Datasets:
id stringlengths 16 16 | problem_id stringlengths 16 16 | style stringclasses 5
values | instruction stringclasses 30
values | input stringlengths 47 2.18k | output stringlengths 3 1.67k | final_answer stringlengths 1 580 | metadata unknown |
|---|---|---|---|---|---|---|---|
5073fa55b1444226 | 14378efbde3f5230 | concise | Solve with minimal but sufficient steps, then provide the final answer. | Solve the quadratic equation over the real numbers:
$x^{2} + \frac{7 x}{3} - 2$ = 0. | Identify coefficients a, b, c in ax^2+bx+c=0.
Here a=$1$, b=$\frac{7}{3}$, c=$-2$.
Compute the discriminant Δ=b^2−4ac.
Final answer: $x \in \{\frac{2}{3}, -3\}$ | x \in \{\frac{2}{3}, -3\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
8e4d8f61ec3bce29 | 14378efbde3f5230 | tutor | Give a detailed derivation, showing intermediate expressions, and then the final answer. | Solve the quadratic equation over the real numbers:
$x^{2} + \frac{7 x}{3} - 2$ = 0. | 1) Identify coefficients a, b, c in ax^2+bx+c=0.
2) Here a=$1$, b=$\frac{7}{3}$, c=$-2$.
3) Compute the discriminant Δ=b^2−4ac.
4) Δ = $\frac{121}{9}$, so √Δ = $\frac{11}{3}$.
5) Apply the quadratic formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}$.
6) x1 = $\frac{2}{3}$.
7) x2 = $-3$.
8) Collect the solutions into a set.
Fina... | x \in \{\frac{2}{3}, -3\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
7f1148038db360c9 | b491c9b52c9a9ab3 | tutor | Show a full solution with explicit computations and a final answer line. | You want 10 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix? | 1) Let x = liters of 30% solution, so 10−x liters is 40% solution.
2) Salt equation: 30x + 40(10−x) = 35·10.
3) (30-40)x = (35-40)·10 ⇒ x = 5.
4) Then 10−x = 5.
Final answer: $5 L of 30% and 5 L of 40%$ | 5 L of 30% and 5 L of 40% | {
"format": "word",
"domain": "arithmetic",
"topic": "mixture",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
45f5e0ac83db0d46 | aaa146a38fe8ea5e | 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:
$f(x)=4 x^{2} + 2 x - 3$.
Student solution:
1) Differentiate term-by-term.
2) Claim: one term $\frac{d}{dx}(4 x^{2}) = 8 x^{2}$.
3) So $f'(x) = 8 x^{2} + 2$. | First incorrect step: 2
The error is differentiating a power: the exponent must decrease by 1.
Correct: $\frac{d}{dx}(4 x^{2}) = 8 x$.
Therefore the correct derivative is $f'(x) = 2 \left(4 x + 1\right)$.
Final answer: $2 \left(4 x + 1\right)$ | 2 \left(4 x + 1\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
} |
b9d7dd51b123395b | 48dbd6030f3b25ea | 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:
$f(x)=3 x^{3} \sin{\left(x \right)} - 2 x^{2} \sin{\left(x \right)} + 2 x \sin{\left(x \right)} + \cos{\left(2 x \right)}$.
Student solution:
1) Differentiate term-by-term.
2) Claim (missing chain rule): $\frac{d}{dx}(2 x \sin{\left(x \right)}) = 2 x \cos{\left(x \right)}$.
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 $\frac{d}{dx}(2 x \sin{\left(x \right)}) = 2 x \cos{\left(x \right)} + 2 \sin{\left(x \right)}$.
Therefore the correct derivative is $f'(x) = 3 x^{3} \cos{\left(x \right)} + 9 x^{2... | 3 x^{3} \cos{\left(x \right)} + 9 x^{2} \sin{\left(x \right)} - 2 x^{2} \cos{\left(x \right)} - 4 x \sin{\left(x \right)} + 2 x \cos{\left(x \right)} + 2 \sin{\left(x \right)} - 2 \sin{\left(2 x \right)} | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0",
"task": "error_localization",
"is_negative": true
} |
089da17c3ce9e100 | 8fe2092ac512f2a4 | tutor | Provide a thorough solution as if teaching, including intermediate computations, then give the final answer. | Two cards are drawn without replacement from a standard 52-card deck. Given that at least one card is an ace, what is the probability both are aces? | 1) Condition on the set of hands with at least one ace.
2) Count hands with ≥1 ace = C(52,2) − C(48,2).
3) Count hands with 2 aces = C(4,2). Probability = C(4,2)/(C(52,2)−C(48,2)).
Final answer: $\frac{1}{33}$ | \frac{1}{33} | {
"format": "symbolic",
"domain": "probability",
"topic": "conditional_probability_cards",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
9e0d75cfe74df2fe | da57f54a86d88836 | verifier | Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer. | Solve the quadratic equation over the real numbers:
$x^{2} - \sqrt{5} x + \frac{5 x}{6} - 1 + \frac{2 \sqrt{5}}{3}$ = 0. | Identify coefficients a, b, c in ax^2+bx+c=0.
Here a=$1$, b=$\frac{5}{6} - \sqrt{5}$, c=$-1 + \frac{2 \sqrt{5}}{3}$.
Compute the discriminant Δ=b^2−4ac.
Δ = $\frac{349}{36} - \frac{13 \sqrt{5}}{3}$, so √Δ = $\sqrt{\frac{349}{36} - \frac{13 \sqrt{5}}{3}}$.
Apply the quadratic formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}$.
x... | x \in \{- \frac{5}{12} + \frac{\sqrt{349 - 156\cdot \sqrt{5}}}{12} + \frac{\sqrt{5}}{2}, - \frac{5}{12} - \frac{\sqrt{349 - 156\cdot \sqrt{5}}}{12} + \frac{\sqrt{5}}{2}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
54f369be46bbd808 | da57f54a86d88836 | answer_only | Final answer only. | Solve the quadratic equation over the real numbers:
$x^{2} - \sqrt{5} x + \frac{5 x}{6} - 1 + \frac{2 \sqrt{5}}{3}$ = 0. | $x \in \{- \frac{5}{12} + \frac{\sqrt{349 - 156\cdot \sqrt{5}}}{12} + \frac{\sqrt{5}}{2}, - \frac{5}{12} - \frac{\sqrt{349 - 156\cdot \sqrt{5}}}{12} + \frac{\sqrt{5}}{2}\}$ | x \in \{- \frac{5}{12} + \frac{\sqrt{349 - 156\cdot \sqrt{5}}}{12} + \frac{\sqrt{5}}{2}, - \frac{5}{12} - \frac{\sqrt{349 - 156\cdot \sqrt{5}}}{12} + \frac{\sqrt{5}}{2}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
021c692573e4b802 | 9bf02002349e16bb | verifier | Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer. | Solve the congruences: x≡2 (mod 5), x≡5 (mod 7). Give the solution modulo 35. | Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.
The solution is unique modulo m1·m2 = 35.
Compute the CRT solution to get x ≡ 12 (mod 35).
Check: 12 mod 5=2 and mod 7=5.
Final answer: $x\equiv 12\pmod{35}$ | x\equiv 12\pmod{35} | {
"format": "symbolic",
"domain": "discrete",
"topic": "chinese_remainder_theorem",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
2e240d68717a67b5 | 9bf02002349e16bb | tutor | Give a detailed derivation, showing intermediate expressions, and then the final answer. | Solve the congruences: x≡2 (mod 5), x≡5 (mod 7). Give the solution modulo 35. | 1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.
2) The solution is unique modulo m1·m2 = 35.
3) Compute the CRT solution to get x ≡ 12 (mod 35).
Final answer: $x\equiv 12\pmod{35}$ | x\equiv 12\pmod{35} | {
"format": "symbolic",
"domain": "discrete",
"topic": "chinese_remainder_theorem",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
fef97135234c6515 | 56037c522cbd4301 | tutor | Solve like a tutor: derive step-by-step and finish with the final answer. | Evaluate the limit:
$\displaystyle \lim_{x\to -3} 2 x^{2} - 4 x + 15$. | 1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).
2) Here p(x) = $2 x^{3} + 2 x^{2} + 3 x$ and a = -3.
3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).
4) Differentiate: p'(x) = $6 x^{2} + 4 x + 3$.
5) Evaluate at a: p'(-3) = $45$.
Final answer: $45$ | 45 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
f58e1c6ee9858cf2 | 56037c522cbd4301 | concise | Give a short solution with the key steps and the final answer. | Evaluate the limit:
$\displaystyle \lim_{x\to -3} 2 x^{2} - 4 x + 15$. | Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).
Here p(x) = $2 x^{3} + 2 x^{2} + 3 x$ and a = -3.
Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).
Final answer: $45$ | 45 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
d21655f5be63df45 | 874a6edcd96a73da | tutor | Show your work in clear steps (with intermediate values) and finish with a final answer. | Solve the congruences: x≡5 (mod 11), x≡12 (mod 13). Give the solution modulo 143. | 1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.
2) The solution is unique modulo m1·m2 = 143.
3) Compute the CRT solution to get x ≡ 38 (mod 143).
Final answer: $x\equiv 38\pmod{143}$ | x\equiv 38\pmod{143} | {
"format": "symbolic",
"domain": "discrete",
"topic": "chinese_remainder_theorem",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
6f16e0f23a0f2af5 | 874a6edcd96a73da | verifier | Show the solution, then validate it with a quick check. End with the final answer. | Solve the congruences: x≡5 (mod 11), x≡12 (mod 13). Give the solution modulo 143. | Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.
