first working version
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howard
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{
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"examId": "linear-algebra-medium-v1",
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"title": "Linear Algebra - Medium Level (Computational)",
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"subject": "Linear Algebra",
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"difficulty": "intermediate",
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"durationMinutes": 50,
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"passingScore": 70,
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"sections": [
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{
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"id": "vectors",
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"title": "Vector Operations & Computations",
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"questions": [
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{
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"id": "q1",
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"type": "single_choice",
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"prompt": "Given vectors u = [3, -2, 1] and v = [1, 4, -2], compute u · v (dot product).",
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"choices": [
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{ "key": "A", "text": "-7" },
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{ "key": "B", "text": "-3" },
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{ "key": "C", "text": "3" },
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{ "key": "D", "text": "7" }
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],
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"answer": "B",
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"allowIDK": true,
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"points": 10
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},
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{
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"id": "q2",
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"type": "single_choice",
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"prompt": "Calculate the magnitude (length) of vector w = [3, 4]. Give ||w|| = ?",
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"choices": [
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{ "key": "A", "text": "5" },
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{ "key": "B", "text": "7" },
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{ "key": "C", "text": "√7" },
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{ "key": "D", "text": "25" }
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],
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"answer": "A",
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"allowIDK": true,
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"points": 10
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},
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{
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"id": "q3",
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"type": "single_choice",
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"prompt": "Find the unit vector in the direction of v = [6, 8]. Which is correct?",
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"choices": [
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{ "key": "A", "text": "[3/5, 4/5]" },
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{ "key": "B", "text": "[6, 8]" },
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{ "key": "C", "text": "[1, 1]" },
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{ "key": "D", "text": "[0.6, 0.8]" }
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],
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"answer": "A",
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"allowIDK": true,
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"points": 10
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}
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]
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},
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{
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"id": "matrices",
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"title": "Matrix Operations & Calculations",
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"questions": [
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{
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"id": "q4",
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"type": "single_choice",
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"prompt": "Compute AB where A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 3]]. What is the element in position (1,1) of AB?",
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"choices": [
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{ "key": "A", "text": "2" },
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{ "key": "B", "text": "4" },
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{ "key": "C", "text": "6" },
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{ "key": "D", "text": "8" }
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],
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"answer": "B",
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"allowIDK": true,
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"points": 10
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},
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{
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"id": "q5",
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"type": "single_choice",
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"prompt": "Calculate the determinant of A = [[2, 1], [4, 3]]. det(A) = ?",
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"choices": [
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{ "key": "A", "text": "1" },
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{ "key": "B", "text": "2" },
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{ "key": "C", "text": "6" },
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{ "key": "D", "text": "10" }
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],
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"answer": "B",
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"allowIDK": true,
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"points": 10
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},
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{
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"id": "q6",
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"type": "single_choice",
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"prompt": "Given matrix A = [[1, 2], [3, 4]], what is the transpose A^T?",
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"choices": [
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{ "key": "A", "text": "[[1, 3], [2, 4]]" },
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{ "key": "B", "text": "[[1, 2], [3, 4]]" },
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{ "key": "C", "text": "[[4, 3], [2, 1]]" },
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{ "key": "D", "text": "[[1, 0], [0, 1]]" }
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],
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"answer": "A",
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"allowIDK": true,
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"points": 10
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}
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]
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},
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{
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"id": "systems",
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"title": "Linear Systems & Solutions",
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"questions": [
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{
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"id": "q7",
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"type": "single_choice",
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"prompt": "Solve the system: x + 2y = 7 and 2x - y = 4. What is x?",
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"choices": [
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{ "key": "A", "text": "x = 1" },
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{ "key": "B", "text": "x = 2" },
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{ "key": "C", "text": "x = 3" },
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{ "key": "D", "text": "x = 4" }
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],
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"answer": "C",
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"allowIDK": true,
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"points": 15
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},
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{
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"id": "q8",
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"type": "single_choice",
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"prompt": "For the inverse of A = [[2, 1], [4, 3]], what is A^(-1)?",
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"choices": [
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{ "key": "A", "text": "[[3/2, -1/2], [-2, 1]]" },
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{ "key": "B", "text": "[[3, -1], [-4, 2]]" },
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{ "key": "C", "text": "[[1/2, 1/2], [1/4, 1/3]]" },
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{ "key": "D", "text": "[[3/2, 1/2], [2, 1]]" }
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],
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"answer": "A",
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"allowIDK": true,
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"points": 15
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}
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]
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},
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{
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"id": "eigenvalues",
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"title": "Eigenvalues & Special Computations",
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"questions": [
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{
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"id": "q9",
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"type": "single_choice",
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"prompt": "Find the eigenvalues of matrix A = [[3, 1], [1, 3]]. What are they?",
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"choices": [
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{ "key": "A", "text": "λ₁ = 2, λ₂ = 4" },
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{ "key": "B", "text": "λ₁ = 3, λ₂ = 3" },
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{ "key": "C", "text": "λ₁ = 4, λ₂ = 2" },
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{ "key": "D", "text": "λ₁ = 1, λ₂ = 5" }
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],
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"answer": "C",
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"allowIDK": true,
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"points": 15
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},
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{
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"id": "q10",
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"type": "single_choice",
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"prompt": "Given matrix A = [[1, 0, 0], [0, 2, 0], [0, 0, 3]], what is the trace of A (sum of diagonal elements)?",
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"choices": [
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{ "key": "A", "text": "3" },
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{ "key": "B", "text": "5" },
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{ "key": "C", "text": "6" },
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{ "key": "D", "text": "0" }
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],
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"answer": "C",
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"allowIDK": true,
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"points": 5
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}
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]
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}
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],
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"metadata": {
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"version": "1.0.0",
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"createdAt": "2025-10-21T12:30:00Z",
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"createdBy": "system",
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"tags": ["linear-algebra", "mathematics", "computational", "intermediate", "auto-graded"],
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"description": "Medium difficulty Linear Algebra exam requiring calculations for vectors, matrices, linear systems, and eigenvalues.",
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"calculationNotes": "Students should show work for calculations. Answers are final results only for auto-grading."
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}
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}
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