problem stringlengths 11 85 | broken_working listlengths 4 6 | error_location stringclasses 6
values | error_explanation stringlengths 48 179 | correct_working listlengths 4 6 | correct_answer stringlengths 1 21 | operation stringclasses 7
values | error_type stringclasses 21
values | has_error bool 2
classes | cot_explanation stringlengths 88 316 | difficulty stringclasses 3
values |
|---|---|---|---|---|---|---|---|---|---|---|
Find the determinant of the matrix [[1,-4],[0,-2]]. | [
"Step 1: For a 2×2 matrix [[a,b],[c,d]], det(A) = a·d - b·c",
"Step 2: Identify: a=1, b=-4, c=0, d=-2",
"Step 3: Compute a·d = 1×-2 = -2",
"Step 4: Compute b·c = -4×0 = 0",
"Step 5: det(A) = -2 - 0 = -2",
"Answer: -2"
] | None | There is no error. The working shown is correct. | [
"Step 1: For a 2×2 matrix [[a,b],[c,d]], det(A) = a·d - b·c",
"Step 2: Identify: a=1, b=-4, c=0, d=-2",
"Step 3: Compute a·d = 1×-2 = -2",
"Step 4: Compute b·c = -4×0 = 0",
"Step 5: det(A) = -2 - 0 = -2",
"Answer: -2"
] | -2 | determinant | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: -2 | easy |
Find lim (x → 3) of (x² - 9) / (x - 3) | [
"Step 1: Direct substitution gives 0/0 — factor and cancel",
"Step 2: Factor: x² - 9 = (x - 3)(x + 3)",
"Step 3: Cancel (x - 3): expression becomes (x + 3)",
"Step 4: Arithmetic error: 3 + 3 = 8",
"Answer: 8"
] | Step 4 | Step 4 has an arithmetic error. 3 + 3 = 6, not 8. The correct answer is 6. | [
"Step 1: Direct substitution gives 0/0 — indeterminate form, must factor",
"Step 2: Factor numerator: x² - 9 = (x - 3)(x + 3)",
"Step 3: Cancel (x - 3): expression becomes (x + 3)",
"Step 4: Substitute x = 3: 3 + 3 = 6",
"Answer: 6"
] | 6 | limits | arithmetic | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: incorrect — Step 4 has an arithmetic error. 3 + 3 = 6, not 8. The correct answer is 6. | Therefore the correct answer is: 6 | medium |
Compute the matrix product A·B where A = [[-3,1],[-4,0]] and B = [[4,-1],[1,2]]. | [
"Step 1: Computed B·A instead of A·B — order is reversed",
"Step 2: D[0][0] = 4×-3 + -1×-4 = -8",
"Step 3: D[0][1] = 4×1 + -1×0 = 4",
"Step 4: D[1][0] = 1×-3 + 2×-4 = -11",
"Step 5: D[1][1] = 1×1 + 2×0 = 1",
"Answer: [[-8,4],[-11,1]]"
] | Step 1 | The multiplication order is reversed in Step 1. Matrix multiplication is not commutative: A·B ≠ B·A. The correct answer for A·B is [[-11,5],[-16,4]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = -3×4 + 1×1 = -11",
"Step 3: C[0][1] = -3×-1 + 1×2 = 5",
"Step 4: C[1][0] = -4×4 + 0×1 = -16",
"Step 5: C[1][1] = -4×-1 + 0×2 = 4",
"Answer: [[-11,5],[-16,4]]"
] | [[-11,5],[-16,4]] | matrix_multiplication | transposed_order | true | Step 1: incorrect — The multiplication order is reversed in Step 1. Matrix multiplication is not commutative: A·B ≠ B·A. The correct answer for A·B is [[-11,5],[-16,4]]. | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: [[-11,5],[-16,4]] | easy |
Solve the system using Gaussian elimination: 1x + 2y = -9 and -3x - 2y = 15. | [
"Step 1: Write augmented matrix: [1 2 | -9] and [-3 -2 | 15]",
"Step 2: Multiply row 1 by -3 and row 2 by 1: [-3 -6 | 27] and [-3 -2 | 15]",
"Step 3: Subtract row 2 from row 1 to eliminate x: [0 -4 | 12]",
"Step 4: Solve for y: -4y = 12 → y = -3",
"Step 5: Back-substitute y = -3 into equation 1: 1x + 2×-3... | None | There is no error. The working shown is correct. | [
"Step 1: Write augmented matrix: [1 2 | -9] and [-3 -2 | 15]",
"Step 2: Multiply row 1 by -3 and row 2 by 1: [-3 -6 | 27] and [-3 -2 | 15]",
"Step 3: Subtract row 2 from row 1 to eliminate x: [0 -4 | 12]",
"Step 4: Solve for y: -4y = 12 → y = -3",
"Step 5: Back-substitute y = -3 into equation 1: 1x + 2×-3... | x=-3,y=-3 | gaussian_elimination | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: x=-3,y=-3 | medium |
Find lim (x → -1) of -3x | [
"Step 1: -3x is continuous everywhere — substitute directly",
"Step 2: Substitute x = -1: -3(-1) + (0)",
"Step 3: Arithmetic error: 3 + 0 = 1",
"Answer: 1"
] | Step 3 | Step 3 has an arithmetic error. 3 + 0 = 3, not 1. The correct answer is 3. | [
"Step 1: -3x is continuous everywhere — substitute directly",
"Step 2: Substitute x = -1: -3(-1) + (0)",
"Step 3: Calculate: 3 + 0 = 3",
"Answer: 3"
] | 3 | limits | arithmetic | true | Step 1: correct | Step 2: correct | Step 3: incorrect — Step 3 has an arithmetic error. 3 + 0 = 3, not 1. The correct answer is 3. | Therefore the correct answer is: 3 | medium |
Differentiate f(x) = -x^2 with respect to x. | [
"Step 1: Apply the power rule: d/dx[ax^n] = an·x^(n-1)",
"Step 2: Sign error: -1 × 2 = 2",
"Step 3: Reduce the exponent by 1: 2 - 1 = 1",
"Answer: f'(x) = 2x"
] | Step 2 | Step 2 flips the sign, giving 2 instead of -2. The derivative is -2x. | [
"Step 1: Apply the power rule: d/dx[ax^n] = an·x^(n-1)",
"Step 2: Multiply coefficient by exponent: -1 × 2 = -2",
"Step 3: Reduce the exponent by 1: 2 - 1 = 1",
"Answer: f'(x) = -2x"
] | -2x | differentiation | sign_error | true | Step 1: correct | Step 2: incorrect — Step 2 flips the sign, giving 2 instead of -2. The derivative is -2x. | Step 3: correct | Therefore the correct answer is: -2x | medium |
Find the determinant of the matrix [[-2,-3],[-4,-4]]. | [
"Step 1: For a 2×2 matrix [[a,b],[c,d]], det(A) = a·d - b·c",
"Step 2: Identify: a=-2, b=-3, c=-4, d=-4",
"Step 3: Compute a·d = -2×-4 = 8",
"Step 4: Compute b·c = -3×-4 = 12",
"Step 5: det(A) = 8 - 12 = -4",
"Answer: -4"
] | None | There is no error. The working shown is correct. | [
"Step 1: For a 2×2 matrix [[a,b],[c,d]], det(A) = a·d - b·c",
"Step 2: Identify: a=-2, b=-3, c=-4, d=-4",
"Step 3: Compute a·d = -2×-4 = 8",
"Step 4: Compute b·c = -3×-4 = 12",
"Step 5: det(A) = 8 - 12 = -4",
"Answer: -4"
] | -4 | determinant | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: -4 | easy |
Find ∫ -2x^3 dx | [
"Step 1: Power rule for integration: ∫x^n dx = x^(n+1)/(n+1) + C",
"Step 2: Raise exponent by 1: 3 + 1 = 4",
"Step 3: Division error: -2 / 4 = (0/2)",
"Step 4: Append constant of integration",
"Answer: (0/2)x^4+C"
] | Step 3 | Step 3 has a division error. -2 / 4 = (-1/2), not (0/2). The correct answer is (-1/2)x^4+C. | [
"Step 1: Power rule for integration: ∫x^n dx = x^(n+1)/(n+1) + C",
"Step 2: Raise exponent by 1: 3 + 1 = 4",
"Step 3: Divide coefficient by new exponent: -2 / 4 = (-1/2)",
"Step 4: Append constant of integration",
"Answer: (-1/2)x^4+C"
] | (-1/2)x^4+C | integration | wrong_coefficient | true | Step 1: correct | Step 2: correct | Step 3: incorrect — Step 3 has a division error. -2 / 4 = (-1/2), not (0/2). The correct answer is (-1/2)x^4+C. | Step 4: correct | Therefore the correct answer is: (-1/2)x^4+C | medium |
Differentiate f(x) = (3x + 1)^2 using the chain rule. | [
"Step 1: Chain rule: d/dx[g(h(x))] = g'(h(x)) · h'(x)",
"Step 2: Outer function g(u) = u^2 → g'(u) = 2u^1",
"Step 3: Inner function h(x) = 3x + 1 → h'(x) = 3",
"Step 4: Combine: 2·(3x + 1)^1 · 3 = 6(3x + 1)^1",
"Answer: f'(x) = 6(3x + 1)^1"
] | None | There is no error. The working shown is correct. | [
"Step 1: Chain rule: d/dx[g(h(x))] = g'(h(x)) · h'(x)",
"Step 2: Outer function g(u) = u^2 → g'(u) = 2u^1",
"Step 3: Inner function h(x) = 3x + 1 → h'(x) = 3",
"Step 4: Combine: 2·(3x + 1)^1 · 3 = 6(3x + 1)^1",
"Answer: f'(x) = 6(3x + 1)^1"
] | 6(3x + 1)^1 | differentiation | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Therefore the correct answer is: 6(3x + 1)^1 | medium |
Differentiate f(x) = 2x^3 with respect to x. | [
"Step 1: Apply the power rule: d/dx[ax^n] = an·x^(n-1)",
"Step 2: Multiply coefficient by exponent: 2 × 3 = 6",
"Step 3: Reduce the exponent by 1: 3 - 1 = 2",
"Answer: f'(x) = 6x^2"
] | None | There is no error. The working shown is correct. | [
"Step 1: Apply the power rule: d/dx[ax^n] = an·x^(n-1)",
"Step 2: Multiply coefficient by exponent: 2 × 3 = 6",
"Step 3: Reduce the exponent by 1: 3 - 1 = 2",
"Answer: f'(x) = 6x^2"
] | 6x^2 | differentiation | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Therefore the correct answer is: 6x^2 | medium |
Find ∫ 3(3x - 2)^2 dx using substitution. | [
"Step 1: Let u = 3x - 2, so du = 3·dx",
"Step 2: Rewrite: (3/3) ∫ u^2 du",
"Step 3: Exponent error: ∫u^2 du = u^2/2 — should be u^3/3",
"Step 4: Back-substitute u = 3x - 2 and add + C",
"Answer: (1/2)(3x - 2)^2+C"
] | Step 3 | Step 3 uses the wrong exponent. ∫u^2 du = u^3/3, not u^2/2. The correct answer is (1/3)(3x - 2)^3+C. | [
"Step 1: Let u = 3x - 2, so du = 3·dx, meaning dx = du/3",
"Step 2: Rewrite: ∫ 3·u^2·(du/3) = (3/3) ∫ u^2 du",
"Step 3: Integrate: (3/3) · u^3/3 = (1/3)u^3",
"Step 4: Back-substitute u = 3x - 2 and add + C",
"Answer: (1/3)(3x - 2)^3+C"
] | (1/3)(3x - 2)^3+C | integration | wrong_exponent | true | Step 1: correct | Step 2: correct | Step 3: incorrect — Step 3 uses the wrong exponent. ∫u^2 du = u^3/3, not u^2/2. The correct answer is (1/3)(3x - 2)^3+C. | Step 4: correct | Therefore the correct answer is: (1/3)(3x - 2)^3+C | medium |
Differentiate f(x) = (2x - 2)^2 using the chain rule. | [
"Step 1: Chain rule: d/dx[g(h(x))] = g'(h(x)) · h'(x)",
"Step 2: Outer function g(u) = u^2 → g'(u) = 2u^1",
"Step 3: Sign error on inner derivative: h'(x) = -2 (should be +2)",
"Step 4: Combine: 2·(2x - 2)^1 · (-2) = -4(2x - 2)^1",
"Answer: f'(x) = -4(2x - 2)^1"
] | Step 3 | Step 3 negates the inner derivative incorrectly. d/dx[2x - 2] = 2. Correct answer: 4(2x - 2)^1. | [
"Step 1: Chain rule: d/dx[g(h(x))] = g'(h(x)) · h'(x)",
"Step 2: Outer function g(u) = u^2 → g'(u) = 2u^1",
"Step 3: Inner function h(x) = 2x - 2 → h'(x) = 2",
"Step 4: Combine: 2·(2x - 2)^1 · 2 = 4(2x - 2)^1",
"Answer: f'(x) = 4(2x - 2)^1"
] | 4(2x - 2)^1 | differentiation | sign_error | true | Step 1: correct | Step 2: correct | Step 3: incorrect — Step 3 negates the inner derivative incorrectly. d/dx[2x - 2] = 2. Correct answer: 4(2x - 2)^1. | Step 4: correct | Therefore the correct answer is: 4(2x - 2)^1 | medium |
Find the determinant of the matrix [[2,-4],[2,-2]]. | [
"Step 1: Error — used det(A) = a·d + b·c (should subtract)",
"Step 2: Identify: a=2, b=-4, c=2, d=-2",
"Step 3: Compute a·d = 2×-2 = -4",
"Step 4: Compute b·c = -4×2 = -8",
"Step 5: det(A) = -4 + -8 = -12",
"Answer: -12"
] | Step 1 | Step 1 uses the wrong formula. The 2×2 determinant is a·d - b·c, not a·d + b·c. The correct answer is 4. | [
"Step 1: For a 2×2 matrix [[a,b],[c,d]], det(A) = a·d - b·c",
"Step 2: Identify: a=2, b=-4, c=2, d=-2",
"Step 3: Compute a·d = 2×-2 = -4",
"Step 4: Compute b·c = -4×2 = -8",
"Step 5: det(A) = -4 - -8 = 4",
"Answer: 4"
] | 4 | determinant | wrong_formula | true | Step 1: incorrect — Step 1 uses the wrong formula. The 2×2 determinant is a·d - b·c, not a·d + b·c. The correct answer is 4. | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: 4 | easy |
Find lim (x → 2) of (x² - 4) / (x - 2) | [
"Step 1: Substitute x = 2: (4 - 4) / (2 - 2) = 0/0",
"Step 2: Conclude the limit does not exist — 0/0 is undefined",
"Step 3: No further work",
"Answer: undefined"
] | Step 2 | 0/0 is an indeterminate form, not proof the limit is undefined. Factor as (x-2)(x+2), cancel (x-2), then substitute to get 4. | [
