3 questions from 2023
Let $P$ be the plane $\sqrt{3}x + 2y + 3z = 16$ and let
$S = \left\{ \alpha\hat{i} + \beta\hat{j} + \gamma\hat{k} : \alpha^2 + \beta^2 + \gamma^2 = 1 \text{ and the distance of } (\alpha, \beta, \gamma) \text{ from the plane } P \text{ is } \frac{7}{2} \right\}.$
Let $\vec{u}, \vec{v}$ and $\vec{w}$ be three distinct vectors in $S$ such that $|\vec{u} - \vec{v}| = |\vec{v} - \vec{w}| = |\vec{w} - \vec{u}|$. Let $V$ be the volume of the parallelepiped determined by vectors $\vec{u}, \vec{v}$ and $\vec{w}$. Then the value of $\frac{80}{\sqrt{3}}V$ is _____.
Let \( Q \) be the cube with the set of vertices \( \{(x_1,x_2,x_3) \in R^3 : x_1,x_2,x_3 \in \{0,1\}\} \). Let \( F \) be the set of all twelve lines containing the diagonals of the six faces of the cube \( Q \). Let \( S \) be the set of all four lines containing the main diagonals of the cube \( Q \); for instance, the line passing through the vertices \( (0,0,0) \) and \( (1,1,1) \) is in \( S \). For lines \( \ell_1 \) and \( \ell_2 \), let \( d(\ell_1,\ell_2) \) denote the shortest distance between them. Then the maximum value of \( d(\ell_1,\ell_2) \), as \( \ell_1 \) varies over \( F \) and \( \ell_2 \) varies over \( S \), is:
Let \( \ell_1 \) and \( \ell_2 \) be the lines \( \vec{r_1}=\lambda(\hat{\imath}+\hat{\jmath}+\hat{k}) \) and \( \vec{r_2}=(\hat{\jmath}-\hat{k})+\mu(\hat{\imath}+\hat{k}) \), respectively. Let \( X \) be the set of all the planes \( H \) that contain the line \( \ell_1 \). For a plane \( H \), let \( d(H) \) denote the smallest possible distance between the points of \( \ell_2 \) and \( H \). Let \( H_0 \) be a plane in \( X \) for which \( d(H_0) \) is the maximum value of \( d(H) \) as \( H \) varies over all planes in \( X \).
Match each entry in List-I to the correct entries in List-II.
The correct option is:
