If $f(x) = \begin{cases} \frac{1-\sqrt{2} \sin x}{\pi-4x} & \text{if } x \neq \frac{\pi}{4} \\ a & \text{if } x = \frac{\pi}{4} \end{cases}$ is continuous at $x = \frac{\pi}{4}$, then $a$ is equal to

$A$ string is vibrating in its fifth overtone between two rigid supports $2.4 ~m$ apart. The distance between successive node and antinode is (in $~m$)

Two simple harmonic motions are represented by $y_{1} = 5[\sin 2 \pi t + \sqrt{3} \cos 2 \pi t]$ and $y_{2} = 5 \sin (2 \pi t + \frac{\pi}{4})$. The ratio of their amplitudes is

The reagent used in the Wolff-Kishner reduction is

$A$ uniform conductor of resistance $R$ is cut into $20$ equal pieces. Half of them are joined in series and the remaining half of them are connected in parallel. If the two combinations are joined in series,the effective resistance of all the pieces is

The plates of a parallel plate capacitor are charged up to $100\, V$. $A$ $2\, mm$ thick plate is inserted between the plates. To maintain the same potential difference,the distance between the capacitor plates is increased by $1.6\, mm$. The dielectric constant of the plate is: