The differential of ${e^{{x^3}}}$ with respect to $\log x$ is

If the line $x+3y=0$ is the tangent at $(0,0)$ to the circle of radius $1$,then the centre of one such circle is

$A$ particle starting from the origin $(0, 0)$ moves in a straight line in the $(x, y)$ plane. Its coordinates at a later time are $(\sqrt{3}, 3)$. The path of the particle makes an angle of $.......^{\circ}$ with the $x$-axis.

Match the following:

$1$. Burning candle	$i$. Line spectrum
$2$. Sodium vapour	$ii$. Continuous spectrum
$3$. Bunsen flame	$iii$. Band spectrum
$4$. Dark lines in solar spectrum	$iv$. Absorption spectrum

One end each of a resistance $r$,a capacitor $C$,and a resistance $2r$ are connected together. The other ends are respectively connected to the positive terminals of batteries $P, Q,$ and $R$ having emf's $E, E,$ and $2E$. The negative terminals of the batteries are then connected together. In this circuit,with steady current,the potential drop across the capacitor is:

Let a random variable $X$ have a Binomial distribution with mean $8$ and variance $4$. If $P(X \leq 2) = \frac{k}{2^{16}}$,then $k$ is equal to: