Mathematics 2023 Paper II 50 marks Prove

Q2

(a) Prove that a non-commutative group of order 2p, where p is an odd prime, must have a subgroup of order p. (15 marks) (b) Using the method of Lagrange's multipliers, find the minimum and maximum distances of the point P(2, 6, 3) from the sphere x² + y² + z² = 4. (15 marks) (c) Evaluate $\int_{0}^{2\pi} \frac{\cos 2\theta}{5+4\cos\theta} d\theta$ using contour integration. (20 marks)

हिंदी में प्रश्न पढ़ें

(a) सिद्ध कीजिए कि 2p कोटि के एक अक्रमविनिमेय समूह, जहाँ p एक विषम अभाज्य संख्या है, में p कोटि का एक उपसमूह होना आवश्यक है। (15 अंक) (b) लग्रांज गुणक विधि के उपयोग से बिंदु P(2, 6, 3) की गोले x² + y² + z² = 4 से न्यूनतम तथा अधिकतम दूरियाँ ज्ञात कीजिए। (15 अंक) (c) कंटूर समाकलन का उपयोग कर $\int_{0}^{2\pi} \frac{\cos 2\theta}{5+4\cos\theta} d\theta$ का मान ज्ञात कीजिए। (20 अंक)

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Directive word: Prove

This question asks you to prove. The directive word signals the depth of analysis expected, the structure of your answer, and the weight of evidence you must bring.

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How this answer will be evaluated

Approach

Begin with a clear statement of what is to be proved for part (a), then solve for (b) and (c). Allocate approximately 30% time to part (a) (group theory proof), 30% to part (b) (Lagrange multipliers), and 40% to part (c) (contour integration, highest marks). Structure as: (a) statement → application of Sylow/Cauchy theorems → conclusion; (b) setup of distance function and constraint → Lagrange equations → solving critical points; (c) contour setup → residue calculation → final evaluation. Conclude with boxed final answers for each part.

Key points expected

  • Part (a): Application of Cauchy's theorem or Sylow theorems to establish existence of element of order p, hence cyclic subgroup of order p
  • Part (a): Explicit use of non-commutativity to rule out cyclic group of order 2p, ensuring the subgroup of order p is proper
  • Part (b): Correct formulation of distance squared function D = (x-2)² + (y-6)² + (z-3)² with constraint g = x² + y² + z² - 4 = 0
  • Part (b): Setting up and solving the system ∇D = λ∇g with geometric interpretation that extrema occur along line through origin and P
  • Part (c): Substitution z = e^(iθ) to convert real integral to complex contour integral over unit circle |z|=1
  • Part (c): Correct identification of poles inside unit circle (at z = -1/2) and calculation of residue
  • Part (c): Evaluation of contour integral using residue theorem and extraction of real part to obtain final answer

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10For (a): correctly identifies group order structure and applies appropriate theorem; for (b): sets up distance function and constraint accurately; for (c): proper substitution z = e^(iθ) with correct dz and limits transformationMinor errors in one part's setup, such as incorrect distance formula or wrong contour parameterization, but other parts correctFundamental errors in multiple setups, such as confusing distance with squared distance, wrong theorem application, or incorrect contour choice
Method choice20%10For (a): elegant use of Cauchy's theorem or Sylow's first theorem; for (b): Lagrange multipliers rather than geometric shortcut; for (c): residue theorem with correct pole selectionCorrect methods chosen but with suboptimal approach in one part, such as using direct geometric argument for (b) instead of requested Lagrange methodWrong methods entirely, such as attempting direct integration for (c) or ignoring the constraint in (b)
Computation accuracy20%10All algebraic manipulations correct: solving Lagrange equations yields x = 2λ/(1-λ) etc., finding λ = ±2/7; contour integral reduces to correct rational function with accurate residue calculationMinor arithmetic slips in one part, such as sign error in residue or one incorrect critical point coordinate, but overall structure recoverableMajor computational errors across multiple parts, such as wrong residues, incorrect distance values, or failure to solve the Lagrange system
Step justification20%10Clear justification for: why non-commutativity matters in (a); why maximum and minimum correspond to specific λ values in (b); why only certain poles are enclosed in (c) with rigorous residue calculationSome steps justified but gaps in logical flow, such as asserting existence of subgroup without citing theorem, or stating residue values without showing calculationUnjustified leaps, missing logical connections, or 'magic' steps with no mathematical basis provided
Final answer & units20%10Precise final answers: (a) complete proof with explicit subgroup identification; (b) minimum distance = 5, maximum distance = 9 (or √25 and √81); (c) integral value = π/6; all properly boxed and dimensionally consistentCorrect answers but poorly presented, missing units where relevant, or incomplete proof statement for (a)Wrong final answers, missing answers for one or more parts, or answers without any derivation support

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