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 + z²/c⁴, when lx + my + nz = 0 and x²/a² + y²/b² + z²/c² = 1. Interpret the result geometrically. (20 marks) (c) Solve the following linear programming problem by the simplex method. Write its dual. Also, write the optimal solution of the dual from the optimal table of the given problem : Maximize Z = x₁ + x₂ + x₃ subject to 2x₁ + x₂ + x₃ ≤ 2 4x₁ + 2x₂ + x₃ ≤ 2 x₁, x₂, x₃ ≥ 0 (15 marks)
हिंदी में प्रश्न पढ़ें
(a) ∫_C (z+4)/(z² + 2z + 5) dz का मान निकालिये, जहाँ C, |z + 1 - i| = 2 है। (15 अंक) (b) x²/a⁴ + y²/b⁴ + z²/c⁴ के अधिकतम तथा न्यूनतम मान निकालिये, जब lx + my + nz = 0 तथा x²/a² + y²/b² + z²/c² = 1 है। परिणाम की ज्यामितीय व्याख्या कीजिए। (20 अंक) (c) निम्नलिखित रैखिक प्रोग्राम समस्या को एकथा विधि द्वारा हल कीजिये। इसकी द्वैती समस्या लिखिये। दी गयी समस्या की इष्टतम सारणी से द्वैती समस्या का इष्टतम हल भी लिखिये : अधिकतमीकरण कीजिये Z = x₁ + x₂ + x₃ बशर्ते कि 2x₁ + x₂ + x₃ ≤ 2 4x₁ + 2x₂ + x₃ ≤ 2 x₁, x₂, x₃ ≥ 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 three-part numerical problem by allocating approximately 30% time to part (a) on complex integration (15 marks), 40% to part (b) on constrained optimization with geometric interpretation (20 marks), and 30% to part (c) on linear programming and duality (15 marks). Begin each part with clear identification of the mathematical technique, show complete computational steps with proper justification, and conclude with verified final answers including geometric interpretation for (b) and dual solution for (c).
Key points expected
- Part (a): Identify poles at z = -1 ± 2i, verify only z = -1 + 2i lies inside circle |z+1-i| = 2, apply Cauchy's residue theorem correctly
- Part (b): Set up Lagrangian with two constraints, derive normal equations, solve for extremal values, identify maximum and minimum as reciprocals of squares of semi-axes of elliptic section
- Part (c): Convert to standard form with slack variables, construct initial simplex tableau, iterate to optimality, verify Z = 1 at (0, 0, 2), formulate dual minimization problem
- Geometric interpretation for (b): Explain that extrema correspond to squares of distances from origin to points on ellipsoid section by plane through center
- Dual solution extraction: Read shadow prices from optimal tableau's z_j - c_j row for slack variables, verify strong duality with primal optimal value
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies poles and their location relative to contour in (a); properly formulates Lagrangian with correct multipliers in (b); accurately converts LP to standard form with appropriate slack variables in (c) | Minor errors in pole identification or contour geometry; Lagrangian setup partially correct with one constraint mishandled; LP standard form has sign or variable errors | Fundamental errors like wrong pole locations, missing constraints in Lagrangian, or incorrect inequality direction in LP conversion |
| Method choice | 20% | 10 | Selects residue theorem for (a) with correct residue calculation; uses Lagrange multipliers efficiently for (b); applies simplex method with proper pivoting rules for (c) | Correct general method but inefficient approach (e.g., direct parameterization for contour integral); attempts Lagrange multipliers but considers substitution instead; simplex method with rule violations | Inappropriate methods like Cauchy's integral formula without residue calculation, or graphical method for 3-variable LP, or ignoring constraints entirely |
| Computation accuracy | 20% | 10 | Accurate residue calculation yielding π(1+i); correct algebraic manipulation of Lagrange equations leading to clean eigenvalue problem; error-free simplex iterations with correct final tableau | Minor arithmetic slips in residue (e.g., sign error); algebraic errors in solving Lagrange system but recoverable; one computational error in simplex pivoting | Major computational failures like incorrect derivative for residue, unsolvable Lagrange system, or cycling/termination at wrong vertex in simplex |
| Step justification | 20% | 10 | Explicitly justifies why only one pole contributes in (a); clearly explains Lagrange multiplier meaning and geometric significance in (b); proves optimality via simplex criterion and explains complementary slackness for dual | Some steps justified but gaps remain (e.g., states residue theorem without verifying conditions, asserts geometric meaning without proof, claims optimality without checking all entries) | Minimal or no justification—bare calculations without theorem citations, missing verification of second-order conditions, or no explanation of why simplex terminates |
| Final answer & units | 20% | 10 | Clear boxed answers: (a) π(1+i); (b) max = 1/a², min = 1/c² (assuming a>b>c) with full geometric interpretation of elliptic section; (c) Z_max = 1, x* = (0,0,2), dual optimal y* = (0,1), W_min = 1 | Correct answers but poorly presented or missing components (e.g., no geometric interpretation, dual stated but solution not extracted, correct value but wrong point) | Missing final answers, incorrect values, or complete omission of dual solution and geometric interpretation despite correct working |
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…