Q4
(a) Let R be a field of real numbers and S, the field of all those polynomials f(x) ∈ R[x] such that f(0) = 0 = f(1). Prove that S is an ideal of R[x]. Is the residue class ring R[x]/S an integral domain? Give justification for your answer. (15 marks) (b) Test for convergence or divergence of the series x + 2²x²/2! + 3³x³/3! + 4⁴x⁴/4! + 5⁵x⁵/5! + ... (x > 0) (15 marks) (c) Find the initial basic feasible solution of the following transportation problem by Vogel's approximation method and use it to find the optimal solution and the transportation cost of the problem : Destination A B C D S₁ 21 16 25 13 11 Source S₂ 17 18 14 23 13 Availability S₃ 32 27 18 41 19 Requirement 6 10 12 15 43 (20 marks)
हिंदी में प्रश्न पढ़ें
(a) मान लीजिये कि R वास्तविक संख्याओं का एक क्षेत्र है तथा S, उन सभी बहुपदों f(x) ∈ R[x], जिनके लिये f(0) = 0 = f(1) है, का क्षेत्र है। सिद्ध कीजिये कि S, R[x] की एक गुणजावली है। क्या अवशेष वर्ग वलय R[x]/S एक पूर्णांकीय प्रांत है? अपने उत्तर का स्पष्टीकरण दीजिये। (15 अंक) (b) श्रेणी x + 2²x²/2! + 3³x³/3! + 4⁴x⁴/4! + 5⁵x⁵/5! + ... (x > 0) के अभिसरण या अपसरण का परीक्षण कीजिये। (15 अंक) (c) वोगेल की संविकलन विधि से निम्नलिखित परिवहन समस्या का आरंभिक आधारी सुसंगत हल ज्ञात कीजिये। इस हल का उपयोग कर समस्या का इष्टतम हल एवं परिवहन लागत ज्ञात कीजिये : गंतव्य A B C D S₁ 21 16 25 13 11 S₂ 17 18 14 23 13 S₃ 32 27 18 41 19 मांग 6 10 12 15 43 उद्गम प्राप्यता (20 अंक)
Directive word: Solve
This question asks you to solve. The directive word signals the depth of analysis expected, the structure of your answer, and the weight of evidence you must bring.
See our UPSC directive words guide for a full breakdown of how to respond to each command word.
How this answer will be evaluated
Approach
Solve this three-part problem by allocating approximately 30% time to part (a) on ideal theory, 30% to part (b) on series convergence, and 40% to part (c) on transportation problem. Begin with clear definitions for (a), apply appropriate convergence tests for (b), and systematically execute VAM followed by optimality test for (c). Present each part with proper mathematical notation and logical flow.
Key points expected
- Part (a): Prove S is an ideal by showing closure under subtraction and absorption under multiplication by R[x] elements; identify R[x]/S ≅ ℝ × ℝ via evaluation maps at 0 and 1, hence not an integral domain as it has zero divisors
- Part (b): Identify general term as nⁿxⁿ/n!; apply Ratio Test or Root Test; show radius of convergence is 1/e; analyze behavior at boundary x = 1/e using Stirling's approximation or comparison
- Part (c): Apply Vogel's Approximation Method (VAM) to obtain initial BFS with m+n-1 = 6 basic variables; calculate row and column penalties correctly
- Part (c): Perform optimality test using MODI/UV method or stepping stone method; verify degeneracy handling if needed
- Part (c): Obtain optimal allocation and compute minimum transportation cost = 743 (or correct value based on calculations)
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies S as kernel of evaluation homomorphism φ: R[x] → ℝ × ℝ, properly defines convergence test for series with general term nⁿxⁿ/n!, and accurately sets up transportation table with supply (11,13,19) and demand (6,10,12,15) constraints | Attempts correct setups but has minor errors in homomorphism definition for (a), misses simplification of general term for (b), or mislabels supply/demand values for (c) | Fundamental misunderstanding of ideal definition, incorrect identification of series terms, or major errors in transportation problem structure |
| Method choice | 20% | 10 | Uses First Isomorphism Theorem for (a); applies Root Test or Ratio Test appropriately for (b) with Stirling's refinement; executes VAM with penalty calculations and MODI method for optimality in (c) | Uses correct but suboptimal methods (e.g., direct ideal verification without homomorphism for (a), Ratio Test without simplification for (b), or stepping stone instead of MODI for (c)) | Inappropriate methods (e.g., tries to prove S is subfield instead of ideal, uses comparison test without justification, or applies North-West Corner instead of VAM) |
| Computation accuracy | 20% | 10 | Flawless calculations: correct limit evaluation for radius of convergence (1/e), accurate penalty computations in VAM iterations, correct opportunity cost calculations in MODI, and precise final cost computation | Minor arithmetic errors in penalty calculations or cost computations, but overall structure and most values correct; correct radius of convergence with minor simplification errors | Major computational errors leading to wrong radius of convergence, incorrect VAM allocations, or wrong optimal cost; calculation errors affecting feasibility |
| Step justification | 20% | 10 | Rigorous justification: proves ideal properties with explicit verification, justifies why R[x]/S ≅ ℝ × ℝ implies zero divisors, explains why series diverges at x = 1/e via Stirling, and validates optimality with non-negative opportunity costs | Adequate justification with some gaps: states key theorems without full verification, asserts convergence behavior without detailed analysis, or shows MODI steps with brief reasoning | Missing crucial justifications: no verification of ideal properties, unjustified convergence claims, or optimality stated without testing; logical gaps throughout |
| Final answer & units | 20% | 10 | Complete precise answers: explicitly states S is ideal but R[x]/S is not integral domain with reason; gives interval of convergence (0, 1/e) with endpoint analysis; presents optimal transportation table and minimum cost = 743 with clear allocation | Correct final answers for most parts but incomplete: misses endpoint analysis for (b), or presents correct VAM solution without confirming optimality, or minor cost calculation error | Missing or wrong final answers: incorrect conclusion about integral domain, wrong interval of convergence, infeasible transportation solution, or no cost stated |
Practice this exact question
Write your answer, then get a detailed evaluation from our AI trained on UPSC's answer-writing standards. Free first evaluation — no signup needed to start.
Evaluate my answer →More from Mathematics 2022 Paper II
- Q1 (a) Show that the multiplicative group G = {1, -1, i, -i}, where i = √(-1), is isomorphic to the group G' = ({0, 1, 2, 3}, +₄). 10 marks (b…
- Q2 (a) Let f(x) = x² on [0, k], k > 0. Show that f is Riemann integrable on the closed interval [0, k] and ∫₀ᵏ f dx = k³/3. 15 marks (b) Prove…
- Q3 (a) Evaluate ∫_C (z+4)/(z² + 2z + 5) dz, where C is |z + 1 - i| = 2. (15 marks) (b) Find the maximum and minimum values of x²/a⁴ + y²/b⁴ +…
- Q4 (a) Let R be a field of real numbers and S, the field of all those polynomials f(x) ∈ R[x] such that f(0) = 0 = f(1). Prove that S is an id…
- Q5 (a) It is given that the equation of any cone with vertex at (a, b, c) is f((x-a)/(z-c), (y-b)/(z-c)) = 0. Find the differential equation o…
- Q6 (a) Solve the heat equation ∂u/∂t = ∂²u/∂x², 0 < x < l, t > 0 subject to the conditions u(0, t) = u(l, t) = 0, u(x, 0) = x(l-x), 0 ≤ x ≤ l.…
- Q7 (a) Find the general solution of the partial differential equation $$(D^2 + DD' - 6D'^2)z = x^2 \sin(x+y)$$ where $D \equiv \frac{\partial}…
- Q8 (a) Reduce the following partial differential equation to a canonical form and hence solve it: $$yu_{xx} + (x+y)u_{xy} + xu_{yy} = 0$$ (15…