instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
value |
|---|---|---|---|---|---|
The functions y=a^(x+1) and y=log_a(x+1), where a>0 and a!=1, are symmetric with respect to a line. Find the equation of this symmetry line. | y=x+1 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Precalculus -> Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Let $n$ be a two-digit integer that is $k$ times the sum of its digits. If the number formed by interchanging the digits of $n$ is $x$ times the sum of its digits, find the expression for $x$ in terms of $k$. Express your answer as a simple algebraic expression. | 11-k | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation (sinθ + cosθ)/(sinθ - cosθ) = 2, find the value of sinθ cosθ. Express your answer as a simplified fraction. | \frac{3}{10} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given points A(3,2,0), B(0,4,0), and C(3,0,2), calculate a normal vector of the plane ABC in the format (x, y, z). | (2, 3, 3) | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression $2\cos 30^{\circ}-\tan 45^{\circ}$ and provide your answer as an exact value. | \sqrt{3}-1 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
In an isosceles triangle, one of the interior angles is $100^{\circ}$. Calculate the measure of the base angle of the isosceles triangle. Express your answer in degrees as a whole number. | 40 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
The diagram shows the eight vertices of an octagon connected by line segments. Jodhvir wants to write one of the integers 1, 2, 3, or 4 at each of the vertices so that the two integers at the ends of every line segment are different. He has already written three integers as shown. How many times will the integer 4 appe... | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Graph Theory"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the expression $log_{0.5}0.125 + log_{2}\left[log_{3}\left(log_{4}64\right)\right]$. Provide your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
The average temperature at the top of Mount Tai is -9°C, and the average temperature at the foot of the mountain is -1°C. Find the temperature difference between the average temperature at the foot of the mountain and the average temperature at the top of the mountain. Express your answer in degrees Celsius (°C) as a w... | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given two vectors $\overset{⇀}{a}$ and $\overset{⇀}{b}$ in a plane with an angle of $60^{\circ}$ between them, and magnitudes $|\overset{→}{a}|=4$ and $|\overset{→}{b}|=1$, find the magnitude of the vector $|\overset{⇀}{a}+2\overset{⇀}{b}|$. Express your answer as a simplified radical expression if necessary. | 2\sqrt{7} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest number among the following: 1, 0, -1, -3. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the real number $m$ is the middle term of an arithmetic sequence with first term $1$ and last term $5$, find the value of $m$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
If point $P\left(m+3,m+1\right)$ lies on the $x$-axis of the rectangular coordinate system, find the coordinates of point $P$. Express your answer as an ordered pair $(x, y)$ with integer coordinates. | (2,0) | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $U=\mathbb{R}$, let set $A=\{x|x^2+x-6>0\}$, and $B=\{y|y=2^x-1, x\leq 2\}$. Calculate the intersection $(\complement_U A) \cap B$ and express your answer as an interval, using parentheses or brackets as necessary. | (-1,2] | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the positive real numbers \(a\), \(b\), and \(c\) that satisfy:
\[ x = \frac{a}{2b + 3c}, \; y = \frac{2b}{3c + a}, \; z = \frac{3c}{a + 2b} \]
Calculate the value of the expression \(\frac{x}{1 + x} + \frac{y}{1 + y} + \frac{z}{1 + z}\). Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Ben has exactly the right number of smaller cubes, each with a side length of 5 cm, to form a larger solid cube with a side length of 1 m. He places these smaller cubes side by side to form a single row. Find the length of this row in meters. | 400 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 3,001,000 in scientific notation. Use the format $a \times 10^{b}$, where $a$ is a number between 1 and 10, and $b$ is an integer. | $3.001 \times 10^{6}$ | big_math | [
"Mathematics -> Other -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence of positive numbers $\{a_n\}$, where $a_1=1$, $a_2=2$, and $2a_n^2 = a_{n+1}^2 + a_{n-1}^2$, find the exact value of $a_6$. Provide your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Paul laid six dominoes in a line as shown in the diagram. Dominoes are said to be arranged correctly if, for each pair of adjacent dominoes, the numbers of spots on the adjacent ends are equal. He can make a move either by swapping the position of any two dominoes (without rotating either domino) or by rotating one dom... | 3 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Find the imaginary part of the complex number (1-i)^2. Express your answer as the coefficient of the imaginary part, a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
You have four cards, each with a different shape drawn on them: a parallelogram, an isosceles right triangle, a regular pentagon, and a circle. The backs of the cards are identical. If you randomly select one card from the set, what is the probability that the shape drawn on the card is a centrally symmetric figure? Ex... | \dfrac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
A foreign investor plans to invest in 3 different projects across 4 candidate cities, with no more than 2 projects in the same city. Calculate the total number of different investment plans the investor can have. | 60 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the vectors $\overrightarrow {a_{1}} = 2\overrightarrow {m} - \overrightarrow {j} + \overrightarrow {k}$, $\overrightarrow {a_{2}} = \overrightarrow {m} + 3\overrightarrow {j} - 2\overrightarrow {k}$, $\overrightarrow {a_{3}} = -2\overrightarrow {m} + \overrightarrow {j} - 3\overrightarrow {k}$, $\overrightarrow ... | (-2, 1, -3) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Solve for x in the equation |-x| = 5. Express your answer using '±' notation if x has multiple values. | \pm 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given that point M is on the parabola x²=4y, calculate the minimum distance from point M to the line y=x-3. Express your answer as a numerical value. | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
The average score of six students in a math exam is 92.5 points, and their scores are distinct integers with the highest being 99 points and the lowest being 76 points. What is the minimum score of the student who ranks third from the top? Express your answer as an integer. | 95 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
If $a,b,c$ are positive integers less than $10$, find the condition for the equation $(10a + b)(10a + c) = 100a(a + 1) + bc$ to hold. Express your answer as an equation or condition using standard algebraic notation. | b+c=10 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \begin{cases} -x^{2}-ax-5, & (x\leqslant 1) \\ \frac {a}{x}, & (x > 1)\end{cases}$ is an increasing function on $(-\infty,+\infty)$, find the range of values for $a$. Express your answer as an interval, e.g., [lower, upper]. | [-3,-2] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given point A(-1,2) in the Cartesian coordinate system, find the coordinates of point B that is symmetrical to point A with respect to the y-axis. Express your answer as a coordinate pair (x, y). | (1,2) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
A factory has three workshops: Workshop A, Workshop B, and Workshop C, which have produced the same type of product in quantities of 120, 80, and 60 pieces, respectively. To determine if there is a significant difference in the quality of their products, a stratified sampling method is used to withdraw a sample of size... | 13 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Barry has 12 coins, including at least one of each of the 5 types of Canadian coins: $2.00, 1.00, 0.25, 0.10, and 0.05 dollars. Calculate the smallest total amount of money that Barry could have. Express your answer in dollars and cents, including the dollar sign and decimal point. | \$3.75 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Estimate the value of $\sqrt{30}$ by comparing it to the square roots of perfect squares close to 30. Express your answer as an inequality or a range, including the specific bounds or values. | 5 < \sqrt{30} < 6 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, if $a\cos A = b\sin B$, calculate the value of $\sin A\cos A + \cos^2B$. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Two individuals, A and B, independently select three courses each from a pool of six elective courses. Let X represent the number of courses that both individuals have in common. Calculate the expected value E(X). Express your answer as a single numerical value. | 1.5 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = |x+1| + |x-a|, where it is known that its graph is symmetric about the line x = 1, find the value of a that satisfies this symmetry condition. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given $f(x) = x^5 + ax^3 + bx - 2$ and $f(-2) = m$, calculate the value of $f(2) + f(-2)$. Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Point A has coordinates (1,-1). After moving point A up by 2 units and then left by 3 units, we get point B. What are the coordinates of point B? Express your answer as an ordered pair (x, y). | (-2, 1) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\alpha$ and $\beta$ are two roots of the equation $x^{2}+2017x+1=0$, find the value of $(1+2020\alpha +\alpha ^{2})(1+2020\beta +\beta ^{2})$. Express your answer as a single numerical value. | 9 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Polynomial Operations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
