answer stringlengths 1 54 | messages listlengths 2 2 |
|---|---|
-10 | [
{
"content": "If det(A) = -2 for a 3x3 matrix, what is the determinant after multiplying one row of A by 5?",
"role": "user"
},
{
"content": "<think> Multiplying one row by 5 multiplies the determinant by 5. </think>\nWhen one row of a matrix is multiplied by 5, the determinant is also multiplied by... |
x = -5/2, 5/2 | [
{
"content": "Solve the quadratic equation: 4x^2 - 25 = 0",
"role": "user"
},
{
"content": "<think> Use the difference of squares formula and solve each linear factor. </think>\nRecognize a difference of squares:\n4x^2 - 25 = (2x - 5)(2x + 5)\n\nSet each factor equal to zero:\n2x - 5 = 0 gives x = 5... |
x = -1, -3/4 | [
{
"content": "Solve the quadratic equation: 4x^2 + 7x + 3 = 0",
"role": "user"
},
{
"content": "<think> Factor the quadratic and solve each linear factor. </think>\nFactor the quadratic:\n4x^2 + 7x + 3 = (4x + 3)(x + 1)\n\nSet each factor equal to zero:\n4x + 3 = 0 gives x = -3/4\nx + 1 = 0 gives x ... |
2x^2 - 1 | [
{
"content": "Divide: (4x^4 - 1) / (2x^2 + 1)",
"role": "user"
},
{
"content": "<think> Recognize the numerator as a difference of squares in terms of 2x^2. </think>\nFactor the numerator: 4x^4 - 1 = (2x^2 - 1)(2x^2 + 1). Dividing by 2x^2 + 1 leaves 2x^2 - 1. Final answer: \\boxed{2x^2 - 1}",
"r... |
(-∞, -3) ∪ (3, ∞) | [
{
"content": "Find the domain of g(x) = ln(x^2 - 9).",
"role": "user"
},
{
"content": "<think> The argument of ln must be positive, so solve x^2 - 9 > 0. </think>\nFor g(x) = ln(x^2 - 9), we need x^2 - 9 > 0. Factor: (x - 3)(x + 3) > 0. This inequality is true when x < -3 or x > 3. So the domain is ... |
-3 < x < -1 | [
{
"content": "Solve the inequality: x^2 + 4x + 3 < 0",
"role": "user"
},
{
"content": "<think> Factor the quadratic and take the interval between the roots. </think>\nSolve:\n\\[x^2+4x+3<0\\]\nFactor:\n\\[(x+1)(x+3)<0\\]\nThe roots are \\(-3\\) and \\(-1\\).\nA quadratic with positive leading coeffi... |
-5 \le x \le 1 | [
{
"content": "Solve the inequality: |x + 2| \\le 3",
"role": "user"
},
{
"content": "<think> Rewrite the absolute value inequality as a compound inequality. </think>\nSolve:\n\\[|x+2|\\le 3\\]\nThis means:\n\\[-3\\le x+2\\le 3\\]\nSubtract 2 from all parts:\n\\[-5\\le x\\le 1\\]\nFinal answer: \\box... |
x = 7 | [
{
"content": "Solve for x: (x + 2)/3 + (x - 1)/6 = 4",
"role": "user"
},
{
"content": "<think> Multiply through by 6 to clear denominators, then solve. </think>\nSolve the equation:\n(x + 2)/3 + (x - 1)/6 = 4\n\nMultiply every term by 6:\n2(x + 2) + (x - 1) = 24\n\nDistribute:\n2x + 4 + x - 1 = 24\n... |
x = 1 | [
{
"content": "Solve the quadratic equation: 7x^2 - 14x + 7 = 0",
"role": "user"
},
{
"content": "<think> Factor out 7 and recognize the remaining quadratic as a perfect square. </think>\nFactor out 7:\n7(x^2 - 2x + 1) = 0\n\nNow factor the trinomial:\n7(x - 1)^2 = 0\n\nSo x - 1 = 0, which gives x = ... |
(3x+1)(x-5) | [
{
"content": "Factor completely: 3x^2 - 14x - 5",
"role": "user"
},
{
"content": "<think> Use the AC method: 3·(-5)=-15, and 1 and -15 add to -14. </think>\nCompute AC: 3·(-5) = -15.\n\nWe need two numbers that multiply to -15 and add to -14: 1 and -15.\n\nRewrite the middle term:\n3x^2 - 14x - 5 = ... |
(-2, 6) | [
{
"content": "What is the reflection of the point (2, -6) across the origin?",
"role": "user"
},
{
"content": "<think> Reflecting across the origin changes both coordinates to their opposites. </think>\nReflecting a point across the origin changes (x, y) to (-x, -y). Applying that rule to (2, -6) gi... |
x = 12 | [
{
"content": "Solve for x: 7 - x/2 = 1",
"role": "user"
},
{
"content": "<think> Subtract 7 from both sides, then multiply by -2. </think>\nSolve the equation:\n7 - x/2 = 1\n\nSubtract 7 from both sides:\n-x/2 = -6\n\nMultiply both sides by -2:\nx = 12\n\nFinal answer: \\boxed{x = 12}",
"role": ... |
(2x-5y)(2x+5y) | [
{
"content": "Factor completely: 4x^2 - 25y^2",
"role": "user"
},
{
