UPSC Mathematics 2025 — Paper I

Solve each sub-part systematically with clear mathematical reasoning. For (a), verify linear independence and extend to basis; for (b), construct the matrix representation and apply rank-nullity; for (c), set up the volume function and optimize using calculus; for (d), use the functional equation to derive the differential equation; for (e), use the cone generator method. Allocate approximately 20% time per sub-part given equal marks distribution, ensuring each part receives complete treatment with proper justification.

• (a) Verify linear independence of the three vectors by checking no non-trivial combination equals zero; extend to basis by adding a fourth vector not in their span, such as (0,0,1,0) or (1,0,0,0)
• (b) Construct the 3×4 matrix representation of T; find rank by row reduction or determinant of 3×3 minors; determine nullity using rank-nullity theorem (nullity = 4 - rank = 2)
• (c) Express volume V = (6-2h)(2-2h)h = 4h(3-h)(1-h); find dV/dh = 0 yielding 3h² - 8h + 3 = 0; select valid root h = (4-√7)/3 ≈ 0.45m within domain 0 < h < 1
• (d) Use f(x+y) = f(x)f(y) to show [f(x+h)-f(x)]/h = f(x)[f(h)-1]/h; as h→0, f'(x) = f(x)f'(0) = 2f(x); solve to get f(x) = e^(2x)
• (e) For cone with vertex (1,1,0) and guiding curve y=0, x²+z²=4: parametrize generator lines (1+(x₀-1)t, 1-t, z₀t) where x₀²+z₀²=4; eliminate parameters to get (x-1)² + z² = 4(y-1)² with y ≤ 1

Solve all four sub-parts systematically, allocating approximately 30% time to part (a) on linear transformation, 30% to part (b) on Mean Value Theorem proof, 20% to part (c)(i) on cylinder equation, and 20% to part (c)(ii) on shortest distance. Begin each part with clear statement of the approach, show complete working with proper mathematical notation, and conclude with boxed final answers.

• For (a): Verify that {(1,1,-1), (4,1,1), (1,-1,2)} forms a basis for ℝ³, then express T as a 2×3 matrix by solving for images of standard basis vectors or using linearity directly
• For (b): Apply MVT to f(x) = sin⁻¹x on [1/2, 3/5], showing f'(c) = 1/√(1-c²) lies between 4/5 and 5/√39, then manipulate to obtain the required bounds
• For (c)(i): Use the condition that for any point (x,y,z) on cylinder, its distance from the axis line x/1=y/2=z/3 equals the radius 4, giving (2x-y)² + (3y-2z)² + (3x-z)² = 16×14 or equivalent simplified form
• For (c)(ii): Verify the lines are skew, then apply SD formula |(a₂-a₁)·(b₁×b₂)|/|b₁×b₂| with correct identification of points and direction vectors, obtaining exact numerical value
• Clear demonstration of linearity properties in (a), proper interval selection in (b), correct generator direction handling in (c)(i), and accurate cross product computation in (c)(ii)

Solve all four sub-parts systematically, allocating approximately 30% time to part (a) matrix reduction, 30% to part (b) sphere equations, 25% to part (c)(i) double integration, and 15% to part (c)(ii) partial derivative proofs. Begin each sub-part with clear identification of the mathematical technique, show complete working with row operations for (a), sphere family parameterization for (b), region sketching and limits for (c)(i), and chain rule applications for (c)(ii). Conclude with boxed final answers for each part.