If the area of the unfolded surface of a cone is $15\pi$, and the length of the generatrix is $5$, calculate the radius of the base of this cone. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given the lines $l_{1}$: $2x+2y-5=0$ and $l_{2}$: $4x+ny+1=0$, if $l_{1}$ is parallel to $l_{2}$, find the value of $n$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Let $a$ be a positive odd number. Find the smallest positive integer that $a^2 - 1$ must be a multiple of. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
The function y = (x - 1)^2 is shifted 3 units to the left. Find the resulting function. Express your answer as a function of x, using the format y = f(x). | y = (x + 2)^2 | big_math | [
"Mathematics -> Precalculus -> Functions -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $$\frac {a+i}{i}=1+bi$$, where $a$, $b$ are real numbers, and $i$ is the imaginary unit, find the value of $a+b$. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, the graph of the linear function y = mx - 1 is shifted down by 2 units. The new line passes through the point (-2, 1). Calculate the value of m. Express your answer as a single integer or fraction. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Functions"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the curve y = x^3 and the tangent line equation y = kx + 2, find the value of k. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z$ that satisfies the equation $z \cdot i = 1 + i$, find the conjugate of $z$ ($ar{z}$). Express your answer as a complex number in the form a + bi. | 1 + i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
The production costs of a PVC industry enterprise increased by 20% each month in August and September. After the fall in international oil prices, the production costs in October and November decreased by 20% each month. Calculate the percentage change in production cost from the beginning of August to the end of Novem... | -8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Find the constant term in the expansion of $((x^{2}+3)(\frac{1}{x^{2}}-1)^{5})$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$. Given that $a_2=3$ and $a_6=11$, find the value of $S_7$. Express your answer as a single numerical value. | 49 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
In the adjoining figure, triangle $ABC$ is such that $AB = 4$ and $AC = 8$. If $M$ is the midpoint of $BC$ and $AM = 3$, what is the length of $BC$? Express your answer as a numerical value with a radical if necessary. | 2\sqrt{31} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangulations"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Joan buys 3 pounds of bologna for $2.80 per pound, and Grant buys 2 pounds of pastrami for $1.80 per pound. What is the average (arithmetic mean) price per pound of all the lunchmeat they buy? Express your answer in dollars per pound, rounded to two decimal places. | 2.40 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Three real numbers \(a\), \(b\), and \(c\) have a sum of 114 and a product of 46656. If \(b = ar\) and \(c = ar^2\) for some real number \(r\), calculate the value of \(a + c\). Express your answer as a single integer. | 78 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Arrange the following statements in the correct sequence according to the pattern of syllogism:
① $Z_1$, $Z_2$ cannot be compared in size; ② $Z_1$, $Z_2$ are imaginary numbers; ③ Imaginary numbers cannot be compared in size.
Express your answer as a sequence of numbers corresponding to the statements in the correct or... | ③②① | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Three rugs have a combined area of \( 90 \, \text{m}^2 \). When they are laid down to cover completely a floor of area \( 60 \, \text{m}^2 \), the area which is covered by exactly two layers of rug is \( 12 \, \text{m}^2 \). Find the area of the floor covered by exactly three layers of rug. Express your answer in squar... | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n}, if a_2 = 10 and a_5 = 1, find the sum of the first 7 terms of {a_n}. Express your answer as a single integer. | 28 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given two unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ on a plane with $\overrightarrow{a} \cdot \overrightarrow{b} = \frac{3}{5}$, determine the minimum value of $|\overrightarrow{a} + m\overrightarrow{b}|$ for $m \in \mathbb{R}$. Express your answer as a fraction. | \frac{4}{5} | big_math | [
"Mathematics -> Geometry -> Other",
"Mathematics -> Algebra -> Other",
"Mathematics -> Applied Mathematics -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
What is the maximum decimal number that can be represented by a four-digit binary number? Express your answer as a decimal number. | 15 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.984375 | 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"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the equation $\sqrt{(x-2)^2+y^2}+\sqrt{(x+2)^2+y^2}=10$ into the standard form of an ellipse. | \frac{x^2}{25}+\frac{y^2}{21}=1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors a = (-1, -1, 1) and b = (-1, 0, 1), calculate the cosine of the angle between vectors a and b using the formula cos < a, b > = (a · b) / (|a| · |b|). Express your answer as a fraction or decimal value. | \frac{\sqrt{6}}{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
The line with equation y = 2x - 6 is reflected in the y-axis. Find the x-intercept of the resulting line. Express your answer as a single number. | -3 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the directrix of the parabola $y^2 = 8x$. Express your answer as an equation of a line (e.g., x = a or y = b). | x = -2 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = A sin(ωx + φ) where A, φ are positive real numbers, find the number of possible values for the positive integer ω such that the smallest positive period T of f(x) is within the interval (1/100, 1/50). Express your answer as a single integer. | 314 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given a quadratic function y=x^{2}-4x+n, where n is a constant, and two points A(x_{1}, y_{1}) and B(x_{2}, y_{2}) on the parabola, if y_{1} > y_{2}, what is the relationship between x_{1} and x_{2} in terms of their distances from the axis of symmetry x=2? Express your answer as an inequality involving absolute values... | |x_{1}-2| > |x_{2}-2| | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions",
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
For the algebraic expression $\frac{x}{x-4}$ to be meaningful, what is the condition on the real number $x$? Express your answer as an inequality or equation using mathematical notation (e.g. $x \neq 4$). | x \neq 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
How many distinct triangles can be formed using three of the dots in the 2x3 grid below as vertices? Exclude any degenerate cases where the three points are collinear.
[asy]dot(origin^^(1,0)^^(2,0)^^(0,1)^^(1,1)^^(2,1));[/asy]
Express your answer as a single integer. | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse C: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{4}=1(a＞2)$$ and the line l: y=x-2 passes through one focus of C, calculate the eccentricity of C. Express your answer as a fraction or decimal, including any necessary square roots. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
One side length of a triangle is 21, the perimeter is 48, and both side lengths and area are integers. Calculate the length of the shortest side. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive integer n that satisfies the inequality \\( \\sqrt{n} - \\sqrt{n-1} < 0.01 \\). Express your answer as a single integer. | 2501 | big_math | [
"Mathematics -> Algebra -> Inequalities"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that there are 2 defective products in 6 items, calculate the probability of selecting exactly 1 defective product when taking 2 items at random. Express your answer as a simplified fraction. | \dfrac{8}{15} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line that passes through point M(1,1) and has a slope angle twice that of the line $2x-y=0$. Express your answer in the standard form of a line equation (Ax + By + C = 0), including all necessary coefficients and constants. | 4x+3y-7=0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given complex numbers $Z_{1}=a+2i$ and $Z_{2}=-2+i$, if $|Z_{1}| < |Z_{2}|$, then find the range of the real number $a$. Express your answer as an interval or inequality. | (-1, 1) | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality (x+m)/2 - 1 > 2m with respect to x, and the solution set x > 5, calculate the value of m. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the largest integer not exceeding $700\pi$, where $\pi$ is the constant pi. Round down to the nearest integer. | 2199 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Anna and Bonnie are racing on a 400-meter elliptical track. They start at the same time, but Anna is 25% faster than Bonnie. Calculate the number of laps it takes for Anna to first overtake Bonnie. Express your answer as a numerical value. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, a_3 = 16 and a_5 = 4. Find the value of a_7. | 1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac{2i}{1+i}$, find the conjugate of $z$. Express your answer as a complex number in the form $a+bi$, where $a$ and $b$ are real numbers. | 1-i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Express the product $89\times 90\times 91\times 92\times \ldots \times 100$ in terms of permutations in the form A_{n}^{r}. | A_{100}^{12} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A chord of a circle has a length equal to the radius of the circle. Find the measure of the central angle corresponding to this chord in radians. | \frac{\pi}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Jack and Jenny start hiking from the trailhead to the fire tower at the top of the mountain. Jenny, carrying