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 |
|---|---|---|---|---|---|---|---|
1bd7720138a03c76 | f6b6d6c059020932 | verifier | Compute the answer and include a short verification step. Give the final answer. | An urn has 17 red balls and 17 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 3 are red? | Use a hypergeometric counting argument.
Total equally likely samples: C(34,11).
Favorable samples: choose k reds and n−k blues: C(17,3)·C(17,8).
Compute totals: C(34,11) = 286097760, and favorable = 16530800.
Divide favorable by total and simplify.
Check: computed probability simplifies to $\frac{935}{16182}$.
Final an... | \frac{935}{16182} | {
"format": "symbolic",
"domain": "probability",
"topic": "hypergeometric",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
7ce725a041c18eb8 | 22a3089edc84880c | 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:
$\displaystyle \lim_{x\to 2} - x - 3$.
Student solution:
1) Substitute x=2 into the expression.
2) (Mistake) Simplify the expression to - 2 x - 3.
3) Then the limit is -7. | 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: -5.
Final answer: $-5$ | -5 | {
"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
} |
155617e06777d376 | 61597971a8e26afc | 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:
$\displaystyle \lim_{x\to 3} - x^{2} - 2 x - 3$.
Student solution:
1) Substitute x=3 into the expression.
2) (Mistake) Simplify the expression to - x^{2} - 3 x - 3.
3) Then the limit is -21. | 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: -18.
Final answer: $-18$ | -18 | {
"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
} |
391a8f24c3b2321a | 935b98dd99668ec4 | tutor | Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer. | Solve the congruences: x≡4 (mod 7), x≡2 (mod 10). Give the solution modulo 70. | 1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.
2) The solution is unique modulo m1·m2 = 70.
3) Compute the CRT solution to get x ≡ 32 (mod 70).
Final answer: $x\equiv 32\pmod{70}$ | x\equiv 32\pmod{70} | {
"format": "symbolic",
"domain": "discrete",
"topic": "chinese_remainder_theorem",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
224f48eaa65e3b7b | b46d5bc7fb935668 | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | Evaluate the limit:
$\displaystyle \lim_{x\to -1} - 2 x^{2} + 5 x - 5$. | 1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).
2) Here p(x) = $- 2 x^{3} + 3 x^{2}$ and a = -1.
3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).
4) Differentiate: p'(x) = $- 6 x^{2} + 6 x$.
5) Evaluate at a: p'(-1) = $-12$.
Final answer: $-12$ | -12 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
b0bdb92fa6e93cb8 | b46d5bc7fb935668 | verifier | Compute the answer and add a brief verification calculation. Provide the final answer. | Evaluate the limit:
$\displaystyle \lim_{x\to -1} - 2 x^{2} + 5 x - 5$. | Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).
Here p(x) = $- 2 x^{3} + 3 x^{2}$ and a = -1.
Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).
Differentiate: p'(x) = $- 6 x^{2} + 6 x$.
Evaluate at a: p'(-1) = $-12$.
Check: the identity gives $-12$.
Final answer: $-12$ | -12 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
075688ff15f375ae | 5d6f611bbe391b5c | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | How many solutions in nonnegative integers satisfy $x_1+\cdots+x_5=27$? | 1) We are counting nonnegative integer solutions to a sum constraint.
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=27 and k=5.
4) Compute C(31,4) = 31465.
Final answer: $31465$ | 31465 | {
"format": "symbolic",
"domain": "discrete",
"topic": "stars_and_bars",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
c08f7224aa575a29 | 5d6f611bbe391b5c | verifier | Compute the answer and include a short verification step. Give the final answer. | How many solutions in nonnegative integers satisfy $x_1+\cdots+x_5=27$? | We are counting nonnegative integer solutions to a sum constraint.
Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).
Here N=27 and k=5.
Compute C(31,4) = 31465.
Check: matches the standard stars-and-bars formula.
Final answer: $31465$ | 31465 | {
"format": "symbolic",
"domain": "discrete",
"topic": "stars_and_bars",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
74fe7fd71082e3d1 | 25ec14d24ded7e9a | verifier | Solve the problem, then verify the result with a quick check. Provide the final answer. | How many binary strings of length 23 contain exactly 9 ones with no two ones adjacent? | Reserve 8 mandatory zeros between the 9 ones.
This leaves n−k+1 = 15 valid slots for the ones.
Choose the slots: C(15,9) = 5005.
Check: standard gap method gives C(n−k+1,k).
Final answer: $5005$ | 5005 | {
"format": "symbolic",
"domain": "discrete",
"topic": "counting_no_adjacent",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
bfc5385c04366b8a | 25ec14d24ded7e9a | tutor | Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer. | How many binary strings of length 23 contain exactly 9 ones with no two ones adjacent? | 1) Reserve 8 mandatory zeros between the 9 ones.
2) This leaves n−k+1 = 15 valid slots for the ones.
3) Choose the slots: C(15,9) = 5005.
Final answer: $5005$ | 5005 | {
"format": "symbolic",
"domain": "discrete",
"topic": "counting_no_adjacent",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
95e8a5b222e7cb9f | 17676e3b922393cb | concise | Provide a brief derivation and the final answer. | Solve the linear system:
$- 3 x - 4 y$ = $-5$
$- 3 y$ = $-6$ | Eliminate x by taking a linear combination of the equations.
Multiply the first equation by $0$: $0$ = $0$.
Multiply the second equation by $-3$: $9 y$ = $18$.
Final answer: $(x,y)=\left(-1, 2\right)$ | (x,y)=\left(-1, 2\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
f55d0fea3a6c25e8 | 17676e3b922393cb | tutor | Explain each step clearly and end with the final answer. | Solve the linear system:
$- 3 x - 4 y$ = $-5$
$- 3 y$ = $-6$ | 1) Eliminate x by taking a linear combination of the equations.
2) Multiply the first equation by $0$: $0$ = $0$.
3) Multiply the second equation by $-3$: $9 y$ = $18$.
4) Subtract to eliminate x: $- 9 y$ = $-18$.
5) Solve for y: y = $2$.
6) Substitute back to find x: x = $-1$.
Final answer: $(x,y)=\left(-1, 2\right)$ | (x,y)=\left(-1, 2\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
8b337a91e612fea8 | abd6a74f32262289 | verifier | Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer. | Solve the linear system:
$- 4 x + 3 y$ = $9$
$- 3 x$ = $9$ | Eliminate x by taking a linear combination of the equations.
Multiply the first equation by $-3$: $12 x - 9 y$ = $-27$.
Multiply the second equation by $-4$: $12 x$ = $-36$.
Subtract to eliminate x: $- 9 y$ = $9$.
Solve for y: y = $-1$.
Substitute back to find x: x = $-3$.
Check: plug (x,y)=(-3,-1) into both equations.... | (x,y)=\left(-3, -1\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
ca508d3b84c8eeec | abd6a74f32262289 | tutor | Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer. | Solve the linear system:
$- 4 x + 3 y$ = $9$
$- 3 x$ = $9$ | 1) Eliminate x by taking a linear combination of the equations.
2) Multiply the first equation by $-3$: $12 x - 9 y$ = $-27$.
3) Multiply the second equation by $-4$: $12 x$ = $-36$.
4) Subtract to eliminate x: $- 9 y$ = $9$.
5) Solve for y: y = $-1$.
6) Substitute back to find x: x = $-3$.
Final answer: $(x,y)=\left(-... | (x,y)=\left(-3, -1\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
c4503a7c5fc990fa | 23c3b00adedb75f4 | tutor | Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer. | Find the equation of the tangent line to $y=- x^{3} + 2 x^{2} + e^{\frac{x}{2}} + \cos{\left(2 x \right)} - 3$ at $x=-1$. | 1) Differentiate: f'(x) = $- 3 x^{2} + 4 x + \frac{e^{\frac{x}{2}}}{2} - 2 \sin{\left(2 x \right)}$.
2) Slope at x=-1: m = f'(-1) = $-7 + \frac{1}{2 e^{\frac{1}{2}}} + 2 \sin{\left(2 \right)}$.
