$A$ block of mass $M$ is pulled along a horizontal frictionless surface by a rope of mass $m$. If a force $P$ is applied at the free end of the rope,the force exerted by the rope on the block is-

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

One junction of a certain thermoelectric couple is at a fixed temperature $T_r$ and the other junction is at temperature $T$. The thermo-electromotive force for this is expressed by $E=k(T-T_r)[T_0-\frac{1}{2}(T+T_r)]$. At temperature $T=\frac{1}{2} T_0$,the thermoelectric power is

Assertion $(A)$: $A$ current of $96.5 \ A$ is passed into aqueous $AgNO_3$ solution for $100 \ s$. The weight of silver deposited is $10.8 \ g$ (At. wt. of $Ag = 108$).
Reason $(R)$: The mass of a substance deposited during the electrolysis of an electrolyte is inversely proportional to the quantity of electricity passing through the electrolyte.
The correct answer is:

Two slits separated by a distance of $1\, mm$ are illuminated with red light of wavelength $6.5 \times 10^{-7}\, m$. The interference fringes are observed on a screen placed $1\, m$ from the slits. Find the distance between the third dark fringe and the fifth bright fringe on the same side of the central maxima.

$A$ point charge $q$ is placed at a distance $a/2$ directly above the centre of a square of side $a$. The electric flux through the square is