a very heavy backpack, walks very slowly. Jack starts at a speed of 4 mph and, upon reaching halfway, where the trail becomes steep, his speed decreases to 2 mph. After reaching the fire tower, Jack immediately ... | \frac{12}{13} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given $a=\log_2 3$, $b=4^{ \frac {1}{3}}$, $c=\log_2 0.3$, determine the ordering of these three numbers from smallest to largest. Express your answer using the variables a, b, and c (e.g., c < a < b). | c < a < b | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the diagonals of parallelogram ABCD intersect at point O, where E is the midpoint of AO, if AE = λAB + μAD, calculate the sum of λ and μ. Express your answer as a single fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
What is the coefficient of the algebraic expression $-8\pi a^{2}b^{3}$? Express your answer as a numerical value, including pi if applicable, and include a negative sign if the coefficient is negative. | -8\pi | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
In an equilateral triangle ABC with a side length of 1, calculate the value of $\overrightarrow {AB}$•$\overrightarrow {AC}$. Express your answer as a single number in the form $\boxed{[answer]}$. | \frac {1}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations of two lines are $x+y+a=0$ and $x+y+b=0$, respectively, where $a$ and $b$ are the two real roots of the equation $x^{2}+x+c=0$, and $0 \leqslant c \leqslant \frac{1}{8}$, find the maximum distance between these two lines. Express your answer as a numerical value. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other",
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $C$: $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ has one of its foci at $(4,0)$, and the two asymptotes of the hyperbola are perpendicular to each other, find the equation of this hyperbola. Express your answer in standard form. | \dfrac{x^{2}}{8} - \dfrac{y^{2}}{8} = 1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the definitions of sets I, M, N, and P, and the four statements regarding these sets:
(1) M ∪ N ∪ P = I;
(2) N ≠ ∅;
(3) M ≠ ∅;
(4) P ≠ ∅.
Determine the number of correct statements. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinates of the vertex of the quadratic function y = (x - 3)^2 + 1. Express your answer as an ordered pair (h, k). | (3, 1) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of the real number m in the equation x^3 - 6x^2 + 9x + m = 0, given that it has exactly three distinct real roots. Express your answer as an interval. | (-4, 0) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that m = √5 - 1, estimate the range of values for m in the format a < m < b, where a and b are specific numerical values. | 1 < m < 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Find the angle of inclination in degrees for the line y = √3x. Express your answer as a whole number. | 60 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the derivative of the function f(x) at x=1 is 6, calculate the value of:
\[\lim_{Δx \to 0} \frac{f(1+Δx) - f(1)}{3Δx}.\]
Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Kai will celebrate his 25th birthday in March 2020. In what year was Kai born? Express your answer as a 4-digit year. | 1995 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
The equation \( y = ax^2 + bx + c \) was used to create the table of values below:
\[
\begin{array}{c|c|c}
x & 0 & 1 \\
\hline
y & 8 & 9
\end{array}
\]
Find the value of \( a + b \). Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence ${a_n}$, if $a_6 + a_{10} = 16$ and $a_4 = 1$, find the value of $a_{12}$. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
In an isosceles triangle, the lengths of two sides are given as $3cm$ and $7cm$. Calculate the perimeter of this triangle. Express your answer in centimeters (cm) as a whole number. | 17 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given the two circles with equations $x^{2}+y^{2}-4x+6y=0$ and $x^{2}+y^{2}-6x=0$, find the equation of their radical axis. Express your answer in the form Ax + By + C = 0. | 3x-y-9=0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the operation $a\otimes b=ab^{2}-b$ and the equation $1\otimes x=k$, find the range of values for $k$ that allows the equation to have two distinct real roots with respect to $x$. Express your answer as an inequality or range of values for $k$. | k > -\frac{1}{4} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $3(4x + 5\pi) = P$, simplify the expression $6(8x + 10\pi)$ in terms of $P$. Express your answer in the form $\boxed{kP}$, where $k$ is an integer. | 4P | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\lim_{\Delta x \to 0} \frac {f(x_{0}+2\Delta x)-f(x_{0})}{\Delta x} =1$, calculate the value of $f'(x_0)$ using the definition of a derivative. Express your answer as a single numerical value in the form $\boxed{[answer]}$. | \frac{1}{2} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
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