UPSC Mathematics 2024

(a) Let H be a subspace of R⁴ spanned by the vectors v₁ = (1, –2, 5, –3), v₂ = (2, 3, 1, –4), v₃ = (3, 8, –3, –5). Then find a basis and dimension of H, and extend the basis of H…

(a) Consider a linear operator T on R³ over R defined by T(x, y, z) = (2x, 4x – y, 2x + 3y – z). Is T invertible? If yes, justify your answer and find T⁻¹. (15 marks) (b) If u = (…

(a) Let V = M₂ₓ₂(ℝ) denote a vector space over the field of real numbers. Find the matrix of the linear mapping φ: V → V given by φ(v) = 1 & 2 3 & -1 v with respect to standard ba…

(a) Let A = 3 & 2 & 4 2 & 0 & 2 4 & 2 & 3 be a 3×3 matrix. Find the eigenvalues and the corresponding eigenvectors of A. Hence find the eigenvalues and the corresponding eigenvect…

(a) Find the orthogonal trajectories of the family of curves r = c(sec θ + tan θ), where c is a parameter. (10 marks) (b) Solve the integral equation y(t) = cos t + ∫₀ᵗ y(x) cos(t…

(a) A regular tetrahedron, formed of six light rods, each of length l, rests on a smooth horizontal plane. A ring of weight W and radius r is supported by the slant sides. Using t…

(a) State uniqueness theorem for the existence of unique solution of the initial value problem dy/dx = f(x, y), y(x₀) = y₀ in the rectangular region R: |x - x₀| ≤ a, |y - y₀| ≤ b.…

(a) Using Laplace transform, solve the initial value problem y'' + 2y' + 5y = δ(t-2), y(0) = 0, y'(0) = 0 where δ(t-2) denotes the Dirac delta function. (15 marks) (b) Using Gauss…

(a) Let G be a finite group of order mn, where m and n are prime numbers with m > n. Show that G has at most one subgroup of order m. 10 marks (b) If w = f(z) is an analytic funct…

(a) Using Cauchy's general principle of convergence, examine the convergence of the sequence < fₙ >, where fₙ = 1 + 1/1! + 1/2! + ... + 1/n!. 15 marks (b) Show that every homomorp…

(a) Locate the poles and their order for the function f(z) = 1/[z(sin πz)(z + 1/2)]. Also, find the residue of f(z) at these poles. (15 marks) (b) Consider the series Σ(n=1 to ∞)…

(a) Consider the polynomial ring Z[x] over the ring Z of integers. Let S be an ideal of Z[x] generated by x. Show that S is prime but not a maximal ideal of Z[x]. (15 marks) (b) F…

(a) Show that if f and g are arbitrary functions of their respective arguments, then u = f(x - kt + iαy) + g(x - kt - iαy), is a solution of ∂²u/∂x² + ∂²u/∂y² = (1/C²)∂²u/∂t², whe…

(a) Show that the solution of the two-dimensional Laplace's equation ∂²φ(x,y)/∂x² + ∂²φ(x,y)/∂y² = 0, x ∈ (-∞, ∞), y ≥ 0 subject to the boundary condition φ(x,0) = f(x), x ∈ (-∞,…

(a) Find the integral surface of the following quasi-linear equation (y - φ) (∂ φ)/(∂ x) + (φ - x) (∂ φ)/(∂ y) = x - y, which passes through the curve φ = 0, xy = 1 and through th…

(a) Solve the partial differential equation (∂)/(∂ y)((∂ φ)/(∂ x) + φ) + 2x^2y((∂ φ)/(∂ x) + φ) = 0 by transforming it to the canonical form. (15 marks) (b) Using Newton's forward…