answer stringlengths 1 72 | messages listlengths 2 2 |
|---|---|
\begin{pmatrix}6 & -7\\ -2 & 4\end{pmatrix} | [
{
"content": "Find the adjugate of \\(\\begin{pmatrix}4 & 7\\\\ 2 & 6\\end{pmatrix}\\).",
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},
{
"content": "<think> For a \\(2\\times2\\) matrix, the adjugate is obtained by swapping the diagonal entries and changing the signs of the off-diagonal entries. </think>\nFor a \\(2\\times2\\... |
x = 2 | [
{
"content": "Solve 27^x = 9^(x+1).",
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},
{
"content": "<think> Rewrite both sides with base 3 and compare exponents. </think>\nRewrite both sides using base 3:\n\n27 = 3^3 and 9 = 3^2\n\nSo:\n\n(3^3)^x = (3^2)^(x+1)\n3^(3x) = 3^(2x+2)\n\nSet the exponents equal:\n\n3x = 2x + 2\nx = 2\... |
(3, 2) | [
{
"content": "Solve the system: 2x + 3y = 12, x - y = 1.",
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},
{
"content": "<think> Use substitution from x - y = 1 to write x = y + 1, then solve for y and x. </think>\nFrom x - y = 1, we get:\nx = y + 1\n\nSubstitute into 2x + 3y = 12:\n2(y + 1) + 3y = 12\n2y + 2 + 3y = 12\n5y = 10\... |
19.3 m | [
{
"content": "A kite string is 30 meters long and makes a 40° angle with the ground. Assuming the string is taut, how high is the kite above the ground? Round to the nearest tenth.",
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{
"content": "<think> The string is the hypotenuse and the height is opposite the 40° angle, so hei... |
3 | [
{
"content": "Solve for x: 5x + 9 = 2x + 18",
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{
"content": "<think> Move variable terms to one side and constants to the other. </think>\nSolve the equation:\n5x + 9 = 2x + 18\n\nSubtract 2x from both sides:\n3x + 9 = 18\n\nSubtract 9 from both sides:\n3x = 9\n\nDivide by 3:\nx = 3... |
5/12 | [
{
"content": "If \\(\\sin\\theta = \\frac{12}{13}\\) and \\(\\theta\\) is acute, find \\(\\cot\\theta\\).",
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{
"content": "<think> Use a 5-12-13 triangle and then compute adjacent over opposite. </think>\nSince\n\\[\n\\sin\\theta = \\frac{12}{13} = \\frac{\\text{opposite}}{\\text{hy... |
12π cm^2/s | [
{
"content": "The radius of a circle is increasing at 2 cm/s. How fast is its area increasing when r = 3 cm?",
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},
{
"content": "<think> Use A=πr^2 and differentiate with respect to time: dA/dt=2πr(dr/dt). </think>\nThe area of a circle is\n\nA = πr^2.\n\nDifferentiate both sides with ... |
4(x+1)/(x^2-4),\ x\ne -2,2 | [
{
"content": "Add and simplify: \\(\\frac{1}{x+2}+\\frac{3}{x-2}\\)",
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},
{
"content": "<think> Use the common denominator \\((x+2)(x-2)=x^2-4\\), combine numerators, and simplify. </think>\nThe least common denominator is \\((x+2)(x-2)=x^2-4\\). Rewrite each fraction:\n\\[\\frac{1}{x+... |
(x-5)(x+4) | [
{
"content": "Factor completely: x^2-x-20",
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},
{
"content": "<think> Find two integers whose product is -20 and whose sum is -1. They are -5 and 4. </think>\nTo factor x^2-x-20, find two numbers that multiply to -20 and add to -1.\n\nThe numbers are -5 and 4, so:\nx^2-x-20=(x-5)(x+4)\... |
-4 ≤ x ≤ 1 | [
{
"content": "Solve: (x - 1)(x + 4) ≤ 0",
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},
{
"content": "<think> Find the zeros and note that an upward-opening product is nonpositive between the roots. </think>\nThe factors are zero at x = -4 and x = 1. A sign analysis shows (x - 1)(x + 4) is negative between the roots and zero a... |
-1 ≤ x < 23/2 | [
{
"content": "Solve the compound inequality: -1 ≤ (2x - 3)/5 < 4",
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},
{
"content": "<think> Multiply all parts by 5, then solve the resulting double inequality. </think>\nStart with -1 ≤ (2x - 3)/5 < 4. Multiply all three parts by 5 to get -5 ≤ 2x - 3 < 20. Add 3 throughout: -2 ≤ 2x <... |
x=0, 6 | [
{
"content": "Solve the equation 2(x-3)^2 = 18.",
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"content": "<think> Divide by 2 to get (x-3)^2=9, then take square roots: x-3=±3. </think>\nSolve 2(x-3)^2 = 18.\n\nFirst divide both sides by 2:\n(x-3)^2=9\n\nTake square roots of both sides:\nx-3=\\pm 3\n\nNow solve each case:\n... |
