UPSC Mathematics 2025 — Paper II

This multi-part question requires proving results in group theory, analyzing convergence, expanding complex functions, and solving linear programming systems. Allocate approximately 15-18 minutes each for parts (a), (b), and (d) which demand rigorous proofs and careful Laurent series construction; spend 10-12 minutes each on (c) and (e). Begin each part with clear statement of what is to be shown, present logical derivation with theorems cited, and conclude with explicit verification of the required result.

• For (a): Apply the product formula |HK| = |H||K|/|H∩K| and use the bound |HK| ≤ |G| to derive contradiction if H∩K = {e}
• For (b): Use the non-Abelian property and orders of x, y to eliminate xy = yx and xy = yx², showing only xy = yx² satisfies group axioms
• For (c): Show Σ1/n diverges (harmonic series) for absolute convergence, then apply Leibniz test for conditional convergence
• For (d): Perform partial fraction decomposition, then expand 1/(z+1) as geometric series in 1/z for |z|>1 and 1/(z+3) as series in z/3 for |z|<3
• For (e): Identify m=2 equations, n=4 variables, so C(4,2)=6 basic solutions; classify each by checking non-negativity and degeneracy
• Correct enumeration of all six basic solutions with proper identification of basic/non-basic variables for part (e)

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.

• 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

Solve all three parts systematically, allocating approximately 30% time to part (a) [15 marks], 40% to part (b) [20 marks], and 30% to part (c) [15 marks]. Begin with identifying singularities and applying Cauchy's residue theorem for (a), then use Lagrange multipliers for constrained optimization in (b), and finally construct the dual LP and solve via simplex method for (c). Present each part with clear headings and show all computational steps.

• Part (a): Identify poles at z=0 (order 2) and z=-1 (order 3) inside |z|=2; apply Cauchy's residue theorem with correct residue formulas for higher-order poles
• Part (a): Compute residues using derivatives: Res(f,0) via first derivative of e^z/(z+1)^3 and Res(f,-1) via second derivative of e^z/z^2
• Part (b): Set up Lagrangian F = 8xyz + λ(1 - x²/a² - y²/b² - z²/c²) for inscribed rectangular parallelopiped with vertices (±x,±y,±z)
• Part (b): Derive critical conditions ∂F/∂x = ∂F/∂y = ∂F/∂z = 0 leading to x²/a² = y²/b² = z²/c² = 1/3, hence maximum volume 8abc/(3√3)
• Part (c): Convert primal (maximization with mixed constraints) to standard form; formulate dual minimization problem with correct variable correspondence
• Part (c): Solve dual using simplex method or verify complementary slackness; recover primal optimal solution x₁=7/2, x₂=1/2 with Z_max=12.5

Begin with (a) by verifying the homomorphism property φ(f+g)=φ(f)+φ(g) and φ(fg)=φ(f)φ(g), then identify ker(φ)=⟨x⟩ and apply the First Isomorphism Theorem to establish primeness via Z[x]/⟨x⟩≅Z (an integral domain but not a field). For (b), construct the proof using uniform continuity on closed intervals and the Darboux criterion, showing upper and lower sums converge. Devote maximum effort to (c): first verify the initial feasible solution (degenerate, with only 5 allocations for 6 rows+columns-1=5, so non-degenerate), compute opportunity costs using u-v method, identify negative Δij, and iterate to optimality. Allocate roughly 30% time to (a), 25% to (b), and 45% to (c) given its higher weight and computational demand.

• For (a): Verify φ preserves addition and multiplication; show ker(φ)=⟨x⟩; apply First Isomorphism Theorem to get Z[x]/⟨x⟩≅Z; conclude ⟨x⟩ is prime (Z is integral domain) but not maximal (Z is not a field)
• For (b): State that continuous functions on [a,b] are uniformly continuous; use this to show for any ε>0, there exists partition P with U(P,f)-L(P,f)<ε; conclude Riemann integrability via Darboux criterion
• For (c): Verify initial solution is feasible (supply=demand=45) but degenerate; compute dual variables ui, vj using occupied cells; calculate opportunity costs Δij=cij-(ui+vj) for unoccupied cells
• For (c): Identify most negative Δij and construct closed loop; determine θ=min allocation at decreasing corners; perform basis change and recompute until all Δij≥0
• For (c): State optimal allocations and minimum total cost with proper units (currency units); verify optimality conditions are satisfied

Solve each of the five independent parts systematically, presenting clear working for PDE solution, iterative numerical method, number system conversions with Boolean algebra, Lagrangian derivation with substitution, and complex potential streamlines. Structure as five distinct sections with proper labeling (a) through (e), showing complete derivations before stating final results.

• Part (a): Factorize operator as (D-D')(D+2D'), find CF using f1(y+x)+f2(y-2x), and PI using 1/(D²+DD'-2D'²) y sin x with proper shifting
• Part (b): Check diagonal dominance, rearrange if needed, apply Gauss-Seidel iteration formula with at least 3-4 iterations showing convergence to x=1, y=-2, z=3
• Part (c)(i): Binary conversion of 3479 using division method giving 110110010111, and hexadecimal to decimal with positional weights for 7AE.9F
• Part (c)(ii): Complete 8-row truth table for F(x,y,z), identify minterms where F=1, and express as Σm(1,2,4,6) or expanded sum of products
• Part (d): Express arc length element ds² = 4a²sin²(θ/2)dθ², form L = 2ma²sin²(θ/2)(dθ/dt)² - mga(1+cosθ), substitute u = cos(θ/2) to obtain SHM equation
• Part (e): Construct complex potential W = m ln[(z-a/2)/(z+a/2)] + m ln[(z-a²/2z̄)/(z+a²/2z̄)] for circle theorem, extract imaginary part for stream function ψ, and manipulate to given form

Solve this three-part numerical problem by allocating approximately 40% time to part (a) given its 20 marks weightage, and roughly 30% each to parts (b) and (c). Begin with clear identification of the mathematical technique for each sub-part: separation of variables for Laplace equation, Boolean algebraic simplification with K-map or algebraic manipulation, and integration for moment of inertia. Present solutions sequentially with proper mathematical derivations, diagrams where requested, and concluding with boxed final answers for each part.

• Part (a): Apply separation of variables u(x,y) = X(x)Y(y), derive eigenvalues λ_n = nπ/a, obtain Fourier sine series coefficients for the non-homogeneous boundary condition u(x,b) = f(x)
• Part (b): Simplify F = yz + xz + xy using Boolean algebra or K-map, identify minimal sum-of-products form, draw AND-OR or NAND-NAND gate network with proper labeling
• Part (c): Set up triple integral in cylindrical coordinates for I_z = ½MR² and I_x = I_y = M(3R² + L²)/12 using ρ = M/(πR²L), minimize I_x by differentiating with respect to L/R ratio to get L/R = √3
• Correct handling of boundary conditions in (a): three homogeneous and one non-homogeneous condition leading to Sturm-Liouville problem
• Proper gate-level diagram in (b) showing input variables, logic gates, and output with standard IEEE/ANSI symbols
• Dimensional consistency check in (c): all moments of inertia having units [ML²] and the optimal ratio being dimensionless

Solve all three sub-parts systematically, allocating approximately 30% time to part (a) on PDE complete integral using Charpit's method, 30% to part (b) on Lagrange interpolation with error bound derivation, and 40% to part (c) on Navier-Stokes verification. Begin with clear identification of the method for each part, show complete derivations with all intermediate steps, and conclude with boxed final answers for each sub-part.

• Part (a): Correct application of Charpit's auxiliary equations to find complete integral of z(p²-q²) = x-y, including proper parameterization and final form with arbitrary constants
• Part (b): Construction of Lagrange interpolation polynomial of degree ≤2 using given data points (0,1), (1,3), (3,55), with explicit polynomial expression
• Part (b): Derivation of truncation error bound using the formula |f(x)-P₂(x)| ≤ M₃|x(x-1)(x-3)|/6 where M₃ bounds |f'''(ξ)|
• Part (c): Verification that u(y) satisfies continuity equation (∂u/∂x + ∂v/∂y = 0) for incompressible flow with v=w=0
• Part (c): Substitution of velocity profile into x-momentum Navier-Stokes equation, showing balance between pressure gradient and viscous terms with dp/dx constant
• Part (c): Verification that y-momentum and z-momentum equations are satisfied with v=w=0 and p=p(x) only

Solve this multi-part problem by allocating approximately 30% time to part (a) on PDE characteristics, 30% to part (b) on numerical integration, 20% to part (c)(i) on Hamiltonian mechanics in spherical coordinates, and 20% to part (c)(ii) on the coupled oscillator system. Begin each part with clear statement of the governing equations, show systematic derivation with intermediate steps, and conclude with boxed final answers for each sub-part.

• For (a): Correctly identify Charpit's equations, solve dx/2p = dy/2q = dz/2(p²+q²) = dp/0 = dq/0, obtain p = a, q = √(2-a²), and find the complete integral z = ax + √(2-a²)y + b; apply initial condition x=0, z=y to determine the integral surface
• For (b): Use Taylor expansion of f(x) about x₀ and x₁ to match coefficients, determine p = 1/12, derive error term as -(h⁵/720)f⁽⁴⁾(ξ), and construct composite rule by summing over N subintervals with endpoint derivative corrections
• For (c)(i): Express kinetic energy T = ½m(ṙ² + r²θ̇² + r²sin²θ φ̇²), construct H = T + V with given potential, derive Hamilton's equations: ṙ = ∂H/∂pᵣ, θ̇ = ∂H/∂p_θ, etc., showing all six canonical equations
• For (c)(ii): Identify non-standard Lagrangian with coupled velocities, perform Legendre transform with pₓ = mẏ, pᵧ = mẺx, obtain H = pₓpᵧ/m + mω₀²xy, derive Hamilton's equations and identify as 2D isotropic oscillator with rotated coordinates
• Demonstrate dimensional consistency throughout: [p] = [q] = L⁰ for (a), [p] = T for (b) constant, [H] = ML²T⁻² for both mechanical parts