3) Point on curve: (-1, f(-1)) = (-1, $\cos{\left(2 \right)} + e^{- \frac{1}{2}}$).
4) Use point-slope form y − y0 = m(x − x0)... | y = - 7\cdot x + \frac{x}{2\cdot e^{\frac{1}{2}}} + 2\cdot x\cdot \sin{\left(2 \right)} - 7 + \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": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
dc5a62ab8ce202d7 | 23c3b00adedb75f4 | verifier | Compute the answer and include a short verification step. Give the final answer. | Find the equation of the tangent line to $y=- x^{3} + 2 x^{2} + e^{\frac{x}{2}} + \cos{\left(2 x \right)} - 3$ at $x=-1$. | Differentiate: f'(x) = $- 3 x^{2} + 4 x + \frac{e^{\frac{x}{2}}}{2} - 2 \sin{\left(2 x \right)}$.
Slope at x=-1: m = f'(-1) = $-7 + \frac{1}{2 e^{\frac{1}{2}}} + 2 \sin{\left(2 \right)}$.
Point on curve: (-1, f(-1)) = (-1, $\cos{\left(2 \right)} + e^{- \frac{1}{2}}$).
Use point-slope form y − y0 = m(x − x0).
Plug in va... | y = - 7\cdot x + \frac{x}{2\cdot e^{\frac{1}{2}}} + 2\cdot x\cdot \sin{\left(2 \right)} - 7 + \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": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
a80c58bd4be1327f | 552ce9f80be21247 | 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:
$f(x)=- 3 x^{2} + 3 x + 1$.
Student solution:
1) Differentiate term-by-term.
2) Claim: one term $\frac{d}{dx}(- 3 x^{2}) = - 6 x^{2}$.
3) So $f'(x) = 3 - 6 x^{2}$. | 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 $f'(x) = 3 \left(1 - 2 x\right)$.
Final answer: $3 \left(1 - 2 x\right)$ | 3 \left(1 - 2 x\right) | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0",
"task": "error_localization",
"is_negative": true
} |
8144633acd3276cf | f03aebff87bb6c61 | tutor | Provide a thorough solution as if teaching, including intermediate computations, then give the final answer. | Find the multiplicative inverse of 13 modulo 23. | 1) We want x such that 13x ≡ 1 (mod 23).
2) Use extended Euclid to find s,t with 13s + 23t = 1.
3) One Bezout identity is: 13·(-7) + 23·(4) = 1.
4) Reduce s modulo 23: x ≡ 16 (mod 23).
Final answer: $x\equiv 16\pmod{23}$ | x\equiv 16\pmod{23} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
f75043d2a05994ba | 81c151717a14e873 | tutor | Show a full solution with explicit computations and a final answer line. | How many solutions in nonnegative integers satisfy $x_1+\cdots+x_7=24$? | 1) We are counting nonnegative integer solutions to a sum constraint.
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=24 and k=7.
4) Compute C(30,6) = 593775.
Final answer: $593775$ | 593775 | {
"format": "symbolic",
"domain": "discrete",
"topic": "stars_and_bars",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
766e3190a226d756 | 81c151717a14e873 | concise | Solve and give a compact solution, then the final answer. | How many solutions in nonnegative integers satisfy $x_1+\cdots+x_7=24$? | We are counting nonnegative integer solutions to a sum constraint.
Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).
Here N=24 and k=7.
Final answer: $593775$ | 593775 | {
"format": "symbolic",
"domain": "discrete",
"topic": "stars_and_bars",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
f787a3de2d2ef95a | 04bc39be437af424 | verifier | Solve the problem, then verify the result with a quick check. Provide the final answer. | How many solutions in integers with $x_i\ge 1$ satisfy $x_1+\cdots+x_5=33$? | Let y_i=x_i-1, then y_i≥0 and y_1+…+y_5=28.
Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).
Here N=28 and k=5.
Compute C(32,4) = 35960.
Check: matches the standard stars-and-bars formula.
Final answer: $35960$ | 35960 | {
"format": "symbolic",
"domain": "discrete",
"topic": "stars_and_bars",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
e02d082aa49912af | 04bc39be437af424 | tutor | Provide a detailed solution with numbered steps and a final answer line. | How many solutions in integers with $x_i\ge 1$ satisfy $x_1+\cdots+x_5=33$? | 1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_5=28.
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=28 and k=5.
4) Compute C(32,4) = 35960.
Final answer: $35960$ | 35960 | {
"format": "symbolic",
"domain": "discrete",
"topic": "stars_and_bars",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
30d42e76ea7a6036 | 211ea4b4bfd17d79 | tutor | Write a complete step-by-step solution and conclude with the final answer. | A cyclist rides at 89 miles per hour for 3/2 hours, then at 25 miles per hour for 2 hours. How many miles does the cyclist travel in total? | 1) First leg distance = 89 × 3/2 = 267/2 miles.
2) Second leg distance = 25 × 2 = 50 miles.
3) Total distance = 267/2 + 50 = 367/2 miles.
Final answer: $\frac{367}{2}$ | \frac{367}{2} | {
"format": "word",
"domain": "arithmetic",
"topic": "rate_time_distance",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
fd45bd21ab5b5a01 | 211ea4b4bfd17d79 | verifier | Solve the problem, then verify the result with a quick check. Provide the final answer. | A cyclist rides at 89 miles per hour for 3/2 hours, then at 25 miles per hour for 2 hours. How many miles does the cyclist travel in total? | First leg distance = 89 × 3/2 = 267/2 miles.
Second leg distance = 25 × 2 = 50 miles.
Total distance = 267/2 + 50 = 367/2 miles.
Check: adding both legs gives 367/2.
Final answer: $\frac{367}{2}$ | \frac{367}{2} | {
"format": "word",
"domain": "arithmetic",
"topic": "rate_time_distance",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
6228b208f500aa32 | c59e89210b4f83d4 | tutor | Provide a detailed solution with numbered steps and a final answer line. | Find the multiplicative inverse of 3 modulo 17. | 1) We want x such that 3x ≡ 1 (mod 17).
2) Use extended Euclid to find s,t with 3s + 17t = 1.
3) One Bezout identity is: 3·(6) + 17·(-1) = 1.
4) Reduce s modulo 17: x ≡ 6 (mod 17).
Final answer: $x\equiv 6\pmod{17}$ | x\equiv 6\pmod{17} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
825f6105b8cc2da8 | c59e89210b4f83d4 | concise | Compute the result with a short explanation and give the final answer. | Find the multiplicative inverse of 3 modulo 17. | We want x such that 3x ≡ 1 (mod 17).
Use extended Euclid to find s,t with 3s + 17t = 1.
One Bezout identity is: 3·(6) + 17·(-1) = 1.
Final answer: $x\equiv 6\pmod{17}$ | x\equiv 6\pmod{17} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
c2c0fb87968f9d26 | 3a366b87b0d01694 | tutor | Solve carefully and do not skip algebra/arithmetic steps. End with the final answer. | A disease has prevalence $\frac{2}{25}$. 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) = $\frac{41}{250}$.
3) Compute P(D|+) = $\frac{18}{41}$.
Final answer: $\frac{18}{41}$ | \frac{18}{41} | {
"format": "symbolic",
"domain": "probability",
"topic": "bayes_theorem",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
05f48d52d8799935 | 3a366b87b0d01694 | concise | Compute the result with a short explanation and give the final answer. | A disease has prevalence $\frac{2}{25}$. 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) = $\frac{41}{250}$.
Compute P(D|+) = $\frac{18}{41}$.
Final answer: $\frac{18}{41}$ | \frac{18}{41} | {
"format": "symbolic",
"domain": "probability",
"topic": "bayes_theorem",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
f7429a6ea65d4462 | b7addf236b39dd38 | tutor | Provide a thorough solution as if teaching, including intermediate computations, then give the final answer. | Differentiate with respect to x:
$f(x)=2 x^{2} - x + 2$. | 1) Differentiate term-by-term:
2) d/dx(2) = 0.
3) d/dx(- x) = -1.
4) d/dx(2 x^{2}) = 4 x.
5) Combine: f'(x) = 4 x - 1.
