Q4
(a) Show that there are infinitely many subgroups of the additive group $\mathbb{Q}$ of rational numbers. (15 marks) (b) Using contour integration, evaluate the integral $\int_{-\infty}^{\infty} \frac{\sin x \, dx}{x(x^2+a^2)}$, $a > 0$. (20 marks) (c) Solve the following linear programming problem using Big M method : Maximize Z = 4x₁ + 5x₂ + 2x₃ subject to 2x₁ + x₂ + x₃ ≥ 10, x₁ + 3x₂ + x₃ ≤ 12, x₁ + x₂ + x₃ = 6, x₁, x₂, x₃ ≥ 0. (15 marks)
हिंदी में प्रश्न पढ़ें
(a) दर्शाइए कि परिमेय संख्याओं के योज्य समूह $\mathbb{Q}$ के अपरिमित रूप से अनेक उपसमूह हैं। (15 अंक) (b) कंटूर समाकलन का उपयोग कर समाकलन $\int_{-\infty}^{\infty} \frac{\sin x \, dx}{x(x^2+a^2)}$, $a > 0$ का मान ज्ञात कीजिए। (20 अंक) (c) बड़ा M (बिग M) विधि का उपयोग करके निम्नलिखित रैखिक प्रोग्राम समस्या को हल कीजिए : अधिकतमीकरण कीजिए $Z = 4x_1 + 5x_2 + 2x_3$ बशर्ते कि $2x_1 + x_2 + x_3 \geq 10$, $x_1 + 3x_2 + x_3 \leq 12$, $x_1 + x_2 + x_3 = 6$, $x_1, x_2, x_3 \geq 0$। (15 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 multi-part problem by allocating time proportionally to marks: approximately 30% (15 minutes) for part (a) on infinite subgroups of (Q,+), 40% (20 minutes) for part (b) on contour integration, and 30% (15 minutes) for part (c) on Big M method. Begin each part with clear statement of approach, show complete working with proper mathematical justification, and conclude with explicit final answers. For (b), explicitly state contour choice and residue calculations; for (c), present the simplex tableaux clearly.
Key points expected
- Part (a): Construct explicit infinite family of subgroups, such as H_n = {m/n^k : m ∈ Z, k ≥ 0} for fixed n > 1, or Z[1/p] for varying primes p, proving closure under addition and inverses
- Part (a): Prove distinctness of infinitely many subgroups by showing H_n ≠ H_m for n ≠ m, or using prime-based constructions
- Part (b): Identify integrand has simple pole at z=0 and simple poles at z=±ia, choose semicircular contour in upper half-plane, handle pole on real axis via principal value
- Part (b): Apply residue theorem correctly: compute Res(f, ia) and half-residue at z=0, combine to get π(1-e^{-a})/a² for the sine integral
- Part (c): Convert to standard form using surplus, slack, and artificial variables with Big M penalty: minimize W = -4x₁-5x₂-2x₃ + M(a₁+a₂) or equivalent
- Part (c): Execute simplex iterations showing entering and leaving variables, pivot operations, until optimality reached with x₁=3, x₂=0, x₃=3, Z=18
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | For (a): correctly defines subgroup structure of Q with proper notation; for (b): identifies all singularities, chooses appropriate contour (semicircle with indentation), and sets up principal value correctly; for (c): converts all constraints accurately to standard form with correct surplus/slack/artificial variables and Big M formulation | Basic setup present but minor errors: vague subgroup definition in (a), contour choice unclear or missing indentation in (b), one constraint conversion error in (c) | Major setup errors: no valid subgroup construction in (a), wrong contour or missed singularities in (b), multiple conversion errors or missing artificial variables in (c) |
| Method choice | 20% | 10 | For (a): elegant constructive proof using Z[1/n] or prime-based approach; for (b): residue theorem with Jordan's lemma justification and proper handling of real pole; for (c): systematic Big M simplex method with correct pivot selection rules | Correct general methods chosen but suboptimal execution: brute-force enumeration in (a), residue calculation without Jordan's lemma mention in (b), simplex with occasional pivot errors in (c) | Inappropriate methods: non-constructive existence argument in (a), real analysis approach instead of contour integration in (b), graphical method or dual simplex instead of Big M in (c) |
| Computation accuracy | 20% | 10 | Flawless calculations: distinctness proof in (a), correct residue values Res(f,ia)=-e^{-a}/(2a²) and πi·Res(f,0) contribution in (b), all simplex tableaux arithmetically correct with final answer Z=18 in (c) | Minor computational slips: one incorrect residue calculation in (b) or single arithmetic error in simplex tableaux in (c), but method recoverable | Major computational errors: wrong formula for residues in (b), multiple tableaux errors leading to wrong optimal solution in (c), or computational gaps without working shown |
| Step justification | 20% | 10 | Rigorous justification at each step: proves H_n are subgroups and distinct in (a), proves contour integral vanishes on semicircular arc using Jordan's lemma in (b), justifies each pivot choice and optimality conditions in (c) | Some steps justified but gaps present: asserts subgroup properties without proof in (a), omits Jordan's lemma or indentation lemma in (b), states pivots without ratio test explanation in (c) | Minimal or no justification: states conclusions without proof in (a), jumps from contour setup to answer in (b), presents final tableaux without showing iterations in (c) |
| Final answer & units | 20% | 10 | Clear explicit answers: infinite family of subgroups explicitly described in (a), boxed answer π(1-e^{-a})/a² for the integral in (b), optimal solution (x₁,x₂,x₃)=(3,0,3) with Z_max=18 clearly stated in (c) | Answers present but unclear: vague description of 'infinitely many' without explicit form in (a), answer buried in working without highlighting in (b) or (c) | Missing or incorrect final answers: no explicit subgroup family in (a), wrong numerical answer or missing factor of π in (b), wrong optimal values or no conclusion in (c) |
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 2021 Paper II
- Q1 (a) Let $m_1, m_2, \cdots, m_k$ be positive integers and $d > 0$ the greatest common divisor of $m_1, m_2, \cdots, m_k$. Show that there ex…
- Q2 (a) Find the maximum and minimum values of f(x) = x³ - 9x² + 26x - 24 for 0 ≤ x ≤ 1. (15 marks) (b) Let F be a field and f(x) ∈ F[x] a poly…
- Q3 (a) Let f be an entire function whose Taylor series expansion with centre z = 0 has infinitely many terms. Show that z = 0 is an essential…
- Q4 (a) Show that there are infinitely many subgroups of the additive group $\mathbb{Q}$ of rational numbers. (15 marks) (b) Using contour inte…
- Q5 (a) Obtain the partial differential equation by eliminating arbitrary function f from the equation f(x+y+z, x²+y²+z²) = 0. (10 marks) (b) F…
- Q6 (a) Solve the wave equation a²∂²u/∂x² = ∂²u/∂t², 0<x<L, t>0 subject to the conditions u(0,t)=0, u(L,t)=0 u(x,0)=(1/4)x(L-x), ∂u/∂t|ₜ₌₀=0 (2…
- Q7 (a) Find the general solution of the partial differential equation (D² - D'² - 3D + 3D')z = xy + e^(x+2y) where D ≡ ∂/∂x and D' ≡ ∂/∂y. 15…
- Q8 (a) Find a complete integral of the partial differential equation p = (z + qy)² by using Charpit's method. 15 marks (b) Derive Newton's bac…