Mathematics 2023 Paper II 50 marks Solve

Q3

(a) Prove that x² + 1 is an irreducible polynomial in Z₃[x]. Further show that the quotient ring $\frac{Z_3[x]}{\langle x^2+1 \rangle}$ is a field of 9 elements. 15 marks (b) Prove that u(x, y) = eˣ(x cos y - y sin y) is harmonic. Find its conjugate harmonic function v(x, y) and express the corresponding analytic function f(z) in terms of z. 15 marks (c) Solve the following linear programming problem by Big M method : Minimize Z = 2x₁ + 3x₂ subject to x₁ + x₂ ≥ 9 x₁ + 2x₂ ≥ 15 2x₁ - 3x₂ ≤ 9 x₁, x₂ ≥ 0 Is the optimal solution unique? Justify your answer. 20 marks

हिंदी में प्रश्न पढ़ें

(a) सिद्ध कीजिए कि x² + 1, Z₃[x] में एक अविभाज्य बहुपद है। यह भी दर्शाइए कि विभाग वलय $\frac{Z_3[x]}{\langle x^2+1 \rangle}$, 9 अवयवों का एक क्षेत्र है। 15 (b) सिद्ध कीजिए कि u(x, y) = eˣ(x cos y - y sin y) प्रसंवादी है। इसका संयुग्मी प्रसंवादी फलन v(x, y) ज्ञात कीजिए तथा संगत विश्लेषिक फलन f(z) को z के पदों में व्यक्त कीजिए। 15 (c) बड़ा M (बिग M) विधि से निम्नलिखित रैखिक प्रोग्रामन समस्या को हल कीजिए : न्यूनतमीकरण कीजिए Z = 2x₁ + 3x₂ बशर्ते कि x₁ + x₂ ≥ 9 x₁ + 2x₂ ≥ 15 2x₁ - 3x₂ ≤ 9 x₁, x₂ ≥ 0 क्या इष्टतम हल अद्वितीय है? अपने उत्तर का तर्क प्रस्तुत कीजिए। 20

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How this answer will be evaluated

Approach

Solve this multi-part problem by proving results in (a) and (b) before applying computational methods in (c). Allocate approximately 30% time to part (a) on irreducibility and field construction, 30% to part (b) on harmonic functions and analytic construction, and 40% to part (c) on the Big M method LP solution including uniqueness analysis. Structure as: direct proofs for (a)-(b), systematic tableau operations for (c), with clear final verification of all claims.

Key points expected

  • For (a): Prove x²+1 has no roots in Z₃ (hence irreducible), then apply the theorem that F[x]/⟨p(x)⟩ is a field when p(x) is irreducible, counting 9 elements explicitly
  • For (b): Verify uₓₓ + uᵧᵧ = 0 using product rule differentiation, then find v(x,y) via Cauchy-Riemann equations or Milne-Thomson method, finally express f(z) = z·e^z
  • For (c): Convert minimization to standard form using surplus and artificial variables with Big M penalty, construct initial simplex tableau
  • For (c): Execute simplex iterations correctly, identifying entering and leaving variables at each pivot
  • For (c): State optimal solution with Z_min = 15 at (x₁, x₂) = (6, 3), and justify uniqueness by checking no alternative optimal solutions exist in final tableau

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10Correctly identifies Z₃ = {0,1,2}, sets up proper quotient ring structure for (a); verifies harmonicity setup with correct partial derivatives for (b); accurately converts all three constraints to standard form with correct surplus, slack and artificial variables for (c)Minor errors in ring element enumeration or constraint conversion; one sign error in standard form or missing one artificial variable identificationFundamental misunderstanding of Z₃ elements, incorrect ring notation, or major errors in LP standard form conversion affecting entire solution
Method choice20%10Uses root test (not factorization) for irreducibility in (a); applies Cauchy-Riemann equations systematically for (b); correctly implements Big M method with appropriate M value handling and simplex pivot rules for (c)Correct method chosen but inefficient approach (e.g., attempting full factorization in (a)); minor deviations in CR equation application; acceptable but messy simplex executionAttempts invalid methods (e.g., rational root test over Q for (a)); uses integration by parts unnecessarily for harmonic conjugate; applies two-phase method instead of Big M or incorrect pivot selection
Computation accuracy20%10All arithmetic in Z₃ verified (0²+1=1, 1²+1=2, 2²+1=2≠0); exact partial derivatives uₓₓ = eˣ(x cos y - y sin y + 2 cos y) etc.; simplex tableaus with correct ratios and no arithmetic errors leading to precise optimal valuesMinor computational slips (e.g., 2²=1 mod 3 error caught later); one derivative component slightly off; one or two tableau arithmetic errors that don't propagate catastrophicallyPersistent modular arithmetic errors; incorrect Laplacian calculation; multiple simplex errors leading to wrong optimal solution or infeasible final answer
Step justification20%10Explicitly cites: 'degree 2 irreducible implies field' theorem for (a); harmonic function definition and CR equations for (b); optimality conditions (all Δⱼ ≥ 0) and uniqueness criterion (non-basic variables have positive reduced costs) for (c)States key theorems but without full hypotheses; mentions CR equations without showing uₓ = vᵧ verification; claims uniqueness without explicit reduced cost checkMissing critical justifications: no theorem cited for field structure, no verification of analyticity, no optimality/uniqueness justification in LP—merely presents final numbers
Final answer & units20%10Clear final answers: explicit 9-element field listing or structure for (a); v(x,y) = eˣ(x sin y + y cos y) and f(z) = z·e^z for (b); Z_min = 15, x₁=6, x₂=3 with definitive uniqueness proof for (c)Correct final answers but poorly formatted or missing one component (e.g., f(z) not simplified); correct optimal values but ambiguous uniqueness conclusionMissing or incorrect final answers; wrong field order, incorrect harmonic conjugate, or wrong LP solution; no attempt at uniqueness analysis despite explicit question demand

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