The temperature-entropy $(T-S)$ diagram of a reversible engine cycle is given in the figure. Its efficiency is

If $z = \sec^{-1}\left(\frac{x^4+y^4-8x^2y^2}{x^2+y^2}\right)$,then $x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y}$ is equal to

$A$ circular disc of radius $R$ is removed from a bigger circular disc of radius $2R$ such that the circumferences of the discs touch. The centre of mass of the new disc is at a distance $\alpha R$ from the centre of the bigger disc. The value of $\alpha$ is

$A$ wire elongates by $l \, mm$ when a load $W$ is hanged from it. If the wire goes over a pulley and two weights $W$ each are hung at the two ends,the elongation of the wire will be (in $mm$):

In a $\triangle ABC$,if $a \cos^2 \frac{C}{2} + c \cos^2 \frac{A}{2} = \frac{3b}{2}$,then the sides of the triangle are in

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