(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 exist integers $x_1, x_2, \cdots, x_k$ such that $$d = x_1m_1 + x_2m_2 + \cdots + x_km_k$$ (10 marks)

(b) Test t

Question

(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 exist integers $x_1, x_2, \cdots, x_k$ such that $$d = x_1m_1 + x_2m_2 + \cdots + x_km_k$$ (10 marks)

(b) Test the uniform convergence of the series $$x^4 + \frac{x^4}{1+x^4} + \frac{x^4}{(1+x^4)^2} + \frac{x^4}{(1+x^4)^3} + \cdots$$ on $[0, 1]$. (10 marks)

(c) If a function $f$ is monotonic in the interval $[a, b]$, then prove that $f$ is Riemann integrable in $[a, b]$. (10 marks)

(d) Let $c : [0, 1] 	o \mathbb{C}$ be the curve, where $c(t) = e^{4\pi it}$, $0 \leq t \leq 1$. Evaluate the contour integral $\displaystyle\int_c \frac{dz}{2z^2 - 5z + 2}$. (10 marks)

(e) A department of a company has five employees with five jobs to be performed. The time (in hours) that each man takes to perform each job is given in the effectiveness matrix. Assign all the jobs to these five employees to minimize the total processing time:

Employees
I II III IV V
A 10 5 13 15 16
B 3 9 18 13 6
Jobs C 10 7 2 2 2
D 7 11 9 7 12
E 7 9 10 4 12

(10 marks)

UPSC Answer Check · Accepted Answer

Solve all five sub-parts systematically, allocating approximately 20% time each since all carry equal marks. For (a), apply the Extended Euclidean Algorithm or ideal-theoretic proof; for (b), identify the geometric series and test convergence at x=0; for (c), use Darboux's theorem or direct ε-δ argument with partition refinement; for (d), apply residue theorem after factorizing the denominator and checking pole locations relative to the curve (unit circle traversed twice); for (e), execute the Hungarian algorithm with row/column reductions. Present each solution with clear theorem citations and boxed final answers.
- (a) Correctly states and applies Bézout's identity/Extended Euclidean Algorithm for k integers, showing d generates the ideal (m₁,...,mₖ)
- (b) Identifies geometric series with ratio 1/(1+x⁴), finds pointwise limit function (0 for x>0, 1 for x=0), and proves non-uniform convergence via supremum norm or discontinuity of limit
- (c) Proves monotonic functions have at most countably many discontinuities (or uses Darboux integrability criterion), establishes upper and lower sums converge
- (d) Factorizes denominator as (2z-1)(z-2), identifies poles at z=½ and z=2, determines only z=½ lies inside |z|=1 (traversed twice), computes residue correctly
- (e) Correctly applies Hungarian algorithm: row reduction, column reduction, minimum lines covering zeros, optimal assignment with minimum total time calculation

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly identifies: (a) the ideal structure and role of GCD; (b) series type and domain issues at x=0; (c) Darboux criterion or discontinuity set properties; (d) contour orientation (twice around unit circle) and pole locations; (e) balanced assignment problem structure with 5×5 matrix	Partially correct setups with minor errors in domain identification for (b), missing contour multiplicity for (d), or incomplete matrix reduction setup for (e)	Major setup errors: wrong theorem choice for (a), misidentifying series type for (b), ignoring discontinuity issue for (c), wrong contour or pole inclusion for (d), incorrect problem type for (e)
Method choice	20%	10	Selects optimal methods: Extended Euclidean/induction for (a), geometric series formula with uniform convergence test for (b), Darboux criterion or oscillation argument for (c), residue theorem with correct winding number for (d), Hungarian algorithm with proper optimization steps for (e)	Acceptable methods with suboptimal efficiency, such as direct ε-δ for (c) without Darboux, or partial fraction without residue theorem for (d)	Inappropriate methods: trial-and-error for (a), ratio test alone for uniform convergence in (b), Riemann's original definition without refinement for (c), direct parametrization without residue calculus for (d), greedy assignment without optimization for (e)
Computation accuracy	20%	10	Flawless calculations: correct Bézout coefficients in (a), precise supremum evaluation showing non-uniform convergence in (b), accurate partition sums in (c), correct residue value 2πi/3 for (d), optimal assignment with minimum total time 21 hours for (e)	Minor computational slips: arithmetic errors in coefficients, incorrect limit evaluation at boundary, residue calculation with sign error, or suboptimal assignment with small time excess	Major computational failures: incorrect GCD expression, wrong sum formula, invalid integral bounds, residue at wrong pole, or completely wrong assignment solution
Step justification	20%	10	Rigorous justification at each step: explicit induction or ideal argument for (a), ε-N argument with supremum attainment for (b), countable discontinuity proof or Darboux theorem citation for (c), winding number calculation and residue formula application for (d), optimality proof via König-Egerváry theorem or equivalent for (e)	Adequate but terse justifications, citing theorems without full verification of hypotheses, or skipping key logical connections	Missing crucial justifications: no proof that d is minimal in (a), no demonstration of non-uniformity in (b), unproven claim about discontinuities in (c), no residue theorem justification in (d), no optimality proof in (e)
Final answer & units	20%	10	Clear, boxed final answers: explicit integer coefficients for (a), definitive 'not uniformly convergent' with reason for (b), completed proof statement for (c), exact value 2πi/3 for (d), optimal assignment matrix with total time 21 hours for (e)	Correct final answers but poorly formatted, missing units where relevant, or incomplete statement of optimal assignment	Missing or wrong final answers, incorrect units, incomplete assignment specification, or answers inconsistent with working shown

Q1

Directive word: Solve

How this answer will be evaluated

Approach

Key points expected

Evaluation rubric

Practice this exact question

More from Mathematics 2021 Paper II