the expansion of $(ax^{2}-3x)(x^{2}-2x-1)$ does not contain the term $x^{3}$. Express your answer as a single value. | -\frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence \( 13, 16, 19, \cdots, 70, 73 \), find the total number of terms. Express your answer as a whole number. | 21 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
The meteorological department has determined that for every increase of 1 kilometer in altitude, the temperature drops by about 5°C. If the ground temperature is 15°C, what is the temperature at an altitude of 4 kilometers? Express your answer in degrees Celsius (°C) as an integer. | -5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = ax^5 + bx^3 + cx - 8, and f(-2) = 10, calculate the value of f(2). | -26 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Among the seven numbers $\frac{π}{2}$, $0$, $\sqrt[3]{8}$, $\frac{22}{7}$, $-\sqrt{5}$, $3.14$, $5.2020020002\ldots$ (with an increasing number of zeros between every two $2$'s), how many are irrational numbers? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Person A can hit a target 8 times out of 10, and person B can hit the target 7 times out of 10. If both persons shoot at the same target simultaneously, what is the probability that they both hit the target? Express your answer as a simplified fraction. | \frac{14}{25} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Xiao Wang needs to create 1000 tables in a Word document using the "copy-paste" method, starting with 1 table. Each "copy-paste" action doubles the number of tables. What is the minimum number of times Xiao Wang needs to use the "copy-paste" method to have at least 1000 tables? Provide your answer as a whole number. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
A square has an area of 144 cm^2. Find the side length of the square. Express your answer in centimeters (cm) as a whole number. | 12 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable $X$ follows a normal distribution $N(2,σ)$, and $P(X < 4)=0.84$, determine the value of $P(X\leqslant 0)$. Express your answer as a decimal value between 0 and 1, rounded to 2 decimal places. | 0.16 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Let $r$ be the result of doubling both the base and exponent of $a^b$, where $b$ does not equal 0. If $r$ equals the product of $a^b$ by $x^b$, find the value of $x$ in terms of $a$. Express your answer as an algebraic expression. | 4a | big_math | [
"Mathematics -> Algebra -> Exponential Functions -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given $z_{1}$, $z_{2} \in \mathbb{C}$, where $|z_{1}|=|z_{2}|=1$ and $|z_{1}+z_{2}|=\sqrt{3}$, calculate the value of $|z_{1}-z_{2}|$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given the monomial $-4mn^{5}$, find its coefficient and degree. Express your answer in the format: "Coefficient: [integer], Degree: [integer]". | Coefficient: -4, Degree: 6 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence $\{a_{n}\}$, $a_{1}=8$, $q= \frac {1}{2}$, and $a_{n}= \frac {1}{2}$. Calculate the sum of the first $n$ terms, $S_{n}$. Express your answer as a simplified fraction. | \frac {31}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = sin(x) + sqrt(3)*cos(x), find the value of varphi such that the graph of y = f(x + varphi) is symmetric about the line x = 0. Express your answer in radians. | \frac{\pi}{6} | big_math | [
"Mathematics -> Trigonometry -> Functions"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the domain of the function $f(\log_2{x})$ is $(2, 4)$, find the domain of the function $f\left( \frac{x}{2}\right)$. Express your answer as an interval in standard notation. | (2, 4) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of the function $y=\log_{\frac{1}{2}}(x^{2}-6x+17)$. Express your answer in interval notation if applicable. | (-\infty, -3] | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given a random variable ξ that follows a normal distribution N(4,5), find the real number a such that P(ξ < 2a-3) = P(ξ > a + 2). Express your answer as a single real number. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Arrange the 5 natural numbers from 1 to 5 in a row from left to right, in such a way that starting from the third number, each number is either the sum or difference of the previous two numbers. How many valid arrangements satisfy these conditions? Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\tan \frac{\pi}{4}$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable X follows a normal distribution with mean 5 and variance σ^2, and P(X > 10-a) = 0.4, calculate the probability P(X > a). Express your answer as a decimal probability value between 0 and 1. | 0.6 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{x|x^{2}+x-2=0,x\in\mathbb{R}\}$ and $N=\{x|x < 0,x\in\mathbb{R}\}$, find the intersection $M\cap N$. Express your answer as a set. | \{-2\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive integer n such that the expansion of the binomial (2x^4 - 1/3x^3)^n contains a non-zero constant term. | 7 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the largest and smallest possible values of \(9 \cos ^{4} x + 12 \sin ^{2} x - 4\). Express your answer as a single integer. | 13 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation cos30°•sinα=$\frac{{\sqrt{3}}}{4}$, find the acute angle α in degrees. Express your answer as a whole number. | 30 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of positive divisors of 2002 that are less than or equal to 100. Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Number Theory -> Factorization"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that all six digits of three 2-digit numbers are different, find the largest possible sum of these three numbers. Express your answer as a single integer. | 255 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In the interval [0, 2π], find the range of x that satisfies the inequality sin(x) > cos(x). Express your answer as an interval in standard notation. | \left( \frac{\pi}{4}, \frac{5\pi}{4} \right) | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of a line with a slope angle of 45° and a y-intercept of 2. Express your answer in slope-intercept form (y = mx + b). | y = x + 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular field has an area of 864 square steps, with a total length and width of 60 steps. Let the length of the field be x steps, where x > 30, and the width be (60 - x) steps. Using the equation x(60 - x) = 864, find the difference between the length and the width. Express your answer in steps as a whole number. | 12 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the odd function f(x) defined on R satisfies f(x-4) = -f(x) and is an increasing function on the interval [0,2], compare the values of f(-25), f(11), and f(80). Express your answer as a direct comparison of the function values, using the symbols <, >, or = as needed. | f(-25) < f(80) < f(11) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the line $3x-\sqrt{3}y+1=0$, find the angle of inclination in degrees. Express your answer in the form $\boxed{[answer]}^\circ$. | 60 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function $y=\frac{\ln^{2} x}{x}$. Express your answer as a single numerical value. | 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the binomial expansion $(2-x)^5 = a_0 + a_1x + a_2x^2 + \cdots + a_5x^5$, calculate the value of $\frac{a_0 + a_2 + a_4}{a_1 + a_3}$. Express your answer as a simplified fraction. | -\frac{61}{60} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z = 1 + i, calculate i times the complex conjugate of z. Express your answer as a complex number in the form a + bi. | 1 + i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
The ancient "Five Elements" theory posits that there are five types of properties in materials: metal, wood, water, fire, and earth. Metal overcomes wood, wood overcomes earth, earth overcomes water, water overcomes fire, and fire overcomes metal. If two different properties are randomly selected from these five, what ... | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Point \(Q\) is on a line segment between \(P\) and \(R\). If \(PR = 12\) and \(PQ = 3\), what is the length of \(QR\)? Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of \((9-2)-(4-1)\). Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that f(x) is an odd function and g(x) is an even function, and it is known that f(-1) + g(1) = 2 and f(1) + g(-1) = 4, calculate the value of g(1). | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = 2x^2 + 4x - 5$, where $x \in [t, t+2]$, the maximum value of the function $f(x)$ forms a function $y = g(t)$. Calculate the minimum value of the function $y = g(t)$. Express your answer as a single integer. | -7 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola equation x^{2}-ky^{2}=1, where one of the foci is at (3,0), find the value of k. Express your answer as a fraction. | \frac{1}{8} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given that vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are both unit vectors, and the angle between them is 120°, calculate the magnitude of $\overrightarrow{a} - 2\overrightarrow{b}$. Express your answer as a single value, including the square root symbol if necessary. | \sqrt{7} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
If moving $2km$ east is denoted as "$+2km$", what is the notation for moving $1km$ west? Express your answer with units (e.g., km). | -1km | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola y = x^2 - 2, translate it 3 units to the right and 1 unit down. What is the analytical expression of the resulting parabola? Express your answer as an equation in the form y = f(x). | y = (x - 3)^2 - 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of \((\sqrt{169} - \sqrt{25})^2\). Provide your answer as a single number. | 64 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
