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|---|---|---|---|---|---|---|---|
89825669d81b5525 | 9da88beb2217dd7d | verifier | Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer. | An urn has 29 red balls and 11 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 1 are red? | Use a hypergeometric counting argument.
Total equally likely samples: C(40,11).
Favorable samples: choose k reds and n−k blues: C(29,1)·C(11,10).
Compute totals: C(40,11) = 2311801440, and favorable = 319.
Divide favorable by total and simplify.
Check: computed probability simplifies to $\frac{319}{2311801440}$.
Final ... | \frac{319}{2311801440} | {
"format": "symbolic",
"domain": "probability",
"topic": "hypergeometric",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
a07ddd80ac479146 | 9da88beb2217dd7d | tutor | Explain each step clearly and end with the final answer. | An urn has 29 red balls and 11 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 1 are red? | 1) Use a hypergeometric counting argument.
2) Total equally likely samples: C(40,11).
3) Favorable samples: choose k reds and n−k blues: C(29,1)·C(11,10).
4) Compute totals: C(40,11) = 2311801440, and favorable = 319.
5) Divide favorable by total and simplify.
Final answer: $\frac{319}{2311801440}$ | \frac{319}{2311801440} | {
"format": "symbolic",
"domain": "probability",
"topic": "hypergeometric",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
bc487c0644a84dcb | 14cfd12df0d5dfb0 | tutor | Solve like a tutor: derive step-by-step and finish with the final answer. | Two friends split $792 in the ratio 7:1. How much does each receive? | 1) Total ratio parts = 7+1 = 8.
2) Each part = 792 ÷ 8 = 99.
3) First share = 7 × 99 = 693.
4) Second share = 1 × 99 = 99.
Final answer: $99$ | 99 | {
"format": "word",
"domain": "arithmetic",
"topic": "ratio_split",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
476a63fc302e105b | 1bea1c540a0aa966 | verifier_negative | A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer. | Differentiate with respect to x:
$f(x)=\left(x + 1\right) e^{x} + \left(- 2 x^{3} + x^{2} - x - 3\right) \cos{\left(x + 2 \right)}$.
Student solution:
1) Differentiate term-by-term.
2) Claim (missing chain rule): $\frac{d}{dx}(x^{2} \cos{\left(x + 2 \right)}) = - x^{2} \sin{\left(x + 2 \right)}$.
3) So $f'(x) = 2 x^{3... | First incorrect step: 2
The error is omitting the chain-rule factor (derivative of the inside function).
Correct derivative of that term is $\frac{d}{dx}(x^{2} \cos{\left(x + 2 \right)}) = - x^{2} \sin{\left(x + 2 \right)} + 2 x \cos{\left(x + 2 \right)}$.
Therefore the correct derivative is $f'(x) = \left(x + 1\right)... | \left(x + 1\right) e^{x} - \left(x \left(x \left(1 - 2 x\right) - 1\right) - 3\right) \sin{\left(x + 2 \right)} + \left(- 6 x^{2} + 2 x - 1\right) \cos{\left(x + 2 \right)} + e^{x} | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0",
"task": "error_localization",
"is_negative": true
} |
3f6741b652f56d1b | 731291ed7828f8a8 | tutor | Write a complete step-by-step solution and conclude with the final answer. | Evaluate the limit:
$\displaystyle \lim_{x\to 2} 2 x + 4$. | 1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).
2) Here p(x) = $2 x^{2} + 3$ and a = 2.
3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).
4) Differentiate: p'(x) = $4 x$.
5) Evaluate at a: p'(2) = $8$.
Final answer: $8$ | 8 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
7e3fe8d06b911e4a | 731291ed7828f8a8 | verifier | Compute the answer and add a brief verification calculation. Provide the final answer. | Evaluate the limit:
$\displaystyle \lim_{x\to 2} 2 x + 4$. | Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).
Here p(x) = $2 x^{2} + 3$ and a = 2.
Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).
Differentiate: p'(x) = $4 x$.
Evaluate at a: p'(2) = $8$.
Check: the identity gives $8$.
Final answer: $8$ | 8 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
79868f6de568c5ee | d425f2e063466925 | tutor | Show a full solution with explicit computations and a final answer line. | Evaluate the limit:
$\displaystyle \lim_{x\to -3} 2 x^{3} - 3 x^{2} + 8 x - 23$. | 1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).
2) Here p(x) = $2 x^{4} + 3 x^{3} - x^{2} + x + 3$ and a = -3.
3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).
4) Differentiate: p'(x) = $8 x^{3} + 9 x^{2} - 2 x + 1$.
5) Evaluate at a: p'(-3) = $-128$.
Final answer: $-128$ | -128 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
66ae4a2a92765b50 | d425f2e063466925 | verifier | Show the solution, then validate it with a quick check. End with the final answer. | Evaluate the limit:
$\displaystyle \lim_{x\to -3} 2 x^{3} - 3 x^{2} + 8 x - 23$. | Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).
Here p(x) = $2 x^{4} + 3 x^{3} - x^{2} + x + 3$ and a = -3.
Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).
Differentiate: p'(x) = $8 x^{3} + 9 x^{2} - 2 x + 1$.
Evaluate at a: p'(-3) = $-128$.
Check: the identity gives $-128$.
Final answer: $-128$ | -128 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
81bafecdbfca293b | 9ac51afd01182727 | answer_only | Provide only the final answer (no steps). | Worker A finishes a job in 22 hours and Worker B finishes it in 19 hours. If they work together at constant rates, how long does it take to finish the job? | $\frac{418}{41}$ | \frac{418}{41} | {
"format": "word",
"domain": "arithmetic",
"topic": "work_rates",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
e5f64eec8ae0193b | 9ac51afd01182727 | tutor | Solve carefully and do not skip algebra/arithmetic steps. End with the final answer. | Worker A finishes a job in 22 hours and Worker B finishes it in 19 hours. If they work together at constant rates, how long does it take to finish the job? | 1) A's rate = 1/22 job/hour. B's rate = 1/19 job/hour.
2) Combined rate = 1/22 + 1/19 = 41/418 job/hour.
3) Time = 1 ÷ (combined rate) = 418/41 hours.
Final answer: $\frac{418}{41}$ | \frac{418}{41} | {
"format": "word",
"domain": "arithmetic",
"topic": "work_rates",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
66feb0aca1aa1e46 | b36b8c88d7868394 | tutor | Give a detailed derivation, showing intermediate expressions, and then the final answer. | Solve the quadratic equation over the real numbers:
$x^{2} - 6 x - 27$ = 0. | 1) Because the roots are simple, factoring is the fastest method.
2) Factor the polynomial: $x^{2} - 6 x - 27$ = $\left(x - 9\right) \left(x + 3\right)$.
3) Set each factor equal to 0 and solve each linear equation.
4) From the first factor: x = $9$.
5) From the second factor: x = $-3$.
Final answer: $x \in \{9, -3\}$ | x \in \{9, -3\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
d75ed60911494b8d | b36b8c88d7868394 | verifier | Compute the answer and include a short verification step. Give the final answer. | Solve the quadratic equation over the real numbers:
$x^{2} - 6 x - 27$ = 0. | Because the roots are simple, factoring is the fastest method.
Factor the polynomial: $x^{2} - 6 x - 27$ = $\left(x - 9\right) \left(x + 3\right)$.
Set each factor equal to 0 and solve each linear equation.
From the first factor: x = $9$.
From the second factor: x = $-3$.
Check: substituting each solution into the poly... | x \in \{9, -3\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
ee7387ab639d7402 | 8d82a0fc214b8c8a | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | Compute the definite integral:
$\displaystyle \int_{- \pi}^{\pi} - x^{4} + 2 x^{3} - 3 x^{2} - 3 x + 3 \sin{\left(2 x \right)} + 2 \cos{\left(x \right)}\,dx$. | 1) Compute an antiderivative F(x) of the integrand $- x^{4} + 2 x^{3} - 3 x^{2} - 3 x + 3 \sin{\left(2 x \right)} + 2 \cos{\left(x \right)}$.
2) Evaluate at the bounds: F(\pi) and F(- \pi).
3) Subtract: F(\pi) − F(- \pi) = $\frac{2 \pi^{3} \left(- \pi^{2} - 5\right)}{5}$.
4) Simplify the result.
