An insulated container contains $4 \, \text{moles}$ of an ideal diatomic gas at temperature $T$. Heat $Q$ is supplied to this gas, due to which $2 \, \text{moles}$ of gas are dissociated into atoms, but the temperature of the gas remains constant. Then:

In a permanent magnet at room temperature,

The coefficient of apparent expansion of mercury in a glass vessel is $153 \times 10^{-6}/^{\circ}C$ and in a steel vessel is $144 \times 10^{-6}/^{\circ}C$. If $\alpha$ for steel is $12 \times 10^{-6}/^{\circ}C$,then $\alpha$ for glass is:

In a playground,there is a merry-go-round of mass $120 \ kg$ and radius $4 \ m$. The radius of gyration is $3 \ m$. $A$ child of mass $30 \ kg$ runs at a speed of $5 \ m/s$ tangent to the rim of the merry-go-round when it is at rest and then jumps on it. Neglecting friction,find the angular velocity of the merry-go-round and child in $rad/s$.

If $f(x)$ satisfies the relation $f\left( \frac{5x - 3y}{2} \right) = \frac{5f(x) - 3f(y)}{2}$ for all $x, y \in R$ and $f(0)=1, f'(0)=2$,then the period of $\sin(f(x))$ is:

$0.16 \ g$ of an organic compound containing sulphur produces $0.233 \ g$ of $BaSO_4$. The percentage of sulphur in the compound is: