UPSC Mathematics 2025

(a) Can the set {(0, 0, 0, 3), (1, 1, 0, 0), (0, 1, –1, 0)} be extended to form a basis of the vector space ℝ⁴? Justify your answer. 10 marks (b) Find the range, rank, kernel and…

(a) Let T : ℝ³ → ℝ² be a linear transformation such that T(1, 1, -1) = (1, 0), T(4, 1, 1) = (0, 1) and T(1, -1, 2) = (1, 1). Find T. 15 marks (b) Using Mean Value Theorem, prove t…

(a) Reduce the following matrix to echelon form: A = 2 & -2 & 2 & 1 -3 & 6 & 0 & -1 1 & -7 & 10 & 2 (15 marks) (b) Find the equations of the spheres which pass through the circle…

(a) Show that there is no tangent plane to the sphere x² + y² + z² - 4x + 2y - 4z + 4 = 0 that can be passed through the straight line (x+6)/2 = y + 3 = z + 1. (15 marks) (b) If f…

(a) Solve (1-y²+(y⁴)/(x²))(dy/dx)²-2y/xdy/dx+(y²)/(x²)=0. 10 marks (b) Form the differential equation of all ellipses whose axes coincide with coordinate axes. 10 marks (c) Prove…

(a) If F(s) and G(s) are Laplace transforms of f(t) and g(t) respectively, then prove that L∫₀^t f(x) g(t-x) dx = F(s) G(s). Using this result, solve the equation y(t) = t + ∫₀^t…

(a) A solid sphere rests inside a fixed rough and hemispherical bowl of twice its radius. If a large amount of weight, whatsoever, is attached to the highest point of the sphere,…

(a) Solve the differential equation (x + 2)(d^2y)/(dx²) - (2x + 5)dy/dx + 2y = (1 + x) e^x by the method of variation of parameters. (15 marks) (b) Verify Gauss's divergence theor…

(a) Let H and K be two subgroups of a group G such that o(H) > √o(G) and o(K) > √o(G). Show that H ∩ K ≠ {e}, where e is the identity element. Here o(H), o(K) and o(G) denote the…

(a) Define Cauchy sequence and prove that every convergent sequence of real numbers is a Cauchy sequence. What is the importance of Cauchy condition? (15 marks) (b) Show that 3 is…

(a) Evaluate the integral ∮_C e^z/(z²(z+1)³) dz, C : |z| = 2. (15 marks) (b) Show that the volume of the greatest rectangular parallelopiped that can be inscribed in the ellipsoid…

(a) Examine whether the mapping φ: Z[x] → Z defined by φ(f(x)) = f(0), for f(x) ∈ Z[x], is a homomorphism. Deduce that the ideal ⟨x⟩ is a prime ideal in Z[x], but not a maximal id…

(a) Find the solution of the equation (D² + DD' - 2D'²)z = ysin x, where D ≡ (∂)/(∂ x) and D' ≡ (∂)/(∂ y). (10 marks) (b) Solve the following system of linear equations by Gauss-S…

(a) Solve (∂² u)/(∂ x²) + (∂² u)/(∂ y²) = 0 for a rectangular plate subject to the boundary conditions u(0,y) = 0, u(a,y) = 0 u(x,0) = 0, u(x,b) = f(x) (20 marks) (b) Simplify the…

(a) Find the complete integral of z(p²-q²) = x-y; p≡∂z/∂x, q≡∂z/∂y. (15 marks) (b) Find the unique polynomial of degree 2 or less which fits the following data: x : 0 1 3 f(x) : 1…

(a) Find the characteristics of the partial differential equation p² + q² = 2; p ≡ ∂z/∂x, q ≡ ∂z/∂y and determine the integral surface which passes through x = 0, z = y. (15 marks…