(a) Let G be a group of order 10 and G′ be a group of order 6. Examine whether there exists a homomorphism of G onto G′. (10 marks)

(b) Express the ideal 4Z + 6Z as a principal ideal in the integral domain Z. (10 marks)

(c) Test the convergence of

Question

(a) Let G be a group of order 10 and G′ be a group of order 6. Examine whether there exists a homomorphism of G onto G′. (10 marks)

(b) Express the ideal 4Z + 6Z as a principal ideal in the integral domain Z. (10 marks)

(c) Test the convergence of the series
$$\sum\limits_{n=1}^{\infty} \frac{1.3.5...(2n-1)}{2.4.6...(2n)} \cdot \frac{x^{2n+1}}{(2n+1)}, \quad x > 0$$
(10 marks)

(d) State the sufficient conditions for a function $f(z) = f(x+iy) = u(x,y) + iv(x,y)$ to be analytic in its domain. Hence, show that $f(z) = \log z$ is analytic in its domain and find $\frac{df}{dz}$ (10 marks)

(e) A person requires 24, 24 and 20 units of chemicals A, B and C respectively for his garden. Product P contains 2, 4 and 1 units of chemicals A, B and C respectively per jar and product Q contains 2, 1 and 5 units of chemicals A, B and C respectively per jar. If a jar of P costs ₹ 30 and a jar of Q costs ₹ 50, then how many jars of each should be purchased in order to minimize the cost and meet the requirements? (10 marks)

UPSC Answer Check · Accepted Answer

Examine each sub-part systematically with equal time allocation (~20% per part) since all carry 10 marks. Begin with clear statements of theorems/conditions for (a), (b), (d); show complete computational steps for (c) and (e). Structure as: direct response to each sub-part labeled (a)-(e), with concise justification for theoretical parts and detailed working for computational parts. No separate introduction or conclusion needed.
- For (a): Apply First Isomorphism Theorem to show no onto homomorphism exists since gcd(10,6)=2 but 6∤10, or use Lagrange's theorem on kernel/image orders
- For (b): Use property that 4Z + 6Z = gcd(4,6)Z = 2Z, proving the sum is principal ideal generated by 2
- For (c): Apply Raabe's test or Gauss test to determine convergence at x=1 (diverges) and x<1 (converges), with careful handling of the limit
- For (d): State Cauchy-Riemann equations as sufficient conditions; verify u=log|z|, v=arg(z) satisfy them; derive df/dz = 1/z
- For (e): Formulate LPP with objective minimize Z=30x+50y subject to 2x+2y≥24, 4x+y≥24, x+5y≥20, x,y≥0; solve by graphical method or simplex to get optimal solution at (8,4)

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly identifies group orders and homomorphism constraints for (a); properly writes ideal sum for (b); sets up correct series form and identifies test applicability for (c); accurately states C-R equations and domain of log z for (d); correctly formulates all constraints and objective function for (e)	Minor errors in stating theorems or constraints, such as missing non-negativity in (e) or imprecise domain specification in (d)	Fundamental setup errors like wrong group properties, incorrect ideal operations, misidentified test for series, or wrong LPP formulation
Method choice	20%	10	Selects optimal methods: First Isomorphism Theorem for (a), gcd property for (b), Raabe/Gauss test for (c), C-R verification for (d), graphical/simplex method for (e); justifies why alternatives are inferior	Correct but suboptimal methods chosen, or correct methods without justification; e.g., using ratio test alone for (c) where it fails	Inappropriate methods selected, such as using element-wise checking for homomorphism in (a) or ignoring integer constraints in (e)
Computation accuracy	20%	10	Precise calculations: correct gcd computation, accurate limit evaluation in Raabe's test with proper series manipulation, exact partial derivatives for C-R equations, correct corner point evaluation in LPP	Minor arithmetic slips that don't affect final conclusion, such as sign errors in derivatives or slight miscalculations in limit evaluation	Major computational errors leading to wrong conclusions, such as incorrect limit values, wrong optimal solution in LPP, or invalid derivative calculation
Step justification	20%	10	Every non-trivial step justified: why kernel order must divide 10, why gcd generates the ideal, rigorous limit justification with series expansion, explicit verification of C-R equations, clear reasoning for optimal vertex selection in LPP	Some steps justified but gaps remain; logical flow present but missing key justifications for critical transitions	Unjustified leaps, missing logical connections, or 'hence proved' without intermediate reasoning; assertion without verification
Final answer & units	20%	10	Precise conclusions: explicit 'no onto homomorphism exists' with reason; 2Z clearly stated; convergence interval (0,1] with behavior at endpoints; df/dz = 1/z; optimal purchase 8 jars of P and 4 jars of Q with minimum cost ₹440	Correct final answers but poorly presented, missing endpoint analysis in (c), or cost value without units	Missing or wrong final answers, incomplete conclusions, or failure to state the principal ideal generator explicitly

Q1

Directive word: Examine

How this answer will be evaluated

Approach

Key points expected

Evaluation rubric

Practice this exact question

More from Mathematics 2023 Paper II