67° | [
{
"content": "In a right triangle, one acute angle measures 23°. Find the measure of the other acute angle.",
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},
{
"content": "<think> The two acute angles in a right triangle add to 90°. So the other angle is 90° - 23° = 67°. </think>\nIn any right triangle, the two acute angles are ... |
(a+b)/(b-a),\ a\ne 0,\ b\ne 0,\ a\ne b | [
{
"content": "Simplify the complex fraction: \\(\\frac{\\frac{1}{a}+\\frac{1}{b}}{\\frac{1}{a}-\\frac{1}{b}}\\)",
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},
{
"content": "<think> Combine the top and bottom using the common denominator \\(ab\\), then divide the resulting fractions. </think>\nCombine the numerator and denomin... |
1/2 | [
{
"content": "A box contains 3 green balls, 3 yellow balls, and 4 purple balls. Three balls are drawn without replacement. What is the probability that exactly 1 of the 3 drawn balls is purple?",
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},
{
"content": "<think> Use combinations: choose 1 purple from 4 and 2 non-purple from 6... |
x = -1, 1 | [
{
"content": "Find the critical numbers of f(x) = x^3 - 3x.",
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{
"content": "<think> Critical numbers occur where f'(x)=0 or undefined. Here f'(x)=3x^2-3, which is zero at x=±1. </think>\nFirst find the derivative:\n\nf'(x) = 3x^2 - 3.\n\nCritical numbers occur where f'(x) = 0 or wh... |
x+y=7 | [
{
"content": "Find the equation of the locus of points equidistant from \\((2,1)\\) and \\((6,5)\\).",
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{
"content": "<think> The locus of points equidistant from two fixed points is the perpendicular bisector of the segment joining them. </think>\nThe locus of points equidistant fr... |
(-2, -7) | [
{
"content": "What is the reflection of the point (-7, -2) across the line y = x?",
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{
"content": "<think> Reflecting across y = x swaps the x- and y-coordinates. </think>\nWhen a point is reflected across the line y = x, its coordinates switch places.\n\nStarting with (-7, -2):\n- ... |
5/13 | [
{
"content": "A right triangle has legs of length 5 and 12. Let θ be the acute angle opposite the side of length 5. Find sin θ.",
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"content": "<think> First find the hypotenuse: √(5² + 12²) = 13. Then sin θ = opposite / hypotenuse = 5/13. </think>\nThe legs are 5 and 12, so the hy... |
5 | [
{
"content": "Evaluate the infinite geometric series \\(\\sum_{n=0}^{\\infty} \\left(\\frac{4}{5}\\right)^n\\).",
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{
"content": "<think> This is an infinite geometric series with first term 1 and ratio 4/5. </think>\nThe series is geometric with first term a = 1 and common ratio r =... |
6√2 cm | [
{
"content": "In a 45°-45°-90° triangle, one leg is 6 cm. Find the hypotenuse.",
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"content": "<think> In a 45°-45°-90° triangle, the hypotenuse equals leg × √2. So the hypotenuse is 6√2 cm. </think>\nA 45°-45°-90° triangle has side ratio\n 1 : 1 : √2.\nSo if each leg is 6 cm, the ... |
x=2 | [
{
"content": "Solve the quadratic equation 3x^2 - 12x + 12 = 0.",
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},
{
"content": "<think> Factor out 3: 3(x^2-4x+4)=3(x-2)^2, so the equation has a repeated root x=2. </think>\nSolve 3x^2 - 12x + 12 = 0.\n\nFirst factor out 3:\n3(x^2 - 4x + 4)=0\n\nNow factor the trinomial:\nx^2 - 4x... |
y = 2*3^x | [
{
"content": "Find the exponential function y = a*b^x if f(1) = 6 and f(3) = 54.",
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},
{
"content": "<think> Set up two equations from the given values, divide to find b, then solve for a. </think>\nLet y = a*b^x.\n\nFrom f(1) = 6:\n\na*b = 6\n\nFrom f(3) = 54:\n\na*b^3 = 54\n\nDivide ... |
1/2 | [
{
"content": "Evaluate the definite integral: ∫ from 0 to 1 of 1/(1 + x)^2 dx.",
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},
{
"content": "<think> Integrate (1+x)^(-2) to get -(1+x)^(-1), then evaluate from 0 to 1. </think>\nCompute\n\n∫ from 0 to 1 of 1/(1 + x)^2 dx = ∫ from 0 to 1 of (1 + x)^(-2) dx.\n\nAn antiderivative i... |
x < 5 | [
{
