instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
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Given the sets $P=\{(x,y) | y=k(x-1)+1, x \in \mathbb{R}, y \in \mathbb{R}\}$ and $Q=\{(x,y) | x^2+y^2-2y=0, x \in \mathbb{R}, y \in \mathbb{R}\}$, find the number of elements in the set $P \cap Q$. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the parametric equations $\begin{cases}x=2+\cos \theta, \\ y=\sin \theta\end{cases}$ (with $\theta$ as the parameter), calculate the maximum value of $\sqrt{(x-5)^{2}+(y+4)^{2}}$. Express your answer as a single numerical value. | 6 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
If the line that passes through the points \((2, 7)\) and \((a, 3a)\) has a slope of 2, what is the value of \(a\)? Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the functional equation f(x) + f(1 / (1-x^3)^(1/3)) = x^3 for every real x ≠ 1, find the value of f(-1). Express your answer as a single fraction or number. | 1/4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = 2⌊x⌋ + 1, where x ∈ [-1, 3) and ⌊x⌋ denotes the largest integer not exceeding x, find the range of y. Express your answer as a set of values, in the format {a, b, c,...}. | {−1, 1, 3, 5} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^2 + 2x (x > 0), and the recursive definition f1(x) = f(x), f(n+1)(x) = f(f(n)(x)), n ∈ N*, find the maximum value of f2019(x) on the interval [1, 2]. Express your answer as a single mathematical expression. | 3^{2^{2019}} - 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A stationery store sold two scientific calculators for 120 yuan each. One was sold at a 20% profit, and the other at a 20% loss. Calculate the overall profit or loss incurred by the stationery store. Express your answer in yuan, clearly indicating whether it is a profit or loss. | -10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $P = \{x | (x - 3)(x - 6) \leq 0, x \in \mathbb{Z}\}$ and $Q = \{5, 7\}$, find the intersection of $P$ and $Q$, denoted as $P \cap Q$. Express your answer as a set, including all elements of the intersection. | \{5\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $| \overrightarrow{a}|=6$, $| \overrightarrow{b}|=3$, and $\overrightarrow{a}\cdot \overrightarrow{b}=-12$, calculate the projection of vector $\overrightarrow{a}$ in the direction of vector $\overrightarrow{b}$. Express your answer as a single number. | -4 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
The current green area in the central urban area of Xunyang is 1000 hm², with a planned annual increase of 4%. After x years (x ∈ ℕ*), the green area is y hm². Express the function relation between x and y that models the exponential growth of the green area, including the condition that x ∈ ℕ*. | y = 1000(1.04)^x | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( a \) is the decimal part of \( \sqrt{3+\sqrt{5}} - \sqrt{3-\sqrt{5}} \), and \( b \) is the decimal part of \( \sqrt{6+3\sqrt{3}} - \sqrt{6-3\sqrt{3}} \), find the value of \( \frac{2}{b} - \frac{1}{a} \). Express your answer as a single algebraic expression. | \sqrt{6} - \sqrt{2} + 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A box contains 3 black balls and 4 white balls. Balls are drawn randomly one at a time without replacement. The process continues until all balls of one color have been drawn. Calculate the probability that the last ball drawn is a black ball. Express your answer as a simplified fraction. | \frac{3}{7} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the plane vectors $\overset{⇀}{a} =(-2,3)$ and $\overset{⇀}{b} =(1,2)$, if the vector $λ\overset{⇀}{a} +\overset{⇀}{b}$ is perpendicular to $\overset{⇀}{b}$, calculate the value of the real number $λ$. Express your answer as a real number, including the negative sign if applicable. | -\dfrac{5}{4} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the expression C_6^3 + C_6^2, where C_n^k denotes the number of combinations of n items taken k at a time. Express your answer as a single integer. | 35 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Combinations"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
A high school with 24 classes in the second grade is conducting a survey using a systematic sampling method. 4 classes are selected for the survey, and the sum of the selected class numbers is 52. Using the systematic sampling method, determine the smallest class number drawn. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Use the completing the square method to solve the equation $x^{2}-8x+1=0$. Express your answer in the form $(x-a)^{2}=b$, where $a$ and $b$ are exact values. | $(x-4)^{2}=15$ | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $(\lg 5)^2 + \lg 50 \cdot \lg 2$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
