(a) Show that the multiplicative group G = {1, -1, i, -i}, where i = √(-1), is isomorphic to the group G' = ({0, 1, 2, 3}, +₄). 10 marks

(b) If f(z) = u + iv is an analytic function of z, and u - v = (cos x + sin x - e⁻ʸ)/(2 cos x - eʸ - e⁻ʸ), then

Question

(a) Show that the multiplicative group G = {1, -1, i, -i}, where i = √(-1), is isomorphic to the group G' = ({0, 1, 2, 3}, +₄). 10 marks

(b) If f(z) = u + iv is an analytic function of z, and u - v = (cos x + sin x - e⁻ʸ)/(2 cos x - eʸ - e⁻ʸ), then find f(z) subject to the condition f(π/2) = 0. 10 marks

(c) Test the convergence of ∫₀^∞ (cos x)/(1+x²) dx. 10 marks

(d) Expand f(z) = 1/((z-1)²(z-3)) in a Laurent series valid for the regions (i) 0 < |z-1| < 2 and (ii) 0 < |z-3| < 2. 10 marks

(e) Use two-phase method to solve the following linear programming problem: Minimize Z = x₁ + x₂ subject to 2x₁ + x₂ ≥ 4, x₁ + 7x₂ ≥ 7, x₁, x₂ ≥ 0. 10 marks

UPSC Answer Check · Accepted Answer

Solve each sub-part systematically with equal time allocation (~20% per part) since all carry 10 marks. Begin with (a) group isomorphism via Cayley table or generator mapping, (b) analytic function using Milne-Thomson method or CR equations, (c) convergence test via comparison/Dirichlet, (d) Laurent series with partial fractions and geometric expansion, and (e) two-phase simplex with artificial variables. Present solutions clearly with headings for each part.
- (a) Construct explicit isomorphism φ: G → G' showing φ(1)=0, φ(-1)=2, φ(i)=1, φ(-i)=3 and verify homomorphism property φ(ab)=φ(a)+₄φ(b)
- (b) Apply Milne-Thomson method: replace x by z and y by 0 in u-v expression to get f(z), then use condition f(π/2)=0 to determine constant
- (c) Establish absolute convergence via |cos x/(1+x²)| ≤ 1/(1+x²) and ∫₀^∞ dx/(1+x²) = π/2, or use Dirichlet test for conditional convergence
- (d) For region (i): write w=z-1, expand 1/(w²(w-2)) = -1/(2w²)·1/(1-w/2) using geometric series; for region (ii): use w=z-3, expand 1/((w+2)²w)
- (e) Phase I: minimize sum of artificial variables A₁+A₂ with constraints 2x₁+x₂-s₁+A₁=4, x₁+7x₂-s₂+A₂=7; Phase II: original objective with feasible basis
- Verify all group properties in (a), check analyticity via CR equations in (b), justify uniform convergence in (c), state radii of convergence in (d), and show optimality via simplex criteria in (e)

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly identifies group structures (cyclic order 4), recognizes u-v as boundary value for Milne-Thomson, chooses appropriate convergence test, sets up correct annular regions for Laurent expansion, and properly converts inequalities to standard form with slack/surplus variables for two-phase method	Identifies most structures correctly but minor errors in region specification or constraint conversion; may confuse additive vs multiplicative notation in (a)	Fundamental misidentification of group types, wrong method for analytic function (e.g., direct CR equations without Milne-Thomson), incorrect convergence test selection, or fails to set up artificial variables in Phase I
Method choice	20%	10	Uses generator-based isomorphism or Cayley table matching for (a); Milne-Thomson for (b); comparison/Dirichlet test for (c); partial fractions with geometric series tailored to each annulus for (d); standard two-phase simplex with proper artificial variable handling for (e)	Correct general methods but suboptimal choices (e.g., direct CR integration instead of Milne-Thomson, making (b) lengthy); may use ratio test incorrectly for (c)	Inappropriate methods such as Lagrange's theorem for isomorphism, Taylor series instead of Laurent, or Big-M method when two-phase is specified
Computation accuracy	20%	10	Error-free calculations: correct mapping values in (a), precise complex integration yielding f(z)=½[(1+i)-(cos z+i sin z)e^(-iz)] or equivalent, correct π/2 evaluation; accurate series coefficients; correct simplex table iterations with optimal Z=11/13 at (21/13, 10/13)	Minor arithmetic slips in coefficients or table entries that don't affect final structure; sign errors in partial fractions correctable in context	Major computational errors: wrong isomorphism values, incorrect f(z) form, divergent series claimed convergent, or wrong optimal solution due to pivot errors
Step justification	20%	10	Explicitly justifies: bijectivity and homomorphism in (a); why Milne-Thomson applies (analyticity) in (b); domination by integrable function in (c); validity of geometric expansion in each annulus for (d); why artificial variables can be eliminated and optimality conditions in (e)	States key theorems but omits verification of hypotheses; assumes convergence without bounding, or asserts optimality without checking all cⱼ-zⱼ ≤ 0	Unjustified leaps: claims isomorphism without verification, asserts convergence without test, expands series without checking \|ratio\|<1, or pivots without ratio test
Final answer & units	20%	10	Complete precise answers: explicit isomorphism formula φ(iᵏ)=k mod 4; f(z)=½[(1+i)-(1-i)e^(-iz)(cos z+i sin z)] with verification; 'absolutely convergent'; two-term Laurent series with explicit general terms; optimal solution x₁=21/13, x₂=10/13, Z_min=31/13 with Phase I ending at zero artificial sum	Correct final forms but missing verification, incomplete series (few terms without general pattern), or correct values without clear statement of optimality	Missing answers, incorrect final forms, or failure to use given condition f(π/2)=0; no convergence conclusion; incomplete simplex stopping at Phase I

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