UPSC Mathematics 2024 — Paper II

(a) Let G be a finite group of order mn, where m and n are prime numbers with m > n. Show that G has at most one subgroup of order m. 10 marks (b) If w = f(z) is an analytic function of z, then show that ((∂²)/(∂ x²) + (∂²)/(∂ y²)) log |f'(z)| = 0. 10 marks (c) Test the convergence of ∫limits₀² (log x)/(√((2-x))) dx . 10 marks (d) If φ and ψ are functions of x and y satisfying Laplace equation, then show that f(z) = p + iq, i = √−1 is an analytic function, where p = (∂ φ)/(∂ y) - (∂ ψ)/(∂ x) and q = (∂ φ)/(∂ x) + (∂ ψ)/(∂ y) . 10 marks (e) Use two phase method to solve the following linear programming problem : Maximize z = x₁ + 2x₂ subject to x₁ - x₂ ≥ 3 2x₁ + x₂ ≤ 10 x₁, x₂ ≥ 0 10 marks

(a) Using Cauchy's general principle of convergence, examine the convergence of the sequence < fₙ >, where fₙ = 1 + 1/1! + 1/2! + ... + 1/n!. 15 marks (b) Show that every homomorphic image of an abelian group is abelian, but the converse is not necessarily true. 15 marks (c) Find the function which is analytic inside and on the circle C : z = e^(iθ), 0 ≤ θ ≤ 2π and has the value (a² - 1) cos θ + i(a² + 1) sin θ ────────────────────────────── a⁴ - 2a² cos 2θ + 1 on the circumference of C, where a² > 1. 20 marks

(a) Locate the poles and their order for the function f(z) = 1/[z(sin πz)(z + 1/2)]. Also, find the residue of f(z) at these poles. (15 marks) (b) Consider the series Σ(n=1 to ∞) U_n(x), 0 ≤ x ≤ 1, the sum of whose first n terms is given by S_n(x) = (1/2n²)log(1 + n⁴x²), x ∈ [0,1]. Show that the given series can be differentiated term-by-term, though Σ(n=1 to ∞) U'_n(x), does not converge uniformly on [0,1]. (20 marks) (c) Using duality principle, solve the following linear programming problem: Minimize z = 4x₁ + 3x₂ + x₃ subject to x₁ + 2x₂ + 4x₃ ≥ 12, 3x₁ + 2x₂ + x₃ ≥ 8, x₁, x₂, x₃ ≥ 0. (15 marks)

(a) Consider the polynomial ring Z[x] over the ring Z of integers. Let S be an ideal of Z[x] generated by x. Show that S is prime but not a maximal ideal of Z[x]. (15 marks) (b) Find the upper and lower Riemann integrals for the function f defined on [0, 1] as follows: f(x) = (1-x²)^½, if x is rational; f(x) = (1-x), if x is irrational. Hence, show that f is not Riemann integrable on [0, 1]. (15 marks) (c) The personnel manager of a company wants to assign officers A, B and C to the regional offices at Delhi, Mumbai, Kolkata and Chennai. The cost of relocation (in thousand Rupees) of the three officers at the four regional offices are given below: | Officer | Delhi | Mumbai | Kolkata | Chennai | |---------|-------|--------|---------|---------| | A | 16 | 22 | 24 | 20 | | B | 10 | 32 | 26 | 16 | | C | 10 | 20 | 46 | 30 | Find the assignment which minimizes the total cost of relocation and also determine the minimum cost. (20 marks)

(a) Show that if f and g are arbitrary functions of their respective arguments, then u = f(x - kt + iαy) + g(x - kt - iαy), is a solution of ∂²u/∂x² + ∂²u/∂y² = (1/C²)∂²u/∂t², where α² = 1 - k²/C². (10 marks) (b) Solve the following system of linear equations by Gauss-Jordan method: (10 marks) 2x + 3y - z = 5 4x + 4y - 3z = 3 2x - 3y + 2z = 2 (c) (i) Determine the decimal equivalent in sign magnitude form of (8D)₁₆ and (FF)₁₆. (ii) Determine the decimal equivalent of (9B2.1A)₁₆. (10 marks) (d) A rough uniform board of mass m and length 2a rests on a smooth horizontal plane and a man of mass M walks on it from one end to the other. Find the distance covered by the board during this time. (10 marks) (e) The velocity potential φ of a flow is given by φ = (1/2)(x² + y² - 2z²). Determine the streamlines. (10 marks)

(a) Show that the solution of the two-dimensional Laplace's equation ∂²φ(x,y)/∂x² + ∂²φ(x,y)/∂y² = 0, x ∈ (-∞, ∞), y ≥ 0 subject to the boundary condition φ(x,0) = f(x), x ∈ (-∞, ∞), along with φ(x,y) → 0 for |x| → ∞ and y → ∞ can be written in the form φ(x,y) = (y/π)∫₋∞^∞ [f(ξ)dξ]/[y² + (x-ξ)²]. (20 marks) (b) Draw the logical circuit for the Boolean expression Y = ABC̄ + BC̄ + ĀB. Also, obtain the output Y (truth table) for the three input bit sequences: A = 10001111, B = 00111100, C = 11000100 (15 marks) (c) Find the moment of inertia of a quadrant of an elliptic disk x²/a² + y²/b² = 1, of mass M about the line passing through its centre and perpendicular to its plane. Given that the density at any point is proportional to xy. (15 marks)

(a) Find the integral surface of the following quasi-linear equation (y - φ) (∂ φ)/(∂ x) + (φ - x) (∂ φ)/(∂ y) = x - y, which passes through the curve φ = 0, xy = 1 and through the circle x + y + φ = 0, x² + y² + φ² = a². (15 marks) (b) Integrate f(x) = 5x³ - 3x² + 2x + 1 from x = -2 to x = 4 using (i) Simpson's 3/8 rule with width h = 1, and (ii) Trapezoidal rule with width h = 1. (15 marks) (c) Let the velocity field u(x, y) = (B(x² - y²))/((x² + y²)²), v(x, y) = 2Bxy/((x² + y²)²), w(x, y) = 0 satisfy the equations of motion for inviscid incompressible flow, where B is a constant. Determine the pressure associated with this velocity field. (20 marks)

(a) Solve the partial differential equation (∂)/(∂ y)((∂ φ)/(∂ x) + φ) + 2x^2y((∂ φ)/(∂ x) + φ) = 0 by transforming it to the canonical form. (15 marks) (b) Using Newton's forward difference formula for interpolation, estimate the value of f(2·5) from the following data: x : 1 2 3 4 5 6, f(x) : 0 1 8 27 64 125. (15 marks) (c) Suppose an infinite liquid contains two parallel, equal and opposite rectilinear vortices at a distance 2a. Show that the streamlines relative to the vortex are given by the equation log (x² + (y-a)²)/(x² + (y+a)²) + y/a = C, where C is a constant, the origin is the middle point of the join, and the line joining the vortices is the axis of y. (20 marks)