(a) Let f(x) = x² on [0, k], k 0. Show that f is Riemann integrable on the closed interval [0, k] and ∫₀ᵏ f dx = k³/3. 15 marks

(b) Prove that every homomorphic image of a group G is isomorphic to some quotient group of G. 15 marks

(c) Apply the

Question

(a) Let f(x) = x² on [0, k], k > 0. Show that f is Riemann integrable on the closed interval [0, k] and ∫₀ᵏ f dx = k³/3. 15 marks

(b) Prove that every homomorphic image of a group G is isomorphic to some quotient group of G. 15 marks

(c) Apply the calculus of residues to evaluate ∫₋∞^∞ (cos x dx)/((x² + a²)(x² + b²)), a > b > 0. 20 marks

UPSC Answer Check · Accepted Answer

Begin with a brief introduction stating the three fundamental results to be established. For part (a), construct upper and lower Riemann sums using uniform partitions and show their convergence to k³/3. For part (b), apply the First Isomorphism Theorem by defining the natural homomorphism and proving kernel normality. For part (c), use contour integration over a semicircular contour in the upper half-plane, identify poles at ia and ib, compute residues, and apply Jordan's lemma. Allocate approximately 25-30% time to (a), 25% to (b), and 45-50% to (c) given its higher weightage and computational complexity.
- Part (a): Verification that f(x)=x² is bounded on [0,k], construction of partition Pₙ with mesh size k/n, calculation of upper sum U(Pₙ,f) and lower sum L(Pₙ,f), demonstration that U(Pₙ,f)-L(Pₙ,f)→0 establishing Riemann integrability, and evaluation of the integral as limit of Riemann sums yielding k³/3
- Part (b): Clear statement that for homomorphism φ:G→H, the image φ(G) is the target; construction of the natural map π:G→G/ker(φ); proof that ker(φ) is normal in G; establishment of the isomorphism φ̄:G/ker(φ)→φ(G) via φ̄(g·ker(φ))=φ(g); verification that φ̄ is well-defined, homomorphism, injective, and surjective
- Part (c): Recognition that the integral equals Re[∫₋∞^∞ e^(ix)/((x²+a²)(x²+b²))dx], choice of semicircular contour C_R consisting of [-R,R] and Γ_R (upper semicircle), identification of simple poles at z=ia and z=ib inside for R>max(a,b)
- Computation of residues: Res(f,ia) = e^(-a)/(2ia(a²-b²)) and Res(f,ib) = -e^(-b)/(2ib(a²-b²)) where f(z)=e^(iz)/((z²+a²)(z²+b²))
- Application of Jordan's lemma to show integral over Γ_R vanishes as R→∞, summation of residues multiplied by 2πi, extraction of real part to obtain final answer π/(a²-b²)[e^(-b)/b - e^(-a)/a]
- Proper handling of the condition a>b>0 ensuring distinct poles and correct ordering in final simplification

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	18%	9	For (a): correctly identifies f as bounded and specifies partition type; for (b): precisely defines homomorphism, kernel, and quotient group structure; for (c): properly constructs complex function f(z)=e^(iz)/((z²+a²)(z²+b²)), identifies correct contour (semicircle in upper half-plane), and verifies pole locations z=±ia, ±ib with only ia, ib enclosed	Basic setup present but missing some verification details—e.g., assumes integrability without showing U-L→0 in (a), states isomorphism theorem without defining the map in (b), or chooses contour without justifying upper vs lower half-plane in (c)	Major setup errors: incorrect interval or function in (a), confuses image with codomain or fails to mention kernel in (b), wrong contour choice or missed poles in (c)
Method choice	22%	11	(a) Uses Darboux/Riemann sum approach with explicit partition rather than invoking FTC; (b) Constructs the canonical isomorphism φ̄ systematically; (c) Applies residue theorem with Jordan's lemma justification, recognizes need for e^(iz) rather than cos(z) directly, and handles real part extraction elegantly	Correct general methods but suboptimal execution—e.g., uses antiderivative in (a) without proving integrability first, states isomorphism theorem without proof details in (b), or computes residues correctly but omits Jordan's lemma verification in (c)	Inappropriate methods: attempts Lebesgue integration in (a), uses Lagrange's theorem instead of isomorphism theorem in (b), or tries real methods for (c) instead of contour integration
Computation accuracy	24%	12	All algebraic manipulations error-free: (a) correct formulas for sum of squares and cubes with proper limit evaluation; (b) no computational steps needed beyond logical structure; (c) accurate partial fraction decomposition, correct residue calculations at both poles, proper limit as R→∞, and exact final simplified form	Minor computational slips: arithmetic errors in Riemann sums for (a), sign errors in residue calculation for (c), or algebraic simplification errors that don't fundamentally derail the answer	Serious computational errors: incorrect summation formulas in (a), wrong residue values (e.g., missing factor of 2πi, incorrect derivative evaluation), or failure to combine terms properly yielding wrong final answer
Step justification	20%	10	Every non-trivial step rigorously justified: (a) proves U(Pₙ,f)-L(Pₙ,f)→0 using explicit bound; (b) proves normality of ker(φ), well-definedness of φ̄, and bijectivity; (c) explicitly states Jordan's lemma conditions and verifies \|f(z)\|→0 uniformly, justifies residue formula application with simple pole verification	Most steps justified but gaps remain—e.g., asserts limit interchange without proof in (a), assumes well-definedness without verification in (b), or states Jordan's lemma without checking hypotheses in (c)	Unjustified leaps: claims integrability without ε-δ argument, asserts isomorphism without checking injectivity/surjectivity, or applies residue theorem without verifying contour encloses correct poles
Final answer & units	16%	8	All three parts yield complete, simplified answers: (a) ∫₀ᵏ x²dx = k³/3 clearly boxed; (b) complete statement of First Isomorphism Theorem with G/ker(φ)≅φ(G); (c) π(e^(-b)/b - e^(-a)/a)/(a²-b²) or equivalent simplified form with correct handling of a>b>0 condition	Correct answers present but not optimally presented—e.g., unsimplified fractions, missing theorem statement in (b), or correct formula with minor sign errors	Missing or incorrect final answers: wrong integral value, incomplete isomorphism statement, or evaluation that doesn't match residue calculation; also includes dimensional inconsistency if units were applicable

Q2

Directive word: Prove

How this answer will be evaluated

Approach

Key points expected

Evaluation rubric

Practice this exact question

More from Mathematics 2022 Paper II