45582e5f59
- 68 exercises from UBA FCE chapters 1-3 - Step-by-step solutions with KaTeX rendering - Theory panels (26 topics) expandable per exercise - Matrix builder (2x2/3x3/4x4) with 7 operations - System solver (Gauss, Gauss-Jordan, Cramer, Rouché-Frobenius) - Glassmorphism UI with dark mode - Canvas particle background - ARIA accessibility (keyboard nav, screen reader) - Zero build step - open index.html directly
150 lines
8.9 KiB
JSON
150 lines
8.9 KiB
JSON
[
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{
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"id": "cap03-01",
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"chapter": 3,
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"topic": "systems",
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"subtopic": "matrix-form",
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"theoryKey": "systems-theory",
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"difficulty": "basic",
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"statement": "Escribir en forma matricial los siguientes sistemas: a) x + y = 5, x - y = 1; b) 2x + y - z = 3, x - y + 2z = 1, 3x + 2y + z = 7; c) x + y + z + t = 4, 2x - y + z - t = 1.",
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"hint": "Ax = b donde A es la matriz de coeficientes",
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"answerType": "expression",
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"answer": { "value": "Formas matriciales escritas", "latex": "a)\\begin{pmatrix}1&1\\\\1&-1\\end{pmatrix}\\begin{pmatrix}x\\\\y\\end{pmatrix}=\\begin{pmatrix}5\\\\1\\end{pmatrix}" },
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"solutionSteps": [
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{ "desc": "a) Sistema 2×2", "expression": "\\begin{pmatrix}1&1\\\\1&-1\\end{pmatrix}\\begin{pmatrix}x\\\\y\\end{pmatrix}=\\begin{pmatrix}5\\\\1\\end{pmatrix}" },
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{ "desc": "b) Sistema 3×3", "expression": "\\begin{pmatrix}2&1&-1\\\\1&-1&2\\\\3&2&1\\end{pmatrix}\\begin{pmatrix}x\\\\y\\\\z\\end{pmatrix}=\\begin{pmatrix}3\\\\1\\\\7\\end{pmatrix}" },
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{ "desc": "c) Sistema 2×4", "expression": "\\begin{pmatrix}1&1&1&1\\\\2&-1&1&-1\\end{pmatrix}\\begin{pmatrix}x\\\\y\\\\z\\\\t\\end{pmatrix}=\\begin{pmatrix}4\\\\1\\end{pmatrix}" }
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]
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},
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{
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"id": "cap03-02",
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"chapter": 3,
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"topic": "systems",
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"subtopic": "classify-solve",
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"theoryKey": "systems-theory",
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"difficulty": "intermediate",
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"statement": "Clasificar los siguientes sistemas y resolver cuando sea posible: a) x + 2y = 5, 3x - y = 1; b) x + y + z = 3, x - y + z = 1, x + y - z = 1; c) x + y + z = 1, 2x + 2y + 2z = 3, x - y + z = 0.",
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"hint": "Usar Rouche-Frobenius para clasificar",
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"answerType": "expression",
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"answer": { "value": "a)CD:(1,2) b)CD:(1,1,1) c)SI", "latex": "a)\\,CD:\\,(1,2),\\quad b)\\,CD:\\,(1,1,1),\\quad c)\\,SI" },
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"solutionSteps": [
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{ "desc": "a) det = -1 - 6 = -7 ≠ 0 → CD", "expression": "\\det\\begin{pmatrix}1&2\\\\3&-1\\end{pmatrix}=-7\\neq 0 \\Rightarrow CD" },
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{ "desc": "Solución a)", "expression": "x=1,\\; y=2" },
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{ "desc": "b) Sistema 3×3", "expression": "\\det\\begin{pmatrix}1&1&1\\\\1&-1&1\\\\1&1&-1\\end{pmatrix}=1(1-1)-1(-1-1)+1(1+1)=4\\neq 0" },
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{ "desc": "Solución b)", "expression": "x=1,\\; y=1,\\; z=1" },
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{ "desc": "c) F₂ = 2F₁ pero 3 ≠ 2 → SI", "expression": "E_2 = 2E_1 \\Rightarrow 2(1)=2\\neq 3 \\Rightarrow \\text{Incompatible}" }
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]
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},
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{
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"id": "cap03-03",
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"chapter": 3,
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"topic": "systems",
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"subtopic": "gauss",
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"theoryKey": "gauss-method",
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"difficulty": "intermediate",
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"statement": "Resolver por eliminación de Gauss: a) x + y + z = 6, 2x - y + z = 3, x + 2y - z = 5; b) x - y + z = 2, 2x + y - z = 3, 3x + 0y + 0z = 5.",
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"hint": "F₂ → F₂ - 2F₁, F₃ → F₃ - F₁, etc.",
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"answerType": "expression",
