instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
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A car starts a journey with 40 liters of fuel in its tank. The car consumes fuel at a rate of 5 liters per hour. Write an equation that describes the relationship between the remaining amount of fuel y (in liters) and the driving time t (in hours). Express your answer in the form y =..., using the variables y for remai... | y = 40 - 5t | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola equation $\dfrac{y^2}{a^2} - \dfrac{x^2}{b^2} = 1 \left( a > 0, b > 0 \right)$ and the circle equation $x^2 + (y - a)^2 = \dfrac{a^2}{9}$, where the asymptote of the hyperbola is tangent to the circle, calculate the eccentricity of the hyperbola. Express your answer as a single value in the form $\b... | \dfrac{3\sqrt{2}}{4} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n} with all real number terms, let S_n denote the sum of the first n terms of this sequence. If S_{10} = 10 and S_{30} = 70, find S_{40}. Express your answer as a single integer. | 150 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ satisfies $z+zi=2$, calculate the modulus of $\overline{z}+i$. Express your answer as a single value, including square root if necessary. | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
A person has 4 keys, of which 2 can open a door. If a key is randomly selected to try to open the door, and the key that cannot open the door is discarded, calculate the probability of opening the door on the second try. Express your answer as a fraction in simplest form. | \frac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of \(a\) such that the distance from the center of the circle \(x^{2}+y^{2}-2x-4y+3=0\) to the line \(x-ay+1=0\) is equal to \(2\). Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given x = √5 - 3, simplify the expression √(x^2 + 6x + 9). | √5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
There are 10 products, among which 3 are defective. If two products are randomly selected, find the probability that the number of defective products selected is less than 2. Express your answer as a simplified fraction. | \frac{14}{15} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ with an angle of 120° between them, and $|\overrightarrow {a}| = |\overrightarrow {b}| = 1$. Vector $\overrightarrow {c}$ is collinear with $\overrightarrow {a} + \overrightarrow {b}$. Calculate the minimum value of $|\overrightarrow {a} + \overrightarrow {c... | \frac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the coefficients of all terms in the expansion of $(C_4+C_4^1x+C_4^2x^2+C_4^3x^3)^2$. Express your answer as a single integer. | 225 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \frac{1}{2}x^2 + 2xf'(2016) - 2016\ln x$, find the value of $f'(2016)$. Express your answer as an integer. | -2015 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
The first AMC 8 was given in 1985 and has been given annually since that time. Samantha turned 12 years old the year that she took the seventh AMC 8. In what year was Samantha born? Express your answer as a four-digit year. | 1979 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$. If $S_2=2$ and $S_4=10$, then find the value of $S_6$. Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Let $z_1=m^2-3m+m^2i$ and $z_2=4+(5m+6)i$, where $m$ is a real number and $i$ is the imaginary unit. If $z_1 - z_2 = 0$, find the value of $m$. Express your answer as a single real number. | -1 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Find the degree of the polynomial $(x^2+1)^4 (x^3+1)^3$. Express your answer as a whole number. | 17 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Polynomials"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression: $\frac{3}{2}+\frac{5}{4}+\frac{9}{8}+\frac{17}{16}+\frac{33}{32}+\frac{65}{64}-7$. Express your answer as a fraction. | -\frac{1}{64} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the line $(a-2)x+ay-1=0$ is perpendicular to the line $2x+3y+5=0$, find the value of $a$. Express your answer as a single numerical value. | \frac{4}{5} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
There are nine coins, including denominations of 1 yuan, 5 yuan, 10 yuan, and 50 yuan, with at least one coin of each denomination. If the total value of these nine coins is 177 yuan, calculate the exact number of 10 yuan coins. Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Transform the equation $7x-3=2x^{2}$ into general form with a constant term of $3$. What is the coefficient of the linear term in the resulting equation? Express your answer as a numerical value. | -7 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(1,1)$, compute $(\overrightarrow{a}+\overrightarrow{b})\cdot(\overrightarrow{a}-2\overrightarrow{b})$. Express your answer as a single integer in the format $\boxed{[answer]}$. | -2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
A cylindrical log has a length of 2 zhang and 4 chi, and a circumference of 5 chi. A kudzu vine starts growing at the base of the cylinder, wraps around the cylinder twice, and ends up even with the top of the log. What is the minimum length of the kudzu vine? Express your answer in chi as a whole number. | 26 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\cos x=\frac{3}{4}$, calculate the value of $\cos 2x$ using the double angle formula. Express your answer as a fraction. | \frac{1}{8} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 2.684 million in scientific notation. Use the format $a \times 10^{b}$. | 2.684 \times 10^{6} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimal positive period of the function $f(x) = \frac{\tan x}{1 + \tan^2 x}$. Express your answer in radians. | \pi | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given $\alpha \in (0, \pi)$, if $\sin \alpha + \cos \alpha = \frac{\sqrt{3}}{3}$, calculate $\cos^2 \alpha - \sin^2 \alpha$. Express your answer as a fraction with a square root if necessary. | -\frac{\sqrt{5}}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Points \(P\) and \(Q\) are two distinct points in the \(xy\)-plane. Find the number of different places in the \(xy\)-plane where a third point, \(R\), can be placed so that \(PQ = QR = PR\). Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, the point P(-1, x^2 + 1) has a negative x-coordinate and a positive y-coordinate for any rational number x. Identify the quadrant in which point P lies. Express your answer as the name of the quadrant (e.g., First, Second, Third, Fourth). | Second | big_math | [
"Mathematics -> Geometry -> Coordinate Geometry -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the graph of the quadratic function y = x^2 - 5x + m (where m is a constant) intersects the x-axis at the point (1,0), find the two real roots of the quadratic equation x^2 - 5x + m = 0. Express your answer as two numerical values. | 1, 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression $(-1)^{5^{2}}+1^{2^{5}}$. Provide your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = sqrt(a^2 - x^2) / (|x + a| - a), find the necessary and sufficient condition on 'a' for f(x) to be an odd function. Express your answer as an inequality or equality condition on 'a'. | a > 0 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given two arithmetic sequences $\{a_n\}$ and $\{b_n\}$, where $a_1=25$ and $b_1=125$, and $a_2+b_2=150$, find the value of the 2006th term of the sequence $\{a_n+b_n\}$. Express your answer as a single numerical value. | 150 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Let \( x = 0.1234567891011 \cdots 998999 \), a decimal formed by concatenating the numbers starting from 1 to 999. Find the digit at the 1983rd position after the decimal point. Express your answer as a single digit. | 7 | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line that passes through the origin and the intersection point of lines $l_1$: $x-3y+4=0$ and $l_2$: $2x+y+5=0$. Express your answer in the standard form of a linear equation (e.g. Ax + By = C). | 3x + 19y = 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given set M = $\{y | y=(\frac{1}{2})^{x}, x \in [0, +\infty)\}$ and set N = $\{y | y=\log_2{x}, x \in (0, 1]\}$, calculate the union of sets M and N. Express your answer as an interval. | (-\infty, 1] | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Compare the values of $\sin 11^\circ$, $\sin 168^\circ$, and $\cos 10^\circ$. Express your answer as an inequality with the correct order of these values. | \sin 11^\circ < \sin 168^\circ < \cos 10^\circ | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinates of the focus of the parabola given by the equation $y^2 = 4x$. Express your answer as an ordered pair (x, y). | (1, 0) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 2^x - sqrt(x) - 14, find the probability that f(x_0) > 0 when a number x_0 is randomly selected from the interval (0, 16). Express your answer as a simplified fraction. | 3/4 | big_math | [
"Mathematics -> Applied Mathematics -> Probability"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given that three individuals, A, B, and C, are attending an interview at a certain company, their respective probabilities of being hired are $$\frac{1}{6}, \frac{1}{4}, \frac{1}{3}$$. The results of their interviews do not affect one another. Calculate the probability that at least one of them will be hired. Express y... | \frac{7}{12} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given two vectors $\overrightarrow{a} = (4,2)$ and $\overrightarrow{b} = (x,3)$, and knowing that vector $\overrightarrow{a}$ is parallel to vector $\overrightarrow{b}$, find the value of the real number $x$. Express your answer as a single real number. | 6 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
