Two cells of e.m.f. $E_1$ and $E_2$ ($E_1 > E_2$) are connected as shown in figure.

When a potentiometer is connected between points A and B the balancing length of potentiometer wire is $412$ cm. When same potentiometer wire is connected between points A and C the balancing length is $103$ cm. The ratio $E_1 : E_2$ is

In the circuit shown charge q varies with time t as $q = t^2 - 5$, where q is in coulomb and t is in second. At time $t = 3$ second, voltage $V_{AB}$ in volt will be

The resistances in the two gaps of a balanced meter bridge are 'X' $\Omega$ and '3X' $\Omega$ respectively. If the resistances are interchanged the balance point shifts by

In a meter bridge experiment, the balance point is obtained at length '$l_1$' cm from left hand when resistances in the left gap and right gap are $15 \ \Omega$ and $R \ \Omega$ respectively. When the resistance $R$ is shunted with equal resistance the new balance point is at $(1.6\,l_1)$. The resistance $R$ in ohm is