Let $P ( 3,2,6 )$ be a point in space and $Q$ be a point on the line
$$\vec { r } = ( \hat { i } - \hat { j } + 2 \hat { k } ) + \mu ( - 3 \hat { i } + \hat { j } + 5 \hat { k } )$$
Then the value of $\mu$ for which the vector $\overrightarrow { P Q }$ is parallel to the plane $x - 4 y + 3 z = 1$ is
(A) $\frac { 1 } { 4 }$
(B) $- \frac { 1 } { 4 }$
(C) $\frac { 1 } { 8 }$
(D) $- \frac { 1 } { 8 }$

Let $P _ { 1 }$ and $P _ { 2 }$ be two planes given by
$$\begin{aligned} & P _ { 1 } : 10 x + 15 y + 12 z - 60 = 0 \\ & P _ { 2 } : \quad - 2 x + 5 y + 4 z - 20 = 0 \end{aligned}$$
Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on $P _ { 1 }$ and $P _ { 2 }$ ?
(A) $\frac { x - 1 } { 0 } = \frac { y - 1 } { 0 } = \frac { z - 1 } { 5 }$
(B) $\frac { x - 6 } { - 5 } = \frac { y } { 2 } = \frac { z } { 3 }$
(C) $\frac { x } { - 2 } = \frac { y - 4 } { 5 } = \frac { z } { 4 }$
(D) $\frac { x } { 1 } = \frac { y - 4 } { - 2 } = \frac { z } { 3 }$

If the line $\frac{x-3}{2} = \frac{y+2}{-1} = \frac{z+4}{3}$ lies in the plane $lx + my - z = 9$, then $l^2 + m^2$ is equal to:
(1) $26$
(2) $18$
(3) $1$
(4) $2$

If the line $\frac{x-3}{2} = \frac{y+2}{-1} = \frac{z+4}{3}$ lies in the plane $lx + my - z = 9$, then $l^2 + m^2$ is equal to: (1) 18 (2) 5 (3) 2 (4) 26

Let $A$ be a point on the line $\vec { r } = ( 1 - 3 \mu ) \hat { i } + ( \mu - 1 ) \hat { j } + ( 2 + 5 \mu ) \hat { k }$ and $B ( 3,2,6 )$ be a point in the space. Then the value of $\mu$ for which the vector $\overrightarrow { A B }$ is parallel to the plane $x - 4 y + 3 z = 1$ is
(1) $\frac { 1 } { 2 }$
(2) $\frac { 1 } { 4 }$
(3) $- \frac { 1 } { 4 }$
(4) $\frac { 1 } { 8 }$

Let $P$ be a plane $l x + m y + n z = 0$ containing the line, $\frac { 1 - x } { 1 } = \frac { y + 4 } { 2 } = \frac { z + 2 } { 3 }$. If plane $P$ divides the line segment $A B$ joining points $A ( - 3 , - 6,1 )$ and $B ( 2,4 , - 3 )$ in ratio $k : 1$ then the value of $k$ is