UPSC Mathematics 2021

(a) If A= 1 & -1 & 1 2 & -1 & 0 1 & 0 & 0 , then show that A² = A⁻¹ (without finding A⁻¹). (10 marks) (b) Find the matrix associated with the linear operator on V₃(R) defined by T…

(a) Show that the planes, which cut the cone ax² + by² + cz² = 0 in perpendicular generators, touch the cone (x²)/(b+c) + (y²)/(c+a) + (z²)/(a+b) = 0. (20 marks) (b) Given that f(…

(a)(i) If u = x² + y², v = x² - y², where x = rcosθ, y = rsinθ, then find (∂(u,v))/(∂(r,θ)). (7 marks) (a)(ii) If ∫limits₀^x f(t) dt = x + ∫limitsₓ¹ tf(t) dt, then find the value…

(a)(i) Reduce the following matrix to a row-reduced echelon form and hence also, find its rank: A = [1 3 2 4 1 0 0 2 2 0 2 6 2 6 2 3 9 1 10 6] (10 marks) (a)(ii) Find the eigen va…

Solve the differential equation: d²y/dx² + 2y = x²e^(3x) + e^x cos 2x (10 marks) Solve the initial value problem: d²y/dx² + 4y = e^(-2x) sin 2x; y(0) = y'(0) = 0 using Laplace tra…

A heavy string, which is not of uniform density, is hung up from two points. Let T₁, T₂, T₃ be the tensions at the intermediate points A, B, C of the catenary respectively where i…

(a) Verify Gauss divergence theorem for F⃗ = 2x^2yî - y^2ĵ + 4xz^2k̂ taken over the region in the first octant bounded by y² + z² = 9 and x = 2. (20 marks) (b) Find all possible…

(a)(i) Find the orthogonal trajectories of the family of confocal conics (x²)/(a²+λ) + (y²)/(b²+λ) = 1; a > b > 0 are constants and λ is a parameter. Show that the given family of…

(a) Let m₁, m₂, …, mₖ be positive integers and d > 0 the greatest common divisor of m₁, m₂, …, mₖ. Show that there exist integers x₁, x₂, …, xₖ such that d = x_1m₁ + x_2m₂ + … + x…

(a) Find the maximum and minimum values of f(x) = x³ - 9x² + 26x - 24 for 0 ≤ x ≤ 1. (15 marks) (b) Let F be a field and f(x) ∈ F[x] a polynomial of degree > 0 over F. Show that t…

(a) Let f be an entire function whose Taylor series expansion with centre z = 0 has infinitely many terms. Show that z = 0 is an essential singularity of f(1/z). (15 marks) (b) Fi…

(a) Show that there are infinitely many subgroups of the additive group Q of rational numbers. (15 marks) (b) Using contour integration, evaluate the integral ∫₋∞^∞ (sin x dx)/(x(…

(a) Obtain the partial differential equation by eliminating arbitrary function f from the equation f(x+y+z, x²+y²+z²) = 0. (10 marks) (b) Find a positive root of the equation 3x =…

(a) Solve the wave equation a²∂²u/∂x² = ∂²u/∂t², 0<x<L, t>0 subject to the conditions u(0,t)=0, u(L,t)=0 u(x,0)=(1/4)x(L-x), ∂u/∂t|ₜ₌₀=0 (20 marks) (b) Obtain the Boolean function…

(a) Find the general solution of the partial differential equation (D² - D'² - 3D + 3D')z = xy + e^(x+2y) where D ≡ ∂/∂x and D' ≡ ∂/∂y. 15 marks (b) Solve the system of equations…

(a) Find a complete integral of the partial differential equation p = (z + qy)² by using Charpit's method. 15 marks (b) Derive Newton's backward difference interpolation formula a…