The number of electrons in the valence shell of the central atom of a molecule is $8$. The molecule is

$A$ solid sphere of mass $M$ and radius $R$ is rotating about its diameter. $A$ solid cylinder of the same mass and radius is also rotating about its geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation ($K_{\text{sphere}}$ to $K_{\text{cylinder}}$) will be

$A$ conducting rod with resistance $r$ per unit length is moving inside a vertical magnetic field $\vec B$ at constant speed $v$ on two horizontal parallel ideal conductor rails. The ends of the rails are connected to a resistor $R$. The separation between the rails is $d$. The rod maintains a tilted angle $\theta$ to the rails. Find the external force required to keep the rod moving.

If $t_n = \frac{1}{4}(n+2)(n+3)$ for $n = 1, 2, 3, \ldots$,then $\frac{1}{t_1} + \frac{1}{t_2} + \ldots + \frac{1}{t_{2003}}$ is equal to

Match the items given in Column $I$ with those in Column $II$ and select the correct option given below:

Column $I$	Column $II$
$(a)$ Tidal volume	$(i)$ $2500 - 3000 \ mL$
$(b)$ Inspiratory Reserve volume	$(ii)$ $1100 - 1200 \ mL$
$(c)$ Expiratory Reserve volume	$(iii)$ $500 - 550 \ mL$
$(d)$ Residual volume	$(iv)$ $1000 - 1100 \ mL$

If the number of $5$-element subsets of the set $A = \{a_1, a_2, \ldots, a_{20}\}$ of $20$ distinct elements is $k$ times the number of $5$-element subsets containing $a_4$,then $k$ is