UPSC Mathematics 2026

(a) Find all the values of a for which the system of homogeneous equations ax + y + z = 0 x + y - z = 0 x + y - az = 0 has non-trivial solutions, and hence determine all the solut…

(a) Let T : P₂[x] → P₂[x] be a linear transformation defined by T(a₀ + a₁x + a₂x²) = –(a₀ + 2a₁ + a₂) + (2a₀ + 3a₁) x – 2(a₀ + a₁) x² where P₂[x] denotes the set of all polynomial…

(a) Given P₃[x] = {a₀ + a₁x + a₂x² + a₃x³ | a₀, a₁, a₂, a₃ ∈ ℝ} as a vector space over the field ℝ. (i) Let H = { p(x) ∈ P₃[x] | ∫₋₁¹ p(x) dx = 0} . Prove that H is a subspace of…

(a) Let F be a subfield of complex numbers and let T be the linear transformation from F³ into F³, defined by T(x, y, z) = (x – y + 2z, 2x + y, – x – 2y + 2z) If (a, b, c) is a ve…

(a) If y₁ and y₂ are two solutions of dy/dx + P(x)y = Q(x), and y₂ = y₁u, then show that u = 1 + a e^{-∫(Q/y₁) dx}, where a is a non-zero constant. (10 marks) (b) Apply Laplace tr…

(a) Solve x² d²y/dx² − 3x dy/dx + y = (sin(log x) + 1)/x log x. (15 marks) (b) (i) A battleship is steaming ahead with velocity u. A gun is mounted on the ship so as to point stra…

(a) A light elastic string of natural length l has one extremity fixed at a point A and the other attached to a stone (mass m) the weight of which in equilibrium would extend the…

(a) Solve \frac{d^2y}{dx^2}+a^2y=\sec ax by the method of variation of parameters. (15 marks) (b) AB is a uniform rod of length l and weight W, which can turn freely about a fixed…

(a) Let G be a group of order pq, where p and q are primes. Show that either G is an abelian group or no non-identity element commutes with every element of G. (10 marks) (b) Show…

(a) Let G be a finite group generated by two elements u and v of order 2 each. Show that G is isomorphic to the dihedral group of order 2n, where O(uv) = n. (15 marks) (b) Prove t…

(a) Prove that cosh ( z + 1/z ) = a₀ + Σ_{n=1}^{∞} aₙ ( zⁿ + 1/zⁿ ) , where aₙ = (1 / 2π) ∫₀²π cos nθ cosh (2 cos θ) dθ, for n = 0, 1, 2, ... . (15 marks) (b) A function f is defi…

(a) Let n be a positive integer. Show that the following statements are equivalent : (i) ℤ/nℤ is an integral domain. (ii) ℤ/nℤ is a field. (iii) n is prime. (15 marks) (b) Prove t…

(a) Find the integral surface of the equations dx/(xz − y) = dy/(yz − x) = dz/(1 − z²) . (10 marks) (b) By using the Newton-Raphson iterative formula, establish the formula x_{i+1…

(a) Reduce the partial differential equation y ∂²z/∂x² + (x + y) ∂²z/∂x∂y + x ∂²z/∂y² = 0 to canonical form and hence solve it. (20 marks) (b) (i) Find the decimal equivalent of t…

(a) Solve the partial differential equation (2D² - 5DD' + 2D'²) z = 24 (y - x) + sin (y - x) where D ≡ ∂/∂x , D' ≡ ∂/∂y , z = z(x, y). (15 marks) (b) Evaluate the integral ∫_0^{0.…

(a) Two-dimensional harmonic equation in plane polar coordinates (r, θ) takes the form : ∂²V/∂r² + (1/r) ∂V/∂r + (1/r²) ∂²V/∂θ² = 0 Deduce that it has solutions of the form (Arⁿ +…