"content": "<think> This is a difference of squares: (2x)^2-(5y)^2. </think>\nRecognize a difference of squares:\n\n4x^2 - 25y^2 = (2x)^2 - (5y)^2\n\nUse a^2-b^2=(a-b)(a+b):\n\n4x^2 - 25y^2 = (2x-5y)(2x+5y)\n\nFinal answer: \\boxed{... |
No | [
{
"content": "Is f(x) = |x| one-to-one on ℝ?",
"role": "user"
},
{
"content": "<think> Different inputs can give the same output, such as 1 and -1. </think>\nA function is one-to-one if no two different inputs have the same output. But |1| = 1 and |-1| = 1, even though 1 ≠ -1. Therefore f(x) = |x| i... |
3/5 | [
{
"content": "A random integer from 1 to 100 is chosen. What is the probability that it is divisible by 2 or by 5?",
"role": "user"
},
{
"content": "<think> Use inclusion-exclusion on the counts of multiples of 2 and 5. </think>\nFrom 1 to 100:\n- Multiples of 2: \\(\\lfloor 100/2\\rfloor=50\\)\n- M... |
(21/5,4/5) | [
{
"content": "Solve the system: x/3 - y/2 = 1, x + y = 5.",
"role": "user"
},
{
"content": "<think> Clear fractions and then substitute using the second equation. </think>\nMultiply x/3 - y/2 = 1 by 6 to get 2x - 3y = 6. From x + y = 5, we get x = 5 - y. Substitute into 2x - 3y = 6: 2(5 - y) - 3y = ... |
0.31 | [
{
"content": "The probability it rains on a given day is 0.3. A person carries an umbrella with probability 0.8 when it rains and 0.1 when it does not rain. What is the probability the person carries an umbrella?",
"role": "user"
},
{
"content": "<think> Use the law of total probability over the cas... |
0.7 | [
{
"content": "Events A and B are mutually exclusive with P(A)=0.4 and P(B)=0.3. What is P(A or B)?",
"role": "user"
},
{
"content": "<think> For mutually exclusive events, add the probabilities directly. </think>\nIf events \\(A\\) and \\(B\\) are mutually exclusive, then\n\\[\nP(A\\cup B)=P(A)+P(B)... |
x=4 | [
{
"content": "Solve 7^(x-2)=49.",
"role": "user"
},
{
"content": "<think> Rewrite 49 as 7^2 and match exponents. </think>\nSince 49=7^2, the equation becomes 7^(x-2)=7^2. Therefore x-2=2, so x=4.\nFinal answer: \\boxed{x=4}",
"role": "assistant"
}
] |
2x + 2 | [
{
"content": "If f(x) = x + 1 and g(x) = 2x, find (g ∘ f)(x).",
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},
{
"content": "<think> Substitute f(x) into g. </think>\nBy definition, (g ∘ f)(x) = g(f(x)). Since f(x) = x + 1, we get g(f(x)) = g(x + 1) = 2(x + 1) = 2x + 2. Final answer: \\boxed{2x + 2}",
"role": "assistant"
... |
x < -2 or x > 1 | [
{
"content": "Solve the inequality: \\frac{x - 1}{x + 2} > 0",
"role": "user"
},
{
"content": "<think> Find where numerator and denominator have the same sign. </think>\nSolve:\n\\[\\frac{x-1}{x+2}>0\\]\nCritical values are \\(x=1\\) and \\(x=-2\\). Note that \\(x\\neq -2\\).\nCheck intervals:\n- Fo... |
[[1,4],[2,5],[3,6]] | [
{
"content": "Find the transpose of the matrix [[1,2,3],[4,5,6]].",
"role": "user"
},
{
"content": "<think> The transpose swaps rows and columns. </think>\nTo transpose a matrix, interchange rows and columns. So the first row [1,2,3] becomes the first column, and the second row [4,5,6] becomes the s... |
x=5/2 | [
{
"content": "Solve 2^(2x)=32.",
"role": "user"
},
{
"content": "<think> Rewrite 32 as a power of 2 and match exponents. </think>\nSince 32=2^5, the equation becomes 2^(2x)=2^5. Equating exponents gives 2x=5, so x=5/2.\nFinal answer: \\boxed{x=5/2}",
"role": "assistant"
}
] |
x = -1/2 | [
{
"content": "Solve the quadratic equation: 4x^2 + 4x + 1 = 0",
"role": "user"
},
{
"content": "<think> Recognize the quadratic as a perfect square trinomial. </think>\nFactor the quadratic:\n4x^2 + 4x + 1 = (2x + 1)^2\n\nSo (2x + 1)^2 = 0.\nThen 2x + 1 = 0, so x = -1/2.\n\nFinal answer: \\boxed{x =... |
25 | [
{
"content": "Solve √x + 1 = 6.",
"role": "user"
},
{
"content": "<think> Isolate the radical, then square both sides. </think>\nSubtract 1 from both sides: √x = 5. Now square both sides to get x = 25. Check: √25 + 1 = 5 + 1 = 6.\nFinal answer: \\boxed{25}",
"role": "assistant"
}
] |
x^2 + 2x - 3 | [
{
"content": "Divide: (x^3 + 4x^2 + x - 6) / (x + 2)",