The solution is unique modulo m1·m2 = 143.
Compute the CRT solution to get x ≡ 38 (mod 143).
Check: 38 mod 11=5 and mod 13=12.
Final answer: $x\equiv 38\pmod{143}$ | x\equiv 38\pmod{143} | {
"format": "symbolic",
"domain": "discrete",
"topic": "chinese_remainder_theorem",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
26e5c435061c0219 | 5546d09580a746d4 | verifier | Compute the answer and add a brief verification calculation. Provide the final answer. | Worker A finishes a job in 17 hours and Worker B finishes it in 7 hours. If they work together at constant rates, how long does it take to finish the job? | A's rate = 1/17 job/hour. B's rate = 1/7 job/hour.
Combined rate = 1/17 + 1/7 = 24/119 job/hour.
Time = 1 ÷ (combined rate) = 119/24 hours.
Check: (combined rate)×(time) = 24/119×119/24 = 1.
Final answer: $\frac{119}{24}$ | \frac{119}{24} | {
"format": "word",
"domain": "arithmetic",
"topic": "work_rates",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
dfc85517d28e273f | 5546d09580a746d4 | tutor | Show a full solution with explicit computations and a final answer line. | Worker A finishes a job in 17 hours and Worker B finishes it in 7 hours. If they work together at constant rates, how long does it take to finish the job? | 1) A's rate = 1/17 job/hour. B's rate = 1/7 job/hour.
2) Combined rate = 1/17 + 1/7 = 24/119 job/hour.
3) Time = 1 ÷ (combined rate) = 119/24 hours.
Final answer: $\frac{119}{24}$ | \frac{119}{24} | {
"format": "word",
"domain": "arithmetic",
"topic": "work_rates",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
0a82cfc1e8b70e79 | 09e3892ab0cb8f2b | concise | Compute and present the key steps only, then the final answer. | Solve the quadratic equation over the real numbers:
$x^{2} - 2 \sqrt{6} x + \frac{x}{2} - \frac{45}{2} + 9 \sqrt{6}$ = 0. | Identify coefficients a, b, c in ax^2+bx+c=0.
Here a=$1$, b=$\frac{1}{2} - 2 \sqrt{6}$, c=$- \frac{45}{2} + 9 \sqrt{6}$.
Compute the discriminant Δ=b^2−4ac.
Final answer: $x \in \{- \frac{1}{4} + \frac{\sqrt{457 - 152\cdot \sqrt{6}}}{4} + \sqrt{6}, - \frac{\sqrt{457 - 152\cdot \sqrt{6}}}{4} - \frac{1}{4} + \sqrt{6}\}$ | x \in \{- \frac{1}{4} + \frac{\sqrt{457 - 152\cdot \sqrt{6}}}{4} + \sqrt{6}, - \frac{\sqrt{457 - 152\cdot \sqrt{6}}}{4} - \frac{1}{4} + \sqrt{6}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
38bd9a2d8fd433d7 | 09e3892ab0cb8f2b | tutor | Provide a thorough solution as if teaching, including intermediate computations, then give the final answer. | Solve the quadratic equation over the real numbers:
$x^{2} - 2 \sqrt{6} x + \frac{x}{2} - \frac{45}{2} + 9 \sqrt{6}$ = 0. | 1) Identify coefficients a, b, c in ax^2+bx+c=0.
2) Here a=$1$, b=$\frac{1}{2} - 2 \sqrt{6}$, c=$- \frac{45}{2} + 9 \sqrt{6}$.
3) Compute the discriminant Δ=b^2−4ac.
4) Δ = $\frac{457}{4} - 38 \sqrt{6}$, so √Δ = $\sqrt{\frac{457}{4} - 38 \sqrt{6}}$.
5) Apply the quadratic formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}$.
6) x... | x \in \{- \frac{1}{4} + \frac{\sqrt{457 - 152\cdot \sqrt{6}}}{4} + \sqrt{6}, - \frac{\sqrt{457 - 152\cdot \sqrt{6}}}{4} - \frac{1}{4} + \sqrt{6}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
1f0c00f2cf8eed17 | b12c3d48a6da4f67 | tutor | Write a complete step-by-step solution and conclude with the final answer. | A store sells a backpack for $42. It is discounted by 40%. You buy 4 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost? | 1) Discounted price = 42 × (1 − 40/100) = 42 × 60/100 = 126/5.
2) Subtotal = 4 × 126/5 = 504/5.
3) Total with tax = 504/5 × 109/100 = 13734/125.
Final answer: $\frac{13734}{125}$ | \frac{13734}{125} | {
"format": "word",
"domain": "arithmetic",
"topic": "percent_discount",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
014ff1560a8210c3 | b12c3d48a6da4f67 | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | A store sells a backpack for $42. It is discounted by 40%. You buy 4 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost? | Discounted price = 42 × (1 − 40/100) = 42 × 60/100 = 126/5.
Subtotal = 4 × 126/5 = 504/5.
Total with tax = 504/5 × 109/100 = 13734/125.
Check: recomputing the formula yields 13734/125.
Final answer: $\frac{13734}{125}$ | \frac{13734}{125} | {
"format": "word",
"domain": "arithmetic",
"topic": "percent_discount",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
c310f0b5f75657a5 | c15028bb2acc83be | tutor | Show your work in clear steps (with intermediate values) and finish with a final answer. | Solve the quadratic equation over the real numbers:
$x^{2} - \frac{21 x}{10} - \frac{\sqrt{2} x}{2} - 1 - \frac{\sqrt{2}}{5}$ = 0. | 1) Identify coefficients a, b, c in ax^2+bx+c=0.
2) Here a=$1$, b=$- \frac{21}{10} - \frac{\sqrt{2}}{2}$, c=$-1 - \frac{\sqrt{2}}{5}$.
3) Compute the discriminant Δ=b^2−4ac.
4) Δ = $\frac{29 \sqrt{2}}{10} + \frac{891}{100}$, so √Δ = $\sqrt{\frac{29 \sqrt{2}}{10} + \frac{891}{100}}$.
5) Apply the quadratic formula $x=\f... | x \in \{\frac{\sqrt{2}}{4} + \frac{21}{20} + \frac{\sqrt{290\cdot \sqrt{2} + 891}}{20}, - \frac{\sqrt{290\cdot \sqrt{2} + 891}}{20} + \frac{\sqrt{2}}{4} + \frac{21}{20}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
4da6abd77940f1c5 | c15028bb2acc83be | verifier | Show the solution, then validate it with a quick check. End with the final answer. | Solve the quadratic equation over the real numbers:
$x^{2} - \frac{21 x}{10} - \frac{\sqrt{2} x}{2} - 1 - \frac{\sqrt{2}}{5}$ = 0. | Identify coefficients a, b, c in ax^2+bx+c=0.
Here a=$1$, b=$- \frac{21}{10} - \frac{\sqrt{2}}{2}$, c=$-1 - \frac{\sqrt{2}}{5}$.
Compute the discriminant Δ=b^2−4ac.
Δ = $\frac{29 \sqrt{2}}{10} + \frac{891}{100}$, so √Δ = $\sqrt{\frac{29 \sqrt{2}}{10} + \frac{891}{100}}$.
Apply the quadratic formula $x=\frac{-b\pm\sqrt{... | x \in \{\frac{\sqrt{2}}{4} + \frac{21}{20} + \frac{\sqrt{290\cdot \sqrt{2} + 891}}{20}, - \frac{\sqrt{290\cdot \sqrt{2} + 891}}{20} + \frac{\sqrt{2}}{4} + \frac{21}{20}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
1a32d2e16ef5bea0 | a10c9ebddd46e043 | tutor | Show a full solution with explicit computations and a final answer line. | Solve the congruences: x≡7 (mod 8), x≡6 (mod 7). Give the solution modulo 56. | 1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.
2) The solution is unique modulo m1·m2 = 56.
3) Compute the CRT solution to get x ≡ 55 (mod 56).
Final answer: $x\equiv 55\pmod{56}$ | x\equiv 55\pmod{56} | {
"format": "symbolic",
"domain": "discrete",
"topic": "chinese_remainder_theorem",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
533c294e94442e1f | a10c9ebddd46e043 | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | Solve the congruences: x≡7 (mod 8), x≡6 (mod 7). Give the solution modulo 56. | Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.
The solution is unique modulo m1·m2 = 56.
Compute the CRT solution to get x ≡ 55 (mod 56).
Check: 55 mod 8=7 and mod 7=6.
Final answer: $x\equiv 55\pmod{56}$ | x\equiv 55\pmod{56} | {
"format": "symbolic",
"domain": "discrete",
"topic": "chinese_remainder_theorem",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
4c275fe42805a034 | 27ff93c1fa6d9223 | verifier | Compute the answer and include a short verification step. Give the final answer. | Solve the linear system:
$- 8 x + 10 y$ = $80$
$10 x - y$ = $-8$ | Eliminate x by taking a linear combination of the equations.
Multiply the first equation by $10$: $- 80 x + 100 y$ = $800$.
Multiply the second equation by $-8$: $- 80 x + 8 y$ = $64$.
Subtract to eliminate x: $92 y$ = $736$.
Solve for y: y = $8$.
Substitute back to find x: x = $0$.
Check: plug (x,y)=(0,8) into both eq... | (x,y)=\left(0, 8\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
494ba24e284d348c | 6116e17f404a4160 | 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:
$\displaystyle \lim_{x\to -4} - x^{2} + 7 x - 29$.
Student solution:
1) Substitute x=-4 into the expression.
