Q2
(a) Define Cauchy sequence and prove that every convergent sequence of real numbers is a Cauchy sequence. What is the importance of Cauchy condition? (15 marks) (b) Show that 3 is an irreducible element in the integral domain Z[i]. (15 marks) (c) Use the method of contour integration to prove that ∫₋∞^∞ (x² - x + 2)/(x⁴ + 10x² + 9) dx = 5π/12. (20 marks)
हिंदी में प्रश्न पढ़ें
(a) कोशी अनुक्रम की परिभाषा दीजिए और सिद्ध कीजिए कि वास्तविक संख्याओं का प्रत्येक अभिसारी अनुक्रम एक कोशी अनुक्रम है। कोशी की शर्त का क्या महत्व है? (15 अंक) (b) दर्शाइए कि पूर्णांकीय प्रांत Z[i] में 3 एक अविभाज्य अवयव है। (15 अंक) (c) कंटूर समाकलन की विधि से सिद्ध कीजिए कि ∫₋∞^∞ (x² - x + 2)/(x⁴ + 10x² + 9) dx = 5π/12 है। (20 अंक)
Directive word: Prove
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How this answer will be evaluated
Approach
Begin with precise definitions for part (a), then construct rigorous proofs for all three sub-parts. Allocate approximately 30% time to (a) covering Cauchy definition, convergence proof and completeness significance; 30% to (b) establishing irreducibility via norm analysis in Z[i]; and 40% to (c) setting up contour integration with semicircular contour, residue calculation at poles i and 3i, and verification of Jordan's lemma applicability. Conclude each part with clear statement of result.
Key points expected
- Part (a): Formal ε-N definition of Cauchy sequence; proof that convergent ⇒ Cauchy using triangle inequality; explanation of completeness (R is Cauchy-complete, Q is not) with significance for constructing real numbers
- Part (a): Importance: Cauchy condition allows convergence testing without knowing limit; foundation for Banach fixed-point theorem; metric space completeness characterization
- Part (b): Definition of irreducible element in integral domain; norm function N(a+bi)=a²+b²; proof that N(3)=9 and if 3=αβ then N(α)N(β)=9; elimination of cases N(α)=1,3,9 showing 3 is not product of non-units
- Part (c): Factorization of denominator (x²+1)(x²+9); identification of poles at z=i, z=3i in upper half-plane; semicircular contour CR with radius R→∞
- Part (c): Calculation of residues: Res(f,i) and Res(f,3i) using simple pole formula; application of residue theorem; Jordan's lemma verification for vanishing arc integral
- Part (c): Final computation yielding 2πi × (sum of residues) = 5π/12 after careful algebraic simplification
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Part (a) gives precise ε-N definition of Cauchy sequence; part (b) correctly defines irreducibility and Gaussian integer norm; part (c) properly factorizes denominator and identifies all singularities in upper half-plane with correct contour selection | Definitions are essentially correct but lack precision (e.g., missing quantifiers in Cauchy definition); norm function stated but not fully exploited; contour described but justification for semicircle over full circle is weak | Incorrect or missing definitions; confuses irreducibility with primality; wrong factorization or missed poles; contour encloses wrong region or includes poles on real axis without indentation |
| Method choice | 20% | 10 | Part (a) uses direct ε-N proof with clear triangle inequality application; part (b) employs norm argument systematically eliminating all factorization cases; part (c) selects semicircular contour with Jordan's lemma for rational function decaying as 1/z² | Correct general methods chosen but execution has gaps; norm method in (b) started but case analysis incomplete; contour integration method correct but residue calculation approach is inefficient | Wrong proof strategy (e.g., attempts converse in (a) without completeness); tries direct factorization in (b) without norm; uses real methods or wrong contour in (c); attempts partial fractions without contour |
| Computation accuracy | 20% | 10 | Part (a): clean algebraic manipulation with correct bound on |a_m - a_n|; part (b): exhaustive case analysis showing no element has norm 3; part (c): accurate residue calculations at i and 3i yielding (1-3i)/16 and (3+9i)/48, correct sum and final value 5π/12 | Minor computational slips: algebraic bounds in (a) slightly loose; one case in (b) analysis not fully resolved; residue at one pole incorrect but other correct, or arithmetic error in final combination | Major computational errors: wrong inequality direction in (a); incorrect norm values or incomplete case elimination in (b); both residues wrong, or 2πi factor omitted, or algebraic errors preventing recognition of 5π/12 |
| Step justification | 20% | 10 | Every logical step explicitly justified: triangle inequality usage in (a); why units have norm 1 and why norm 3 is impossible in Z[i]; Jordan's lemma conditions verified in (c) with limit argument for ML-estimate; residue theorem hypotheses checked | Key steps justified but some gaps: assumes without proof that Cauchy sequences are bounded; states norm multiplicativity without proof; applies residue theorem without verifying simple poles or contour orientation | Missing crucial justifications: no mention of completeness being essential for converse; asserts 3 is irreducible without norm argument; applies residue theorem without checking pole location or uses Jordan's lemma for non-decaying integrand |
| Final answer & units | 20% | 10 | Clear concluding statements for each part: explicit statement that convergent sequences are Cauchy with completeness caveat; definitive conclusion that 3 is irreducible in Z[i]; boxed final answer 5π/12 with verification that imaginary parts cancel and result is real as expected | Answers present but presentation weak: conclusion in (a) mentions importance without clarity; (b) conclusion present but tentative; (c) reaches numerical answer but doesn't confirm it matches expected real value or leaves answer as complex expression | Missing or wrong conclusions: no importance discussion in (a); fails to conclude irreducibility in (b); wrong numerical answer, or answer left as unevaluated residue sum, or forgets 2πi factor giving wrong magnitude |
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