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},
{
"content": "<think> Use synthetic division with -2. </think>\nUse synthetic division with -2 on coefficients 1, 4, 1, -6. Bring down 1; multiply by -2 to get -2 and add to 4 to get 2; multiply by -2 to get -4 and add to 1 to ... |
112 | [
{
"content": "Evaluate Σ from k=1 to 6 of k(k+1).",
"role": "user"
},
{
"content": "<think> Expand k(k+1) as k^2 + k, then use known formulas for Σk and Σk^2. </think>\nExpand the summand:\nk(k+1) = k^2 + k.\nSo\nΣ from k=1 to 6 of k(k+1) = Σ from k=1 to 6 of k^2 + Σ from k=1 to 6 of k.\nNow,\nΣ k^2... |
x = 4 | [
{
"content": "Solve for x: 15 - (2x + 1)/3 = 12",
"role": "user"
},
{
"content": "<think> Subtract 15 from both sides or move the fraction term first, then solve. </think>\nSolve the equation:\n15 - (2x + 1)/3 = 12\n\nSubtract 12 from 15:\n15 - 12 = (2x + 1)/3\n\nSo:\n3 = (2x + 1)/3\n\nMultiply both... |
y = -1/2 x + 2 | [
{
"content": "Find an equation of the line with x-intercept 4 and y-intercept 2.",
"role": "user"
},
{
"content": "<think> The line passes through (4, 0) and (0, 2), so find the slope and write the equation. </think>\nThe intercepts give two points: (4, 0) and (0, 2). The slope is m = (0 - 2)/(4 - 0... |
(x+y)^2 | [
{
"content": "Factor completely: x^2 + 2xy + y^2",
"role": "user"
},
{
"content": "<think> This matches the perfect square pattern a^2+2ab+b^2. </think>\nNotice that\n\nx^2 + 2xy + y^2 = x^2 + 2(x)(y) + y^2\n\nThis is exactly the pattern a^2+2ab+b^2=(a+b)^2.\n\nSo,\nx^2 + 2xy + y^2 = (x+y)^2\n\nFina... |
0.8 | [
{
"content": "Using the same survey data, what is P(coffee | female)?",
"role": "user"
},
{
"content": "<think> Conditional probability within females is female coffee count divided by total females. </think>\nAmong females, 10 prefer tea and 40 prefer coffee, so there are 50 females total. The cond... |
(2x+3)^2 | [
{
"content": "Factor completely: 4x^2 + 12x + 9",
"role": "user"
},
{
"content": "<think> This is a perfect square trinomial because 4x^2=(2x)^2, 9=3^2, and the middle term is 2(2x)(3). </think>\nCheck whether the trinomial is a perfect square:\n\n4x^2 = (2x)^2, 9 = 3^2, and 12x = 2(2x)(3)\n\nSo it ... |
6\sqrt{2} | [
{
"content": "A 45-45-90 triangle has a leg of length 6. Find the hypotenuse.",
"role": "user"
},
{
"content": "<think> In a 45-45-90 triangle, hypotenuse = leg × sqrt(2). </think>\nFor a 45-45-90 triangle, the side ratios are 1 : 1 : sqrt(2). With leg 6, the hypotenuse is 6sqrt(2). Final answer: \\... |
(0, 0) | [
{
"content": "What are the coordinates of the origin?",
"role": "user"
},
{
"content": "<think> The origin is where the x-axis and y-axis intersect. </think>\nThe origin is the point where the horizontal x-axis and vertical y-axis cross. That point has coordinates (0, 0).\nFinal answer: \\boxed{(0, ... |
103 | [
{
"content": "Find the 25th term of the arithmetic sequence 7, 11, 15, 19, ...",
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},
{
"content": "<think> This is an arithmetic sequence with first term 7 and common difference 4, so use a_n = a_1 + (n-1)d. </think>\nThe sequence is arithmetic with a_1 = 7 and d = 4.\nUsing the nth-te... |
1/3 | [
{
"content": "A random digit from 0 to 9 is chosen. Given that it is not prime, what is the probability that it is odd?",
"role": "user"
},
{
"content": "<think> Among the non-prime digits, count how many are odd. </think>\nThe prime digits are \\(2,3,5,7\\).\nSo the non-prime digits are\n\\[\n0,1,4... |
arctan(3) | [
{
"content": "Find the acute angle between the lines y=2x+1 and y=-x+4.",
"role": "user"
},
{
"content": "<think> Using tanθ=|(m2-m1)/(1+m1m2)| with m1=2 and m2=-1 gives tanθ=3, so θ=arctan(3). </think>\nThe slopes are m1 = 2 and m2 = -1. The angle θ between two lines satisfies tanθ = |(m2 - m1)/(1 ... |
x(x+4)^2 | [
{
"content": "Factor completely: x^3 + 8x^2 + 16x",
"role": "user"
},
{
"content": "<think> Factor out x, then recognize the remaining trinomial as a perfect square. </think>\nFirst factor out the common factor x:\n\nx^3 + 8x^2 + 16x = x(x^2+8x+16)\n\nNow factor the trinomial:\n\nx^2+8x+16 = (x+4)^2... |
