Let $u$ and $v$ be two vectors. Then,$|u-v|=||u|-|v||$ if and only if

Identify the reagent used in the following reaction.
Benzoic acid $\stackrel{\text{Reagent}}{\Delta}$ Benzoyl chloride + Phosphorous oxychloride + Hydrogen chloride

The time in seconds required to produce a potential difference of $20 \ V$ across a capacitor of $1000 \ \mu F$ when it is charged at a steady rate of $200 \ \mu C/s$ is:

An insect crawls up a hemispherical surface very slowly (see the figure). The coefficient of friction between the insect and the surface is $\mu = 1/3$. If the line joining the centre of the hemispherical surface to the insect makes an angle $\alpha$ with the vertical,the maximum possible value of $\alpha$ is given by

$A$ ring of radius $4a$ is rigidly fixed in a vertical position on a table. $A$ small disc of mass $m$ and radius $a$ is released as shown in the figure. When the disc rolls down,without slipping,to the lowest point of the ring,then its speed will be

In a meter bridge,the null point is found at a distance of $25\, cm$ from $A$. If now a resistance of $10\,\Omega$ is connected in parallel with $S$,the null point occurs at the midpoint of $AB$. The value of $R$ is .............. $\Omega$. (in $.67$)