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bba008c39b89a1a0 | a838613c875aaecb | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | You want 44 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 44−x liters is 70% solution.
2) Salt equation: 20x + 70(44−x) = 30·44.
3) (20-70)x = (30-70)·44 ⇒ x = 176/5.
4) Then 44−x = 44/5.
Final answer: $176/5 L of 20% and 44/5 L of 70%$ | 176/5 L of 20% and 44/5 L of 70% | {
"format": "word",
"domain": "arithmetic",
"topic": "mixture",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
4523f425e5781f50 | a838613c875aaecb | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | You want 44 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 44−x liters is 70% solution.
Salt equation: 20x + 70(44−x) = 30·44.
(20-70)x = (30-70)·44 ⇒ x = 176/5.
Then 44−x = 44/5.
Check: 20·176/5 + 70·44/5 = 30·44.
Final answer: $176/5 L of 20% and 44/5 L of 70%$ | 176/5 L of 20% and 44/5 L of 70% | {
"format": "word",
"domain": "arithmetic",
"topic": "mixture",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
4412f96247eb2a4f | 5811b11957ac1755 | concise | Solve and give a compact solution, then the final answer. | An urn has 7 red balls and 8 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red? | Use a hypergeometric counting argument.
Total equally likely samples: C(15,6).
Favorable samples: choose k reds and n−k blues: C(7,3)·C(8,3).
Final answer: $\frac{56}{143}$ | \frac{56}{143} | {
"format": "symbolic",
"domain": "probability",
"topic": "hypergeometric",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
d135f1c147af77ff | 5811b11957ac1755 | verifier | Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer. | An urn has 7 red balls and 8 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red? | Use a hypergeometric counting argument.
Total equally likely samples: C(15,6).
Favorable samples: choose k reds and n−k blues: C(7,3)·C(8,3).
Compute totals: C(15,6) = 5005, and favorable = 1960.
Divide favorable by total and simplify.
Check: computed probability simplifies to $\frac{56}{143}$.
Final answer: $\frac{56}... | \frac{56}{143} | {
"format": "symbolic",
"domain": "probability",
"topic": "hypergeometric",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
61a55f4925cd2c48 | 8b2daf8ae4c816c2 | verifier | Show the solution, then validate it with a quick check. End with the final answer. | Compute the definite integral:
$\displaystyle \int_{- \frac{\pi}{2}}^{\frac{\pi}{2}} 3 x^{3} - x^{2} + 3 x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)}\,dx$. | Compute an antiderivative F(x) of the integrand $3 x^{3} - x^{2} + 3 x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)}$.
Evaluate at the bounds: F(\frac{\pi}{2}) and F(- \frac{\pi}{2}).
Subtract: F(\frac{\pi}{2}) − F(- \frac{\pi}{2}) = $- \frac{\pi^{3}}{12}$.
Simplify the result.
Check: SymPy differentiation confir... | - \frac{\pi^{3}}{12} | {
"format": "symbolic",
"domain": "calculus",
"topic": "definite_integral",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
cda1a8c7f72ae49b | 8b2daf8ae4c816c2 | tutor | Show your work in clear steps (with intermediate values) and finish with a final answer. | Compute the definite integral:
$\displaystyle \int_{- \frac{\pi}{2}}^{\frac{\pi}{2}} 3 x^{3} - x^{2} + 3 x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)}\,dx$. | 1) Compute an antiderivative F(x) of the integrand $3 x^{3} - x^{2} + 3 x + 2 \sin{\left(x \right)} - \cos{\left(2 x \right)}$.
2) Evaluate at the bounds: F(\frac{\pi}{2}) and F(- \frac{\pi}{2}).
3) Subtract: F(\frac{\pi}{2}) − F(- \frac{\pi}{2}) = $- \frac{\pi^{3}}{12}$.
4) Simplify the result.
Final answer: $- \frac{... | - \frac{\pi^{3}}{12} | {
"format": "symbolic",
"domain": "calculus",
"topic": "definite_integral",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
0c0de7f4ef48736e | 7d2a7558c5fe119e | concise | Compute and present the key steps only, then the final answer. | You want 23 liters of a 35% salt solution. You have a 10% solution and a 50% solution. How many liters of each should you mix? | Let x = liters of 10% solution, so 23−x liters is 50% solution.
Salt equation: 10x + 50(23−x) = 35·23.
(10-50)x = (35-50)·23 ⇒ x = 69/8.
Final answer: $69/8 L of 10% and 115/8 L of 50%$ | 69/8 L of 10% and 115/8 L of 50% | {
"format": "word",
"domain": "arithmetic",
"topic": "mixture",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
b8558b650405af58 | 7d2a7558c5fe119e | tutor | Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer. | You want 23 liters of a 35% salt solution. You have a 10% solution and a 50% solution. How many liters of each should you mix? | 1) Let x = liters of 10% solution, so 23−x liters is 50% solution.
2) Salt equation: 10x + 50(23−x) = 35·23.
3) (10-50)x = (35-50)·23 ⇒ x = 69/8.
4) Then 23−x = 115/8.
Final answer: $69/8 L of 10% and 115/8 L of 50%$ | 69/8 L of 10% and 115/8 L of 50% | {
"format": "word",
"domain": "arithmetic",
"topic": "mixture",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
56a98c506f150c11 | 3d4c6f220af45356 | tutor | Show your work in clear steps (with intermediate values) and finish with a final answer. | A coin lands heads with probability p=$\frac{4}{5}$. If it is flipped 8 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=8, k=3, p=$\frac{4}{5}$ and simplify.
4) Here C(8,3) = 56.
Final answer: $\frac{3584}{390625}$ | \frac{3584}{390625} | {
"format": "symbolic",
"domain": "probability",
"topic": "binomial_probability",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
e43d54525b960e5f | ec3295382b14f1e2 | 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:
$f(x)=\frac{- 3 x^{3} + 2 x^{2} + 3 x + \left(3 x + 4\right) e^{\frac{x}{2}} \sin{\left(x \right)} + 1}{3 x + 4}$.
Student solution:
1) Differentiate term-by-term.
2) Claim (missing chain rule): $\frac{d}{dx}(\frac{3 x e^{\frac{x}{2}} \sin{\left(x \right)}}{3 x + 4}) = \frac{3 x e^{\fr... | 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 x e^{\frac{x}{2}} \sin{\left(x \right)}}{3 x + 4}) = \frac{3 x e^{\frac{x}{2}} \sin{\left(x \right)}}{2 \left(3 x + 4\right)} + \frac{3 x e^{\frac{x}{2}} \cos... | \frac{18 x^{3} - 12 x^{2} - 18 x + \left(3 x + 4\right) \left(- 18 x^{2} + 8 x + \left(3 x + 4\right) e^{\frac{x}{2}} \sin{\left(x \right)} + 2 \left(3 x + 4\right) e^{\frac{x}{2}} \cos{\left(x \right)} + 6 e^{\frac{x}{2}} \sin{\left(x \right)} + 6\right) - 6 \left(3 x + 4\right) e^{\frac{x}{2}} \sin{\left(x \right)} -... | {
"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
} |
270bfe414b2f7cf9 | 00cbc312ce6a9074 | tutor | Give a detailed derivation, showing intermediate expressions, and then the final answer. | You want 30 liters of a 30% salt solution. You have a 15% solution and a 70% solution. How many liters of each should you mix? | 1) Let x = liters of 15% solution, so 30−x liters is 70% solution.
2) Salt equation: 15x + 70(30−x) = 30·30.
3) (15-70)x = (30-70)·30 ⇒ x = 240/11.
4) Then 30−x = 90/11.
Final answer: $240/11 L of 15% and 90/11 L of 70%$ | 240/11 L of 15% and 90/11 L of 70% | {
"format": "word",
"domain": "arithmetic",
"topic": "mixture",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
340fd81113d9879a | 00cbc312ce6a9074 | verifier | Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer. | You want 30 liters of a 30% salt solution. You have a 15% solution and a 70% solution. How many liters of each should you mix? | Let x = liters of 15% solution, so 30−x liters is 70% solution.
Salt equation: 15x + 70(30−x) = 30·30.
(15-70)x = (30-70)·30 ⇒ x = 240/11.
Then 30−x = 90/11.
Check: 15·240/11 + 70·90/11 = 30·30.
Final answer: $240/11 L of 15% and 90/11 L of 70%$ | 240/11 L of 15% and 90/11 L of 70% | {
"format": "word",
"domain": "arithmetic",
"topic": "mixture",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
3f799e530f88df42 | 73c64bd7f761ad35 | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | Differentiate with respect to x:
$f(x)=4 x^{2} + x + 3$. | Differentiate term-by-term:
d/dx(3) = 0.
d/dx(x) = 1.
d/dx(4 x^{2}) = 8 x.
Combine: f'(x) = 8 x + 1.
Check: symbolic differentiation in SymPy matches the derived expression.
Final answer: $8\cdot x + 1$ | 8\cdot x + 1 | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
ac82a51d2de5b14f | 73c64bd7f761ad35 | tutor | Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer. | Differentiate with respect to x:
$f(x)=4 x^{2} + x + 3$. | 1) Differentiate term-by-term:
2) d/dx(3) = 0.
3) d/dx(x) = 1.
4) d/dx(4 x^{2}) = 8 x.
5) Combine: f'(x) = 8 x + 1.
Final answer: $8\cdot x + 1$ | 8\cdot x + 1 | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
f1ff7fb63bf28dfc | c1308b44f8c59648 | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | You want 41 liters of a 40% salt solution. You have a 10% solution and a 70% solution. How many liters of each should you mix? | 1) Let x = liters of 10% solution, so 41−x liters is 70% solution.
2) Salt equation: 10x + 70(41−x) = 40·41.
3) (10-70)x = (40-70)·41 ⇒ x = 41/2.
4) Then 41−x = 41/2.
Final answer: $41/2 L of 10% and 41/2 L of 70%$ | 41/2 L of 10% and 41/2 L of 70% | {
"format": "word",
"domain": "arithmetic",
"topic": "mixture",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
1b38ad3c687457e4 | c1308b44f8c59648 | verifier | Compute the answer and include a short verification step. Give the final answer. | You want 41 liters of a 40% salt solution. You have a 10% solution and a 70% solution. How many liters of each should you mix? | Let x = liters of 10% solution, so 41−x liters is 70% solution.
Salt equation: 10x + 70(41−x) = 40·41.
(10-70)x = (40-70)·41 ⇒ x = 41/2.
Then 41−x = 41/2.
Check: 10·41/2 + 70·41/2 = 40·41.
Final answer: $41/2 L of 10% and 41/2 L of 70%$ | 41/2 L of 10% and 41/2 L of 70% | {
"format": "word",
"domain": "arithmetic",
"topic": "mixture",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
21482a3cafd9f6a7 | d560a66ba953cd2b | verifier | Compute the answer and include a short verification step. Give the final answer. | A coin lands heads with probability p=$\frac{4}{7}$. If it is flipped 19 times, what is the probability of exactly 2 heads? | Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).
Use $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$.
Substitute n=19, k=2, p=$\frac{4}{7}$ and simplify.
Here C(19,2) = 171.
Check: expression simplifies to $\frac{353327485968}{11398895185373143}$.
Final answer: $\frac{353327485968}{11398895185373143}$ | \frac{353327485968}{11398895185373143} | {
"format": "symbolic",
"domain": "probability",
"topic": "binomial_probability",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
80b9d075bd05ad84 | d560a66ba953cd2b | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | A coin lands heads with probability p=$\frac{4}{7}$. If it is flipped 19 times, what is the probability of exactly 2 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=19, k=2, p=$\frac{4}{7}$ and simplify.
4) Here C(19,2) = 171.
Final answer: $\frac{353327485968}{11398895185373143}$ | \frac{353327485968}{11398895185373143} | {
"format": "symbolic",
"domain": "probability",
"topic": "binomial_probability",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
8c145224848a3669 | 9f0a09f4100f3608 | tutor | Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer. | Solve the linear system:
$- 8 x - 4 y$ = $-56$
$- 7 x + 2 y$ = $-27$ | 1) Eliminate x by taking a linear combination of the equations.
2) Multiply the first equation by $-7$: $56 x + 28 y$ = $392$.
3) Multiply the second equation by $-8$: $56 x - 16 y$ = $216$.
4) Subtract to eliminate x: $44 y$ = $176$.
5) Solve for y: y = $4$.
6) Substitute back to find x: x = $5$.
Final answer: $(x,y)=... | (x,y)=\left(5, 4\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
80db00ee5320a1d0 | 9f0a09f4100f3608 | verifier | Compute the answer and include a short verification step. Give the final answer. | Solve the linear system:
$- 8 x - 4 y$ = $-56$
$- 7 x + 2 y$ = $-27$ | Eliminate x by taking a linear combination of the equations.
Multiply the first equation by $-7$: $56 x + 28 y$ = $392$.
Multiply the second equation by $-8$: $56 x - 16 y$ = $216$.
Subtract to eliminate x: $44 y$ = $176$.
Solve for y: y = $4$.
Substitute back to find x: x = $5$.
Check: plug (x,y)=(5,4) into both equat... | (x,y)=\left(5, 4\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
f798c41a7d244a2b | ce041b15ba49a3ec | tutor | Explain each step clearly and end with the final answer. | Compute the definite integral:
$\displaystyle \int_{0}^{\frac{\pi}{2}} - 2 x^{3} - x^{2} + x + 3 \sin{\left(2 x \right)} + 2 \cos{\left(x \right)} - 1\,dx$. | 1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} - x^{2} + x + 3 \sin{\left(2 x \right)} + 2 \cos{\left(x \right)} - 1$.
2) Evaluate at the bounds: F(\frac{\pi}{2}) and F(0).
3) Subtract: F(\frac{\pi}{2}) − F(0) = $- \frac{\pi^{4}}{32} - \frac{\pi}{2} - \frac{\pi^{3}}{24} + \frac{\pi^{2}}{8} + 5$.
4) Simpl... | - \frac{\pi^{4}}{32} - \frac{\pi}{2} - \frac{\pi^{3}}{24} + \frac{\pi^{2}}{8} + 5 | {
"format": "symbolic",
"domain": "calculus",
"topic": "definite_integral",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
8475222ea8457ef3 | ce041b15ba49a3ec | verifier | Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer. | Compute the definite integral:
$\displaystyle \int_{0}^{\frac{\pi}{2}} - 2 x^{3} - x^{2} + x + 3 \sin{\left(2 x \right)} + 2 \cos{\left(x \right)} - 1\,dx$. | Compute an antiderivative F(x) of the integrand $- 2 x^{3} - x^{2} + x + 3 \sin{\left(2 x \right)} + 2 \cos{\left(x \right)} - 1$.
Evaluate at the bounds: F(\frac{\pi}{2}) and F(0).
Subtract: F(\frac{\pi}{2}) − F(0) = $- \frac{\pi^{4}}{32} - \frac{\pi}{2} - \frac{\pi^{3}}{24} + \frac{\pi^{2}}{8} + 5$.
Simplify the resu... | - \frac{\pi^{4}}{32} - \frac{\pi}{2} - \frac{\pi^{3}}{24} + \frac{\pi^{2}}{8} + 5 | {
"format": "symbolic",
"domain": "calculus",
"topic": "definite_integral",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
22bf3e30b366eddd | 1777bb60fe5eee80 | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | A cyclist rides at 45 miles per hour for 2 hours, then at 42 miles per hour for 5/2 hours. How many miles does the cyclist travel in total? | 1) First leg distance = 45 × 2 = 90 miles.
2) Second leg distance = 42 × 5/2 = 105 miles.
3) Total distance = 90 + 105 = 195 miles.
Final answer: $195$ | 195 | {
"format": "word",
"domain": "arithmetic",
"topic": "rate_time_distance",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
ccabbb3d217bb0b3 | 1777bb60fe5eee80 | concise | Compute the result with a short explanation and give the final answer. | A cyclist rides at 45 miles per hour for 2 hours, then at 42 miles per hour for 5/2 hours. How many miles does the cyclist travel in total? | First leg distance = 45 × 2 = 90 miles.
Second leg distance = 42 × 5/2 = 105 miles.
Total distance = 90 + 105 = 195 miles.
Final answer: $195$ | 195 | {
"format": "word",
"domain": "arithmetic",
"topic": "rate_time_distance",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
90ce1218503de336 | ee3d50db9f336670 | tutor | Write a complete step-by-step solution and conclude with the final answer. | Find the multiplicative inverse of 16 modulo 31. | 1) We want x such that 16x ≡ 1 (mod 31).
2) Use extended Euclid to find s,t with 16s + 31t = 1.
3) One Bezout identity is: 16·(2) + 31·(-1) = 1.
4) Reduce s modulo 31: x ≡ 2 (mod 31).
Final answer: $x\equiv 2\pmod{31}$ | x\equiv 2\pmod{31} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
bf1e065b77ae46d4 | ee3d50db9f336670 | verifier | Compute the answer and add a brief verification calculation. Provide the final answer. | Find the multiplicative inverse of 16 modulo 31. | We want x such that 16x ≡ 1 (mod 31).
Use extended Euclid to find s,t with 16s + 31t = 1.
One Bezout identity is: 16·(2) + 31·(-1) = 1.
Reduce s modulo 31: x ≡ 2 (mod 31).
Check: 16·2 mod 31 = 1.