x = 2, 3 | [
{
"content": "Solve the quadratic equation: x^2 - 5x + 6 = 0",
"role": "user"
},
{
"content": "<think> Factor the trinomial into two binomials and set each factor equal to zero. </think>\nFactor the quadratic:\n(x - 2)(x - 3) = 0\n\nBy the zero-product property, either x - 2 = 0 or x - 3 = 0.\nSo x ... |
69184/270725 | [
{
"content": "Four cards are drawn without replacement from a standard 52-card deck. What is the probability that exactly one of them is an ace?",
"role": "user"
},
{
"content": "<think> Choose 1 ace from 4 and 3 non-aces from 48, then divide by the total number of 4-card hands. </think>\nTo get exa... |
\text{Quadrant IV} | [
{
"content": "In which quadrant is the point (6, -3)?",
"role": "user"
},
{
"content": "<think> A positive x-coordinate and negative y-coordinate place a point in Quadrant IV. </think>\nFor the point (6, -3), the x-coordinate is positive and the y-coordinate is negative. Points with x > 0 and y < 0 ... |
x = (-1 ± √5)/2 | [
{
"content": "Solve the quadratic equation: x^2 + x - 1 = 0",
"role": "user"
},
{
"content": "<think> This quadratic does not factor over the integers, so use the quadratic formula. </think>\nUse the quadratic formula with a = 1, b = 1, c = -1.\n\nx = [-1 ± √(1^2 - 4(1)(-1))]/2\n= [-1 ± √(1 + 4)]/2\... |
[[4,-1],[-3,-2]] | [
{
"content": "Find X if A + X = B, where A = [[1,2],[3,4]] and B = [[5,1],[0,2]].",
"role": "user"
},
{
"content": "<think> Rearrange to X=B−A and subtract entrywise. </think>\nFrom A + X = B, we have X = B - A. So\nX = [[5,1],[0,2]] - [[1,2],[3,4]] = [[5-1,1-2],[0-3,2-4]] = [[4,-1],[-3,-2]].\nFinal... |
x \ge 4 | [
{
"content": "Solve the inequality: 6x - 4 \\ge 2x + 12",
"role": "user"
},
{
"content": "<think> Collect x-terms on one side and constants on the other. </think>\nSolve:\n\\[6x-4\\ge 2x+12\\]\nSubtract \\(2x\\):\n\\[4x-4\\ge 12\\]\nAdd 4:\n\\[4x\\ge 16\\]\nDivide by 4:\n\\[x\\ge 4\\]\nFinal answer:... |
5x^3 + 2x^2 + 5x - 4 | [
{
"content": "Simplify: (3x^2 - 2x + 4) + (5x^3 + x - 7) - (x^2 - 6x + 1)",
"role": "user"
},
{
"content": "<think> Distribute the negative sign to the last polynomial, then combine like terms. </think>\nRewrite the expression: 3x^2 - 2x + 4 + 5x^3 + x - 7 - x^2 + 6x - 1. Combine like terms: 5x^3 + ... |
5(x-2)^2 | [
{
"content": "Factor completely: 5x^2 - 20x + 20",
"role": "user"
},
{
"content": "<think> Factor out 5, then the trinomial is a perfect square. </think>\nFirst factor out 5:\n\n5x^2 - 20x + 20 = 5(x^2-4x+4)\n\nNow factor the trinomial inside:\n\nx^2-4x+4 = (x-2)^2\n\nTherefore,\n5x^2 - 20x + 20 = 5... |
-64, 64 | [
{
"content": "Solve x^(2/3) = 16 for real x.",
"role": "user"
},
{
"content": "<think> Let y = x^(1/3). Then y^2 = 16, so y = ±4 and x = y^3. </think>\nLet y = x^(1/3). Then x^(2/3) = y^2, so y^2 = 16. Thus y = 4 or y = -4. Cubing gives x = 4^3 = 64 or x = (-4)^3 = -64.\nFinal answer: \\boxed{-64, 6... |
x = 8 | [
{
"content": "Solve (x-2)/(x+2) = 3/5.",
"role": "user"
},
{
"content": "<think> Cross-multiply and solve the resulting linear equation. </think>\nCross-multiply:\n\n5(x-2) = 3(x+2).\n\nExpand:\n\n5x - 10 = 3x + 6.\n\nSolve:\n\n2x = 16\nx = 8.\n\nRestriction: x ≠ -2, and 8 is valid.\n\nFinal answer:... |
12 | [
{
"content": "Find the area of the rectangle with vertices (1, 1), (5, 1), (5, 4), and (1, 4).",
"role": "user"
},
{
"content": "<think> The side lengths are the horizontal distance 4 and the vertical distance 3, so multiply them. </think>\nThe width is the distance from x = 1 to x = 5, which is 4. ... |
16 | [
{
"content": "Find the length of the latus rectum of the parabola x^2=16y.",
"role": "user"
},
{
"content": "<think> Compare x^2=16y with x^2=4ay, so a=4. The latus rectum length is 4a=16. </think>\nThe standard form of a parabola opening upward is x^2 = 4ay. Comparing x^2 = 16y with this form, we g... |
x | [
{
"content": "Assuming x ≥ 0, simplify (x^(2/3))^(3/2).",
"role": "user"
},
{
