If two events $A$ and $B$ are such that $P(A^c) = 0.3$,$P(B) = 0.4$,and $P(A \cap B^c) = 0.5$,then $P\left[ \frac{B}{A \cup B^c} \right]$ is equal to

$\int_0^{\pi / 2} \frac{200 \sin x+100 \cos x}{\sin x+\cos x} d x$ is equal to (in $\pi$)

The angle of elevation of the top of a $T.V.$ tower from three points $A, B, C$ in a straight line through the foot of the tower are $\alpha, 2\alpha, 3\alpha$ respectively. If $AB = a$,the height of the tower is

$A$ metal sphere of radius $r$ and specific heat $S$ is rotated about an axis passing through its centre at a speed of $n$ rotations per second. It is suddenly stopped and $50 \%$ of its energy is used in increasing its temperature. Then,the rise in temperature of the sphere is

As an electron makes a transition from an excited state to the ground state of a hydrogen-like atom/ion:

$A$ plano-convex lens is made of glass with a refractive index of $1.5$. The radius of curvature of its convex surface is $R$. What is its focal length?