2) (Mistake) Simplify the expression to - x^{2} + 6 x - 29.
3) Then the limit is -69. | First incorrect step: 2
The error is dropping/altering terms before taking the limit.
Use the original expression and substitute x=-4 (polynomial is continuous).
Correct limit value: -73.
Final answer: $-73$ | -73 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0",
"task": "error_localization",
"is_negative": true
} |
3e6c63e8191de2f1 | 13ea56c2c58b5710 | tutor | Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer. | Find the multiplicative inverse of 34 modulo 53. | 1) We want x such that 34x ≡ 1 (mod 53).
2) Use extended Euclid to find s,t with 34s + 53t = 1.
3) One Bezout identity is: 34·(-14) + 53·(9) = 1.
4) Reduce s modulo 53: x ≡ 39 (mod 53).
Final answer: $x\equiv 39\pmod{53}$ | x\equiv 39\pmod{53} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
23bd891b828dda83 | 13ea56c2c58b5710 | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | Find the multiplicative inverse of 34 modulo 53. | We want x such that 34x ≡ 1 (mod 53).
Use extended Euclid to find s,t with 34s + 53t = 1.
One Bezout identity is: 34·(-14) + 53·(9) = 1.
Reduce s modulo 53: x ≡ 39 (mod 53).
Check: 34·39 mod 53 = 1.
Final answer: $x\equiv 39\pmod{53}$ | x\equiv 39\pmod{53} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
19e36082324900cb | 49f21da7ba95c98b | 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:
$f(x)=\left(x + 2\right) e^{x} + \left(x^{2} - 2 x + 3\right) \cos{\left(x^{2} + x - 3 \right)}$.
Student solution:
1) Differentiate term-by-term.
2) Claim (missing chain rule): $\frac{d}{dx}(x e^{x}) = x e^{x}$.
3) So $f'(x) = - 2 x^{3} \sin{\left(x^{2} + x - 3 \right)} + 3 x^{2} \sin... | First incorrect step: 2
The error is omitting the chain-rule factor (derivative of the inside function).
Correct derivative of that term is $\frac{d}{dx}(x e^{x}) = x e^{x} + e^{x}$.
Therefore the correct derivative is $f'(x) = 2 \left(x - 1\right) \cos{\left(\left(x^{2} + x\right) - 3 \right)} + \left(x + 2\right) e^{... | 2 \left(x - 1\right) \cos{\left(\left(x^{2} + x\right) - 3 \right)} + \left(x + 2\right) e^{x} - \left(2 x + 1\right) \left(x \left(x - 2\right) + 3\right) \sin{\left(x^{2} + 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
} |
2a933b1d7b29af75 | a36349e25388376a | tutor | Give a detailed derivation, showing intermediate expressions, and then the final answer. | An urn has 12 red balls and 10 blue balls. If 4 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(22,4).
3) Favorable samples: choose k reds and n−k blues: C(12,2)·C(10,2).
4) Compute totals: C(22,4) = 7315, and favorable = 2970.
5) Divide favorable by total and simplify.
Final answer: $\frac{54}{133}$ | \frac{54}{133} | {
"format": "symbolic",
"domain": "probability",
"topic": "hypergeometric",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
5a88e583806f4de5 | 0cf69d286051546d | concise | Compute the result with a short explanation and give the final answer. | A disease has prevalence $\frac{1}{200}$. A test has sensitivity $\frac{9}{10}$ and specificity $\frac{9}{10}$. If a person tests positive, what is P(disease | positive)? | Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).
P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\frac{13}{125}$.
Compute P(D|+) = $\frac{9}{208}$.
Final answer: $\frac{9}{208}$ | \frac{9}{208} | {
"format": "symbolic",
"domain": "probability",
"topic": "bayes_theorem",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
c718ec41ac60f853 | 0cf69d286051546d | tutor | Solve carefully and do not skip algebra/arithmetic steps. End with the final answer. | A disease has prevalence $\frac{1}{200}$. A test has sensitivity $\frac{9}{10}$ and specificity $\frac{9}{10}$. If a person tests positive, what is P(disease | positive)? | 1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).
2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\frac{13}{125}$.
3) Compute P(D|+) = $\frac{9}{208}$.
Final answer: $\frac{9}{208}$ | \frac{9}{208} | {
"format": "symbolic",
"domain": "probability",
"topic": "bayes_theorem",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
6a682fcbb3230904 | f018355b7ee08b26 | 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:
$\displaystyle \lim_{x\to -1} - 3 x^{2} - 1$.
Student solution:
1) Substitute x=-1 into the expression.
2) (Mistake) Simplify the expression to - 3 x^{2} - x - 1.
3) Then the limit is -3. | 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: -4.
Final answer: $-4$ | -4 | {
"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
} |
468f2ed6510fe6b6 | 948c9f90cce495a2 | tutor | Provide a thorough solution as if teaching, including intermediate computations, then give the final answer. | Compute the indefinite integral:
$\displaystyle \int \left(x + 3\right) e^{x} + \left(2 x + 1\right) \cos{\left(x \right)}\,dx$. | 1) Identify an integration method (term-by-term, substitution, or a known pattern).
2) Integrand: $\left(x + 3\right) e^{x} + \left(2 x + 1\right) \cos{\left(x \right)}$.
3) Compute an antiderivative: F(x) = $2 x \sin{\left(x \right)} + \left(x + 2\right) e^{x} + \sin{\left(x \right)} + 2 \cos{\left(x \right)}$.
4) Inc... | C + 2\cdot x\cdot \sin{\left(x \right)} + \left(x + 2\right)\cdot e^{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"
} |
afe1e32eba4f4a92 | 1a63eda3cf2df06c | concise | Compute and present the key steps only, then the final answer. | Compute the definite integral:
$\displaystyle \int_{- \pi}^{\frac{\pi}{2}} 2 x^{3} - 3 x^{2} - x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} - 3\,dx$. | Compute an antiderivative F(x) of the integrand $2 x^{3} - 3 x^{2} - x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} - 3$.
Evaluate at the bounds: F(\frac{\pi}{2}) and F(- \pi).
Subtract: F(\frac{\pi}{2}) − F(- \pi) = $- \frac{15 \pi^{4}}{32} - \frac{9 \pi^{3}}{8} - \frac{9 \pi}{2} - 2 + \frac{3 \pi^{2}}{8}$.
Fin... | - \frac{15\cdot \pi^{4}}{32} - \frac{9\cdot \pi^{3}}{8} - \frac{9\cdot \pi}{2} - 2 + \frac{3\cdot \pi^{2}}{8} | {
"format": "symbolic",
"domain": "calculus",
"topic": "definite_integral",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
cc0876e5d45b29a7 | 1a63eda3cf2df06c | tutor | Give a detailed derivation, showing intermediate expressions, and then the final answer. | Compute the definite integral:
$\displaystyle \int_{- \pi}^{\frac{\pi}{2}} 2 x^{3} - 3 x^{2} - x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} - 3\,dx$. | 1) Compute an antiderivative F(x) of the integrand $2 x^{3} - 3 x^{2} - x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} - 3$.
2) Evaluate at the bounds: F(\frac{\pi}{2}) and F(- \pi).
3) Subtract: F(\frac{\pi}{2}) − F(- \pi) = $- \frac{15 \pi^{4}}{32} - \frac{9 \pi^{3}}{8} - \frac{9 \pi}{2} - 2 + \frac{3 \pi^{2}}... | - \frac{15\cdot \pi^{4}}{32} - \frac{9\cdot \pi^{3}}{8} - \frac{9\cdot \pi}{2} - 2 + \frac{3\cdot \pi^{2}}{8} | {
"format": "symbolic",
"domain": "calculus",
"topic": "definite_integral",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
044a1dafb13d1857 | 1eb1289b8232fd66 | concise | Compute and present the key steps only, then the final answer. | Solve the linear system:
$- x - 6 y$ = $-31$
$- 8 x$ = $40$ | Eliminate x by taking a linear combination of the equations.
Multiply the first equation by $-8$: $8 x + 48 y$ = $248$.
Multiply the second equation by $-1$: $8 x$ = $-40$.
Final answer: $(x,y)=\left(-5, 6\right)$ | (x,y)=\left(-5, 6\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
28105609ee6aaab3 | 1eb1289b8232fd66 | verifier | Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer. | Solve the linear system:
$- x - 6 y$ = $-31$
$- 8 x$ = $40$ | Eliminate x by taking a linear combination of the equations.
Multiply the first equation by $-8$: $8 x + 48 y$ = $248$.
Multiply the second equation by $-1$: $8 x$ = $-40$.
Subtract to eliminate x: $48 y$ = $288$.
Solve for y: y = $6$.
Substitute back to find x: x = $-5$.
Check: plug (x,y)=(-5,6) into both equations.
F... | (x,y)=\left(-5, 6\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
7a29c0011e2ba7d8 | 271e4dbb47e11c50 | tutor | Show a full solution with explicit computations and a final answer line. | Compute the indefinite integral:
$\displaystyle \int 3 x^{4} + 2 x^{3} + x^{2} - 4 x + 2\,dx$. | 1) Identify an integration method (term-by-term, substitution, or a known pattern).
2) Integrand: $3 x^{4} + 2 x^{3} + x^{2} - 4 x + 2$.
3) Compute an antiderivative: F(x) = $\frac{3 x^{5}}{5} + \frac{x^{4}}{2} + \frac{x^{3}}{3} - 2 x^{2} + 2 x$.
4) Include the constant of integration C.
