(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 singularity of f(1/z). (15 marks)

(b) Find the stationary values of $x^2 + y^2 + z^2$ subject to the condition

Question

(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 singularity of f(1/z). (15 marks)

(b) Find the stationary values of $x^2 + y^2 + z^2$ subject to the conditions $ax^2 + by^2 + cz^2 = 1$ and $lx + my + nz = 0$. Interpret the result geometrically. (20 marks)

(c) Convert the following LPP into dual LPP : Minimize $Z = x_1 - 3x_2 - 2x_3$ subject to $$3x_1 - x_2 + 2x_3 \leq 7$$ $$2x_1 - 4x_2 \geq 12$$ $$-4x_1 + 3x_2 + 8x_3 = 10$$ where $x_1, x_2 \geq 0$ and $x_3$ is unrestricted in sign. (15 marks)

UPSC Answer Check · Accepted Answer

The directive 'prove' for part (a) and 'find' for parts (b)-(c) demand rigorous mathematical demonstration. Allocate approximately 30% time to part (a) establishing the essential singularity via Laurent series analysis, 40% to part (b) solving the constrained optimization with Lagrange multipliers and geometric interpretation, and 30% to part (c) systematically converting the primal LPP to dual form. Structure as: brief statement of key theorems → step-by-step derivation for each part → concluding verification of results.
- Part (a): Taylor series of f(z) with infinitely many terms implies Laurent series of f(1/z) has infinitely many negative powers, proving z=0 is essential singularity via Casorati-Weierstrass or definition
- Part (b): Correct formulation of Lagrangian with two multipliers λ and μ; derivation of 5 equations from partial derivatives; elimination leading to characteristic equation for stationary values
- Part (b): Geometric interpretation as finding extremal distances from origin to intersection of ellipsoid and plane (ellipse), yielding maximum and minimum distance squared values
- Part (c): Conversion of unrestricted x₃ to x₃⁺ - x₃⁻; transformation of ≥ constraint to ≤ by sign reversal; proper assignment of dual variables (y₁ ≥ 0, y₂ ≤ 0, y₃ unrestricted)
- Part (c): Correct dual objective (maximize) with right-hand side coefficients and proper constraint coefficients from primal transpose, verifying weak duality structure

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly identifies f(1/z) has Laurent series with infinite principal part for (a); properly constructs Lagrangian L = x²+y²+z² - λ(ax²+by²+cz²-1) - μ(lx+my+nz) for (b); accurately transforms all primal constraints including unrestricted variable handling for (c)	Basic setup present but minor errors in Lagrangian construction or dual variable assignment; may miss sign conventions in constraint conversion	Fundamental setup errors: confuses Taylor/Laurent series, wrong Lagrangian formulation, or incorrect primal-to-dual transformation rules
Method choice	20%	10	Uses classification of singularities via principal part for (a); applies Lagrange multipliers method systematically for (b); employs standard primal-dual conversion algorithm with proper handling of unrestricted variables for (c)	Appropriate methods chosen but execution lacks elegance; may use alternative valid approaches but with unnecessary complexity	Inappropriate methods: attempts Casorati-Weierstrass without Laurent analysis, uses substitution instead of multipliers, or fails to recognize dual transformation rules
Computation accuracy	20%	10	Flawless algebraic manipulation in deriving characteristic equation for stationary values; correct determinant conditions; accurate dual constraint coefficients and RHS values throughout	Minor computational slips in algebra or arithmetic that don't fundamentally derail the solution; partial credit for correct approach with calculation errors	Major computational errors: incorrect elimination of variables, wrong characteristic equation, or systematic errors in dual constraint formation
Step justification	20%	10	Explicitly justifies why infinite Taylor series implies infinite principal part; clearly explains geometric meaning of Lagrange conditions as orthogonality; states duality theorems used and verifies constraint correspondence	Steps shown but with gaps in logical connections; geometric interpretation mentioned superficially; dual conversion steps listed without theorem references	Missing crucial justifications: asserts essential singularity without definition, omits why stationary points correspond to extremal distances, or presents dual without explaining transformation rules
Final answer & units	20%	10	Clear statement that z=0 is essential singularity with precise reasoning; explicit stationary values formula in terms of a,b,c,l,m,n with geometric interpretation as semi-axes of ellipse; complete dual LPP with correct objective, constraints, and variable conditions	Correct final answers but poorly organized or missing explicit boxed conclusions; geometric interpretation incomplete; dual present but with minor constraint errors	Missing or incorrect final answers; no geometric interpretation for (b); incomplete dual formulation with wrong objective direction or variable restrictions

Q3

Directive word: Prove

How this answer will be evaluated

Approach

Key points expected

Evaluation rubric

Practice this exact question

More from Mathematics 2021 Paper II