Final answer: $4\cdot x - 1$ | 4\cdot x - 1 | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
ede31683e70aa639 | b7addf236b39dd38 | concise | Solve with minimal but sufficient steps, then provide the final answer. | Differentiate with respect to x:
$f(x)=2 x^{2} - x + 2$. | Differentiate term-by-term:
d/dx(2) = 0.
d/dx(- x) = -1.
Final answer: $4\cdot x - 1$ | 4\cdot x - 1 | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
aaa9b48044a17118 | 8e73a5c19917465b | tutor | Write a complete step-by-step solution and conclude with the final answer. | Find the equation of the tangent line to $y=x^{2} - 3 x + 2 \sin{\left(x \right)} + 3$ at $x=-2$. | 1) Differentiate: f'(x) = $2 x + 2 \cos{\left(x \right)} - 3$.
2) Slope at x=-2: m = f'(-2) = $-7 + 2 \cos{\left(2 \right)}$.
3) Point on curve: (-2, f(-2)) = (-2, $13 - 2 \sin{\left(2 \right)}$).
4) Use point-slope form y − y0 = m(x − x0).
5) Plug in values: y − 13 - 2 \sin{\left(2 \right)} = -7 + 2 \cos{\left(2 \righ... | y = - 7\cdot x + 2\cdot x\cdot \cos{\left(2 \right)} - 2\cdot \sin{\left(2 \right)} + 4\cdot \cos{\left(2 \right)} - 1 | {
"format": "symbolic",
"domain": "calculus",
"topic": "tangent_line",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
e09b520d78e5af0e | d903d488291fd67c | 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:
$f(x)=\frac{- 2 x^{2} + 2 x + \left(x^{2} + 3 x - 5\right) e^{\frac{x}{2}} \sin{\left(x \right)} - 2}{x^{2} + 3 x - 5}$.
Student solution:
1) Differentiate term-by-term.
2) Claim (missing chain rule): $\frac{d}{dx}(\frac{x^{2} e^{\frac{x}{2}} \sin{\left(x \right)}}{\left(x^{2} + 3 x\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}(\frac{x^{2} e^{\frac{x}{2}} \sin{\left(x \right)}}{\left(x^{2} + 3 x\right) - 5}) = \frac{x^{2} \left(- 2 x - 3\right) e^{\frac{x}{2}} \sin{\left(x \right)}}{\left(\l... | \frac{2 \left(- 2 x - 3\right) \left(- 2 x^{2} + 2 x + \left(x \left(x + 3\right) - 5\right) e^{\frac{x}{2}} \sin{\left(x \right)} - 2\right) + \left(x \left(x + 3\right) - 5\right) \left(- 8 x + 2 \left(2 x + 3\right) e^{\frac{x}{2}} \sin{\left(x \right)} + \left(x \left(x + 3\right) - 5\right) e^{\frac{x}{2}} \sin{\l... | {
"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
} |
d731983712c8bbac | 961ca3c143b686d9 | tutor | Show a full solution with explicit computations and a final answer line. | Compute the definite integral:
$\displaystyle \int_{- \pi}^{\pi} 4 x^{4} + 3 x^{3} + x^{2} - 4 x + 3 \sin{\left(2 x \right)} + 2 \cos{\left(x \right)} - 3\,dx$. | 1) Compute an antiderivative F(x) of the integrand $4 x^{4} + 3 x^{3} + x^{2} - 4 x + 3 \sin{\left(2 x \right)} + 2 \cos{\left(x \right)} - 3$.
2) Evaluate at the bounds: F(\pi) and F(- \pi).
3) Subtract: F(\pi) − F(- \pi) = $\frac{2 \pi \left(-45 + 5 \pi^{2} + 12 \pi^{4}\right)}{15}$.
4) Simplify the result.
Final ans... | \frac{2\cdot \pi\cdot \left(-45 + 5\cdot \pi^{2} + 12\cdot \pi^{4}\right)}{15} | {
"format": "symbolic",
"domain": "calculus",
"topic": "definite_integral",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
8bfc634a4f7e566f | 961ca3c143b686d9 | verifier | Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer. | Compute the definite integral:
$\displaystyle \int_{- \pi}^{\pi} 4 x^{4} + 3 x^{3} + x^{2} - 4 x + 3 \sin{\left(2 x \right)} + 2 \cos{\left(x \right)} - 3\,dx$. | Compute an antiderivative F(x) of the integrand $4 x^{4} + 3 x^{3} + x^{2} - 4 x + 3 \sin{\left(2 x \right)} + 2 \cos{\left(x \right)} - 3$.
Evaluate at the bounds: F(\pi) and F(- \pi).
Subtract: F(\pi) − F(- \pi) = $\frac{2 \pi \left(-45 + 5 \pi^{2} + 12 \pi^{4}\right)}{15}$.
Simplify the result.
Check: SymPy differen... | \frac{2\cdot \pi\cdot \left(-45 + 5\cdot \pi^{2} + 12\cdot \pi^{4}\right)}{15} | {
"format": "symbolic",
"domain": "calculus",
"topic": "definite_integral",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
86b6c4ab6cd316f3 | 4d2fd72c18a669c1 | tutor | Give a detailed derivation, showing intermediate expressions, and then the final answer. | Find the multiplicative inverse of 7 modulo 11. | 1) We want x such that 7x ≡ 1 (mod 11).
2) Use extended Euclid to find s,t with 7s + 11t = 1.
3) One Bezout identity is: 7·(-3) + 11·(2) = 1.
4) Reduce s modulo 11: x ≡ 8 (mod 11).
Final answer: $x\equiv 8\pmod{11}$ | x\equiv 8\pmod{11} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
41a57e3574826d12 | 4d2fd72c18a669c1 | verifier | Solve the problem, then verify the result with a quick check. Provide the final answer. | Find the multiplicative inverse of 7 modulo 11. | We want x such that 7x ≡ 1 (mod 11).
Use extended Euclid to find s,t with 7s + 11t = 1.
One Bezout identity is: 7·(-3) + 11·(2) = 1.
Reduce s modulo 11: x ≡ 8 (mod 11).
Check: 7·8 mod 11 = 1.
Final answer: $x\equiv 8\pmod{11}$ | x\equiv 8\pmod{11} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
676e6f7c74132b8d | 8a31666b19b66863 | tutor | Provide a thorough solution as if teaching, including intermediate computations, then give the final answer. | Compute the definite integral:
$\displaystyle \int_{0}^{\frac{\pi}{2}} x^{4} + 3 x^{3} + 3 x^{2} + 4 x + 3 \sin{\left(2 x \right)} + 2 \cos{\left(x \right)} + 4\,dx$. | 1) Compute an antiderivative F(x) of the integrand $x^{4} + 3 x^{3} + 3 x^{2} + 4 x + 3 \sin{\left(2 x \right)} + 2 \cos{\left(x \right)} + 4$.
2) Evaluate at the bounds: F(\frac{\pi}{2}) and F(0).
3) Subtract: F(\frac{\pi}{2}) − F(0) = $\frac{\pi^{5}}{160} + \frac{\pi^{3}}{8} + \frac{3 \pi^{4}}{64} + \frac{\pi^{2}}{2}... | \frac{\pi^{5}}{160} + \frac{\pi^{3}}{8} + \frac{3\cdot \pi^{4}}{64} + \frac{\pi^{2}}{2} + 5 + 2\cdot \pi | {
"format": "symbolic",
"domain": "calculus",
"topic": "definite_integral",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
d28217e1f9ad430f | 8a31666b19b66863 | verifier | Compute the answer and add a brief verification calculation. Provide the final answer. | Compute the definite integral:
$\displaystyle \int_{0}^{\frac{\pi}{2}} x^{4} + 3 x^{3} + 3 x^{2} + 4 x + 3 \sin{\left(2 x \right)} + 2 \cos{\left(x \right)} + 4\,dx$. | Compute an antiderivative F(x) of the integrand $x^{4} + 3 x^{3} + 3 x^{2} + 4 x + 3 \sin{\left(2 x \right)} + 2 \cos{\left(x \right)} + 4$.
