Find the value of the current through the $2\,\Omega$ resistance for the given circuit. (in $,A$)

$A$ particle moves in a straight line with retardation proportional to its displacement. Its loss of kinetic energy for any displacement $x$ is proportional to:

For any integer $n \geq 2$,let $I_n = \int \tan^n x \, dx$. If $I_n = \frac{1}{a} \tan^{n-1} x - b I_{n-2}$ for $n \geq 2$,then the ordered pair $(a, b)$ equals to

$A$ small metal sphere of radius $a$ is falling with a velocity $v$ through a vertical column of a viscous liquid. If the coefficient of viscosity of the liquid is $\eta$,then the sphere encounters an opposing force of

If $f(x)=|x|+|\sin x|$ for $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, then its left hand derivative at $x=0$ is

The equation of the normal to the curve $y=x \log x$,which is parallel to the line $2x-2y+3=0$,is