"Step 1: Direct substitution gives 0/0 — indeterminate form, must factor",
"Step 2: Factor numerator: x² - 4 = (x - 2)(x + 2)",
"Step 3: Cancel (x - 2): expression becomes (x + 2)",
"Step 4: Substitute x = 2: 2 + 2 = 4",
"Answer: 4"
] | 4 | limits | forgot_cancel | true | Step 1: correct | Step 2: incorrect — 0/0 is an indeterminate form, not proof the limit is undefined. Factor as (x-2)(x+2), cancel (x-2), then substitute to get 4. | Step 3: correct | Therefore the correct answer is: 4 | medium |
Find the determinant of the matrix [[0,4],[-2,-1]]. | [
"Step 1: For a 2×2 matrix [[a,b],[c,d]], det(A) = a·d - b·c",
"Step 2: Identify: a=0, b=4, c=-2, d=-1",
"Step 3: Arithmetic error — a·d = 1 (correct: 0)",
"Step 4: Compute b·c = 4×-2 = -8",
"Step 5: det(A) = 1 - -8 = 9",
"Answer: 9"
] | Step 3 | Step 3 has an arithmetic error. 0×-1 = 0, not 1. The correct answer is 8. | [
"Step 1: For a 2×2 matrix [[a,b],[c,d]], det(A) = a·d - b·c",
"Step 2: Identify: a=0, b=4, c=-2, d=-1",
"Step 3: Compute a·d = 0×-1 = 0",
"Step 4: Compute b·c = 4×-2 = -8",
"Step 5: det(A) = 0 - -8 = 8",
"Answer: 8"
] | 8 | determinant | arithmetic | true | Step 1: correct | Step 2: correct | Step 3: incorrect — Step 3 has an arithmetic error. 0×-1 = 0, not 1. The correct answer is 8. | Step 4: correct | Step 5: correct | Therefore the correct answer is: 8 | easy |
Compute the matrix product A·B where A = [[4,-4],[-4,0]] and B = [[1,1],[1,-4]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = 4×1 + -4×1 = 1",
"Step 3: C[0][1] = 4×1 + -4×-4 = 20",
"Step 4: C[1][0] = -4×1 + 0×1 = -4",
"Step 5: C[1][1] = -4×1 + 0×-4 = -4",
"Answer: [[1,20],[-4,-4]]"
] | Step 2 | Entry C[0][0] is wrong. The correct value is 0, not 1. The correct result is [[0,20],[-4,-4]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = 4×1 + -4×1 = 0",
"Step 3: C[0][1] = 4×1 + -4×-4 = 20",
"Step 4: C[1][0] = -4×1 + 0×1 = -4",
"Step 5: C[1][1] = -4×1 + 0×-4 = -4",
"Answer: [[0,20],[-4,-4]]"
] | [[0,20],[-4,-4]] | matrix_multiplication | wrong_entry | true | Step 1: correct | Step 2: incorrect — Entry C[0][0] is wrong. The correct value is 0, not 1. The correct result is [[0,20],[-4,-4]]. | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: [[0,20],[-4,-4]] | easy |
Find the eigenvalues of the matrix [[-3,0],[0,-5]]. | [
"Step 1: Form characteristic equation: det(A - λI) = 0",
"Step 2: trace(A) = -8",
"Step 3: det(A) = 15",
"Step 4: Characteristic equation: λ² - -8λ + 15 = 0",
"Step 5: Factoring error — wrong roots: -4 and -2",
"Answer: λ1=-4,λ2=-2"
] | Step 5 | Step 5 factors the characteristic equation incorrectly. The correct factors are (λ - -5)(λ - -3) = 0. The correct answer is λ1=-5,λ2=-3. | [
"Step 1: Form characteristic equation: det(A - λI) = 0",
"Step 2: trace(A) = -3 + -5 = -8",
"Step 3: det(A) = -3×-5 - 0×0 = 15",
"Step 4: Characteristic equation: λ² - -8λ + 15 = 0",
"Step 5: Factorise: (λ - -5)(λ - -3) = 0",
"Answer: λ1=-5,λ2=-3"
] | λ1=-5,λ2=-3 | eigenvalues | arithmetic_in_quadratic | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: incorrect — Step 5 factors the characteristic equation incorrectly. The correct factors are (λ - -5)(λ - -3) = 0. The correct answer is λ1=-5,λ2=-3. | Therefore the correct answer is: λ1=-5,λ2=-3 | hard |
Compute the matrix product A·B where A = [[4,-3],[-3,1]] and B = [[-2,-4],[1,-4]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = 4×-2 + -3×1 = -11",
"Step 3: C[0][1] = 4×-4 + -3×-4 = -4",
"Step 4: C[1][0] = -3×-2 + 1×1 = 7",
"Step 5: C[1][1] = -3×-4 + 1×-4 = 8",
"Answer: [[-11,-4],[7,8]]"
] | None | There is no error. The working shown is correct. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = 4×-2 + -3×1 = -11",
"Step 3: C[0][1] = 4×-4 + -3×-4 = -4",
"Step 4: C[1][0] = -3×-2 + 1×1 = 7",
"Step 5: C[1][1] = -3×-4 + 1×-4 = 8",
"Answer: [[-11,-4],[7,8]]"
] | [[-11,-4],[7,8]] | matrix_multiplication | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: [[-11,-4],[7,8]] | easy |
Find the eigenvalues of the matrix [[0,1],[0,3]]. | [
"Step 1: Form characteristic equation: det(A - λI) = 0",
"Step 2: Trace error — trace(A) computed as 4 instead of 3",
"Step 3: det(A) = 0",
"Step 4: Wrong equation: λ² - 4λ + 0 = 0",
"Step 5: Solve: eigenvalues are 0 and 4",
"Answer: λ1=0,λ2=4"
] | Step 2 | Step 2 computes the trace incorrectly. trace(A) = 0 + 3 = 3, not 4. The correct eigenvalues are λ1=0,λ2=3. | [
"Step 1: Form characteristic equation: det(A - λI) = 0",
"Step 2: trace(A) = 0 + 3 = 3",
"Step 3: det(A) = 0×3 - 1×0 = 0",
"Step 4: Characteristic equation: λ² - 3λ + 0 = 0",
"Step 5: Factorise: (λ - 0)(λ - 3) = 0",
"Answer: λ1=0,λ2=3"
] | λ1=0,λ2=3 | eigenvalues | wrong_trace | true | Step 1: correct | Step 2: incorrect — Step 2 computes the trace incorrectly. trace(A) = 0 + 3 = 3, not 4. The correct eigenvalues are λ1=0,λ2=3. | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: λ1=0,λ2=3 | hard |
Solve the system using Gaussian elimination: -2x + 4y = -10 and -5x - 2y = -25. | [
"Step 1: Write augmented matrix: [-2 4 | -10] and [-5 -2 | -25]",
"Step 2: Multiply row 1 by -5 and row 2 by -2: [10 -20 | 50] and [10 4 | 50]",
"Step 3: Arithmetic error in RHS: [0 -24 | 24] (correct RHS: 0)",
"Step 4: Solve for y: -24y = 24 → y = -1",
"Step 5: Back-substitute y = -1: -2x + 4×-1 = -10 ... | Step 3 | Step 3 has an arithmetic error in the RHS after elimination. The correct RHS is 0, not 24. The correct solution is x=5,y=0. | [
"Step 1: Write augmented matrix: [-2 4 | -10] and [-5 -2 | -25]",
"Step 2: Multiply row 1 by -5 and row 2 by -2: [10 -20 | 50] and [10 4 | 50]",
"Step 3: Subtract row 2 from row 1 to eliminate x: [0 -24 | 0]",
"Step 4: Solve for y: -24y = 0 → y = 0",
"Step 5: Back-substitute y = 0 into equation 1: -2x + 4... | x=5,y=0 | gaussian_elimination | arithmetic | true | Step 1: correct | Step 2: correct | Step 3: incorrect — Step 3 has an arithmetic error in the RHS after elimination. The correct RHS is 0, not 24. The correct solution is x=5,y=0. | Step 4: correct | Step 5: correct | Therefore the correct answer is: x=5,y=0 | medium |
Compute the matrix product A·B where A = [[1,-2],[3,-2]] and B = [[-4,-4],[-3,2]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = 1×-4 + -2×-3 = 2",
"Step 3: C[0][1] = 1×-4 + -2×2 = -8",
"Step 4: C[1][0] = 3×-4 + -2×-3 = -6",
"Step 5: C[1][1] = 3×-4 + -2×2 = -16",
"Answer: [[2,-8],[-6,-16]]"
] | None | There is no error. The working shown is correct. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = 1×-4 + -2×-3 = 2",
"Step 3: C[0][1] = 1×-4 + -2×2 = -8",
"Step 4: C[1][0] = 3×-4 + -2×-3 = -6",
"Step 5: C[1][1] = 3×-4 + -2×2 = -16",
"Answer: [[2,-8],[-6,-16]]"
] | [[2,-8],[-6,-16]] | matrix_multiplication | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: [[2,-8],[-6,-16]] | easy |
Compute the matrix product A·B where A = [[3,1],[0,4]] and B = [[0,0],[1,-3]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: Arithmetic error — C[0][0] = 3 (correct: 1)",
"Step 3: C[0][1] = -3",
"Step 4: C[1][0] = 4",
"Step 5: C[1][1] = -12",
"Answer: [[3,-3],[4,-12]]"
] | Step 2 | Step 2 has an arithmetic error in C[0][0]. The correct value is 1, not 3. The correct result is [[1,-3],[4,-12]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = 3×0 + 1×1 = 1",
"Step 3: C[0][1] = 3×0 + 1×-3 = -3",
"Step 4: C[1][0] = 0×0 + 4×1 = 4",
"Step 5: C[1][1] = 0×0 + 4×-3 = -12",
"Answer: [[1,-3],[4,-12]]"
] | [[1,-3],[4,-12]] | matrix_multiplication | arithmetic | true | Step 1: correct | Step 2: incorrect — Step 2 has an arithmetic error in C[0][0]. The correct value is 1, not 3. The correct result is [[1,-3],[4,-12]]. | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: [[1,-3],[4,-12]] | easy |
Solve the system using Gaussian elimination: 4x - 2y = -12 and 5x + 2y = -6. | [
"Step 1: Write augmented matrix: [4 -2 | -12] and [5 2 | -6]",
"Step 2: Multiply row 1 by 5 and row 2 by 4: [20 -10 | -60] and [20 8 | -24]",
"Step 3: Error — added rows instead of subtracting: [0 -2 | -84]",
"Step 4: Solve for y: -2y = -84 → y = 42",
"Step 5: Back-substitute y = 42: 4x + -2×42 = -12 → ... | Step 3 | Step 3 adds the scaled rows instead of subtracting. Subtracting to eliminate x gives [0 -18 | -36], not [0 -2 | -84]. The correct solution is x=-2,y=2. | [
"Step 1: Write augmented matrix: [4 -2 | -12] and [5 2 | -6]",
"Step 2: Multiply row 1 by 5 and row 2 by 4: [20 -10 | -60] and [20 8 | -24]",
"Step 3: Subtract row 2 from row 1 to eliminate x: [0 -18 | -36]",
"Step 4: Solve for y: -18y = -36 → y = 2",
"Step 5: Back-substitute y = 2 into equation 1: 4x + -... | x=-2,y=2 | gaussian_elimination | wrong_row_op | true | Step 1: correct | Step 2: correct | Step 3: incorrect — Step 3 adds the scaled rows instead of subtracting. Subtracting to eliminate x gives [0 -18 | -36], not [0 -2 | -84]. The correct solution is x=-2,y=2. | Step 4: correct | Step 5: correct | Therefore the correct answer is: x=-2,y=2 | medium |
Solve the system using Gaussian elimination: -5x - 3y = 19 and -5x + 5y = -5. | [
"Step 1: Write augmented matrix: [-5 -3 | 19] and [-5 5 | -5]",
"Step 2: Multiply row 1 by -5 and row 2 by -5: [25 15 | -95] and [25 -25 | 25]",
"Step 3: Subtract row 2 from row 1: [0 40 | -120]",
"Step 4: Solve for y: 40y = -120 → y = -3",
"Step 5: Back-substitution arithmetic error: -5x + -3×-3 = 19 → ... | Step 5 | The elimination in Steps 3–4 correctly gives y = -3, but Step 5 makes an arithmetic error in back-substitution, giving x = -5 instead of x = -2. The correct solution is x=-2,y=-3. | [
"Step 1: Write augmented matrix: [-5 -3 | 19] and [-5 5 | -5]",
"Step 2: Multiply row 1 by -5 and row 2 by -5: [25 15 | -95] and [25 -25 | 25]",
"Step 3: Subtract row 2 from row 1 to eliminate x: [0 40 | -120]",
"Step 4: Solve for y: 40y = -120 → y = -3",
"Step 5: Back-substitute y = -3 into equation 1: -... | x=-2,y=-3 | gaussian_elimination | wrong_backsub | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: incorrect — The elimination in Steps 3–4 correctly gives y = -3, but Step 5 makes an arithmetic error in back-substitution, giving x = -5 instead of x = -2. The correct solution is x=-2,y=-3. | Therefore the correct answer is: x=-2,y=-3 | medium |
Differentiate f(x) = (5x - 3)^2 using the chain rule. | [
"Step 1: Chain rule: d/dx[g(h(x))] = g'(h(x)) · h'(x)",
"Step 2: Outer function g(u) = u^2 → g'(u) = 2u^1",
"Step 3: Inner function h(x) = 5x - 3 → h'(x) = 5",
"Step 4: Combine: 2·(5x - 3)^1 · 5 = 10(5x - 3)^1",
"Answer: f'(x) = 10(5x - 3)^1"
] | None | There is no error. The working shown is correct. | [
"Step 1: Chain rule: d/dx[g(h(x))] = g'(h(x)) · h'(x)",
"Step 2: Outer function g(u) = u^2 → g'(u) = 2u^1",
"Step 3: Inner function h(x) = 5x - 3 → h'(x) = 5",
"Step 4: Combine: 2·(5x - 3)^1 · 5 = 10(5x - 3)^1",
"Answer: f'(x) = 10(5x - 3)^1"
] | 10(5x - 3)^1 | differentiation | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Therefore the correct answer is: 10(5x - 3)^1 | medium |
Find lim (x → 0) of sin(7x) / x using L'Hôpital's rule. | [
"Step 1: Direct substitution gives 0/0 — indeterminate form",
"Step 2: Apply L'Hôpital's rule",
"Step 3: d/dx[sin(7x)] = 7cos(7x), d/dx[x] = 1",
"Step 4: Error — used cos(0) = 0 instead of 1: 7·0 = 0",
"Answer: 0"
] | Step 4 | Step 4 incorrectly evaluates cos(0) as 0. cos(0) = 1, so the limit is 7·1 = 7. | [
"Step 1: Direct substitution gives sin(0)/0 = 0/0 — indeterminate form",
"Step 2: Apply L'Hôpital's rule: differentiate numerator and denominator separately",
"Step 3: d/dx[sin(7x)] = 7cos(7x), d/dx[x] = 1",
"Step 4: Evaluate new limit at x = 0: 7·cos(0) = 7·1 = 7",
"Answer: 7"
] | 7 | limits | cos_zero_error | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: incorrect — Step 4 incorrectly evaluates cos(0) as 0. cos(0) = 1, so the limit is 7·1 = 7. | Therefore the correct answer is: 7 | medium |
Find lim (x → -2) of -2x - 1 | [
"Step 1: -2x - 1 is continuous everywhere — substitute directly",
"Step 2: Substitution error — used x = -1 instead of x = -2",
"Step 3: Calculate: 2 + -1 = 1",
"Answer: 1"
] | Step 2 | Step 2 uses the wrong value. The limit is as x → -2, so substitute x = -2. The correct answer is 3. | [
"Step 1: -2x - 1 is continuous everywhere — substitute directly",
"Step 2: Substitute x = -2: -2(-2) + (-1)",
"Step 3: Calculate: 4 + -1 = 3",
"Answer: 3"
] | 3 | limits | wrong_substitution | true | Step 1: correct | Step 2: incorrect — Step 2 uses the wrong value. The limit is as x → -2, so substitute x = -2. The correct answer is 3. | Step 3: correct | Therefore the correct answer is: 3 | medium |
Compute the matrix product A·B where A = [[3,2],[0,0]] and B = [[2,0],[2,2]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: Arithmetic error — C[0][0] = 11 (correct: 10)",
"Step 3: C[0][1] = 4",
"Step 4: C[1][0] = 0",
"Step 5: C[1][1] = 0",
"Answer: [[11,4],[0,0]]"
] | Step 2 | Step 2 has an arithmetic error in C[0][0]. The correct value is 10, not 11. The correct result is [[10,4],[0,0]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = 3×2 + 2×2 = 10",
"Step 3: C[0][1] = 3×0 + 2×2 = 4",
"Step 4: C[1][0] = 0×2 + 0×2 = 0",
"Step 5: C[1][1] = 0×0 + 0×2 = 0",
"Answer: [[10,4],[0,0]]"
] | [[10,4],[0,0]] | matrix_multiplication | arithmetic | true | Step 1: correct | Step 2: incorrect — Step 2 has an arithmetic error in C[0][0]. The correct value is 10, not 11. The correct result is [[10,4],[0,0]]. | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: [[10,4],[0,0]] | easy |
Solve the system using Gaussian elimination: 5x + 4y = 45 and -3x + 2y = -5. | [
"Step 1: Write augmented matrix: [5 4 | 45] and [-3 2 | -5]",
"Step 2: Multiply row 1 by -3 and row 2 by 5: [-15 -12 | -135] and [-15 10 | -25]",
"Step 3: Error — added rows instead of subtracting: [0 -2 | -160]",
"Step 4: Solve for y: -2y = -160 → y = 80",
"Step 5: Back-substitute y = 80: 5x + 4×80 = 45 ... | Step 3 | Step 3 adds the scaled rows instead of subtracting. Subtracting to eliminate x gives [0 -22 | -110], not [0 -2 | -160]. The correct solution is x=5,y=5. | [
"Step 1: Write augmented matrix: [5 4 | 45] and [-3 2 | -5]",
"Step 2: Multiply row 1 by -3 and row 2 by 5: [-15 -12 | -135] and [-15 10 | -25]",
"Step 3: Subtract row 2 from row 1 to eliminate x: [0 -22 | -110]",
"Step 4: Solve for y: -22y = -110 → y = 5",
"Step 5: Back-substitute y = 5 into equation 1: ... | x=5,y=5 | gaussian_elimination | wrong_row_op | true | Step 1: correct | Step 2: correct | Step 3: incorrect — Step 3 adds the scaled rows instead of subtracting. Subtracting to eliminate x gives [0 -22 | -110], not [0 -2 | -160]. The correct solution is x=5,y=5. | Step 4: correct | Step 5: correct | Therefore the correct answer is: x=5,y=5 | medium |
Find lim (x → 0) of 4x - 3 | [
"Step 1: 4x - 3 is continuous everywhere — substitute directly",
"Step 2: Substitute x = 0: 4(0) + (-3)",
"Step 3: Calculate: 0 + -3 = -3",
"Answer: -3"
] | None | There is no error. The working shown is correct. | [
"Step 1: 4x - 3 is continuous everywhere — substitute directly",
"Step 2: Substitute x = 0: 4(0) + (-3)",
"Step 3: Calculate: 0 + -3 = -3",
"Answer: -3"
] | -3 | limits | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Therefore the correct answer is: -3 | medium |
Find ∫ 4x dx | [
"Step 1: Power rule for integration: ∫x^n dx = x^(n+1)/(n+1) + C",
"Step 2: Exponent error — kept as 1 instead of raising to 2",
"Step 3: Divide by original exponent: 4 / 1 = 4",
"Step 4: Append constant of integration",
"Answer: 4x^1+C"
] | Step 2 | Step 2 fails to raise the exponent. Integration requires adding 1: 1 + 1 = 2. The correct answer is 2x^2+C. | [
"Step 1: Power rule for integration: ∫x^n dx = x^(n+1)/(n+1) + C",
"Step 2: Raise exponent by 1: 1 + 1 = 2",
"Step 3: Divide coefficient by new exponent: 4 / 2 = 2",
"Step 4: Append constant of integration",
"Answer: 2x^2+C"
] | 2x^2+C | integration | wrong_exponent | true | Step 1: correct | Step 2: incorrect — Step 2 fails to raise the exponent. Integration requires adding 1: 1 + 1 = 2. The correct answer is 2x^2+C. | Step 3: correct | Step 4: correct | Therefore the correct answer is: 2x^2+C | medium |
Differentiate f(x) = (5x + 4)^5 using the chain rule. | [
"Step 1: Chain rule: d/dx[g(h(x))] = g'(h(x)) · h'(x)",
"Step 2: Outer function g(u) = u^5 → g'(u) = 5u^4",
"Step 3: Inner function h(x) = 5x + 4 → h'(x) = 5",
"Step 4: Inner derivative h'(x) omitted from product: 5(5x + 4)^4",
"Answer: f'(x) = 5(5x + 4)^4"
] | Step 4 | Step 4 forgets to multiply by h'(x) = 5, so the coefficient should be 25 not 5. Answer: 25(5x + 4)^4. | [
"Step 1: Chain rule: d/dx[g(h(x))] = g'(h(x)) · h'(x)",
"Step 2: Outer function g(u) = u^5 → g'(u) = 5u^4",
"Step 3: Inner function h(x) = 5x + 4 → h'(x) = 5",
"Step 4: Combine: 5·(5x + 4)^4 · 5 = 25(5x + 4)^4",
"Answer: f'(x) = 25(5x + 4)^4"
] | 25(5x + 4)^4 | differentiation | forgot_step | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: incorrect — Step 4 forgets to multiply by h'(x) = 5, so the coefficient should be 25 not 5. Answer: 25(5x + 4)^4. | Therefore the correct answer is: 25(5x + 4)^4 | medium |
Solve the system using Gaussian elimination: -1x - 3y = 9 and 2x + 4y = -10. | [
"Step 1: Write augmented matrix: [-1 -3 | 9] and [2 4 | -10]",
"Step 2: Multiply row 1 by 2 and row 2 by -1: [-2 -6 | 18] and [-2 -4 | 10]",
"Step 3: Arithmetic error in RHS: [0 -2 | 10] (correct RHS: 8)",
"Step 4: Solve for y: -2y = 10 → y = -5",
"Step 5: Back-substitute y = -5: -1x + -3×-5 = 9 → x = ... | Step 3 | Step 3 has an arithmetic error in the RHS after elimination. The correct RHS is 8, not 10. The correct solution is x=3,y=-4. | [
"Step 1: Write augmented matrix: [-1 -3 | 9] and [2 4 | -10]",
"Step 2: Multiply row 1 by 2 and row 2 by -1: [-2 -6 | 18] and [-2 -4 | 10]",
"Step 3: Subtract row 2 from row 1 to eliminate x: [0 -2 | 8]",
"Step 4: Solve for y: -2y = 8 → y = -4",
"Step 5: Back-substitute y = -4 into equation 1: -1x + -3×-4... | x=3,y=-4 | gaussian_elimination | arithmetic | true | Step 1: correct | Step 2: correct | Step 3: incorrect — Step 3 has an arithmetic error in the RHS after elimination. The correct RHS is 8, not 10. The correct solution is x=3,y=-4. | Step 4: correct | Step 5: correct | Therefore the correct answer is: x=3,y=-4 | medium |
Differentiate f(x) = -2x^3 with respect to x. | [
"Step 1: Apply the power rule: d/dx[ax^n] = an·x^(n-1)",
"Step 2: Multiply coefficient by exponent: -2 × 3 = -6",
"Step 3: Reduce the exponent by 1: 3 - 1 = 2",
"Answer: f'(x) = -6x^2"
] | None | There is no error. The working shown is correct. | [
"Step 1: Apply the power rule: d/dx[ax^n] = an·x^(n-1)",
"Step 2: Multiply coefficient by exponent: -2 × 3 = -6",
"Step 3: Reduce the exponent by 1: 3 - 1 = 2",
"Answer: f'(x) = -6x^2"
] | -6x^2 | differentiation | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Therefore the correct answer is: -6x^2 | medium |
Solve the system using Gaussian elimination: 2x - 5y = 21 and -3x + 2y = -4. | [
"Step 1: Write augmented matrix: [2 -5 | 21] and [-3 2 | -4]",
"Step 2: Multiply row 1 by -3 and row 2 by 2: [-6 15 | -63] and [-6 4 | -8]",
"Step 3: Subtract row 2 from row 1: [0 11 | -55]",
"Step 4: Solve for y: 11y = -55 → y = -5",
"Step 5: Back-substitution arithmetic error: 2x + -5×-5 = 21 → x = 1 ... | Step 5 | The elimination in Steps 3–4 correctly gives y = -5, but Step 5 makes an arithmetic error in back-substitution, giving x = 1 instead of x = -2. The correct solution is x=-2,y=-5. | [
"Step 1: Write augmented matrix: [2 -5 | 21] and [-3 2 | -4]",
"Step 2: Multiply row 1 by -3 and row 2 by 2: [-6 15 | -63] and [-6 4 | -8]",
"Step 3: Subtract row 2 from row 1 to eliminate x: [0 11 | -55]",
"Step 4: Solve for y: 11y = -55 → y = -5",
"Step 5: Back-substitute y = -5 into equation 1: 2x + -5... | x=-2,y=-5 | gaussian_elimination | wrong_backsub | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: incorrect — The elimination in Steps 3–4 correctly gives y = -5, but Step 5 makes an arithmetic error in back-substitution, giving x = 1 instead of x = -2. The correct solution is x=-2,y=-5. | Therefore the correct answer is: x=-2,y=-5 | medium |
Find lim (x → 0) of sin(6x) / x using L'Hôpital's rule. | [
"Step 1: Direct substitution gives 0/0 — indeterminate form",
"Step 2: Apply L'Hôpital's rule",
"Step 3: d/dx[sin(6x)] = 6cos(6x), d/dx[x] = 1",
"Step 4: Error — used cos(0) = 0 instead of 1: 6·0 = 0",
"Answer: 0"
] | Step 4 | Step 4 incorrectly evaluates cos(0) as 0. cos(0) = 1, so the limit is 6·1 = 6. | [
"Step 1: Direct substitution gives sin(0)/0 = 0/0 — indeterminate form",
"Step 2: Apply L'Hôpital's rule: differentiate numerator and denominator separately",
"Step 3: d/dx[sin(6x)] = 6cos(6x), d/dx[x] = 1",
"Step 4: Evaluate new limit at x = 0: 6·cos(0) = 6·1 = 6",
"Answer: 6"
] | 6 | limits | cos_zero_error | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: incorrect — Step 4 incorrectly evaluates cos(0) as 0. cos(0) = 1, so the limit is 6·1 = 6. | Therefore the correct answer is: 6 | medium |
Find lim (x → 5) of (x² - 25) / (x - 5) | [
"Step 1: Direct substitution gives 0/0 — indeterminate form, must factor",
"Step 2: Factor numerator: x² - 25 = (x - 5)(x + 5)",
"Step 3: Cancel (x - 5): expression becomes (x + 5)",
"Step 4: Substitute x = 5: 5 + 5 = 10",
"Answer: 10"
] | None | There is no error. The working shown is correct. | [
"Step 1: Direct substitution gives 0/0 — indeterminate form, must factor",
"Step 2: Factor numerator: x² - 25 = (x - 5)(x + 5)",
"Step 3: Cancel (x - 5): expression becomes (x + 5)",
"Step 4: Substitute x = 5: 5 + 5 = 10",
"Answer: 10"
] | 10 | limits | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Therefore the correct answer is: 10 | medium |
Differentiate f(x) = -5x^6 with respect to x. | [
"Step 1: Apply the power rule: d/dx[ax^n] = an·x^(n-1)",
"Step 2: Sign error: -5 × 6 = 30",
"Step 3: Reduce the exponent by 1: 6 - 1 = 5",
"Answer: f'(x) = 30x^5"
] | Step 2 | Step 2 has a sign error. -5 × 6 = -30, not 30. The correct derivative is -30x^5. | [
"Step 1: Apply the power rule: d/dx[ax^n] = an·x^(n-1)",
"Step 2: Multiply coefficient by exponent: -5 × 6 = -30",
"Step 3: Reduce the exponent by 1: 6 - 1 = 5",
"Answer: f'(x) = -30x^5"