Evaluate at the bounds: F(\frac{\pi}{2}) and F(0).
Subtract: F(\frac{\pi}{2}) − F(0) = $\frac{\pi^{5}}{160} + \frac{\pi^{3}}{8} + \frac{3 \pi^{4}}{64} + \frac{\pi^{2}}{2} + 5 + 2 ... | \frac{\pi^{5}}{160} + \frac{\pi^{3}}{8} + \frac{3\cdot \pi^{4}}{64} + \frac{\pi^{2}}{2} + 5 + 2\cdot \pi | {
"format": "symbolic",
"domain": "calculus",
"topic": "definite_integral",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
bbbbbf6abf63038b | 1a985e2d9e4dcec9 | concise | Provide a compact reasoning and the final answer. | Solve the linear system:
$- x + 7 y$ = $-63$
$7 x + 6 y$ = $-54$ | Eliminate x by taking a linear combination of the equations.
Multiply the first equation by $7$: $- 7 x + 49 y$ = $-441$.
Multiply the second equation by $-1$: $- 7 x - 6 y$ = $54$.
Final answer: $(x,y)=\left(0, -9\right)$ | (x,y)=\left(0, -9\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
043f2ed5bee39917 | 1a985e2d9e4dcec9 | tutor | Provide a detailed solution with numbered steps and a final answer line. | Solve the linear system:
$- x + 7 y$ = $-63$
$7 x + 6 y$ = $-54$ | 1) Eliminate x by taking a linear combination of the equations.
2) Multiply the first equation by $7$: $- 7 x + 49 y$ = $-441$.
3) Multiply the second equation by $-1$: $- 7 x - 6 y$ = $54$.
4) Subtract to eliminate x: $55 y$ = $-495$.
5) Solve for y: y = $-9$.
6) Substitute back to find x: x = $0$.
Final answer: $(x,y... | (x,y)=\left(0, -9\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
cbad449b6643c7e3 | d14866f0ac52d8c3 | concise | Solve the problem and keep the explanation concise but complete. Provide the final answer. | Differentiate with respect to x:
$f(x)=2 \left(x + 1\right) e^{x} + \left(3 x^{2} + x - 2\right) \cos{\left(3 x + 3 \right)}$. | Let f(x) = $2 \left(x + 1\right) e^{x} + \left(3 x^{2} + x - 2\right) \cos{\left(3 x + 3 \right)}$.
Differentiate using product/chain/quotient rules as needed.
Simplify the derivative to get f'(x) = $2 \left(x + 1\right) e^{x} + \left(6 x + 1\right) \cos{\left(3 x + 3 \right)} - 3 \left(3 x^{2} + x - 2\right) \sin{\lef... | \left(2\cdot x + 2\right)\cdot e^{x} + \left(6\cdot x + 1\right)\cdot \cos{\left(3\cdot x + 3 \right)} - \left(9\cdot x^{2} + 3\cdot x - 6\right)\cdot \sin{\left(3\cdot x + 3 \right)} + 2\cdot e^{x} | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
edda8d4cc5d45954 | d14866f0ac52d8c3 | tutor | Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer. | Differentiate with respect to x:
$f(x)=2 \left(x + 1\right) e^{x} + \left(3 x^{2} + x - 2\right) \cos{\left(3 x + 3 \right)}$. | 1) Let f(x) = $2 \left(x + 1\right) e^{x} + \left(3 x^{2} + x - 2\right) \cos{\left(3 x + 3 \right)}$.
2) Differentiate using product/chain/quotient rules as needed.
3) Simplify the derivative to get f'(x) = $2 \left(x + 1\right) e^{x} + \left(6 x + 1\right) \cos{\left(3 x + 3 \right)} - 3 \left(3 x^{2} + x - 2\right) ... | \left(2\cdot x + 2\right)\cdot e^{x} + \left(6\cdot x + 1\right)\cdot \cos{\left(3\cdot x + 3 \right)} - \left(9\cdot x^{2} + 3\cdot x - 6\right)\cdot \sin{\left(3\cdot x + 3 \right)} + 2\cdot e^{x} | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
e90fc0f1fe55a8bf | 0bfe1547e47bdb0e | verifier | Compute the answer and include a short verification step. Give the final answer. | A disease has prevalence $\frac{1}{50}$. A test has sensitivity $\frac{17}{20}$ and specificity $\frac{97}{100}$. 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) = $\frac{29}{625}$.
Compute P(D|+) = $\frac{85}{232}$.
Check: value is between 0 and 1 and increases with prevalence/sensitivity.
Final answer: $\frac{85}{232}$ | \frac{85}{232} | {
"format": "symbolic",
"domain": "probability",
"topic": "bayes_theorem",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
f286d7d74ee54873 | 0bfe1547e47bdb0e | tutor | Provide a thorough solution as if teaching, including intermediate computations, then give the final answer. | A disease has prevalence $\frac{1}{50}$. A test has sensitivity $\frac{17}{20}$ and specificity $\frac{97}{100}$. 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) = $\frac{29}{625}$.
3) Compute P(D|+) = $\frac{85}{232}$.
Final answer: $\frac{85}{232}$ | \frac{85}{232} | {
"format": "symbolic",
"domain": "probability",
"topic": "bayes_theorem",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
d3c2546e3f4785d3 | cb77b073bb8e944c | tutor | Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer. | Solve the linear system:
$2 x$ = $-4$
$- x + 4 y$ = $18$ | 1) Eliminate x by taking a linear combination of the equations.
2) Multiply the first equation by $-1$: $- 2 x$ = $4$.
3) Multiply the second equation by $2$: $- 2 x + 8 y$ = $36$.
4) Subtract to eliminate x: $- 8 y$ = $-32$.
5) Solve for y: y = $4$.
6) Substitute back to find x: x = $-2$.
Final answer: $(x,y)=\left(-2... | (x,y)=\left(-2, 4\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
f2b76908591d26c3 | 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): $\frac{d}{dx}(x^{2} \cos{\left(x - 3 \right)}) = - x^{2} \sin{\left(x - 3 \right)}$.
3) So $f'(x) = 3 x^... | First incorrect step: 2
The error is omitting the chain-rule factor (derivative of the inside function).
Correct derivative of that term is $\frac{d}{dx}(x^{2} \cos{\left(x - 3 \right)}) = - x^{2} \sin{\left(x - 3 \right)} + 2 x \cos{\left(x - 3 \right)}$.
Therefore the correct derivative is $f'(x) = \left(x + 2\right)... | \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
} |
a459d1a2f7a1624c | ded5f5ab78bb3fa0 | concise | Compute the result with a short explanation and give the final answer. | An urn has 15 red balls and 28 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 1 are red? | Use a hypergeometric counting argument.
Total equally likely samples: C(43,9).
Favorable samples: choose k reds and n−k blues: C(15,1)·C(28,8).
Final answer: $\frac{102465}{1239389}$ | \frac{102465}{1239389} | {
"format": "symbolic",
"domain": "probability",
"topic": "hypergeometric",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
8c1ca3d2fa1691b7 | ded5f5ab78bb3fa0 | tutor | Write a complete step-by-step solution and conclude with the final answer. | An urn has 15 red balls and 28 blue balls. If 9 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(43,9).
3) Favorable samples: choose k reds and n−k blues: C(15,1)·C(28,8).
4) Compute totals: C(43,9) = 563921995, and favorable = 46621575.
5) Divide favorable by total and simplify.
Final answer: $\frac{102465}{1239389}$ | \frac{102465}{1239389} | {
"format": "symbolic",
"domain": "probability",
"topic": "hypergeometric",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
2b3cbadc0725d94f | cf186e9e958657cb | tutor | Explain each step clearly and end with the final answer. | Solve the congruences: x≡3 (mod 4), x≡6 (mod 11). Give the solution modulo 44. | 1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.
2) The solution is unique modulo m1·m2 = 44.
3) Compute the CRT solution to get x ≡ 39 (mod 44).
Final answer: $x\equiv 39\pmod{44}$ | x\equiv 39\pmod{44} | {
"format": "symbolic",
"domain": "discrete",
"topic": "chinese_remainder_theorem",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
51ee4255fe45d938 | d559cfa8e57a172d | 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:
$f(x)=3 x^{2} \sin{\left(x \right)} + 3 x \sin{\left(x \right)} + 3 \sin{\left(x \right)} + 3 \cos{\left(2 x \right)}$.
Student solution:
1) Differentiate term-by-term.
2) Claim (missing chain rule): $\frac{d}{dx}(3 \cos{\left(2 x \right)}) = - 3 \sin{\left(2 x \right)}$.
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 $\frac{d}{dx}(3 \cos{\left(2 x \right)}) = - 6 \sin{\left(2 x \right)}$.
Therefore the correct derivative is $f'(x) = 3 \left(x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)... | 3 \left(x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)} + x \cos{\left(x \right)} + \sin{\left(x \right)} - 2 \sin{\left(2 x \right)} + \cos{\left(x \right)}\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
} |
7ae06c4d318e2e45 | 9b79e1058bb96e39 | tutor | Show a full solution with explicit computations and a final answer line. | You want 47 liters of a 35% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix? | 1) Let x = liters of 20% solution, so 47−x liters is 40% solution.
2) Salt equation: 20x + 40(47−x) = 35·47.
3) (20-40)x = (35-40)·47 ⇒ x = 47/4.
4) Then 47−x = 141/4.
Final answer: $47/4 L of 20% and 141/4 L of 40%$ | 47/4 L of 20% and 141/4 L of 40% | {
"format": "word",
"domain": "arithmetic",
"topic": "mixture",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
478e34cb86030e67 | 59c32a17a05252ed | 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:
$\displaystyle \lim_{x\to -3} - 3 x^{3} + 11 x^{2} - 31 x + 92$.
Student solution:
1) Substitute x=-3 into the expression.
2) (Mistake) Simplify the expression to - 3 x^{3} + 11 x^{2} - 32 x + 92.
3) Then the limit is 368. | 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: 365.
Final answer: $365$ | 365 | {
"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
} |
7a0508fd678299dc | d5c83e10cdef30ac | tutor | Give a detailed derivation, showing intermediate expressions, and then the final answer. | Solve the congruences: x≡1 (mod 11), x≡2 (mod 10). Give the solution modulo 110. | 1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.
2) The solution is unique modulo m1·m2 = 110.
3) Compute the CRT solution to get x ≡ 12 (mod 110).
Final answer: $x\equiv 12\pmod{110}$ | x\equiv 12\pmod{110} | {
"format": "symbolic",
"domain": "discrete",
"topic": "chinese_remainder_theorem",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
f8801999618e5d03 | d5c83e10cdef30ac | concise | Solve and give a compact solution, then the final answer. | Solve the congruences: x≡1 (mod 11), x≡2 (mod 10). Give the solution modulo 110. | Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.
The solution is unique modulo m1·m2 = 110.
Compute the CRT solution to get x ≡ 12 (mod 110).
Final answer: $x\equiv 12\pmod{110}$ | x\equiv 12\pmod{110} | {
"format": "symbolic",
"domain": "discrete",
"topic": "chinese_remainder_theorem",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
ca0dfcb2b7b45d3d | a636c7de333ca721 | tutor | Show a full solution with explicit computations and a final answer line. | A store sells a backpack for $840. It is discounted by 15%. You buy 9 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost? | 1) Discounted price = 840 × (1 − 15/100) = 840 × 85/100 = 714.
2) Subtotal = 9 × 714 = 6426.
3) Total with tax = 6426 × 110/100 = 35343/5.
Final answer: $\frac{35343}{5}$ | \frac{35343}{5} | {
"format": "word",
"domain": "arithmetic",
"topic": "percent_discount",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
863dab9d78583859 | a636c7de333ca721 | verifier | Compute the answer and add a brief verification calculation. Provide the final answer. | A store sells a backpack for $840. It is discounted by 15%. You buy 9 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost? | Discounted price = 840 × (1 − 15/100) = 840 × 85/100 = 714.
Subtotal = 9 × 714 = 6426.
Total with tax = 6426 × 110/100 = 35343/5.
Check: recomputing the formula yields 35343/5.
Final answer: $\frac{35343}{5}$ | \frac{35343}{5} | {
"format": "word",
"domain": "arithmetic",
"topic": "percent_discount",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
d2ccd40d628236d1 | 2c8d970f33f1adea | verifier | Compute the answer and add a brief verification calculation. Provide the final answer. | Compute the definite integral:
$\displaystyle \int_{- \pi}^{2 \pi} 2 x^{3} + 3 x^{2} - 3 x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} - 1\,dx$. | Compute an antiderivative F(x) of the integrand $2 x^{3} + 3 x^{2} - 3 x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} - 1$.
Evaluate at the bounds: F(2 \pi) and F(- \pi).
Subtract: F(2 \pi) − F(- \pi) = $- \frac{9 \pi^{2}}{2} - 3 \pi - 4 + 9 \pi^{3} + \frac{15 \pi^{4}}{2}$.
Simplify the result.
Check: SymPy diff... | - \frac{9\cdot \pi^{2}}{2} - 3\cdot \pi - 4 + 9\cdot \pi^{3} + \frac{15\cdot \pi^{4}}{2} | {
"format": "symbolic",
"domain": "calculus",
"topic": "definite_integral",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
80bd4226cf40e586 | 2c8d970f33f1adea | tutor | Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer. | Compute the definite integral:
$\displaystyle \int_{- \pi}^{2 \pi} 2 x^{3} + 3 x^{2} - 3 x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} - 1\,dx$. | 1) Compute an antiderivative F(x) of the integrand $2 x^{3} + 3 x^{2} - 3 x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} - 1$.
2) Evaluate at the bounds: F(2 \pi) and F(- \pi).
3) Subtract: F(2 \pi) − F(- \pi) = $- \frac{9 \pi^{2}}{2} - 3 \pi - 4 + 9 \pi^{3} + \frac{15 \pi^{4}}{2}$.
4) Simplify the result.
Final... | - \frac{9\cdot \pi^{2}}{2} - 3\cdot \pi - 4 + 9\cdot \pi^{3} + \frac{15\cdot \pi^{4}}{2} | {
"format": "symbolic",
"domain": "calculus",
"topic": "definite_integral",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
e8be039a49d8ef40 | e73242e102210611 | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | Compute the definite integral:
$\displaystyle \int_{- \frac{\pi}{2}}^{\pi} 2 x^{2} + 3 x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} - 2\,dx$. | 1) Compute an antiderivative F(x) of the integrand $2 x^{2} + 3 x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)} - 2$.
2) Evaluate at the bounds: F(\pi) and F(- \frac{\pi}{2}).
3) Subtract: F(\pi) − F(- \frac{\pi}{2}) = $- 3 \pi + 2 + \frac{9 \pi^{2}}{8} + \frac{3 \pi^{3}}{4}$.
4) Simplify the result.
Final answer... | - 3\cdot \pi + 2 + \frac{9\cdot \pi^{2}}{8} + \frac{3\cdot \pi^{3}}{4} | {
"format": "symbolic",
"domain": "calculus",
"topic": "definite_integral",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
df202bb723a04fcc | 3787fad3e50f603c | 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:
$f(x)=3 x^{2} + 4$.
Student solution:
1) Differentiate term-by-term.
2) Claim: one term $\frac{d}{dx}(3 x^{2}) = 6 x^{2}$.
3) So $f'(x) = 6 x^{2}$. | 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 $f'(x) = 6 x$.
Final answer: $6 x$ | 6 x | {
"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
} |
438ea462874d7ad4 | e1373520055de1b4 | concise | Compute the result with a short explanation and give the final answer. | Solve the linear system:
$- 7 x + 8 y$ = $2$
$- 2 x - 3 y$ = $27$ | Eliminate x by taking a linear combination of the equations.
Multiply the first equation by $-2$: $14 x - 16 y$ = $-4$.
Multiply the second equation by $-7$: $14 x + 21 y$ = $-189$.
