UPSC Mathematics 2022 — Paper II

Solve each sub-part systematically with equal time allocation (~20% per part) since all carry 10 marks. Begin with (a) group isomorphism via Cayley table or generator mapping, (b) analytic function using Milne-Thomson method or CR equations, (c) convergence test via comparison/Dirichlet, (d) Laurent series with partial fractions and geometric expansion, and (e) two-phase simplex with artificial variables. Present solutions clearly with headings for each part.

• (a) Construct explicit isomorphism φ: G → G' showing φ(1)=0, φ(-1)=2, φ(i)=1, φ(-i)=3 and verify homomorphism property φ(ab)=φ(a)+₄φ(b)
• (b) Apply Milne-Thomson method: replace x by z and y by 0 in u-v expression to get f(z), then use condition f(π/2)=0 to determine constant
• (c) Establish absolute convergence via |cos x/(1+x²)| ≤ 1/(1+x²) and ∫₀^∞ dx/(1+x²) = π/2, or use Dirichlet test for conditional convergence
• (d) For region (i): write w=z-1, expand 1/(w²(w-2)) = -1/(2w²)·1/(1-w/2) using geometric series; for region (ii): use w=z-3, expand 1/((w+2)²w)
• (e) Phase I: minimize sum of artificial variables A₁+A₂ with constraints 2x₁+x₂-s₁+A₁=4, x₁+7x₂-s₂+A₂=7; Phase II: original objective with feasible basis
• Verify all group properties in (a), check analyticity via CR equations in (b), justify uniform convergence in (c), state radii of convergence in (d), and show optimality via simplex criteria in (e)

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

Solve this three-part numerical problem by allocating approximately 30% time to part (a) on complex integration (15 marks), 40% to part (b) on constrained optimization with geometric interpretation (20 marks), and 30% to part (c) on linear programming and duality (15 marks). Begin each part with clear identification of the mathematical technique, show complete computational steps with proper justification, and conclude with verified final answers including geometric interpretation for (b) and dual solution for (c).

• Part (a): Identify poles at z = -1 ± 2i, verify only z = -1 + 2i lies inside circle |z+1-i| = 2, apply Cauchy's residue theorem correctly
• Part (b): Set up Lagrangian with two constraints, derive normal equations, solve for extremal values, identify maximum and minimum as reciprocals of squares of semi-axes of elliptic section
• Part (c): Convert to standard form with slack variables, construct initial simplex tableau, iterate to optimality, verify Z = 1 at (0, 0, 2), formulate dual minimization problem
• Geometric interpretation for (b): Explain that extrema correspond to squares of distances from origin to points on ellipsoid section by plane through center
• Dual solution extraction: Read shadow prices from optimal tableau's z_j - c_j row for slack variables, verify strong duality with primal optimal value

Solve this three-part problem by allocating approximately 30% time to part (a) on ideal theory, 30% to part (b) on series convergence, and 40% to part (c) on transportation problem. Begin with clear definitions for (a), apply appropriate convergence tests for (b), and systematically execute VAM followed by optimality test for (c). Present each part with proper mathematical notation and logical flow.

• Part (a): Prove S is an ideal by showing closure under subtraction and absorption under multiplication by R[x] elements; identify R[x]/S ≅ ℝ × ℝ via evaluation maps at 0 and 1, hence not an integral domain as it has zero divisors
• Part (b): Identify general term as nⁿxⁿ/n!; apply Ratio Test or Root Test; show radius of convergence is 1/e; analyze behavior at boundary x = 1/e using Stirling's approximation or comparison
• Part (c): Apply Vogel's Approximation Method (VAM) to obtain initial BFS with m+n-1 = 6 basic variables; calculate row and column penalties correctly
• Part (c): Perform optimality test using MODI/UV method or stepping stone method; verify degeneracy handling if needed
• Part (c): Obtain optimal allocation and compute minimum transportation cost = 743 (or correct value based on calculations)

Solve each sub-part systematically with clear section headings. For (a), set up the cone equation and eliminate the arbitrary function f to get the PDE; for (b), perform Gauss elimination with back substitution; for (c)(i), convert decimal to octal/hexadecimal using successive division/multiplication, and for (c)(ii), expand to canonical POS form using Boolean algebra or K-map; for (d), construct Lagrangian in polar coordinates and derive Euler-Lagrange equations; for (e), verify irrotational flow condition, integrate for velocity potential, and solve streamline equations. Allocate approximately 15% time each to (a), (b), (c)(i), (c)(ii), and 20% each to (d) and (e) due to their derivational complexity.

