Mathematics UPSC PYQ 2014

Evaluate by Stokes' theorem \int_{\Gamma} (y dx + z dy + x dz) where \Gamma is the curve given by x^2 + y^2 + z^2 - 2ax - 2ay = 0, x + y = 2a, starting from (2a, 0, 0) and then going below the z-plane.

Find the equations of the two generating lines through any point (a cos \theta, b \sin \theta, 0), of the principal elliptic section \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, z = 0, of the hyperboloid by the plane z = 0.

Two equal uniform rods AB and AC, each of length l, are freely jointed at A and rest on a smooth fixed vertical circle of radius r. If 2θ is the angle between the rods, then find the relation between l, r and θ, by using the principle of virtual work.