"content": "Solve: 3x - 5 < 10",
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},
{
"content": "<think> Add 5 to both sides and divide by 3. </think>\nStart with 3x - 5 < 10. Add 5 to both sides to get 3x < 15. Divide both sides by 3, which does not change the inequality direction, to get x < 5.\nFinal answer: \\boxed{x < 5}",
... |
20 is an outlier | [
{
"content": "Determine whether 20 is an outlier in the data set 2, 3, 4, 5, 6, 7, 20 using the 1.5×IQR rule.",
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{
"content": "<think> Find Q1 and Q3, compute the fences, and check whether 20 lies beyond them. </think>\nThe ordered data are 2, 3, 4, 5, 6, 7, 20.\nThe median is 5, so... |
log_5(12) | [
{
"content": "Solve 5^x = 12.",
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},
{
"content": "<think> Take logarithm base 5 of both sides to isolate x. </think>\nGiven\n5^x = 12,\ntake log base 5 of both sides:\nx = log_5(12).\nEquivalently, x = ln(12)/ln(5), but the exact logarithmic answer is log_5(12).\nFinal answer: \\boxed{... |
17.7 ft | [
{
"content": "A tree breaks so that the top touches the ground 10 feet from the base. If the stump left standing is 6 feet tall, what was the original height of the tree? Round to the nearest tenth.",
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},
{
"content": "<think> The broken top is the hypotenuse of a right triangle with l... |
729/2500 | [
{
"content": "A factory produces items with a 10% defect rate. If 4 items are chosen independently, what is the probability that exactly 1 is defective?",
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{
"content": "<think> Use the binomial formula: C(4,1)(0.1)^1(0.9)^3. </think>\nThis is a binomial probability with n = 4 and p... |
2x/(x^2 + 1) | [
{
"content": "Find the derivative of y = ln(x^2 + 1).",
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{
"content": "<think> Use the chain rule: derivative of ln(u) is u'/u with u=x^2+1. </think>\nLet\n\nu = x^2 + 1.\n\nThen y = ln(u), so by the chain rule,\n\ny' = u'/u.\n\nSince\n\nu' = 2x,\n\nwe get\n\ny' = 2x/(x^2 + 1).\n\nF... |
1275 | [
{
"content": "Evaluate \\(\\sum_{n=1}^{50} n\\).",
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},
{
"content": "<think> Use the formula for the sum of the first n positive integers: n(n+1)/2. </think>\nThe sum of the first n positive integers is n(n+1)/2. With n = 50, we get \\sum_{n=1}^{50} n = 50\\cdot 51/2 = 25\\cdot 51 = 12... |
\ln 2 | [
{
"content": "Evaluate the series \\(\\sum_{n=1}^{\\infty} \\frac{(-1)^{n+1}}{n}\\).",
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},
{
"content": "<think> Recognize it as the alternating harmonic series, whose sum is ln(2). </think>\nThe series 1 - 1/2 + 1/3 - 1/4 + \\cdots is the alternating harmonic series. From the Maclauri... |
(3 + sqrt(9 + 4e^2))/2 | [
{
"content": "Solve ln(x) + ln(x - 3) = 2.",
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},
{
"content": "<think> Combine the logs: ln(x(x-3)) = 2, so x(x-3) = e^2. Solving the quadratic and using x > 3 leaves the positive branch. </think>\nUse the product rule:\nln(x) + ln(x - 3) = ln(x(x - 3)).\nSo,\nln(x(x - 3)) = 2.\nExpone... |
31 | [
{
"content": "Evaluate \\(\\sum_{n=0}^{4} 2^n\\).",
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},
{
"content": "<think> Add the geometric terms directly or use the finite geometric sum formula. </think>\nCompute the terms: 2^0 + 2^1 + 2^2 + 2^3 + 2^4 = 1 + 2 + 4 + 8 + 16 = 31. Equivalently, using the geometric sum formula also... |
2 | [
{
"content": "Find the slope of the line passing through \\((2,5)\\) and \\((-4,-7)\\).",
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},
{
"content": "<think> Use the slope formula \\(m=\\frac{y_2-y_1}{x_2-x_1}\\). Both numerator and denominator are negative, so the slope is positive. </think>\nThe slope of a line through \\((x... |
7/25 | [
{
"content": "A wheelchair ramp is 25 feet long and rises 7 feet. Let \\(\\theta\\) be the angle of inclination of the ramp. Find \\(\\sin\\theta\\).",
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{
"content": "<think> Sine equals rise over ramp length. </think>\nThe ramp and ground form a right triangle where:\n- opposite si... |
7 | [
{
"content": "In a right triangle, one acute angle measures 30° and the hypotenuse is 14. Find the side opposite the 30° angle.",
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{
"content": "<think> Use sine: sin 30° = opposite / 14. Since sin 30° = 1/2, the opposite side is 14 × 1/2 = 7. </think>\nLet the opposite side be x.\n... |
23 | [
{
"content": "A sequence is defined recursively by a_1 = 2 and a_n = 2a_{n-1} + 1 for n \\ge 2. Find a_4.",
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},
{
"content": "<think> Compute terms one by one from the recurrence. </think>\nStart with a_1 = 2. Then a_2 = 2(2) + 1 = 5, a_3 = 2(5) + 1 = 11, and a_4 = 2(11) + 1 = 23.\nFin... |
440 | [
{
"content": "Evaluate \\(\\sum_{n=1}^{10} n(n+1)\\).",
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},
{
"content": "<think> Expand n(n+1) into n^2 + n and use standard summation formulas. </think>\nFirst expand the summand: n(n+1) = n^2 + n. So \\sum_{n=1}^{10} n(n+1) = \\sum_{n=1}^{10} n^2 + \\sum_{n=1}^{10} n. Using formulas... |
x-axis | [
{
"content": "On what part of the coordinate plane does the point (-3, 0) lie?",
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},
{
"content": "<think> Any point with y = 0 lies on the x-axis. </think>\nA point lies on the x-axis whenever its y-coordinate is 0.\n\nFor the point (-3, 0):\n- y = 0\n\nSo the point lies on the x-axis... |
640 | [
{
"content": "A bacteria culture starts with 80 cells and doubles every 6 hours. How many cells are there after 18 hours?",
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},
{
"content": "<think> Use the doubling model N = 80*2^(t/6) and substitute t = 18. </think>\nUse the exponential model for doubling:\n\nN = 80*2^(t/6)\n\nSubs... |
x=3, 7 | [
{
"content": "Solve the quadratic equation x^2 - 10x + 21 = 0.",
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},
{
"content": "<think> Factor: x^2 - 10x + 21 = (x-3)(x-7), so the roots are 3 and 7. </think>\nTo solve x^2 - 10x + 21 = 0, factor the quadratic.\n\nWe need two numbers that multiply to 21 and add to -10. Those number... |
cos(x) + 3x^2 | [
{
"content": "Differentiate y = sin(x) + x^3.",
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},
{
"content": "<think> Differentiate each term separately: d/dx[sin x]=cos x and d/dx[x^3]=3x^2. </think>\nDifferentiate term by term:\n\nd/dx[sin(x)] = cos(x),\n\nd/dx[x^3] = 3x^2.\n\nSo,\n\ny' = cos(x) + 3x^2.\n\nFinal answer: \\boxe... |
ab | [
{
"content": "Assume \\(a>0\\) and \\(b>0\\). Simplify \\(\\frac{a^{1/2}b^{3/2}}{a^{-1/2}b^{1/2}}\\).",
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},
{
"content": "<think> Subtract exponents when dividing powers with the same base. </think>\nUse the quotient rule for exponents:\n\\[\n\\frac{a^{1/2}b^{3/2}}{a^{-1/2}b^{1/2}}=a^{... |
2 | [
{
"content": "Evaluate the limit: lim(x→∞) (2x^2 + 1)/(x^2 - 3).",
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},
{
"content": "<think> For a rational function with equal highest powers, the limit is the ratio of leading coefficients. </think>\nConsider\n\nlim(x→∞) (2x^2 + 1)/(x^2 - 3).\n\nThe numerator and denominator have the... |
1/2 ≤ x ≤ 13 | [
{
"content": "Solve: √(2x - 1) ≤ 5",
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},
{
"content": "<think> Use the domain of the square root and then square both sides. </think>\nFor the square root to exist, 2x - 1 ≥ 0, so x ≥ 1/2. Now square both sides of √(2x - 1) ≤ 5 to get 2x - 1 ≤ 25. Then 2x ≤ 26, so x ≤ 13. Combining bot... |
[46.08, 53.92] | [
{
"content": "A random sample of size 25 from a population with known standard deviation 10 has sample mean 50. Find the 95% confidence interval for the population mean.",
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},
{
"content": "<think> Use x-bar ± z*(sigma/sqrt(n)) with z = 1.96 for 95% confidence. </think>\nUse the 95% co... |
(3y+4)^2 | [
{
"content": "Factor completely: 9y^2+24y+16",
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},
{
"content": "<think> This matches the perfect-square trinomial pattern a^2+2ab+b^2 with a=3y and b=4. </think>\nCheck whether the trinomial is a perfect square:\n9y^2=(3y)^2,\n16=4^2,\nand 24y=2(3y)(4)\n\nSo it fits a^2+2ab+b^2=(a+b)^... |
4/3 | [
{
"content": "If \\(\\cos\\theta = \\frac{3}{5}\\) and \\(\\theta\\) is acute, find \\(\\tan\\theta\\).",
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},
{
"content": "<think> Use a 3-4-5 triangle, then tangent is opposite over adjacent. </think>\nGiven\n\\[\n\\cos\\theta = \\frac{3}{5} = \\frac{\\text{adjacent}}{\\text{hypotenu... |
decay | [
{
"content": "Does the function y = 7*(0.8)^x represent exponential growth or exponential decay?",
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},
{
"content": "<think> If the base is between 0 and 1, the function is exponential decay. </think>\nIn the function y = 7*(0.8)^x, the base is 0.8.\n\nBecause 0 < 0.8 < 1, the function... |
(x-4)(x-1)(x+1) | [
{
"content": "Factor completely: x^3-4x^2-x+4",
"role": "user"
},
{
"content": "<think> Factor by grouping: group the first two terms and the last two terms, then factor the common binomial. </think>\nGroup the terms:\nx^3-4x^2-x+4=(x^3-4x^2)+(-x+4)\n\nFactor each group:\n=x^2(x-4)-1(x-4)\n\nNow fac... |
m=8 or m=-8 | [
{
"content": "Find all values of m such that the equation 2x^2 + mx + 8 = 0 has one real solution.",
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},
{
"content": "<think> One real solution means the discriminant is zero: m^2-4(2)(8)=0, so m^2-64=0 and m=±8. </think>\nFor the quadratic\n2x^2 + mx + 8 = 0\n\nto have one real solut... |
114.3 m | [
{
"content": "From the top of an 80-meter cliff, the angle of depression to a boat is 35°. How far is the boat from the base of the cliff horizontally? Round to the nearest tenth.",
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},
{
"content": "<think> The angle of depression equals the angle of elevation, so tan 35° = 80 / x. So... |
6 | [
{
"content": "Find the area of the triangle with vertices \\((0,0)\\), \\((4,0)\\), and \\((1,3)\\).",
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},
{
"content": "<think> Take the segment from \\((0,0)\\) to \\((4,0)\\) as the base. Its length is 4, and the height to \\((1,3)\\) is 3. </think>\nUse the triangle area formula:\n... |
17 | [
{
"content": "Solve for x: 3(x - 4) + 2 = 2(x + 1) + 5",
"role": "user"
},
{
"content": "<think> Expand both sides and then isolate x. </think>\nSolve the equation:\n3(x - 4) + 2 = 2(x + 1) + 5\n\nDistribute:\n3x - 12 + 2 = 2x + 2 + 5\n\nSimplify both sides:\n3x - 10 = 2x + 7\n\nSubtract 2x from bot... |
6 | [
{
"content": "Evaluate the limit: lim(x→3) (x^2 - 9)/(x - 3).",
"role": "user"
},
{
"content": "<think> Factor the numerator as (x-3)(x+3), cancel x-3, and then substitute x=3. </think>\nTo evaluate\n\nlim(x→3) (x^2 - 9)/(x - 3),\n\nfirst factor the numerator:\n\nx^2 - 9 = (x - 3)(x + 3).\n\nSo for ... |
3x^3 - 14x^2 + 9x - 4 | [
{
"content": "Expand: (3x^2 - 2x + 1)(x - 4)",
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},
{
"content": "<think> Distribute each term in the trinomial across the binomial and combine like terms. </think>\nMultiply each term in (3x^2 - 2x + 1) by each term in (x - 4):\n(3x^2 - 2x + 1)(x - 4)\n= 3x^2(x - 4) - 2x(x - 4) + 1(x -... |
y = 3*3^x - 2 | [
{
"content": "Find the exponential function of the form y = a*b^x + k with horizontal asymptote y = -2 that passes through (0, 1) and (2, 25).",
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},
{
"content": "<think> Use the asymptote to identify k, then substitute the two points to solve for a and b. </think>\nSince the horizonta... |
2870 | [
{
"content": "Evaluate \\(\\sum_{n=1}^{20} n^2\\).",
"role": "user"
},
{
"content": "<think> Use the formula for the sum of squares: n(n+1)(2n+1)/6. </think>\nUse the formula \\sum_{n=1}^{N} n^2 = N(N+1)(2N+1)/6. For N = 20, this gives \\sum_{n=1}^{20} n^2 = 20\\cdot 21\\cdot 41/6. Simplifying, 20/6... |
2 | [
{
"content": "Solve ln(x - 1) = 0.",
"role": "user"
},
{
"content": "<think> Exponentiating gives x - 1 = e^0 = 1, so x = 2. </think>\nStart with\nln(x - 1) = 0.\nExponentiate both sides using base e:\nx - 1 = e^0 = 1.\nSo,\nx = 2.\nCheck: x - 1 = 1 > 0, so the solution is valid.\nFinal answer: \\bo... |
ln(x^2/(x + 1)) | [
{
"content": "Condense 2ln(x) - ln(x + 1) into a single logarithm.",
"role": "user"
},
{
"content": "<think> Use the power rule 2ln(x) = ln(x^2), then the quotient rule to combine the logs. </think>\nUse the power rule first:\n2ln(x) = ln(x^2).\nNow apply the quotient rule:\nln(x^2) - ln(x + 1) = ln... |
Yes | [
{
"content": "On the integers, define aRb if a - b is even. Is this an equivalence relation?",
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},
{
"content": "<think> It is reflexive because a-a=0 is even, symmetric because if a-b is even then b-a is even, and transitive because sums of even numbers are even. </think>\nTo be an eq... |
\text{Domain: } x \ne 2;\ \text{vertical asymptote: } x = 2 | [
{
"content": "For f(x) = 1/(x - 2), find the domain and the vertical asymptote.",
"role": "user"
},
{
"content": "<think> The denominator cannot be zero, so x≠2. That excluded value is also the vertical asymptote. </think>\nThe function is\n\nf(x) = \\frac{1}{x - 2}\n\nThe denominator cannot be zero... |
x=\pm3 | [
{
"content": "Solve the equation x^2 = 9.",
"role": "user"
},
{
"content": "<think> Take square roots of both sides: x=±√9=±3. </think>\nSolve x^2 = 9.\n\nTake the square root of both sides:\nx=\\pm\\sqrt{9}\n\nSince \\sqrt{9}=3,\n\nx=\\pm 3\n\nSo the two solutions are x=3 and x=-3.\nFinal answer: \... |
x=2 | [
{
"content": "Solve: \\(\\frac{3}{x+1}=1\\)",
"role": "user"
},
{
"content": "<think> Multiply both sides by \\(x+1\\) and solve the linear equation. </think>\nMultiply both sides by \\(x+1\\):\n\\[3=x+1.\\]\nSubtract 1:\n\\[x=2.\\]\nThis is valid because \\(x\\neq -1\\).\nFinal answer: \\boxed{x=2}... |
(3,0) | [
{
"content": "Find the focus of the parabola \\(y^2=12x\\).",
"role": "user"
},
{
"content": "<think> Compare with the standard form \\(y^2=4ax\\). Then the focus is \\((a,0)\\). </think>\nThe standard form of a parabola opening right is\n\\[\ny^2=4ax.\n\\]\nGiven\n\\[\ny^2=12x,\n\\]\nwe compare and... |
16.9 | [
{
"content": "In a right triangle, an acute angle measures 28° and the side opposite that angle is 9. Find the adjacent side, rounded to the nearest tenth.",
"role": "user"
},
{
"content": "<think> Use tangent: tan 28° = 9 / adjacent. So adjacent = 9 / tan 28° ≈ 16.9. </think>\nLet the adjacent side... |
x > 5/2 | [
{
"content": "State the domain of f(x) = ln(2x - 5).",
"role": "user"
},
{
"content": "<think> The argument of a natural log must be positive, so 2x - 5 > 0. </think>\nFor ln(2x - 5) to be defined, its argument must be positive:\n2x - 5 > 0.\nSolve this inequality:\n2x > 5\nx > 5/2.\nSo the domain i... |
1/5 | [
{
"content": "Events A and B are independent with P(A)=0.4 and P(B)=0.5. What is P(A and B)?",
"role": "user"
},
{
"content": "<think> For independent events, P(A and B) = P(A)P(B). </think>\nBecause A and B are independent, the probability that both occur is the product of their probabilities: P(A ... |
(-4, -3) | [
{
"content": "What is the reflection of the point (-4, 3) across the x-axis?",
"role": "user"
},
{
"content": "<think> Reflecting across the x-axis keeps x the same and changes the sign of y. </think>\nWhen a point is reflected across the x-axis:\n- the x-coordinate stays the same\n- the y-coordinat... |
9.1 ft | [
{
"content": "A 10-foot ladder leans against a wall and makes a 65° angle with the ground. How high up the wall does the ladder reach? Round to the nearest tenth.",
"role": "user"
},
{
"content": "<think> The height is opposite the 65° angle, with hypotenuse 10. So height = 10 sin 65° ≈ 9.1 ft. </th... |
5s^3 + 18s^2 - 13s + 2 | [
{
"content": "Expand: (5s - 2)(s^2 + 4s - 1)",
"role": "user"
},
{
"content": "<think> Distribute each term in the binomial across the trinomial and combine like terms. </think>\nDistribute:\n(5s - 2)(s^2 + 4s - 1)\n= 5s(s^2 + 4s - 1) - 2(s^2 + 4s - 1)\n= 5s^3 + 20s^2 - 5s - 2s^2 - 8s + 2\nCombine l... |
17.9 | [
{
"content": "In a right triangle, an acute angle measures 52° and the side adjacent to the angle is 11. Find the hypotenuse, rounded to the nearest tenth.",
"role": "user"
},
{
"content": "<think> Use cosine: cos 52° = 11 / hypotenuse. Solving gives hypotenuse = 11 / cos 52° ≈ 17.9. </think>\nLet t... |
0.25 | [
{
"content": "In a frequency table, a category appears 15 times out of 60 observations. What is its relative frequency?",
"role": "user"
},
{
"content": "<think> Relative frequency is count divided by total observations. </think>\nRelative frequency = 15/60 = 1/4 = 0.25.\nFinal answer: \\boxed{0.25}... |
2 | [
{
"content": "A test score of 85 comes from a normal distribution with mean 75 and standard deviation 5. Find the z-score.",
"role": "user"
},
{