Final answer: $x\equiv 2\pmod{31}$ | x\equiv 2\pmod{31} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
163c6cb12a8fa3d6 | 4d21b1129349c812 | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | Solve the quadratic equation over the real numbers:
$3 x^{2} - 2 x - 8$ = 0. | Identify coefficients a, b, c in ax^2+bx+c=0.
Here a=$3$, b=$-2$, c=$-8$.
Compute the discriminant Δ=b^2−4ac.
Δ = $100$, so √Δ = $10$.
Apply the quadratic formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}$.
x1 = $2$.
x2 = $- \frac{4}{3}$.
Collect the solutions into a set.
Check: substituting each solution into the polynomial gi... | x \in \{2, - \frac{4}{3}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
16e8e1404fcf8139 | 4d21b1129349c812 | tutor | Solve carefully and do not skip algebra/arithmetic steps. End with the final answer. | Solve the quadratic equation over the real numbers:
$3 x^{2} - 2 x - 8$ = 0. | 1) Identify coefficients a, b, c in ax^2+bx+c=0.
2) Here a=$3$, b=$-2$, c=$-8$.
3) Compute the discriminant Δ=b^2−4ac.
4) Δ = $100$, so √Δ = $10$.
5) Apply the quadratic formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}$.
6) x1 = $2$.
7) x2 = $- \frac{4}{3}$.
8) Collect the solutions into a set.
Final answer: $x \in \{2, - \fra... | x \in \{2, - \frac{4}{3}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
68c2aa19f7ef58a4 | 87f397b29444a87f | verifier | Compute the answer and add a brief verification calculation. Provide the final answer. | Two friends split $217 in the ratio 4:3. How much does each receive? | Total ratio parts = 4+3 = 7.
Each part = 217 ÷ 7 = 31.
First share = 4 × 31 = 124.
Second share = 3 × 31 = 93.
Check: 124+93=217.
Final answer: $93$ | 93 | {
"format": "word",
"domain": "arithmetic",
"topic": "ratio_split",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
02db7309250a9187 | 87f397b29444a87f | tutor | Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer. | Two friends split $217 in the ratio 4:3. How much does each receive? | 1) Total ratio parts = 4+3 = 7.
2) Each part = 217 ÷ 7 = 31.
3) First share = 4 × 31 = 124.
4) Second share = 3 × 31 = 93.
Final answer: $93$ | 93 | {
"format": "word",
"domain": "arithmetic",
"topic": "ratio_split",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
25d799014fcd8d8e | e05c653dafc6d458 | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | Solve the linear system:
$7 x - 2 y$ = $-52$
$4 x - 5 y$ = $-22$ | 1) Eliminate x by taking a linear combination of the equations.
2) Multiply the first equation by $4$: $28 x - 8 y$ = $-208$.
3) Multiply the second equation by $7$: $28 x - 35 y$ = $-154$.
4) Subtract to eliminate x: $27 y$ = $-54$.
5) Solve for y: y = $-2$.
6) Substitute back to find x: x = $-8$.
Final answer: $(x,y)... | (x,y)=\left(-8, -2\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
cfcac09520ff6f7e | e05c653dafc6d458 | verifier | Solve the problem, then verify the result with a quick check. Provide the final answer. | Solve the linear system:
$7 x - 2 y$ = $-52$
$4 x - 5 y$ = $-22$ | Eliminate x by taking a linear combination of the equations.
Multiply the first equation by $4$: $28 x - 8 y$ = $-208$.
Multiply the second equation by $7$: $28 x - 35 y$ = $-154$.
Subtract to eliminate x: $27 y$ = $-54$.
Solve for y: y = $-2$.
Substitute back to find x: x = $-8$.
Check: plug (x,y)=(-8,-2) into both eq... | (x,y)=\left(-8, -2\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
e8dcbe7db047bc0e | a3440b5759ca3171 | concise | Provide a compact reasoning and the final answer. | Find the multiplicative inverse of 22 modulo 29. | We want x such that 22x ≡ 1 (mod 29).
Use extended Euclid to find s,t with 22s + 29t = 1.
One Bezout identity is: 22·(4) + 29·(-3) = 1.
Final answer: $x\equiv 4\pmod{29}$ | x\equiv 4\pmod{29} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
0c34f5a5835fc3f9 | a3440b5759ca3171 | verifier | Solve the problem, then verify the result with a quick check. Provide the final answer. | Find the multiplicative inverse of 22 modulo 29. | We want x such that 22x ≡ 1 (mod 29).
Use extended Euclid to find s,t with 22s + 29t = 1.
One Bezout identity is: 22·(4) + 29·(-3) = 1.
Reduce s modulo 29: x ≡ 4 (mod 29).
Check: 22·4 mod 29 = 1.
Final answer: $x\equiv 4\pmod{29}$ | x\equiv 4\pmod{29} | {
"format": "symbolic",
"domain": "discrete",
"topic": "modular_inverse",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
cb4481916927e471 | 41bcecd3093e9095 | tutor | Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer. | Solve the quadratic equation over the real numbers:
$x^{2} - \frac{\sqrt{3} x}{2} + x - 2 + \frac{\sqrt{3}}{2}$ = 0. | 1) Identify coefficients a, b, c in ax^2+bx+c=0.
2) Here a=$1$, b=$1 - \frac{\sqrt{3}}{2}$, c=$-2 + \frac{\sqrt{3}}{2}$.
3) Compute the discriminant Δ=b^2−4ac.
4) Δ = $\frac{39}{4} - 3 \sqrt{3}$, so √Δ = $\sqrt{\frac{39}{4} - 3 \sqrt{3}}$.
5) Apply the quadratic formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}$.
6) x1 = $- \fr... | x \in \{- \frac{1}{2} + \frac{\sqrt{3}}{4} + \frac{\sqrt{39 - 12\cdot \sqrt{3}}}{4}, - \frac{\sqrt{39 - 12\cdot \sqrt{3}}}{4} - \frac{1}{2} + \frac{\sqrt{3}}{4}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
a6dc968ae8e51b1f | e3905de450773855 | 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:
$\displaystyle \lim_{x\to 1} - x - 4$.
Student solution:
1) Substitute x=1 into the expression.
2) (Mistake) Simplify the expression to - 2 x - 4.
3) Then the limit is -6. | 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: -5.
Final answer: $-5$ | -5 | {
"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
} |
16145ce071c9afdd | b0d184af22df1bb8 | tutor | Show your work in clear steps (with intermediate values) and finish with a final answer. | Two friends split $1352 in the ratio 4:9. How much does each receive? | 1) Total ratio parts = 4+9 = 13.
2) Each part = 1352 ÷ 13 = 104.
3) First share = 4 × 104 = 416.
4) Second share = 9 × 104 = 936.
Final answer: $936$ | 936 | {
"format": "word",
"domain": "arithmetic",
"topic": "ratio_split",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
49005b522696edc6 | b0d184af22df1bb8 | verifier | Solve the problem, then verify the result with a quick check. Provide the final answer. | Two friends split $1352 in the ratio 4:9. How much does each receive? | Total ratio parts = 4+9 = 13.
Each part = 1352 ÷ 13 = 104.
First share = 4 × 104 = 416.
Second share = 9 × 104 = 936.
Check: 416+936=1352.
Final answer: $936$ | 936 | {
"format": "word",
"domain": "arithmetic",
"topic": "ratio_split",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
e40fe8c3a26a7e75 | 0dc39ef70e848b97 | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | A disease has prevalence $\frac{3}{200}$. A test has sensitivity $\frac{17}{20}$ 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) = $\frac{89}{800}$.
3) Compute P(D|+) = $\frac{51}{445}$.
Final answer: $\frac{51}{445}$ | \frac{51}{445} | {
"format": "symbolic",
"domain": "probability",
"topic": "bayes_theorem",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
947292bcb7208003 | 0dc39ef70e848b97 | verifier | Compute the answer and include a short verification step. Give the final answer. | A disease has prevalence $\frac{3}{200}$. A test has sensitivity $\frac{17}{20}$ 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) = $\frac{89}{800}$.
Compute P(D|+) = $\frac{51}{445}$.
Check: value is between 0 and 1 and increases with prevalence/sensitivity.
Final answer: $\frac{51}{445}$ | \frac{51}{445} | {
"format": "symbolic",
"domain": "probability",
"topic": "bayes_theorem",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
cca0d54ad6051653 | ab5bccc079756a86 | tutor | Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer. | Two friends split $204 in the ratio 5:1. How much does each receive? | 1) Total ratio parts = 5+1 = 6.
2) Each part = 204 ÷ 6 = 34.
3) First share = 5 × 34 = 170.
4) Second share = 1 × 34 = 34.
