$A$ conducting rod $AC$ of length $4l$ is rotated about a point $O$ in a uniform magnetic field $\vec{B}$ directed into the paper. $AO = l$ and $OC = 3l$. Then:

$A$ body is projected vertically upwards at time $t=0$ and it is seen at a height $H$ at time $t_1$ and $t_2$ seconds during its flight. The maximum height attained is ($g$ is acceleration due to gravity).

Which one of the following elements,when present as an impurity in silicon,makes it a $p$-type semiconductor?

If the circle $x^2 + y^2 + 6x - 2y + k = 0$ bisects the circumference of the circle $x^2 + y^2 + 2x - 6y - 15 = 0$,then $k =$

Two identical vessels $A$ and $B$ with frictionless pistons contain the same ideal gas at the same temperature and the same volume $V$. The masses of gas in $A$ and $B$ are $m_A$ and $m_B$,respectively. The gases are allowed to expand isothermally to the same final volume $2V$. The changes in pressures of the gas in $A$ and $B$ are found to be $\Delta P$ and $1.5 \Delta P$,respectively. Then:

When a current of $2\, A$ flows in a battery from negative to positive terminal,the potential difference across it is $12\, V$. If a current of $3\, A$ flows in the opposite direction,the potential difference across the terminals of the battery is $15\, V$. The $emf$ of the battery is ................ $V$.