Final answer: $\frac{2\... | \frac{2\cdot \pi^{3}\cdot \left(- \pi^{2} - 5\right)}{5} | {
"format": "symbolic",
"domain": "calculus",
"topic": "definite_integral",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
18c39939b704c596 | 8d82a0fc214b8c8a | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | Compute the definite integral:
$\displaystyle \int_{- \pi}^{\pi} - x^{4} + 2 x^{3} - 3 x^{2} - 3 x + 3 \sin{\left(2 x \right)} + 2 \cos{\left(x \right)}\,dx$. | Compute an antiderivative F(x) of the integrand $- x^{4} + 2 x^{3} - 3 x^{2} - 3 x + 3 \sin{\left(2 x \right)} + 2 \cos{\left(x \right)}$.
Evaluate at the bounds: F(\pi) and F(- \pi).
Subtract: F(\pi) − F(- \pi) = $\frac{2 \pi^{3} \left(- \pi^{2} - 5\right)}{5}$.
Simplify the result.
Check: SymPy differentiation confir... | \frac{2\cdot \pi^{3}\cdot \left(- \pi^{2} - 5\right)}{5} | {
"format": "symbolic",
"domain": "calculus",
"topic": "definite_integral",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
a45d63473c36776c | 46f9e5f88447c37d | answer_only | Solve the problem and give only the final answer. | A coin lands heads with probability p=$\frac{2}{7}$. If it is flipped 14 times, what is the probability of exactly 3 heads? | $\frac{20312500000}{96889010407}$ | \frac{20312500000}{96889010407} | {
"format": "symbolic",
"domain": "probability",
"topic": "binomial_probability",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
81bbfd93b554d452 | 46f9e5f88447c37d | tutor | Show a full solution with explicit computations and a final answer line. | A coin lands heads with probability p=$\frac{2}{7}$. If it is flipped 14 times, what is the probability of exactly 3 heads? | 1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).
2) Use $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$.
3) Substitute n=14, k=3, p=$\frac{2}{7}$ and simplify.
4) Here C(14,3) = 364.
Final answer: $\frac{20312500000}{96889010407}$ | \frac{20312500000}{96889010407} | {
"format": "symbolic",
"domain": "probability",
"topic": "binomial_probability",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
da6ab5c873a559e9 | 53e2d97f7f5d58dd | tutor | Write a complete step-by-step solution and conclude with the final answer. | Differentiate with respect to x:
$f(x)=3 x^{4} + x^{3} - 2 x^{2} + 4 x + 3$. | 1) Differentiate term-by-term:
2) d/dx(3) = 0.
3) d/dx(x^{3}) = 3 x^{2}.
4) d/dx(- 2 x^{2}) = - 4 x.
5) d/dx(3 x^{4}) = 12 x^{3}.
6) d/dx(4 x) = 4.
7) Combine: f'(x) = 12 x^{3} + 3 x^{2} - 4 x + 4.
Final answer: $12\cdot x^{3} + 3\cdot x^{2} - 4\cdot x + 4$ | 12\cdot x^{3} + 3\cdot x^{2} - 4\cdot x + 4 | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
e39a5bda9442751c | 53e2d97f7f5d58dd | concise | Provide a compact reasoning and the final answer. | Differentiate with respect to x:
$f(x)=3 x^{4} + x^{3} - 2 x^{2} + 4 x + 3$. | Differentiate term-by-term:
d/dx(3) = 0.
d/dx(x^{3}) = 3 x^{2}.
Final answer: $12\cdot x^{3} + 3\cdot x^{2} - 4\cdot x + 4$ | 12\cdot x^{3} + 3\cdot x^{2} - 4\cdot x + 4 | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
6b03cee1274354f1 | 05190806a6b1cd79 | tutor | Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer. | Solve the quadratic equation over the real numbers:
$3 x^{2} + \frac{36 x}{5} + \frac{21}{5}$ = 0. | 1) Identify coefficients a, b, c in ax^2+bx+c=0.
2) Here a=$3$, b=$\frac{36}{5}$, c=$\frac{21}{5}$.
3) Compute the discriminant Δ=b^2−4ac.
4) Δ = $\frac{36}{25}$, so √Δ = $\frac{6}{5}$.
5) Apply the quadratic formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}$.
6) x1 = $-1$.
7) x2 = $- \frac{7}{5}$.
8) Collect the solutions into... | x \in \{-1, - \frac{7}{5}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
af5c41fc28413b5d | 05190806a6b1cd79 | verifier | Solve the problem, then verify the result with a quick check. Provide the final answer. | Solve the quadratic equation over the real numbers:
$3 x^{2} + \frac{36 x}{5} + \frac{21}{5}$ = 0. | Identify coefficients a, b, c in ax^2+bx+c=0.
Here a=$3$, b=$\frac{36}{5}$, c=$\frac{21}{5}$.
Compute the discriminant Δ=b^2−4ac.
Δ = $\frac{36}{25}$, so √Δ = $\frac{6}{5}$.
Apply the quadratic formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}$.
x1 = $-1$.
x2 = $- \frac{7}{5}$.
Collect the solutions into a set.
Check: substitut... | x \in \{-1, - \frac{7}{5}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
2c1a1adaf5e07551 | 10871827149b2f65 | tutor | Solve like a tutor: derive step-by-step and finish with the final answer. | Compute the indefinite integral:
$\displaystyle \int \left(2 x + 1\right) \cos{\left(x \right)} + \left(2 x + 3\right) e^{2 x}\,dx$. | 1) Identify an integration method (term-by-term, substitution, or a known pattern).
2) Integrand: $\left(2 x + 1\right) \cos{\left(x \right)} + \left(2 x + 3\right) e^{2 x}$.
3) Compute an antiderivative: F(x) = $2 x \sin{\left(x \right)} + \left(x + 1\right) e^{2 x} + \sin{\left(x \right)} + 2 \cos{\left(x \right)}$.
... | C + 2\cdot x\cdot \sin{\left(x \right)} + \left(x + 1\right)\cdot e^{2\cdot x} + \sin{\left(x \right)} + 2\cdot \cos{\left(x \right)} | {
"format": "symbolic",
"domain": "calculus",
"topic": "indefinite_integral",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
7977d4a7d332a382 | a4898b82e7b4d45a | verifier_negative | A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer. | Differentiate with respect to x:
$f(x)=2 \left(x + 2\right) e^{x} + \left(- 2 x^{3} - 2 x + 3\right) \cos{\left(2 x + 1 \right)}$.
Student solution:
1) Differentiate term-by-term.
2) Claim (missing chain rule): $\frac{d}{dx}(3 \cos{\left(2 x + 1 \right)}) = - 3 \sin{\left(2 x + 1 \right)}$.
3) So $f'(x) = 4 x^{3} \sin... | First incorrect step: 2
The error is omitting the chain-rule factor (derivative of the inside function).
Correct derivative of that term is $\frac{d}{dx}(3 \cos{\left(2 x + 1 \right)}) = - 6 \sin{\left(2 x + 1 \right)}$.
Therefore the correct derivative is $f'(x) = 2 \left(\left(x + 2\right) e^{x} + \left(- 3 x^{2} - 1... | 2 \left(\left(x + 2\right) e^{x} + \left(- 3 x^{2} - 1\right) \cos{\left(2 x + 1 \right)} - \left(2 x \left(- x^{2} - 1\right) + 3\right) \sin{\left(2 x + 1 \right)} + e^{x}\right) | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0",
"task": "error_localization",
"is_negative": true
} |
67221dccbfa3f98a | 4b8ede2c06626f86 | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | An urn has 16 red balls and 18 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 5 are red? | 1) Use a hypergeometric counting argument.
2) Total equally likely samples: C(34,11).
3) Favorable samples: choose k reds and n−k blues: C(16,5)·C(18,6).
4) Compute totals: C(34,11) = 286097760, and favorable = 81087552.
5) Divide favorable by total and simplify.
Final answer: $\frac{1274}{4495}$ | \frac{1274}{4495} | {
"format": "symbolic",
"domain": "probability",
"topic": "hypergeometric",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
4514169482c73294 | 4b8ede2c06626f86 | verifier | Compute the answer and include a short verification step. Give the final answer. | An urn has 16 red balls and 18 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 5 are red? | Use a hypergeometric counting argument.
Total equally likely samples: C(34,11).
Favorable samples: choose k reds and n−k blues: C(16,5)·C(18,6).
Compute totals: C(34,11) = 286097760, and favorable = 81087552.