How many non-empty subsets does the set {1, 2, 3} have? Provide your answer as a single integer. | 7 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sin 74^\circ = m$, use the cofunction identity and the half-angle identity for cosine to find $\cos 8^\circ$. Express your answer as a mathematical expression involving'm'. | \sqrt{\frac{1+m}{2}} | big_math | [
"Mathematics -> Precalculus -> Functions -> Trigonometric Functions"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
The graph \( F_1 \) of the function \( y = x^2 \) is translated by the vector \( \vec{a} = (m, 1) \) to form the graph \( F_2 \). The tangent lines to \( F_1 \) and \( F_2 \) at \( x = 1 \) are perpendicular to each other. Calculate the value of the real number \( m \) that satisfies this condition. Express your answer... | \frac{5}{4} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the circumference of a sector is 6 and the area is 2, find the measure of the central angle of the sector in radians. Express your answer(s) in radians, and note if there are multiple solutions. | 1, 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $A=\{1,2,3\}$ and $B=\{3,4\}$, calculate the intersection $A \cap B$. Express your answer as a set. | \{3\} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that {a_n} is an arithmetic sequence, and the sum a_2 + a_3 + a_{10} + a_{11} = 48, find the value of a_6 + a_7. Express your answer as a single numerical value. | 24 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of the term x^3 in the expansion of (1-3x)^5(3-x). Express your answer as a single integer, including a negative sign if applicable. | -900 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the line $l: x-2y+3=0$ intersects the circle $C: x^{2}+y^{2}-2x-6y+6=0$ at points $A$ and $B$, calculate the length of the segment $AB$. Express your answer as a fraction with a radical. | \frac{8\sqrt{5}}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Bret, Abby, Carl, and Dana are seated in a row of four seats numbered #1 to #4. Joe looks at them and says:
"Bret is next to Carl."
"Abby is between Bret and Carl."
However, each one of Joe's statements is false. Given that Bret is actually sitting in seat #3, who is sitting in seat #2? Provide the name of the person... | Dana | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the coefficients of all terms except the constant term in the expansion of $(\\frac{2}{\\sqrt{x}}-x)^{9}$. Express your answer as a single integer. | 5377 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence ${a_n}$, the arithmetic mean of $a_3$ and $a_8$ is 10. Find the value of $a_1 + a_{10}$. Express your answer as a single integer. | 20 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Let $f(x) = \cos x - \sin x$. The graph of $f(x)$ is translated by vector $\overrightarrow{a}(m, 0)$ to exactly match the graph of $-f'(x)$. Calculate the value of $m$ that satisfies this condition. Express your answer as a single value, including any necessary mathematical notation. | \frac{\pi}{2} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given $x+y=7$ and $xy=10$, calculate the value of $3x^2+3y^2$. Express your answer as a single integer. | 87 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Among the angles -20°, -400°, -2000°, 1600°, how many are in the fourth quadrant? Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Find the angle between the line $x+3=0$ and the line $x+y-3=0$. Express your answer in degrees as a whole number. | 45 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Among the real numbers $\frac{7}{8}$, $\sqrt{36}$, $-3\pi$, $\sqrt{7}$, and $1.41414141$, how many are rational numbers? Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $A=\{x|-3<x<6\}$ and $B=\{x|2<x<7\}$, find the intersection of set $A$ and the complement of set $B$, denoted as $A\cap (\complement_{\mathbb{R}} B)$. Express your answer in interval notation, using brackets or parentheses to denote the type of interval. | (-3,2] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
When written out in full, the number \((10^{2020}+2020)^{2}\) has 4041 digits. Calculate the sum of the digits of this 4041-digit number. Express your answer as a single integer. | 25 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Xiao Wang and Xiao Li play a game of "Rock, Paper, Scissors". The rules of the game are: win once to get 3 points, tie to get 1 point, lose to get 0 points. A total of 7 games are played, and the one with the higher score at the end of the games wins. If Xiao Wang scores 10 points at the end of the games, what is the m... | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
An 18 inch by 24 inch painting is mounted in a wooden frame where the width of the wood at the top and bottom of the frame is twice the width of the wood at the sides. If the area of the frame is equal to the area of the painting, calculate the ratio of the shorter side to the longer side of this frame. Express your an... | 2:3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Let event $A=\{\text{the numbers on both throws are odd}\}$, and $B=\{\text{the sum of the numbers on the two throws is less than }7\}$. Calculate the conditional probability $P(B|A)$. Express your answer as a simplified fraction. | \dfrac{2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