"content": "<think> Multiply the exponents: (2/3)(3/2) = 1. </think>\nUsing the power-of-a-power rule, (x^(2/3))^(3/2) = x^((2/3)(3/2)) = x^1 = x. The condition x ≥ 0 avoids issues with principal even roots.\nFinal answer:... |
30 | [
{
"content": "If Q1 = 10 and Q3 = 18, find the upper fence for identifying outliers using the 1.5 IQR rule.",
"role": "user"
},
{
"content": "<think> First find IQR, then compute Q3 + 1.5 times IQR. </think>\nThe interquartile range is IQR = Q3 - Q1 = 18 - 10 = 8. The upper fence is Q3 + 1.5(IQR) = ... |
{x ∈ ℤ : x ≡ 3 (mod 4)} | [
{
"content": "What is the equivalence class of 3 modulo 4?",
"role": "user"
},
{
"content": "<think> It contains all integers differing from 3 by a multiple of 4. </think>\nTwo integers are equivalent modulo 4 when their difference is divisible by 4. So the class of 3 consists of all integers of the... |
2, x ≠ -2 | [
{
"content": "Subtract (3x)/(x+2) - (x-4)/(x+2).",
"role": "user"
},
{
"content": "<think> Since the denominators match, subtract the numerators directly and simplify. </think>\nThe denominators are the same, so combine the numerators:\n\n(3x)/(x+2) - (x-4)/(x+2) = [3x - (x-4)]/(x+2).\n\nSimplify th... |
x = 3 | [
{
"content": "Solve for x: 2.5x + 1.5 = 9",
"role": "user"
},
{
"content": "<think> Subtract 1.5, then divide by 2.5. </think>\nSolve the equation:\n2.5x + 1.5 = 9\n\nSubtract 1.5 from both sides:\n2.5x = 7.5\n\nDivide both sides by 2.5:\nx = 3\n\nFinal answer: \\boxed{x = 3}",
"role": "assistan... |
2 | [
{
"content": "Solve for x: log_7(log_2(x)) = 0.",
"role": "user"
},
{
"content": "<think> If log_7(A)=0, then A=1; so solve log_2(x)=1. </think>\nFrom log_7(log_2(x)) = 0, the inside must equal 1 because log_7(1) = 0. So log_2(x) = 1. Converting to exponential form gives x = 2^1 = 2.\nFinal answer: ... |
(x-1)/(x+1), x ≠ -1, 0 | [
{
"content": "Simplify (1 - 1/x)/(1 + 1/x).",
"role": "user"
},
{
"content": "<think> Rewrite each part with denominator x, then divide the two resulting fractions. </think>\nRewrite the numerator and denominator:\n\n1 - 1/x = (x-1)/x\n1 + 1/x = (x+1)/x\n\nNow divide:\n\n[(x-1)/x] / [(x+1)/x] = (x-1... |
8/15 | [
{
"content": "A committee of 2 is chosen at random from 6 men and 4 women. What is the probability that the committee contains exactly 1 woman?",
"role": "user"
},
{
"content": "<think> Count the number of committees with 1 man and 1 woman and divide by the total number of 2-person committees. </thi... |
\text{Undefined} | [
{
"content": "What is the slope of the vertical line x = -4?",
"role": "user"
},
{
"content": "<think> A vertical line has zero run, so the slope would involve division by zero. </think>\nSlope is rise divided by run. On a vertical line, the x-values do not change, so the run is 0. Division by 0 is ... |
x = (5 ± √13)/6 | [
{
"content": "Solve the quadratic equation: 3x^2 - 5x + 1 = 0",
"role": "user"
},
{
"content": "<think> Use the quadratic formula because the quadratic does not factor nicely over the integers. </think>\nUse the quadratic formula with a = 3, b = -5, c = 1.\n\nx = [5 ± √((-5)^2 - 4(3)(1))]/6\n= [5 ± ... |
x \le 2 or x \ge 3 | [
{
"content": "Solve the inequality: x^2 - 5x + 6 \\ge 0",
"role": "user"
},
{
"content": "<think> Factor the quadratic and use a sign chart. </think>\nSolve:\n\\[x^2-5x+6\\ge 0\\]\nFactor:\n\\[(x-2)(x-3)\\ge 0\\]\nThe critical points are \\(x=2\\) and \\(x=3\\).\nThe product is nonnegative outside t... |
2 | [
{
"content": "Find \\(\\sec 60^\\circ\\).",
"role": "user"
},
{
"content": "<think> Secant is the reciprocal of cosine, and \\(\\cos 60^\\circ=\\frac{1}{2}\\), so \\(\\sec 60^\\circ=2\\). </think>\nSince secant is the reciprocal of cosine,\n\\[\n\\sec 60^\\circ=\\frac{1}{\\cos 60^\\circ}\n\\]\nBecau... |
18 | [
{
"content": "Find the determinant of the matrix [[1,2,3],[0,-1,4],[2,1,0]].",
"role": "user"
},
{
"content": "<think> Expand along the first row: 1((−1)(0)−4·1) − 2(0·0−4·2) + 3(0·1−(−1)·2) = -4 + 16 + 6 = 18. </think>\nExpanding along the first row,\ndet = 1\\begin{vmatrix}-1&4\\\\1&0\\end{vmatrix... |
x = (-3 ± √65)/4 | [
{
"content": "Solve the quadratic equation: 2x^2 + 3x - 7 = 0",
"role": "user"
},
{
"content": "<think> This quadratic does not factor nicely, so use the quadratic formula. </think>\nUse the quadratic formula x = [-b ± √(b^2 - 4ac)]/(2a) with a = 2, b = 3, c = -7.\n\nx = [-3 ± √(3^2 - 4(2)(-7))]/(2·... |
(x+3)/(x+1), x ≠ -1, -1/2, 1 | [
{
"content": "Simplify ((2x^2 + 7x + 3)/(x^2 - 1)) * ((x-1)/(2x+1)).",
"role": "user"
},
{
"content": "<think> Factor the quadratic numerator and denominator, then cancel x-1 and 2x+1. </think>\nFactor:\n\n2x^2 + 7x + 3 = (2x+1)(x+3)\nx^2 - 1 = (x-1)(x+1)\n\nNow simplify:\n\n[(2x+1)(x+3)]/[(x-1)(x+1... |
x = 2 or x = -4/3 | [
{
"content": "Solve 1/x + 1/(x+2) = 3/4.",
"role": "user"
},
{
"content": "<think> Clear denominators by multiplying by 4x(x+2), solve the quadratic, and reject any excluded values. </think>\nStart with\n\n1/x + 1/(x+2) = 3/4.\n\nMultiply both sides by 4x(x+2):\n\n4(x+2) + 4x = 3x(x+2).\n\nSimplify:... |
ln(7)/ln(4) | [
{
"content": "Express log_4(7) using the change-of-base formula.",
"role": "user"
},
{
"content": "<think> Use log_a(b) = ln(b)/ln(a). </think>\nBy the change-of-base formula, log_4(7) = ln(7)/ln(4). An equivalent form is log(7)/log(4), but using natural logs gives ln(7)/ln(4).\nFinal answer: \\boxe... |
y = 3x - 8 | [
{
"content": "Find the equation of the line passing through the intersection of x+y=4 and x-y=2 and parallel to y=3x+1.",
"role": "user"
},
{
"content": "<think> Solve the two lines to get the intersection point (3,1). A parallel line has slope 3, so through (3,1): y-1=3(x-3), hence y=3x-8. </think>... |
\frac{8}{15} | [
{
"content": "In an 8-15-17 right triangle, let θ be the acute angle opposite the side of length 8. Find tan(θ).",
"role": "user"
},
{
"content": "<think> Tangent is opposite over adjacent. For this angle, opposite = 8 and adjacent = 15. </think>\nFor angle θ, the opposite side is 8 and the adjacent... |
(13, 0) | [
{
"content": "Find the x-intercept of y = log_3(x - 4) - 2.",
"role": "user"
},
{
"content": "<think> Set y = 0 and solve the resulting logarithmic equation. </think>\nAt the x-intercept, y = 0. So 0 = log_3(x - 4) - 2, which gives log_3(x - 4) = 2. Rewrite in exponential form: x - 4 = 3^2 = 9. Thus... |
61/60 | [
{
"content": "In a right triangle, for angle \\(\\theta\\), the opposite side is 60 and the hypotenuse is 61. Find \\(\\csc\\theta\\).",
"role": "user"
},
{
"content": "<think> Cosecant is the reciprocal of sine, so \\(\\csc\\theta=\\frac{\\text{hypotenuse}}{\\text{opposite}}=\\frac{61}{60}\\). </th... |
π^2/6 | [
{
"content": "Find the sum of the infinite series Σ from n=1 to infinity of 1/n^2.",
"role": "user"
},
{
"content": "<think> This is the Basel series, whose exact sum is a standard result. </think>\nThe series\nΣ from n=1 to infinity of 1/n^2\nis a famous convergent series called the Basel problem.\... |
x | [
{
"content": "Assuming x > 0, simplify (x^(3/2) · x^(1/2)) / x.",
"role": "user"
},
{
"content": "<think> Add exponents in the numerator and subtract the denominator exponent. </think>\nCombine exponents in the numerator: x^(3/2)·x^(1/2) = x^(4/2) = x^2. Then divide by x: x^2/x = x.\nFinal answer: \... |
2870 | [
{
"content": "Find the sum 1^2 + 2^2 + 3^2 + ... + 20^2.",
"role": "user"
},
{
"content": "<think> Use the formula for the sum of the first n squares: n(n+1)(2n+1)/6. </think>\nUse the formula\n1^2 + 2^2 + ... + n^2 = n(n+1)(2n+1)/6.\nWith n = 20,\nS = 20·21·41 / 6.\nSimplify: 20/6 = 10/3, so\nS = 1... |
y = -2x + 5 | [
{
"content": "Find the image of the line y=2x-5 after reflection in the x-axis.",
"role": "user"
},
{
"content": "<think> Reflection in the x-axis changes y to -y. So -y=2x-5, hence y=-2x+5. </think>\nReflecting across the x-axis changes each point (x,y) to (x,-y). So to find the image of the line y... |
x = 5 | [
{
"content": "Solve for x: 3x + 5 = 20",
"role": "user"
},
{
