Two large parallel plates are connected to a $100\, V$ power supply. These plates have a fine hole at the centre. An electron having energy $200\, eV$ is directed such that it passes through the holes. When it emerges,its de-Broglie wavelength is ............. $\mathring{A}$.

The mean of four observations is $3$. If the sum of the squares of these observations is $48$,then their standard deviation is

Two cylinders $A$ and $B$ fitted with pistons contain equal number of moles of an ideal monoatomic gas at $400 ~K$. The piston of $A$ is free to move while that of $B$ is held fixed. Same amount of heat energy is given to the gas in each cylinder. If the rise in temperature of the gas in $A$ is $42 ~K$,the rise in temperature of the gas in $B$ is (in $~K$)

Let $M$ be the foot of the perpendicular from a point $P$ on the parabola $y^2=8(x-3)$ onto its directrix and let $S$ be the focus of the parabola. If $\triangle SPM$ is an equilateral triangle,then $P$ is equal to

The result after adding $3.8 \times 10^{-6}$ and $4.2 \times 10^{-5}$ with due regard to significant figures is:

$\cos \alpha \sin (\beta-\gamma)+\cos \beta \sin (\gamma-\alpha)+\cos \gamma \sin (\alpha-\beta)$ is equal to