Mathematics UPSC PYQ 2022

A cable of weight w per unit length and length 2l hangs from two points P and Q in the same horizontal line. Show that the span of the cable is 2l * [ 1 - (2h^2) / (3l^2) ], where h is the sag in the middle of the tightly stretched position.

Prove that any set of n linearly independent vectors in a vector space V of dimension n constitutes a basis for V.

Evaluate lim (ex + x)^(1/x) as x -> infinity.