UPSC Mathematics 2023

Solve each sub-part systematically with equal time allocation (~20% per part) since all carry 10 marks each. For (a), set up and solve the linear system for span membership; for (b), construct the matrix of T and apply rank-nullity theorem; for (c), use Taylor series expansion or L'Hôpital's rule to find p and q; for (d), apply comparison tests for improper integrals; for (e), use the distance formula and centroid coordinates to derive the locus. Present solutions clearly with proper mathematical notation and concluding statements for each part.

• Part (a): Set up the augmented matrix [V₁ V₂ V₃ | (3,-1,0,-1)] and determine consistency via row reduction; conclude whether the target vector lies in the span
• Part (b): Construct the 3×3 matrix of T, compute its rank via determinant or row reduction, then apply rank-nullity theorem (rank + nullity = 3)
• Part (c): Expand cos x and sin x using Taylor series up to x³ terms, equate coefficients to ensure limit exists and equals 1, solving for p = -5/2 and q = -3/2
• Part (d): Identify the singularity at x=0, use limit comparison test with x^(-1/2) or direct evaluation showing |log x|/(1+x) is integrable, proving convergence
• Part (e): Use the plane equation x/a + y/b + z/c = 1 with distance condition 1/√(1/a²+1/b²+1/c²) = 3p, find centroid (a/4, b/4, c/4), eliminate parameters to obtain 9(1/x²+1/y²+1/z²) = 16/p²

Solve all three parts systematically, allocating approximately 30% time to part (a) on change of basis matrices, 30% to part (b) on triple integration using cylindrical coordinates, and 40% to part (c) covering both the elliptic paraboloid identification and the cone equation derivation. Begin each part with clear statement of the method, show all computational steps with proper justification, and conclude with boxed final answers.

• Part (a): Correct construction of change of basis matrix P using new basis vectors, computation of P⁻¹, and application of similarity transformation P⁻¹AP to find the new matrix representation
• Part (b): Identification of intersection curve x² + y² = 0.5, correct setup of cylindrical coordinate bounds (r: 0 to 1/√2, θ: 0 to 2π, z: 5r² to 6-7r²), and accurate evaluation of the triple integral
• Part (c)(i): Completion of squares to standard form, verification of elliptic paraboloid by identifying one linear term and two squared terms with same sign, identification of principal axis (z-axis direction) and principal planes
• Part (c)(ii): Correct determination of coordinates A(a,0,0), B(0,b,0), C(0,0,c), equation of sphere through OABC, intersection with plane to get circle ABC, and homogenization to derive the cone equation
• Proper matrix notation and determinant calculations for part (a) with verification of non-singularity
• Clear geometric interpretation in part (b) showing the paraboloid intersection forms a bounded solid
• Systematic algebraic manipulation in (c)(ii) showing the homogenization process with λ = 0 substitution

Solve this multi-part problem by allocating approximately 40% time to part (a) covering Cayley-Hamilton verification and recurrence relation (20 marks), 30% to part (b) on extreme point analysis using Hessian and higher-order tests (15 marks), and 30% to part (c) on sphere equations through given circle with two conditions (15 marks). Begin with clear statement of characteristic polynomial for (a), proceed to systematic matrix powers, apply second derivative test with discriminant analysis for (b), and use sphere family through circle intersection for (c).

• For (a)(i): Correct computation of characteristic polynomial det(A - λI) = -λ³ + λ² + λ - 1 and verification that A³ = A² + A - I
• For (a)(ii): Proof of recurrence Aⁿ = Aⁿ⁻² + A² - I using Cayley-Hamilton, and efficient computation of A⁴⁰ via pattern or binary exponentiation
• For (b): Computation of first partials fx = 8x³ - 6xy, fy = -3x² + 2y, verification that (0,0) is critical point, and application of discriminant D = fxx·fyy - (fxy)² with higher-order analysis showing saddle point
• For (c)(i): Formation of sphere family S: x²+y²+z²-4x-6y+2z-16 + λ(3x+y+3z-4) = 0 and substitution of (1,0,-3) to find λ
• For (c)(ii): Condition that given circle is great circle requires sphere center to lie on plane 3x+y+3z-4=0, yielding center (-4+3λ)/2, (-6+λ)/2, (2+3λ)/2 satisfying plane equation
• Explicit final answers: A⁴⁰ expression, conclusion that (0,0) is not extreme point (saddle), and two sphere equations with specific λ values

Solve all three parts systematically, allocating approximately 30% time to part (a) matrix rank reduction, 40% to part (b) curve tracing as it carries highest marks, and 30% to part (c) 3D geometry proof. Begin with clear identification of each part, show complete row operations for (a), detailed curve analysis with asymptotes and intercepts for (b), and rigorous parametric derivation for (c).

• Part (a): Correct reduction of 4×4 matrix to row-reduced echelon form using elementary row operations, identification of pivot positions, and accurate rank determination
• Part (b): Complete curve tracing of y²(x²-1)=2x-1 including symmetry analysis, asymptotes (vertical at x=±1), intercepts, domain restrictions, and sketch of two branches
• Part (c): Proper parameterization of lines meeting given skew lines and circle, elimination of parameters to derive the stated quartic locus surface
• Verification of rank by checking linear dependence of rows/columns or using determinant test for part (a)
• Analysis of singular points and behavior near asymptotes for part (b) curve

Solve all five sub-parts systematically, allocating approximately 20% time to each part given equal 10-mark weighting. For (a), apply integrating factor method for linear ODE; for (b), use Laplace transform properties with Fubini's theorem; for (c), draw free-body diagram and apply equilibrium conditions; for (d), use SHM equations with phase analysis; for (e), apply vector triple product identity before differentiation. Present solutions with clear intermediate steps and boxed final answers.

• Part (a): Identify integrating factor e^(-x²), solve using standard formula, apply initial condition y(0)=1, and manipulate to express using error function erf(x)
• Part (b): Prove the identity using Laplace definition and changing order of integration (Fubini/Tonelli), then apply to f(t)=e^(-t)-e^(-3t) with F(p)=1/(p+1)-1/(p+3) to get ln(3)
• Part (c): Construct geometry with 45° beam, locate contact points, write three equilibrium equations (ΣFx=0, ΣFy=0, ΣM=0), eliminate reactions to find a/l ratio, then back-substitute for reaction magnitudes
• Part (d): Use SHM equation x = Asin(ωt+φ), apply conditions at point P to find phase, determine return time using periodicity and symmetry, evaluate p when elapsed time is T/2
• Part (e): Apply vector identity ā×(b̄×c̄) = (ā·c̄)b̄ - (ā·b̄)c̄, compute scalar products, differentiate resultant vector component-wise, evaluate at θ=π/2 and θ=π

Solve this multi-part numerical problem by allocating approximately 30% time to part (a) on third-order linear ODE, 30% to part (b) on projectile motion derivation, and 40% to part (c) on surface integral evaluation and Gauss-Divergence theorem verification. Begin each part with clear problem identification, show complete working with intermediate steps, and conclude with boxed final answers. For part (c), explicitly compute both the surface integral directly and the volume integral via divergence theorem to demonstrate verification.

• Part (a): Correctly find complementary function by solving characteristic equation m³ - 3m² + 4m - 2 = 0 with roots m = 1, 1±i, and construct particular integrals for both eˣ and cos x terms using appropriate methods
• Part (a): Apply correct operator methods or undetermined coefficients to obtain PI for eˣ (resonance case) and PI for cos x, then combine for complete general solution
• Part (b): Set up projectile equations from sea level O₁: derive range R₁ = v²sin(2θ)/g using standard trajectory equations y = xtanθ - gx²/(2v²cos²θ)
• Part (b): Set up modified equations from height h at O₂, find new range by solving when projectile hits sea level (y = -h), derive R₂ = (v²cosθ/g)[sinθ + √(sin²θ + 2gh/v²)], prove R₂ > R₁ and establish required ratio
• Part (c): Identify S as upper hemisphere 4x² + 4y² + 4z² = 1, z ≥ 0 with radius 1/2, correctly compute divergence ∇·F = 2yz² + 2z²x + 2zy² for verification
• Part (c): Evaluate surface integral directly using spherical coordinates or projection method, then compute volume integral ∭(∇·F)dV over hemisphere, show equality to verify Gauss-Divergence theorem

Solve this multi-part problem by allocating approximately 40% time to part (a) covering both differential equations (20 marks), 30% to the mechanics problem (b) involving equilibrium of a hemisphere (15 marks), and 30% to the differential geometry proof in part (c) on curvature and torsion (15 marks). Begin with clear identification of the solution method for each sub-part, execute calculations systematically with proper justification, and conclude with verification of results including the singular solution in (a)(ii) and the constant angle property in (c).

• For (a)(i): Recognize the equation as exact after rewriting in differential form Mdx + Ndy = 0, verify ∂M/∂y = ∂N/∂x, and find the potential function ψ(x,y) = x²y³ + 2x + 2e^(4y) = c
• For (a)(ii): Apply substitution y = u, xy = v to transform to Clairaut's form v = pu + f(p) where p = dv/du, obtain general solution v = cu + c²/(c+1), and verify y + 4x = 0 as singular solution via c-discriminant or envelope method
• For (b): Draw correct free-body diagram with three forces (weight W at center of mass, tension T along string, normal reaction R perpendicular to wall), apply equilibrium conditions ΣFx = 0, ΣFy = 0, ΣM = 0 about appropriate point, use geometry relating θ and φ through hemisphere radius a, and derive tanφ - tanθ = 1/2
• For (c): Use Frenet-Serret formulas with fixed direction making angle θ with tangent, express d𝐭/ds = κ𝐧 and d𝐛/ds = -τ𝐧, show that 𝐭·𝐚 = cosθ implies 𝐧·𝐚 = 0 and 𝐛·𝐚 = sinθ, derive τ/κ = tanθ, and prove converse by showing constant τ/κ implies 𝐭·𝐚 is constant
• Cross-cutting: Maintain dimensional consistency, state all assumptions explicitly (smooth wall, uniform hemisphere, inextensible string), and verify boundary conditions or special cases where applicable

Solve this three-part numerical problem by allocating approximately 30% time to part (a) Laplace transform IVP, 40% to part (b) central force motion and orbit determination, and 30% to part (c) vector calculus identity proof and verification. Begin with clear statement of given conditions, apply appropriate mathematical techniques systematically, show all computational steps explicitly, and conclude with precise final answers for each sub-part.

• Part (a): Correct application of Laplace transform to second-order ODE with proper handling of initial conditions y(0)=1, y'(0)=0 and symbolic f(t)
• Part (a): Accurate partial fraction decomposition and inverse Laplace transform to obtain complete solution y(t)
• Part (b): Correct formulation of central force problem using Binet's equation or energy equation, identifying the given acceleration λ(r⁵ - c²r)
• Part (b): Derivation of orbital equation and verification whether the path satisfies x⁴ + y⁴ = c² through polar to Cartesian conversion
• Part (c): Rigorous proof of vector identity ∇·(φf⃗) = ∇φ·f⃗ + φ(∇·f⃗) using component-wise expansion or index notation
• Part (c): Correct computation of ∇·(f(r)/r · r⃗) using spherical/polar coordinate formulas and explicit verification of the identity

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)

Begin with a clear statement of what is to be proved for part (a), then solve for (b) and (c). Allocate approximately 30% time to part (a) (group theory proof), 30% to part (b) (Lagrange multipliers), and 40% to part (c) (contour integration, highest marks). Structure as: (a) statement → application of Sylow/Cauchy theorems → conclusion; (b) setup of distance function and constraint → Lagrange equations → solving critical points; (c) contour setup → residue calculation → final evaluation. Conclude with boxed final answers for each part.

• Part (a): Application of Cauchy's theorem or Sylow theorems to establish existence of element of order p, hence cyclic subgroup of order p
• Part (a): Explicit use of non-commutativity to rule out cyclic group of order 2p, ensuring the subgroup of order p is proper
• Part (b): Correct formulation of distance squared function D = (x-2)² + (y-6)² + (z-3)² with constraint g = x² + y² + z² - 4 = 0
• Part (b): Setting up and solving the system ∇D = λ∇g with geometric interpretation that extrema occur along line through origin and P
• Part (c): Substitution z = e^(iθ) to convert real integral to complex contour integral over unit circle |z|=1
• Part (c): Correct identification of poles inside unit circle (at z = -1/2) and calculation of residue
• Part (c): Evaluation of contour integral using residue theorem and extraction of real part to obtain final answer

Solve this multi-part problem by proving results in (a) and (b) before applying computational methods in (c). Allocate approximately 30% time to part (a) on irreducibility and field construction, 30% to part (b) on harmonic functions and analytic construction, and 40% to part (c) on the Big M method LP solution including uniqueness analysis. Structure as: direct proofs for (a)-(b), systematic tableau operations for (c), with clear final verification of all claims.