] | -30x^5 | differentiation | sign_error | true | Step 1: correct | Step 2: incorrect — Step 2 has a sign error. -5 × 6 = -30, not 30. The correct derivative is -30x^5. | Step 3: correct | Therefore the correct answer is: -30x^5 | medium |
Find the eigenvalues of the matrix [[0,1],[0,3]]. | [
"Step 1: Form characteristic equation: det(A - λI) = 0",
"Step 2: trace(A) = 3",
"Step 3: det(A) = 0",
"Step 4: Characteristic equation: λ² - 3λ + 0 = 0",
"Step 5: Factoring error — wrong roots: 1 and 2",
"Answer: λ1=1,λ2=2"
] | Step 5 | Step 5 factors the characteristic equation incorrectly. The correct factors are (λ - 0)(λ - 3) = 0. The correct answer is λ1=0,λ2=3. | [
"Step 1: Form characteristic equation: det(A - λI) = 0",
"Step 2: trace(A) = 0 + 3 = 3",
"Step 3: det(A) = 0×3 - 1×0 = 0",
"Step 4: Characteristic equation: λ² - 3λ + 0 = 0",
"Step 5: Factorise: (λ - 0)(λ - 3) = 0",
"Answer: λ1=0,λ2=3"
] | λ1=0,λ2=3 | eigenvalues | arithmetic_in_quadratic | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: incorrect — Step 5 factors the characteristic equation incorrectly. The correct factors are (λ - 0)(λ - 3) = 0. The correct answer is λ1=0,λ2=3. | Therefore the correct answer is: λ1=0,λ2=3 | hard |
Solve the system using Gaussian elimination: 2x - 1y = 1 and -2x + 5y = 11. | [
"Step 1: Write augmented matrix: [2 -1 | 1] and [-2 5 | 11]",
"Step 2: Multiply row 1 by -2 and row 2 by 2: [-4 2 | -2] and [-4 10 | 22]",
"Step 3: Subtract row 2 from row 1: [0 -8 | -24]",
"Step 4: Solve for y: -8y = -24 → y = 3",
"Step 5: Back-substitution arithmetic error: 2x + -1×3 = 1 → x = 4 (sho... | Step 5 | The elimination in Steps 3–4 correctly gives y = 3, but Step 5 makes an arithmetic error in back-substitution, giving x = 4 instead of x = 2. The correct solution is x=2,y=3. | [
"Step 1: Write augmented matrix: [2 -1 | 1] and [-2 5 | 11]",
"Step 2: Multiply row 1 by -2 and row 2 by 2: [-4 2 | -2] and [-4 10 | 22]",
"Step 3: Subtract row 2 from row 1 to eliminate x: [0 -8 | -24]",
"Step 4: Solve for y: -8y = -24 → y = 3",
"Step 5: Back-substitute y = 3 into equation 1: 2x + -1×3 =... | x=2,y=3 | gaussian_elimination | wrong_backsub | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: incorrect — The elimination in Steps 3–4 correctly gives y = 3, but Step 5 makes an arithmetic error in back-substitution, giving x = 4 instead of x = 2. The correct solution is x=2,y=3. | Therefore the correct answer is: x=2,y=3 | medium |
Solve the system using Gaussian elimination: 2x - 1y = 4 and 4x + 2y = -8. | [
"Step 1: Write augmented matrix: [2 -1 | 4] and [4 2 | -8]",
"Step 2: Multiply row 1 by 4 and row 2 by 2: [8 -4 | 16] and [8 4 | -16]",
"Step 3: Subtract row 2 from row 1 to eliminate x: [0 -8 | 32]",
"Step 4: Solve for y: -8y = 32 → y = -4",
"Step 5: Back-substitute y = -4 into equation 1: 2x + -1×-4 = 4... | None | There is no error. The working shown is correct. | [
"Step 1: Write augmented matrix: [2 -1 | 4] and [4 2 | -8]",
"Step 2: Multiply row 1 by 4 and row 2 by 2: [8 -4 | 16] and [8 4 | -16]",
"Step 3: Subtract row 2 from row 1 to eliminate x: [0 -8 | 32]",
"Step 4: Solve for y: -8y = 32 → y = -4",
"Step 5: Back-substitute y = -4 into equation 1: 2x + -1×-4 = 4... | x=0,y=-4 | gaussian_elimination | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: x=0,y=-4 | medium |
Find lim (x → -5) of (x² - 25) / (x - -5) | [
"Step 1: Substitute x = -5: (25 - 25) / (-5 - -5) = 0/0",
"Step 2: Conclude the limit does not exist — 0/0 is undefined",
"Step 3: No further work",
"Answer: undefined"
] | Step 2 | 0/0 is an indeterminate form, not proof the limit is undefined. Factor as (x--5)(x+-5), cancel (x--5), then substitute to get -10. | [
"Step 1: Direct substitution gives 0/0 — indeterminate form, must factor",
"Step 2: Factor numerator: x² - 25 = (x - -5)(x + -5)",
"Step 3: Cancel (x - -5): expression becomes (x + -5)",
"Step 4: Substitute x = -5: -5 + -5 = -10",
"Answer: -10"
] | -10 | limits | forgot_cancel | true | Step 1: correct | Step 2: incorrect — 0/0 is an indeterminate form, not proof the limit is undefined. Factor as (x--5)(x+-5), cancel (x--5), then substitute to get -10. | Step 3: correct | Therefore the correct answer is: -10 | medium |
Find the eigenvalues of the matrix [[-1,-3],[0,-2]]. | [
"Step 1: Form characteristic equation: det(A - λI) = 0",
"Step 2: trace(A) = -3",
"Step 3: det(A) = 2",
"Step 4: Characteristic equation: λ² - -3λ + 2 = 0",
"Step 5: Factoring error — wrong roots: -1 and 0",
"Answer: λ1=-1,λ2=0"
] | Step 5 | Step 5 factors the characteristic equation incorrectly. The correct factors are (λ - -2)(λ - -1) = 0. The correct answer is λ1=-2,λ2=-1. | [
"Step 1: Form characteristic equation: det(A - λI) = 0",
"Step 2: trace(A) = -1 + -2 = -3",
"Step 3: det(A) = -1×-2 - -3×0 = 2",
"Step 4: Characteristic equation: λ² - -3λ + 2 = 0",
"Step 5: Factorise: (λ - -2)(λ - -1) = 0",
"Answer: λ1=-2,λ2=-1"
] | λ1=-2,λ2=-1 | eigenvalues | arithmetic_in_quadratic | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: incorrect — Step 5 factors the characteristic equation incorrectly. The correct factors are (λ - -2)(λ - -1) = 0. The correct answer is λ1=-2,λ2=-1. | Therefore the correct answer is: λ1=-2,λ2=-1 | hard |
Compute the matrix product A·B where A = [[-1,-2],[0,1]] and B = [[3,2],[-2,-1]]. | [
"Step 1: Computed B·A instead of A·B — order is reversed",
"Step 2: D[0][0] = 3×-1 + 2×0 = -3",
"Step 3: D[0][1] = 3×-2 + 2×1 = -4",
"Step 4: D[1][0] = -2×-1 + -1×0 = 2",
"Step 5: D[1][1] = -2×-2 + -1×1 = 3",
"Answer: [[-3,-4],[2,3]]"
] | Step 1 | The multiplication order is reversed in Step 1. Matrix multiplication is not commutative: A·B ≠ B·A. The correct answer for A·B is [[1,0],[-2,-1]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = -1×3 + -2×-2 = 1",
"Step 3: C[0][1] = -1×2 + -2×-1 = 0",
"Step 4: C[1][0] = 0×3 + 1×-2 = -2",
"Step 5: C[1][1] = 0×2 + 1×-1 = -1",
"Answer: [[1,0],[-2,-1]]"
] | [[1,0],[-2,-1]] | matrix_multiplication | transposed_order | true | Step 1: incorrect — The multiplication order is reversed in Step 1. Matrix multiplication is not commutative: A·B ≠ B·A. The correct answer for A·B is [[1,0],[-2,-1]]. | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: [[1,0],[-2,-1]] | easy |
Differentiate f(x) = (2x - 4)^4 using the chain rule. | [
"Step 1: Chain rule: d/dx[g(h(x))] = g'(h(x)) · h'(x)",
"Step 2: Outer function g(u) = u^4 → g'(u) = 4u^3",
"Step 3: Inner function h(x) = 2x - 4 → h'(x) = 2",
"Step 4: Multiplication error: 4 × 2 = 6 (should be 8)",
"Answer: f'(x) = 6(2x - 4)^3"
] | Step 4 | Step 4 has a multiplication error. 4 × 2 = 8, not 6. The correct answer is 8(2x - 4)^3. | [
"Step 1: Chain rule: d/dx[g(h(x))] = g'(h(x)) · h'(x)",
"Step 2: Outer function g(u) = u^4 → g'(u) = 4u^3",
"Step 3: Inner function h(x) = 2x - 4 → h'(x) = 2",
"Step 4: Combine: 4·(2x - 4)^3 · 2 = 8(2x - 4)^3",
"Answer: f'(x) = 8(2x - 4)^3"
] | 8(2x - 4)^3 | differentiation | wrong_coefficient | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: incorrect — Step 4 has a multiplication error. 4 × 2 = 8, not 6. The correct answer is 8(2x - 4)^3. | Therefore the correct answer is: 8(2x - 4)^3 | medium |
Find ∫ -2x^3 dx | [
"Step 1: Power rule for integration: ∫x^n dx = x^(n+1)/(n+1) + C",
"Step 2: Exponent error — kept as 3 instead of raising to 4",
"Step 3: Divide by original exponent: -2 / 3 = (-2/3)",
"Step 4: Append constant of integration",
"Answer: (-2/3)x^3+C"
] | Step 2 | Step 2 fails to raise the exponent. Integration requires adding 1: 3 + 1 = 4. The correct answer is (-1/2)x^4+C. | [
"Step 1: Power rule for integration: ∫x^n dx = x^(n+1)/(n+1) + C",
"Step 2: Raise exponent by 1: 3 + 1 = 4",
"Step 3: Divide coefficient by new exponent: -2 / 4 = (-1/2)",
"Step 4: Append constant of integration",
"Answer: (-1/2)x^4+C"
] | (-1/2)x^4+C | integration | wrong_exponent | true | Step 1: correct | Step 2: incorrect — Step 2 fails to raise the exponent. Integration requires adding 1: 3 + 1 = 4. The correct answer is (-1/2)x^4+C. | Step 3: correct | Step 4: correct | Therefore the correct answer is: (-1/2)x^4+C | medium |
Evaluate ∫ from 3 to 6 of x^3 dx | [
"Step 1: Antiderivative of x^3 is x^4/4",
"Step 2: Evaluate at upper limit 6: 6^4/4 = 324",
"Step 3: Evaluate at lower limit 3: 3^4/4 = (81/4)",
"Step 4: Arithmetic error: 324 - (81/4) = (1211/4)",
"Answer: (1211/4)"
] | Step 4 | Step 4 has an arithmetic error. 324 - (81/4) = (1215/4), not (1211/4). The correct answer is (1215/4). | [
"Step 1: Antiderivative of x^3 is x^4/4",
"Step 2: Evaluate at upper limit 6: 6^4/4 = 324",
"Step 3: Evaluate at lower limit 3: 3^4/4 = (81/4)",
"Step 4: F(6) - F(3) = 324 - (81/4) = (1215/4)",
"Answer: (1215/4)"
] | (1215/4) | integration | arithmetic | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: incorrect — Step 4 has an arithmetic error. 324 - (81/4) = (1215/4), not (1211/4). The correct answer is (1215/4). | Therefore the correct answer is: (1215/4) | medium |
Compute the matrix product A·B where A = [[-4,2],[-3,-4]] and B = [[4,4],[-3,2]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = -4×4 + 2×-3 = -22",
"Step 3: C[0][1] = -4×4 + 2×2 = -12",
"Step 4: C[1][0] = -3×4 + -4×-3 = 0",
"Step 5: C[1][1] = -3×4 + -4×2 = -20",
"Answer: [[-22,-12],[0,-20]]"
] | None | There is no error. The working shown is correct. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = -4×4 + 2×-3 = -22",
"Step 3: C[0][1] = -4×4 + 2×2 = -12",
"Step 4: C[1][0] = -3×4 + -4×-3 = 0",
"Step 5: C[1][1] = -3×4 + -4×2 = -20",
"Answer: [[-22,-12],[0,-20]]"
] | [[-22,-12],[0,-20]] | matrix_multiplication | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: [[-22,-12],[0,-20]] | easy |
Differentiate f(x) = 2x^5 with respect to x. | [
"Step 1: Apply the power rule: d/dx[ax^n] = an·x^(n-1)",
"Step 2: Multiply coefficient by exponent: 2 × 5 = 10",
"Step 3: Reduce the exponent by 1: 5 - 1 = 4",
"Answer: f'(x) = 10x^4"
] | None | There is no error. The working shown is correct. | [
"Step 1: Apply the power rule: d/dx[ax^n] = an·x^(n-1)",
"Step 2: Multiply coefficient by exponent: 2 × 5 = 10",
"Step 3: Reduce the exponent by 1: 5 - 1 = 4",
"Answer: f'(x) = 10x^4"
] | 10x^4 | differentiation | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Therefore the correct answer is: 10x^4 | medium |
Compute the matrix product A·B where A = [[-4,2],[1,3]] and B = [[3,2],[-3,-3]]. | [
"Step 1: Computed B·A instead of A·B — order is reversed",
"Step 2: D[0][0] = 3×-4 + 2×1 = -10",
"Step 3: D[0][1] = 3×2 + 2×3 = 12",
"Step 4: D[1][0] = -3×-4 + -3×1 = 9",
"Step 5: D[1][1] = -3×2 + -3×3 = -15",
"Answer: [[-10,12],[9,-15]]"
] | Step 1 | The multiplication order is reversed in Step 1. Matrix multiplication is not commutative: A·B ≠ B·A. The correct answer for A·B is [[-18,-14],[-6,-7]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = -4×3 + 2×-3 = -18",
"Step 3: C[0][1] = -4×2 + 2×-3 = -14",
"Step 4: C[1][0] = 1×3 + 3×-3 = -6",
"Step 5: C[1][1] = 1×2 + 3×-3 = -7",
"Answer: [[-18,-14],[-6,-7]]"
