The angle between the straight lines, whose direction cosines $l , m , n$ are given by the equations $2 l + 2 m - n = 0$ and $m n + n l + \operatorname { lm } = 0$, is: (1) $\frac { \pi } { 3 }$ (2) $\frac { \pi } { 2 }$ (3) $\cos ^ { - 1 } \left( \frac { 8 } { 9 } \right)$ (4) $\pi - \cos ^ { - 1 } \left( \frac { 4 } { 9 } \right)$

Let the equation of the plane, that passes through the point $( 1,4 , - 3 )$ and contains the line of intersection of the planes $3 x - 2 y + 4 z - 7 = 0$ and $x + 5 y - 2 z + 9 = 0$, be $\alpha x + \beta y + \gamma z + 3 = 0$, then $\alpha + \beta + \gamma$ is equal to :
(1) $- 15$
(2) 15
(3) $- 23$
(4) 23

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

The square of the distance of the point of intersection of the line $\frac { x - 1 } { 2 } = \frac { y - 2 } { 3 } = \frac { z + 1 } { 6 }$ and the plane $2 x - y + z = 6$ from the point $( - 1 , - 1,2 )$ is

If the shortest distance between the lines $\frac { x - 1 } { 2 } = \frac { y - 2 } { 3 } = \frac { z - 3 } { \lambda }$ and $\frac { x - 2 } { 1 } = \frac { y - 4 } { 4 } = \frac { z - 5 } { 5 }$ is $\frac { 1 } { \sqrt { 3 } }$, then the sum of all possible values of $\lambda$ is:
(1) 16
(2) 6
(3) 12
(4) 15

Let $P$ be the plane containing the straight line $\frac { x - 3 } { 9 } = \frac { y + 4 } { - 1 } = \frac { z - 7 } { - 5 }$ and perpendicular to the plane containing the straight lines $\frac { x } { 2 } = \frac { y } { 3 } = \frac { z } { 5 }$ and $\frac { x } { 3 } = \frac { y } { 7 } = \frac { z } { 8 }$. If $d$ is the distance of $P$ from the point $(2,-5,11)$, then $d ^ { 2 }$ is equal to
(1) $\frac { 147 } { 2 }$
(2) 96
(3) $\frac { 32 } { 3 }$
(4) 54

The length of the perpendicular from the point $( 1 , - 2,5 )$ on the line passing through $( 1,2,4 )$ and parallel to the line $x + y - z = 0 = x - 2 y + 3 z - 5$ is:
(1) $\sqrt { \frac { 21 } { 2 } }$
(2) $\sqrt { \frac { 9 } { 2 } }$
(3) $\sqrt { \frac { 73 } { 2 } }$
(4) 1

If the line of intersection of the planes $a x + b y = 3$ and $a x + b y + c z = 0 , a > 0$ makes an angle $30 ^ { \circ }$ with the plane $y - z + 2 = 0$, then the direction cosines of the line are
(1) $\frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } , 0$
(2) $\frac { 1 } { \sqrt { 2 } } , \frac { - 1 } { \sqrt { 2 } } , 0$
(3) $\frac { 1 } { \sqrt { 5 } } , - \frac { 2 } { \sqrt { 5 } } , 0$
(4) $\frac { 1 } { 2 } , - \frac { \sqrt { 3 } } { 2 } , 0$

Let the lines $\frac { x - 1 } { \lambda } = \frac { y - 2 } { 1 } = \frac { z - 3 } { 2 }$ and $\frac { x + 26 } { - 2 } = \frac { y + 18 } { 3 } = \frac { z + 28 } { \lambda }$ be coplanar and $P$ be the plane containing these two lines. Then which of the following points does NOT lie on $P$?
(1) $( 0 , - 2 , - 2 )$
(2) $( - 5,0 , - 1 )$
(3) $( 3 , - 1,0 )$
(4) $( 0,4,5 )$

Let the points on the plane $P$ be equidistant from the points $( - 4,2,1 )$ and $( 2 , - 2,3 )$. Then the acute angle between the plane $P$ and the plane $2 x + y + 3 z = 1$ is
(1) $\frac { \pi } { 6 }$
(2) $\frac { \pi } { 4 }$
(3) $\frac { \pi } { 3 }$
(4) $\frac { 5 \pi } { 12 }$

If the plane $2 x + y - 5 z = 0$ is rotated about its line of intersection with the plane $3 x - y + 4 z - 7 = 0$ by an angle of $\frac { \pi } { 2 }$, then the plane after the rotation passes through the point
(1) $( 2 , - 2,0 )$
(2) $( - 2,2,0 )$
(3) $( 1,0,2 )$
(4) $( - 1,0 , - 2 )$

The shortest distance between the lines $\frac{x+7}{-6} = \frac{y-6}{7} = z$ and $\frac{7-x}{2} = y-2 = z-6$ is
(1) $2\sqrt{29}$
(2) 1
(3) $\sqrt{\frac{37}{2}}$
(4) (truncated)

If the plane $P$ passes through the intersection of two mutually perpendicular planes $2 x + k y - 5 z = 1$ and $3 k x - k y + z = 5 , k < 3$ and intercepts a unit length on positive $x$-axis, then the intercept made by the plane $P$ on the $y$-axis is
(1) $\frac { 1 } { 11 }$
(2) $\frac { 5 } { 11 }$
(3) 6
(4) 7