Final answer: $C + \frac{3\cdot... | C + \frac{3\cdot x^{5}}{5} + \frac{x^{4}}{2} + \frac{x^{3}}{3} - 2\cdot x^{2} + 2\cdot x | {
"format": "symbolic",
"domain": "calculus",
"topic": "indefinite_integral",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
d9e1a5ed7d69d642 | 683f51bf6c7a9c8e | 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:
$f(x)=x^{4} - x^{3} - x^{2} - 2 x - 3$.
Student solution:
1) Differentiate term-by-term.
2) Claim: one term $\frac{d}{dx}(- 2 x) = - 2 x$.
3) So $f'(x) = 4 x^{3} - 3 x^{2} - 4 x$. | First incorrect step: 2
The error is differentiating a power: the exponent must decrease by 1.
Correct: $\frac{d}{dx}(- 2 x) = -2$.
Therefore the correct derivative is $f'(x) = 4 x^{3} - 3 x^{2} - 2 x - 2$.
Final answer: $4 x^{3} - 3 x^{2} - 2 x - 2$ | 4 x^{3} - 3 x^{2} - 2 x - 2 | {
"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
} |
0afa5cd49c8da620 | f66393517368d768 | 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:
$\displaystyle \lim_{x\to 2} - 2 x^{2} - 2 x - 2$.
Student solution:
1) Substitute x=2 into the expression.
2) (Mistake) Simplify the expression to - 2 x^{2} - 3 x - 2.
3) Then the limit is -16. | 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: -14.
Final answer: $-14$ | -14 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0",
"task": "error_localization",
"is_negative": true
} |
909c48038086aef5 | 0e556ce228c5b409 | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | A cyclist rides at 24 miles per hour for 2 hours, then at 27 miles per hour for 3 hours. How many miles does the cyclist travel in total? | 1) First leg distance = 24 × 2 = 48 miles.
2) Second leg distance = 27 × 3 = 81 miles.
3) Total distance = 48 + 81 = 129 miles.
Final answer: $129$ | 129 | {
"format": "word",
"domain": "arithmetic",
"topic": "rate_time_distance",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
67fce362fd5fafde | 02dc49021ae9421e | tutor | Show a full solution with explicit computations and a final answer line. | Find the equation of the tangent line to $y=- 3 x^{3} - x^{2} - 2 x + e^{\frac{x}{2}} + \cos{\left(2 x \right)} - 2$ at $x=-2$. | 1) Differentiate: f'(x) = $- 9 x^{2} - 2 x + \frac{e^{\frac{x}{2}}}{2} - 2 \sin{\left(2 x \right)} - 2$.
2) Slope at x=-2: m = f'(-2) = $-34 + 2 \sin{\left(4 \right)} + \frac{1}{2 e}$.
3) Point on curve: (-2, f(-2)) = (-2, $\cos{\left(4 \right)} + e^{-1} + 22$).
4) Use point-slope form y − y0 = m(x − x0).
5) Plug in va... | y = - 34\cdot x + 2\cdot x\cdot \sin{\left(4 \right)} + \frac{x}{2\cdot e} - 46 + 4\cdot \sin{\left(4 \right)} + \cos{\left(4 \right)} + \frac{2}{e} | {
"format": "symbolic",
"domain": "calculus",
"topic": "tangent_line",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
441279144d6ada4a | 02dc49021ae9421e | verifier | Compute the answer and include a short verification step. Give the final answer. | Find the equation of the tangent line to $y=- 3 x^{3} - x^{2} - 2 x + e^{\frac{x}{2}} + \cos{\left(2 x \right)} - 2$ at $x=-2$. | Differentiate: f'(x) = $- 9 x^{2} - 2 x + \frac{e^{\frac{x}{2}}}{2} - 2 \sin{\left(2 x \right)} - 2$.
Slope at x=-2: m = f'(-2) = $-34 + 2 \sin{\left(4 \right)} + \frac{1}{2 e}$.
Point on curve: (-2, f(-2)) = (-2, $\cos{\left(4 \right)} + e^{-1} + 22$).
Use point-slope form y − y0 = m(x − x0).
Plug in values: y − \cos{... | y = - 34\cdot x + 2\cdot x\cdot \sin{\left(4 \right)} + \frac{x}{2\cdot e} - 46 + 4\cdot \sin{\left(4 \right)} + \cos{\left(4 \right)} + \frac{2}{e} | {
"format": "symbolic",
"domain": "calculus",
"topic": "tangent_line",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
8fb96196602ac267 | 93c560e9cdb2555e | verifier | Compute the answer and include a short verification step. Give the final answer. | Evaluate the limit:
$\displaystyle \lim_{x\to 2} 2 x^{3} + 4 x^{2} + 10 x + 17$. | Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).
Here p(x) = $2 x^{4} + 2 x^{2} - 3 x + 2$ and a = 2.
Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).
Differentiate: p'(x) = $8 x^{3} + 4 x - 3$.
Evaluate at a: p'(2) = $69$.
Check: the identity gives $69$.
Final answer: $69$ | 69 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
5f0203c53d0d38a0 | 93c560e9cdb2555e | tutor | Show your work in clear steps (with intermediate values) and finish with a final answer. | Evaluate the limit:
$\displaystyle \lim_{x\to 2} 2 x^{3} + 4 x^{2} + 10 x + 17$. | 1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).
2) Here p(x) = $2 x^{4} + 2 x^{2} - 3 x + 2$ and a = 2.
3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).
4) Differentiate: p'(x) = $8 x^{3} + 4 x - 3$.
5) Evaluate at a: p'(2) = $69$.
Final answer: $69$ | 69 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
6635e299c23bbf25 | 3ffd222b20b72af7 | concise | Solve and give a compact solution, then the final answer. | Compute the definite integral:
$\displaystyle \int_{0}^{\frac{\pi}{2}} x^{2} - 3 x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} + 3\,dx$. | Compute an antiderivative F(x) of the integrand $x^{2} - 3 x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} + 3$.
Evaluate at the bounds: F(\frac{\pi}{2}) and F(0).
Subtract: F(\frac{\pi}{2}) − F(0) = $- \frac{3 \pi^{2}}{8} + \frac{\pi^{3}}{24} + 2 + \frac{3 \pi}{2}$.
Final answer: $- \frac{3\cdot \pi^{2}}{8} + \f... | - \frac{3\cdot \pi^{2}}{8} + \frac{\pi^{3}}{24} + 2 + \frac{3\cdot \pi}{2} | {
"format": "symbolic",
"domain": "calculus",
"topic": "definite_integral",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
96b300c59c28f927 | 3ffd222b20b72af7 | tutor | Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer. | Compute the definite integral:
$\displaystyle \int_{0}^{\frac{\pi}{2}} x^{2} - 3 x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} + 3\,dx$. | 1) Compute an antiderivative F(x) of the integrand $x^{2} - 3 x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} + 3$.
2) Evaluate at the bounds: F(\frac{\pi}{2}) and F(0).
3) Subtract: F(\frac{\pi}{2}) − F(0) = $- \frac{3 \pi^{2}}{8} + \frac{\pi^{3}}{24} + 2 + \frac{3 \pi}{2}$.
4) Simplify the result.
Final answer:... | - \frac{3\cdot \pi^{2}}{8} + \frac{\pi^{3}}{24} + 2 + \frac{3\cdot \pi}{2} | {
"format": "symbolic",
"domain": "calculus",
"topic": "definite_integral",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
13657d58ca75def9 | 9cb8d66629ce6baa | tutor | Solve like a tutor: derive step-by-step and finish with the final answer. | An urn has 7 red balls and 8 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(15,5).
3) Favorable samples: choose k reds and n−k blues: C(7,1)·C(8,4).
4) Compute totals: C(15,5) = 3003, and favorable = 490.
5) Divide favorable by total and simplify.
Final answer: $\frac{70}{429}$ | \frac{70}{429} | {
"format": "symbolic",
"domain": "probability",
"topic": "hypergeometric",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
848d454ad8d79c0e | 9cb8d66629ce6baa | verifier | Solve the problem, then verify the result with a quick check. Provide the final answer. | An urn has 7 red balls and 8 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(15,5).
Favorable samples: choose k reds and n−k blues: C(7,1)·C(8,4).
Compute totals: C(15,5) = 3003, and favorable = 490.
Divide favorable by total and simplify.
Check: computed probability simplifies to $\frac{70}{429}$.
Final answer: $\frac{70}{... | \frac{70}{429} | {
"format": "symbolic",
"domain": "probability",
"topic": "hypergeometric",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
9795cc40fe84c9f0 | 9c1dbbda3b656743 | tutor | Show a full solution with explicit computations and a final answer line. | How many binary strings of length 9 contain exactly 4 ones with no two ones adjacent? | 1) Reserve 3 mandatory zeros between the 4 ones.
2) This leaves n−k+1 = 6 valid slots for the ones.
3) Choose the slots: C(6,4) = 15.
Final answer: $15$ | 15 | {
"format": "symbolic",
"domain": "discrete",
"topic": "counting_no_adjacent",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
6bccac9769b29120 | 9c1dbbda3b656743 | verifier | Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer. | How many binary strings of length 9 contain exactly 4 ones with no two ones adjacent? | Reserve 3 mandatory zeros between the 4 ones.
This leaves n−k+1 = 6 valid slots for the ones.
Choose the slots: C(6,4) = 15.
Check: standard gap method gives C(n−k+1,k).
Final answer: $15$ | 15 | {
"format": "symbolic",
"domain": "discrete",
"topic": "counting_no_adjacent",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
0929941a9f5c7ea5 | 16bbaf902fbfb4bf | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | Worker A finishes a job in 11 hours and Worker B finishes it in 4 hours. If they work together at constant rates, how long does it take to finish the job? | A's rate = 1/11 job/hour. B's rate = 1/4 job/hour.