If \(x=2\), calculate the value of the expression \((x+2-x)(2-x-2)\). Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given the set A = {x|1≤x≤2}, find the necessary and sufficient condition on 'a' such that the proposition ∀x∈A, x^2-a≤0 is true. Express your answer as an inequality involving 'a'. | a ≥ 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In an equilateral triangle ABC with side length 1, let vectors a, b, and c be defined as follows: a is along BC, b is along CA, and c is along AB. Calculate the sum of the dot products of these vectors, i.e., a · b + b · c + c · a. Express your answer as a single numerical value. | -\frac{3}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given a function f′(x) = x^{2} + 3x - 4, find the interval where the function y = f(x + 1) is decreasing. Express your answer in standard interval notation. | (-5, 0) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Ari and Bri are riding their bicycles on a 30 km trip. Ari rides at an average speed of 20 km/h, and Bri rides at an average speed of 15 km/h. If they begin at the same time, how many minutes after Ari finishes the trip will Bri finish? Express your answer in minutes as a whole number. | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression \(1-(2-(3-4-(5-6)))\) to find its value. Provide your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the set M = {1, 2, 3, 4} and set N defined by the condition y = sqrt(x - 3), calculate the intersection of sets M and N. Express your answer using proper set notation. | M ∩ N = {3, 4} | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $(-2)+(-5)$. Express your answer as a single integer. | -7 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence $\{a_n\}$ with all terms positive, where $a_1=1$ and $a_1+a_3+a_5=21$, find the value of $a_2+a_4+a_6$. Express your answer as a single integer. | 42 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the direction vector of line $l$ is $\left(-1,2,1\right)$, and the normal vector of plane $\alpha$ is $\left(m,n,3\right)$, and knowing that $l \perp \alpha$, find the sum of $m$ and $n$. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Let the function $f(x)$ be differentiable. Calculate the limit $$\lim_{\Delta x \to 0} \frac {f(1)-f(1+\Delta x)}{3\Delta x}$$ in terms of $f'(1)$. | -\frac{1}{3}f'(1) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $a=\sqrt{5}$, $c=2$, and $\cos A=\frac{2}{3}$, calculate the value of $b$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
How many distinct arrangements are possible for wearing five different rings in the five fingers of the right hand, considering that multiple rings can be worn on one finger? Express your answer as a single integer. | 3125 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors a=(8, 1/2x) and b=(x, 1), where x > 0, if (a-2b) || (2a+b), calculate the value of x. | 4 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence {a_n} whose sum of the first n terms is S_n, and S_n = n^2 + 1, calculate the value of a_5. Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Let $a, b \in \mathbb{R}^{+}$ and $a+b=1$. Find the supremum of the expression $-\\frac{1}{2a} - \\frac{2}{b}$. Express your answer as a single number. | -\frac{9}{2} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^3 - 3x + m, find the value of the real number m such that the maximum and minimum values of f(x) in the interval [-3, 0] sum up to -14. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
A line that passes through point P(2, 4) and has exactly one common point with the parabola $y^2=8x$ can be either a tangent line at point P or a line parallel to the axis of symmetry. Find the total number of such lines. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given a hyperbola $C: \\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ with foci ${F_1}$ and ${F_2}$, where ${F_1}$ is the left focus and ${F_2}$ is the right focus, and the point symmetric to ${F_2}$ with respect to the asymptote lies on the circle with center ${F_1}$ and radius $|OF_1|$, calculate the eccentricity of the hyp... | 2 | big_math | [
