UPSC Mathematics 2023

(a) Let V₁ = (2, -1, 3, 2), V₂ = (-1, 1, 1, -3) and V₃ = (1, 1, 9, -5) be three vectors of the space ℝ⁴. Does (3, -1, 0, -1) ∈ span {V₁, V₂, V₃} ? Justify your answer. (10 marks)…

(a) If the matrix of a linear transformation T : IR³→IR³ relative to the basis (1, 0, 0), (0, 1, 0), (0, 0, 1) is 1 & 1 & 2 -1 & 2 & 1 0 & 1 & 3 , then find the matrix of T relati…

Let A = 1 & 0 & 0 1 & 0 & 1 0 & 1 & 0 (i) Verify the Cayley-Hamilton theorem for the matrix A. (ii) Show that Aⁿ = Aⁿ⁻² + A² - I for n ≥ 3, where I is the identity matrix of order…

Find the rank of the matrix A = 1 & 2 & -1 & 0 -1 & 3 & 0 & -4 2 & 1 & 3 & -2 1 & 1 & 1 & -1 by reducing it to row-reduced echelon form. 15 (b) Trace the curve y²(x² - 1) = 2x - 1…

(a) Obtain the solution of the initial-value problem dy/dx - 2xy = 2, y(0) = 1 in the form y = eˣ²[1 + √π erf(x)]. (10 marks) (b) Given that L{f(t); p} = F(p). Show that ∫₀^∞ f(t)…

(a) Solve the differential equation: d³y/dx³ - 3d²y/dx² + 4dy/dx - 2y = eˣ + cos x. (15 marks) (b) When a particle is projected from a point O₁ on the sea level with a velocity v…

(a)(i) Find the solution of the differential equation : dy/dx=-(2xy³+2)/(3x^2y²+8e^4y) 10 (a)(ii) Reduce the equation x^2p²+y(2x+y)p+y²=0 to Clairaut's form by the substitution y=…

(a) Solve the following initial value problem by using Laplace transform technique : (d^2y)/(dt²) - 4dy/dt + 3y(t) = f(t), y(0) = 1, y'(0) = 0 and f(t) is a given function of t. 1…

(a) Let G be a group of order 10 and G′ be a group of order 6. Examine whether there exists a homomorphism of G onto G′. (10 marks) (b) Express the ideal 4Z + 6Z as a principal id…

(a) Prove that a non-commutative group of order 2p, where p is an odd prime, must have a subgroup of order p. (15 marks) (b) Using the method of Lagrange's multipliers, find the m…

(a) Prove that x² + 1 is an irreducible polynomial in Z₃[x]. Further show that the quotient ring (Z₃[x])/(⟨ x²+1 ⟩) is a field of 9 elements. 15 marks (b) Prove that u(x, y) = eˣ(…

(a) Prove that the oscillation of a real-valued bounded function f defined on [a, b] is the supremum of the set {|f(x₁)-f(x₂)| : x₁, x₂ ∈ [a, b]}. 15 marks (b) Classify the singul…

(a) By eliminating the arbitrary functions f and g from z = f(x² - y) + g(x² + y), form partial differential equation. (10 marks) (b) Given dy/dx = (y² - x)/(y² + x) with initial…

(a) Find the surface passing through the two lines z = x = 0 and z-1 = x-y = 0, and satisfying the partial differential equation ∂²z/∂x² - 4∂²z/∂x∂y + 4∂²z/∂y² = 0. (15 marks) (b)…

(a) (i) Find the conjunctive normal form (CNF) of the following Boolean function: f(x, y, z, t) = x · y · z + x̄ · y · (t + z̄) (15 marks) (ii) Express the Boolean function f(x, y…

(a) Reduce the partial differential equation ∂²z/∂y² - ∂²z/∂x∂y + ∂z/∂x - ∂z/∂y(1+1/x) + z/x = 0 to canonical form. (15 marks) (b) Compute a root of the equation log₁₀(2x+1) - x²…