UPSC Mathematics 2024 — Paper I

(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 to a basis of R⁴. (10 marks) (b) Let T : R³ → R³ be a linear operator and B = {v₁, v₂, v₃} be a basis of R³ over R. Suppose that Tv₁ = (1, 1, 0), Tv₂ = (1, 0, –1), Tv₃ = (2, 1, –1). Find a basis for the range space and null space of T. (10 marks) (c) Discuss the continuity of the function f(x) = { 1/(1–e^(–1/x)), x ≠ 0 { 0, x = 0 for all values of x. (10 marks) (d) Expand ln(x) in powers of (x–1) by Taylor's theorem and hence find the value of ln(1·1) correct up to four decimal places. (10 marks) (e) Find the equation of the right circular cylinder which passes through the circle x² + y² + z² = 9, x – y + z = 3. (10 marks)

(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 = (x+y)/(1-xy) and v = tan⁻¹x + tan⁻¹y, then find ∂(u, v)/∂(x, y). Are u and v functionally related? If yes, find the relationship. (15 marks) (c) Find the image of the line x = 3-6t, y = 2t, z = 3+2t in the plane 3x+4y-5z+26 = 0. (20 marks)

(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 basis of M₂ₓ₂(ℝ), and hence find the rank of φ. Is φ invertible? Justify your answer. (15 marks) (b) Find the volume of the greatest cylinder which can be inscribed in a cone of height h and semi-vertical angle α. (20 marks) (c) Find the vertex of the cone 4x² - y² + 2z² + 2xy - 3yz + 12x - 11y + 6z + 4 = 0. (15 marks)

(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 eigenvectors of A⁻¹⁵, where A⁻¹⁵ = (A⁻¹)¹⁵. (20 marks) (b) Using double integration, find the area lying inside the cardioid r = a(1+cos θ) and outside the circle r = a. (15 marks) (c) Find the equation of the sphere which touches the plane 3x+2y-z+2=0 at the point (1, -2, 1) and cuts orthogonally the sphere x²+y²+z²-4x+6y+4=0. (15 marks)

(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-x)dx using Laplace transform. (10 marks) (c) At any time t (in seconds), the coterminous edges of a variable parallelepiped are represented by the vectors ᾱ = tî + (t+1)ĵ + (2t+1)k̂ β̄ = 2tî + (3t-1)ĵ + tk̂ γ̄ = î + 3tĵ + k̂ What is the rate of change of the vectorial area of the parallelogram, whose coterminous edges are ᾱ and γ̄? Also find the rate of change of the volume of the parallelepiped at t = 1 second. (10 marks) (d) A solid hemisphere rests in equilibrium on a solid sphere of equal radius. Determine the stability of the equilibrium in the two situations—(i) when the curved surface and (ii) when the flat surface of the hemisphere rests on the sphere. (10 marks) (e) (i) Let C be a plane curve r̄(t) = f(t)î + g(t)ĵ, where f and g have second-order derivatives. Show that the curvature at a point is given by κ = |f'(t)g''(t) - g'(t)f''(t)| / ([f'(t)]² + [g'(t)]²)^(3/2) What is the value of torsion τ at any point of this curve? (5 marks) (ii) Show that the principal normals at two consecutive points of a curve do not intersect unless torsion τ is zero. (5 marks)

(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 the principle of virtual work, find the stress in any of the horizontal sides. (15 marks) (b) A particle executes simple harmonic motion such that in two of its positions, velocities are u and v, and the two corresponding accelerations are f₁ and f₂. For what value(s) of k, the distance between the two positions is k(v² - u²)? Show also that the amplitude of the motion is 1/(f₂² - f₁²) [(u² - v²)(u²f₂² - v²f₁²)]^(1/2) (15 marks) (c) (i) Find the second solution of the differential equation xy'' + (x-1)y' - y = 0 using u(x) = -e^-x as one of the solutions. (10 marks) (ii) Find the general solution of the differential equation x²y'' - 2xy' + 2y = x³ sin x by the method of variation of parameters. (10 marks)

(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. Test the existence and uniqueness of the solution of the initial value problem dy/dx = 2√y, y(1) = 0, in a suitable rectangle R. If more than one solution exist, then find all the solutions. (15 marks) (b) A heavy particle hanging vertically from a fixed point by a light inextensible string of length l starts to move with initial velocity u in a circle so as to make a complete revolution in a vertical plane. Show that the sum of tensions at the ends of any diameter is constant. (15 marks) (c) State Stokes' theorem and verify it for the vector field F⃗ = xyî + yzĵ + zxk̂ over the surface S, which is the upwardly oriented part of the cylinder z = 1 - x², for 0 ≤ x ≤ 1, -2 ≤ y ≤ 2. (20 marks)

(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 divergence theorem, evaluate the integral ∬_S (y²î + xz³ĵ + (z-1)²k̂) · n̂ dS over the region bounded by the cylinder x² + y² = 16 and the planes z = 1 and z = 5. (15 marks) (c) A particle moves with a central acceleration μ(3/r³ + d²/r⁵) being projected from a distance d at an angle 45° with a velocity equal to that in a circle at the same distance. Prove that the time it takes to reach the centre of force is d²/√(2μ) (2 - π/2). (20 marks)