When $MnO_2$ is fused with $KOH$ in the presence of air,a coloured compound is formed. Identify the product and its colour.

$A$ circular disc of radius $R$ is removed from a bigger uniform circular disc of radius $2R$ such that the circumferences of the discs coincide. The centre of mass of the new disc is $\alpha R$ from the centre of the bigger disc. The value of $\alpha$ is

$A$ small square loop of wire of side $l$ is placed inside a large square loop of side $L$ $(L \gg l)$. If the loops are coplanar and their centres coincide,the mutual inductance of the system is directly proportional to :

In the circuit shown,a potential difference of $60\,V$ is applied across $AB$. The potential difference between the points $M$ and $N$ is.....$V$

$\int_0^{\pi / 2} \frac{d x}{1+\tan ^3 x}$ is equal to :

The reaction $2A_{(g)} + B_{(g)} \rightleftharpoons 3C_{(g)} + D_{(g)}$ is begun with the concentrations of $A$ and $B$ both at an initial value of $1.00 \ M$. When equilibrium is reached,the concentration of $D$ is measured and found to be $0.25 \ M$. The value for the equilibrium constant for this reaction is given by the expression