• For (a): Eliminate arbitrary function f by differentiating the cone equation with respect to x, y, z and eliminating f_u, f_v to obtain the PDE (z-c)p + (z-c)q = (x-a) + (y-b) where p=∂z/∂x, q=∂z/∂y
• For (b): Form augmented matrix, perform row operations to achieve upper triangular form, back-substitute to get x=1, y=2, z=3 with verification
• For (c)(i): Convert (1093.21875)₁₀ = (2105.16)₈ and (1693.0628)₁₀ = (69D.101)₁₆ showing integer and fractional part conversions separately
• For (c)(ii): Express F(x,y,z) = xy + x'z in product of maxterms as ΠM(0,2,4,6) or (x+y+z)(x+y'+z)(x'+y+z)(x'+y'+z) using truth table or algebraic expansion
• For (d): Construct Lagrangian L = ½m(ṙ² + r²θ̇²) + k/r in polar coordinates, derive equations: m(r̈ - rθ̇²) = -k/r² and d/dt(mr²θ̇) = 0 (conservation of angular momentum)
• For (e): Verify ∇×v = 0, obtain velocity potential φ = -Mr⁻²cosθ, and derive streamline equations as r²sin²θ = constant, ψ = constant

Solve requires systematic derivation of exact solutions with complete mathematical rigor. Allocate approximately 40% of effort to part (a) as it carries 20 marks—apply separation of variables, determine Fourier coefficients for the parabolic initial condition, and write the complete series solution. Spend roughly 30% each on (b) and (c): for (b), construct the logic circuit using AND/OR/NOT gates and enumerate all 8 input combinations; for (c), set up proper coordinate system with cone vertex at origin, use parallel axis theorem or direct integration about slant generator, and express result purely in terms of M, h, a. Present each part distinctly with clear labeling.

• Part (a): Correct application of separation of variables u(x,t) = X(x)T(t), eigenvalues λ_n = n²π²/l², and Fourier sine series coefficients a_n = (4l²/n³π³)[1-(-1)ⁿ] for the initial condition x(l-x)
• Part (a): Final solution expressed as u(x,t) = Σ_{n=1}^∞ (8l²/n³π³)sin(nπx/l)exp(-n²π²t/l²) for odd n, with explicit recognition that even terms vanish
• Part (b): Correct Boolean simplification to f = xz + xȳ + y, or equivalent form, followed by circuit diagram with proper gate symbols (AND, OR, NOT) showing input/output flow
• Part (b): Complete truth table with all 8 rows (000 through 111) showing intermediate values ȳ, (ȳ+z), x·(ȳ+z), and final output f
• Part (c): Proper coordinate setup with cone geometry, density ρ = 3M/πa²h, and integration limits; correct application of perpendicular axis theorem or direct moment calculation about slant generator
• Part (c): Final expression I = (3M/20)(a² + 4h²) or equivalent simplified form involving slant height l = √(a²+h²), with dimensional verification [ML²]

Solve this three-part numerical problem by allocating approximately 30% time to part (a) PDE solution, 30% to part (b) numerical integration, and 40% to part (c) vortex dynamics. Begin with the complementary function and particular integral for (a), apply Simpson's 1/3rd rule with proper unit conversion for (b), and establish complex potential, velocity field, and streamline equations for (c). Conclude each part with verified final answers and proper dimensional analysis.

• Part (a): Factorize the operator (D² + DD' - 6D'²) = (D + 3D')(D - 2D'), find complementary function φ₁(y - 3x) + φ₂(y + 2x), and use correct particular integral method for x²sin(x+y)
• Part (b): Convert time interval to hours (h = 2/60 = 1/30 hour), verify even number of intervals for Simpson's 1/3rd rule, and compute distance = ∫v dt with proper unit handling
• Part (c): Construct complex potential w = -ik/2π[ln(z-a) + ln(z+a) - ½ln(z)], show dw/dz = 0 at stagnation points, derive streamline equations ψ = constant
• Part (c): Locate stagnation points on x-axis by solving velocity equations, find streamline passing through them, and verify the given algebraic relation 3√3(b²-a²)² = 16a³b
• Correct handling of units throughout: km/hour to km for distance, consistent time conversions, and dimensionless verification in part (c)

Solve all three sub-parts systematically, allocating approximately 30% time to part (a) on PDE canonical reduction, 30% to part (b) on RK4 numerical computation, and 40% to part (c) on complex potential verification and force calculation. Begin with identifying the PDE type and characteristic equations for (a), execute precise iterative calculations for (b), and apply Blasius theorem or residue calculus for force in (c). Present each part with clear headings and final boxed answers.

• For (a): Correct classification of PDE type (hyperbolic/parabolic/elliptic) via discriminant B²-4AC = (x+y)²-4xy = (x-y)² ≥ 0, identifying hyperbolic nature with characteristic curves ξ = x-y and η = x+y or similar
• For (a): Proper reduction to canonical form u_ξη = 0 or equivalent, followed by general solution u = f(x-y) + g(x+y) or variant
• For (b): Correct RK4 formulas with k₁, k₂, k₃, k₄ calculations at each step, showing two complete steps (h=0.1) to reach x=0.2 with y(0.2) ≈ 1.2734
• For (c): Verification that w = ik log[(z-ia)/(z+ia)] represents flow past cylinder |z|=a with y=0 as streamline, using conformal mapping properties and image system
• For (c): Application of Blasius theorem or pressure integration to find force, showing zero drag (d'Alembert paradox) and lift calculation using circulation Γ = 2πk
• For (c): Final force expression as (0, 2πρk²) or equivalent per unit length, with proper physical interpretation