$A$ body of mass $m$ falls from a height $R$ above the surface of the earth,where $R$ is the radius of the earth. What is the velocity attained by the body on reaching the ground? (Acceleration due to gravity on the surface of the earth is $g$)

$A$ mercury drop of radius $1 ~cm$ is sprayed into $10^6$ drops of equal size. The energy expended in joules is (Surface tension of mercury is $460 \times 10^{-3} ~N/m$)

Five masses,each of $2 \, kg$,are placed on a horizontal circular disc that can rotate about a vertical axis passing through its centre. All the masses are equidistant from the axis at a distance of $10 \, cm$. The moment of inertia of the whole system (in $g \cdot cm^2$) is (Assume the disc has negligible mass).

If $f: R \rightarrow R$ is defined by $f(x) = \begin{cases} \frac{x+2}{x^2+3x+2} & \text{if } x \in R-\{-1,-2\} \\ -1 & \text{if } x=-2 \\ 0 & \text{if } x=-1 \end{cases}$,then $f$ is continuous on the set

Six moles of an ideal gas perform a cycle shown in the figure. If the temperatures are $T_A = 600\,K, T_B = 800\,K, T_C = 2200\,K$ and $T_D = 1200\,K,$ the work done per cycle is...... $kJ$

Let $\alpha, \beta$ be such that $\pi < \alpha - \beta < 3\pi$. If $\sin \alpha + \sin \beta = -\frac{21}{65}$ and $\cos \alpha + \cos \beta = -\frac{27}{65}$,then the value of $\cos \frac{\alpha - \beta}{2}$ is