Mathematics UPSC PYQ 2025

Let H and K be two subgroups of a group G such that o(H) > √o(G) and o(K) > √o(G). Show that H ∩ K ≠ {e}, where e is the identity element. Here o(H), o(K) and o(G) denote the order of H, K and G respectively. Let G = {e, x, x², y, yx, yx²} be a non-Abelian group with o(x) = 3 and o(y) = 2. Show that xy = yx² (where e is the identity element of G and o(x), o(y) denote the order of the elements x, y respectively). Examine whether the series Σ_{n=1}^{∞} (–1)^{n-1}/n is absolutely or conditionally convergent. Expand f(z) = 1/[(z+1)(z+3)] in a Laurent series valid for 1 < |z| < 3. How many basic solutions are there for the following system of equations? 2x₁ – x₂ + 3x₃ + x₄ = 6 4x₁ – 2x₂ – x₃ + 2x₄ = 10 Find all of them. Furthermore, find the number of basic solutions, which are feasible/non-feasible/non-degenerate. <!--qid:MAINS_2025_Mathematics-II_Q1-->

Define Cauchy sequence and prove that every convergent sequence of real numbers is a Cauchy sequence. What is the importance of Cauchy condition? Show that 3 is an irreducible element in the integral domain Z[i]. Use the method of contour integration to prove that ∫_{–∞}^{∞} (x² – x + 2)/(x⁴ + 10x² + 9) dx = 5π/12. <!--qid:MAINS_2025_Mathematics-II_Q2-->

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]. Prove that every continuous function is Riemann integrable. 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 Warehouse A 5 2 4 3 22 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. <!--qid:MAINS_2025_Mathematics-II_Q4-->

Evaluate the integral ∮_C e^z / [z² (z+1)³] dz, C : |z| = 2. Show that the volume of the greatest rectangular parallelepiped that can be inscribed in the ellipsoid x²/a² + y²/b² + z²/c² = 1 is 8abc / (3√3). Apply the principle of duality to solve the following linear programming problem : Maximize Z = 3x₁ + 4x₂ subject to x₁ – x₂ ≤ 1, x₁ + x₂ ≥ 4, x₁ – 3x₂ ≤ 3, x₁, x₂ ≥ 0. <!--qid:MAINS_2025_Mathematics-II_Q3-->

Find the solution of the equation (D² + DD′ – 2D′²) z = y sin x, where D = ∂/∂x and D′ = ∂/∂y. Solve the following system of linear equations by Gauss-Seidel method: 10x + 2y + z = 9 2x + 20y – 2z = –44 –2x + 3y +10z = 22 (i) Convert the number (3479)₁₀ into binary system and the number (7AE·9F)₁₆ into decimal system. (ii) Determine the truth table for the Boolean function F(x, y, z) = (x + y + z′)(x′ + y′). Also derive the full disjunctive normal form of F(x, y, z) from the truth table. <!--qid:MAINS_2025_Mathematics-II_Q5-->