$A$ black body at a temperature of $227^{\circ}C$ radiates heat energy at the rate of $5 \ cal/cm^2-sec$. At a temperature of $727^{\circ}C$,the rate of heat radiated per unit area in $cal/cm^2-sec$ will be:

The molar heat capacity in a process of a diatomic gas,if it does a work of $\frac{Q}{4}$ when a heat of $Q$ is supplied to it,is:

For $x > 1$,if $(2x)^{2y} = 4e^{2x - 2y}$,then $(1 + \log_e 2x)^2 \frac{dy}{dx}$ is equal to

$A$ steel meter scale is to be ruled so that millimeter intervals are accurate within about $5 \times 10^{-5} \ m$ at a certain temperature. The maximum temperature variation allowable during the ruling is (Coefficient of linear expansion of steel $= 10 \times 10^{-6} \ K^{-1}$) (in $^{\circ} C$)

$A$ $10 \mu F$ capacitor is charged by a $100 \text{ V}$ battery. It is disconnected from the battery and is connected to another uncharged capacitor of capacitance $30 \mu F$. During this process,the electrostatic energy lost by the first capacitor is

Seven white balls and three black balls are randomly arranged in a row. The probability that no two black balls are placed adjacently is