Combined rate = 1/11 + 1/4 = 15/44 job/hour.
Time = 1 ÷ (combined rate) = 44/15 hours.
Check: (combined rate)×(time) = 15/44×44/15 = 1.
Final answer: $\frac{44}{15}$ | \frac{44}{15} | {
"format": "word",
"domain": "arithmetic",
"topic": "work_rates",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
495a558e34221746 | 16bbaf902fbfb4bf | tutor | Give a detailed derivation, showing intermediate expressions, and then the final answer. | Worker A finishes a job in 11 hours and Worker B finishes it in 4 hours. If they work together at constant rates, how long does it take to finish the job? | 1) A's rate = 1/11 job/hour. B's rate = 1/4 job/hour.
2) Combined rate = 1/11 + 1/4 = 15/44 job/hour.
3) Time = 1 ÷ (combined rate) = 44/15 hours.
Final answer: $\frac{44}{15}$ | \frac{44}{15} | {
"format": "word",
"domain": "arithmetic",
"topic": "work_rates",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
a522cf6befd025b0 | 835d3645f761103d | tutor | Write a complete step-by-step solution and conclude with the final answer. | Solve the congruences: x≡0 (mod 7), x≡2 (mod 13). Give the solution modulo 91. | 1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.
2) The solution is unique modulo m1·m2 = 91.
3) Compute the CRT solution to get x ≡ 28 (mod 91).
Final answer: $x\equiv 28\pmod{91}$ | x\equiv 28\pmod{91} | {
"format": "symbolic",
"domain": "discrete",
"topic": "chinese_remainder_theorem",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
163faec88c9fd6c2 | 145328d3f21fe968 | 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:
$f(x)=\left(x + 1\right) e^{x} + \left(- 3 x^{2} - 2 x - 2\right) \cos{\left(2 x^{2} - x + 1 \right)}$.
Student solution:
1) Differentiate term-by-term.
2) Claim (missing chain rule): $\frac{d}{dx}(- 2 \cos{\left(\left(2 x^{2} - x\right) + 1 \right)}) = 2 \sin{\left(2 x^{2} - x + 1 \ri... | First incorrect step: 2
The error is omitting the chain-rule factor (derivative of the inside function).
Correct derivative of that term is $\frac{d}{dx}(- 2 \cos{\left(\left(2 x^{2} - x\right) + 1 \right)}) = 2 \left(4 x - 1\right) \sin{\left(\left(2 x^{2} - x\right) + 1 \right)}$.
Therefore the correct derivative is ... | 2 \left(- 3 x - 1\right) \cos{\left(\left(2 x^{2} - x\right) + 1 \right)} + \left(x + 1\right) e^{x} - \left(4 x - 1\right) \left(x \left(- 3 x - 2\right) - 2\right) \sin{\left(\left(2 x^{2} - x\right) + 1 \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
} |
436a9630cc73a02f | a2ba257433a1a56b | verifier | Compute the answer and add a brief verification calculation. Provide the final answer. | Two friends split $1920 in the ratio 9:6. How much does each receive? | Total ratio parts = 9+6 = 15.
Each part = 1920 ÷ 15 = 128.
First share = 9 × 128 = 1152.
Second share = 6 × 128 = 768.
Check: 1152+768=1920.
Final answer: $768$ | 768 | {
"format": "word",
"domain": "arithmetic",
"topic": "ratio_split",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
a51955115bc30c75 | a2ba257433a1a56b | tutor | Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer. | Two friends split $1920 in the ratio 9:6. How much does each receive? | 1) Total ratio parts = 9+6 = 15.
2) Each part = 1920 ÷ 15 = 128.
3) First share = 9 × 128 = 1152.
4) Second share = 6 × 128 = 768.
Final answer: $768$ | 768 | {
"format": "word",
"domain": "arithmetic",
"topic": "ratio_split",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
0323e1cfaccc557a | 1cad086fbb1211a7 | verifier | Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer. | Compute the indefinite integral:
$\displaystyle \int \frac{\sin{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{2}{2 x + 3}\,dx$. | Identify an integration method (term-by-term, substitution, or a known pattern).
Integrand: $\frac{\sin{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{2}{2 x + 3}$.
Compute an antiderivative: F(x) = $\log{\left(2 x + 3 \right)} - \log{\left(\cos{\left(x \right)} + 1 \right)}$.
Include the constant of integration ... | C + \log{\left(2\cdot x + 3 \right)} - \log{\left(\cos{\left(x \right)} + 1 \right)} | {
"format": "symbolic",
"domain": "calculus",
"topic": "indefinite_integral",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
8999c27e70875d2d | 1cad086fbb1211a7 | tutor | Show your work in clear steps (with intermediate values) and finish with a final answer. | Compute the indefinite integral:
$\displaystyle \int \frac{\sin{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{2}{2 x + 3}\,dx$. | 1) Identify an integration method (term-by-term, substitution, or a known pattern).
2) Integrand: $\frac{\sin{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{2}{2 x + 3}$.
3) Compute an antiderivative: F(x) = $\log{\left(2 x + 3 \right)} - \log{\left(\cos{\left(x \right)} + 1 \right)}$.
4) Include the constant of ... | C + \log{\left(2\cdot x + 3 \right)} - \log{\left(\cos{\left(x \right)} + 1 \right)} | {
"format": "symbolic",
"domain": "calculus",
"topic": "indefinite_integral",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
4f47dfe9390940ea | 1c9aaca61a26f70f | tutor | Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer. | Compute the definite integral:
$\displaystyle \int_{- \pi}^{2 \pi} - 2 x^{3} + 2 x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} - 2\,dx$. | 1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} + 2 x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} - 2$.
2) Evaluate at the bounds: F(2 \pi) and F(- \pi).
3) Subtract: F(2 \pi) − F(- \pi) = $- \frac{15 \pi^{4}}{2} - 6 \pi - 4 + 3 \pi^{2}$.
4) Simplify the result.
Final answer: $- \frac{15\cdot \pi^... | - \frac{15\cdot \pi^{4}}{2} - 6\cdot \pi - 4 + 3\cdot \pi^{2} | {
"format": "symbolic",
"domain": "calculus",
"topic": "definite_integral",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
2f319366f79f6d62 | 1c9aaca61a26f70f | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | Compute the definite integral:
$\displaystyle \int_{- \pi}^{2 \pi} - 2 x^{3} + 2 x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} - 2\,dx$. | Compute an antiderivative F(x) of the integrand $- 2 x^{3} + 2 x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} - 2$.
Evaluate at the bounds: F(2 \pi) and F(- \pi).
Subtract: F(2 \pi) − F(- \pi) = $- \frac{15 \pi^{4}}{2} - 6 \pi - 4 + 3 \pi^{2}$.
Simplify the result.
Check: SymPy differentiation confirms the antid... | - \frac{15\cdot \pi^{4}}{2} - 6\cdot \pi - 4 + 3\cdot \pi^{2} | {
"format": "symbolic",
"domain": "calculus",
"topic": "definite_integral",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
f2f6052b4d637f36 | 2309eed682f1a030 | tutor | Provide a thorough solution as if teaching, including intermediate computations, then give the final answer. | You want 40 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix? | 1) Let x = liters of 30% solution, so 40−x liters is 40% solution.
2) Salt equation: 30x + 40(40−x) = 35·40.
3) (30-40)x = (35-40)·40 ⇒ x = 20.
4) Then 40−x = 20.
Final answer: $20 L of 30% and 20 L of 40%$ | 20 L of 30% and 20 L of 40% | {
"format": "word",
"domain": "arithmetic",
"topic": "mixture",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
e35662aed3501c5b | b82e99176332d679 | 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:
$\displaystyle \lim_{x\to -1} x \left(- x - 2\right)$.
Student solution:
1) Substitute x=-1 into the expression.
2) (Mistake) Simplify the expression to - x^{2} - 3 x.
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: 1.
Final answer: $1$ | 1 | {
"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
} |
9c6150b007dea59a | 673e550974014bef | verifier | Solve the problem, then verify the result with a quick check. Provide the final answer. | Find the equation of the tangent line to $y=- 3 x^{3} + x^{2} + x + 2 \sin{\left(x \right)}$ at $x=1$. | Differentiate: f'(x) = $- 9 x^{2} + 2 x + 2 \cos{\left(x \right)} + 1$.
Slope at x=1: m = f'(1) = $-6 + 2 \cos{\left(1 \right)}$.
Point on curve: (1, f(1)) = (1, $-1 + 2 \sin{\left(1 \right)}$).
Use point-slope form y − y0 = m(x − x0).
Plug in values: y − -1 + 2 \sin{\left(1 \right)} = -6 + 2 \cos{\left(1 \right)}(x − ... | y = - 6\cdot x + 2\cdot x\cdot \cos{\left(1 \right)} - 2\cdot \cos{\left(1 \right)} + 2\cdot \sin{\left(1 \right)} + 5 | {
"format": "symbolic",
"domain": "calculus",
"topic": "tangent_line",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
9ffa34bff4837b4a | 6f3a478956ce613a | 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:
$f(x)=x^{3} \sin{\left(x \right)} + x^{2} \sin{\left(x \right)} + 2 x \sin{\left(x \right)} - 3 \sin{\left(x \right)} + 2 \cos{\left(2 x \right)}$.
Student solution:
1) Differentiate term-by-term.
