Let $\overline{\mathrm{A}}=2 \hat{\mathrm{i}}+\hat{\mathrm{k}}, \overline{\mathrm{B}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overline{\mathrm{C}}=4 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+7 \hat{\mathrm{k}}$. If a vector $\bar{R}$ satisfies $\bar{R} \times \bar{B}=\bar{C} \times \bar{B}$ and $\bar{R} \cdot \overline{\mathrm{~A}}=0$, then $\overline{\mathrm{R}}$ is given by

Distance between the parallel lines $\frac{x}{3}=\frac{y-1}{-2}=\frac{z}{1}$ and $\frac{x+4}{3}=\frac{y-3}{-2}=\frac{z+2}{1}$ is

If $\mathrm{f}(x)=\log _{x^2}\left(\log _{\mathrm{e}} x\right)$, then $\mathrm{f}^{\prime}(x)$ at $x=\mathrm{e}$ is

A body cools according to Newton's law of cooling from $100^{\circ} \mathrm{C}$ to $60^{\circ} \mathrm{C}$ in 15 minutes. If the temperature of the surrounding is $20^{\circ} \mathrm{C}$, then the temperature of the body after cooling down for one hour is