If the population grows at the rate of $8 \%$ per year,then the time taken for the population to be doubled is (Given $\log 2=0.6912$ ) (in $\text{ years}$)

If $\omega$ is a complex cube root of unity,then $\sin \left\{\left(\omega^{10}+\omega^{23}\right) \pi-\frac{\pi}{4}\right\}$ is equal to

Select the correct pair from the following.

$A$ black body radiates energy at the rate of $E \text{ W/m}^2$ at a high temperature $T \text{ K}$. When the temperature is reduced to $\left(\frac{T}{2}\right) \text{ K}$,the radiant energy is

The solution of $\frac{dy}{dx} = (\frac{x}{y})^{-1/3}$ is

$A$ plane passes through $(2,3,-1)$ and is perpendicular to the line having direction ratios $3,-4,7$. The perpendicular distance from the origin to this plane is