Q1
(a) Let G be a finite group of order mn, where m and n are prime numbers with m > n. Show that G has at most one subgroup of order m. 10 marks (b) If w = f(z) is an analytic function of z, then show that $$(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}) \log |f'(z)| = 0.$$ 10 marks (c) Test the convergence of $$\int\limits_{0}^{2} \frac{\log x}{\sqrt{(2-x)}} dx$$ . 10 marks (d) If φ and ψ are functions of x and y satisfying Laplace equation, then show that f(z) = p + iq, i = √−1 is an analytic function, where p = $$\frac{\partial \phi}{\partial y} - \frac{\partial \psi}{\partial x}$$ and q = $$\frac{\partial \phi}{\partial x} + \frac{\partial \psi}{\partial y}$$ . 10 marks (e) Use two phase method to solve the following linear programming problem : Maximize z = x₁ + 2x₂ subject to x₁ - x₂ ≥ 3 2x₁ + x₂ ≤ 10 x₁, x₂ ≥ 0 10 marks
हिंदी में प्रश्न पढ़ें
(a) मान लीजिए कि कोटि mn का एक परिमित समूह G है, जहाँ m और n, (m > n) अभाज्य संख्याएँ हैं । दर्शाइए कि G का कोटि m का अधिक-से-अधिक एक उपसमूह है । 10 अंक (b) यदि w = f(z), z का एक विसलेषिक फलन है, तब दर्शाइए कि $$(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}) \log |f'(z)| = 0$$ है । 10 अंक (c) $$\int\limits_{0}^{2} \frac{\log x}{\sqrt{(2-x)}} dx$$ के अभिसरण का परीक्षण कीजिए । 10 अंक (d) यदि x तथा y के फलन φ और ψ लाप्लास समीकरण को सन्तुष्ट करते हैं, तो दर्शाइए कि f(z) = p + iq, i = √−1 एक विसलेषिक फलन है, जहाँ p = $$\frac{\partial \phi}{\partial y} - \frac{\partial \psi}{\partial x}$$ तथा q = $$\frac{\partial \phi}{\partial x} + \frac{\partial \psi}{\partial y}$$ हैं । 10 अंक (e) निम्नलिखित रैखिक प्रोग्रामन समस्या को हल करने के लिए द्विचरण विधि का उपयोग कीजिए : अधिकतमीकरण कीजिए z = x₁ + 2x₂ बशर्ते कि x₁ - x₂ ≥ 3 2x₁ + x₂ ≤ 10 x₁, x₂ ≥ 0 10 अंक
Directive word: Prove
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How this answer will be evaluated
Approach
Prove the five mathematical statements systematically, allocating approximately 20% time to each sub-part since all carry equal marks. For (a), apply Sylow's theorems or Lagrange's theorem with counting arguments; for (b), use Cauchy-Riemann equations and harmonic function properties; for (c), apply limit comparison test or Dirichlet's test for improper integrals; for (d), verify Cauchy-Riemann equations using the given harmonic functions; for (e), execute Phase I (artificial variables) and Phase II (simplex method) with clear tableau presentations. Structure as five distinct proofs with clear headings, showing all logical steps without skipping to conclusions.
Key points expected
- For (a): Apply Sylow's third theorem or direct counting argument using Lagrange's theorem to show uniqueness of subgroup of order m, noting that number of such subgroups divides n and is ≡ 1 (mod m), forcing exactly one subgroup
- For (b): Express log|f'(z)| in terms of real and imaginary parts, apply Laplacian operator, and use analyticity of f(z) (Cauchy-Riemann equations) to show the sum of second partial derivatives vanishes identically
- For (c): Identify singularities at x=0 and x=2, split integral, apply limit comparison test with appropriate standard integrals (e.g., 1/x^α near 0 and 1/(2-x)^β near 2) to establish convergence
- For (d): Verify that p and q satisfy Cauchy-Riemann equations (∂p/∂x = ∂q/∂y and ∂p/∂y = -∂q/∂x) using the given Laplace equations for φ and ψ, and equality of mixed partials
- For (e): Phase I: Introduce surplus and slack variables, add artificial variables for ≥ constraint, minimize sum of artificials; Phase II: Remove artificials, use final Phase I tableau to maximize original objective, obtaining optimal solution x₁=13/3, x₂=4/3, z=7
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies all hypotheses for each sub-part: group order conditions for (a), analyticity of f(z) for (b), improper integral type and singular points for (c), harmonic function properties for (d), and standard form conversion with artificial/surplus variables for (e) | Identifies most hypotheses but misses subtle conditions like m>n implication in (a), or misidentifies singularity behavior in (c), or makes minor errors in constraint conversion for (e) | Fundamental setup errors: wrong theorem choice for (a), fails to recognize f'(z)≠0 requirement in (b), misses one or both singularities in (c), ignores Laplace equation usage in (d), or wrong inequality direction in (e) |
| Method choice | 20% | 10 | Selects optimal methods: Sylow theorems or counting argument for (a), Laplacian in complex form or direct CR verification for (b), appropriate convergence tests for (c), direct CR verification for (d), and correct two-phase simplex execution for (e) | Uses correct but suboptimal methods (e.g., brute force calculation in (b), comparison test with weaker bounds in (c), or big-M method instead of two-phase for (e)) | Wrong methods entirely: attempts direct subgroup listing for large m,n in (a), uses real analysis incorrectly in (b), applies wrong convergence test in (c), or uses graphical method for (e) |
| Computation accuracy | 20% | 10 | All calculations precise: correct arithmetic in Sylow counting for (a), accurate partial derivatives and Laplacian computation in (b), correct limit evaluations in (c), error-free partial derivative chain in (d), and accurate simplex pivot operations yielding x₁=13/3, x₂=4/3, z=7 for (e) | Minor computational slips: one arithmetic error in derivative calculation, or single pivot error in simplex, or incorrect limit value but correct convergence conclusion | Major computational errors: wrong derivative formulas, incorrect limit evaluations leading to wrong convergence conclusion, or multiple pivot errors giving infeasible or suboptimal solution |
| Step justification | 20% | 10 | Every logical step explicitly justified: cites Lagrange's theorem and Sylow theorems with theorem numbers for (a), invokes harmonicity and CR equations with clear logical flow for (b), states comparison functions and limit existence for (c), shows equality of mixed partials for (d), and explains pivot selection and optimality conditions for (e) | Most steps justified but some gaps: assumes without stating that harmonic functions are infinitely differentiable, or skips limit existence verification, or omits optimality condition check in simplex | Major logical gaps: asserts conclusions without justification, circular reasoning, or presents calculations as proof without connecting to what was required |
| Final answer & units | 20% | 10 | Clear precise conclusions: 'at most one subgroup' explicitly stated for (a), Laplacian equals zero clearly shown for (b), 'convergent' or 'divergent' with value if computable for (c), analyticity of f(z) verified for (d), and complete optimal solution with z_max=7, x₁=13/3, x₂=4/3 for (e) | Correct conclusions but poorly presented: buried in text, or missing one component (e.g., z-value in (e)), or convergence stated without specifying absolute/conditional | Missing or wrong conclusions: fails to state final result, or states opposite of what was proved, or gives numerical answer without verification of constraints in (e) |
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