Mathematics UPSC PYQ 2023

Find the solution of the differential equation: dy/dx = −(2xy³ + 2)/(3x²y² + 8e^{4y}). <!--qid:MAINS_2023_Mathematics-I_Q7-->

Solve the following initial value problem by using Laplace transform technique: d²y/dt² − 4 dy/dt + 3y(t) = f(t), y(0) = 1, y′(0) = 0 and f(t) is a given function of t. <!--qid:MAINS_2023_Mathematics-I_Q8-->

(a) (i) Find the conjunctive normal form (CNF) of the following Boolean function: f(x, y, z, t) = x·y·z + x̄·y·(t + z) (ii) Express the Boolean function f(x, y, z) = x + (x̄·ȳ + x̄·z̄) + z in disjunctive normal form (DNF) and construct the truth table for the function. (b) A perfectly rough ball is at rest within a hollow cylindrical roller. The roller is drawn along a level path with uniform velocity V. Let a and b be the radii of the ball and the roller respectively. If V² > (27/7) g (b − a), then show that the ball will roll completely round the inside of the roller. (c) Solve the partial differential equation a² ∂²u/∂x² = ∂²u/∂t², 0 < x < L, t > 0 subject to the conditions u(0, t) = 0, u(L, t) = 0, t > 0 u(x, 0) = x, (∂u/∂t)|_{t=0} = 1, 0 < x < L. <!--qid:MAINS_2023_Mathematics-II_Q7-->

When a particle is projected from a point O1 on the sea level with a velocity v and angle of projection θ with the horizon in a vertical plane, its horizontal range is R1. If it is further projected from a point O2, which is vertically above O1 at a height h in the same vertical plane, with the same velocity v and same angle θ with the horizon, its horizontal range is R2. Prove that R2 > R1 and (R2 − R1) : R1 is equal to 1/2 [√(1 + 2gh / (v² sin²θ)) − 1] : 1. <!--qid:MAINS_2023_Mathematics-I_Q6-->

(a) Reduce the partial differential equation ∂²z/∂y² − ∂²z/∂x∂y + ∂z/∂x − ∂z/∂y (1 + 1/x) + z/x = 0 to canonical form. (b) Compute a root of the equation log₁₀(2x + 1) − x² + 3 = 0, in the interval [0, 3], by the Regula-Falsi method, correct to 6 decimal places. (c) Determine under what conditions the velocity field u = c(x² − y²), v = −2cxy, w = 0 is a solution to the Navier-Stokes momentum equations. Assuming that the conditions are met, determine the resulting pressure distribution when z is up and the external body forces are Bₓ = 0 = B_y, B_z = −g. <!--qid:MAINS_2023_Mathematics-II_Q8-->