2) Claim (missing chain rule): $\frac{d}{dx}(x^{2} \sin{\left(x \right)}) = x^{2} \cos{\l... | First incorrect step: 2
The error is omitting the chain-rule factor (derivative of the inside function).
Correct derivative of that term is $\frac{d}{dx}(x^{2} \sin{\left(x \right)}) = x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)}$.
Therefore the correct derivative is $f'(x) = x^{3} \cos{\left(x \right)} + 3 ... | x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)} + x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)} + 2 x \cos{\left(x \right)} + 2 \sin{\left(x \right)} - 4 \sin{\left(2 x \right)} - 3 \cos{\left(x \right)} | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0",
"task": "error_localization",
"is_negative": true
} |
85482d4de34881f6 | 9a9f431d99569750 | tutor | Give a detailed derivation, showing intermediate expressions, and then the final answer. | Find the multiplicative inverse of 8 modulo 11. | 1) We want x such that 8x ≡ 1 (mod 11).
2) Use extended Euclid to find s,t with 8s + 11t = 1.
3) One Bezout identity is: 8·(-4) + 11·(3) = 1.
4) Reduce s modulo 11: x ≡ 7 (mod 11).
Final answer: $x\equiv 7\pmod{11}$ | x\equiv 7\pmod{11} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
5d9bd6ebef6bfde5 | 9a9f431d99569750 | verifier | Compute the answer and add a brief verification calculation. Provide the final answer. | Find the multiplicative inverse of 8 modulo 11. | We want x such that 8x ≡ 1 (mod 11).
Use extended Euclid to find s,t with 8s + 11t = 1.
One Bezout identity is: 8·(-4) + 11·(3) = 1.
Reduce s modulo 11: x ≡ 7 (mod 11).
Check: 8·7 mod 11 = 1.
Final answer: $x\equiv 7\pmod{11}$ | x\equiv 7\pmod{11} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
a998314ced062289 | e62155e03a5ae2aa | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | You want 36 liters of a 35% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix? | 1) Let x = liters of 15% solution, so 36−x liters is 50% solution.
2) Salt equation: 15x + 50(36−x) = 35·36.
3) (15-50)x = (35-50)·36 ⇒ x = 108/7.
4) Then 36−x = 144/7.
Final answer: $108/7 L of 15% and 144/7 L of 50%$ | 108/7 L of 15% and 144/7 L of 50% | {
"format": "word",
"domain": "arithmetic",
"topic": "mixture",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
d6ae2f00c9ba4da8 | e62155e03a5ae2aa | verifier | Compute the answer and include a short verification step. Give the final answer. | You want 36 liters of a 35% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix? | Let x = liters of 15% solution, so 36−x liters is 50% solution.
Salt equation: 15x + 50(36−x) = 35·36.
(15-50)x = (35-50)·36 ⇒ x = 108/7.
Then 36−x = 144/7.
Check: 15·108/7 + 50·144/7 = 35·36.
Final answer: $108/7 L of 15% and 144/7 L of 50%$ | 108/7 L of 15% and 144/7 L of 50% | {
"format": "word",
"domain": "arithmetic",
"topic": "mixture",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
d722790abf016db6 | bad330931ec058df | tutor | Show your work in clear steps (with intermediate values) and finish with a final answer. | An urn has 16 red balls and 28 blue balls. If 10 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(44,10).
3) Favorable samples: choose k reds and n−k blues: C(16,5)·C(28,5).
4) Compute totals: C(44,10) = 2481256778, and favorable = 429287040.
5) Divide favorable by total and simplify.
Final answer: $\frac{2358720}{13633279}$ | \frac{2358720}{13633279} | {
"format": "symbolic",
"domain": "probability",
"topic": "hypergeometric",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
3cb71d9d84f90683 | 49578745ecf6750f | 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:
$\displaystyle \lim_{x\to 3} - 2 x - 4$.
Student solution:
1) Substitute x=3 into the expression.
2) (Mistake) Simplify the expression to - 3 x - 4.
3) Then the limit is -13. | First incorrect step: 2
The error is dropping/altering terms before taking the limit.
Use the original expression and substitute x=3 (polynomial is continuous).
Correct limit value: -10.
Final answer: $-10$ | -10 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0",
"task": "error_localization",
"is_negative": true
} |
5cd3c1294a32347b | 54d401eb1276e4cd | tutor | Show your work in clear steps (with intermediate values) and finish with a final answer. | Two friends split $924 in the ratio 6:6. How much does each receive? | 1) Total ratio parts = 6+6 = 12.
2) Each part = 924 ÷ 12 = 77.
3) First share = 6 × 77 = 462.
4) Second share = 6 × 77 = 462.
Final answer: $462$ | 462 | {
"format": "word",
"domain": "arithmetic",
"topic": "ratio_split",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
522742b50d475216 | 54d401eb1276e4cd | verifier | Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer. | Two friends split $924 in the ratio 6:6. How much does each receive? | Total ratio parts = 6+6 = 12.
Each part = 924 ÷ 12 = 77.
First share = 6 × 77 = 462.
Second share = 6 × 77 = 462.
Check: 462+462=924.
Final answer: $462$ | 462 | {
"format": "word",
"domain": "arithmetic",
"topic": "ratio_split",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
5049af6e0f61596e | fa7b526602d4102b | tutor | Solve carefully and do not skip algebra/arithmetic steps. End with the final answer. | A store sells a backpack for $167. It is discounted by 10%. You buy 3 backpacks. What is the total cost? | 1) Discounted price = 167 × (1 − 10/100) = 167 × 90/100 = 1503/10.
2) Subtotal = 3 × 1503/10 = 4509/10.
Final answer: $\frac{4509}{10}$ | \frac{4509}{10} | {
"format": "word",
"domain": "arithmetic",
"topic": "percent_discount",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
5a0da5e1c1cc4805 | fa7b526602d4102b | concise | Give a short solution with the key steps and the final answer. | A store sells a backpack for $167. It is discounted by 10%. You buy 3 backpacks. What is the total cost? | Discounted price = 167 × (1 − 10/100) = 167 × 90/100 = 1503/10.
Subtotal = 3 × 1503/10 = 4509/10.
Final answer: $\frac{4509}{10}$ | \frac{4509}{10} | {
"format": "word",
"domain": "arithmetic",
"topic": "percent_discount",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
c02678ffac7c7b2a | 07199a47c3ef5867 | tutor | Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer. | Compute the definite integral:
$\displaystyle \int_{0}^{\frac{\pi}{2}} 2 x^{2} - 2 x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} + 2\,dx$. | 1) Compute an antiderivative F(x) of the integrand $2 x^{2} - 2 x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} + 2$.
2) Evaluate at the bounds: F(\frac{\pi}{2}) and F(0).
3) Subtract: F(\frac{\pi}{2}) − F(0) = $- \frac{\pi^{2}}{4} + 2 + \frac{\pi^{3}}{12} + \pi$.
4) Simplify the result.
Final answer: $- \frac{\p... | - \frac{\pi^{2}}{4} + 2 + \frac{\pi^{3}}{12} + \pi | {
"format": "symbolic",
"domain": "calculus",
"topic": "definite_integral",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
307cffbdb5534202 | 07199a47c3ef5867 | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | Compute the definite integral:
$\displaystyle \int_{0}^{\frac{\pi}{2}} 2 x^{2} - 2 x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} + 2\,dx$. | Compute an antiderivative F(x) of the integrand $2 x^{2} - 2 x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} + 2$.
Evaluate at the bounds: F(\frac{\pi}{2}) and F(0).
Subtract: F(\frac{\pi}{2}) − F(0) = $- \frac{\pi^{2}}{4} + 2 + \frac{\pi^{3}}{12} + \pi$.
Simplify the result.
Check: SymPy differentiation confirms... | - \frac{\pi^{2}}{4} + 2 + \frac{\pi^{3}}{12} + \pi | {
"format": "symbolic",
"domain": "calculus",
"topic": "definite_integral",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
bb56c26fe6796679 | d143c011702f5f25 | tutor | Show a full solution with explicit computations and a final answer line. | Find the multiplicative inverse of 18 modulo 23. | 1) We want x such that 18x ≡ 1 (mod 23).
2) Use extended Euclid to find s,t with 18s + 23t = 1.
3) One Bezout identity is: 18·(9) + 23·(-7) = 1.
4) Reduce s modulo 23: x ≡ 9 (mod 23).
Final answer: $x\equiv 9\pmod{23}$ | x\equiv 9\pmod{23} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
831152b9669a8b83 | 6854d03417f22b81 | 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:
$f(x)=x^{2} + 4 x + 1$.
Student solution:
1) Differentiate term-by-term.
2) Claim: one term $\frac{d}{dx}(x^{2}) = 2 x^{2}$.
3) So $f'(x) = 2 x^{2} + 4$. | First incorrect step: 2
The error is differentiating a power: the exponent must decrease by 1.
Correct: $\frac{d}{dx}(x^{2}) = 2 x$.
Therefore the correct derivative is $f'(x) = 2 \left(x + 2\right)$.
Final answer: $2 \left(x + 2\right)$ | 2 \left(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
} |
c9914dd4f7beb05a | 50ae1a8874323499 | tutor | Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer. | A cyclist rides at 75 miles per hour for 1/2 hours, then at 87 miles per hour for 5/2 hours. How many miles does the cyclist travel in total? | 1) First leg distance = 75 × 1/2 = 75/2 miles.
2) Second leg distance = 87 × 5/2 = 435/2 miles.
3) Total distance = 75/2 + 435/2 = 255 miles.