] | [[-18,-14],[-6,-7]] | matrix_multiplication | transposed_order | true | Step 1: incorrect — The multiplication order is reversed in Step 1. Matrix multiplication is not commutative: A·B ≠ B·A. The correct answer for A·B is [[-18,-14],[-6,-7]]. | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: [[-18,-14],[-6,-7]] | easy |
Differentiate f(x) = (3x + 4)^3 using the chain rule. | [
"Step 1: Chain rule: d/dx[g(h(x))] = g'(h(x)) · h'(x)",
"Step 2: Outer function g(u) = u^3 → g'(u) = 3u^2",
"Step 3: Inner function h(x) = 3x + 4 → h'(x) = 3",
"Step 4: Combine: 3·(3x + 4)^2 · 3 = 9(3x + 4)^2",
"Answer: f'(x) = 9(3x + 4)^2"
] | None | There is no error. The working shown is correct. | [
"Step 1: Chain rule: d/dx[g(h(x))] = g'(h(x)) · h'(x)",
"Step 2: Outer function g(u) = u^3 → g'(u) = 3u^2",
"Step 3: Inner function h(x) = 3x + 4 → h'(x) = 3",
"Step 4: Combine: 3·(3x + 4)^2 · 3 = 9(3x + 4)^2",
"Answer: f'(x) = 9(3x + 4)^2"
] | 9(3x + 4)^2 | differentiation | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Therefore the correct answer is: 9(3x + 4)^2 | medium |
Find the determinant of the matrix [[-2,3],[-4,1]]. | [
"Step 1: For a 2×2 matrix [[a,b],[c,d]], det(A) = a·d - b·c",
"Step 2: Identify: a=-2, b=3, c=-4, d=1",
"Step 3: Compute a·d = -2×1 = -2",
"Step 4: Compute b·c = 3×-4 = -12",
"Step 5: det(A) = -2 - -12 = 10",
"Answer: 10"
] | None | There is no error. The working shown is correct. | [
"Step 1: For a 2×2 matrix [[a,b],[c,d]], det(A) = a·d - b·c",
"Step 2: Identify: a=-2, b=3, c=-4, d=1",
"Step 3: Compute a·d = -2×1 = -2",
"Step 4: Compute b·c = 3×-4 = -12",
"Step 5: det(A) = -2 - -12 = 10",
"Answer: 10"
] | 10 | determinant | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: 10 | easy |
Differentiate f(x) = -4x^4 with respect to x. | [
"Step 1: Apply the power rule: d/dx[ax^n] = an·x^(n-1)",
"Step 2: Multiply coefficient by exponent: -4 × 4 = -16",
"Step 3: Exponent reduction error: 4 - 1 = 4",
"Answer: f'(x) = -16x^4"
] | Step 3 | After applying the power rule, Step 3 should give exponent 3, not 4. Answer: -16x^3. | [
"Step 1: Apply the power rule: d/dx[ax^n] = an·x^(n-1)",
"Step 2: Multiply coefficient by exponent: -4 × 4 = -16",
"Step 3: Reduce the exponent by 1: 4 - 1 = 3",
"Answer: f'(x) = -16x^3"
] | -16x^3 | differentiation | wrong_exponent | true | Step 1: correct | Step 2: correct | Step 3: incorrect — After applying the power rule, Step 3 should give exponent 3, not 4. Answer: -16x^3. | Therefore the correct answer is: -16x^3 | medium |
Solve the system using Gaussian elimination: -5x - 5y = -20 and 3x - 5y = -12. | [
"Step 1: Write augmented matrix: [-5 -5 | -20] and [3 -5 | -12]",
"Step 2: Multiply row 1 by 3 and row 2 by -5: [-15 -15 | -60] and [-15 25 | 60]",
"Step 3: Arithmetic error in RHS: [0 -40 | -80] (correct RHS: -120)",
"Step 4: Solve for y: -40y = -80 → y = 2",
"Step 5: Back-substitute y = 2: -5x + -5×2 =... | Step 3 | Step 3 has an arithmetic error in the RHS after elimination. The correct RHS is -120, not -80. The correct solution is x=1,y=3. | [
"Step 1: Write augmented matrix: [-5 -5 | -20] and [3 -5 | -12]",
"Step 2: Multiply row 1 by 3 and row 2 by -5: [-15 -15 | -60] and [-15 25 | 60]",
"Step 3: Subtract row 2 from row 1 to eliminate x: [0 -40 | -120]",
"Step 4: Solve for y: -40y = -120 → y = 3",
"Step 5: Back-substitute y = 3 into equation 1... | x=1,y=3 | gaussian_elimination | arithmetic | true | Step 1: correct | Step 2: correct | Step 3: incorrect — Step 3 has an arithmetic error in the RHS after elimination. The correct RHS is -120, not -80. The correct solution is x=1,y=3. | Step 4: correct | Step 5: correct | Therefore the correct answer is: x=1,y=3 | medium |
Find lim (x → -1) of 3x - 2 | [
"Step 1: 3x - 2 is continuous everywhere — substitute directly",
"Step 2: Substitute x = -1: 3(-1) + (-2)",
"Step 3: Arithmetic error: -3 + -2 = -7",
"Answer: -7"
] | Step 3 | Step 3 has an arithmetic error. -3 + -2 = -5, not -7. The correct answer is -5. | [
"Step 1: 3x - 2 is continuous everywhere — substitute directly",
"Step 2: Substitute x = -1: 3(-1) + (-2)",
"Step 3: Calculate: -3 + -2 = -5",
"Answer: -5"
] | -5 | limits | arithmetic | true | Step 1: correct | Step 2: correct | Step 3: incorrect — Step 3 has an arithmetic error. -3 + -2 = -5, not -7. The correct answer is -5. | Therefore the correct answer is: -5 | medium |
Find ∫ 2(3x + 2)^3 dx using substitution. | [
"Step 1: Let u = 3x + 2, so du = 3·dx, meaning dx = du/3",
"Step 2: Rewrite: ∫ 2·u^3·(du/3) = (2/3) ∫ u^3 du",
"Step 3: Integrate: (2/3) · u^4/4 = (1/6)u^4",
"Step 4: Back-substitute u = 3x + 2 and add + C",
"Answer: (1/6)(3x + 2)^4+C"
] | None | There is no error. The working shown is correct. | [
"Step 1: Let u = 3x + 2, so du = 3·dx, meaning dx = du/3",
"Step 2: Rewrite: ∫ 2·u^3·(du/3) = (2/3) ∫ u^3 du",
"Step 3: Integrate: (2/3) · u^4/4 = (1/6)u^4",
"Step 4: Back-substitute u = 3x + 2 and add + C",
"Answer: (1/6)(3x + 2)^4+C"
] | (1/6)(3x + 2)^4+C | integration | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Therefore the correct answer is: (1/6)(3x + 2)^4+C | medium |
Find the eigenvalues of the matrix [[4,3],[0,1]]. | [
"Step 1: Form characteristic equation: det(A - λI) = 0",
"Step 2: trace(A) = 4 + 1 = 5",
"Step 3: det(A) = 4×1 - 3×0 = 4",
"Step 4: Characteristic equation: λ² - 5λ + 4 = 0",
"Step 5: Factorise: (λ - 1)(λ - 4) = 0",
"Answer: λ1=1,λ2=4"
] | None | There is no error. The working shown is correct. | [
"Step 1: Form characteristic equation: det(A - λI) = 0",
"Step 2: trace(A) = 4 + 1 = 5",
"Step 3: det(A) = 4×1 - 3×0 = 4",
"Step 4: Characteristic equation: λ² - 5λ + 4 = 0",
"Step 5: Factorise: (λ - 1)(λ - 4) = 0",
"Answer: λ1=1,λ2=4"
] | λ1=1,λ2=4 | eigenvalues | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: λ1=1,λ2=4 | hard |
Evaluate ∫ from 1 to 3 of x^3 dx | [
"Step 1: Antiderivative of x^3 is x^4/4",
"Step 2: Evaluate at upper limit 3: 3^4/4 = (81/4)",
"Step 3: Evaluate at lower limit 1: 1^4/4 = (1/4)",
"Step 4: F(3) - F(1) = (81/4) - (1/4) = 20",
"Answer: 20"
] | None | There is no error. The working shown is correct. | [
"Step 1: Antiderivative of x^3 is x^4/4",
"Step 2: Evaluate at upper limit 3: 3^4/4 = (81/4)",
"Step 3: Evaluate at lower limit 1: 1^4/4 = (1/4)",
"Step 4: F(3) - F(1) = (81/4) - (1/4) = 20",
"Answer: 20"
] | 20 | integration | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Therefore the correct answer is: 20 | medium |
Evaluate ∫ from 2 to 6 of x^3 dx | [
"Step 1: Antiderivative of x^3 is x^4/4",
"Step 2: Evaluate at upper limit 6: 6^4/4 = 324",
"Step 3: Evaluate at lower limit 2: 2^4/4 = 4",
"Step 4: Arithmetic error: 324 - 4 = 321",
"Answer: 321"
] | Step 4 | Step 4 has an arithmetic error. 324 - 4 = 320, not 321. The correct answer is 320. | [
"Step 1: Antiderivative of x^3 is x^4/4",
"Step 2: Evaluate at upper limit 6: 6^4/4 = 324",
"Step 3: Evaluate at lower limit 2: 2^4/4 = 4",
"Step 4: F(6) - F(2) = 324 - 4 = 320",
"Answer: 320"
] | 320 | integration | arithmetic | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: incorrect — Step 4 has an arithmetic error. 324 - 4 = 320, not 321. The correct answer is 320. | Therefore the correct answer is: 320 | medium |
Find lim (x → 4) of (x² - 16) / (x - 4) | [
"Step 1: Direct substitution gives 0/0 — factor and cancel",
"Step 2: Factor: x² - 16 = (x - 4)(x + 4)",
"Step 3: Cancel (x - 4): expression becomes (x + 4)",
"Step 4: Arithmetic error: 4 + 4 = 10",
"Answer: 10"
] | Step 4 | Step 4 has an arithmetic error. 4 + 4 = 8, not 10. The correct answer is 8. | [
"Step 1: Direct substitution gives 0/0 — indeterminate form, must factor",
"Step 2: Factor numerator: x² - 16 = (x - 4)(x + 4)",
"Step 3: Cancel (x - 4): expression becomes (x + 4)",
"Step 4: Substitute x = 4: 4 + 4 = 8",
"Answer: 8"
] | 8 | limits | arithmetic | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: incorrect — Step 4 has an arithmetic error. 4 + 4 = 8, not 10. The correct answer is 8. | Therefore the correct answer is: 8 | medium |
Find lim (x → 0) of sin(6x) / x using L'Hôpital's rule. | [
"Step 1: Direct substitution gives 0/0 — indeterminate form",
"Step 2: Apply L'Hôpital's rule",
"Step 3: Chain rule error: d/dx[sin(6x)] = 4cos(6x) (should be 6cos(6x))",
"Step 4: New limit at x = 0: 4·cos(0) = 4",
"Answer: 4"
] | Step 3 | Step 3 has the wrong chain-rule coefficient. d/dx[sin(6x)] = 6cos(6x), not 4cos(6x). The correct answer is 6. | [
"Step 1: Direct substitution gives sin(0)/0 = 0/0 — indeterminate form",
"Step 2: Apply L'Hôpital's rule: differentiate numerator and denominator separately",
"Step 3: d/dx[sin(6x)] = 6cos(6x), d/dx[x] = 1",
"Step 4: Evaluate new limit at x = 0: 6·cos(0) = 6·1 = 6",
"Answer: 6"
] | 6 | limits | wrong_derivative | true | Step 1: correct | Step 2: correct | Step 3: incorrect — Step 3 has the wrong chain-rule coefficient. d/dx[sin(6x)] = 6cos(6x), not 4cos(6x). The correct answer is 6. | Step 4: correct | Therefore the correct answer is: 6 | medium |
Find lim (x → 0) of sin(5x) / x using L'Hôpital's rule. | [
"Step 1: Direct substitution gives 0/0 — indeterminate form",
"Step 2: Apply L'Hôpital's rule",
"Step 3: Chain rule error: d/dx[sin(5x)] = 6cos(5x) (should be 5cos(5x))",
"Step 4: New limit at x = 0: 6·cos(0) = 6",
"Answer: 6"
] | Step 3 | Step 3 has the wrong chain-rule coefficient. d/dx[sin(5x)] = 5cos(5x), not 6cos(5x). The correct answer is 5. | [
"Step 1: Direct substitution gives sin(0)/0 = 0/0 — indeterminate form",
"Step 2: Apply L'Hôpital's rule: differentiate numerator and denominator separately",
"Step 3: d/dx[sin(5x)] = 5cos(5x), d/dx[x] = 1",
"Step 4: Evaluate new limit at x = 0: 5·cos(0) = 5·1 = 5",
"Answer: 5"
] | 5 | limits | wrong_derivative | true | Step 1: correct | Step 2: correct | Step 3: incorrect — Step 3 has the wrong chain-rule coefficient. d/dx[sin(5x)] = 5cos(5x), not 6cos(5x). The correct answer is 5. | Step 4: correct | Therefore the correct answer is: 5 | medium |
Solve the system using Gaussian elimination: 5x - 2y = -18 and -3x - 2y = 14. | [
"Step 1: Write augmented matrix: [5 -2 | -18] and [-3 -2 | 14]",
"Step 2: Multiply row 1 by -3 and row 2 by 5: [-15 6 | 54] and [-15 -10 | 70]",
"Step 3: Subtract row 2 from row 1: [0 16 | -16]",
"Step 4: Solve for y: 16y = -16 → y = -1",
"Step 5: Back-substitution arithmetic error: 5x + -2×-1 = -18 → x... | Step 5 | The elimination in Steps 3–4 correctly gives y = -1, but Step 5 makes an arithmetic error in back-substitution, giving x = -3 instead of x = -4. The correct solution is x=-4,y=-1. | [
"Step 1: Write augmented matrix: [5 -2 | -18] and [-3 -2 | 14]",
"Step 2: Multiply row 1 by -3 and row 2 by 5: [-15 6 | 54] and [-15 -10 | 70]",
"Step 3: Subtract row 2 from row 1 to eliminate x: [0 16 | -16]",
"Step 4: Solve for y: 16y = -16 → y = -1",
"Step 5: Back-substitute y = -1 into equation 1: 5x ... | x=-4,y=-1 | gaussian_elimination | wrong_backsub | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: incorrect — The elimination in Steps 3–4 correctly gives y = -1, but Step 5 makes an arithmetic error in back-substitution, giving x = -3 instead of x = -4. The correct solution is x=-4,y=-1. | Therefore the correct answer is: x=-4,y=-1 | medium |
Solve the system using Gaussian elimination: 1x - 2y = -1 and 5x - 3y = 16. | [
"Step 1: Write augmented matrix: [1 -2 | -1] and [5 -3 | 16]",
"Step 2: Multiply row 1 by 5 and row 2 by 1: [5 -10 | -5] and [5 -3 | 16]",
"Step 3: Arithmetic error in RHS: [0 -7 | -28] (correct RHS: -21)",
"Step 4: Solve for y: -7y = -28 → y = 4",
"Step 5: Back-substitute y = 4: 1x + -2×4 = -1 → x = 7... | Step 3 | Step 3 has an arithmetic error in the RHS after elimination. The correct RHS is -21, not -28. The correct solution is x=5,y=3. | [
"Step 1: Write augmented matrix: [1 -2 | -1] and [5 -3 | 16]",
"Step 2: Multiply row 1 by 5 and row 2 by 1: [5 -10 | -5] and [5 -3 | 16]",
"Step 3: Subtract row 2 from row 1 to eliminate x: [0 -7 | -21]",
"Step 4: Solve for y: -7y = -21 → y = 3",
"Step 5: Back-substitute y = 3 into equation 1: 1x + -2×3 =... | x=5,y=3 | gaussian_elimination | arithmetic | true | Step 1: correct | Step 2: correct | Step 3: incorrect — Step 3 has an arithmetic error in the RHS after elimination. The correct RHS is -21, not -28. The correct solution is x=5,y=3. | Step 4: correct | Step 5: correct | Therefore the correct answer is: x=5,y=3 | medium |
Evaluate ∫ from 0 to 4 of x^2 dx | [
"Step 1: Antiderivative error — wrote x^2/3 instead of x^3/3",
"Step 2: Evaluate at upper limit: 4^2/3 = (16/3)",
"Step 3: Evaluate at lower limit: 0^2/3 = 0",
"Step 4: (16/3) - 0 = (16/3)",
"Answer: (16/3)"
] | Step 1 | Step 1 uses the wrong exponent in the antiderivative. ∫x^2 dx = x^3/3, not x^2/3. The correct answer is (64/3). | [
"Step 1: Antiderivative of x^2 is x^3/3",
"Step 2: Evaluate at upper limit 4: 4^3/3 = (64/3)",
"Step 3: Evaluate at lower limit 0: 0^3/3 = 0",
"Step 4: F(4) - F(0) = (64/3) - 0 = (64/3)",
"Answer: (64/3)"
] | (64/3) | integration | wrong_exponent | true | Step 1: incorrect — Step 1 uses the wrong exponent in the antiderivative. ∫x^2 dx = x^3/3, not x^2/3. The correct answer is (64/3). | Step 2: correct | Step 3: correct | Step 4: correct | Therefore the correct answer is: (64/3) | medium |
Differentiate f(x) = x^4 · x^3 using the product rule. | [
"Step 1: Product rule: d/dx[u·v] = u'v + uv'",
"Step 2: u = x^4, v = x^3 → u' = 4x^3, v' = 3x^2",
"Step 3: u'v + uv' = 4x^3·x^3 + x^4·3x^2",
"Step 4: Index law error: exponents add to 5 instead of 6: 7x^5",
"Answer: f'(x) = 7x^5"
] | Step 4 | Step 4 uses the wrong exponent. x^3·x^3 = x^6 = x^6, not x^5. The correct answer is 7x^6. | [
"Step 1: Product rule: d/dx[u·v] = u'v + uv'",
"Step 2: u = x^4, v = x^3 → u' = 4x^3, v' = 3x^2",
"Step 3: u'v + uv' = 4x^3·x^3 + x^4·3x^2",
"Step 4: Simplify using index laws: 4x^6 + 3x^6 = 7x^6",
"Answer: f'(x) = 7x^6"
] | 7x^6 | differentiation | wrong_coefficient | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: incorrect — Step 4 uses the wrong exponent. x^3·x^3 = x^6 = x^6, not x^5. The correct answer is 7x^6. | Therefore the correct answer is: 7x^6 | medium |
Find the determinant of the matrix [[3,1],[-3,1]]. | [
"Step 1: For a 2×2 matrix [[a,b],[c,d]], det(A) = a·d - b·c",
"Step 2: Identify: a=3, b=1, c=-3, d=1",
"Step 3: Compute a·d = 3×1 = 3",
"Step 4: Compute b·c = 1×-3 = -3",
"Step 5: Wrong order — b·c - a·d = -3 - 3 = -6",
"Answer: -6"
] | Step 5 | Step 5 subtracts in the wrong order. det(A) = a·d - b·c = 3 - -3 = 6. The correct answer is 6. | [
"Step 1: For a 2×2 matrix [[a,b],[c,d]], det(A) = a·d - b·c",
"Step 2: Identify: a=3, b=1, c=-3, d=1",
"Step 3: Compute a·d = 3×1 = 3",
"Step 4: Compute b·c = 1×-3 = -3",
"Step 5: det(A) = 3 - -3 = 6",
"Answer: 6"
] | 6 | determinant | sign_error | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: incorrect — Step 5 subtracts in the wrong order. det(A) = a·d - b·c = 3 - -3 = 6. The correct answer is 6. | Therefore the correct answer is: 6 | easy |
Compute the matrix product A·B where A = [[3,-4],[2,2]] and B = [[-3,4],[-4,-2]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = 3×-3 + -4×-4 = 7",
"Step 3: C[0][1] = 3×4 + -4×-2 = 20",
"Step 4: C[1][0] = 2×-3 + 2×-4 = -14",
"Step 5: C[1][1] = 2×4 + 2×-2 = 4",
"Answer: [[7,20],[-14,4]]"
] | None | There is no error. The working shown is correct. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = 3×-3 + -4×-4 = 7",
"Step 3: C[0][1] = 3×4 + -4×-2 = 20",
"Step 4: C[1][0] = 2×-3 + 2×-4 = -14",
"Step 5: C[1][1] = 2×4 + 2×-2 = 4",
"Answer: [[7,20],[-14,4]]"
] | [[7,20],[-14,4]] | matrix_multiplication | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: [[7,20],[-14,4]] | easy |
Solve the system using Gaussian elimination: 2x - 4y = 18 and 1x + 5y = -12. | [
"Step 1: Write augmented matrix: [2 -4 | 18] and [1 5 | -12]",
"Step 2: Multiply row 1 by 1 and row 2 by 2: [2 -4 | 18] and [2 10 | -24]",
"Step 3: Arithmetic error in RHS: [0 -14 | 28] (correct RHS: 42)",
"Step 4: Solve for y: -14y = 28 → y = -2",
"Step 5: Back-substitute y = -2: 2x + -4×-2 = 18 → x =... | Step 3 | Step 3 has an arithmetic error in the RHS after elimination. The correct RHS is 42, not 28. The correct solution is x=3,y=-3. | [
"Step 1: Write augmented matrix: [2 -4 | 18] and [1 5 | -12]",
"Step 2: Multiply row 1 by 1 and row 2 by 2: [2 -4 | 18] and [2 10 | -24]",
"Step 3: Subtract row 2 from row 1 to eliminate x: [0 -14 | 42]",
"Step 4: Solve for y: -14y = 42 → y = -3",
"Step 5: Back-substitute y = -3 into equation 1: 2x + -4×-... | x=3,y=-3 | gaussian_elimination | arithmetic | true | Step 1: correct | Step 2: correct | Step 3: incorrect — Step 3 has an arithmetic error in the RHS after elimination. The correct RHS is 42, not 28. The correct solution is x=3,y=-3. | Step 4: correct | Step 5: correct | Therefore the correct answer is: x=3,y=-3 | medium |
Find the eigenvalues of the matrix [[4,3],[0,1]]. | [
"Step 1: Form characteristic equation: det(A - λI) = 0",
"Step 2: trace(A) = 5",
"Step 3: det(A) = 4",
"Step 4: Characteristic equation: λ² - 5λ + 4 = 0",
"Step 5: Factoring error — wrong roots: 2 and 3",
"Answer: λ1=2,λ2=3"
] | Step 5 | Step 5 factors the characteristic equation incorrectly. The correct factors are (λ - 1)(λ - 4) = 0. The correct answer is λ1=1,λ2=4. | [
"Step 1: Form characteristic equation: det(A - λI) = 0",
"Step 2: trace(A) = 4 + 1 = 5",
"Step 3: det(A) = 4×1 - 3×0 = 4",
"Step 4: Characteristic equation: λ² - 5λ + 4 = 0",
"Step 5: Factorise: (λ - 1)(λ - 4) = 0",
"Answer: λ1=1,λ2=4"
] | λ1=1,λ2=4 | eigenvalues | arithmetic_in_quadratic | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: incorrect — Step 5 factors the characteristic equation incorrectly. The correct factors are (λ - 1)(λ - 4) = 0. The correct answer is λ1=1,λ2=4. | Therefore the correct answer is: λ1=1,λ2=4 | hard |
Compute the matrix product A·B where A = [[-2,4],[0,0]] and B = [[-4,4],[4,-2]]. | [
"Step 1: Computed B·A instead of A·B — order is reversed",
"Step 2: D[0][0] = -4×-2 + 4×0 = 8",
"Step 3: D[0][1] = -4×4 + 4×0 = -16",
"Step 4: D[1][0] = 4×-2 + -2×0 = -8",
"Step 5: D[1][1] = 4×4 + -2×0 = 16",
"Answer: [[8,-16],[-8,16]]"
] | Step 1 | The multiplication order is reversed in Step 1. Matrix multiplication is not commutative: A·B ≠ B·A. The correct answer for A·B is [[24,-16],[0,0]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = -2×-4 + 4×4 = 24",
"Step 3: C[0][1] = -2×4 + 4×-2 = -16",
"Step 4: C[1][0] = 0×-4 + 0×4 = 0",
"Step 5: C[1][1] = 0×4 + 0×-2 = 0",
"Answer: [[24,-16],[0,0]]"
] | [[24,-16],[0,0]] | matrix_multiplication | transposed_order | true | Step 1: incorrect — The multiplication order is reversed in Step 1. Matrix multiplication is not commutative: A·B ≠ B·A. The correct answer for A·B is [[24,-16],[0,0]]. | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: [[24,-16],[0,0]] | easy |
Find the eigenvalues of the matrix [[1,-3],[0,-4]]. | [
"Step 1: Form characteristic equation: det(A - λI) = 0",
"Step 2: trace(A) = -3",
"Step 3: det(A) = -4",
"Step 4: Characteristic equation: λ² - -3λ + -4 = 0",
"Step 5: Factoring error — wrong roots: -5 and 0",
"Answer: λ1=-5,λ2=0"
] | Step 5 | Step 5 factors the characteristic equation incorrectly. The correct factors are (λ - -4)(λ - 1) = 0. The correct answer is λ1=-4,λ2=1. | [
"Step 1: Form characteristic equation: det(A - λI) = 0",
"Step 2: trace(A) = 1 + -4 = -3",
"Step 3: det(A) = 1×-4 - -3×0 = -4",
"Step 4: Characteristic equation: λ² - -3λ + -4 = 0",
"Step 5: Factorise: (λ - -4)(λ - 1) = 0",
"Answer: λ1=-4,λ2=1"
] | λ1=-4,λ2=1 | eigenvalues | arithmetic_in_quadratic | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: incorrect — Step 5 factors the characteristic equation incorrectly. The correct factors are (λ - -4)(λ - 1) = 0. The correct answer is λ1=-4,λ2=1. | Therefore the correct answer is: λ1=-4,λ2=1 | hard |
Differentiate f(x) = (2x + 4)^3 using the chain rule. | [
"Step 1: Chain rule: d/dx[g(h(x))] = g'(h(x)) · h'(x)",
"Step 2: Outer function g(u) = u^3 → g'(u) = 3u^2",
"Step 3: Inner function h(x) = 2x + 4 → h'(x) = 2",
"Step 4: Multiplication error: 3 × 2 = 4 (should be 6)",
"Answer: f'(x) = 4(2x + 4)^2"
] | Step 4 | The coefficient in Step 4 should be 3 × 2 = 6, not 4. Answer: 6(2x + 4)^2. | [
"Step 1: Chain rule: d/dx[g(h(x))] = g'(h(x)) · h'(x)",
"Step 2: Outer function g(u) = u^3 → g'(u) = 3u^2",
"Step 3: Inner function h(x) = 2x + 4 → h'(x) = 2",
"Step 4: Combine: 3·(2x + 4)^2 · 2 = 6(2x + 4)^2",
"Answer: f'(x) = 6(2x + 4)^2"
] | 6(2x + 4)^2 | differentiation | wrong_coefficient | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: incorrect — The coefficient in Step 4 should be 3 × 2 = 6, not 4. Answer: 6(2x + 4)^2. | Therefore the correct answer is: 6(2x + 4)^2 | medium |
Solve the system using Gaussian elimination: -5x + 1y = -1 and -5x + 5y = 15. | [
"Step 1: Write augmented matrix: [-5 1 | -1] and [-5 5 | 15]",
"Step 2: Multiply row 1 by -5 and row 2 by -5: [25 -5 | 5] and [25 -25 | -75]",
"Step 3: Subtract row 2 from row 1: [0 20 | 80]",
"Step 4: Solve for y: 20y = 80 → y = 4",
"Step 5: Back-substitution arithmetic error: -5x + 1×4 = -1 → x = -1 ... | Step 5 | The elimination in Steps 3–4 correctly gives y = 4, but Step 5 makes an arithmetic error in back-substitution, giving x = -1 instead of x = 1. The correct solution is x=1,y=4. | [
"Step 1: Write augmented matrix: [-5 1 | -1] and [-5 5 | 15]",
"Step 2: Multiply row 1 by -5 and row 2 by -5: [25 -5 | 5] and [25 -25 | -75]",
"Step 3: Subtract row 2 from row 1 to eliminate x: [0 20 | 80]",
"Step 4: Solve for y: 20y = 80 → y = 4",
"Step 5: Back-substitute y = 4 into equation 1: -5x + 1×4... | x=1,y=4 | gaussian_elimination | wrong_backsub | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: incorrect — The elimination in Steps 3–4 correctly gives y = 4, but Step 5 makes an arithmetic error in back-substitution, giving x = -1 instead of x = 1. The correct solution is x=1,y=4. | Therefore the correct answer is: x=1,y=4 | medium |
Find the determinant of the matrix [[4,3],[3,4]]. | [
"Step 1: For a 2×2 matrix [[a,b],[c,d]], det(A) = a·d - b·c",
"Step 2: Identify: a=4, b=3, c=3, d=4",
"Step 3: Compute a·d = 4×4 = 16",
"Step 4: Compute b·c = 3×3 = 9",
"Step 5: det(A) = 16 - 9 = 7",
"Answer: 7"
] | None | There is no error. The working shown is correct. | [