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"answer": { "value": "a)(2;1;3) b)(5/3;1/3;2/3)", "latex": "a)\\,(2;1;3),\\quad b)\\,(5/3;\\;-2/3;\\;-1/3)" },
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"solutionSteps": [
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{ "desc": "a) Matriz aumentada", "expression": "\\begin{pmatrix}1&1&1&|&6\\\\2&-1&1&|&3\\\\1&2&-1&|&5\\end{pmatrix}" },
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{ "desc": "F₂ → F₂ - 2F₁", "expression": "\\begin{pmatrix}1&1&1&|&6\\\\0&-3&-1&|&-9\\\\0&1&-2&|&-1\\end{pmatrix}" },
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{ "desc": "F₃ → F₃ + F₂/3", "expression": "\\begin{pmatrix}1&1&1&|&6\\\\0&-3&-1&|&-9\\\\0&0&-7/3&|&-4\\end{pmatrix}" },
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{ "desc": "Sustitución regresiva", "expression": "z = 3,\\; y = 1,\\; x = 2" },
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{ "desc": "b) 3x = 5 → x = 5/3", "expression": "3x = 5 \\Rightarrow x = \\frac{5}{3}" },
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{ "desc": "Sustituir", "expression": "y = \\frac{1}{3},\\; z = \\frac{2}{3}" }
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]
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},
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{
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"id": "cap03-04",
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"chapter": 3,
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"topic": "systems",
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"subtopic": "gauss-jordan",
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"theoryKey": "gauss-jordan",
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"difficulty": "intermediate",
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"statement": "Resolver por Gauss-Jordan: a) 2x + y = 5, x + 3y = 10; b) x + y + z = 3, 2x - y + z = 2, x + 2y - z = 4.",
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"hint": "Reducir a forma reducida por filas (identidad)",
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"answerType": "expression",
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"answer": { "value": "a)(1;3) b)(2;1;0)", "latex": "a)\\,(1;3),\\quad b)\\,(2;1;0)" },
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"solutionSteps": [
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{ "desc": "a) Matriz aumentada", "expression": "\\begin{pmatrix}2&1&|&5\\\\1&3&|&10\\end{pmatrix}" },
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{ "desc": "F₁ ↔ F₂, F₂ → F₂ - 2F₁", "expression": "\\begin{pmatrix}1&3&|&10\\\\0&-5&|&-15\\end{pmatrix}" },
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{ "desc": "F₂ → F₂/(-5), F₁ → F₁ - 3F₂", "expression": "\\begin{pmatrix}1&0&|&1\\\\0&1&|&3\\end{pmatrix}" },
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{ "desc": "Solución: x=1, y=3", "expression": "x = 1,\\; y = 3" },
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{ "desc": "b) Gauss-Jordan en 3×3", "expression": "x = 2,\\; y = 1,\\; z = 0" }
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]
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},
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{
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"id": "cap03-05",
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"chapter": 3,
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"topic": "systems",
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"subtopic": "cramer",
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"theoryKey": "cramer-rule",
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"difficulty": "intermediate",
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"statement": "Resolver por la regla de Cramer: a) 3x + 2y = 12, x - y = 1; b) x + y + z = 6, x - y + z = 2, x + y - z = 0.",
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"hint": "x = Δ₁/Δ, y = Δ₂/Δ, etc.",
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"answerType": "expression",
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"answer": { "value": "a)(2.8;1.8) b)(1;3;2)", "latex": "a)\\,(\\frac{14}{5};\\;\\frac{9}{5}),\\quad b)\\,(1;\\;3;\\;2)" },
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"solutionSteps": [
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{ "desc": "a) Δ = (3)(-1)-(2)(1) = -5", "expression": "\\Delta = -5" },
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{ "desc": "Δ₁ = (12)(-1)-(2)(1) = -14", "expression": "\\Delta_1 = -14" },
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{ "desc": "Δ₂ = (3)(1)-(12)(1) = -9", "expression": "\\Delta_2 = -9" },
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{ "desc": "x = 14/5, y = 9/5", "expression": "x = \\frac{14}{5},\\; y = \\frac{9}{5}" },
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{ "desc": "b) Δ = det coeficientes", "expression": "\\Delta = \\begin{vmatrix}1&1&1\\\\1&-1&1\\\\1&1&-1\\end{vmatrix} = 1(1-1)-1(-1-1)+1(1+1) = 4" },
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{ "desc": "Δ₁, Δ₂, Δ₃", "expression": "x = \\frac{4}{4}=1,\\; y = \\frac{12}{4}=3,\\; z = \\frac{8}{4}=2" }
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]
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},
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{
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"id": "cap03-06",
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"chapter": 3,
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"topic": "systems",
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"subtopic": "rouche-frobenius",
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"theoryKey": "rouche-frobenius",
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"difficulty": "advanced",
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"statement": "Aplicar el teorema de Rouche-Frobenius para clasificar y resolver: a) x + y + z = 1, x - y + z = 0, 2x + 0y + 2z = 2; b) x + 2y + 3z = 1, 2x + 4y + 6z = 2, x + y + z = 1; c) x + y + z = 1, x + y + z = 2, 2x + 2y + 2z = 3.",
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"hint": "Calcular rg(A) y rg(A|b), comparar",
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"answerType": "expression",
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"answer": { "value": "a)CI b)CI c)SI", "latex": "a)\\,CI,\\quad b)\\,CI,\\quad c)\\,SI" },
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"solutionSteps": [
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{ "desc": "a) F₃ = F₁ + F₂ → rg(A) = 2, rg(A|b) = 2", "expression": "\\text{rg}(A)=\\text{rg}(A|b)=2 < 3 \\Rightarrow CI" },
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{ "desc": "b) F₂ = 2F₁ → rg(A) = 2, rg(A|b) = 2", "expression": "\\text{rg}(A)=\\text{rg}(A|b)=2 < 3 \\Rightarrow CI" },
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{ "desc": "c) E₁: x+y+z=1, E₂: x+y+z=2 (contradicción)", "expression": "1 \\neq 2 \\Rightarrow \\text{rg}(A)=1,\\;\\text{rg}(A|b)=2 \\Rightarrow SI" }
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]
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},
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{
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"id": "cap03-07",
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"chapter": 3,
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"topic": "systems",
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"subtopic": "homogeneous",
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"theoryKey": "homogeneous",
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"difficulty": "intermediate",
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"statement": "Resolver los siguientes sistemas homogéneos: a) x + y = 0, x - y = 0; b) x + 2y - z = 0, 2x + 4y - 2z = 0, 3x + 6y - 3z = 0; c) x + y + z = 0, 2x - y + z = 0, x + 2y - z = 0.",
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"hint": "Ax = 0 siempre tiene solución trivial. No trivial ↔ rg(A) < n",
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"answerType": "expression",
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"answer": { "value": "a)trivial:(0,0) b)CI c)trivial:(0,0,0)", "latex": "a)\\,(0;0),\\quad b)\\,CI,\\quad c)\\,(0;0;0)" },
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"solutionSteps": [
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{ "desc": "a) det = -2 ≠ 0 → solución trivial x=y=0", "expression": "\\det = -2 \\neq 0 \\Rightarrow x=y=0" },
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{ "desc": "b) Todas las ecuaciones proporcionales → rg=1 < 3 → CI", "expression": "E_2=2E_1,\\;E_3=3E_1 \\Rightarrow \\text{rg}=1,\\;\\dim(\\ker)=2" },
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{ "desc": "Solución paramétrica b)", "expression": "(x,y,z) = \\alpha(-2,1,0) + \\beta(1,0,1)" },
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{ "desc": "c) det ≠ 0 → trivial", "expression": "\\det = \\begin{vmatrix}1&1&1\\\\2&-1&1\\\\1&2&-1\\end{vmatrix} = 7 \\neq 0 \\Rightarrow (0;0;0)" }
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]
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},
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{
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"id": "cap03-08",
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"chapter": 3,
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"topic": "systems",
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"subtopic": "parameter",
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"theoryKey": "rouche-frobenius",
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"difficulty": "advanced",
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"statement": "Determinar el valor de k para que el sistema sea: a) Compatible determinado, b) Compatible indeterminado, c) Incompatible. Sistema: x + y + z = 1, 2x - y + z = k, x + 2y - z = 3.",
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"hint": "Calcular det de coeficientes. Si ≠ 0 → CD. Si = 0 → analizar según k.",
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"answerType": "expression",
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"answer": { "value": "k≠4:CD, k=4:CI, nunca SI", "latex": "k \\neq 4: CD,\\; k = 4: CI,\\; \\text{nunca SI}" },
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"solutionSteps": [
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{ "desc": "Calcular det(A)", "expression": "\\det(A) = \\begin{vmatrix}1&1&1\\\\2&-1&1\\\\1&2&-1\\end{vmatrix} = 1(1-2)-1(-2-1)+1(4+1) = -1+3+5 = 7" },
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{ "desc": "Como det(A) = 7 ≠ 0 siempre → siempre CD", "expression": "\\det(A) = 7 \\neq 0 \\Rightarrow \\text{Siempre CD para todo } k" },
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{ "desc": "Solución para cada k", "expression": "x = \\frac{7-2k}{7},\\; y = \\frac{k+2}{7},\\; z = \\frac{k+4}{7}" }
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]
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}
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]
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