Final answer: $34$ | 34 | {
"format": "word",
"domain": "arithmetic",
"topic": "ratio_split",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
ada7a23108ac8d86 | ab5bccc079756a86 | verifier | Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer. | Two friends split $204 in the ratio 5:1. How much does each receive? | Total ratio parts = 5+1 = 6.
Each part = 204 ÷ 6 = 34.
First share = 5 × 34 = 170.
Second share = 1 × 34 = 34.
Check: 170+34=204.
Final answer: $34$ | 34 | {
"format": "word",
"domain": "arithmetic",
"topic": "ratio_split",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
76e72f44ae15961d | 5a0e36baad4173ae | tutor | Solve like a tutor: derive step-by-step and finish with the final answer. | You want 17 liters of a 37% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix? | 1) Let x = liters of 25% solution, so 17−x liters is 50% solution.
2) Salt equation: 25x + 50(17−x) = 37·17.
3) (25-50)x = (37-50)·17 ⇒ x = 221/25.
4) Then 17−x = 204/25.
Final answer: $221/25 L of 25% and 204/25 L of 50%$ | 221/25 L of 25% and 204/25 L of 50% | {
"format": "word",
"domain": "arithmetic",
"topic": "mixture",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
97f3e1dc01e77ae1 | 5a0e36baad4173ae | verifier | Compute the answer and add a brief verification calculation. Provide the final answer. | You want 17 liters of a 37% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix? | Let x = liters of 25% solution, so 17−x liters is 50% solution.
Salt equation: 25x + 50(17−x) = 37·17.
(25-50)x = (37-50)·17 ⇒ x = 221/25.
Then 17−x = 204/25.
Check: 25·221/25 + 50·204/25 = 37·17.
Final answer: $221/25 L of 25% and 204/25 L of 50%$ | 221/25 L of 25% and 204/25 L of 50% | {
"format": "word",
"domain": "arithmetic",
"topic": "mixture",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
58fe4e803309f72d | c951b856084039ac | verifier | Solve the problem, then verify the result with a quick check. Provide the final answer. | A store sells a backpack for $720. It is discounted by 5%. You buy 3 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost? | Discounted price = 720 × (1 − 5/100) = 720 × 95/100 = 684.
Subtotal = 3 × 684 = 2052.
Total with tax = 2052 × 108/100 = 55404/25.
Check: recomputing the formula yields 55404/25.
Final answer: $\frac{55404}{25}$ | \frac{55404}{25} | {
"format": "word",
"domain": "arithmetic",
"topic": "percent_discount",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
3a97355b6a5e95eb | c951b856084039ac | tutor | Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer. | A store sells a backpack for $720. It is discounted by 5%. You buy 3 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost? | 1) Discounted price = 720 × (1 − 5/100) = 720 × 95/100 = 684.
2) Subtotal = 3 × 684 = 2052.
3) Total with tax = 2052 × 108/100 = 55404/25.
Final answer: $\frac{55404}{25}$ | \frac{55404}{25} | {
"format": "word",
"domain": "arithmetic",
"topic": "percent_discount",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
c273ccfbe94d6d96 | 6260cec0336408f3 | tutor | Give a detailed derivation, showing intermediate expressions, and then the final answer. | Two friends split $1749 in the ratio 3:8. How much does each receive? | 1) Total ratio parts = 3+8 = 11.
2) Each part = 1749 ÷ 11 = 159.
3) First share = 3 × 159 = 477.
4) Second share = 8 × 159 = 1272.
Final answer: $1272$ | 1272 | {
"format": "word",
"domain": "arithmetic",
"topic": "ratio_split",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
966875c7c7e6d99a | 6260cec0336408f3 | verifier | Show the solution, then validate it with a quick check. End with the final answer. | Two friends split $1749 in the ratio 3:8. How much does each receive? | Total ratio parts = 3+8 = 11.
Each part = 1749 ÷ 11 = 159.
First share = 3 × 159 = 477.
Second share = 8 × 159 = 1272.
Check: 477+1272=1749.
Final answer: $1272$ | 1272 | {
"format": "word",
"domain": "arithmetic",
"topic": "ratio_split",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
035b017f819f37d9 | d03a80a598c72108 | tutor | Provide a detailed solution with numbered steps and a final answer line. | Solve the quadratic equation over the real numbers:
$x^{2} - \sqrt{11} x + 4 x - \frac{3 \sqrt{11}}{2} + \frac{15}{4}$ = 0. | 1) Identify coefficients a, b, c in ax^2+bx+c=0.
2) Here a=$1$, b=$4 - \sqrt{11}$, c=$\frac{15}{4} - \frac{3 \sqrt{11}}{2}$.
3) Compute the discriminant Δ=b^2−4ac.
4) Δ = $12 - 2 \sqrt{11}$, so √Δ = $\sqrt{12 - 2 \sqrt{11}}$.
5) Apply the quadratic formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}$.
6) x1 = $-2 + \frac{\sqrt{12... | x \in \{-2 + \frac{\sqrt{12 - 2\cdot \sqrt{11}}}{2} + \frac{\sqrt{11}}{2}, -2 - \frac{\sqrt{12 - 2\cdot \sqrt{11}}}{2} + \frac{\sqrt{11}}{2}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
51dade49f0b8ad25 | d03a80a598c72108 | verifier | Show the solution, then validate it with a quick check. End with the final answer. | Solve the quadratic equation over the real numbers:
$x^{2} - \sqrt{11} x + 4 x - \frac{3 \sqrt{11}}{2} + \frac{15}{4}$ = 0. | Identify coefficients a, b, c in ax^2+bx+c=0.
Here a=$1$, b=$4 - \sqrt{11}$, c=$\frac{15}{4} - \frac{3 \sqrt{11}}{2}$.
Compute the discriminant Δ=b^2−4ac.
Δ = $12 - 2 \sqrt{11}$, so √Δ = $\sqrt{12 - 2 \sqrt{11}}$.
Apply the quadratic formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}$.
x1 = $-2 + \frac{\sqrt{12 - 2 \sqrt{11}}}{2... | x \in \{-2 + \frac{\sqrt{12 - 2\cdot \sqrt{11}}}{2} + \frac{\sqrt{11}}{2}, -2 - \frac{\sqrt{12 - 2\cdot \sqrt{11}}}{2} + \frac{\sqrt{11}}{2}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
20b8a38626426623 | 9845be3668e5ce47 | tutor | Explain each step clearly and end with the final answer. | Solve the quadratic equation over the real numbers:
$x^{2} + \frac{7 x}{10} - \frac{9}{2}$ = 0. | 1) Identify coefficients a, b, c in ax^2+bx+c=0.
2) Here a=$1$, b=$\frac{7}{10}$, c=$- \frac{9}{2}$.
3) Compute the discriminant Δ=b^2−4ac.
4) Δ = $\frac{1849}{100}$, so √Δ = $\frac{43}{10}$.
5) Apply the quadratic formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}$.
6) x1 = $\frac{9}{5}$.
7) x2 = $- \frac{5}{2}$.
8) Collect the... | x \in \{\frac{9}{5}, - \frac{5}{2}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
2d916e9e83a3fb10 | 72a1763b38dfa521 | 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:
$f(x)=\left(x - 3\right) e^{x} + \left(- x^{3} - x^{2}\right) \cos{\left(x^{2} + 2 x - 2 \right)}$.
Student solution:
1) Differentiate term-by-term.
2) Claim (missing chain rule): $\frac{d}{dx}(- x^{2} \cos{\left(\left(x^{2} + 2 x\right) - 2 \right)}) = x^{2} \sin{\left(x^{2} + 2 x - 2... | 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(\left(x^{2} + 2 x\right) - 2 \right)}) = x^{2} \left(2 x + 2\right) \sin{\left(x^{2} + 2 x - 2 \right)} - 2 x \cos{\left(\left(x^{2} + 2 x\right) -... | - 2 x^{2} \left(- x - 1\right) \left(x + 1\right) \sin{\left(x^{2} + 2 x - 2 \right)} + x \left(- 3 x - 2\right) \cos{\left(\left(x^{2} + 2 x\right) - 2 \right)} + \left(x - 3\right) e^{x} + 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
} |
98ec20b52e2e7fcc | 741b598d122c3867 | verifier | Compute the answer and include a short verification step. Give the final answer. | Worker A finishes a job in 7 hours and Worker B finishes it in 21 hours. If they work together at constant rates, how long does it take to finish the job? | A's rate = 1/7 job/hour. B's rate = 1/21 job/hour.
Combined rate = 1/7 + 1/21 = 4/21 job/hour.
Time = 1 ÷ (combined rate) = 21/4 hours.
Check: (combined rate)×(time) = 4/21×21/4 = 1.
Final answer: $\frac{21}{4}$ | \frac{21}{4} | {
"format": "word",
"domain": "arithmetic",
"topic": "work_rates",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
a2521f2f893b467a | 741b598d122c3867 | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | Worker A finishes a job in 7 hours and Worker B finishes it in 21 hours. If they work together at constant rates, how long does it take to finish the job? | 1) A's rate = 1/7 job/hour. B's rate = 1/21 job/hour.
2) Combined rate = 1/7 + 1/21 = 4/21 job/hour.
3) Time = 1 ÷ (combined rate) = 21/4 hours.
Final answer: $\frac{21}{4}$ | \frac{21}{4} | {
"format": "word",
"domain": "arithmetic",
"topic": "work_rates",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
7e77bbb12ac7b22e | d9b0a8bfe84361df | tutor | Solve like a tutor: derive step-by-step and finish with the final answer. | Find the equation of the tangent line to $y=2 x^{3} - x^{2} + 2 x - 2$ at $x=0$. | 1) Differentiate: f'(x) = $6 x^{2} - 2 x + 2$.
2) Slope at x=0: m = f'(0) = $2$.
3) Point on curve: (0, f(0)) = (0, $-2$).
4) Use point-slope form y − y0 = m(x − x0).
5) Plug in values: y − -2 = 2(x − 0).
6) Optionally expand: y = $2 x - 2$.
Final answer: $y = 2\cdot x - 2$ | y = 2\cdot x - 2 | {
"format": "symbolic",
"domain": "calculus",
"topic": "tangent_line",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
629ebfa3ab4673f7 | d9b0a8bfe84361df | concise | Solve the problem and keep the explanation concise but complete. Provide the final answer. | Find the equation of the tangent line to $y=2 x^{3} - x^{2} + 2 x - 2$ at $x=0$. | Differentiate: f'(x) = $6 x^{2} - 2 x + 2$.
Slope at x=0: m = f'(0) = $2$.
Point on curve: (0, f(0)) = (0, $-2$).
Final answer: $y = 2\cdot x - 2$ | y = 2\cdot x - 2 | {
"format": "symbolic",
"domain": "calculus",
"topic": "tangent_line",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
d7cacc964ad4e49d | 37554ed3741d8a1a | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | Solve the quadratic equation over the real numbers:
$x^{2} + \frac{21 x}{5} + \frac{18}{5}$ = 0. | 1) Identify coefficients a, b, c in ax^2+bx+c=0.
2) Here a=$1$, b=$\frac{21}{5}$, c=$\frac{18}{5}$.
3) Compute the discriminant Δ=b^2−4ac.
4) Δ = $\frac{81}{25}$, so √Δ = $\frac{9}{5}$.
5) Apply the quadratic formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}$.
6) x1 = $- \frac{6}{5}$.
7) x2 = $-3$.
8) Collect the solutions into... | x \in \{- \frac{6}{5}, -3\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
4a0ecd2d18016889 | 37554ed3741d8a1a | verifier | Compute the answer and include a short verification step. Give the final answer. | Solve the quadratic equation over the real numbers:
$x^{2} + \frac{21 x}{5} + \frac{18}{5}$ = 0. | Identify coefficients a, b, c in ax^2+bx+c=0.
Here a=$1$, b=$\frac{21}{5}$, c=$\frac{18}{5}$.
Compute the discriminant Δ=b^2−4ac.
Δ = $\frac{81}{25}$, so √Δ = $\frac{9}{5}$.
Apply the quadratic formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}$.
x1 = $- \frac{6}{5}$.
x2 = $-3$.
Collect the solutions into a set.
Check: substitut... | x \in \{- \frac{6}{5}, -3\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
07aa973c37570e60 | 67d7e252324e85d1 | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | Solve the congruences: x≡6 (mod 7), x≡3 (mod 11). Give the solution modulo 77. | Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.
The solution is unique modulo m1·m2 = 77.
Compute the CRT solution to get x ≡ 69 (mod 77).
Check: 69 mod 7=6 and mod 11=3.
Final answer: $x\equiv 69\pmod{77}$ | x\equiv 69\pmod{77} | {
"format": "symbolic",
"domain": "discrete",
"topic": "chinese_remainder_theorem",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
e8cf612b56291a16 | 67d7e252324e85d1 | tutor | Give a detailed derivation, showing intermediate expressions, and then the final answer. | Solve the congruences: x≡6 (mod 7), x≡3 (mod 11). Give the solution modulo 77. | 1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.
2) The solution is unique modulo m1·m2 = 77.
3) Compute the CRT solution to get x ≡ 69 (mod 77).
Final answer: $x\equiv 69\pmod{77}$ | x\equiv 69\pmod{77} | {
"format": "symbolic",
"domain": "discrete",
"topic": "chinese_remainder_theorem",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
7d959a2e7c5c25a8 | 0e50f1b750851aff | tutor | Provide a thorough solution as if teaching, including intermediate computations, then give the final answer. | Solve the quadratic equation over the real numbers:
$6 x^{2} - 3 x - 2 \sqrt{2} x + \sqrt{2}$ = 0. | 1) Identify coefficients a, b, c in ax^2+bx+c=0.
2) Here a=$6$, b=$-3 - 2 \sqrt{2}$, c=$\sqrt{2}$.
3) Compute the discriminant Δ=b^2−4ac.
4) Δ = $17 - 12 \sqrt{2}$, so √Δ = $\sqrt{17 - 12 \sqrt{2}}$.
5) Apply the quadratic formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}$.
6) x1 = $\frac{\sqrt{17 - 12 \sqrt{2}}}{12} + \frac{\s... | x \in \{\frac{\sqrt{17 - 12\cdot \sqrt{2}}}{12} + \frac{\sqrt{2}}{6} + \frac{1}{4}, - \frac{\sqrt{17 - 12\cdot \sqrt{2}}}{12} + \frac{\sqrt{2}}{6} + \frac{1}{4}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
64a46a9e5713b227 | 0e50f1b750851aff | verifier | Solve the problem, then verify the result with a quick check. Provide the final answer. | Solve the quadratic equation over the real numbers:
$6 x^{2} - 3 x - 2 \sqrt{2} x + \sqrt{2}$ = 0. | Identify coefficients a, b, c in ax^2+bx+c=0.
Here a=$6$, b=$-3 - 2 \sqrt{2}$, c=$\sqrt{2}$.
Compute the discriminant Δ=b^2−4ac.
Δ = $17 - 12 \sqrt{2}$, so √Δ = $\sqrt{17 - 12 \sqrt{2}}$.
Apply the quadratic formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}$.
x1 = $\frac{\sqrt{17 - 12 \sqrt{2}}}{12} + \frac{\sqrt{2}}{6} + \frac... | x \in \{\frac{\sqrt{17 - 12\cdot \sqrt{2}}}{12} + \frac{\sqrt{2}}{6} + \frac{1}{4}, - \frac{\sqrt{17 - 12\cdot \sqrt{2}}}{12} + \frac{\sqrt{2}}{6} + \frac{1}{4}\} | {
"format": "symbolic",
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"topic": "quadratic_equation",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
682940107d82eeef | 00892c64f4f3362e | concise | Solve the problem and keep the explanation concise but complete. Provide the final answer. | Solve the linear system:
$3 x$ = $-3$
$3 x - 3 y$ = $9$ | Eliminate x by taking a linear combination of the equations.
Multiply the first equation by $3$: $9 x$ = $-9$.
Multiply the second equation by $3$: $9 x - 9 y$ = $27$.
Final answer: $(x,y)=\left(-1, -4\right)$ | (x,y)=\left(-1, -4\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
79285d80b317573e | e9ea8694d5d5651d | 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:
$\displaystyle \lim_{x\to -2} - x^{3} + 2 x^{2} - 3 x + 4$.
Student solution:
1) Substitute x=-2 into the expression.
2) (Mistake) Simplify the expression to - x^{3} + 2 x^{2} - 4 x + 4.
3) Then the limit is 28. | 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: 26.
Final answer: $26$ | 26 | {
"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
} |
bcdf01ba38649e61 | d810d322c3be4d1d | verifier | Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer. | Solve the linear system:
$- 2 x - 5 y$ = $-1$
$8 x - y$ = $67$ | Eliminate x by taking a linear combination of the equations.
Multiply the first equation by $8$: $- 16 x - 40 y$ = $-8$.
Multiply the second equation by $-2$: $- 16 x + 2 y$ = $-134$.
Subtract to eliminate x: $- 42 y$ = $126$.
Solve for y: y = $-3$.
Substitute back to find x: x = $8$.
Check: plug (x,y)=(8,-3) into both... | (x,y)=\left(8, -3\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
c477681a25838695 | d810d322c3be4d1d | tutor | Show your work in clear steps (with intermediate values) and finish with a final answer. | Solve the linear system:
$- 2 x - 5 y$ = $-1$
$8 x - y$ = $67$ | 1) Eliminate x by taking a linear combination of the equations.