If $b$ and $c$ are constants and $(x + 2)(x + b) = x^2 + cx + 6$, then what is the value of $c$? Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation $x^{2}-mx+2=0$ with one root as $x=1$, find the value of $m$. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation ${(m+3)^2}+\sqrt{4-n}=0$, find the exact coordinates of point $P(m,n)$. Express your answer as a coordinate pair $(m,n)$. | (-3,4) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
What is the opposite number of the real number 7? Express your answer as a single integer. | -7 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = (x+1)(x-a), find the value of 'a' such that the function is even. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given that θ ∈ (3π/4, π) and sin θ cos θ = -√3/2, find the value of cos θ - sin θ. Express your answer as a single value, including the negative sign if applicable. | -\sqrt{1 + \sqrt{3}} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the polynomial p = x^4 + 6x^3 + 11x^2 + 3x + 31, find the number of integer values of x for which p is a perfect square. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola y = 3x^2, translate it down 2 units. Express the resulting equation in the form y = f(x). | y = 3x^2 - 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given that {a_n} is a geometric sequence with a_n > 0, and the equation a_2a_4 + 2a_3a_5 + a_4a_6 = 25 holds, find the value of a_3 + a_5. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the coordinates of vector $\overrightarrow{p}$ in a unit orthogonal basis $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ in space are $\left(1,3,2
ight)$, find the coordinates of $\overrightarrow{p}$ in the basis $\overrightarrow{a}+\overrightarrow{b}$, $\overrightarrow{a}-\overrightarrow{b... | (2, -1, 2) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
If $f(x)=\sin \alpha - \cos x$, find the value of $f'(α)$. Express your answer as a trigonometric function of α. | \sin α | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number equation (a-i)(1+ai) = -4 + 3i, solve for the real number a. Express your answer as a single real number. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $3x-2y-5=0$ and the constraint $1 \leqslant x \leqslant 2$, calculate the range of $\frac{y}{x}$. Express your answer as an interval, including the endpoints. | [-1,\frac{1}{4}] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Azarah, Charlize, and Greg run at different but constant speeds. Each pair ran a race on a track that measured 100 meters from start to finish. In the first race, when Azarah crossed the finish line, Charlize was 20 meters behind. In the second race, when Charlize crossed the finish line, Greg was 10 meters behind. In ... | 28 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function f(x) = 2x^3 - 6x^2 + 3 within the interval [-2, 2]. Express your answer as a single numerical value in the form \boxed{[answer]}. | -37 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
A parabolic arch has a height of 16 inches and a span of 40 inches. Using a quadratic equation to model the arch, calculate the height of the arch at a point 5 inches from the center. Express your answer in inches, rounded to the nearest whole number if necessary. | 15 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given that \(144 \times 177 = 25488\), calculate the result of \(254.88 \div 0.177\). Express your answer as a numerical value. | 1440 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
The Fibonacci sequence is defined as follows: ${a}_{1}={a}_{2}=1,{a}_{n}={a}_{n-1}+{a}_{n-2}\phantom{\rule{0.2em}{0ex}}\left(n\ge 3,n\in {\mathbb{N}}^{*}\right)$. Calculate the exact value of the 8th term of this sequence (${a}_{8}$). Express your answer as a single integer. | 21 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 132,000,000 in scientific notation. Use the format $a \\times 10^{b}$, where $a$ is the significant part and $b$ is the exponent. | 1.32 \times 10^{8} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Convert the binary number $(11011)_{(2)}$ into a decimal number. Express your answer as a single integer. | 27 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Arrange five different products in a row on a shelf such that products A and B must be together, while products C and D must not be next to each other. Calculate the total number of different arrangements possible. Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ that satisfies $\frac{{1-i}}{{1+i}}-z=1$, find the magnitude of $2+i\overline{z}$. Express your answer as a numerical value. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
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