• For (a): Prove x²+1 has no roots in Z₃ (hence irreducible), then apply the theorem that F[x]/⟨p(x)⟩ is a field when p(x) is irreducible, counting 9 elements explicitly
• For (b): Verify uₓₓ + uᵧᵧ = 0 using product rule differentiation, then find v(x,y) via Cauchy-Riemann equations or Milne-Thomson method, finally express f(z) = z·e^z
• For (c): Convert minimization to standard form using surplus and artificial variables with Big M penalty, construct initial simplex tableau
• For (c): Execute simplex iterations correctly, identifying entering and leaving variables at each pivot
• For (c): State optimal solution with Z_min = 15 at (x₁, x₂) = (6, 3), and justify uniqueness by checking no alternative optimal solutions exist in final tableau

Begin with a clear statement of definitions for oscillation, supremum, and bounded variation. For part (a), establish the equivalence through careful set-theoretic arguments showing ω(f) = sup|f(x₁)-f(x₂)|. For part (b), use Taylor series expansion to identify the order of the pole at z=0 and extract coefficients for the principal part. For part (c), apply the Hungarian algorithm to solve the 5×5 assignment problem, then re-optimize with the additional constraint. Allocate approximately 30% time to (a), 30% to (b), and 40% to (c) given its higher weightage and computational demand.

• For (a): Correct definition of oscillation ω(f,[a,b]) = M-m where M=sup f, m=inf f; proof that this equals sup{|f(x₁)-f(x₂)|} through showing both inequalities
• For (a): Demonstration that for any ε>0, there exist x₁,x₂ with f(x₁)>M-ε/2 and f(x₂)<m+ε/2, establishing the supremum attainment
• For (b): Taylor expansion of sin z = z - z³/6 + z⁵/120 - ... and e^z = 1 + z + z²/2 + ... to determine z-sin z = z³/6 - z⁵/120 + ...
• For (b): Identification of z=0 as a pole of order 2 (simple pole cancellation leaves z³ in denominator, e^z≈1), computation of Laurent series principal part coefficients a₋₁, a₋₂
• For (c): Correct setup of cost matrix, application of Hungarian algorithm (row/column reduction, minimum lines covering zeros, matrix adjustments)
• For (c): Optimal assignment without constraints and verification of optimality; modified solution with constraint IV≠C using branch-and-bound or re-optimization
• For (c): Clear statement of minimum total time in both cases with proper units (hours)

Solve each sub-part systematically with clear working: for (a) apply partial differentiation to eliminate arbitrary functions; for (b) execute Euler's method with 4 iterations; for (c)(i)-(ii) perform octal and hexadecimal subtraction with borrowing; for (d) construct Hamiltonian from T-V and derive canonical equations; for (e) use complex potential for source-sink superposition. Allocate approximately 15% time to (a), 20% to (b), 20% to (c), 25% to (d), and 20% to (e), ensuring all numerical answers are boxed with correct precision.

• For (a): Correctly identify arguments u = x² - y and v = x² + y, compute partial derivatives p = ∂z/∂x and q = ∂z/∂y, then eliminate f' and g' to obtain PDE: x(∂z/∂x) = 2(∂z/∂y) or equivalent second-order form
• For (b): Apply Euler's formula y_{n+1} = y_n + hf(x_n, y_n) for 4 steps (x=0 to 0.4), showing each iteration with h=0.1, final answer y(0.4) ≈ 1.4615 (to 4 decimal places)
• For (c)(i): Perform octal subtraction (634.235)₈ - (132.223)₈ = (502.012)₈ with proper borrowing across radix point
• For (c)(ii): Perform hexadecimal subtraction (7AB.432)₁₆ - (5CA.D61)₁₆ = (1E0.871)₁₆ handling borrow from 16's complement
• For (d): Form Hamiltonian H = T + V = ½m(p_r²/m² + p_θ²/(m²r²)) + GMm(1/2a - 1/r), derive ṙ = ∂H/∂p_r, θ̇ = ∂H/∂p_θ, ṗ_r = -∂H/∂r, ṗ_θ = -∂H/∂θ = 0
• For (e): Construct complex potential W = 2m ln(z-2) - m ln(z-2-i) - m ln(z-2+i), extract stream function ψ = Im(W), obtain streamline equation or implicit form

Solve all three sub-parts systematically: for (a) use Monge's auxiliary equations to find the general solution and apply boundary conditions; for (b) apply Gauss-Seidel iteration with proper convergence check to 4 significant figures; for (c) transform coordinates, eliminate cross terms by choosing θ = ½ tan⁻¹(2k/(m(w₁²-w₂²))), then derive uncoupled Lagrange equations. Allocate time proportionally: ~30% to (a), ~30% to (b), and ~40% to (c) given mark distribution.