In the unit circle, the terminal side of angle alpha intersects the unit circle at point P(1/2, sqrt(3)/2). If the terminal side of angle alpha is rotated counterclockwise by pi/3, find the exact coordinates of the point Q where the terminal side of angle alpha intersects the unit circle after rotation. Express your an... | (-1/2, sqrt(3)/2) | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation $3x^{2}+2x-2=0$ with roots $a$ and $b$, simplify the expression $\frac{2a}{a^2-b^2}-\frac{1}{a-b}$. Express your answer as a fraction. | -\frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
For the fraction $\frac{1}{3-x}$ to be meaningful, what condition must be met regarding the value of $x$? Express your answer in terms of what value $x$ cannot equal. | x \neq 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
The temperature in Vancouver is 22°C. The temperature in Calgary is 19°C colder than the temperature in Vancouver. The temperature in Quebec City is 11°C colder than the temperature in Calgary. Calculate the temperature in Quebec City. Express your answer in degrees Celsius (°C). | -8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Three students each choose one day from Saturday or Sunday to participate in a charity event. Calculate the probability that there will be students participating on both Saturday and Sunday. Express your answer as a simplified fraction. | \frac{3}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given that sin(α) - cos(α) = 1/5 and 0 < α < 180°, find the value of tan(α). Express your answer as a fraction. | \frac{4}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} where the common difference and the first term are both non-zero, and a_1, a_2, a_4 form a geometric sequence, calculate the value of (a_1 + a_14) / a_3. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the parabola y = (x+2)^2 - 1, identify the coordinates (h, k) of its vertex. Express your answer as an ordered pair. | (-2, -1) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
The equation of a hyperbola is given by $\frac{y^{2}}{4}-x^{2}=1$. Find the distance from point P(0, 1) to the asymptotes of the hyperbola. Express your answer as a simplified fraction or decimal. | \frac{\sqrt{5}}{5} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given that tan θ = 2, simplify the expression ((sin θ + cos θ) cos 2θ) / sin θ. Express your answer as a fraction. | -\frac{9}{10} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
The line represented by the equation $(a-1)x-y+2a+1=0$ ($a \in \mathbb{R}$) always passes through a fixed point. Find the coordinates of this fixed point. | (-2, 3) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $2^{m}=3$ and $2^{n}=4$, calculate the value of $2^{3m-2n}$. Express your answer as a fraction in the form $\boxed{\frac{a}{b}}$. | \frac{27}{16} | big_math | [
"Mathematics -> Algebra -> Exponential Functions"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 0.000815 in scientific notation. Ensure that the coefficient is between 1 and 10, and include the exponent. | 8.15 \times 10^{-4} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the angles in the triangle are related as follows: \( 2x^\circ + 3x^\circ + x^\circ = 180^\circ \). Additionally, \( x \) and \( z \) are supplementary angles. Find the value of \( z \) in degrees. | 150 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Express 496 as \( 2^m - 2^n \), where \( m \) and \( n \) are integers. Calculate the sum of \( m \) and \( n \). Provide your answer as a single integer. | 13 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(-2,m)$, if $\overrightarrow{a}$ and $\overrightarrow{b}$ are collinear, find the real number $m$. | -4 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Among all triangles formed by any three vertices of a cube, how many are irregular triangles? Express your answer as a whole number. | 24 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A bag contains only green, yellow, and red marbles. The ratio of green marbles to yellow marbles to red marbles in the bag is \(3:4:2\). If 63 of the marbles in the bag are not red, find the number of red marbles in the bag. Express your answer as a whole number. | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Let vectors $\vec{a}$, $\vec{b}$, $\vec{c}$ satisfy $\vec{a}\parallel \vec{b}$ and $\vec{a}\perp \vec{c}$. Calculate the value of $\vec{c} \cdot (\vec{a} + 2\vec{b})$. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Let $P$ be a point on the circle $(x-5)^{2}+(y-3)^{2}=9$. Calculate the maximum distance from point $P$ to the line $3x+4y-2=0$. Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinates of the vertex of the quadratic function y = -3x^2 - 6x + 5. Express your answer as an ordered pair (x, y). | (-1, 8) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the number of ways to distribute 5 volunteers into 3 different Olympic venues for reception duties, given that each venue must have at least one volunteer. Express your answer as a single integer. | 150 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Observe the following equations: Given $a+b=1$, $a^{2}+b^{2}=3$, $a^{3}+b^{3}=4$, $a^{4}+b^{4}=7$, $a^{5}+b^{5}=11$, then find the value of $a^{7}+b^{7}$ based on the pattern in the sequence. | 29 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the custom operation ⊕ defined as a ⊕ b = a when a ≥ b, and a ⊕ b = b^2 when a < b, find the maximum value of the function f(x) = (1 ⊕ x) * x - (2 ⊕ x) over the domain x ∈ [-2, 2]. Express your answer as a single number. | 6 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the solution set of the inequality $\frac{x^{2}+2x+2}{x+2} > 1$. Express your answer in interval notation. | (-2,-1)\cup(0,\infty) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the curve equation y = x^2 + ax + b, find the values of a and b such that the tangent line to the curve at the point (0, b) is given by the equation x - y + 1 = 0. Express your answer as a pair of values (a, b), including negative signs if applicable. | (-1, 1) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number $z$ satisfies $z^{2}=-4$, find the real part of the complex number $z$. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
In the twelve-tone equal temperament, 11 positive numbers are inserted between 1 and 2 to form an increasing geometric sequence with 13 numbers. Find the value of the third term of this sequence. Express your answer as a power of 2. | 2^{\frac{1}{6}} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given that July 1, 1997, was a Tuesday, determine the day of the week for June 9, 1898. Express your answer as the full day of the week (e.g., Monday, Tuesday, etc.). | Thursday | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the polynomial P(x) = 8x^3 - 4x^2 - 42x + 45, find the value of r such that P(x) can be divided by (x - r)^2. Express your answer as a decimal value, rounded to 2 decimal places. | 1.50 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given point (2,4) on the parabola y^2 = 8x, find the number of lines that intersect the parabola at exactly one point. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z = 1 - 2i, calculate the value of the expression (z^2 + 3) / (z - 1). Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
If \( x + 6y = 17 \), what is the value of \( 7x + 42y \)? Express your answer as a single integer. | 119 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Solve the system of linear inequalities $\left\{\begin{array}{l}{x-2＞1}\\{x＜4}\end{array}\right.$ and express the solution set as an interval. | (3, 4) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
A rectangle is divided into two vertical strips of equal width. The strip on the left is divided into three equal parts and the strip on the right is divided into four equal parts. Parts of the rectangle are then shaded as shown. Calculate the fraction of the original rectangle that is shaded. Express your answer as a ... | \frac{7}{12} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Find the approximate value of the product \(0.000315 \times 7928564\). Round your answer to the nearest whole number and express it in the format \(\boxed{[answer]}\). | 2400 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $$a=( \sqrt {3},1),b=(0,1),c=(k, \sqrt {3})$$, if $$\overrightarrow {a}$$+2 $$\overrightarrow {b}$$ is perpendicular to $$\overrightarrow {c}$$, calculate the value of k. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $\left(-4\right)\times \left[-\left(-\dfrac{1}{2}\right)\right]$. Provide your answer as a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that 54□9607502 is approximately 5.5 billion, what is the largest digit that can fill in □ to make the number as close to 5.5 billion as possible? Provide your answer as a single digit. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation x + 4y - xy = 0, where x and y are positive numbers, find the minimum value of x + y. Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality $\log{x} \ge \log{2} + \frac{1}{2}\log{x}$ and knowing that $x$ is positive, find the minimum value of $x$ that satisfies this inequality. | 4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
