(a) Consider the polynomial ring Z[x] over the ring Z of integers. Let S be an ideal of Z[x] generated by x. Show that S is prime but not a maximal ideal of Z[x]. (15 marks)

(b) Find the upper and lower Riemann integrals for the function f defined o

Question

(a) Consider the polynomial ring Z[x] over the ring Z of integers. Let S be an ideal of Z[x] generated by x. Show that S is prime but not a maximal ideal of Z[x]. (15 marks)

(b) Find the upper and lower Riemann integrals for the function f defined on [0, 1] as follows: f(x) = (1-x²)^½, if x is rational; f(x) = (1-x), if x is irrational. Hence, show that f is not Riemann integrable on [0, 1]. (15 marks)

(c) The personnel manager of a company wants to assign officers A, B and C to the regional offices at Delhi, Mumbai, Kolkata and Chennai. The cost of relocation (in thousand Rupees) of the three officers at the four regional offices are given below:

| Officer | Delhi | Mumbai | Kolkata | Chennai |
|---------|-------|--------|---------|---------|
| A | 16 | 22 | 24 | 20 |
| B | 10 | 32 | 26 | 16 |
| C | 10 | 20 | 46 | 30 |

Find the assignment which minimizes the total cost of relocation and also determine the minimum cost. (20 marks)

UPSC Answer Check · Accepted Answer

Prove the three mathematical claims systematically: for (a) establish primality via quotient ring isomorphism to Z and non-maximality by embedding in (x,2); for (b) compute upper and lower Darboux sums using density of rationals and irrationals, showing unequal integrals; for (c) solve the unbalanced assignment problem by adding a dummy officer with zero costs, then apply Hungarian algorithm. Allocate approximately 30% time to (a), 35% to (b), and 35% to (c) reflecting their computational demands.
- For (a): Prove S=(x) is prime by showing Z[x]/(x) ≅ Z is an integral domain; prove not maximal by showing (x) ⊂ (x,2) ⊂ Z[x] or noting Z is not a field
- For (a): Correctly identify that maximality would require Z[x]/(x) to be a field, which Z is not
- For (b): Establish that on any subinterval, sup f = √(1-x²) (achieved at rationals dense in [0,1]) and inf f = (1-x) (achieved at irrationals), using density arguments
- For (b): Compute upper integral ∫₀¹ √(1-x²)dx = π/4 and lower integral ∫₀¹ (1-x)dx = 1/2, showing π/4 ≠ 1/2 hence non-integrability
- For (c): Convert unbalanced 3×4 problem to balanced 4×4 by adding dummy officer D with zero costs; apply Hungarian algorithm steps correctly
- For (c): Identify optimal assignment (e.g., A-Chennai, B-Delhi, C-Mumbai or equivalent) with minimum cost calculation in thousand Rupees

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly defines S=(x) in Z[x], properly sets up quotient ring isomorphism for (a); correctly identifies sup/inf on arbitrary partitions for (b); properly constructs 4×4 cost matrix with dummy for (c)	Basic setup correct but misses subtlety like explicit isomorphism in (a) or density justification in (b); adds dummy but with errors in cost matrix	Misidentifies S as maximal or confuses Z[x] with Q[x]; confuses function definitions or uses wrong supremum/infimum; fails to balance the assignment problem
Method choice	20%	10	Uses isomorphism theorem for prime ideals, proper containment for non-maximality; applies Darboux criterion with density arguments; employs Hungarian algorithm with row/column reductions systematically	Correct general methods but inefficient approach or missing key theorems; uses Riemann sum definition without Darboux; solves by enumeration rather than algorithm	Attempts element-wise verification of prime ideal; uses Lebesgue criterion incorrectly; brute force without systematic optimization
Computation accuracy	20%	10	Accurate integration yielding π/4 and 1/2 with exact inequality; correct Hungarian algorithm iterations with proper covering lines and zero assignments; arithmetic error-free throughout	Minor arithmetic errors in integration constants or final cost calculation; correct algorithm but slips in matrix updates	Major errors in evaluating ∫√(1-x²)dx; incorrect minimum cost due to wrong assignment; computational breakdown in algorithm
Step justification	20%	10	Rigorous justification of why sup/inf are achieved on dense sets; explicit appeal to First Isomorphism Theorem; clear proof that π/4 ≠ 1/2; step-by-step Hungarian algorithm justification with optimality proof	States key steps but skips density justification or assumes without proof; mentions theorems without full hypotheses; algorithm steps shown but optimality briefly justified	Unjustified claims about function behavior; missing crucial logical steps; algorithm applied without explaining why solution is optimal
Final answer & units	20%	10	Clear conclusion: S prime not maximal with explicit counterexample; explicit statement that f not Riemann integrable due to unequal upper/lower integrals; optimal assignment stated with officer-office mapping and minimum cost in thousand Rupees	Correct conclusions but poorly organized or missing explicit non-integrability statement; assignment correct but cost unit unclear	Missing final conclusions; wrong assignment or cost; no units specified

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