Mathematics UPSC PYQ 2023

A solid hemisphere is supported by a string fixed to a point on its rim and to a point on a smooth vertical wall with which the curved surface is in contact. If \theta is the angle of inclination of the string with vertical and \phi is the angle of inclination of the plane base of the hemisphere to the vertical, then find the value of (\tan \phi - \tan \theta).

If \vec{a} = \sin \theta \hat{i} + \cos \theta \hat{j} + \theta \hat{k}, \vec{b} = \cos \theta \hat{i} - \sin \theta \hat{j} - 3\hat{k}, \vec{c} = 2\hat{i} + 3\hat{j} - 3\hat{k}, then find the values of the derivative of the vector function \vec{a} \times (\vec{b} \times \vec{c}) w.r.t. \theta at \theta = \frac{\pi}{2} and \theta = \pi.

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 V^2 > \frac{27}{7} g(b - a), then show that the ball will roll completely round the inside of the roller.