Q3
(a) Evaluate the integral ∮_C e^z/(z²(z+1)³) dz, C : |z| = 2. (15 marks) (b) Show that the volume of the greatest rectangular parallelopiped that can be inscribed in the ellipsoid (x²/a²) + (y²/b²) + (z²/c²) = 1 is 8abc/(3√3). (20 marks) (c) Apply the principle of duality to solve the following linear programming problem : Maximize Z = 3x₁ + 4x₂ subject to the constraints x₁ - x₂ ≤ 1 x₁ + x₂ ≥ 4 x₁ - 3x₂ ≤ 3 x₁, x₂ ≥ 0 (15 marks)
हिंदी में प्रश्न पढ़ें
(a) समाकल ∮_C e^z/(z²(z+1)³) dz, C : |z| = 2 का मान ज्ञात कीजिए। (15 अंक) (b) सिद्ध कीजिए कि दीर्घवृत्ताख (x²/a²) + (y²/b²) + (z²/c²) = 1 के अंतर्गत सबसे बड़े समकोणिक समांतरपृष्ठक का आयतन 8abc/(3√3) है। (20 अंक) (c) द्वैतता (ड्युअलिटी) के सिद्धांत का उपयोग कर निम्न रैखिक प्रोग्रामन समस्या को हल कीजिए : अधिकतमीकरण कीजिए Z = 3x₁ + 4x₂ बशर्ते कि x₁ - x₂ ≤ 1 x₁ + x₂ ≥ 4 x₁ - 3x₂ ≤ 3 x₁, x₂ ≥ 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 all three parts systematically, allocating approximately 30% time to part (a) [15 marks], 40% to part (b) [20 marks], and 30% to part (c) [15 marks]. Begin with identifying singularities and applying Cauchy's residue theorem for (a), then use Lagrange multipliers for constrained optimization in (b), and finally construct the dual LP and solve via simplex method for (c). Present each part with clear headings and show all computational steps.
Key points expected
- Part (a): Identify poles at z=0 (order 2) and z=-1 (order 3) inside |z|=2; apply Cauchy's residue theorem with correct residue formulas for higher-order poles
- Part (a): Compute residues using derivatives: Res(f,0) via first derivative of e^z/(z+1)^3 and Res(f,-1) via second derivative of e^z/z^2
- Part (b): Set up Lagrangian F = 8xyz + λ(1 - x²/a² - y²/b² - z²/c²) for inscribed rectangular parallelopiped with vertices (±x,±y,±z)
- Part (b): Derive critical conditions ∂F/∂x = ∂F/∂y = ∂F/∂z = 0 leading to x²/a² = y²/b² = z²/c² = 1/3, hence maximum volume 8abc/(3√3)
- Part (c): Convert primal (maximization with mixed constraints) to standard form; formulate dual minimization problem with correct variable correspondence
- Part (c): Solve dual using simplex method or verify complementary slackness; recover primal optimal solution x₁=7/2, x₂=1/2 with Z_max=12.5
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies all singularities and their orders for (a); properly sets up Lagrangian with constraint for (b); accurately converts primal to standard form and constructs dual for (c) | Identifies most singularities correctly but misses order or location for one pole; Lagrangian setup partially correct; dual formulation has minor sign or inequality errors | Fails to identify correct singularities or their nature; incorrect Lagrangian setup; fundamentally wrong dual construction or constraint conversion |
| Method choice | 20% | 10 | Selects Cauchy's residue theorem with appropriate higher-order pole formulas for (a); Lagrange multipliers for constrained optimization in (b); duality principle with simplex method for (c) | Uses correct general methods but applies suboptimal techniques (e.g., partial fractions unnecessarily); attempts Lagrange multipliers with algebraic errors; uses graphical method instead of duality for (c) | Attempts direct integration for (a); uses substitution without Lagrange multipliers for (b); ignores duality and solves primal directly or fails to solve (c) |
| Computation accuracy | 20% | 10 | Accurate residue calculations yielding 2πi(3e⁻¹-2) for (a); correct algebraic manipulation leading to x=a/√3, y=b/√3, z=c/√3 for (b); precise simplex iterations giving Z_max=25/2 for (c) | Minor arithmetic errors in residue derivatives; algebraic slips in solving Lagrange equations but correct final form; one or two simplex table errors but correct final answer | Major computational errors in residues; incorrect critical point calculation; simplex method abandoned or grossly miscalculated with wrong optimal value |
| Step justification | 20% | 10 | Explicitly states residue formulas for poles of order m; justifies why critical point gives maximum via Hessian or boundary analysis; proves dual feasibility and strong duality verification | Shows key steps but omits justification for residue formula application; asserts maximum without second-order test; states dual solution without verifying complementary slackness | Jumps to answers without showing residue calculations; no justification for maximum; no explanation of duality principle application or verification |
| Final answer & units | 20% | 10 | Presents exact answers: (a) 2πi(3/e - 2), (b) 8abc/(3√3) with clear derivation, (c) x₁=7/2, x₂=1/2, Z_max=25/2; includes 2πi factor for contour integral and dimensional consistency | Correct final forms but missing 2πi in (a) or unsimplified radicals in (b); correct numerical values in (c) but fractional form not simplified | Missing final answers or incorrect boxed results; no units or factors; incomplete solution for one or more parts |
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 2025 Paper II
- Q1 (a) Let H and K be two subgroups of a group G such that o(H) > √o(G) and o(K) > √o(G). Show that H ∩ K ≠ {e}, where e is the identity eleme…
- Q2 (a) Define Cauchy sequence and prove that every convergent sequence of real numbers is a Cauchy sequence. What is the importance of Cauchy…
- Q3 (a) Evaluate the integral ∮_C e^z/(z²(z+1)³) dz, C : |z| = 2. (15 marks) (b) Show that the volume of the greatest rectangular parallelopipe…
- Q4 (a) Examine whether the mapping φ: Z[x] → Z defined by φ(f(x)) = f(0), for f(x) ∈ Z[x], is a homomorphism. Deduce that the ideal ⟨x⟩ is a p…
- Q5 (a) Find the solution of the equation $(D^2 + DD' - 2D'^2)z = y\sin x$, where $D \equiv \frac{\partial}{\partial x}$ and $D' \equiv \frac{\…
- Q6 (a) Solve $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ for a rectangular plate subject to the boundary condi…
- Q7 (a) Find the complete integral of z(p²-q²) = x-y; p≡∂z/∂x, q≡∂z/∂y. (15 marks) (b) Find the unique polynomial of degree 2 or less which fit…
- Q8 (a) Find the characteristics of the partial differential equation p² + q² = 2; p ≡ ∂z/∂x, q ≡ ∂z/∂y and determine the integral surface whic…