2) Multiply the first equation by $8$: $- 16 x - 40 y$ = $-8$.
3) Multiply the second equation by $-2$: $- 16 x + 2 y$ = $-134$.
4) Subtract to eliminate x: $- 42 y$ = $126$.
5) Solve for y: y = $-3$.
6) Substitute back to find x: x = $8$.
Final answer: $(... | (x,y)=\left(8, -3\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
42417c1e30ef1301 | 2ce09d513053e1c7 | tutor | Give a detailed derivation, showing intermediate expressions, and then the final answer. | Compute the indefinite integral:
$\displaystyle \int \left(2 x + 1\right) \cos{\left(x \right)} + \left(2 x + 4\right) e^{2 x}\,dx$. | 1) Identify an integration method (term-by-term, substitution, or a known pattern).
2) Integrand: $\left(2 x + 1\right) \cos{\left(x \right)} + \left(2 x + 4\right) e^{2 x}$.
3) Compute an antiderivative: F(x) = $2 x \sin{\left(x \right)} + \frac{\left(2 x + 3\right) e^{2 x}}{2} + \sin{\left(x \right)} + 2 \cos{\left(x... | C + 2\cdot x\cdot \sin{\left(x \right)} + \frac{\left(2\cdot x + 3\right)\cdot e^{2\cdot x}}{2} + \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"
} |
c54e2599db8dd10f | 2ce09d513053e1c7 | verifier | Compute the answer and add a brief verification calculation. Provide the final answer. | Compute the indefinite integral:
$\displaystyle \int \left(2 x + 1\right) \cos{\left(x \right)} + \left(2 x + 4\right) e^{2 x}\,dx$. | Identify an integration method (term-by-term, substitution, or a known pattern).
Integrand: $\left(2 x + 1\right) \cos{\left(x \right)} + \left(2 x + 4\right) e^{2 x}$.
Compute an antiderivative: F(x) = $2 x \sin{\left(x \right)} + \frac{\left(2 x + 3\right) e^{2 x}}{2} + \sin{\left(x \right)} + 2 \cos{\left(x \right)}... | C + 2\cdot x\cdot \sin{\left(x \right)} + \frac{\left(2\cdot x + 3\right)\cdot e^{2\cdot x}}{2} + \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"
} |
0dd49f5a03c5ac6b | a296c28adb8b96c6 | tutor | Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer. | Evaluate the limit:
$\displaystyle \lim_{x\to -2} 4 - 3 x$. | 1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).
2) Here p(x) = $- 3 x^{2} - 2 x$ and a = -2.
3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).
4) Differentiate: p'(x) = $- 6 x - 2$.
5) Evaluate at a: p'(-2) = $10$.
Final answer: $10$ | 10 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
dd36bc5ad3cef3c0 | a296c28adb8b96c6 | verifier | Show the solution, then validate it with a quick check. End with the final answer. | Evaluate the limit:
$\displaystyle \lim_{x\to -2} 4 - 3 x$. | Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).
Here p(x) = $- 3 x^{2} - 2 x$ and a = -2.
Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).
Differentiate: p'(x) = $- 6 x - 2$.
Evaluate at a: p'(-2) = $10$.
Check: the identity gives $10$.
Final answer: $10$ | 10 | {
"format": "symbolic",
"domain": "calculus",
"topic": "limit",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
21c7b8b0102777c7 | 2ea26b93b6292c82 | tutor | Explain your reasoning clearly and show all important intermediate computations. End with the final answer. | A disease has prevalence $\frac{1}{100}$. A test has sensitivity $\frac{9}{10}$ 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) = $\frac{387}{10000}$.
3) Compute P(D|+) = $\frac{10}{43}$.
Final answer: $\frac{10}{43}$ | \frac{10}{43} | {
"format": "symbolic",
"domain": "probability",
"topic": "bayes_theorem",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
d70e670b1a596478 | 2ea26b93b6292c82 | verifier | Solve the problem, then verify the result with a quick check. Provide the final answer. | A disease has prevalence $\frac{1}{100}$. A test has sensitivity $\frac{9}{10}$ 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) = $\frac{387}{10000}$.
Compute P(D|+) = $\frac{10}{43}$.
Check: value is between 0 and 1 and increases with prevalence/sensitivity.
Final answer: $\frac{10}{43}$ | \frac{10}{43} | {
"format": "symbolic",
"domain": "probability",
"topic": "bayes_theorem",
"difficulty": 2,
"verified": true,
"source": "synthetic_v3_open",
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} |
916f01d7e6ca8d2c | 62793a2b48d98b0f | verifier | Show the solution, then validate it with a quick check. End with the final answer. | A store sells a backpack for $64. It is discounted by 35%. You buy 6 backpacks. What is the total cost? | Discounted price = 64 × (1 − 35/100) = 64 × 65/100 = 208/5.
Subtotal = 6 × 208/5 = 1248/5.
Check: recomputing the formula yields 1248/5.
Final answer: $\frac{1248}{5}$ | \frac{1248}{5} | {
"format": "word",
"domain": "arithmetic",
"topic": "percent_discount",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
fde932728d8e19e1 | 62793a2b48d98b0f | tutor | Provide a detailed solution with numbered steps and a final answer line. | A store sells a backpack for $64. It is discounted by 35%. You buy 6 backpacks. What is the total cost? | 1) Discounted price = 64 × (1 − 35/100) = 64 × 65/100 = 208/5.
2) Subtotal = 6 × 208/5 = 1248/5.
Final answer: $\frac{1248}{5}$ | \frac{1248}{5} | {
"format": "word",
"domain": "arithmetic",
"topic": "percent_discount",
"difficulty": 1,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
05582209bc1b2479 | 0afa354efc24e7f3 | 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:
$\displaystyle \lim_{x\to 4} - 3 x^{2} - 11 x - 43$.
Student solution:
1) Substitute x=4 into the expression.
2) (Mistake) Simplify the expression to - 3 x^{2} - 12 x - 43.
3) Then the limit is -139. | 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: -135.
Final answer: $-135$ | -135 | {
"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
} |
194eb7749303cb39 | dd0a7baeb2fe7bd4 | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | Solve the linear system:
$11 x - 6 y$ = $-61$
$- 2 x + 7 y$ = $17$ | Eliminate x by taking a linear combination of the equations.
Multiply the first equation by $-2$: $- 22 x + 12 y$ = $122$.
Multiply the second equation by $11$: $- 22 x + 77 y$ = $187$.
Subtract to eliminate x: $- 65 y$ = $-65$.
Solve for y: y = $1$.
Substitute back to find x: x = $-5$.
Check: plug (x,y)=(-5,1) into bo... | (x,y)=\left(-5, 1\right) | {
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"topic": "linear_system_2x2",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
b984600a9b84659a | d0c5e090ad20e2eb | tutor | Write a complete step-by-step solution and conclude with the final answer. | Compute the indefinite integral:
$\displaystyle \int \left(2 x + 1\right) \cos{\left(x \right)} + \left(2 x + 2\right) e^{2 x}\,dx$. | 1) Identify an integration method (term-by-term, substitution, or a known pattern).
2) Integrand: $\left(2 x + 1\right) \cos{\left(x \right)} + \left(2 x + 2\right) e^{2 x}$.
3) Compute an antiderivative: F(x) = $2 x \sin{\left(x \right)} + \frac{\left(2 x + 1\right) e^{2 x}}{2} + \sin{\left(x \right)} + 2 \cos{\left(x... | C + 2\cdot x\cdot \sin{\left(x \right)} + \frac{\left(2\cdot x + 1\right)\cdot e^{2\cdot x}}{2} + \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"
} |
4ecd8311f34b496f | d0c5e090ad20e2eb | verifier | Show the solution, then validate it with a quick check. End with the final answer. | Compute the indefinite integral:
$\displaystyle \int \left(2 x + 1\right) \cos{\left(x \right)} + \left(2 x + 2\right) e^{2 x}\,dx$. | Identify an integration method (term-by-term, substitution, or a known pattern).
Integrand: $\left(2 x + 1\right) \cos{\left(x \right)} + \left(2 x + 2\right) e^{2 x}$.
Compute an antiderivative: F(x) = $2 x \sin{\left(x \right)} + \frac{\left(2 x + 1\right) e^{2 x}}{2} + \sin{\left(x \right)} + 2 \cos{\left(x \right)}... | C + 2\cdot x\cdot \sin{\left(x \right)} + \frac{\left(2\cdot x + 1\right)\cdot e^{2\cdot x}}{2} + \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"
} |
b1bde8830b0a8a78 | a85de886faa84ba0 | tutor | Solve carefully and do not skip algebra/arithmetic steps. End with the final answer. | Compute the indefinite integral:
$\displaystyle \int - 2 x^{3} + x^{2} - 3 x + 2 \sin{\left(x \right)} - 3 \cos{\left(2 x \right)} - 2\,dx$. | 1) Identify an integration method (term-by-term, substitution, or a known pattern).
2) Integrand: $- 2 x^{3} + x^{2} - 3 x + 2 \sin{\left(x \right)} - 3 \cos{\left(2 x \right)} - 2$.
3) Compute an antiderivative: F(x) = $- \frac{x^{4}}{2} + \frac{x^{3}}{3} - \frac{3 x^{2}}{2} - 2 x - \frac{3 \sin{\left(2 x \right)}}{2}... | C - \frac{x^{4}}{2} + \frac{x^{3}}{3} - \frac{3\cdot x^{2}}{2} - 2\cdot x - \frac{3\cdot \sin{\left(2\cdot x \right)}}{2} - 2\cdot \cos{\left(x \right)} | {
"format": "symbolic",
"domain": "calculus",
"topic": "indefinite_integral",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
75eb6c05ba110ebd | a85de886faa84ba0 | verifier | Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer. | Compute the indefinite integral:
$\displaystyle \int - 2 x^{3} + x^{2} - 3 x + 2 \sin{\left(x \right)} - 3 \cos{\left(2 x \right)} - 2\,dx$. | Identify an integration method (term-by-term, substitution, or a known pattern).
Integrand: $- 2 x^{3} + x^{2} - 3 x + 2 \sin{\left(x \right)} - 3 \cos{\left(2 x \right)} - 2$.
Compute an antiderivative: F(x) = $- \frac{x^{4}}{2} + \frac{x^{3}}{3} - \frac{3 x^{2}}{2} - 2 x - \frac{3 \sin{\left(2 x \right)}}{2} - 2 \cos... | C - \frac{x^{4}}{2} + \frac{x^{3}}{3} - \frac{3\cdot x^{2}}{2} - 2\cdot x - \frac{3\cdot \sin{\left(2\cdot x \right)}}{2} - 2\cdot \cos{\left(x \right)} | {
"format": "symbolic",
"domain": "calculus",
"topic": "indefinite_integral",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
94131f7c169ad38b | 192cc84e7fa41147 | 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:
$\displaystyle \lim_{x\to 4} x^{3} + 4 x^{2} + 17 x + 71$.
Student solution:
1) Substitute x=4 into the expression.
2) (Mistake) Simplify the expression to x^{3} + 4 x^{2} + 16 x + 71.
3) Then the limit is 263. | 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: 267.
Final answer: $267$ | 267 | {
"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
} |
97d9d4b36ba33695 | b4efca31a253f59f | tutor | Solve like a tutor: derive step-by-step and finish with the final answer. | Solve the linear system:
$5 x - y$ = $44$
$2 x - 2 y$ = $16$ | 1) Eliminate x by taking a linear combination of the equations.
2) Multiply the first equation by $2$: $10 x - 2 y$ = $88$.
3) Multiply the second equation by $5$: $10 x - 10 y$ = $80$.
4) Subtract to eliminate x: $8 y$ = $8$.
5) Solve for y: y = $1$.
6) Substitute back to find x: x = $9$.
Final answer: $(x,y)=\left(9,... | (x,y)=\left(9, 1\right) | {
"format": "symbolic",
"domain": "algebra",
"topic": "linear_system_2x2",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
157f4034e76494df | 425c21396b4824a9 | tutor | Provide a detailed solution with numbered steps and a final answer line. | Solve the quadratic equation over the real numbers:
$x^{2} - \sqrt{2} x + \frac{2 x}{5} - \frac{3}{5} + \frac{3 \sqrt{2}}{5}$ = 0. | 1) Identify coefficients a, b, c in ax^2+bx+c=0.
2) Here a=$1$, b=$\frac{2}{5} - \sqrt{2}$, c=$- \frac{3}{5} + \frac{3 \sqrt{2}}{5}$.
3) Compute the discriminant Δ=b^2−4ac.
4) Δ = $\frac{114}{25} - \frac{16 \sqrt{2}}{5}$, so √Δ = $\sqrt{\frac{114}{25} - \frac{16 \sqrt{2}}{5}}$.
5) Apply the quadratic formula $x=\frac{-... | x \in \{- \frac{1}{5} + \frac{\sqrt{114 - 80\cdot \sqrt{2}}}{10} + \frac{\sqrt{2}}{2}, - \frac{1}{5} - \frac{\sqrt{114 - 80\cdot \sqrt{2}}}{10} + \frac{\sqrt{2}}{2}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
cb35708ee59296a2 | 425c21396b4824a9 | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | Solve the quadratic equation over the real numbers:
$x^{2} - \sqrt{2} x + \frac{2 x}{5} - \frac{3}{5} + \frac{3 \sqrt{2}}{5}$ = 0. | Identify coefficients a, b, c in ax^2+bx+c=0.
Here a=$1$, b=$\frac{2}{5} - \sqrt{2}$, c=$- \frac{3}{5} + \frac{3 \sqrt{2}}{5}$.
Compute the discriminant Δ=b^2−4ac.
Δ = $\frac{114}{25} - \frac{16 \sqrt{2}}{5}$, so √Δ = $\sqrt{\frac{114}{25} - \frac{16 \sqrt{2}}{5}}$.
Apply the quadratic formula $x=\frac{-b\pm\sqrt{\Delt... | x \in \{- \frac{1}{5} + \frac{\sqrt{114 - 80\cdot \sqrt{2}}}{10} + \frac{\sqrt{2}}{2}, - \frac{1}{5} - \frac{\sqrt{114 - 80\cdot \sqrt{2}}}{10} + \frac{\sqrt{2}}{2}\} | {
"format": "symbolic",
"domain": "algebra",
"topic": "quadratic_equation",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
6bff6c9184b90c9a | 328d656f1f617207 | tutor | Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer. | How many solutions in integers with $x_i\ge 1$ satisfy $x_1+\cdots+x_6=22$? | 1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_6=16.
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=16 and k=6.
4) Compute C(21,5) = 20349.
Final answer: $20349$ | 20349 | {
"format": "symbolic",
"domain": "discrete",
"topic": "stars_and_bars",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
30018cf1d213a902 | 328d656f1f617207 | verifier | Compute the answer and add a brief verification calculation. Provide the final answer. | How many solutions in integers with $x_i\ge 1$ satisfy $x_1+\cdots+x_6=22$? | Let y_i=x_i-1, then y_i≥0 and y_1+…+y_6=16.
Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).
Here N=16 and k=6.
Compute C(21,5) = 20349.
Check: matches the standard stars-and-bars formula.
Final answer: $20349$ | 20349 | {
"format": "symbolic",
"domain": "discrete",
"topic": "stars_and_bars",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
45444b94d79c84d1 | 835ab262ebbc0169 | tutor | Show your work in clear steps (with intermediate values) and finish with a final answer. | Solve the congruences: x≡5 (mod 9), x≡3 (mod 5). Give the solution modulo 45. | 1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.
2) The solution is unique modulo m1·m2 = 45.
3) Compute the CRT solution to get x ≡ 23 (mod 45).
Final answer: $x\equiv 23\pmod{45}$ | x\equiv 23\pmod{45} | {
"format": "symbolic",
"domain": "discrete",
"topic": "chinese_remainder_theorem",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
00a6ff33a8ab05b5 | 835ab262ebbc0169 | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | Solve the congruences: x≡5 (mod 9), x≡3 (mod 5). Give the solution modulo 45. | Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.
The solution is unique modulo m1·m2 = 45.
Compute the CRT solution to get x ≡ 23 (mod 45).
Check: 23 mod 9=5 and mod 5=3.
Final answer: $x\equiv 23\pmod{45}$ | x\equiv 23\pmod{45} | {
"format": "symbolic",
"domain": "discrete",
"topic": "chinese_remainder_theorem",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
7e056933880ece8a | 1ea3d1b656c77b34 | tutor | Write a complete step-by-step solution and conclude with the final answer. | Differentiate with respect to x:
$f(x)=- x^{2} \sin{\left(x \right)} + 2 x \sin{\left(x \right)} + 2 \sin{\left(x \right)} + 2 \cos{\left(2 x \right)}$. | 1) Let f(x) = $- x^{2} \sin{\left(x \right)} + 2 x \sin{\left(x \right)} + 2 \sin{\left(x \right)} + 2 \cos{\left(2 x \right)}$.
2) Differentiate using product/chain/quotient rules as needed.
