$\sum_{k=1}^{2n+1} (-1)^{k-1} k^2$ is equal to

$AB$ is an ionic solid. The ionic radii of $A^{+}$ and $B^{-}$ are respectively $r_c$ and $r_a$. Lattice energy of $AB$ is proportional to :

The coordinates of the point $P$ on the curve $x=a(\theta+\sin \theta), y=a(1-\cos \theta)$,where the tangent is inclined at an angle $\frac{\pi}{4}$ to the $x$-axis,are

Two cylinders $A$ and $B$ fitted with pistons contain equal amounts of an ideal diatomic gas at $300 \ K$. The piston of $A$ is free to move while that of $B$ is held fixed. The same amount of heat is given to the gas in each cylinder. If the rise in temperature of the gas in $A$ is $30 \ K$,then the rise in temperature of the gas in $B$ is ..... $K$

In maize,hybrid vigour is exploited by

The figure shows the vertical section of a frictionless surface. $A$ block of mass $2\, kg$ is released from position $A$. Its kinetic energy $(KE)$ as it reaches position $C$ is ................ $J$.