If the angles of a triangle are in the ratio $4:1:1$,then the ratio of the longest side to the perimeter is

The solution of the differential equation $\frac{dy}{dx} - 2y \tan 2x = e^x \sec 2x$ is

If $\overrightarrow{a} = 2\widehat{i} - \widehat{j} + \widehat{k}$,$\overrightarrow{b} = \widehat{i} + \widehat{j} - 2\widehat{k}$ and $\overrightarrow{c} = \widehat{i} + 3\widehat{j} - (\lambda^2 + 3\lambda)\widehat{k}$ (where $\lambda$ is a constant) and $\overrightarrow{a}$ is perpendicular to $\overrightarrow{c} - \lambda\overrightarrow{b}$,then the sum of different values of $\lambda$ is:

$A$ non-uniformly shaped conductor is charged. Then,at its sharpest point:

In the arrangement shown in the figure,the ends $P$ and $Q$ of an unstretchable string move downwards with a uniform speed $U$. Pulleys $A$ and $B$ are fixed. The mass $M$ moves upwards with a speed:

One end of a string of length $l$ is connected to a particle of mass $m$ and the other to a small peg on a smooth horizontal table. If the particle moves in a circle with speed $v$ and $T$ is the tension in the string,then the net force on the particle (directed towards the centre) is