"Step 1: For a 2×2 matrix [[a,b],[c,d]], det(A) = a·d - b·c",
"Step 2: Identify: a=4, b=3, c=3, d=4",
"Step 3: Compute a·d = 4×4 = 16",
"Step 4: Compute b·c = 3×3 = 9",
"Step 5: det(A) = 16 - 9 = 7",
"Answer: 7"
] | 7 | determinant | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: 7 | easy |
Compute the matrix product A·B where A = [[2,3],[2,0]] and B = [[3,-3],[-2,-3]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = 2×3 + 3×-2 = 0",
"Step 3: C[0][1] = 2×-3 + 3×-3 = -15",
"Step 4: C[1][0] = 2×3 + 0×-2 = 6",
"Step 5: C[1][1] = 2×-3 + 0×-3 = -6",
"Answer: [[0,-15],[6,-6]]"
] | None | There is no error. The working shown is correct. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = 2×3 + 3×-2 = 0",
"Step 3: C[0][1] = 2×-3 + 3×-3 = -15",
"Step 4: C[1][0] = 2×3 + 0×-2 = 6",
"Step 5: C[1][1] = 2×-3 + 0×-3 = -6",
"Answer: [[0,-15],[6,-6]]"
] | [[0,-15],[6,-6]] | matrix_multiplication | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: [[0,-15],[6,-6]] | easy |
Find the determinant of the matrix [[3,-1],[3,-3]]. | [
"Step 1: Error — used det(A) = a·d + b·c (should subtract)",
"Step 2: Identify: a=3, b=-1, c=3, d=-3",
"Step 3: Compute a·d = 3×-3 = -9",
"Step 4: Compute b·c = -1×3 = -3",
"Step 5: det(A) = -9 + -3 = -12",
"Answer: -12"
] | Step 1 | Step 1 uses the wrong formula. The 2×2 determinant is a·d - b·c, not a·d + b·c. The correct answer is -6. | [
"Step 1: For a 2×2 matrix [[a,b],[c,d]], det(A) = a·d - b·c",
"Step 2: Identify: a=3, b=-1, c=3, d=-3",
"Step 3: Compute a·d = 3×-3 = -9",
"Step 4: Compute b·c = -1×3 = -3",
"Step 5: det(A) = -9 - -3 = -6",
"Answer: -6"
] | -6 | determinant | wrong_formula | true | Step 1: incorrect — Step 1 uses the wrong formula. The 2×2 determinant is a·d - b·c, not a·d + b·c. The correct answer is -6. | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: -6 | easy |
Compute the matrix product A·B where A = [[0,-3],[3,3]] and B = [[-1,-4],[-1,2]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = 0×-1 + -3×-1 = 3",
"Step 3: C[0][1] = 0×-4 + -3×2 = -6",
"Step 4: C[1][0] = 3×-1 + 3×-1 = -6",
"Step 5: C[1][1] = 3×-4 + 3×2 = -5",
"Answer: [[3,-6],[-6,-5]]"
] | Step 5 | Entry C[1][1] is wrong. The correct value is -6, not -5. The correct result is [[3,-6],[-6,-6]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = 0×-1 + -3×-1 = 3",
"Step 3: C[0][1] = 0×-4 + -3×2 = -6",
"Step 4: C[1][0] = 3×-1 + 3×-1 = -6",
"Step 5: C[1][1] = 3×-4 + 3×2 = -6",
"Answer: [[3,-6],[-6,-6]]"
] | [[3,-6],[-6,-6]] | matrix_multiplication | wrong_entry | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: incorrect — Entry C[1][1] is wrong. The correct value is -6, not -5. The correct result is [[3,-6],[-6,-6]]. | Therefore the correct answer is: [[3,-6],[-6,-6]] | easy |
Differentiate f(x) = 2x^4 with respect to x. | [
"Step 1: Apply the power rule: d/dx[ax^n] = an·x^(n-1)",
"Step 2: Sign error: 2 × 4 = -8",
"Step 3: Reduce the exponent by 1: 4 - 1 = 3",
"Answer: f'(x) = -8x^3"
] | Step 2 | Step 2 has a sign error. 2 × 4 = 8, not -8. The correct derivative is 8x^3. | [
"Step 1: Apply the power rule: d/dx[ax^n] = an·x^(n-1)",
"Step 2: Multiply coefficient by exponent: 2 × 4 = 8",
"Step 3: Reduce the exponent by 1: 4 - 1 = 3",
"Answer: f'(x) = 8x^3"
] | 8x^3 | differentiation | sign_error | true | Step 1: correct | Step 2: incorrect — Step 2 has a sign error. 2 × 4 = 8, not -8. The correct derivative is 8x^3. | Step 3: correct | Therefore the correct answer is: 8x^3 | medium |
Find lim (x → 0) of sin(6x) / x using L'Hôpital's rule. | [
"Step 1: Direct substitution gives 0/0 — indeterminate form",
"Step 2: Apply L'Hôpital's rule",
"Step 3: Chain rule error: d/dx[sin(6x)] = 7cos(6x) (should be 6cos(6x))",
"Step 4: New limit at x = 0: 7·cos(0) = 7",
"Answer: 7"
] | Step 3 | Step 3 has the wrong chain-rule coefficient. d/dx[sin(6x)] = 6cos(6x), not 7cos(6x). The correct answer is 6. | [
"Step 1: Direct substitution gives sin(0)/0 = 0/0 — indeterminate form",
"Step 2: Apply L'Hôpital's rule: differentiate numerator and denominator separately",
"Step 3: d/dx[sin(6x)] = 6cos(6x), d/dx[x] = 1",
"Step 4: Evaluate new limit at x = 0: 6·cos(0) = 6·1 = 6",
"Answer: 6"
] | 6 | limits | wrong_derivative | true | Step 1: correct | Step 2: correct | Step 3: incorrect — Step 3 has the wrong chain-rule coefficient. d/dx[sin(6x)] = 6cos(6x), not 7cos(6x). The correct answer is 6. | Step 4: correct | Therefore the correct answer is: 6 | medium |
Find the eigenvalues of the matrix [[4,-2],[0,2]]. | [
"Step 1: Form characteristic equation: det(A - λI) = 0",
"Step 2: trace(A) = 4 + 2 = 6",
"Step 3: det(A) = 4×2 - -2×0 = 8",
"Step 4: Characteristic equation: λ² - 6λ + 8 = 0",
"Step 5: Factorise: (λ - 2)(λ - 4) = 0",
"Answer: λ1=2,λ2=4"
] | None | There is no error. The working shown is correct. | [
"Step 1: Form characteristic equation: det(A - λI) = 0",
"Step 2: trace(A) = 4 + 2 = 6",
"Step 3: det(A) = 4×2 - -2×0 = 8",
"Step 4: Characteristic equation: λ² - 6λ + 8 = 0",
"Step 5: Factorise: (λ - 2)(λ - 4) = 0",
"Answer: λ1=2,λ2=4"
] | λ1=2,λ2=4 | eigenvalues | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: λ1=2,λ2=4 | hard |
Differentiate f(x) = (2x + 2)^2 using the chain rule. | [
"Step 1: Chain rule: d/dx[g(h(x))] = g'(h(x)) · h'(x)",
"Step 2: Outer function g(u) = u^2 → g'(u) = 2u^1",
"Step 3: Inner function h(x) = 2x + 2 → h'(x) = 2",
"Step 4: Combine: 2·(2x + 2)^1 · 2 = 4(2x + 2)^1",
"Answer: f'(x) = 4(2x + 2)^1"
] | None | There is no error. The working shown is correct. | [
"Step 1: Chain rule: d/dx[g(h(x))] = g'(h(x)) · h'(x)",
"Step 2: Outer function g(u) = u^2 → g'(u) = 2u^1",
"Step 3: Inner function h(x) = 2x + 2 → h'(x) = 2",
"Step 4: Combine: 2·(2x + 2)^1 · 2 = 4(2x + 2)^1",
"Answer: f'(x) = 4(2x + 2)^1"
] | 4(2x + 2)^1 | differentiation | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Therefore the correct answer is: 4(2x + 2)^1 | medium |
Compute the matrix product A·B where A = [[4,3],[1,-4]] and B = [[1,-2],[4,0]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = 4×1 + 3×4 = 16",
"Step 3: C[0][1] = 4×-2 + 3×0 = -8",
"Step 4: C[1][0] = 1×1 + -4×4 = -15",
"Step 5: C[1][1] = 1×-2 + -4×0 = -3",
"Answer: [[16,-8],[-15,-3]]"
] | Step 5 | Entry C[1][1] is wrong. The correct value is -2, not -3. The correct result is [[16,-8],[-15,-2]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = 4×1 + 3×4 = 16",
"Step 3: C[0][1] = 4×-2 + 3×0 = -8",
"Step 4: C[1][0] = 1×1 + -4×4 = -15",
"Step 5: C[1][1] = 1×-2 + -4×0 = -2",
"Answer: [[16,-8],[-15,-2]]"
] | [[16,-8],[-15,-2]] | matrix_multiplication | wrong_entry | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: incorrect — Entry C[1][1] is wrong. The correct value is -2, not -3. The correct result is [[16,-8],[-15,-2]]. | Therefore the correct answer is: [[16,-8],[-15,-2]] | easy |
Find the eigenvalues of the matrix [[-4,2],[0,1]]. | [
"Step 1: Form characteristic equation: det(A - λI) = 0",
"Step 2: trace(A) = -3",
"Step 3: det(A) = -4",
"Step 4: Characteristic equation: λ² - -3λ + -4 = 0",
"Step 5: Factoring error — wrong roots: -5 and 2",
"Answer: λ1=-5,λ2=2"
] | Step 5 | Step 5 factors the characteristic equation incorrectly. The correct factors are (λ - -4)(λ - 1) = 0. The correct answer is λ1=-4,λ2=1. | [
"Step 1: Form characteristic equation: det(A - λI) = 0",
"Step 2: trace(A) = -4 + 1 = -3",
"Step 3: det(A) = -4×1 - 2×0 = -4",
"Step 4: Characteristic equation: λ² - -3λ + -4 = 0",
"Step 5: Factorise: (λ - -4)(λ - 1) = 0",
"Answer: λ1=-4,λ2=1"
] | λ1=-4,λ2=1 | eigenvalues | arithmetic_in_quadratic | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: incorrect — Step 5 factors the characteristic equation incorrectly. The correct factors are (λ - -4)(λ - 1) = 0. The correct answer is λ1=-4,λ2=1. | Therefore the correct answer is: λ1=-4,λ2=1 | hard |
Solve the system using Gaussian elimination: 4x - 2y = 8 and 4x - 3y = 12. | [
"Step 1: Write augmented matrix: [4 -2 | 8] and [4 -3 | 12]",
"Step 2: Multiply row 1 by 4 and row 2 by 4: [16 -8 | 32] and [16 -12 | 48]",
"Step 3: Subtract row 2 from row 1: [0 4 | -16]",
"Step 4: Solve for y: 4y = -16 → y = -4",
"Step 5: Back-substitution arithmetic error: 4x + -2×-4 = 8 → x = -1 (s... | Step 5 | The elimination in Steps 3–4 correctly gives y = -4, but Step 5 makes an arithmetic error in back-substitution, giving x = -1 instead of x = 0. The correct solution is x=0,y=-4. | [
"Step 1: Write augmented matrix: [4 -2 | 8] and [4 -3 | 12]",
"Step 2: Multiply row 1 by 4 and row 2 by 4: [16 -8 | 32] and [16 -12 | 48]",
"Step 3: Subtract row 2 from row 1 to eliminate x: [0 4 | -16]",
"Step 4: Solve for y: 4y = -16 → y = -4",
"Step 5: Back-substitute y = -4 into equation 1: 4x + -2×-4... | x=0,y=-4 | gaussian_elimination | wrong_backsub | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: incorrect — The elimination in Steps 3–4 correctly gives y = -4, but Step 5 makes an arithmetic error in back-substitution, giving x = -1 instead of x = 0. The correct solution is x=0,y=-4. | Therefore the correct answer is: x=0,y=-4 | medium |
Differentiate f(x) = (4x + 1)^3 using the chain rule. | [
"Step 1: Chain rule: d/dx[g(h(x))] = g'(h(x)) · h'(x)",
"Step 2: Outer function g(u) = u^3 → g'(u) = 3u^2",
"Step 3: Inner function h(x) = 4x + 1 → h'(x) = 4",
"Step 4: Multiplication error: 3 × 4 = 9 (should be 12)",
"Answer: f'(x) = 9(4x + 1)^2"
] | Step 4 | The coefficient in Step 4 should be 3 × 4 = 12, not 9. Answer: 12(4x + 1)^2. | [
"Step 1: Chain rule: d/dx[g(h(x))] = g'(h(x)) · h'(x)",
"Step 2: Outer function g(u) = u^3 → g'(u) = 3u^2",
"Step 3: Inner function h(x) = 4x + 1 → h'(x) = 4",
"Step 4: Combine: 3·(4x + 1)^2 · 4 = 12(4x + 1)^2",
"Answer: f'(x) = 12(4x + 1)^2"
] | 12(4x + 1)^2 | differentiation | wrong_coefficient | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: incorrect — The coefficient in Step 4 should be 3 × 4 = 12, not 9. Answer: 12(4x + 1)^2. | Therefore the correct answer is: 12(4x + 1)^2 | medium |
Find lim (x → 2) of (x² - 4) / (x - 2) | [
"Step 1: Substitute x = 2: (4 - 4) / (2 - 2) = 0/0",
"Step 2: Conclude the limit does not exist — 0/0 is undefined",
"Step 3: No further work",
"Answer: undefined"
] | Step 2 | 0/0 is an indeterminate form, not proof the limit is undefined. Factor as (x-2)(x+2), cancel (x-2), then substitute to get 4. | [
"Step 1: Direct substitution gives 0/0 — indeterminate form, must factor",
"Step 2: Factor numerator: x² - 4 = (x - 2)(x + 2)",
"Step 3: Cancel (x - 2): expression becomes (x + 2)",
"Step 4: Substitute x = 2: 2 + 2 = 4",
"Answer: 4"
] | 4 | limits | forgot_cancel | true | Step 1: correct | Step 2: incorrect — 0/0 is an indeterminate form, not proof the limit is undefined. Factor as (x-2)(x+2), cancel (x-2), then substitute to get 4. | Step 3: correct | Therefore the correct answer is: 4 | medium |
Compute the matrix product A·B where A = [[-2,2],[3,-2]] and B = [[-1,4],[-2,3]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = -2×-1 + 2×-2 = -2",
"Step 3: C[0][1] = -2×4 + 2×3 = -2",
"Step 4: C[1][0] = 3×-1 + -2×-2 = 1",
"Step 5: C[1][1] = 3×4 + -2×3 = 7",
"Answer: [[-2,-2],[1,7]]"