• For (a): Correct application of elementary row operations to achieve row echelon form with leading entries 1, 2, 0 in successive rows and proper identification of pivot positions
• For (b): Formation of sphere family equation S + λP = 0, correct application of tangency condition using distance from center to plane equals radius, yielding two valid sphere equations
• For (c)(i): Accurate sketch of region R bounded by line y=x and parabola y=4x-x², correct intersection points (0,0) and (3,3), proper order of integration with limits 0 to 3 for x and x to 4x-x² for y
• For (c)(ii): Verification of Euler's homogeneous function theorem for part I using substitution v=y/x and chain rule, and confirmation of part II as second-degree Euler equation for homogeneous functions
• Clear presentation of all row operation steps in (a) with explicit notation (R₂ → R₂ + ³⁄₂R₁ etc.) to enable partial credit tracing

Prove the geometric impossibility in (a), calculate mixed partial derivatives in (b), and solve the eigenvalue problem with quotient space verification in (c). Allocate approximately 30% time to part (a) as it requires conceptual rigour, 30% to part (b) for careful limit analysis, 25% to (c)(i) for eigenvalue computation, and 15% to (c)(ii) for the dimension theorem verification. Begin with sphere-line analysis, proceed to partial derivative limits, then matrix characteristic polynomial, and conclude with basis construction for the quotient space.

• For (a): Complete the square to find sphere centre (2,-1,2) and radius 3; verify the given line does not intersect the sphere or lies entirely outside; show the distance from centre to line exceeds radius, making tangent plane impossible
• For (b): Compute f_x(0,y) using limit definition, then f_xy(0,0); compute f_y(x,0) then f_yx(0,0); demonstrate f_xy(0,0) = -1 and f_yx(0,0) = 1, showing inequality of mixed partials
• For (c)(i): Find characteristic equation det(A-λI)=0 yielding eigenvalues λ=3,3,-2; find eigenvectors for λ=3 (geometric multiplicity 1) and λ=-2; handle the defective case for repeated eigenvalue appropriately
• For (c)(ii): Identify basis {1,x,x²,x³,x⁴} for P₄ and {1,x,x²} for P₂; construct coset basis {x³+P₂, x⁴+P₂} for P₄/P₂; verify dim(P₄/P₂)=2=5-3=dim P₄-dim P₂
• Cross-part rigour: Use ε-δ arguments or explicit limit calculations in (b); justify why tangent plane condition fails via distance formula or system inconsistency in (a)

Solve each sub-part systematically, allocating approximately equal time (~20%) to each 10-mark section. For (a), identify the substitution v = y/x to reduce to Clairaut's form; for (b), eliminate parameters from the standard ellipse equation; for (c), apply Kepler's second law with area integration; for (d), use catenary boundary conditions at symmetric points; for (e), compute gradients and verify scalar triple product vanishes. Present solutions with clear headings for each part.

• Part (a): Recognize homogeneous structure, substitute v = y/x, transform to Clairaut's equation v = px + f(p), obtain complete primitive and singular solution
• Part (b): Start with ellipse equation x²/a² + y²/b² = 1, eliminate two arbitrary constants a and b to get second-order differential equation xy(d²y/dx²) + x(dy/dx)² - y(dy/dx) = 0
• Part (c): Apply Kepler's second law (equal areas in equal times), compute sector area from minor axis to aphelion using polar ellipse equation r = a(1-e²)/(1+e cos θ), integrate and compare with half-year
• Part (d): Set up symmetric catenary y = c cosh(x/c) with origin at lowest point O, apply boundary conditions at x = ±a with y = c + d, eliminate parameter to derive required relation
• Part (e): Compute ∇u, ∇v, ∇w explicitly, form scalar triple product [∇u ∇v ∇w] or show linear dependence via ∇v = x∇u + z∇w-type relation, verify coplanarity condition

Begin with the proof of the convolution theorem in part (a) using Fubini's theorem and the definition of Laplace transform, then apply it to solve the integral equation via algebraic manipulation in s-domain and partial fractions. For part (b), establish the equation of motion in two phases (stretched string SHM and free fall), carefully handling the transition conditions at natural length. Part (c)(i) requires computing the gradient and unit tangent vector before taking the dot product, while (c)(ii) demands vector calculus identities to derive the wave equations. Allocate approximately 30% time to (a), 35% to (b), 20% to (c)(i), and 15% to (c)(ii) based on mark distribution and computational complexity.