• For (a): Recognize PDE as parabolic type (B²-AC=0), use characteristic coordinates ξ=y+2x, η=y to reduce to canonical form and obtain general solution z = f(y+2x) + x·g(y+2x)
• For (a): Apply boundary conditions on lines z=x=0 and z-1=x-y=0 to determine arbitrary functions f and g, yielding specific surface equation
• For (b): Verify diagonal dominance of coefficient matrix, rewrite equations in iterative form, execute Gauss-Seidel iterations until convergence to 4 significant figures
• For (c): Transform kinetic and potential energy terms using rotation matrix, collect coefficients of q₁q₂ and set to zero to find tan(2θ) = 2k/(m(w₁²-w₂²))
• For (c): Obtain diagonalized Lagrangian L = ½m(Ω₁²q̇₁² + Ω₂²q̇₂²) - ½m(Ω₁²q₁² + Ω₂²q₂²) and derive two independent harmonic oscillator equations
• For (c): Express final frequencies Ω₁² = ω₁²cos²θ + ω₂²sin²θ - (k/m)sin2θ and Ω₂² = ω₁²sin²θ + ω₂²cos²θ + (k/m)sin2θ in terms of original parameters

Solve this multi-part numerical problem by allocating approximately 30% time to part (a)(i) CNF conversion, 30% to part (a)(ii) DNF and truth table, 20% to part (b) mechanics proof, and 20% to part (c) wave equation solution. Begin with Boolean algebra manipulations using De Morgan's laws and canonical forms, then proceed to the constrained motion analysis with energy and angular momentum conservation, and finally apply separation of variables with Fourier series for the PDE. Present each sub-part clearly with proper mathematical notation and logical flow.

• For (a)(i): Apply De Morgan's laws and distributive laws to convert f(x,y,z,t) = xyz + x̄y(t+z̄) into CNF; identify maxterms where f=0 and express as product of sums
• For (a)(ii): Simplify f(x,y,z) = x + (x̄ȳ + x̄z) + z using Boolean identities, convert to DNF as sum of minterms, and construct complete 8-row truth table
• For (b): Set up Lagrangian for ball in rolling cylinder with constraint of rolling without slipping; derive energy equation and condition for complete loop using effective potential
• For (c): Apply separation of variables u(x,t) = X(x)T(t) to wave equation; determine eigenvalues λn = nπ/L and eigenfunctions sin(nπx/L); apply initial conditions u(x,0)=x and ut(x,0)=1 to find Fourier coefficients
• For (c): Compute Fourier sine series coefficients bn = (2/L)∫₀ᴸ x sin(nπx/L)dx and cn = (2/nπa)∫₀ᴸ sin(nπx/L)dx for the complete solution

Solve this three-part numerical problem by allocating approximately 30% time to part (a) on PDE canonical reduction, 30% to part (b) on Regula-Falsi root-finding, and 40% to part (c) on Navier-Stokes verification and pressure determination. Begin with clear identification of equation types, proceed through systematic derivations and iterative calculations, and conclude with boxed final answers for each sub-part.

• Part (a): Correct classification of the second-order PDE and identification of characteristic curves to transform to canonical form
• Part (a): Proper substitution of new variables and reduction to standard canonical form (parabolic/hyperbolic/elliptic)
• Part (b): Verification that f(0)·f(3) < 0 for root existence and correct Regula-Falsi iteration formula setup
• Part (b): Iterative computation showing convergence to 6 decimal places with clear tabulation of iterations
• Part (c): Verification of continuity equation (∇·u = 0) as necessary condition for Navier-Stokes solution
• Part (c): Substitution into momentum equations to determine constraints on c and fluid properties
• Part (c): Integration of pressure gradients to obtain p(x,y,z) with proper incorporation of body force B_z = -g