A circle M with its center on the x-axis and a radius of sqrt(5) is located on the left side of the y-axis. It is tangent to the line x + 2y = 0. Determine the equation of circle M in standard form. | (x + 5)^2 + y^2 = 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given that z and w are complex numbers with a modulus of 1, and 1 ≤ |z + w| ≤ √2, find the minimum value of |z - w|. Express your answer as a single value, including the square root symbol if necessary. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Factor the quadratic expression $a^{2}+4a-21$ from left to right. Express your answer as a product of two binomials. | (a-3)(a+7) | big_math | [
"Mathematics -> Algebra -> Algebra -> Polynomial Operations"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x=1$ is a root of the equation $x^{2}+ax+2=0$, find the other root of the equation. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Determine the minimum value of the function f(x) = sin x cos x - cos^2 x + 1/2 in the interval [0, pi/2]. Express your answer as a single numerical value. | -\frac{1}{2} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
A quadrilateral is inscribed in a circle. If angles are inscribed in the four arcs cut off by the sides of the quadrilateral, what is the sum of these inscribed angles? Express your answer in degrees as a whole number. | 180^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given a function y=f(x) defined on (0,+∞) that satisfies f(x)=[f''(x)-1]x and f(1)=0, find the minimum value of the function y=f(x). Express your answer as a single value, including the negative sign if the value is negative. | -\frac{1}{e} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A line passing through the right focus $F$ of the hyperbola $C$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) and perpendicular to the $x$-axis intersects the hyperbola $C$ and its asymptote in the first quadrant at points $A$ and $B$, respectively. If $A$ is the midpoint of $BF$, calculate the eccentricity... | \frac{2\sqrt{3}}{3} | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the derivative of the function y = (5x - 4)^3 with respect to x. Express your answer as a mathematical expression. | 15(5x - 4)^2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
If the ratio of volumes of two spheres is 8:27, calculate the ratio of their surface areas. Express your answer as a ratio in the form a:b. | 4:9 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence \(2, 2^2, \cdots, 2^{2015}\), how many numbers give the same remainder when divided by 9 and have the last digit as 2? Express your answer as a single integer. | 168 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the curve equation y = x^2 + ax + b, find the values of a and b such that the tangent line to the curve at the point (0, b) has the equation x - y + 1 = 0. Express your answer as a pair of values (a, b), including negative signs if applicable. | (-1, 1) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given $\sin (\pi-a)=-2\sin \left( \frac{\pi}{2}+a
ight)$, find the value of $\sin a\cos a$. Express your answer as a fraction, including the negative sign if applicable. | -\frac{2}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given that one solution of the equation $kx+y=5$ is $\left\{\begin{array}{l}{x=2}\\{y=1}\end{array}\right.$, calculate the value of $k$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
If $x$ is real and positive and grows beyond all bounds, find the limit of the expression $\log_3{(6x-5)}-\log_3{(2x+1)}$ as $x$ approaches infinity. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
The sequence {a_n} is a geometric sequence. Given that a_1 = 1 and a_7 = 8a_4, find the sum of the first 5 terms of the sequence {1/a_n}. Express your answer as a fraction. | 31/16 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions",
"Mathematics -> Algebra -> Sequences and Series"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
For all non-zero real numbers $x$ and $y$ such that $x-y=xy$, simplify the expression $\frac{1}{x}-\frac{1}{y}$. Express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle equation $x^{2}+y^{2}-2x-3=0$ and a point of intersection $P(0,1)$, find the equation of the line that intersects with the circle at point $P$ and forms the longest chord within the circle. Express your answer as an equation of a line in standard form (e.g., $Ax + By + C = 0$). | x+y-1=0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
In the polar coordinate system, find the distance between point A (1, π/6) and point B (3, 5π/6). Express your answer as a square root. | \sqrt{13} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
The temperature in the morning was $-2^{\circ}\mathrm{C}$, it rose by $12^{\circ}\mathrm{C}$ at noon, and then dropped by $8^{\circ}\mathrm{C}$ at midnight. What was the temperature at midnight? Express your answer in degrees Celsius ($^{\circ}\mathrm{C}$) as an integer. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
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