"Mathematics -> Geometry -> Differential Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the number of days required by team A working alone is \( a \) times the number of days required by teams B and C working together; the number of days required by team B working alone is \( b \) times the number of days required by teams A and C working together; and the number of days required by team C wor... | 1 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Find the difference between the reciprocal of -4 and its opposite. Express your answer as a fraction. | -\frac{17}{4} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets {a^{2}, 0, -1} = {a, b, 0}, calculate the value of a^{2023} + b^{2023}. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$, $b$, and $c$ are all integers and satisfy $\left(a-b\right)^{10}+\left(a-c\right)^{10}=1$, find the value of $|a-b|+|b-c|+|c-a|$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
From the 20 integers ranging from 1 to 20, if any 11 numbers are selected, what is the sum that must always be achievable by adding two of the selected numbers? Provide your answer as a single integer. | 21 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
On a circle, fifteen points A₁, A₂, A₃, …, A₁₅ are equally spaced. Find the measure of angle ∠A₁A₃A₇. Express your answer in degrees as a whole number. | 108 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets M = {2, 3, 4} and N = {0, 2, 3, 5}, find the intersection of M and N, denoted as M ∩ N. Express your answer as a set with elements separated by commas and enclosed in curly brackets. | {2, 3} | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given $a=\left( \frac{2}{5} \right)^{\frac{3}{5}}, b=\left( \frac{2}{5} \right)^{\frac{2}{5}}, c=\left( \frac{3}{5} \right)^{\frac{2}{5}}$, determine the relationship between the sizes of $a$, $b$, and $c$. Express your answer as an inequality in the format $a \\square b \\square c$ where $\\square$ represents $<$, $>$... | a < b < c | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given that cos(α + π/6) = 4/5 and α is an acute angle, calculate the value of sin(α). Express your answer as a fraction or exact decimal value. | \frac{3\sqrt{3} - 4}{10} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Express 400 billion in scientific notation, using the format $a \times 10^n$. Provide your answer in the form $\boxed{a \times 10^n}$. | 4 \times 10^{11} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
In a regular quadrilateral prism $ABCDA'A'B'C'D'$ with vertices on the same sphere, $AB = 1$ and $AA' = \sqrt{2}$. Calculate the spherical distance between points $A$ and $C$. Express your answer in radians. | \frac{\pi}{2} | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Spherical Geometry"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the derivative of the function f(x) at x=a is A, evaluate the limit: $$\lim_{\Delta x \to 0} \frac {f(a+4\Delta x)-f(a+5\Delta x)}{\Delta x}$$ Express your answer as a single value. | -A | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
In the acute-angled triangle $ABC$ with all three sides being unequal, the three altitudes $AD$, $BE$, and $CF$ intersect at point $H$. Count the number of different shapes (non-similar triangles) present in the figure. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangulations"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=a+i$ (where $a>0$ and $i$ is the imaginary unit), if $|z|= \sqrt {5}$, find the imaginary part of $\frac {1}{z}$. Express your answer as a single number, including a negative sign if applicable. | -\frac{1}{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Three balls labeled $A$, $B$, and $C$ are randomly placed into four boxes numbered $1$, $2$, $3$, and $4$. Find the total number of different ways in which box $1$ has no balls and all three balls are placed in boxes $2$, $3$, and $4$. Express your answer as a single integer. | 27 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Randomly select two numbers within the interval [-π,π], denoted as a and b, respectively. Calculate the probability that the function f(x) = x^2 + 2ax - b^2 + π has a root. Express your answer as a simplified fraction. | 3/4 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Please fold a long rope in half, then fold it in half again along the middle of the folded rope, and continue to fold it in half 6 times in total. Finally, cut the rope along the middle after it has been folded 6 times. Calculate the total number of segments the rope will be cut into. Express your answer as a whole num... | 65 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $\frac {5i}{2-i}$. Express your answer as a complex number in the form a + bi. | -1+2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Find the distance between the two intersection points of the graphs of $x^2 + y = 10$ and $x + y = 10$. Express your answer as a single numerical value. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the arithmetic square root of the integer 100. Express your answer as a single positive number. | 10 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Two arithmetic sequences, {a_n} and {b_n}, have initial terms a_1 = 25 and b_1 = 75, respectively. Given that a_2 + b_2 = 100, find the 37th term of the sequence formed by a_n + b_n. Express your answer as a single numerical value. | 100 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given that \(abc = 1\), \(a + b + c = 2\), and \(a^2 + b^2 + c^2 = 3\), calculate the value of \(\frac{1}{ab + c - 1} + \frac{1}{bc + a - 1} + \frac{1}{ca + b - 1}\). Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given a point $P(x,y)$ on the ellipse $\frac{(x-2)^2}{4}+(y-1)^2=1$, find the maximum value of $x+y$. Express your answer using any necessary mathematical notation (e.g. square roots). | 3+\sqrt{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the conjugate of the complex number $\frac{2i}{1+i}$. Express your answer as a complex number in the form $a+bi$. | 1-i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given points A(1,1,0), B(0,3,0), and C(2,2,3), calculate the projection vector of vector AC onto AB. Express your answer as a vector in the format (x, y, z). | (-\frac{1}{5}, \frac{2}{5}, 0) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that P and Q are the midpoints of the sides AB and AC of triangle ABC, respectively, and vector PQ = (2,3), and point B has coordinates (-1,-2), calculate the coordinates of point C. Express your answer as a coordinate pair (x, y). | (3,4) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
If the area of a sphere's great circle is doubled, calculate the factor by which its volume is increased. Express your answer as a numerical factor. | 2\sqrt{2} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ with the general term formula $a_n = n^2 - 2n - 8$ ($n \in \mathbb{N}^*$), calculate $a_4$. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Based on the circle graph, calculate the percentage of Mr. Auckland's Grade 9 students who do at least one hour of homework per day. Express your answer as a percentage. | 75\% | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A car uses 8.4 liters of gas for every 100 km it is driven. A mechanic can modify the car's engine for a cost of $400 so that it will only use 6.3 liters of gas per 100 km. If gas costs $0.80 per liter, calculate the minimum distance the owner would have to drive to recover the cost of the modifications. Express your a... | 23809.52 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
The six edges of a tetrahedron \(ABCD\) have lengths \(7,\ 13,\ 18,\ 27,\ 36,\ 41\). Given that \(AB = 41\), what is the length of \(CD\)? Express your answer as a whole number. | 13 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \( AD < BC \). The lengths of the sides are given as follows: \( AD = 4, AB = 5, BC = 10, CD = 7 \). Find the perimeter of \( ABCD \). Express your answer as a whole number. | 26 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Find the common factor of each term in the polynomial $2x^{2}+6x^{3}$. Express your answer as a mathematical expression in the form $\boxed{ext{common factor}}$. | 2x^{2} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, point A has coordinates (2,-8). Point B is symmetric to point A with respect to the y-axis. Calculate the coordinates of point B. Express your answer as an ordered pair (x, y). | (-2,-8) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
There are 5 different balls to be placed into 4 different boxes, with each box containing at least one ball. Calculate the total number of different ways to do this. | 240 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression \( 6a - 5a + 4a - 3a + 2a - a \). Express your answer as a simplified algebraic expression. | 3a | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
For the system of equations \(x^{2} + x^{2} y^{2} + x^{2} y^{4} = 525\) and \(x + xy + xy^{2} = 35\), calculate the sum of the real \(y\) values that satisfy the equations. Express your answer as a single numerical value. | \frac{5}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $P=\{x|\log_{2}x < -1\}$ and $Q=\{x||x| < 1\}$, determine the set $P\cap Q$. Express your answer in interval notation. | (0, \frac {1}{2}) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Solve for \( n \) in the equation \( 3 \times n = 6 \times 2 \). Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
The radius of a certain sector is 1cm, and its arc length is 2cm. Find the central angle of this sector in radians. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate 25% of 60. Express your answer as a whole number. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the one-variable linear equation $2x + m = 5$ and knowing that $x = 1$ is a solution, find the value of $m$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given that \(S_{n}\) is the sum of the first \(n\) terms of an arithmetic sequence \(\{ a_{n}\}\), and \(2(a_{1}+a_{3}+a_{5})+3(a_{8}+a_{10})=36\), calculate the value of \(S_{11}\). Express your answer as a single integer. | 33 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
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