Divide favorable by total and simplify.
Check: computed probability simplifies to $\frac{1274}{4495}$.
Final an... | \frac{1274}{4495} | {
"format": "symbolic",
"domain": "probability",
"topic": "hypergeometric",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
91fccd95faebde27 | 5f59fcb81066bf31 | tutor | Provide a thorough solution as if teaching, including intermediate computations, then give the final answer. | Solve the quadratic equation over the real numbers:
$3 x^{2} - \frac{x}{2} - 6$ = 0. | 1) Identify coefficients a, b, c in ax^2+bx+c=0.
2) Here a=$3$, b=$- \frac{1}{2}$, c=$-6$.
3) Compute the discriminant Δ=b^2−4ac.
4) Δ = $\frac{289}{4}$, so √Δ = $\frac{17}{2}$.
5) Apply the quadratic formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}$.
6) x1 = $\frac{3}{2}$.
7) x2 = $- \frac{4}{3}$.
8) Collect the solutions int... | x \in \{\frac{3}{2}, - \frac{4}{3}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
65d89cc7fa7324be | 5f59fcb81066bf31 | verifier | Compute the answer and add a brief verification calculation. Provide the final answer. | Solve the quadratic equation over the real numbers:
$3 x^{2} - \frac{x}{2} - 6$ = 0. | Identify coefficients a, b, c in ax^2+bx+c=0.
Here a=$3$, b=$- \frac{1}{2}$, c=$-6$.
Compute the discriminant Δ=b^2−4ac.
Δ = $\frac{289}{4}$, so √Δ = $\frac{17}{2}$.
Apply the quadratic formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}$.
x1 = $\frac{3}{2}$.
x2 = $- \frac{4}{3}$.
Collect the solutions into a set.
Check: substitu... | x \in \{\frac{3}{2}, - \frac{4}{3}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
475cc78c267d0d72 | 9d4cd9c05c99030c | verifier_negative | A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer. | Evaluate the limit:
$\displaystyle \lim_{x\to 2} x^{3} + 4 x^{2} + 7 x + 13$.
Student solution:
1) Substitute x=2 into the expression.
2) (Mistake) Simplify the expression to x^{3} + 4 x^{2} + 6 x + 13.
3) Then the limit is 49. | First incorrect step: 2
The error is dropping/altering terms before taking the limit.
Use the original expression and substitute x=2 (polynomial is continuous).
Correct limit value: 51.
Final answer: $51$ | 51 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0",
"task": "error_localization",
"is_negative": true
} |
602afd9fd5425cbd | 56ad6eb37304924e | tutor | Show your work in clear steps (with intermediate values) and finish with a final answer. | Differentiate with respect to x:
$f(x)=- 2 x^{2} \sin{\left(x \right)} - 3 \sin{\left(x \right)} + 2 \cos{\left(2 x \right)}$. | 1) Let f(x) = $- 2 x^{2} \sin{\left(x \right)} - 3 \sin{\left(x \right)} + 2 \cos{\left(2 x \right)}$.
2) Differentiate using product/chain/quotient rules as needed.
3) Simplify the derivative to get f'(x) = $- 2 x^{2} \cos{\left(x \right)} - 4 x \sin{\left(x \right)} - 4 \sin{\left(2 x \right)} - 3 \cos{\left(x \right... | - 2\cdot x^{2}\cdot \cos{\left(x \right)} - 4\cdot x\cdot \sin{\left(x \right)} - 4\cdot \sin{\left(2\cdot x \right)} - 3\cdot \cos{\left(x \right)} | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
e7e517f9ab6b0da9 | 1a77ef5ac64de3da | tutor | Explain each step clearly and end with the final answer. | Solve the linear system:
$- 3 x + 8 y$ = $69$
$- 2 x - 11 y$ = $-52$ | 1) Eliminate x by taking a linear combination of the equations.
2) Multiply the first equation by $-2$: $6 x - 16 y$ = $-138$.
3) Multiply the second equation by $-3$: $6 x + 33 y$ = $156$.
4) Subtract to eliminate x: $- 49 y$ = $-294$.
5) Solve for y: y = $6$.
6) Substitute back to find x: x = $-7$.
Final answer: $(x,... | (x,y)=\left(-7, 6\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
d9eee6978f635459 | 1a77ef5ac64de3da | answer_only | Solve the problem and give only the final answer. | Solve the linear system:
$- 3 x + 8 y$ = $69$
$- 2 x - 11 y$ = $-52$ | $(x,y)=\left(-7, 6\right)$ | (x,y)=\left(-7, 6\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
1a291187768940bf | 42206942bab5f088 | tutor | Show a full solution with explicit computations and a final answer line. | A store sells a backpack for $490. It is discounted by 15%. You buy 5 backpacks. What is the total cost? | 1) Discounted price = 490 × (1 − 15/100) = 490 × 85/100 = 833/2.
2) Subtotal = 5 × 833/2 = 4165/2.
Final answer: $\frac{4165}{2}$ | \frac{4165}{2} | {
"format": "word",
"domain": "arithmetic",
"topic": "percent_discount",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
c655f627c5cbc29a | 42206942bab5f088 | concise | Solve with minimal but sufficient steps, then provide the final answer. | A store sells a backpack for $490. It is discounted by 15%. You buy 5 backpacks. What is the total cost? | Discounted price = 490 × (1 − 15/100) = 490 × 85/100 = 833/2.
Subtotal = 5 × 833/2 = 4165/2.
Final answer: $\frac{4165}{2}$ | \frac{4165}{2} | {
"format": "word",
"domain": "arithmetic",
"topic": "percent_discount",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
4a0227c19dea6e31 | 0a37779126447415 | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | A coin lands heads with probability p=$\frac{2}{3}$. If it is flipped 20 times, what is the probability of exactly 8 heads? | Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).
Use $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$.
Substitute n=20, k=8, p=$\frac{2}{3}$ and simplify.
Here C(20,8) = 125970.
Check: expression simplifies to $\frac{10749440}{1162261467}$.
Final answer: $\frac{10749440}{1162261467}$ | \frac{10749440}{1162261467} | {
"format": "symbolic",
"domain": "probability",
"topic": "binomial_probability",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
0e154b18a07c94c3 | 0a37779126447415 | concise | Give a short solution with the key steps and the final answer. | A coin lands heads with probability p=$\frac{2}{3}$. If it is flipped 20 times, what is the probability of exactly 8 heads? | Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).
Use $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$.
Substitute n=20, k=8, p=$\frac{2}{3}$ and simplify.
Final answer: $\frac{10749440}{1162261467}$ | \frac{10749440}{1162261467} | {
"format": "symbolic",
"domain": "probability",
"topic": "binomial_probability",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
ccf4a3f16e1567ea | 83a3b0f783dfd312 | verifier_negative | A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer. | Differentiate with respect to x:
$f(x)=- 4 x^{3} + 3 x^{2} - 4 x - 1$.
Student solution:
1) Differentiate term-by-term.
2) Claim: one term $\frac{d}{dx}(3 x^{2}) = 6 x^{2}$.
3) So $f'(x) = - 6 x^{2} - 4$. | First incorrect step: 2
The error is differentiating a power: the exponent must decrease by 1.
Correct: $\frac{d}{dx}(3 x^{2}) = 6 x$.
Therefore the correct derivative is $f'(x) = 2 \left(- 6 x^{2} + 3 x - 2\right)$.
Final answer: $2 \left(- 6 x^{2} + 3 x - 2\right)$ | 2 \left(- 6 x^{2} + 3 x - 2\right) | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0",
"task": "error_localization",
"is_negative": true
} |
d0261690cf8ab883 | f3d7d9357bcb6775 | verifier | Compute the answer and include a short verification step. Give the final answer. | Two friends split $901 in the ratio 8:9. How much does each receive? | Total ratio parts = 8+9 = 17.
Each part = 901 ÷ 17 = 53.
First share = 8 × 53 = 424.
Second share = 9 × 53 = 477.
Check: 424+477=901.
Final answer: $477$ | 477 | {
"format": "word",
"domain": "arithmetic",
"topic": "ratio_split",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
131ad5b6d776ea09 | 1ac3439e75f7ba4f | verifier | Compute the answer and add a brief verification calculation. Provide the final answer. | Solve the linear system:
$7 x - 8 y$ = $-32$
$- 6 x - 5 y$ = $63$ | Eliminate x by taking a linear combination of the equations.
Multiply the first equation by $-6$: $- 42 x + 48 y$ = $192$.
Multiply the second equation by $7$: $- 42 x - 35 y$ = $441$.
Subtract to eliminate x: $83 y$ = $-249$.
Solve for y: y = $-3$.
Substitute back to find x: x = $-8$.
Check: plug (x,y)=(-8,-3) into bo... | (x,y)=\left(-8, -3\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
d3f9489d51e8d6eb | 1ac3439e75f7ba4f | tutor | Explain each step clearly and end with the final answer. | Solve the linear system:
$7 x - 8 y$ = $-32$
$- 6 x - 5 y$ = $63$ | 1) Eliminate x by taking a linear combination of the equations.
2) Multiply the first equation by $-6$: $- 42 x + 48 y$ = $192$.
3) Multiply the second equation by $7$: $- 42 x - 35 y$ = $441$.
4) Subtract to eliminate x: $83 y$ = $-249$.
5) Solve for y: y = $-3$.
6) Substitute back to find x: x = $-8$.
Final answer: $... | (x,y)=\left(-8, -3\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
44f2a51b1fa556cb | 1e5973de65beda58 | verifier | Solve the problem, then verify the result with a quick check. Provide the final answer. | Find the multiplicative inverse of 4 modulo 13. | We want x such that 4x ≡ 1 (mod 13).
Use extended Euclid to find s,t with 4s + 13t = 1.
One Bezout identity is: 4·(-3) + 13·(1) = 1.
Reduce s modulo 13: x ≡ 10 (mod 13).
Check: 4·10 mod 13 = 1.
Final answer: $x\equiv 10\pmod{13}$ | x\equiv 10\pmod{13} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
bd73828b8b308686 | 1e5973de65beda58 | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | Find the multiplicative inverse of 4 modulo 13. | 1) We want x such that 4x ≡ 1 (mod 13).
2) Use extended Euclid to find s,t with 4s + 13t = 1.
3) One Bezout identity is: 4·(-3) + 13·(1) = 1.
4) Reduce s modulo 13: x ≡ 10 (mod 13).
Final answer: $x\equiv 10\pmod{13}$ | x\equiv 10\pmod{13} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
1fbd896220c5eae9 | dcabcacd885a0374 | tutor | Explain each step clearly and end with the final answer. | Solve the linear system:
$- 8 x + 3 y$ = $77$
$7 x + 4 y$ = $-21$ | 1) Eliminate x by taking a linear combination of the equations.
2) Multiply the first equation by $7$: $- 56 x + 21 y$ = $539$.
3) Multiply the second equation by $-8$: $- 56 x - 32 y$ = $168$.
4) Subtract to eliminate x: $53 y$ = $371$.
5) Solve for y: y = $7$.
6) Substitute back to find x: x = $-7$.
Final answer: $(x... | (x,y)=\left(-7, 7\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
d7b9350712c4729f | dcabcacd885a0374 | verifier | Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer. | Solve the linear system:
$- 8 x + 3 y$ = $77$
$7 x + 4 y$ = $-21$ | Eliminate x by taking a linear combination of the equations.
Multiply the first equation by $7$: $- 56 x + 21 y$ = $539$.
Multiply the second equation by $-8$: $- 56 x - 32 y$ = $168$.
Subtract to eliminate x: $53 y$ = $371$.
Solve for y: y = $7$.
Substitute back to find x: x = $-7$.
Check: plug (x,y)=(-7,7) into both ... | (x,y)=\left(-7, 7\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
017854fce77cb7e5 | 96464306420da28b | verifier | Compute the answer and include a short verification step. Give the final answer. | Solve the quadratic equation over the real numbers:
$10 x^{2} - 5 \sqrt{5} x + 18 x - 9 \sqrt{5}$ = 0. | Identify coefficients a, b, c in ax^2+bx+c=0.
Here a=$10$, b=$18 - 5 \sqrt{5}$, c=$- 9 \sqrt{5}$.
Compute the discriminant Δ=b^2−4ac.
Δ = $180 \sqrt{5} + 449$, so √Δ = $\sqrt{180 \sqrt{5} + 449}$.
Apply the quadratic formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}$.
x1 = $- \frac{9}{10} + \frac{\sqrt{5}}{4} + \frac{\sqrt{180 ... | x \in \{- \frac{9}{10} + \frac{\sqrt{5}}{4} + \frac{\sqrt{180\cdot \sqrt{5} + 449}}{20}, - \frac{\sqrt{180\cdot \sqrt{5} + 449}}{20} - \frac{9}{10} + \frac{\sqrt{5}}{4}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
f844a9efd6afaf97 | 96464306420da28b | concise | Solve with minimal but sufficient steps, then provide the final answer. | Solve the quadratic equation over the real numbers:
$10 x^{2} - 5 \sqrt{5} x + 18 x - 9 \sqrt{5}$ = 0. | Identify coefficients a, b, c in ax^2+bx+c=0.
Here a=$10$, b=$18 - 5 \sqrt{5}$, c=$- 9 \sqrt{5}$.
Compute the discriminant Δ=b^2−4ac.
Final answer: $x \in \{- \frac{9}{10} + \frac{\sqrt{5}}{4} + \frac{\sqrt{180\cdot \sqrt{5} + 449}}{20}, - \frac{\sqrt{180\cdot \sqrt{5} + 449}}{20} - \frac{9}{10} + \frac{\sqrt{5}}{4}\}$ | x \in \{- \frac{9}{10} + \frac{\sqrt{5}}{4} + \frac{\sqrt{180\cdot \sqrt{5} + 449}}{20}, - \frac{\sqrt{180\cdot \sqrt{5} + 449}}{20} - \frac{9}{10} + \frac{\sqrt{5}}{4}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
ac7f90297923c4bf | 3b64fdac755f4e5e | tutor | Show a full solution with explicit computations and a final answer line. | How many solutions in integers with $x_i\ge 2$ satisfy $x_1+\cdots+x_6=32$? | 1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_6=20.
2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).
3) Here N=20 and k=6.
4) Compute C(25,5) = 53130.
Final answer: $53130$ | 53130 | {
"format": "symbolic",
"domain": "discrete",
"topic": "stars_and_bars",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
8eaf8935d675f8dd | 3b64fdac755f4e5e | verifier | Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer. | How many solutions in integers with $x_i\ge 2$ satisfy $x_1+\cdots+x_6=32$? | Let y_i=x_i-2, then y_i≥0 and y_1+…+y_6=20.
Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).
Here N=20 and k=6.
Compute C(25,5) = 53130.
Check: matches the standard stars-and-bars formula.
Final answer: $53130$ | 53130 | {
"format": "symbolic",
"domain": "discrete",
"topic": "stars_and_bars",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
9de457133f059b48 | 2593213492596b00 | verifier | Compute the answer and add a brief verification calculation. Provide the final answer. | An urn has 13 red balls and 10 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 1 are red? | Use a hypergeometric counting argument.
Total equally likely samples: C(23,5).
Favorable samples: choose k reds and n−k blues: C(13,1)·C(10,4).
Compute totals: C(23,5) = 33649, and favorable = 2730.
Divide favorable by total and simplify.
Check: computed probability simplifies to $\frac{390}{4807}$.
Final answer: $\fra... | \frac{390}{4807} | {
"format": "symbolic",
"domain": "probability",
"topic": "hypergeometric",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
124f8c96f3e1ee02 | 2593213492596b00 | tutor | Explain each step clearly and end with the final answer. | An urn has 13 red balls and 10 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 1 are red? | 1) Use a hypergeometric counting argument.
2) Total equally likely samples: C(23,5).
3) Favorable samples: choose k reds and n−k blues: C(13,1)·C(10,4).
4) Compute totals: C(23,5) = 33649, and favorable = 2730.
5) Divide favorable by total and simplify.
Final answer: $\frac{390}{4807}$ | \frac{390}{4807} | {
"format": "symbolic",
"domain": "probability",
"topic": "hypergeometric",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
4d3158e49546dd0e | 135d7ab4d032896a | concise | Compute and present the key steps only, then the final answer. | Worker A finishes a job in 10 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job? | A's rate = 1/10 job/hour. B's rate = 1/16 job/hour.
Combined rate = 1/10 + 1/16 = 13/80 job/hour.
Time = 1 ÷ (combined rate) = 80/13 hours.
Final answer: $\frac{80}{13}$ | \frac{80}{13} | {
"format": "word",
"domain": "arithmetic",
"topic": "work_rates",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
280e6e0907f51954 | 135d7ab4d032896a | tutor | Explain each step clearly and end with the final answer. | Worker A finishes a job in 10 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job? | 1) A's rate = 1/10 job/hour. B's rate = 1/16 job/hour.
2) Combined rate = 1/10 + 1/16 = 13/80 job/hour.
3) Time = 1 ÷ (combined rate) = 80/13 hours.
Final answer: $\frac{80}{13}$ | \frac{80}{13} | {
"format": "word",
"domain": "arithmetic",
"topic": "work_rates",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
c249e7a8c3542764 | 28221eb396be6a59 | tutor | Provide a detailed solution with numbered steps and a final answer line. | Solve the congruences: x≡0 (mod 9), x≡1 (mod 13). Give the solution modulo 117. | 1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.
2) The solution is unique modulo m1·m2 = 117.
3) Compute the CRT solution to get x ≡ 27 (mod 117).
Final answer: $x\equiv 27\pmod{117}$ | x\equiv 27\pmod{117} | {
"format": "symbolic",
"domain": "discrete",
"topic": "chinese_remainder_theorem",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
5624b0a61960a69b | 28221eb396be6a59 | answer_only | Provide only the final answer (no steps). | Solve the congruences: x≡0 (mod 9), x≡1 (mod 13). Give the solution modulo 117. | $x\equiv 27\pmod{117}$ | x\equiv 27\pmod{117} | {
"format": "symbolic",
"domain": "discrete",
"topic": "chinese_remainder_theorem",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
eeef86826ca21dde | f0f142984a0a177e | tutor | Solve carefully and do not skip algebra/arithmetic steps. End with the final answer. | Solve the linear system:
$- x$ = $3$
$- 2 x + 3 y$ = $12$ | 1) Eliminate x by taking a linear combination of the equations.
2) Multiply the first equation by $-2$: $2 x$ = $-6$.
3) Multiply the second equation by $-1$: $2 x - 3 y$ = $-12$.
4) Subtract to eliminate x: $3 y$ = $6$.
5) Solve for y: y = $2$.
6) Substitute back to find x: x = $-3$.
Final answer: $(x,y)=\left(-3, 2\r... | (x,y)=\left(-3, 2\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
05f7e5681479855b | f0f142984a0a177e | concise | Provide a compact reasoning and the final answer. | Solve the linear system:
$- x$ = $3$
$- 2 x + 3 y$ = $12$ | Eliminate x by taking a linear combination of the equations.
Multiply the first equation by $-2$: $2 x$ = $-6$.
Multiply the second equation by $-1$: $2 x - 3 y$ = $-12$.
Final answer: $(x,y)=\left(-3, 2\right)$ | (x,y)=\left(-3, 2\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
ba21ed71685809c8 | cc19d6803382c812 | concise | Compute and present the key steps only, then the final answer. | Compute the indefinite integral:
$\displaystyle \int \left(2 x + 1\right) e^{2 x} + \left(2 x + 1\right) \cos{\left(x \right)}\,dx$. | Identify an integration method (term-by-term, substitution, or a known pattern).
Integrand: $\left(2 x + 1\right) e^{2 x} + \left(2 x + 1\right) \cos{\left(x \right)}$.
Compute an antiderivative: F(x) = $x e^{2 x} + 2 x \sin{\left(x \right)} + \sin{\left(x \right)} + 2 \cos{\left(x \right)}$.
Final answer: $C + x\cdot ... | C + x\cdot e^{2\cdot x} + 2\cdot x\cdot \sin{\left(x \right)} + \sin{\left(x \right)} + 2\cdot \cos{\left(x \right)} | {
"format": "symbolic",
"domain": "calculus",
"topic": "indefinite_integral",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
0b2929502164c088 | cc19d6803382c812 | tutor | Provide a detailed solution with numbered steps and a final answer line. | Compute the indefinite integral:
$\displaystyle \int \left(2 x + 1\right) e^{2 x} + \left(2 x + 1\right) \cos{\left(x \right)}\,dx$. | 1) Identify an integration method (term-by-term, substitution, or a known pattern).
2) Integrand: $\left(2 x + 1\right) e^{2 x} + \left(2 x + 1\right) \cos{\left(x \right)}$.
3) Compute an antiderivative: F(x) = $x e^{2 x} + 2 x \sin{\left(x \right)} + \sin{\left(x \right)} + 2 \cos{\left(x \right)}$.
4) Include the co... | C + x\cdot e^{2\cdot x} + 2\cdot x\cdot \sin{\left(x \right)} + \sin{\left(x \right)} + 2\cdot \cos{\left(x \right)} | {
"format": "symbolic",
"domain": "calculus",
"topic": "indefinite_integral",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
a9f0e738bc63fd7e | 7b9354e4d818564f | concise | Give a short solution with the key steps and the final answer. | Solve the linear system:
$8 x + 3 y$ = $-43$
$7 x$ = $-56$ | Eliminate x by taking a linear combination of the equations.
Multiply the first equation by $7$: $56 x + 21 y$ = $-301$.
Multiply the second equation by $8$: $56 x$ = $-448$.
Final answer: $(x,y)=\left(-8, 7\right)$ | (x,y)=\left(-8, 7\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
8ae7e965d13cb22e | 7b9354e4d818564f | tutor | Solve like a tutor: derive step-by-step and finish with the final answer. | Solve the linear system:
$8 x + 3 y$ = $-43$
$7 x$ = $-56$ | 1) Eliminate x by taking a linear combination of the equations.
2) Multiply the first equation by $7$: $56 x + 21 y$ = $-301$.
3) Multiply the second equation by $8$: $56 x$ = $-448$.
4) Subtract to eliminate x: $21 y$ = $147$.
5) Solve for y: y = $7$.
6) Substitute back to find x: x = $-8$.
Final answer: $(x,y)=\left(... | (x,y)=\left(-8, 7\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
1febcfcb15f4bc77 | 52119ff68cb17c7e | verifier_negative | A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer. | Differentiate with respect to x:
$f(x)=\left(x + 2\right) e^{x} + \left(- 3 x^{3} + x^{2} + 2 x - 2\right) \cos{\left(x - 3 \right)}$.
Student solution:
1) Differentiate term-by-term.
2) Claim (missing chain rule): $\frac{d}{dx}(- 3 x^{3} \cos{\left(x - 3 \right)}) = 3 x^{3} \sin{\left(x - 3 \right)}$.
3) So $f'(x) = ... | First incorrect step: 2
The error is omitting the chain-rule factor (derivative of the inside function).
Correct derivative of that term is $\frac{d}{dx}(- 3 x^{3} \cos{\left(x - 3 \right)}) = 3 x^{3} \sin{\left(x - 3 \right)} - 9 x^{2} \cos{\left(x - 3 \right)}$.
Therefore the correct derivative is $f'(x) = \left(x + ... | \left(x + 2\right) e^{x} - \left(x \left(x \left(1 - 3 x\right) + 2\right) - 2\right) \sin{\left(x - 3 \right)} + \left(- 9 x^{2} + 2 x + 2\right) \cos{\left(x - 3 \right)} + e^{x} | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0",
"task": "error_localization",
"is_negative": true
} |
89883c08271bc84d | 4377d7c79ac41493 | verifier | Show the solution, then validate it with a quick check. End with the final answer. | A cyclist rides at 73 miles per hour for 1/2 hours, then at 64 miles per hour for 2 hours. How many miles does the cyclist travel in total? | First leg distance = 73 × 1/2 = 73/2 miles.
Second leg distance = 64 × 2 = 128 miles.
Total distance = 73/2 + 128 = 329/2 miles.
Check: adding both legs gives 329/2.
