$A$ pendulum clock keeps correct time at $0\ ^oC$. The thermal coefficient of linear expansion of the material of the pendulum is $\alpha$. If the temperature rises to $t\ ^oC$,then the clock loses per day by (in seconds):

Which of the following is paramagnetic?

If the distance between the points $(a \cos \theta, a \sin \theta)$ and $(a \cos \phi, a \sin \phi)$ is $2a$,then $\theta$ is equal to

Some inert gas is added at constant volume to the following reaction at equilibrium:
$NH_4HS_{(s)} \rightleftharpoons NH_{3(g)} + H_2S_{(g)}$
Predict the effect of adding the inert gas.

If $\lim _{x \rightarrow 0}\left(\frac{\cos 4 x+a \cos 2 x+b}{x^4}\right)$ is finite,then the values of $a, b$ are respectively :

$A$ launching vehicle carrying an artificial satellite of mass $m$ is set for launch on the surface of the earth of mass $M$ and radius $R$. If the satellite is intended to move in a circular orbit of radius $7R$,the minimum energy required to be spent by the launching vehicle on the satellite is (Gravitational constant $= G$)