If $|a|=1, |b|=2$ and the angle between $a$ and $b$ is $120^{\circ}$,then ${(a+3b) \times (3a-b)}^2$ is equal to

Two particles $A$ and $B$ initially at rest move towards each other under a mutual force of attraction. At the instant when the speed of $A$ is $v$ and the speed of $B$ is $2v$,the speed of the centre of mass of the system is:

If three points $A, B$ and $C$ have position vectors $(1, x, 3), (3, 4, 7)$ and $(y, -2, -5)$ respectively and if they are collinear,then $(x, y)$ is

$A$ simple pendulum has a periodic time $T_1$ when it is on the surface of the Earth of radius $R$. Its periodic time is $T_2$ when it is taken to a height $R$ above the Earth's surface. The value of $\frac{T_2}{T_1}$ is

$A$ conducting ring is placed around the core of an electromagnet as shown in the figure. When key $K$ is pressed,the ring

Two spherical conductors each of capacity $C$ are charged to potentials $V$ and $-V$. These are then connected by means of a fine conducting wire. The loss of energy will be