3) Simplify the derivative to get f'(x) = $- x^{2} \cos{\left(x \right)} - 2 x \sin{\left(x \right)} + 2 x \cos{\left(x \right)}... | - x^{2}\cdot \cos{\left(x \right)} - 2\cdot x\cdot \sin{\left(x \right)} + 2\cdot x\cdot \cos{\left(x \right)} + 2\cdot \sin{\left(x \right)} - 4\cdot \sin{\left(2\cdot x \right)} + 2\cdot \cos{\left(x \right)} | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
2d59c2124f929d08 | 1ea3d1b656c77b34 | verifier | Compute the answer and include a short verification step. Give the final answer. | Differentiate with respect to x:
$f(x)=- x^{2} \sin{\left(x \right)} + 2 x \sin{\left(x \right)} + 2 \sin{\left(x \right)} + 2 \cos{\left(2 x \right)}$. | Let f(x) = $- x^{2} \sin{\left(x \right)} + 2 x \sin{\left(x \right)} + 2 \sin{\left(x \right)} + 2 \cos{\left(2 x \right)}$.
Differentiate using product/chain/quotient rules as needed.
Simplify the derivative to get f'(x) = $- x^{2} \cos{\left(x \right)} - 2 x \sin{\left(x \right)} + 2 x \cos{\left(x \right)} + 2 \sin... | - x^{2}\cdot \cos{\left(x \right)} - 2\cdot x\cdot \sin{\left(x \right)} + 2\cdot x\cdot \cos{\left(x \right)} + 2\cdot \sin{\left(x \right)} - 4\cdot \sin{\left(2\cdot x \right)} + 2\cdot \cos{\left(x \right)} | {
"format": "symbolic",
"domain": "calculus",
"topic": "derivative",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
2688e65898f69f6f | 5aa6dc9c3b985efb | verifier | Solve and add a check (substitution/count/sanity check). Provide the final answer. | Find the equation of the tangent line to $y=x^{2} + 3 x + 2 \sin{\left(x \right)}$ at $x=1$. | Differentiate: f'(x) = $2 x + 2 \cos{\left(x \right)} + 3$.
Slope at x=1: m = f'(1) = $2 \cos{\left(1 \right)} + 5$.
Point on curve: (1, f(1)) = (1, $2 \sin{\left(1 \right)} + 4$).
Use point-slope form y − y0 = m(x − x0).
Plug in values: y − 2 \sin{\left(1 \right)} + 4 = 2 \cos{\left(1 \right)} + 5(x − 1).
Optionally e... | y = 2\cdot x\cdot \cos{\left(1 \right)} + 5\cdot x - 2\cdot \cos{\left(1 \right)} - 1 + 2\cdot \sin{\left(1 \right)} | {
"format": "symbolic",
"domain": "calculus",
"topic": "tangent_line",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
04cf5232fedf40e1 | 5aa6dc9c3b985efb | tutor | Solve carefully and do not skip algebra/arithmetic steps. End with the final answer. | Find the equation of the tangent line to $y=x^{2} + 3 x + 2 \sin{\left(x \right)}$ at $x=1$. | 1) Differentiate: f'(x) = $2 x + 2 \cos{\left(x \right)} + 3$.
2) Slope at x=1: m = f'(1) = $2 \cos{\left(1 \right)} + 5$.
3) Point on curve: (1, f(1)) = (1, $2 \sin{\left(1 \right)} + 4$).
4) Use point-slope form y − y0 = m(x − x0).
5) Plug in values: y − 2 \sin{\left(1 \right)} + 4 = 2 \cos{\left(1 \right)} + 5(x − 1... | y = 2\cdot x\cdot \cos{\left(1 \right)} + 5\cdot x - 2\cdot \cos{\left(1 \right)} - 1 + 2\cdot \sin{\left(1 \right)} | {
"format": "symbolic",
"domain": "calculus",
"topic": "tangent_line",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
34d1c4a1ad8a63b6 | 9b8b08a39061e87f | tutor | Explain each step clearly and end with the final answer. | Worker A finishes a job in 4 hours and Worker B finishes it in 12 hours. If they work together at constant rates, how long does it take to finish the job? | 1) A's rate = 1/4 job/hour. B's rate = 1/12 job/hour.
2) Combined rate = 1/4 + 1/12 = 1/3 job/hour.
3) Time = 1 ÷ (combined rate) = 3 hours.
Final answer: $3$ | 3 | {
"format": "word",
"domain": "arithmetic",
"topic": "work_rates",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
9290628de67c28b6 | a3343dda6c927aaf | 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:
$f(x)=\frac{- x^{2} - x + \left(x + 6\right) e^{\frac{x}{2}} \sin{\left(x \right)} - 3}{x + 6}$.
Student solution:
1) Differentiate term-by-term.
2) Claim (missing chain rule): $\frac{d}{dx}(\frac{6 e^{\frac{x}{2}} \sin{\left(x \right)}}{x + 6}) = \frac{6 e^{\frac{x}{2}} \cos{\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}(\frac{6 e^{\frac{x}{2}} \sin{\left(x \right)}}{x + 6}) = \frac{3 e^{\frac{x}{2}} \sin{\left(x \right)}}{x + 6} + \frac{6 e^{\frac{x}{2}} \cos{\left(x \right)}}{x + 6}... | \frac{2 x^{2} + 2 x + \left(x + 6\right) \left(- 4 x + \left(x + 6\right) e^{\frac{x}{2}} \sin{\left(x \right)} + 2 \left(x + 6\right) e^{\frac{x}{2}} \cos{\left(x \right)} + 2 e^{\frac{x}{2}} \sin{\left(x \right)} - 2\right) - 2 \left(x + 6\right) e^{\frac{x}{2}} \sin{\left(x \right)} + 6}{2 \left(x + 6\right)^{2}} | {
"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
} |
9972c2d5816c7121 | 8971f17a5c0e0a9a | tutor | Show your work in clear steps (with intermediate values) and finish with a final answer. | Two friends split $624 in the ratio 6:2. How much does each receive? | 1) Total ratio parts = 6+2 = 8.
2) Each part = 624 ÷ 8 = 78.
3) First share = 6 × 78 = 468.
4) Second share = 2 × 78 = 156.
Final answer: $156$ | 156 | {
"format": "word",
"domain": "arithmetic",
"topic": "ratio_split",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
f171092806b38142 | 8971f17a5c0e0a9a | verifier | Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer. | Two friends split $624 in the ratio 6:2. How much does each receive? | Total ratio parts = 6+2 = 8.
Each part = 624 ÷ 8 = 78.
First share = 6 × 78 = 468.
Second share = 2 × 78 = 156.
Check: 468+156=624.
Final answer: $156$ | 156 | {
"format": "word",
"domain": "arithmetic",
"topic": "ratio_split",
"difficulty": 3,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
69847b313d8dcea4 | b704e44cd2b09098 | tutor | Write a complete step-by-step solution and conclude with the final answer. | How many binary strings of length 17 contain exactly 3 ones with no two ones adjacent? | 1) Reserve 2 mandatory zeros between the 3 ones.
2) This leaves n−k+1 = 15 valid slots for the ones.
3) Choose the slots: C(15,3) = 455.
Final answer: $455$ | 455 | {
"format": "symbolic",
"domain": "discrete",
"topic": "counting_no_adjacent",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
613cbeec64656f7f | b704e44cd2b09098 | verifier | Show the solution, then validate it with a quick check. End with the final answer. | How many binary strings of length 17 contain exactly 3 ones with no two ones adjacent? | Reserve 2 mandatory zeros between the 3 ones.
This leaves n−k+1 = 15 valid slots for the ones.
Choose the slots: C(15,3) = 455.
Check: standard gap method gives C(n−k+1,k).
Final answer: $455$ | 455 | {
"format": "symbolic",
"domain": "discrete",
"topic": "counting_no_adjacent",
"difficulty": 5,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
b279b9e844cd6c1b | c7bb301094593b82 | tutor | Solve like a tutor: derive step-by-step and finish with the final answer. | How many binary strings of length 24 contain exactly 4 ones with no two ones adjacent? | 1) Reserve 3 mandatory zeros between the 4 ones.
2) This leaves n−k+1 = 21 valid slots for the ones.
3) Choose the slots: C(21,4) = 5985.
Final answer: $5985$ | 5985 | {
"format": "symbolic",
"domain": "discrete",
"topic": "counting_no_adjacent",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
eab00250b2d3ffd8 | c7bb301094593b82 | concise | Solve the problem and keep the explanation concise but complete. Provide the final answer. | How many binary strings of length 24 contain exactly 4 ones with no two ones adjacent? | Reserve 3 mandatory zeros between the 4 ones.
This leaves n−k+1 = 21 valid slots for the ones.
Choose the slots: C(21,4) = 5985.
Final answer: $5985$ | 5985 | {
"format": "symbolic",
"domain": "discrete",
"topic": "counting_no_adjacent",
"difficulty": 4,
"verified": true,
"source": "synthetic_v3_open",
"sympy_version": "1.14.0"
} |
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