Final answer: $255$ | 255 | {
"format": "word",
"domain": "arithmetic",
"topic": "rate_time_distance",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
916ff6a6ad368116 | 50ae1a8874323499 | verifier | Show the solution, then validate it with a quick check. End with the final answer. | A cyclist rides at 75 miles per hour for 1/2 hours, then at 87 miles per hour for 5/2 hours. How many miles does the cyclist travel in total? | First leg distance = 75 × 1/2 = 75/2 miles.
Second leg distance = 87 × 5/2 = 435/2 miles.
Total distance = 75/2 + 435/2 = 255 miles.
Check: adding both legs gives 255.
Final answer: $255$ | 255 | {
"format": "word",
"domain": "arithmetic",
"topic": "rate_time_distance",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
b110a2d8b25fca7e | 2dc0051335f633be | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | Solve the quadratic equation over the real numbers:
$x^{2} - \sqrt{11} x + \frac{23 x}{12} - \frac{\sqrt{11}}{4} + \frac{5}{12}$ = 0. | Identify coefficients a, b, c in ax^2+bx+c=0.
Here a=$1$, b=$\frac{23}{12} - \sqrt{11}$, c=$\frac{5}{12} - \frac{\sqrt{11}}{4}$.
Compute the discriminant Δ=b^2−4ac.
Δ = $\frac{1873}{144} - \frac{17 \sqrt{11}}{6}$, so √Δ = $\sqrt{\frac{1873}{144} - \frac{17 \sqrt{11}}{6}}$.
Apply the quadratic formula $x=\frac{-b\pm\sqr... | x \in \{- \frac{23}{24} + \frac{\sqrt{1873 - 408\cdot \sqrt{11}}}{24} + \frac{\sqrt{11}}{2}, - \frac{23}{24} - \frac{\sqrt{1873 - 408\cdot \sqrt{11}}}{24} + \frac{\sqrt{11}}{2}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
c27711d87bc45627 | 2dc0051335f633be | tutor | Provide a detailed solution with numbered steps and a final answer line. | Solve the quadratic equation over the real numbers:
$x^{2} - \sqrt{11} x + \frac{23 x}{12} - \frac{\sqrt{11}}{4} + \frac{5}{12}$ = 0. | 1) Identify coefficients a, b, c in ax^2+bx+c=0.
2) Here a=$1$, b=$\frac{23}{12} - \sqrt{11}$, c=$\frac{5}{12} - \frac{\sqrt{11}}{4}$.
3) Compute the discriminant Δ=b^2−4ac.
4) Δ = $\frac{1873}{144} - \frac{17 \sqrt{11}}{6}$, so √Δ = $\sqrt{\frac{1873}{144} - \frac{17 \sqrt{11}}{6}}$.
5) Apply the quadratic formula $x=... | x \in \{- \frac{23}{24} + \frac{\sqrt{1873 - 408\cdot \sqrt{11}}}{24} + \frac{\sqrt{11}}{2}, - \frac{23}{24} - \frac{\sqrt{1873 - 408\cdot \sqrt{11}}}{24} + \frac{\sqrt{11}}{2}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
f283b64bb476c675 | a9836c54ff5c5377 | 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:
$f(x)=\frac{x^{3} + 3 x^{2} + 2 x + \left(2 x - 3\right) e^{\frac{x}{2}} \sin{\left(x \right)} + 2}{2 x - 3}$.
Student solution:
1) Differentiate term-by-term.
2) Claim (missing chain rule): $\frac{d}{dx}(- \frac{3 e^{\frac{x}{2}} \sin{\left(x \right)}}{2 x - 3}) = - \frac{3 e^{\frac{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}(- \frac{3 e^{\frac{x}{2}} \sin{\left(x \right)}}{2 x - 3}) = - \frac{3 e^{\frac{x}{2}} \sin{\left(x \right)}}{2 \left(2 x - 3\right)} - \frac{3 e^{\frac{x}{2}} \cos{\... | \frac{- 4 x^{3} - 12 x^{2} - 8 x + \left(2 x - 3\right) \left(6 x^{2} + 12 x + \left(2 x - 3\right) e^{\frac{x}{2}} \sin{\left(x \right)} + 2 \left(2 x - 3\right) e^{\frac{x}{2}} \cos{\left(x \right)} + 4 e^{\frac{x}{2}} \sin{\left(x \right)} + 4\right) - 4 \left(2 x - 3\right) e^{\frac{x}{2}} \sin{\left(x \right)} - 8... | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0",
"task": "error_localization",
"is_negative": true
} |
01dca67b33c1a600 | 383f8185b5358276 | tutor | Solve carefully and do not skip algebra/arithmetic steps. End with the final answer. | Solve the linear system:
$2 x + 4 y$ = $-34$
$- 7 x + 9 y$ = $-19$ | 1) Eliminate x by taking a linear combination of the equations.
2) Multiply the first equation by $-7$: $- 14 x - 28 y$ = $238$.
3) Multiply the second equation by $2$: $- 14 x + 18 y$ = $-38$.
4) Subtract to eliminate x: $- 46 y$ = $276$.
5) Solve for y: y = $-6$.
6) Substitute back to find x: x = $-5$.
Final answer: ... | (x,y)=\left(-5, -6\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
23f69048aba81d86 | 05feeace1c3d8a0b | tutor | Give a detailed derivation, showing intermediate expressions, and then the final answer. | Solve the quadratic equation over the real numbers:
$x^{2} - \frac{25 x}{6} - \sqrt{2} x + \frac{8 \sqrt{2}}{3} + 4$ = 0. | 1) Identify coefficients a, b, c in ax^2+bx+c=0.
2) Here a=$1$, b=$- \frac{25}{6} - \sqrt{2}$, c=$\frac{8 \sqrt{2}}{3} + 4$.
3) Compute the discriminant Δ=b^2−4ac.
4) Δ = $\frac{121}{36} - \frac{7 \sqrt{2}}{3}$, so √Δ = $\sqrt{\frac{121}{36} - \frac{7 \sqrt{2}}{3}}$.
5) Apply the quadratic formula $x=\frac{-b\pm\sqrt{\... | x \in \{\frac{\sqrt{121 - 84\cdot \sqrt{2}}}{12} + \frac{\sqrt{2}}{2} + \frac{25}{12}, - \frac{\sqrt{121 - 84\cdot \sqrt{2}}}{12} + \frac{\sqrt{2}}{2} + \frac{25}{12}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
1e9b7a0b4e408a3c | 05feeace1c3d8a0b | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | Solve the quadratic equation over the real numbers:
$x^{2} - \frac{25 x}{6} - \sqrt{2} x + \frac{8 \sqrt{2}}{3} + 4$ = 0. | Identify coefficients a, b, c in ax^2+bx+c=0.
Here a=$1$, b=$- \frac{25}{6} - \sqrt{2}$, c=$\frac{8 \sqrt{2}}{3} + 4$.
Compute the discriminant Δ=b^2−4ac.
Δ = $\frac{121}{36} - \frac{7 \sqrt{2}}{3}$, so √Δ = $\sqrt{\frac{121}{36} - \frac{7 \sqrt{2}}{3}}$.
Apply the quadratic formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}$.
x... | x \in \{\frac{\sqrt{121 - 84\cdot \sqrt{2}}}{12} + \frac{\sqrt{2}}{2} + \frac{25}{12}, - \frac{\sqrt{121 - 84\cdot \sqrt{2}}}{12} + \frac{\sqrt{2}}{2} + \frac{25}{12}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
ee46575000eda9de | fb66368f2415621d | tutor | Show a full solution with explicit computations and a final answer line. | Differentiate with respect to x:
$f(x)=2 \left(x + 2\right) e^{x} + \left(- 3 x^{3} - x + 1\right) \cos{\left(- x^{2} + x + 2 \right)}$. | 1) Let f(x) = $2 \left(x + 2\right) e^{x} + \left(- 3 x^{3} - x + 1\right) \cos{\left(- x^{2} + x + 2 \right)}$.
2) Differentiate using product/chain/quotient rules as needed.
3) Simplify the derivative to get f'(x) = $- \left(1 - 2 x\right) \left(- 3 x^{3} - x + 1\right) \sin{\left(- x^{2} + x + 2 \right)} + 2 \left(x... | \left(2\cdot x - 1\right)\cdot \left(- 3\cdot x^{3} - x + 1\right)\cdot \sin{\left(- x^{2} + x + 2 \right)} + \left(2\cdot x + 4\right)\cdot e^{x} + \left(- 9\cdot x^{2} - 1\right)\cdot \cos{\left(- x^{2} + x + 2 \right)} + 2\cdot e^{x} | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
8ad2791bddb39eb8 | fb66368f2415621d | concise | Compute and present the key steps only, then the final answer. | Differentiate with respect to x:
$f(x)=2 \left(x + 2\right) e^{x} + \left(- 3 x^{3} - x + 1\right) \cos{\left(- x^{2} + x + 2 \right)}$. | Let f(x) = $2 \left(x + 2\right) e^{x} + \left(- 3 x^{3} - x + 1\right) \cos{\left(- x^{2} + x + 2 \right)}$.
Differentiate using product/chain/quotient rules as needed.
Simplify the derivative to get f'(x) = $- \left(1 - 2 x\right) \left(- 3 x^{3} - x + 1\right) \sin{\left(- x^{2} + x + 2 \right)} + 2 \left(x + 2\righ... | \left(2\cdot x - 1\right)\cdot \left(- 3\cdot x^{3} - x + 1\right)\cdot \sin{\left(- x^{2} + x + 2 \right)} + \left(2\cdot x + 4\right)\cdot e^{x} + \left(- 9\cdot x^{2} - 1\right)\cdot \cos{\left(- x^{2} + x + 2 \right)} + 2\cdot e^{x} | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
08b889a2e6625ec2 | c0fbe97de884fdee | tutor | Provide a detailed solution with numbered steps and a final answer line. | Differentiate with respect to x:
$f(x)=x^{3} + 4 x + 1$. | 1) Differentiate term-by-term:
2) d/dx(1) = 0.
3) d/dx(x^{3}) = 3 x^{2}.
4) d/dx(4 x) = 4.
5) Combine: f'(x) = 3 x^{2} + 4.
Final answer: $3\cdot x^{2} + 4$ | 3\cdot x^{2} + 4 | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
2e0dd62264260ce1 | c0fbe97de884fdee | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | Differentiate with respect to x:
$f(x)=x^{3} + 4 x + 1$. | Differentiate term-by-term:
d/dx(1) = 0.
d/dx(x^{3}) = 3 x^{2}.
d/dx(4 x) = 4.
Combine: f'(x) = 3 x^{2} + 4.
Check: symbolic differentiation in SymPy matches the derived expression.
Final answer: $3\cdot x^{2} + 4$ | 3\cdot x^{2} + 4 | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
cf9a413f903267d8 | 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): $\frac{d}{dx}(- 3 \cos{\left(x + 2 \right)}) = 3 \sin{\left(x + 2 \right)}$.
3) So $f'(x) = 2 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 $\frac{d}{dx}(- 3 \cos{\left(x + 2 \right)}) = 3 \sin{\left(x + 2 \right)}$.
Therefore the correct derivative is $f'(x) = \left(x + 1\right) e^{x} - \left(x \left(x \left(1 - 2 x\r... | \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
} |
cfd3b47c1dc07d52 | 28f9fbe3d798596d | tutor | Provide a detailed solution with numbered steps and a final answer line. | Solve the quadratic equation over the real numbers:
$3 x^{2} + \frac{33 x}{4} + \frac{21}{4}$ = 0. | 1) Identify coefficients a, b, c in ax^2+bx+c=0.
2) Here a=$3$, b=$\frac{33}{4}$, c=$\frac{21}{4}$.
3) Compute the discriminant Δ=b^2−4ac.
4) Δ = $\frac{81}{16}$, so √Δ = $\frac{9}{4}$.
5) Apply the quadratic formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}$.
6) x1 = $-1$.
7) x2 = $- \frac{7}{4}$.
8) Collect the solutions into... | x \in \{-1, - \frac{7}{4}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
ec9ff43afc123a0b | 3ef96dfabadc66d8 | 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:
$\displaystyle \lim_{x\to -1} - 2 x^{3} + 5 x^{2} - 6 x + 8$.
Student solution:
1) Substitute x=-1 into the expression.
2) (Mistake) Simplify the expression to - 2 x^{3} + 5 x^{2} - 7 x + 8.
3) Then the limit is 22. | 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: 21.
Final answer: $21$ | 21 | {
"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
} |
2747f0c74b1c6093 | 9788f6ca403b6b1e | tutor | Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer. | A coin lands heads with probability p=$\frac{1}{4}$. If it is flipped 8 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=8, k=5, p=$\frac{1}{4}$ and simplify.
4) Here C(8,5) = 56.
Final answer: $\frac{189}{8192}$ | \frac{189}{8192} | {
"format": "symbolic",
"domain": "probability",
"topic": "binomial_probability",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
90813d1b22a2569d | 89aaf14324e98359 | tutor | Solve carefully and do not skip algebra/arithmetic steps. End with the final answer. | An urn has 7 red balls and 7 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red? | 1) Use a hypergeometric counting argument.
2) Total equally likely samples: C(14,7).
3) Favorable samples: choose k reds and n−k blues: C(7,3)·C(7,4).
4) Compute totals: C(14,7) = 3432, and favorable = 1225.
5) Divide favorable by total and simplify.
Final answer: $\frac{1225}{3432}$ | \frac{1225}{3432} | {
"format": "symbolic",
"domain": "probability",
"topic": "hypergeometric",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
e72f4de8a186432f | 220b1d991a18d9da | 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:
$f(x)=\left(x + 1\right) e^{x} + \left(- 3 x^{2} - 3 x - 2\right) \cos{\left(x - 3 \right)}$.
Student solution:
1) Differentiate term-by-term.
2) Claim (missing chain rule): $\frac{d}{dx}(- 2 \cos{\left(x - 3 \right)}) = 2 \sin{\left(x - 3 \right)}$.
3) So $f'(x) = 3 x^{2} \sin{\left(x... | First incorrect step: 2
The error is omitting the chain-rule factor (derivative of the inside function).
Correct derivative of that term is $\frac{d}{dx}(- 2 \cos{\left(x - 3 \right)}) = 2 \sin{\left(x - 3 \right)}$.
Therefore the correct derivative is $f'(x) = 3 \left(- 2 x - 1\right) \cos{\left(x - 3 \right)} + \left... | 3 \left(- 2 x - 1\right) \cos{\left(x - 3 \right)} + \left(x + 1\right) e^{x} - \left(3 x \left(- x - 1\right) - 2\right) \sin{\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
} |
f47652d5ae49b0d2 | 139e7461e8f5f798 | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | Two friends split $1920 in the ratio 4:8. How much does each receive? | 1) Total ratio parts = 4+8 = 12.
2) Each part = 1920 ÷ 12 = 160.
3) First share = 4 × 160 = 640.
4) Second share = 8 × 160 = 1280.
Final answer: $1280$ | 1280 | {
"format": "word",
"domain": "arithmetic",
"topic": "ratio_split",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
a5f0a605e25f9b55 | 139e7461e8f5f798 | answer_only | Provide only the final answer (no steps). | Two friends split $1920 in the ratio 4:8. How much does each receive? | $1280$ | 1280 | {
"format": "word",
"domain": "arithmetic",
"topic": "ratio_split",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
End of preview. Expand in Data Studio
Dataset Card for Math-HQ-10k
Dataset Description
Math-HQ-10k is a high-quality, synthetic, programmatic math instruction dataset designed to train large language models on mathematical reasoning, problem solving, and self-correction.
The dataset mixes symbolic/LaTeX-style math problems and plain-language word problems. Ground-truth computations were generated and verified programmatically with SymPy (v1.14.0) to reduce arithmetic and algebraic errors in the targets.
- Curated by: x0root
- Language(s): English
- License: MIT
- Format: JSONL
- Total Examples: 10,000
- Unique problem IDs: 4,995
Dataset Structure
Data Instances
Each row in the dataset corresponds to a specific instruction style applied to a mathematical problem.
{
"id": "5073fa55b1444226",
"problem_id": "14378efbde3f5230",
"style": "concise",
"instruction": "Solve with minimal but sufficient steps, then provide the final answer.",
"input": "Solve the quadratic equation over the real numbers:\n$x^{2} + \\frac{7 x}{3} - 2$ = 0.",
"output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{7}{3}$, c=$-2$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{2}{3}, -3\\}$",
"final_answer": "x \\in \\{\\frac{2}{3}, -3\\}",
"metadata": {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
}
}
Data Fields
id(string): Identifier for the prompt-completion pair. In this raw export, someidvalues repeat because there are repeated rows.problem_id(string): Shared identifier for the base mathematical problem. Use this to group style variants of the same problem.style(string): The response style requested, such astutor,concise,verifier,answer_only, orverifier_negative.instruction(string): The instruction that defines the desired response behavior.input(string): The math problem, either in LaTeX-heavy form or plain-language word-problem form.output(string): The target response.final_answer(string): The answer extracted for evaluation.metadata(dictionary):format: Output format type, currentlysymbolicorword.domain: Broad mathematical field such asalgebra,calculus,arithmetic,discrete, orprobability.topic: More specific problem type, such asquadratic_equation,derivative, ormixture.difficulty: Integer from 1 to 5.verified: Boolean indicating deterministic verification.source: Generator source.sympy_version: SymPy version used for validation.
Key Features & Supported Tasks
1. Error Localization (verifier_negative)
The dataset includes negative examples where a model must identify the first incorrect step in a flawed solution, explain the error, and give the corrected reasoning.
2. Multi-Style Instruction Tuning
The same base problem is represented with multiple response styles:
tutor: detailed, pedagogical solutionsconcise: minimal but sufficient derivationsverifier: solutions with explicit checksanswer_only: final-answer-focused responsesverifier_negative: critique and correction examples
3. Curriculum Learning Metadata
Every row includes domain, topic, and difficulty, which makes the dataset useful for curriculum learning and difficulty-based sampling.
Dataset Creation
The data was generated using a structured synthetic pipeline (synthetic_v3_open). Problems and target solutions were derived and verified programmatically with SymPy to reduce hallucinated arithmetic and algebraic mistakes.
Considerations for Using the Data
- The raw export contains repeated rows, so deduplication is recommended before training or evaluation if you need strict uniqueness.
- Split train/test sets by
problem_idto avoid leakage across different style variants of the same problem. - The dataset contains both symbolic/LaTeX-style inputs and plain-language word problems, so the tokenizer and preprocessing pipeline should handle both.
Citation
If you use this dataset in your research or for training models, please cite:
@misc{MathHQ10k,
author = {x0root},
title = {Math-HQ-10k: Programmatic Math Instruction Dataset},
year = {2026},
publisher = {Hugging Face},
journal = {Hugging Face repository},
howpublished = {\url{https://huggingface.co/datasets/x0root/math-hq-10k}}
}
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