Final answer: $(x,y)=\left(-6, -5\right)$ | (x,y)=\left(-6, -5\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
79ce3afc14068a8e | e1373520055de1b4 | tutor | Give a detailed derivation, showing intermediate expressions, and then the final answer. | Solve the linear system:
$- 7 x + 8 y$ = $2$
$- 2 x - 3 y$ = $27$ | 1) Eliminate x by taking a linear combination of the equations.
2) Multiply the first equation by $-2$: $14 x - 16 y$ = $-4$.
3) Multiply the second equation by $-7$: $14 x + 21 y$ = $-189$.
4) Subtract to eliminate x: $- 37 y$ = $185$.
5) Solve for y: y = $-5$.
6) Substitute back to find x: x = $-6$.
Final answer: $(x... | (x,y)=\left(-6, -5\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
a07ca6de1a36e01a | b18dc57c389a574a | concise | Compute and present the key steps only, then the final answer. | Solve the quadratic equation over the real numbers:
$3 x^{2} - \frac{19 x}{4} + 1$ = 0. | Identify coefficients a, b, c in ax^2+bx+c=0.
Here a=$3$, b=$- \frac{19}{4}$, c=$1$.
Compute the discriminant Δ=b^2−4ac.
Final answer: $x \in \{\frac{4}{3}, \frac{1}{4}\}$ | x \in \{\frac{4}{3}, \frac{1}{4}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
09beafd33fc5bf93 | b18dc57c389a574a | tutor | Show your work in clear steps (with intermediate values) and finish with a final answer. | Solve the quadratic equation over the real numbers:
$3 x^{2} - \frac{19 x}{4} + 1$ = 0. | 1) Identify coefficients a, b, c in ax^2+bx+c=0.
2) Here a=$3$, b=$- \frac{19}{4}$, c=$1$.
3) Compute the discriminant Δ=b^2−4ac.
4) Δ = $\frac{169}{16}$, so √Δ = $\frac{13}{4}$.
5) Apply the quadratic formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}$.
6) x1 = $\frac{4}{3}$.
7) x2 = $\frac{1}{4}$.
8) Collect the solutions into... | x \in \{\frac{4}{3}, \frac{1}{4}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
dc783090ffb9f377 | c1bc557687aa8004 | verifier | Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer. | You want 16 liters of a 30% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix? | Let x = liters of 20% solution, so 16−x liters is 70% solution.
Salt equation: 20x + 70(16−x) = 30·16.
(20-70)x = (30-70)·16 ⇒ x = 64/5.
Then 16−x = 16/5.
Check: 20·64/5 + 70·16/5 = 30·16.
Final answer: $64/5 L of 20% and 16/5 L of 70%$ | 64/5 L of 20% and 16/5 L of 70% | {
"format": "word",
"domain": "arithmetic",
"topic": "mixture",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
fd08b9a64247db5a | c1bc557687aa8004 | tutor | Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer. | You want 16 liters of a 30% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix? | 1) Let x = liters of 20% solution, so 16−x liters is 70% solution.
2) Salt equation: 20x + 70(16−x) = 30·16.
3) (20-70)x = (30-70)·16 ⇒ x = 64/5.
4) Then 16−x = 16/5.
Final answer: $64/5 L of 20% and 16/5 L of 70%$ | 64/5 L of 20% and 16/5 L of 70% | {
"format": "word",
"domain": "arithmetic",
"topic": "mixture",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
c0ecd50ad317fc5c | 64fcb233dd7aee28 | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | Solve the linear system:
$- y$ = $-1$
$x + 3 y$ = $-2$ | Eliminate x by taking a linear combination of the equations.
Multiply the first equation by $1$: $- y$ = $-1$.
Multiply the second equation by $0$: $0$ = $0$.
Subtract to eliminate x: $- y$ = $-1$.
Solve for y: y = $1$.
Substitute back to find x: x = $-5$.
Check: plug (x,y)=(-5,1) into both equations.
Final answer: $(x... | (x,y)=\left(-5, 1\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
d4983708fafc9b1f | 64fcb233dd7aee28 | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | Solve the linear system:
$- y$ = $-1$
$x + 3 y$ = $-2$ | 1) Eliminate x by taking a linear combination of the equations.
2) Multiply the first equation by $1$: $- y$ = $-1$.
3) Multiply the second equation by $0$: $0$ = $0$.
4) Subtract to eliminate x: $- y$ = $-1$.
5) Solve for y: y = $1$.
6) Substitute back to find x: x = $-5$.
Final answer: $(x,y)=\left(-5, 1\right)$ | (x,y)=\left(-5, 1\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
1151a94a98c51a42 | 8cb2c612c6b331f3 | tutor | Solve carefully and do not skip algebra/arithmetic steps. End with the final answer. | Find the multiplicative inverse of 10 modulo 19. | 1) We want x such that 10x ≡ 1 (mod 19).
2) Use extended Euclid to find s,t with 10s + 19t = 1.
3) One Bezout identity is: 10·(2) + 19·(-1) = 1.
4) Reduce s modulo 19: x ≡ 2 (mod 19).
Final answer: $x\equiv 2\pmod{19}$ | x\equiv 2\pmod{19} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
6c16cd338cd9efdc | 5c107b6a4bb1aea3 | 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:
$\displaystyle \lim_{x\to 4} - 2 x^{2} - 10 x - 42$.
Student solution:
1) Substitute x=4 into the expression.
2) (Mistake) Simplify the expression to - 2 x^{2} - 11 x - 42.
3) Then the limit is -118. | 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: -114.
Final answer: $-114$ | -114 | {
"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
} |
cd0cb0c9e36b566e | 115b862cd75cf87e | tutor | Solve carefully and do not skip algebra/arithmetic steps. End with the final answer. | Find the multiplicative inverse of 10 modulo 37. | 1) We want x such that 10x ≡ 1 (mod 37).
2) Use extended Euclid to find s,t with 10s + 37t = 1.
3) One Bezout identity is: 10·(-11) + 37·(3) = 1.
4) Reduce s modulo 37: x ≡ 26 (mod 37).
Final answer: $x\equiv 26\pmod{37}$ | x\equiv 26\pmod{37} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
013d3da8a5949188 | 115b862cd75cf87e | concise | Solve with minimal but sufficient steps, then provide the final answer. | Find the multiplicative inverse of 10 modulo 37. | We want x such that 10x ≡ 1 (mod 37).
Use extended Euclid to find s,t with 10s + 37t = 1.
One Bezout identity is: 10·(-11) + 37·(3) = 1.
Final answer: $x\equiv 26\pmod{37}$ | x\equiv 26\pmod{37} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
9590a11e1254f30b | 53dc42542e8e3cbe | tutor | Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer. | A coin lands heads with probability p=$\frac{1}{4}$. If it is flipped 10 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=10, k=3, p=$\frac{1}{4}$ and simplify.
4) Here C(10,3) = 120.
Final answer: $\frac{32805}{131072}$ | \frac{32805}{131072} | {
"format": "symbolic",
"domain": "probability",
"topic": "binomial_probability",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
4c433fad79b6ee9c | 43199a59849d810a | 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:
$f(x)=\left(x + 2\right) e^{x} + \left(3 x^{3} - 3 x^{2} + 2\right) \cos{\left(x - 3 \right)}$.
Student solution:
1) Differentiate term-by-term.
2) Claim (missing chain rule): $\frac{d}{dx}(- 3 x^{2} \cos{\left(x - 3 \right)}) = 3 x^{2} \sin{\left(x - 3 \right)}$.
3) So $f'(x) = - 3 x^... | First incorrect step: 2
The error is omitting the chain-rule factor (derivative of the inside function).
Correct derivative of that term is $\frac{d}{dx}(- 3 x^{2} \cos{\left(x - 3 \right)}) = 3 x^{2} \sin{\left(x - 3 \right)} - 6 x \cos{\left(x - 3 \right)}$.
Therefore the correct derivative is $f'(x) = 3 x \left(3 x ... | 3 x \left(3 x - 2\right) \cos{\left(x - 3 \right)} + \left(x + 2\right) e^{x} - \left(3 x^{2} \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
} |
55742f2f50fc41fd | 99b06a6bb87dc2b6 | 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:
$\displaystyle \lim_{x\to 2} x$.
