Find the $g.c.d.$ and $l.c.m.$ using the fundamental theorem of arithmetic for the numbers: $24$,$36$,and $60$.

The number of integral terms in the expansion of $(\sqrt{3}+\sqrt[8]{5})^{256}$ is

$A$ man of mass $m$ starts falling towards a planet of mass $M$ and radius $R$. As he reaches near to the surface,he realizes that he will pass through a small hole in the planet. As he enters the hole,he sees that the planet is really made of two pieces: a spherical shell of negligible thickness of mass $2M/3$ and a point mass $M/3$ at the centre. What is the change in the force of gravity experienced by the man?

The differential equation of the family of parabolas with vertex at $(0, -1)$ and having axis along the $Y$-axis is

Pulses are obtained from which family?

$A \rightarrow D$ is an endothermic reaction occurring in $3$ steps (elementary).
$(i)$ $A \rightarrow B$ $\Delta H_{i} = +ve$
$(ii)$ $B \rightarrow C$ $\Delta H_{ii} = -ve$
$(iii)$ $C \rightarrow D$ $\Delta H_{iii} = -ve$
Which of the following graphs between potential energy ($y$-axis) vs reaction coordinate ($x$-axis) correctly represents the reaction profile of $A \rightarrow D$?