(a) Prove that the oscillation of a real-valued bounded function f defined on [a, b] is the supremum of the set {|f(x₁)-f(x₂)| : x₁, x₂ ∈ [a, b]}. 15 marks

(b) Classify the singular point z = 0 of the function f(z) = e^z/(z-sin z) and obtain the pri

Question

(a) Prove that the oscillation of a real-valued bounded function f defined on [a, b] is the supremum of the set {|f(x₁)-f(x₂)| : x₁, x₂ ∈ [a, b]}. 15 marks

(b) Classify the singular point z = 0 of the function f(z) = e^z/(z-sin z) and obtain the principal part of its Laurent series expansion. 15 marks

(c) A department head has 5 subordinates and 5 jobs to be performed. The time (in hours) that each subordinate will take to perform each job is given in the matrix below :

How should the jobs be assigned, one to each subordinate, so as to minimize the total time? Also, obtain the total minimum time to perform all the jobs if the subordinate IV cannot be assigned job C. 20 marks

UPSC Answer Check · Accepted Answer

Begin with a clear statement of definitions for oscillation, supremum, and bounded variation. For part (a), establish the equivalence through careful set-theoretic arguments showing ω(f) = sup|f(x₁)-f(x₂)|. For part (b), use Taylor series expansion to identify the order of the pole at z=0 and extract coefficients for the principal part. For part (c), apply the Hungarian algorithm to solve the 5×5 assignment problem, then re-optimize with the additional constraint. Allocate approximately 30% time to (a), 30% to (b), and 40% to (c) given its higher weightage and computational demand.
- For (a): Correct definition of oscillation ω(f,[a,b]) = M-m where M=sup f, m=inf f; proof that this equals sup{|f(x₁)-f(x₂)|} through showing both inequalities
- For (a): Demonstration that for any ε>0, there exist x₁,x₂ with f(x₁)>M-ε/2 and f(x₂)<m+ε/2, establishing the supremum attainment
- For (b): Taylor expansion of sin z = z - z³/6 + z⁵/120 - ... and e^z = 1 + z + z²/2 + ... to determine z-sin z = z³/6 - z⁵/120 + ...
- For (b): Identification of z=0 as a pole of order 2 (simple pole cancellation leaves z³ in denominator, e^z≈1), computation of Laurent series principal part coefficients a₋₁, a₋₂
- For (c): Correct setup of cost matrix, application of Hungarian algorithm (row/column reduction, minimum lines covering zeros, matrix adjustments)
- For (c): Optimal assignment without constraints and verification of optimality; modified solution with constraint IV≠C using branch-and-bound or re-optimization
- For (c): Clear statement of minimum total time in both cases with proper units (hours)

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Precise definitions: oscillation as M-m, correct identification of pole order 2 in (b), proper cost matrix setup with 5×5 dimensions in (c); all notations and domains clearly stated	Definitions present but somewhat imprecise; minor errors in identifying singularity type or matrix dimensions; missing some domain specifications	Missing or incorrect definitions; wrong identification of singularity (essential vs pole) or incorrect matrix setup; fundamental misunderstanding of boundedness conditions
Method choice	20%	10	For (a): direct supremum argument with ε-δ precision; for (b): systematic Laurent expansion via Taylor series; for (c): Hungarian algorithm with proper optimization steps, constraint handling via re-optimization	Correct general methods but suboptimal execution (e.g., using limit definition for oscillation without supremum properties; partial Hungarian algorithm; brute force for constrained case)	Incorrect methods: attempting pointwise limits for oscillation, residue formula misuse for Laurent series, trial-and-error or incomplete enumeration for assignment problem
Computation accuracy	20%	10	Exact coefficients in Laurent expansion: principal part 6/z² - 6/z + ...; correct Hungarian algorithm iterations with proper matrix reductions; accurate final minimum times (unconstrained and constrained)	Minor arithmetic errors in series coefficients or matrix operations; correct method but one computational slip affecting final answer; partial credit for correct setup	Major computational errors: wrong pole order, incorrect Laurent coefficients, systematic errors in assignment algorithm leading to wrong allocations; missing or wrong final numerical answers
Step justification	20%	10	Rigorous ε-δ arguments for supremum attainment; explicit justification of pole order via lowest power term; proof of optimality in Hungarian algorithm (minimum lines = matrix rank); logical flow connecting all three parts	Steps shown but gaps in rigorous justification; assertions without proof (e.g., 'clearly the supremum'); algorithm steps listed without explaining why optimality holds; missing connections between sub-parts	Unjustified leaps, 'it can be shown that' without demonstration; missing crucial steps; circular reasoning; no explanation of why methods work or why answers are optimal
Final answer & units	20%	10	Clear boxed/concluded answers: oscillation formula proved with equality established; principal part explicitly stated as a₋₂/z² + a₋₁/z with numerical coefficients; optimal assignment clearly stated (e.g., I→A, II→B,...) and minimum times in hours for both cases	Answers present but format unclear; missing units; incomplete statement of principal part (only coefficients, not assembled); assignment stated without clear mapping	Missing final answers; answers without units; wrong answers due to earlier errors; incomplete or illegible presentation; no distinction between constrained and unconstrained cases

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