• Part (a): Correct proof of convolution theorem using change of order of integration and identification of Laplace kernel; proper setup of subsidiary equation ȳ(s) = 1/s² + ȳ(s)/(s²+1) and its solution yielding y(t) = t + t³/6
• Part (b): Correct derivation of SHM equation ẍ = -g(x-a)/a for x > a with solution x = a + 3a cos(√(g/a)t); determination of time to reach natural length t₁ = √(a/g)·π/3 and velocity v₁ = 3√(ag)/2
• Part (b) continued: Analysis of free motion phase with initial conditions, time to reach maximum height and return, and total time calculation showing the required expression √(a/g)(2√3 + 4π/3)
• Part (c)(i): Computation of ∇φ = (2xy²z², 2x²yz², 2x²y²z) = (2, 2, -2) at (1,1,-1); tangent vector (1, 2, 1) at t=0; unit vector and directional derivative magnitude |8/√6| = 4√(2/3) or equivalent simplified form
• Part (c)(ii): Application of curl to Faraday's and Ampère's equations, use of vector identity ∇×(∇×H) = ∇(∇·H) - ∇²H with divergence-free condition to obtain wave equations for both E and H fields

Prove the stability condition in part (a) by analyzing the potential energy and center of mass displacement; verify Green's theorem in part (b) by computing both line integral and double integral separately; solve the Clairaut-type equation in (c)(i) for general and singular solutions; and apply the substitution x = e^t to reduce (c)(ii) to a linear equation with constant coefficients. Allocate approximately 30% time to (a), 25% to (b), 25% to (c)(i), and 20% to (c)(ii) based on mark distribution.

• Part (a): Correct geometric setup with sphere radius r and bowl radius 2r, identification of contact point and angle θ, calculation of new center of mass position after adding weight W, and proof that potential energy minimum exists for any W
• Part (b): Proper identification of region bounded by y=x and y=x² with intersection points (0,0) and (1,1), correct application of Green's theorem with P=xy+y² and Q=x², accurate computation of both ∮(Pdx+Qdy) and ∬(∂Q/∂x-∂P/∂y)dA
• Part (c)(i): Substitution u=x+y to transform equation, recognition of Clairaut's equation form, derivation of general solution (x+y-c)³(1+8a/27c³)=0 and singular solution envelope
• Part (c)(ii): Substitution x=e^t to convert to Cauchy-Euler form, reduction to linear ODE with constant coefficients, finding complementary function and particular integral for RHS e^t·t
• Correct handling of rough constraint in (a) ensuring no slipping condition is satisfied

Solve all three parts systematically, allocating approximately 30% time to part (a) on variation of parameters, 30% to part (b) on divergence theorem verification, and 40% to part (c) on particle dynamics which carries the highest marks. Begin each part with clear identification of the method/theorem being applied, show complete working with intermediate steps, and conclude with explicit verification or derived conditions. For part (c), clearly distinguish the three energy regimes with proper energy conservation and normal reaction analysis.

• Part (a): Reduce to standard form, find complementary function by solving characteristic equation, apply variation of parameters correctly with Wronskian calculation, obtain particular integral and general solution
• Part (b): Compute divergence of F correctly as 2(x+y+z), evaluate volume integral over rectangular parallelopiped, calculate surface integral over all six faces with proper orientation, show equality of both integrals
• Part (c): Apply energy conservation between lowest point and arbitrary position, derive expression for normal reaction R in terms of angle and velocity, analyze R=0 condition for leaving the curve, establish critical velocity thresholds at u²=2ag and u²=5ag
• Part (c)(i): Show particle cannot reach horizontal diameter when u²≤2ag, prove oscillatory motion in lower half with amplitude determined by initial energy
• Part (c)(ii): Demonstrate complete circular motion requires R≥0 throughout, show u²≥5ag ensures positive normal reaction at topmost point
• Part (c)(iii): Derive leaving condition R=0, obtain cos α = (u²-2ag)/(3ag) by simultaneous solution of energy and force equations