Student solution:
1) Substitute x=2 into the expression.
2) (Mistake) Simplify the expression to 0.
3) Then the limit is 0. | 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: 2.
Final answer: $2$ | 2 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0",
"task": "error_localization",
"is_negative": true
} |
254bc9debb013864 | 7e3a2f6f278e814c | tutor | Show a full solution with explicit computations and a final answer line. | Worker A finishes a job in 5 hours and Worker B finishes it in 14 hours. If they work together at constant rates, how long does it take to finish the job? | 1) A's rate = 1/5 job/hour. B's rate = 1/14 job/hour.
2) Combined rate = 1/5 + 1/14 = 19/70 job/hour.
3) Time = 1 ÷ (combined rate) = 70/19 hours.
Final answer: $\frac{70}{19}$ | \frac{70}{19} | {
"format": "word",
"domain": "arithmetic",
"topic": "work_rates",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
dc5459bef9af3515 | 11024044ec7ec250 | tutor | Show your work in clear steps (with intermediate values) and finish with a final answer. | Evaluate the limit:
$\displaystyle \lim_{x\to 2} 2 x^{3} + 4 x^{2} + 9 x + 20$. | 1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).
2) Here p(x) = $2 x^{4} + x^{2} + 2 x - 1$ and a = 2.
3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).
4) Differentiate: p'(x) = $8 x^{3} + 2 x + 2$.
5) Evaluate at a: p'(2) = $70$.
Final answer: $70$ | 70 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
a72daf2cd9d18587 | 11024044ec7ec250 | verifier | Solve the problem, then verify the result with a quick check. Provide the final answer. | Evaluate the limit:
$\displaystyle \lim_{x\to 2} 2 x^{3} + 4 x^{2} + 9 x + 20$. | Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).
Here p(x) = $2 x^{4} + x^{2} + 2 x - 1$ and a = 2.
Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).
Differentiate: p'(x) = $8 x^{3} + 2 x + 2$.
Evaluate at a: p'(2) = $70$.
Check: the identity gives $70$.
Final answer: $70$ | 70 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
c8c66cf932b7b4dc | 2826f3434c0f2591 | verifier | Compute the answer and add a brief verification calculation. Provide the final answer. | Worker A finishes a job in 15 hours and Worker B finishes it in 10 hours. If they work together at constant rates, how long does it take to finish the job? | A's rate = 1/15 job/hour. B's rate = 1/10 job/hour.
Combined rate = 1/15 + 1/10 = 1/6 job/hour.
Time = 1 ÷ (combined rate) = 6 hours.
Check: (combined rate)×(time) = 1/6×6 = 1.
Final answer: $6$ | 6 | {
"format": "word",
"domain": "arithmetic",
"topic": "work_rates",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
40dc3284a8b6dd7c | b46d5bc7fb935668 | 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:
$\displaystyle \lim_{x\to -1} - 2 x^{2} + 5 x - 5$.
Student solution:
1) Substitute x=-1 into the expression.
2) (Mistake) Simplify the expression to - 2 x^{2} + 4 x - 5.
3) Then the limit is -11. | 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: -12.
Final answer: $-12$ | -12 | {
"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
} |
742b2410a9777aea | fcb9bd0865def3fa | tutor | Provide a thorough solution as if teaching, including intermediate computations, then give the final answer. | Evaluate the limit:
$\displaystyle \lim_{x\to -1} 3 x - 3$. | 1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).
2) Here p(x) = $3 x^{2} - 3$ and a = -1.
3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).
4) Differentiate: p'(x) = $6 x$.
5) Evaluate at a: p'(-1) = $-6$.
Final answer: $-6$ | -6 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
52e5b6d4149a1a82 | 06e4bc2eebfcb779 | 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:
$\displaystyle \lim_{x\to 1} 2 x^{3} + x^{2} + 4 x + 1$.
Student solution:
1) Substitute x=1 into the expression.
2) (Mistake) Simplify the expression to 2 x^{3} + x^{2} + 3 x + 1.
3) Then the limit is 7. | 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: 8.
Final answer: $8$ | 8 | {
"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
} |
6b9d0a117feb7952 | eec3e433000893c7 | tutor | Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer. | Compute the indefinite integral:
$\displaystyle \int - x^{3} - 2 x^{2} + 3 x + 2 \sin{\left(x \right)} - 3 \cos{\left(2 x \right)}\,dx$. | 1) Identify an integration method (term-by-term, substitution, or a known pattern).
2) Integrand: $- x^{3} - 2 x^{2} + 3 x + 2 \sin{\left(x \right)} - 3 \cos{\left(2 x \right)}$.
3) Compute an antiderivative: F(x) = $- \frac{x^{4}}{4} - \frac{2 x^{3}}{3} + \frac{3 x^{2}}{2} - \frac{3 \sin{\left(2 x \right)}}{2} - 2 \co... | C - \frac{x^{4}}{4} - \frac{2\cdot x^{3}}{3} + \frac{3\cdot x^{2}}{2} - \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"
} |
675b4349cc6a01a8 | eec3e433000893c7 | concise | Give a short solution with the key steps and the final answer. | Compute the indefinite integral:
$\displaystyle \int - x^{3} - 2 x^{2} + 3 x + 2 \sin{\left(x \right)} - 3 \cos{\left(2 x \right)}\,dx$. | Identify an integration method (term-by-term, substitution, or a known pattern).
Integrand: $- x^{3} - 2 x^{2} + 3 x + 2 \sin{\left(x \right)} - 3 \cos{\left(2 x \right)}$.
Compute an antiderivative: F(x) = $- \frac{x^{4}}{4} - \frac{2 x^{3}}{3} + \frac{3 x^{2}}{2} - \frac{3 \sin{\left(2 x \right)}}{2} - 2 \cos{\left(x... | C - \frac{x^{4}}{4} - \frac{2\cdot x^{3}}{3} + \frac{3\cdot x^{2}}{2} - \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"
} |
841fbd3763458865 | f4bb53485da72686 | 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:
$f(x)=4 x^{4} - 3 x^{2} + x - 4$.
Student solution:
1) Differentiate term-by-term.
2) Claim: one term $\frac{d}{dx}(x) = x$.
3) So $f'(x) = 16 x^{3} - 5 x$. | First incorrect step: 2
The error is differentiating a power: the exponent must decrease by 1.
Correct: $\frac{d}{dx}(x) = 1$.
Therefore the correct derivative is $f'(x) = 16 x^{3} - 6 x + 1$.
Final answer: $16 x^{3} - 6 x + 1$ | 16 x^{3} - 6 x + 1 | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0",
"task": "error_localization",
"is_negative": true
} |
cd02b58e093e21c7 | bc8fc1d01cf40016 | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | How many solutions in integers with $x_i\ge 3$ satisfy $x_1+\cdots+x_6=18$? | 1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_6=0.
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=0 and k=6.
4) Compute C(5,5) = 1.
Final answer: $1$ | 1 | {
"format": "symbolic",
"domain": "discrete",
"topic": "stars_and_bars",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
e83168f96dea9876 | 8fd197769ad52a9f | 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 15 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 = 1/15 job/hour.
2) Combined rate = 1/10 + 1/15 = 1/6 job/hour.
3) Time = 1 ÷ (combined rate) = 6 hours.
Final answer: $6$ | 6 | {
"format": "word",
"domain": "arithmetic",
"topic": "work_rates",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
d9d3c660c2d6a904 | 8fd197769ad52a9f | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | Worker A finishes a job in 10 hours and Worker B finishes it in 15 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 = 1/15 job/hour.
Combined rate = 1/10 + 1/15 = 1/6 job/hour.
Time = 1 ÷ (combined rate) = 6 hours.
Check: (combined rate)×(time) = 1/6×6 = 1.
Final answer: $6$ | 6 | {
"format": "word",
"domain": "arithmetic",
"topic": "work_rates",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
2cc1e1f3e29067f9 | 0477b9f26d5b5b77 | tutor | Solve carefully and do not skip algebra/arithmetic steps. End with the final answer. | A store sells a backpack for $75. It is discounted by 10%. You buy 9 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost? | 1) Discounted price = 75 × (1 − 10/100) = 75 × 90/100 = 135/2.
2) Subtotal = 9 × 135/2 = 1215/2.
3) Total with tax = 1215/2 × 108/100 = 6561/10.
Final answer: $\frac{6561}{10}$ | \frac{6561}{10} | {
"format": "word",
"domain": "arithmetic",
"topic": "percent_discount",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
18b45799c5ebb24d | 0477b9f26d5b5b77 | concise | Give a short solution with the key steps and the final answer. | A store sells a backpack for $75. It is discounted by 10%. You buy 9 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost? | Discounted price = 75 × (1 − 10/100) = 75 × 90/100 = 135/2.
Subtotal = 9 × 135/2 = 1215/2.
Total with tax = 1215/2 × 108/100 = 6561/10.
Final answer: $\frac{6561}{10}$ | \frac{6561}{10} | {
"format": "word",
"domain": "arithmetic",
"topic": "percent_discount",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
b2f0a823160cc6a3 | a53bad13f9097225 | 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:
$f(x)=- 3 x^{3} + 3 x^{2} - 4 x$.
Student solution:
1) Differentiate term-by-term.
2) Claim: one term $\frac{d}{dx}(- 3 x^{3}) = - 9 x^{3}$.
3) So $f'(x) = - 9 x^{3} + 6 x - 4$. | First incorrect step: 2
The error is differentiating a power: the exponent must decrease by 1.
Correct: $\frac{d}{dx}(- 3 x^{3}) = - 9 x^{2}$.
Therefore the correct derivative is $f'(x) = - 9 x^{2} + 6 x - 4$.
Final answer: $- 9 x^{2} + 6 x - 4$ | - 9 x^{2} + 6 x - 4 | {
"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
} |
5821489d3d63d1f8 | 6b0e3c2fd06ac29e | tutor | Show your work in clear steps (with intermediate values) and finish with a final answer. | Differentiate with respect to x:
$f(x)=- 3 x^{2} \sin{\left(x \right)} + x \sin{\left(x \right)} - \sin{\left(x \right)} + 3 \cos{\left(2 x \right)}$. | 1) Let f(x) = $- 3 x^{2} \sin{\left(x \right)} + x \sin{\left(x \right)} - \sin{\left(x \right)} + 3 \cos{\left(2 x \right)}$.
2) Differentiate using product/chain/quotient rules as needed.
3) Simplify the derivative to get f'(x) = $- 3 x^{2} \cos{\left(x \right)} - 6 x \sin{\left(x \right)} + x \cos{\left(x \right)} +... | - 3\cdot x^{2}\cdot \cos{\left(x \right)} - 6\cdot x\cdot \sin{\left(x \right)} + x\cdot \cos{\left(x \right)} + \sin{\left(x \right)} - 6\cdot \sin{\left(2\cdot x \right)} - \cos{\left(x \right)} | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
b33ba2381b31321b | ab726daf6318eaad | tutor | Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer. | A coin lands heads with probability p=$\frac{1}{7}$. If it is flipped 22 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=22, k=7, p=$\frac{1}{7}$ and simplify.
4) Here C(22,7) = 170544.
Final answer: $\frac{80187228009529344}{3909821048582988049}$ | \frac{80187228009529344}{3909821048582988049} | {
"format": "symbolic",
"domain": "probability",
"topic": "binomial_probability",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
acc18d8209764610 | 208ef6ba63abd874 | 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:
$f(x)=2 x^{3} - 2 x^{2} + 3 x - 4$.
Student solution:
1) Differentiate term-by-term.
2) Claim: one term $\frac{d}{dx}(- 2 x^{2}) = - 4 x^{2}$.
3) So $f'(x) = 2 x^{2} + 3$. | First incorrect step: 2
The error is differentiating a power: the exponent must decrease by 1.
Correct: $\frac{d}{dx}(- 2 x^{2}) = - 4 x$.
Therefore the correct derivative is $f'(x) = 6 x^{2} - 4 x + 3$.
Final answer: $6 x^{2} - 4 x + 3$ | 6 x^{2} - 4 x + 3 | {
"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
} |
f7acb8d86710913d | dfbce3242d325829 | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | Solve the linear system:
$5 x + 8 y$ = $-37$
$- 3 x - 6 y$ = $21$ | Eliminate x by taking a linear combination of the equations.
Multiply the first equation by $-3$: $- 15 x - 24 y$ = $111$.
Multiply the second equation by $5$: $- 15 x - 30 y$ = $105$.
Subtract to eliminate x: $6 y$ = $6$.
Solve for y: y = $1$.
Substitute back to find x: x = $-9$.
Check: plug (x,y)=(-9,1) into both equ... | (x,y)=\left(-9, 1\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
c2315d9d949b105c | dfbce3242d325829 | tutor | Give a detailed derivation, showing intermediate expressions, and then the final answer. | Solve the linear system:
$5 x + 8 y$ = $-37$
$- 3 x - 6 y$ = $21$ | 1) Eliminate x by taking a linear combination of the equations.
2) Multiply the first equation by $-3$: $- 15 x - 24 y$ = $111$.
3) Multiply the second equation by $5$: $- 15 x - 30 y$ = $105$.
4) Subtract to eliminate x: $6 y$ = $6$.
5) Solve for y: y = $1$.
6) Substitute back to find x: x = $-9$.
Final answer: $(x,y)... | (x,y)=\left(-9, 1\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
1e918ffb9c50cbb1 | aed1a0eaae1d9cef | 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:
$\displaystyle \lim_{x\to 2} 2 x^{3} + x^{2} + 4 x + 6$.
Student solution:
1) Substitute x=2 into the expression.
2) (Mistake) Simplify the expression to 2 x^{3} + x^{2} + 3 x + 6.
3) Then the limit is 32. | 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: 34.
Final answer: $34$ | 34 | {
"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
} |
fcac51c5b05fa947 | bdefc566f0b334af | tutor | Solve like a tutor: derive step-by-step and finish with the final answer. | You want 43 liters of a 27% salt solution. You have a 15% solution and a 40% solution. How many liters of each should you mix? | 1) Let x = liters of 15% solution, so 43−x liters is 40% solution.
2) Salt equation: 15x + 40(43−x) = 27·43.
3) (15-40)x = (27-40)·43 ⇒ x = 559/25.
4) Then 43−x = 516/25.
Final answer: $559/25 L of 15% and 516/25 L of 40%$ | 559/25 L of 15% and 516/25 L of 40% | {
"format": "word",
"domain": "arithmetic",
"topic": "mixture",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
1ea6238a88159506 | f88fcd176d3a4708 | tutor | Explain each step clearly and end with the final answer. | Find the equation of the tangent line to $y=2 x^{3} - x^{2} + 4 x - 4$ at $x=-2$. | 1) Differentiate: f'(x) = $6 x^{2} - 2 x + 4$.
2) Slope at x=-2: m = f'(-2) = $32$.
3) Point on curve: (-2, f(-2)) = (-2, $-32$).
4) Use point-slope form y − y0 = m(x − x0).
5) Plug in values: y − -32 = 32(x − -2).
6) Optionally expand: y = $32 x + 32$.
Final answer: $y = 32\cdot x + 32$ | y = 32\cdot x + 32 | {
"format": "symbolic",
"domain": "calculus",
"topic": "tangent_line",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
811e63c57b60fa52 | f88fcd176d3a4708 | concise | Compute the result with a short explanation and give the final answer. | Find the equation of the tangent line to $y=2 x^{3} - x^{2} + 4 x - 4$ at $x=-2$. | Differentiate: f'(x) = $6 x^{2} - 2 x + 4$.
Slope at x=-2: m = f'(-2) = $32$.
Point on curve: (-2, f(-2)) = (-2, $-32$).
Final answer: $y = 32\cdot x + 32$ | y = 32\cdot x + 32 | {
"format": "symbolic",
"domain": "calculus",
"topic": "tangent_line",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.