"content": "<think> Subtract 5 from both sides, then divide by 3. </think>\nSolve the equation:\n3x + 5 = 20\n\nSubtract 5 from both sides:\n3x = 15\n\nDivide both sides by 3:\nx = 5\n\nFinal answer: \\boxed{x = 5}",
"role": "assistant... |
2 | [
{
"content": "Find \\(\\csc 30^\\circ\\).",
"role": "user"
},
{
"content": "<think> Cosecant is the reciprocal of sine, and \\(\\sin 30^\\circ=\\frac{1}{2}\\), so \\(\\csc 30^\\circ=2\\). </think>\nSince cosecant is the reciprocal of sine,\n\\[\n\\csc 30^\\circ=\\frac{1}{\\sin 30^\\circ}\n\\]\nBecau... |
7 | [
{
"content": "If f(x) = 2x - 4, solve f(x) = 10.",
"role": "user"
},
{
"content": "<think> Set 2x - 4 equal to 10 and solve. </think>\nWe solve 2x - 4 = 10. Adding 4 to both sides gives 2x = 14. Dividing by 2 gives x = 7. Final answer: \\boxed{7}",
"role": "assistant"
}
] |
x = 13/2 | [
{
"content": "Solve for x: 7(x - 3) + 2 = 5x - 6",
"role": "user"
},
{
"content": "<think> Expand the left side, then solve for x. </think>\nSolve the equation:\n7(x - 3) + 2 = 5x - 6\n\nDistribute 7:\n7x - 21 + 2 = 5x - 6\n\nSimplify:\n7x - 19 = 5x - 6\n\nSubtract 5x from both sides:\n2x - 19 = -6\... |
y = ±(3/4)x | [
{
"content": "Find the asymptotes of the hyperbola 9x^2-16y^2=144.",
"role": "user"
},
{
"content": "<think> Write the equation as x^2/16-y^2/9=1. For this form, asymptotes are y=±(b/a)x=±(3/4)x. </think>\nDivide the equation 9x^2 - 16y^2 = 144 by 144 to get x^2/16 - y^2/9 = 1. This is a hyperbola o... |
Focus (3,0), directrix x = -3 | [
{
"content": "For the parabola y^2=12x, find the focus and directrix.",
"role": "user"
},
{
"content": "<think> Compare y^2=12x with y^2=4ax, so 4a=12 and a=3. Then focus is (a,0) and directrix is x=-a. </think>\nThe standard form of a parabola opening rightward is y^2 = 4ax. Comparing y^2 = 12x wit... |
a_n = n(n+1) - 1 | [
{
"content": "Given a_1 = 1 and a_n = a_(n-1) + 2n for n ≥ 2, find an explicit formula for a_n.",
"role": "user"
},
{
"content": "<think> Write a_n as 1 plus the sum of 2k from k = 2 to n, then simplify. </think>\nFrom the recursion,\na_n = 1 + Σ from k=2 to n of 2k.\nSo\na_n = 1 + 2(Σ from k=1 to n... |
20/3 | [
{
"content": "Find the sum of the infinite geometric series Σ from n=0 to infinity of 5(1/4)^n.",
"role": "user"
},
{
"content": "<think> Use the infinite geometric series formula S = a/(1-r) with a = 5 and r = 1/4. </think>\nThis is an infinite geometric series with first term a = 5 and ratio r = 1... |
-3 | [
{
"content": "Evaluate log_3(1/27).",
"role": "user"
},
{
"content": "<think> Because 3^-3 = 1/27, the logarithm equals -3. </think>\nWe want the exponent n such that 3^n = 1/27. Since 27 = 3^3, we have 1/27 = 3^-3. Therefore log_3(1/27) = -3.\nFinal answer: \\boxed{-3}",
"role": "assistant"
}... |
9 | [
{
"content": "If A is a 3x3 matrix with det(A) = -3, find det(adj(A)).",
"role": "user"
},
{
"content": "<think> For an n x n matrix, det(adj(A)) = det(A)^(n−1). With n=3, this is (−3)^2 = 9. </think>\nFor an invertible 3x3 matrix, det(adj(A)) = det(A)^{3-1} = det(A)^2. Since det(A) = -3, we get det... |
2600 | [
{
"content": "If a_n = 2n + 1, find Σ from n=1 to 50 of a_n.",
"role": "user"
},
{
"content": "<think> This sums the arithmetic sequence 3, 5, 7, ..., 101. </think>\nSince a_n = 2n + 1,\nthe terms from n = 1 to 50 are\n3, 5, 7, ..., 101.\nThis is an arithmetic series with 50 terms, first term 3, and... |
4x^2 - 9 | [
{
"content": "Expand: (2x + 3)(2x - 3)",
"role": "user"
},
{
"content": "<think> Use the difference of squares formula or distribute directly. </think>\nMultiply: (2x)(2x) + (2x)(-3) + 3(2x) + 3(-3) = 4x^2 - 6x + 6x - 9. The middle terms cancel, so the result is 4x^2 - 9. Final answer: \\boxed{4x^2 ... |
5 | [
{
"content": "Find the first quartile Q1 of the ordered data set: 2, 4, 6, 8, 10, 12, 14, 16.",
"role": "user"
},
{
"content": "<think> Q1 is the median of the lower half of the ordered data. </think>\nThere are 8 values, so split the data into the lower half 2, 4, 6, 8 and the upper half 10, 12, 14... |
x \ge -3 | [
{
"content": "Solve the inequality: 2 - 5x \\le 17",
"role": "user"
},
{
"content": "<think> Subtract 2, then divide by -5 and reverse the inequality sign. </think>\nSolve:\n\\[2-5x\\le 17\\]\nSubtract 2 from both sides:\n\\[-5x\\le 15\\]\nDivide by \\(-5\\). Since we divide by a negative number, re... |
(2,1,3) | [
{
"content": "Solve the system: x + 2y - z = 1, 2x - y + z = 6, 3x + y + 2z = 13.",
"role": "user"
},
{
"content": "<think> Use elimination with pairs of equations to reduce to two variables. </think>\nAdd the first two equations: (x + 2y - z) + (2x - y + z) = 1 + 6, so 3x + y = 7. Compare this with... |
6\% | [
{
"content": "In the model y=120(1.06)^t, what is the percent growth rate?",
"role": "user"
},
{
"content": "<think> The growth factor is 1+r, so r=0.06. </think>\nFor an exponential growth model y=a(1+r)^t, the number added to 1 is the growth rate. Here 1+r=1.06, so r=0.06, which is 6%.\nFinal answ... |
(3,2,-2) | [
{
"content": "Solve the system: 2x + y - z = 10, x - 2y + z = -3, 3x + y + z = 9.",
"role": "user"
},
{
"content": "<think> Add the first two equations to eliminate z, then use another equation to finish. </think>\nAdd the first two equations: (2x + y - z) + (x - 2y + z) = 10 + (-3), so 3x - y = 7. ... |
-2 | [
{
"content": "Find the trace of the matrix [[3,1],[2,-5]].",
"role": "user"
},
{
"content": "<think> The trace is the sum of the diagonal entries: 3 + (−5) = −2. </think>\nThe trace of a square matrix is the sum of its diagonal entries. For [[3,1],[2,-5]], the trace is 3 + (-5) = -2.\nFinal answer: ... |
0 | [
{
"content": "Find the determinant of [[4,2],[6,3]].",
"role": "user"
},
{
"content": "<think> Use ad−bc: 4·3−2·6 = 12−12 = 0. </think>\nFor the matrix [[4,2],[6,3]], the determinant is 4\\cdot3 - 2\\cdot6 = 12 - 12 = 0.\nFinal answer: \\boxed{0}",
"role": "assistant"
}
] |
f^{-1}(x) = 2^(x-3) + 1 | [
{
"content": "Find the inverse of y = log_2(x - 1) + 3.",
"role": "user"
},
{
"content": "<think> Solve for x in terms of y, then swap x and y. </think>\nStart with y = log_2(x - 1) + 3. Subtract 3 to get y - 3 = log_2(x - 1). Rewrite in exponential form: x - 1 = 2^(y - 3). So x = 2^(y - 3) + 1. Swa... |
x = 0, 4 | [
{
"content": "Solve the quadratic equation: 2x^2 - 8x = 0",
"role": "user"
},
{
"content": "<think> Factor out the greatest common factor and use the zero-product property. </think>\nFirst factor out the GCF:\n2x(x - 4) = 0\n\nNow set each factor equal to zero:\n2x = 0 gives x = 0\nx - 4 = 0 gives x... |
45/53 | [
{
"content": "In a right triangle, for angle \\(\\theta\\), the opposite side is 28, the adjacent side is 45, and the hypotenuse is 53. Find \\(\\cos\\theta\\).",
"role": "user"
},
{
"content": "<think> Cosine is adjacent over hypotenuse, so \\(\\cos\\theta=\\frac{45}{53}\\). </think>\nUse the cosin... |
(6,4) | [
{
"content": "Solve the system: 0.5x + 0.25y = 4, x - y = 2.",
"role": "user"
},
{
"content": "<think> Clear decimals first and then solve the resulting linear system. </think>\nMultiply 0.5x + 0.25y = 4 by 4 to get 2x + y = 16. From x - y = 2, we have y = x - 2. Substitute into 2x + y = 16: 2x + (x... |
x = -2, 1/5 | [
{
"content": "Solve the quadratic equation: 5x^2 + 9x - 2 = 0",
"role": "user"
},
{
"content": "<think> Factor the quadratic and solve the resulting linear equations. </think>\nFactor the quadratic:\n5x^2 + 9x - 2 = (5x - 1)(x + 2)\n\nSet each factor equal to zero:\n5x - 1 = 0 gives x = 1/5\nx + 2 =... |
2x^2 + x - 3 | [
{
"content": "Divide: (2x^4 + 3x^3 - 6x^2 - 5x + 6) / (x^2 + x - 2)",
"role": "user"
},
{
"content": "<think> Use polynomial long division; the division is exact. </think>\nDivide leading terms: 2x^4 / x^2 = 2x^2. Multiply back: 2x^2(x^2 + x - 2) = 2x^4 + 2x^3 - 4x^2. Subtract to get x^3 - 2x^2 - 5x... |
End of preview. Expand in Data Studio
High School Math Reasoning
Synthetic high school math SFT data with train/test splits.
Format
Each row contains:
messages: chat-style user/assistant turns for SFTanswer: 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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