UPSC Mathematics 2023 — Paper I

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