Two conducting circular loops of radii $R_1$ and $R_2$ are placed in the same plane with their centres coinciding. If $R_1 > R_2$,the mutual inductance $M$ between them will be directly proportional to

The solution of the differential equation $\frac{dy}{dx} = \frac{x - y + 3}{2(x - y) + 5}$ is

The pulleys in the diagram are all smooth and light. The acceleration of $A$ is $a$ upwards and the acceleration of $C$ is $f$ downwards. The acceleration of $B$ is

The $pH$ of a buffer solution made by mixing $25 \ mL$ of $0.02 \ M \ NH_4OH$ and $25 \ mL$ of $0.2 \ M \ NH_4Cl$ at $25^{\circ}C$ is ($pK_b$ of $NH_4OH = 4.8$).

$A$ sonometer wire resonates with a given tuning fork,forming standing waves with five antinodes between the two bridges when a mass of $9 \, kg$ is suspended from the wire. When this mass is replaced by a mass $M$,the wire resonates with the same tuning fork,forming three antinodes for the same positions of the bridges. The value of $M$ is .... $kg$.

Identify the conformer of $2-$methylpentane.