CSS 2021 Applied Mathematics Paper

If r is the position vector of any point (x, y, z), prove that div(r^n r) = (n+3)r^n and curl(r^n r) = 0.

Show that the length of a heavy uniform string which can be hung across two smooth pegs in the same horizontal line, so that the sagging part between the pegs is a catenary, and the ends are vertical, is L = l(1 + 2/sinhθ) where 2l is the span.

Discuss the motion of a particle performing Simple Harmonic Motion (SHM). Find the time period and the maximum velocity.

A particle moves in a straight line with an acceleration kv³. If the initial velocity is u, find the distance traveled and the time taken to reduce the velocity to u/2.

Use the method of separation of variables to solve Laplace's equation ∂²u/∂x² + ∂²u/∂y² = 0 in a rectangle 0 < x < a, 0 < y < b.

Find the real root of x³ - x - 1 = 0 using the Bisection Method after 4 iterations.

Show that every tensor can be expressed as the sum of a symmetric and an anti-symmetric tensor.