Final answer: $\frac{329}{2}$ | \frac{329}{2} | {
"format": "word",
"domain": "arithmetic",
"topic": "rate_time_distance",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
e1708093f02d31ed | 35e10b4b38ad0d8f | answer_only | Final answer only. | Compute the indefinite integral:
$\displaystyle \int - 3 x^{3} - 2 x^{2} + 2 x + 2 \sin{\left(x \right)} - 3 \cos{\left(2 x \right)} + 1\,dx$. | $C - \frac{3\cdot x^{4}}{4} - \frac{2\cdot x^{3}}{3} + x^{2} + x - \frac{3\cdot \sin{\left(2\cdot x \right)}}{2} - 2\cdot \cos{\left(x \right)}$ | C - \frac{3\cdot x^{4}}{4} - \frac{2\cdot x^{3}}{3} + x^{2} + x - \frac{3\cdot \sin{\left(2\cdot x \right)}}{2} - 2\cdot \cos{\left(x \right)} | {
"format": "symbolic",
"domain": "calculus",
"topic": "indefinite_integral",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
52a1e963f343586a | 35e10b4b38ad0d8f | concise | Solve the problem and keep the explanation concise but complete. Provide the final answer. | Compute the indefinite integral:
$\displaystyle \int - 3 x^{3} - 2 x^{2} + 2 x + 2 \sin{\left(x \right)} - 3 \cos{\left(2 x \right)} + 1\,dx$. | Identify an integration method (term-by-term, substitution, or a known pattern).
Integrand: $- 3 x^{3} - 2 x^{2} + 2 x + 2 \sin{\left(x \right)} - 3 \cos{\left(2 x \right)} + 1$.
Compute an antiderivative: F(x) = $- \frac{3 x^{4}}{4} - \frac{2 x^{3}}{3} + x^{2} + x - \frac{3 \sin{\left(2 x \right)}}{2} - 2 \cos{\left(x... | C - \frac{3\cdot x^{4}}{4} - \frac{2\cdot x^{3}}{3} + x^{2} + x - \frac{3\cdot \sin{\left(2\cdot x \right)}}{2} - 2\cdot \cos{\left(x \right)} | {
"format": "symbolic",
"domain": "calculus",
"topic": "indefinite_integral",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
5d467dd64d51802a | 77059cff0e0e9a9a | verifier_negative | A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer. | Evaluate the limit:
$\displaystyle \lim_{x\to 1} x^{2} + 2$.
Student solution:
1) Substitute x=1 into the expression.
2) (Mistake) Simplify the expression to x^{2} - x + 2.
3) Then the limit is 2. | First incorrect step: 2
The error is dropping/altering terms before taking the limit.
Use the original expression and substitute x=1 (polynomial is continuous).
Correct limit value: 3.
Final answer: $3$ | 3 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0",
"task": "error_localization",
"is_negative": true
} |
9f9886ad2370aae2 | 6d47e8d1d7158db0 | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | A cyclist rides at 64 miles per hour for 3/2 hours, then at 46 miles per hour for 3 hours. How many miles does the cyclist travel in total? | 1) First leg distance = 64 × 3/2 = 96 miles.
2) Second leg distance = 46 × 3 = 138 miles.
3) Total distance = 96 + 138 = 234 miles.
Final answer: $234$ | 234 | {
"format": "word",
"domain": "arithmetic",
"topic": "rate_time_distance",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
d63715bbff48f8e6 | 2f8d8b07695f4e0f | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | A coin lands heads with probability p=$\frac{1}{5}$. If it is flipped 14 times, what is the probability of exactly 5 heads? | 1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).
2) Use $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$.
3) Substitute n=14, k=5, p=$\frac{1}{5}$ and simplify.
4) Here C(14,5) = 2002.
Final answer: $\frac{524812288}{6103515625}$ | \frac{524812288}{6103515625} | {
"format": "symbolic",
"domain": "probability",
"topic": "binomial_probability",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
447f92ed55db83f6 | 2f8d8b07695f4e0f | verifier | Show the solution, then validate it with a quick check. End with the final answer. | A coin lands heads with probability p=$\frac{1}{5}$. If it is flipped 14 times, what is the probability of exactly 5 heads? | Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).
Use $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$.
Substitute n=14, k=5, p=$\frac{1}{5}$ and simplify.
Here C(14,5) = 2002.
Check: expression simplifies to $\frac{524812288}{6103515625}$.
Final answer: $\frac{524812288}{6103515625}$ | \frac{524812288}{6103515625} | {
"format": "symbolic",
"domain": "probability",
"topic": "binomial_probability",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
321563dc8ef350eb | 68de494a4b261d7d | verifier | Compute the answer and add a brief verification calculation. Provide the final answer. | A coin lands heads with probability p=$\frac{3}{8}$. If it is flipped 13 times, what is the probability of exactly 7 heads? | Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).
Use $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$.
Substitute n=13, k=7, p=$\frac{3}{8}$ and simplify.
Here C(13,7) = 1716.
Check: expression simplifies to $\frac{14659734375}{137438953472}$.
Final answer: $\frac{14659734375}{137438953472}$ | \frac{14659734375}{137438953472} | {
"format": "symbolic",
"domain": "probability",
"topic": "binomial_probability",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
f32fcbcfac791aac | 68de494a4b261d7d | tutor | Solve carefully and do not skip algebra/arithmetic steps. End with the final answer. | A coin lands heads with probability p=$\frac{3}{8}$. If it is flipped 13 times, what is the probability of exactly 7 heads? | 1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).
2) Use $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$.
3) Substitute n=13, k=7, p=$\frac{3}{8}$ and simplify.
4) Here C(13,7) = 1716.
Final answer: $\frac{14659734375}{137438953472}$ | \frac{14659734375}{137438953472} | {
"format": "symbolic",
"domain": "probability",
"topic": "binomial_probability",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
6e9f95ce01a26231 | 0360536a8d694ff9 | tutor | Solve carefully and do not skip algebra/arithmetic steps. End with the final answer. | You want 22 liters of a 25% salt solution. You have a 10% solution and a 70% solution. How many liters of each should you mix? | 1) Let x = liters of 10% solution, so 22−x liters is 70% solution.
2) Salt equation: 10x + 70(22−x) = 25·22.
3) (10-70)x = (25-70)·22 ⇒ x = 33/2.
4) Then 22−x = 11/2.
Final answer: $33/2 L of 10% and 11/2 L of 70%$ | 33/2 L of 10% and 11/2 L of 70% | {
"format": "word",
"domain": "arithmetic",
"topic": "mixture",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
1f588e30d5c9a267 | 0360536a8d694ff9 | answer_only | Provide only the final answer (no steps). | You want 22 liters of a 25% salt solution. You have a 10% solution and a 70% solution. How many liters of each should you mix? | 33/2 L of 10% and 11/2 L of 70% | 33/2 L of 10% and 11/2 L of 70% | {
"format": "word",
"domain": "arithmetic",
"topic": "mixture",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
e83ade3e0ddbd3f8 | d485cfd828d2b148 | tutor | Explain each step clearly and end with the final answer. | A cyclist rides at 62 miles per hour for 2 hours, then at 39 miles per hour for 2 hours. How many miles does the cyclist travel in total? | 1) First leg distance = 62 × 2 = 124 miles.
2) Second leg distance = 39 × 2 = 78 miles.
3) Total distance = 124 + 78 = 202 miles.
Final answer: $202$ | 202 | {
"format": "word",
"domain": "arithmetic",
"topic": "rate_time_distance",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
2d1a67313eaff8cf | d485cfd828d2b148 | verifier | Solve the problem, then verify the result with a quick check. Provide the final answer. | A cyclist rides at 62 miles per hour for 2 hours, then at 39 miles per hour for 2 hours. How many miles does the cyclist travel in total? | First leg distance = 62 × 2 = 124 miles.
Second leg distance = 39 × 2 = 78 miles.
Total distance = 124 + 78 = 202 miles.
Check: adding both legs gives 202.
Final answer: $202$ | 202 | {
"format": "word",
"domain": "arithmetic",
"topic": "rate_time_distance",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
bef4c6b89519ed74 | accb830cfaf893ae | tutor | Explain each step clearly and end with the final answer. | Find the multiplicative inverse of 5 modulo 13. | 1) We want x such that 5x ≡ 1 (mod 13).
2) Use extended Euclid to find s,t with 5s + 13t = 1.
3) One Bezout identity is: 5·(-5) + 13·(2) = 1.
4) Reduce s modulo 13: x ≡ 8 (mod 13).
Final answer: $x\equiv 8\pmod{13}$ | x\equiv 8\pmod{13} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
e76034cd9c57fae4 | accb830cfaf893ae | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | Find the multiplicative inverse of 5 modulo 13. | We want x such that 5x ≡ 1 (mod 13).
Use extended Euclid to find s,t with 5s + 13t = 1.
One Bezout identity is: 5·(-5) + 13·(2) = 1.
Reduce s modulo 13: x ≡ 8 (mod 13).
Check: 5·8 mod 13 = 1.
Final answer: $x\equiv 8\pmod{13}$ | x\equiv 8\pmod{13} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
7ccfeff5f33d988e | 6bbf3d3938c041e5 | concise | Compute the result with a short explanation and give the final answer. | Evaluate the limit:
$\displaystyle \lim_{x\to 4} x^{3} + x^{2} + 7 x + 25$. | Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).
Here p(x) = $x^{4} - 3 x^{3} + 3 x^{2} - 3 x - 2$ and a = 4.
Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).
Final answer: $133$ | 133 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
8a052d668876d558 | 6bbf3d3938c041e5 | tutor | Write a complete step-by-step solution and conclude with the final answer. | Evaluate the limit:
$\displaystyle \lim_{x\to 4} x^{3} + x^{2} + 7 x + 25$. | 1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).
2) Here p(x) = $x^{4} - 3 x^{3} + 3 x^{2} - 3 x - 2$ and a = 4.
3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).
4) Differentiate: p'(x) = $4 x^{3} - 9 x^{2} + 6 x - 3$.
5) Evaluate at a: p'(4) = $133$.
Final answer: $133$ | 133 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
1d88f8aab0ce907a | de0e5b46ef7f5f12 | tutor | Provide a thorough solution as if teaching, including intermediate computations, then give the final answer. | An urn has 8 red balls and 7 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 2 are red? | 1) Use a hypergeometric counting argument.
2) Total equally likely samples: C(15,5).
3) Favorable samples: choose k reds and n−k blues: C(8,2)·C(7,3).
4) Compute totals: C(15,5) = 3003, and favorable = 980.
5) Divide favorable by total and simplify.
Final answer: $\frac{140}{429}$ | \frac{140}{429} | {
"format": "symbolic",
"domain": "probability",
"topic": "hypergeometric",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
c00bf1ff706a0058 | de0e5b46ef7f5f12 | verifier | Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer. | An urn has 8 red balls and 7 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 2 are red? | Use a hypergeometric counting argument.
Total equally likely samples: C(15,5).
Favorable samples: choose k reds and n−k blues: C(8,2)·C(7,3).
Compute totals: C(15,5) = 3003, and favorable = 980.
Divide favorable by total and simplify.
Check: computed probability simplifies to $\frac{140}{429}$.
Final answer: $\frac{140... | \frac{140}{429} | {
"format": "symbolic",
"domain": "probability",
"topic": "hypergeometric",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
f2f9afdac1e0d440 | f494e02783d9eb10 | tutor | Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer. | Two friends split $1400 in the ratio 7:3. How much does each receive? | 1) Total ratio parts = 7+3 = 10.
2) Each part = 1400 ÷ 10 = 140.
3) First share = 7 × 140 = 980.
4) Second share = 3 × 140 = 420.
Final answer: $420$ | 420 | {
"format": "word",
"domain": "arithmetic",
"topic": "ratio_split",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
c5bfc094c08d8812 | f494e02783d9eb10 | verifier | Show the solution, then validate it with a quick check. End with the final answer. | Two friends split $1400 in the ratio 7:3. How much does each receive? | Total ratio parts = 7+3 = 10.
Each part = 1400 ÷ 10 = 140.
First share = 7 × 140 = 980.
Second share = 3 × 140 = 420.
Check: 980+420=1400.
Final answer: $420$ | 420 | {
"format": "word",
"domain": "arithmetic",
"topic": "ratio_split",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
e97090a2ea6e31cd | 9e2fc5a48c97023f | tutor | Provide a detailed solution with numbered steps and a final answer line. | Solve the congruences: x≡2 (mod 3), x≡1 (mod 13). Give the solution modulo 39. | 1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.
2) The solution is unique modulo m1·m2 = 39.
3) Compute the CRT solution to get x ≡ 14 (mod 39).
Final answer: $x\equiv 14\pmod{39}$ | x\equiv 14\pmod{39} | {
"format": "symbolic",
"domain": "discrete",
"topic": "chinese_remainder_theorem",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
9bf0f047f7cde07c | 9e2fc5a48c97023f | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | Solve the congruences: x≡2 (mod 3), x≡1 (mod 13). Give the solution modulo 39. | Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.
The solution is unique modulo m1·m2 = 39.
Compute the CRT solution to get x ≡ 14 (mod 39).
Check: 14 mod 3=2 and mod 13=1.
Final answer: $x\equiv 14\pmod{39}$ | x\equiv 14\pmod{39} | {
"format": "symbolic",
"domain": "discrete",
"topic": "chinese_remainder_theorem",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
d0e0f50467251f26 | e9ea8694d5d5651d | concise | Compute and present the key steps only, then the final answer. | Evaluate the limit:
$\displaystyle \lim_{x\to -2} - x^{3} + 2 x^{2} - 3 x + 4$. | Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).
Here p(x) = $- x^{4} + x^{2} - 2 x - 3$ and a = -2.
Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).
Final answer: $26$ | 26 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
0de00ca57378acec | e9ea8694d5d5651d | tutor | Explain each step clearly and end with the final answer. | Evaluate the limit:
$\displaystyle \lim_{x\to -2} - x^{3} + 2 x^{2} - 3 x + 4$. | 1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).
2) Here p(x) = $- x^{4} + x^{2} - 2 x - 3$ and a = -2.
3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).
4) Differentiate: p'(x) = $- 4 x^{3} + 2 x - 2$.
5) Evaluate at a: p'(-2) = $26$.
Final answer: $26$ | 26 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
132a27476b0cb323 | e8b4cfaf06219beb | answer_only | Provide only the final answer (no steps). | Find the multiplicative inverse of 8 modulo 17. | $x\equiv 15\pmod{17}$ | x\equiv 15\pmod{17} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
1b43e9067d31050f | e8b4cfaf06219beb | tutor | Solve like a tutor: derive step-by-step and finish with the final answer. | Find the multiplicative inverse of 8 modulo 17. | 1) We want x such that 8x ≡ 1 (mod 17).
2) Use extended Euclid to find s,t with 8s + 17t = 1.
3) One Bezout identity is: 8·(-2) + 17·(1) = 1.
4) Reduce s modulo 17: x ≡ 15 (mod 17).
Final answer: $x\equiv 15\pmod{17}$ | x\equiv 15\pmod{17} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
b13e098e5c04dc14 | 2818d624c62634e4 | verifier | Compute the answer and include a short verification step. Give the final answer. | Solve the congruences: x≡3 (mod 9), x≡5 (mod 8). Give the solution modulo 72. | Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.
The solution is unique modulo m1·m2 = 72.
Compute the CRT solution to get x ≡ 21 (mod 72).
Check: 21 mod 9=3 and mod 8=5.
Final answer: $x\equiv 21\pmod{72}$ | x\equiv 21\pmod{72} | {
"format": "symbolic",
"domain": "discrete",
"topic": "chinese_remainder_theorem",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
9bfbc4fe13e0c42d | 2818d624c62634e4 | tutor | Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer. | Solve the congruences: x≡3 (mod 9), x≡5 (mod 8). Give the solution modulo 72. | 1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.
2) The solution is unique modulo m1·m2 = 72.
3) Compute the CRT solution to get x ≡ 21 (mod 72).
Final answer: $x\equiv 21\pmod{72}$ | x\equiv 21\pmod{72} | {
"format": "symbolic",
"domain": "discrete",
"topic": "chinese_remainder_theorem",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
223379514a753e83 | ff6fb2f4132718dc | answer_only | Provide only the final answer (no steps). | Two friends split $513 in the ratio 4:5. How much does each receive? | $285$ | 285 | {
"format": "word",
"domain": "arithmetic",
"topic": "ratio_split",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
8b04fabff8dc2b26 | ff6fb2f4132718dc | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | Two friends split $513 in the ratio 4:5. How much does each receive? | 1) Total ratio parts = 4+5 = 9.
2) Each part = 513 ÷ 9 = 57.
3) First share = 4 × 57 = 228.
4) Second share = 5 × 57 = 285.
Final answer: $285$ | 285 | {
"format": "word",
"domain": "arithmetic",
"topic": "ratio_split",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
c64728304b856ecb | b06da15d51312a7d | tutor | Provide a detailed solution with numbered steps and a final answer line. | Find the equation of the tangent line to $y=3 x^{3} + 2 x + e^{\frac{x}{2}} + \cos{\left(2 x \right)} + 3$ at $x=-1$. | 1) Differentiate: f'(x) = $9 x^{2} + \frac{e^{\frac{x}{2}}}{2} - 2 \sin{\left(2 x \right)} + 2$.
