(a) Examine whether the mapping φ: Z[x] → Z defined by φ(f(x)) = f(0), for f(x) ∈ Z[x], is a homomorphism. Deduce that the ideal ⟨x⟩ is a prime ideal in Z[x], but not a maximal ideal in Z[x]. (15 marks)

(b) Prove that every continuous function is Ri

Question

(a) Examine whether the mapping φ: Z[x] → Z defined by φ(f(x)) = f(0), for f(x) ∈ Z[x], is a homomorphism. Deduce that the ideal ⟨x⟩ is a prime ideal in Z[x], but not a maximal ideal in Z[x]. (15 marks)

(b) Prove that every continuous function is Riemann integrable. (15 marks)

(c) The following table shows all the necessary information on the available supply to each warehouse, the requirement of each market and the unit transportation cost from each warehouse to each market :

Market

I II III IV Supply

A 5 2 4 3 22

Warehouse B 4 8 1 6 15

C 4 6 7 5 8

Requirement 7 12 17 9

The shipping clerk has worked out the following schedule from experience :

12 units from A to II, 1 unit from A to III, 9 units from A to IV, 15 units from B to III, 7 units from C to I and 1 unit from C to III

Find the optimal schedule and minimum total shipping cost. (20 marks)

UPSC Answer Check · Accepted Answer

Begin with (a) by verifying the homomorphism property φ(f+g)=φ(f)+φ(g) and φ(fg)=φ(f)φ(g), then identify ker(φ)=⟨x⟩ and apply the First Isomorphism Theorem to establish primeness via Z[x]/⟨x⟩≅Z (an integral domain but not a field). For (b), construct the proof using uniform continuity on closed intervals and the Darboux criterion, showing upper and lower sums converge. Devote maximum effort to (c): first verify the initial feasible solution (degenerate, with only 5 allocations for 6 rows+columns-1=5, so non-degenerate), compute opportunity costs using u-v method, identify negative Δij, and iterate to optimality. Allocate roughly 30% time to (a), 25% to (b), and 45% to (c) given its higher weight and computational demand.
- For (a): Verify φ preserves addition and multiplication; show ker(φ)=⟨x⟩; apply First Isomorphism Theorem to get Z[x]/⟨x⟩≅Z; conclude ⟨x⟩ is prime (Z is integral domain) but not maximal (Z is not a field)
- For (b): State that continuous functions on [a,b] are uniformly continuous; use this to show for any ε>0, there exists partition P with U(P,f)-L(P,f)<ε; conclude Riemann integrability via Darboux criterion
- For (c): Verify initial solution is feasible (supply=demand=45) but degenerate; compute dual variables ui, vj using occupied cells; calculate opportunity costs Δij=cij-(ui+vj) for unoccupied cells
- For (c): Identify most negative Δij and construct closed loop; determine θ=min allocation at decreasing corners; perform basis change and recompute until all Δij≥0
- For (c): State optimal allocations and minimum total cost with proper units (currency units); verify optimality conditions are satisfied

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly defines φ for (a) as evaluation homomorphism, states domain/codomain precisely; for (b) specifies closed interval [a,b] and continuity hypothesis; for (c) verifies supply=demand=45, constructs correct cost matrix, and identifies degeneracy in initial solution	Basic definitions present but misses evaluation homomorphism terminology for (a), omits closed interval specification for (b), or fails to check balance condition for (c)	Incorrect definition of homomorphism for (a), wrong theorem statement for (b), or fundamental error in transportation problem setup such as unbalanced problem not recognized
Method choice	20%	10	Uses First Isomorphism Theorem elegantly for (a); applies uniform continuity + Darboux criterion (not just Riemann's original definition) for (b); employs MODI/u-v method with proper handling of degeneracy for (c)	Correct but circuitous approach for (a); uses only Riemann's definition without uniform continuity for (b); uses stepping stone or Vogel's approximation without MODI for optimality test in (c)	Attempts direct ideal-theoretic proof without isomorphism theorem for (a); confuses Riemann with Lebesgue integrability for (b); uses intuitive trial-and-error without systematic optimization method for (c)
Computation accuracy	20%	10	Flawless arithmetic in isomorphism proof for (a); correct ε-δ constructions for (b); accurate dual variable calculations, loop tracing, and θ-determination leading to correct optimal cost for (c)	Minor algebraic slips in (a) or (b); one or two calculation errors in opportunity costs or loop adjustments for (c) that don't fundamentally derail the solution	Serious errors: wrong kernel identification for (a), incorrect inequality direction for (b), or sign errors in Δij leading to wrong optimal solution or non-terminating iterations for (c)
Step justification	20%	10	Explicitly cites: ring homomorphism properties and ideal correspondence theorem for (a); Heine-Cantor theorem and Darboux integrability criterion for (b); optimality condition (all Δij≥0) and why degeneracy requires special handling for (c)	States key theorems without full explanation of why they apply; logical flow present but gaps in connecting steps to conclusions	Unjustified leaps between steps; missing theorem citations; asserts conclusions without proof (e.g., claims optimality without showing all opportunity costs non-negative)
Final answer & units	20%	10	Clear statement: ⟨x⟩ is prime but not maximal with explicit justification for (a); precise integrability conclusion for (b); complete optimal transportation table with all allocations and minimum total cost (e.g., Rs. 143 or correct computed value) with units for (c)	Correct conclusions stated but without full supporting detail; optimal cost correct but allocation table incomplete or units missing	Missing or wrong final answers; incorrect ideal classification for (a); false claim about integrability for (b); wrong optimal cost or infeasible final solution for (c)

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