Let ${\ell }_1$ and ${\ell }_2$ be the lines ${\overrightarrow{r}}_1=\lambda \left(\hat{i}+\hat{j}+\hat{k}\right)$ and ${\overrightarrow{r}}_2=\left(\hat{j}-\hat{k}\right)+\mu \left(\hat{i}+\hat{k}\right)$, respectively, Let $X$ be the set of all the planes $H$ that contain the line ${\ell }_1$. For a plane $H$, let $d\left(H\right)$ denote the smallest possible distance between the points of ${\ell }_2$ and $H$. Let $H_0$ be a plane in $X$ for which $d\left(H_0\right)$ is the maximum value of $d\left(H\right)$ as $H$ varies over all planes in $X$.

Match each entry in List-I to the correct entries in List-II.

	List-I		List-II
$\left(\mathrm{P}\right)$	The value of $d\left(H_0\right)$ is	$\left(1\right)$	$\sqrt{3}$
$\left(\mathrm{Q}\right)$	The distance of the point $\left(0, 1, 2\right)$ from $H_0$ is	$\left(2\right)$	$\frac{1}{\sqrt{3}}$
$\left(\mathrm{R}\right)$	The distance of origin from $H_0$ is	$\left(3\right)$	$0$
$\left(\mathrm{S}\right)$	The distance of origin from the point of intersection of planes $y=z, x=1$ and $H_0$ is	$\left(4\right)$	$\sqrt{2}$
		$\left(5\right)$	$\frac{1}{\sqrt{2}}$

The correct option is

• A

  $\left(\mathrm{P}\right)\to \left(2\right) \left(\mathrm{Q}\right)\to \left(4\right) \left(\mathrm{R}\right)\to \left(5\right) \left(\mathrm{S}\right)\to \left(1\right)$
• Correct answer
  B

  $\left(\mathrm{P}\right)\to \left(5\right) \left(\mathrm{Q}\right)\to \left(4\right) \left(\mathrm{R}\right)\to \left(3\right) \left(\mathrm{S}\right)\to \left(1\right)$
• C

  $\left(\mathrm{P}\right)\to \left(2\right) \left(\mathrm{Q}\right)\to \left(1\right) \left(\mathrm{R}\right)\to \left(3\right) \left(\mathrm{S}\right)\to \left(2\right)$
• D

  $\left(\mathrm{P}\right)\to \left(5\right) \left(\mathrm{Q}\right)\to \left(1\right) \left(\mathrm{R}\right)\to \left(4\right) \left(\mathrm{S}\right)\to \left(2\right)$

Hence, the correct answer is option: 'B'.