2) Slope at x=-1: m = f'(-1) = $\frac{1}{2 e^{\frac{1}{2}}} + 2 \sin{\left(2 \right)} + 11$.
3) Point on curve: (-1, f(-1)) = (-1, $-2 + \cos{\left(2 \right)} + e^{- \frac{1}{2}}$).
4) Use point-slope form y − y0 = m(x − x0... | y = \frac{x}{2\cdot e^{\frac{1}{2}}} + 2\cdot x\cdot \sin{\left(2 \right)} + 11\cdot x + \cos{\left(2 \right)} + \frac{3}{2\cdot e^{\frac{1}{2}}} + 2\cdot \sin{\left(2 \right)} + 9 | {
"format": "symbolic",
"domain": "calculus",
"topic": "tangent_line",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
e9edf903b1692f06 | 15a878ae69b5201a | verifier_negative | A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer. | Differentiate with respect to x:
$f(x)=2 \left(x - 3\right) e^{x} + \left(3 x^{3} - 3 x^{2} + 3 x\right) \cos{\left(3 x + 2 \right)}$.
Student solution:
1) Differentiate term-by-term.
2) Claim (missing chain rule): $\frac{d}{dx}(3 x^{3} \cos{\left(3 x + 2 \right)}) = - 3 x^{3} \sin{\left(3 x + 2 \right)}$.
3) So $f'(x... | First incorrect step: 2
The error is omitting the chain-rule factor (derivative of the inside function).
Correct derivative of that term is $\frac{d}{dx}(3 x^{3} \cos{\left(3 x + 2 \right)}) = - 9 x^{3} \sin{\left(3 x + 2 \right)} + 9 x^{2} \cos{\left(3 x + 2 \right)}$.
Therefore the correct derivative is $f'(x) = - 9 ... | - 9 x \left(x \left(x - 1\right) + 1\right) \sin{\left(3 x + 2 \right)} + 2 \left(x - 3\right) e^{x} + 3 \left(3 x^{2} - 2 x + 1\right) \cos{\left(3 x + 2 \right)} + 2 e^{x} | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0",
"task": "error_localization",
"is_negative": true
} |
43ea7caef10b3e7d | 6113ed9bdfbc176e | tutor | Write a complete step-by-step solution and conclude with the final answer. | Find the equation of the tangent line to $y=- x^{3} + x^{2} - x + e^{\frac{x}{2}} + \cos{\left(2 x \right)} - 2$ at $x=-1$. | 1) Differentiate: f'(x) = $- 3 x^{2} + 2 x + \frac{e^{\frac{x}{2}}}{2} - 2 \sin{\left(2 x \right)} - 1$.
2) Slope at x=-1: m = f'(-1) = $-6 + \frac{1}{2 e^{\frac{1}{2}}} + 2 \sin{\left(2 \right)}$.
3) Point on curve: (-1, f(-1)) = (-1, $\cos{\left(2 \right)} + e^{- \frac{1}{2}} + 1$).
4) Use point-slope form y − y0 = m... | y = - 6\cdot x + \frac{x}{2\cdot e^{\frac{1}{2}}} + 2\cdot x\cdot \sin{\left(2 \right)} - 5 + \cos{\left(2 \right)} + \frac{3}{2\cdot e^{\frac{1}{2}}} + 2\cdot \sin{\left(2 \right)} | {
"format": "symbolic",
"domain": "calculus",
"topic": "tangent_line",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
3d18af0db70b41f6 | 6113ed9bdfbc176e | verifier | Compute the answer and add a brief verification calculation. Provide the final answer. | Find the equation of the tangent line to $y=- x^{3} + x^{2} - x + e^{\frac{x}{2}} + \cos{\left(2 x \right)} - 2$ at $x=-1$. | Differentiate: f'(x) = $- 3 x^{2} + 2 x + \frac{e^{\frac{x}{2}}}{2} - 2 \sin{\left(2 x \right)} - 1$.
Slope at x=-1: m = f'(-1) = $-6 + \frac{1}{2 e^{\frac{1}{2}}} + 2 \sin{\left(2 \right)}$.
Point on curve: (-1, f(-1)) = (-1, $\cos{\left(2 \right)} + e^{- \frac{1}{2}} + 1$).
Use point-slope form y − y0 = m(x − x0).
Pl... | y = - 6\cdot x + \frac{x}{2\cdot e^{\frac{1}{2}}} + 2\cdot x\cdot \sin{\left(2 \right)} - 5 + \cos{\left(2 \right)} + \frac{3}{2\cdot e^{\frac{1}{2}}} + 2\cdot \sin{\left(2 \right)} | {
"format": "symbolic",
"domain": "calculus",
"topic": "tangent_line",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
33c4c55dfa332021 | f66f8ac891977545 | tutor | Explain each step clearly and end with the final answer. | Solve the linear system:
$- 8 x + 6 y$ = $26$
$- x - 8 y$ = $-23$ | 1) Eliminate x by taking a linear combination of the equations.
2) Multiply the first equation by $-1$: $8 x - 6 y$ = $-26$.
3) Multiply the second equation by $-8$: $8 x + 64 y$ = $184$.
4) Subtract to eliminate x: $- 70 y$ = $-210$.
5) Solve for y: y = $3$.
6) Substitute back to find x: x = $-1$.
Final answer: $(x,y)... | (x,y)=\left(-1, 3\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
5f0e15346cdb2a26 | f66f8ac891977545 | concise | Provide a brief derivation and the final answer. | Solve the linear system:
$- 8 x + 6 y$ = $26$
$- x - 8 y$ = $-23$ | Eliminate x by taking a linear combination of the equations.
Multiply the first equation by $-1$: $8 x - 6 y$ = $-26$.
Multiply the second equation by $-8$: $8 x + 64 y$ = $184$.
Final answer: $(x,y)=\left(-1, 3\right)$ | (x,y)=\left(-1, 3\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
06a624e5475c010b | 2f6e612e17a48446 | verifier | Solve the problem, then verify the result with a quick check. Provide the final answer. | You want 32 liters of a 45% salt solution. You have a 30% solution and a 60% solution. How many liters of each should you mix? | Let x = liters of 30% solution, so 32−x liters is 60% solution.
Salt equation: 30x + 60(32−x) = 45·32.
(30-60)x = (45-60)·32 ⇒ x = 16.
Then 32−x = 16.
Check: 30·16 + 60·16 = 45·32.
Final answer: $16 L of 30% and 16 L of 60%$ | 16 L of 30% and 16 L of 60% | {
"format": "word",
"domain": "arithmetic",
"topic": "mixture",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
577ee5c482f30bf3 | 2f6e612e17a48446 | tutor | Provide a detailed solution with numbered steps and a final answer line. | You want 32 liters of a 45% salt solution. You have a 30% solution and a 60% solution. How many liters of each should you mix? | 1) Let x = liters of 30% solution, so 32−x liters is 60% solution.
2) Salt equation: 30x + 60(32−x) = 45·32.
3) (30-60)x = (45-60)·32 ⇒ x = 16.
4) Then 32−x = 16.
Final answer: $16 L of 30% and 16 L of 60%$ | 16 L of 30% and 16 L of 60% | {
"format": "word",
"domain": "arithmetic",
"topic": "mixture",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
ac6c79530b412cf8 | 5c564341da0fb8b0 | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | Find the multiplicative inverse of 7 modulo 47. | 1) We want x such that 7x ≡ 1 (mod 47).
2) Use extended Euclid to find s,t with 7s + 47t = 1.
3) One Bezout identity is: 7·(-20) + 47·(3) = 1.
4) Reduce s modulo 47: x ≡ 27 (mod 47).
Final answer: $x\equiv 27\pmod{47}$ | x\equiv 27\pmod{47} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
f41f7d81570f18f6 | 5c564341da0fb8b0 | concise | Compute the result with a short explanation and give the final answer. | Find the multiplicative inverse of 7 modulo 47. | We want x such that 7x ≡ 1 (mod 47).
Use extended Euclid to find s,t with 7s + 47t = 1.
One Bezout identity is: 7·(-20) + 47·(3) = 1.
Final answer: $x\equiv 27\pmod{47}$ | x\equiv 27\pmod{47} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
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