"content": "<think> Use z = (x - mu) / sigma. </think>\nUse the z-score formula:\nz = (x - mu)/sigma = (85 - 75)/5 = 10/5 = 2.\nSo the score is 2 standard ... |
12 | [
{
"content": "A substance is modeled by N = 200*(1/2)^(t/4). Find t when N = 25.",
"role": "user"
},
{
"content": "<think> Set the model equal to 25, simplify the fraction, and match powers of 1/2. </think>\nSet N = 25:\n\n25 = 200*(1/2)^(t/4)\n\nDivide both sides by 200:\n\n25/200 = (1/2)^(t/4)\n1/... |
\frac{10}{11} | [
{
"content": "Evaluate the telescoping sum \\(\\sum_{n=1}^{10} \\frac{1}{n(n+1)}\\).",
"role": "user"
},
{
"content": "<think> Rewrite each term using partial fractions so the sum telescopes. </think>\nDecompose the general term: 1/[n(n+1)] = 1/n - 1/(n+1). Then the sum becomes (1 - 1/2) + (1/2 - 1/... |
x/(2(x-1)),\ x\ne -1,1 | [
{
"content": "Divide and simplify: \\(\\frac{x}{x^2-1}\\div\\frac{2}{x+1}\\)",
"role": "user"
},
{
"content": "<think> Multiply by the reciprocal, factor the denominator, and cancel the common factor \\((x+1)\\). </think>\nRewrite division as multiplication by the reciprocal:\n\\[\\frac{x}{x^2-1}\\d... |
y = 5*3^x | [
{
"content": "Find the exponential function of the form y = a*b^x that passes through (0, 5) and (2, 45).",
"role": "user"
},
{
"content": "<think> Use x = 0 to find a, then substitute the second point to solve for b. </think>\nAssume the function has the form y = a*b^x.\n\nUsing the point (0, 5):\n... |
5 | [
{
"content": "Find the mode of the data set: 2, 4, 4, 5, 5, 5, 7.",
"role": "user"
},
{
"content": "<think> The mode is the value that occurs most often. </think>\nCount the occurrences of each value:\n2 appears once, 4 appears twice, 5 appears three times, and 7 appears once.\nThe most frequent val... |
6 | [
{
"content": "Solve for x: 2(x + 5) - 3(x - 1) = 7",
"role": "user"
},
{
"content": "<think> Distribute both terms, combine like terms, and solve for x. </think>\nSolve the equation:\n2(x + 5) - 3(x - 1) = 7\n\nDistribute:\n2x + 10 - 3x + 3 = 7\n\nCombine like terms:\n-x + 13 = 7\n\nSubtract 13 from... |
258 | [
{
"content": "Find the sum of the first 12 terms of the arithmetic sequence 5, 8, 11, ... .",
"role": "user"
},
{
"content": "<think> Find the 12th term, then use the arithmetic series sum formula. </think>\nThis arithmetic sequence has a_1 = 5 and d = 3. First find the 12th term: a_{12} = 5 + 11(3)... |
(3,2) | [
{
"content": "Find the vertex of the parabola \\(y=x^2-6x+11\\).",
"role": "user"
},
{
"content": "<think> Complete the square to rewrite the quadratic in vertex form. </think>\nStart with\n\\[\ny=x^2-6x+11.\n\\]\nComplete the square:\n\\[\ny=(x^2-6x+9)+11-9.\n\\]\nSo,\n\\[\ny=(x-3)^2+2.\n\\]\nIn ve... |
7/10 | [
{
"content": "An urn contains 3 white balls and 2 black balls. Two balls are drawn without replacement. What is the probability of drawing at least one black ball?",
"role": "user"
},
{
"content": "<think> Use the complement: subtract the probability of drawing no black balls, i.e. both white, from ... |
\begin{pmatrix}3 & -1\\ -5 & 2\end{pmatrix} | [
{
"content": "Find the inverse of \\(\\begin{pmatrix}2 & 1\\\\ 5 & 3\\end{pmatrix}\\).",
"role": "user"
},
{
"content": "<think> Use the \\(2\\times2\\) inverse formula \\(A^{-1}=\\frac1{ad-bc}\\begin{pmatrix}d&-b\\\\-c&a\\end{pmatrix}\\). </think>\nLet\n\\[\nA=\\begin{pmatrix}2 & 1\\\\ 5 & 3\\end{p... |
(x+2)/(x-2),\ x\ne -3,2 | [
{
"content": "Multiply and simplify: \\(\\frac{x^2-4}{x^2+x-6}\\cdot\\frac{x+3}{x-2}\\)",
"role": "user"
},
{
"content": "<think> Factor both quadratics, cancel common factors, and keep the original domain restrictions. </think>\nFactor:\n\\[x^2-4=(x-2)(x+2),\\qquad x^2+x-6=(x+3)(x-2).\\]\nThen\n\\[... |
20.0 | [
{
"content": "In a right triangle, an acute angle measures 37° and the hypotenuse is 25. Find the side adjacent to the angle, rounded to the nearest tenth.",
"role": "user"
},
{
"content": "<think> Use cosine: adjacent = 25 cos 37°. This gives about 19.97, which rounds to 20.0. </think>\nLet the adj... |
5 < x < 14 | [
{
"content": "Solve the compound inequality: 2 < (x + 1)/3 < 5",
"role": "user"
},
{
"content": "<think> Multiply all parts by 3, then subtract 1. </think>\nStart with 2 < (x + 1)/3 < 5. Multiply every part by 3 to get 6 < x + 1 < 15. Subtract 1 throughout, yielding 5 < x < 14.\nFinal answer: \\boxe... |
