If $\cos ^{-1}\left(\frac{y}{b}\right)=2 \log \left(\frac{x}{2}\right)$, where $x>0$, then $x^2 \frac{d^2 y}{d x^2}+x \frac{d y}{d x}$ is equal to

$A$ parallel plate capacitor of area $A$,plate separation $d$ and capacitance $C$ is filled with three different dielectric materials having dielectric constants $K_1, K_2$ and $K_3$ as shown in the figure. If a single dielectric material is to be used to have the same capacitance $C$ in this capacitor,then its dielectric constant $K$ is given by: ($A =$ Area of plates)

If one of the lines of the pair of straight lines $ax^2 + 2hxy + by^2 = 0$ bisects the angle between the coordinate axes,then:

If $\frac{d}{d x}\left[(x+1)\left(x^2+1\right)\left(x^4+1\right)\left(x^8+1\right)\right] = \left(15 x^p-16 x^q+1\right)(x-1)^{-2}$,then $(p, q)$ is equal to

The component of vector $\overrightarrow{A} = a_x \hat{i} + a_y \hat{j} + a_z \hat{k}$ along the direction of $\hat{i} - \hat{j}$ is

Hairs of a shaving brush cling together when it is removed from water due to: