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JEE Advance | PYQ

Let \( P \) be the plane such that it contains the straight line \( \dfrac{x-1}{2} = \dfrac{y-3}{3} = \dfrac{z+2}{1} \) and is perpendicular to the plane \( x + 2y + 3z = 4 \). Let \( P_1 \) be the plane which passes through the point \( (4, 2, 2) \) and is parallel to \( P \). Then which of the following statements is (are) TRUE?

A straight line drawn from the point \( (1,3,2) \), \( P \) parallel to the line \( \dfrac{x-2}{1}=\dfrac{y-4}{2}=\dfrac{z-6}{1} \) intersects the plane \( L_1: x-y+3z=6 \) at the point \( Q \). Another straight line which passes through \( Q \) and is perpendicular to the plane \( L_1 \) intersects the plane \( L_2: 2x-y+z=-4 \) at the point \( R \). Then which of the following statements is (are) TRUE?

Let \( \mathbb{R}^3 \) denote the three-dimensional space. Take two points \( P = (1, 2, 3) \) and \( Q = (4, 2, 7) \).
Let \( \operatorname{dist}(X, Y) \) denote the distance between two points \( X \) and \( Y \) in \( \mathbb{R}^3 \). Let
\( S = \left\{ X \in \mathbb{R}^3 : \big(\operatorname{dist}(X, P)\big)^2 - \big(\operatorname{dist}(X, Q)\big)^2 = 50 \right\} \) and
\( T = \left\{ Y \in \mathbb{R}^3 : \big(\operatorname{dist}(Y, Q)\big)^2 - \big(\operatorname{dist}(Y, P)\big)^2 = 50 \right\} \).
Then which of the following statements is (are) TRUE?

Let \( \gamma \in \mathbb{R} \) be such that the lines \( L_1 : \dfrac{x+11}{1} = \dfrac{y+21}{2} = \dfrac{z+29}{3} \) and \( L_2 : \dfrac{x+16}{3} = \dfrac{y+11}{2} = \dfrac{z+4}{\gamma} \) intersect. Let \( R_1 \) be the point of intersection of \( L_1 \) and \( L_2 \). Let \( O = (0, 0, 0) \), and \( \hat{n} \) denote a unit normal vector to the plane containing both the lines \( L_1 \) and \( L_2 \).
Match each entry in List-I to the correct entry in List-II.

The correct option 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: