CSS 2023 Applied Mathematics Paper

State and prove the Divergence Theorem of Gauss and verify it for F = 4xi - 2y^2j + z^2k taken over the region bounded by x^2 + y^2 = 4, z = 0 and z = 3.

A uniform chain of length L is suspended from two points A and B in the same horizontal line. If the tension at the lowest point is equal to the weight of a length 'a' of the chain, find the span AB.

State the principle of Virtual Work. A ladder of weight W rests with one end against a smooth vertical wall and the other on a smooth horizontal floor. It is kept from slipping by a string attached to the wall and the foot of the ladder. Find the tension in the string using the principle of virtual work.

A particle moves in a plane with constant speed v. If its radius vector from a fixed point O rotates with constant angular velocity w, show that the path is a circle and find its radius.

Find the law of force to the pole when the path is the cardioid r = a(1 + cosθ).

Solve the one-dimensional heat equation ∂u/∂t = k(∂²u/∂x²) subject to the boundary conditions u(0, t) = 0, u(L, t) = 0 and the initial condition u(x, 0) = sin(πx/L).

Solve the equation x³ - 2x - 5 = 0 by using the Newton-Raphson method correct to three decimal places.