] | Step 5 | Entry C[1][1] is wrong. The correct value is 6, not 7. The correct result is [[-2,-2],[1,6]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = -2×-1 + 2×-2 = -2",
"Step 3: C[0][1] = -2×4 + 2×3 = -2",
"Step 4: C[1][0] = 3×-1 + -2×-2 = 1",
"Step 5: C[1][1] = 3×4 + -2×3 = 6",
"Answer: [[-2,-2],[1,6]]"
] | [[-2,-2],[1,6]] | matrix_multiplication | wrong_entry | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: incorrect — Entry C[1][1] is wrong. The correct value is 6, not 7. The correct result is [[-2,-2],[1,6]]. | Therefore the correct answer is: [[-2,-2],[1,6]] | easy |
Solve the system using Gaussian elimination: -1x + 1y = 2 and -4x - 4y = -16. | [
"Step 1: Write augmented matrix: [-1 1 | 2] and [-4 -4 | -16]",
"Step 2: Multiply row 1 by -4 and row 2 by -1: [4 -4 | -8] and [4 4 | 16]",
"Step 3: Subtract row 2 from row 1 to eliminate x: [0 -8 | -24]",
"Step 4: Solve for y: -8y = -24 → y = 3",
"Step 5: Back-substitute y = 3 into equation 1: -1x + 1×3 ... | None | There is no error. The working shown is correct. | [
"Step 1: Write augmented matrix: [-1 1 | 2] and [-4 -4 | -16]",
"Step 2: Multiply row 1 by -4 and row 2 by -1: [4 -4 | -8] and [4 4 | 16]",
"Step 3: Subtract row 2 from row 1 to eliminate x: [0 -8 | -24]",
"Step 4: Solve for y: -8y = -24 → y = 3",
"Step 5: Back-substitute y = 3 into equation 1: -1x + 1×3 ... | x=1,y=3 | gaussian_elimination | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: x=1,y=3 | medium |
Compute the matrix product A·B where A = [[-3,-3],[2,1]] and B = [[3,-4],[1,2]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: Arithmetic error — C[0][0] = -13 (correct: -12)",
"Step 3: C[0][1] = 6",
"Step 4: C[1][0] = 7",
"Step 5: C[1][1] = -6",
"Answer: [[-13,6],[7,-6]]"
] | Step 2 | Step 2 has an arithmetic error in C[0][0]. The correct value is -12, not -13. The correct result is [[-12,6],[7,-6]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = -3×3 + -3×1 = -12",
"Step 3: C[0][1] = -3×-4 + -3×2 = 6",
"Step 4: C[1][0] = 2×3 + 1×1 = 7",
"Step 5: C[1][1] = 2×-4 + 1×2 = -6",
"Answer: [[-12,6],[7,-6]]"
] | [[-12,6],[7,-6]] | matrix_multiplication | arithmetic | true | Step 1: correct | Step 2: incorrect — Step 2 has an arithmetic error in C[0][0]. The correct value is -12, not -13. The correct result is [[-12,6],[7,-6]]. | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: [[-12,6],[7,-6]] | easy |
Differentiate f(x) = x^4 · x^5 using the product rule. | [
"Step 1: Product rule: d/dx[u·v] = u'v + uv'",
"Step 2: u = x^4, v = x^5 → u' = 4x^3, v' = 5x^4",
"Step 3: u'v + uv' = 4x^3·x^5 + x^4·5x^4",
"Step 4: Index law error: exponents add to 7 instead of 8: 9x^7",
"Answer: f'(x) = 9x^7"
] | Step 4 | Step 4 uses the wrong exponent. x^3·x^5 = x^8 = x^8, not x^7. The correct answer is 9x^8. | [
"Step 1: Product rule: d/dx[u·v] = u'v + uv'",
"Step 2: u = x^4, v = x^5 → u' = 4x^3, v' = 5x^4",
"Step 3: u'v + uv' = 4x^3·x^5 + x^4·5x^4",
"Step 4: Simplify using index laws: 4x^8 + 5x^8 = 9x^8",
"Answer: f'(x) = 9x^8"
] | 9x^8 | differentiation | wrong_exponent | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: incorrect — Step 4 uses the wrong exponent. x^3·x^5 = x^8 = x^8, not x^7. The correct answer is 9x^8. | Therefore the correct answer is: 9x^8 | medium |
Evaluate ∫ from 1 to 2 of x^3 dx | [
"Step 1: Antiderivative error — wrote x^3/4 instead of x^4/4",
"Step 2: Evaluate at upper limit: 2^3/4 = 2",
"Step 3: Evaluate at lower limit: 1^3/4 = (1/4)",
"Step 4: 2 - (1/4) = (7/4)",
"Answer: (7/4)"
] | Step 1 | Step 1 uses the wrong exponent in the antiderivative. ∫x^3 dx = x^4/4, not x^3/4. The correct answer is (15/4). | [
"Step 1: Antiderivative of x^3 is x^4/4",
"Step 2: Evaluate at upper limit 2: 2^4/4 = 4",
"Step 3: Evaluate at lower limit 1: 1^4/4 = (1/4)",
"Step 4: F(2) - F(1) = 4 - (1/4) = (15/4)",
"Answer: (15/4)"
] | (15/4) | integration | wrong_exponent | true | Step 1: incorrect — Step 1 uses the wrong exponent in the antiderivative. ∫x^3 dx = x^4/4, not x^3/4. The correct answer is (15/4). | Step 2: correct | Step 3: correct | Step 4: correct | Therefore the correct answer is: (15/4) | medium |
Differentiate f(x) = -x^4 with respect to x. | [
"Step 1: Apply the power rule: d/dx[ax^n] = an·x^(n-1)",
"Step 2: Multiplication error: -1 × 4 = -2",
"Step 3: Reduce the exponent by 1: 4 - 1 = 3",
"Answer: f'(x) = -2x^3"
] | Step 2 | The multiplication in Step 2 is wrong: -1 × 4 should give -4, but -2 was used. Correct answer: -4x^3. | [
"Step 1: Apply the power rule: d/dx[ax^n] = an·x^(n-1)",
"Step 2: Multiply coefficient by exponent: -1 × 4 = -4",
"Step 3: Reduce the exponent by 1: 4 - 1 = 3",
"Answer: f'(x) = -4x^3"
] | -4x^3 | differentiation | wrong_coefficient | true | Step 1: correct | Step 2: incorrect — The multiplication in Step 2 is wrong: -1 × 4 should give -4, but -2 was used. Correct answer: -4x^3. | Step 3: correct | Therefore the correct answer is: -4x^3 | medium |
Evaluate ∫ from 2 to 4 of x^1 dx | [
"Step 1: Antiderivative of x^1 is x^2/2",
"Step 2: Evaluate at upper limit 4: 4^2/2 = 8",
"Step 3: Evaluate at lower limit 2: 2^2/2 = 2",
"Step 4: Arithmetic error: 8 - 2 = 5",
"Answer: 5"
] | Step 4 | Step 4 has an arithmetic error. 8 - 2 = 6, not 5. The correct answer is 6. | [
"Step 1: Antiderivative of x^1 is x^2/2",
"Step 2: Evaluate at upper limit 4: 4^2/2 = 8",
"Step 3: Evaluate at lower limit 2: 2^2/2 = 2",
"Step 4: F(4) - F(2) = 8 - 2 = 6",
"Answer: 6"
] | 6 | integration | arithmetic | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: incorrect — Step 4 has an arithmetic error. 8 - 2 = 6, not 5. The correct answer is 6. | Therefore the correct answer is: 6 | medium |
Solve the system using Gaussian elimination: 5x + 4y = 21 and 1x + 4y = 1. | [
"Step 1: Write augmented matrix: [5 4 | 21] and [1 4 | 1]",
"Step 2: Multiply row 1 by 1 and row 2 by 5: [5 4 | 21] and [5 20 | 5]",
"Step 3: Subtract row 2 from row 1: [0 -16 | 16]",
"Step 4: Solve for y: -16y = 16 → y = -1",
"Step 5: Back-substitution arithmetic error: 5x + 4×-1 = 21 → x = 3 (should ... | Step 5 | The elimination in Steps 3–4 correctly gives y = -1, but Step 5 makes an arithmetic error in back-substitution, giving x = 3 instead of x = 5. The correct solution is x=5,y=-1. | [
"Step 1: Write augmented matrix: [5 4 | 21] and [1 4 | 1]",
"Step 2: Multiply row 1 by 1 and row 2 by 5: [5 4 | 21] and [5 20 | 5]",
"Step 3: Subtract row 2 from row 1 to eliminate x: [0 -16 | 16]",
"Step 4: Solve for y: -16y = 16 → y = -1",
"Step 5: Back-substitute y = -1 into equation 1: 5x + 4×-1 = 21 ... | x=5,y=-1 | gaussian_elimination | wrong_backsub | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: correct | Step 5: incorrect — The elimination in Steps 3–4 correctly gives y = -1, but Step 5 makes an arithmetic error in back-substitution, giving x = 3 instead of x = 5. The correct solution is x=5,y=-1. | Therefore the correct answer is: x=5,y=-1 | medium |
Find ∫ 3(4x + 3)^3 dx using substitution. | [
"Step 1: Let u = 4x + 3, so du = 4·dx",
"Step 2: Rewrite: (3/4) ∫ u^3 du",
"Step 3: Integrate: (3/16)u^4",
"Step 4: Back-substitute u = 4x + 3 — omits + C",
"Answer: (3/16)(4x + 3)^4"
] | Step 4 | Step 4 omits + C. The correct answer is (3/16)(4x + 3)^4+C. | [
"Step 1: Let u = 4x + 3, so du = 4·dx, meaning dx = du/4",
"Step 2: Rewrite: ∫ 3·u^3·(du/4) = (3/4) ∫ u^3 du",
"Step 3: Integrate: (3/4) · u^4/4 = (3/16)u^4",
"Step 4: Back-substitute u = 4x + 3 and add + C",
"Answer: (3/16)(4x + 3)^4+C"
] | (3/16)(4x + 3)^4+C | integration | forgot_plus_c | true | Step 1: correct | Step 2: correct | Step 3: correct | Step 4: incorrect — Step 4 omits + C. The correct answer is (3/16)(4x + 3)^4+C. | Therefore the correct answer is: (3/16)(4x + 3)^4+C | medium |
Differentiate f(x) = 4x^4 with respect to x. | [
"Step 1: Apply the power rule: d/dx[ax^n] = an·x^(n-1)",
"Step 2: Multiply coefficient by exponent: 4 × 4 = 16",
"Step 3: Reduce the exponent by 1: 4 - 1 = 3",
"Answer: f'(x) = 16x^3"
] | None | There is no error. The working shown is correct. | [
"Step 1: Apply the power rule: d/dx[ax^n] = an·x^(n-1)",
"Step 2: Multiply coefficient by exponent: 4 × 4 = 16",
"Step 3: Reduce the exponent by 1: 4 - 1 = 3",
"Answer: f'(x) = 16x^3"
] | 16x^3 | differentiation | no_error | false | Step 1: correct | Step 2: correct | Step 3: correct | Therefore the correct answer is: 16x^3 | medium |
Compute the matrix product A·B where A = [[2,-4],[1,3]] and B = [[-4,-3],[3,-2]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: Arithmetic error — C[0][0] = -19 (correct: -20)",
"Step 3: C[0][1] = 2",
"Step 4: C[1][0] = 5",
"Step 5: C[1][1] = -9",
"Answer: [[-19,2],[5,-9]]"
] | Step 2 | Step 2 has an arithmetic error in C[0][0]. The correct value is -20, not -19. The correct result is [[-20,2],[5,-9]]. | [
"Step 1: Row × Column rule: C[i][j] = Σ A[i][k]·B[k][j]",
"Step 2: C[0][0] = 2×-4 + -4×3 = -20",
"Step 3: C[0][1] = 2×-3 + -4×-2 = 2",
"Step 4: C[1][0] = 1×-4 + 3×3 = 5",
"Step 5: C[1][1] = 1×-3 + 3×-2 = -9",
"Answer: [[-20,2],[5,-9]]"
] | [[-20,2],[5,-9]] | matrix_multiplication | arithmetic | true | Step 1: correct | Step 2: incorrect — Step 2 has an arithmetic error in C[0][0]. The correct value is -20, not -19. The correct result is [[-20,2],[5,-9]]. | Step 3: correct | Step 4: correct | Step 5: correct | Therefore the correct answer is: [[-20,2],[5,-9]] | easy |
Find the determinant of the matrix [[1,2],[0,2]]. | [
"Step 1: For a 2×2 matrix [[a,b],[c,d]], det(A) = a·d - b·c",
"Step 2: Identify: a=1, b=2, c=0, d=2",
"Step 3: Arithmetic error — a·d = 0 (correct: 2)",
"Step 4: Compute b·c = 2×0 = 0",
"Step 5: det(A) = 0 - 0 = 0",
"Answer: 0"
] | Step 3 | Step 3 has an arithmetic error. 1×2 = 2, not 0. The correct answer is 2. | [
"Step 1: For a 2×2 matrix [[a,b],[c,d]], det(A) = a·d - b·c",
"Step 2: Identify: a=1, b=2, c=0, d=2",
"Step 3: Compute a·d = 1×2 = 2",
"Step 4: Compute b·c = 2×0 = 0",
"Step 5: det(A) = 2 - 0 = 2",
"Answer: 2"
] | 2 | determinant | arithmetic | true | Step 1: correct | Step 2: correct | Step 3: incorrect — Step 3 has an arithmetic error. 1×2 = 2, not 0. The correct answer is 2. | Step 4: correct | Step 5: correct | Therefore the correct answer is: 2 | easy |
Evaluate ∫ from 3 to 5 of x^2 dx | [
"Step 1: Antiderivative error — wrote x^2/3 instead of x^3/3",
"Step 2: Evaluate at upper limit: 5^2/3 = (25/3)",
"Step 3: Evaluate at lower limit: 3^2/3 = 3",
"Step 4: (25/3) - 3 = (16/3)",
"Answer: (16/3)"
] | Step 1 | Step 1 uses the wrong exponent in the antiderivative. ∫x^2 dx = x^3/3, not x^2/3. The correct answer is (98/3). | [
"Step 1: Antiderivative of x^2 is x^3/3",
"Step 2: Evaluate at upper limit 5: 5^3/3 = (125/3)",
"Step 3: Evaluate at lower limit 3: 3^3/3 = 9",
"Step 4: F(5) - F(3) = (125/3) - 9 = (98/3)",
"Answer: (98/3)"
] | (98/3) | integration | wrong_exponent | true | Step 1: incorrect — Step 1 uses the wrong exponent in the antiderivative. ∫x^2 dx = x^3/3, not x^2/3. The correct answer is (98/3). | Step 2: correct | Step 3: correct | Step 4: correct | Therefore the correct answer is: (98/3) | medium |
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