6 | [
{
"content": "A geometric sequence has a_3 = 24 and a_5 = 96. If the common ratio is positive, find a_1.",
"role": "user"
},
{
"content": "<think> Use a_n = a_1 r^{n-1}; divide a_5 by a_3 to find r, then solve for a_1. </think>\nFor a geometric sequence, a_n = a_1 r^{n-1}. We have a_3 = a_1 r^2 = 24... |
x < 2 or x > 3 | [
{
"content": "Solve: x^2 - 5x + 6 > 0",
"role": "user"
},
{
"content": "<think> Factor the quadratic and use a sign chart. </think>\nFactor the expression: x^2 - 5x + 6 = (x - 2)(x - 3). The product is positive when both factors have the same sign, which occurs for x < 2 or x > 3. The roots are not ... |
47/100 | [
{
"content": "A random integer from 1 to 100 inclusive is chosen. What is the probability that it is divisible by 3 or 5?",
"role": "user"
},
{
"content": "<think> Use inclusion-exclusion: count multiples of 3, add multiples of 5, then subtract multiples of 15. </think>\nFrom 1 to 100, there are flo... |
21/5 | [
{
"content": "Solve for x: 4(2x - 1) - 3(x + 5) = 2",
"role": "user"
},
{
"content": "<think> Distribute both terms, combine like terms, and solve. </think>\nSolve the equation:\n4(2x - 1) - 3(x + 5) = 2\n\nDistribute:\n8x - 4 - 3x - 15 = 2\n\nCombine like terms:\n5x - 19 = 2\n\nAdd 19 to both sides... |
(t-5)^2 | [
{
"content": "Factor completely: t^2-10t+25",
"role": "user"
},
{
"content": "<think> This is a perfect-square trinomial because 25=5^2 and -10t=2(t)(-5). </think>\nCheck the perfect-square pattern:\nt^2-10t+25=t^2-2(t)(5)+5^2\n\nThis matches a^2-2ab+b^2=(a-b)^2, so:\nt^2-10t+25=(t-5)^2\n\nFinal ans... |
9 | [
{
"content": "Evaluate \\(27^{2/3}\\).",
"role": "user"
},
{
"content": "<think> Take the cube root of 27 and then square it. </think>\nRewrite the exponent:\n\\[\n27^{2/3}=\\left(27^{1/3}\\right)^2.\n\\]\nThe cube root of \\(27\\) is \\(3\\), so\n\\[\n27^{2/3}=3^2=9.\n\\]\nFinal answer: \\boxed{9}"... |
x^2 | [
{
"content": "Simplify \\(x^{3/2}\\cdot x^{1/2}\\).",
"role": "user"
},
{
"content": "<think> When multiplying powers with the same base, add the exponents. </think>\nApply the product rule for exponents:\n\\[\nx^{3/2}\\cdot x^{1/2}=x^{3/2+1/2}=x^{4/2}=x^2.\n\\]\nFinal answer: \\boxed{x^2}",
"ro... |
8 | [
{
"content": "Evaluate \\(16^{3/4}\\).",
"role": "user"
},
{
"content": "<think> Take the fourth root of 16 first, then cube the result. </think>\nUse the meaning of a rational exponent:\n\\[\n16^{3/4}=\\left(16^{1/4}\\right)^3.\n\\]\nSince the fourth root of \\(16\\) is \\(2\\),\n\\[\n16^{3/4}=2^3=... |
(6, 4) | [
{
"content": "Solve the system: 0.5x + 0.5y = 5, x - y = 2.",
"role": "user"
},
{
"content": "<think> Rewrite the first equation as x + y = 10, then solve by elimination. </think>\nMultiply the first equation by 2:\n2(0.5x + 0.5y) = 2(5)\nx + y = 10\n\nNow the system is:\nx + y = 10\nx - y = 2\n\nAd... |
(-1, 5] | [
{
"content": "Solve: (x - 5)/(x + 1) ≤ 0",
"role": "user"
},
{
"content": "<think> Use the zeros of the numerator and denominator to make a sign chart. </think>\nThe numerator is zero at x = 5, and the denominator is zero at x = -1. Check the intervals determined by -1 and 5: the fraction is positiv... |
2/5 | [
{
"content": "A spinner has 5 equal sectors labeled 1 through 5. What is the probability of landing on an even number?",
"role": "user"
},
{
"content": "<think> The even labels are 2 and 4, so there are 2 favorable sectors out of 5. </think>\nThe possible outcomes are 1, 2, 3, 4, and 5, all equally ... |
-2 < x < 6 | [
{
"content": "Solve: (x - 2)^2 < 16",
"role": "user"
},
{
"content": "<think> Take square roots through the equivalent absolute value form. </think>\nRewrite (x - 2)^2 < 16 as |x - 2| < 4. This gives the compound inequality -4 < x - 2 < 4. Add 2 to every part to get -2 < x < 6.\nFinal answer: \\boxe... |
End of preview. Expand in Data Studio
High School Math Reasoning RL
Synthetic high school math reasoning data with train/test splits for SFT-style chat training.
Splits
train: 400 examplestest: 100 examples
Format
Each row contains:
messages: chat-style user and assistant turnsanswer: a separate short-answer column
The assistant message contains the reasoning inside <think> ... </think> tags followed by the worked full_answer, which ends with Final answer: \boxed{...}.
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