$A$ uniform rod of mass $M$ and length $L$ is placed on a smooth horizontal surface. $A$ horizontal force $F$ is applied at one end of the rod perpendicular to its length. The acceleration of the centre of mass of the rod is:

Let $f:[-1, 3] \to R$ be defined as $f(x) = \begin{cases} |x| + [x], & -1 \le x < 1 \\ x + |x|, & 1 \le x < 2 \\ x + [x], & 2 \le x \le 3 \end{cases}$ where $[t]$ denotes the greatest integer function. Then $f$ is discontinuous at:

The atomic weights of two elements $A$ and $B$ are $40$ and $80$ respectively. If $x \ g$ of $A$ contains $y$ atoms,how many atoms are present in $2x \ g$ of $B$?

An inductor $(L = 100 \, mH)$,a resistor $(R = 100 \, \Omega)$,and a battery $(E = 100 \, V)$ are initially connected in series as shown in the figure. After a long time,the battery is disconnected by short-circuiting the points $A$ and $B$. The current in the circuit $1 \, ms$ after the short circuit is:

If $\cot ^{-1}(\sqrt{\cos \alpha})-\tan ^{-1}(\sqrt{\cos \alpha})=x$,then the value of $\sin x$ is

The ratio of the magnetic field at the centre of a current-carrying circular wire and the magnetic field at the centre of a square coil made from the same length of wire will be