A plane $P$ is parallel to two lines whose direction ratios are $- 2,1 , - 3$ and $- 1,2 , - 2$ and it contains the point $( 2,2 , - 2 )$. Let $P$ intersect the co-ordinate axes at the points $A , B , C$ making the intercepts $\alpha , \beta , \gamma$. If $V$ is the volume of the tetrahedron $O A B C$, where $O$ is the origin and $p = \alpha + \beta + \gamma$, then the ordered pair $( V , p )$ is equal to
(1) $( 48 , - 13 )$
(2) $( 24 , - 13 )$
(3) $( 48,11 )$
(4) $( 24 , - 5 )$

If the lines $\vec { r } = ( \hat { i } - \hat { j } + \widehat { k } ) + \lambda ( 3 \hat { j } - \widehat { k } )$ and $\vec { r } = ( \alpha \hat { i } - \hat { j } ) + \mu ( 2 \hat { i } - 3 \widehat { k } )$ are co-planar, then the distance of the plane containing these two lines from the point $( \alpha , 0,0 )$ is
(1) $\frac { 2 } { 9 }$
(2) $\frac { 2 } { 11 }$
(3) $\frac { 4 } { 11 }$
(4) 2

The line of shortest distance between the lines $\frac { x - 2 } { 0 } = \frac { y - 1 } { 1 } = \frac { z } { 1 }$ and $\frac { x - 3 } { 2 } = \frac { y - 5 } { 2 } = \frac { z - 1 } { 1 }$ makes an angle of $\sin ^ { - 1 } \sqrt { \frac { 2 } { 27 } }$ with the plane $P : ax - y - z = 0$, $a > 0$. If the image of the point $(1,1,-5)$ in the plane $P$ is $(\alpha, \beta, \gamma)$, then $\alpha + \beta - \gamma$ is equal to $\_\_\_\_$.

Let the line $\frac { x - 3 } { 7 } = \frac { y - 2 } { - 1 } = \frac { z - 3 } { - 4 }$ intersect the plane containing the lines $\frac { x - 4 } { 1 } = \frac { y + 1 } { - 2 } = \frac { z } { 1 }$ and $4 a x - y + 5 z - 7 a = 0 = 2 x - 5 y - z - 3 , a \in \mathbb { R }$ at the point $P ( \alpha , \beta , \gamma )$. Then the value of $\alpha + \beta + \gamma$ equals $\_\_\_\_$ .

Let $P$ be the plane passing through the intersection of the planes $\vec{r}\cdot(\hat{i}+3\hat{j}-\hat{k}) = 5$ and $\vec{r}\cdot(2\hat{i}-\hat{j}+\hat{k}) = 3$, and the point $(2, 1, -2)$. Let the position vectors of the points $X$ and $Y$ be $\hat{i} - 2\hat{j} + 4\hat{k}$ and $5\hat{i} - \hat{j} + 2\hat{k}$ respectively. Then the points $X$ and $Y$ with respect to the plane $P$ are
(1) on the same side
(2) on opposite sides
(3) $X$ lies on $P$
(4) $Y$ lies on $P$

The plane, passing through the points $( 0 , - 1 , 2 )$ and $( - 1 , 2 , 1 )$ and parallel to the line passing through $( 5 , 1 , - 7 )$ and $( 1 , - 1 , - 1 )$, also passes through the point
(1) $( - 2 , 5 , 0 )$
(2) $( 1 , - 2 , 1 )$
(3) $( 2 , 0 , 1 )$
(4) $( 0 , 5 , - 2 )$

The line, that is coplanar to the line $\frac { x + 3 } { - 3 } = \frac { y - 1 } { 1 } = \frac { z - 5 } { 5 }$, is
(1) $\frac { x + 1 } { - 1 } = \frac { y - 2 } { 2 } = \frac { z - 5 } { 4 }$
(2) $\frac { x + 1 } { - 1 } = \frac { y - 2 } { 2 } = \frac { z - 5 } { 5 }$
(3) $\frac { x - 1 } { - 1 } = \frac { y - 2 } { 2 } = \frac { z - 5 } { 5 }$
(4) $\frac { x + 1 } { 1 } = \frac { y - 2 } { 2 } = \frac { z - 5 } { 5 }$

Let the foot of perpendicular of the point $P ( 3 , - 2 , - 9 )$ on the plane passing through the points $( - 1 , - 2 , - 3 ) , ( 9,3,4 ) , ( 9 , - 2,1 )$ be $Q ( \alpha , \beta , \gamma )$. Then the distance of $Q$ from the origin is
(1) $\sqrt { 42 }$
(2) $\sqrt { 38 }$
(3) $\sqrt { 35 }$
(4) $\sqrt { 29 }$

The shortest distance between the lines $\frac{x+2}{1} = \frac{y}{-2} = \frac{z-5}{2}$ and $\frac{x-4}{1} = \frac{y-1}{2} = \frac{z+3}{0}$ is
(1) 8
(2) 6
(3) 7
(4) 9

Let $N$ be the foot of perpendicular from the point $P ( 1 , - 2 , 3 )$ on the line passing through the points $( 4 , 5 , 8 )$ and $( 1 , - 7 , 5 )$. Then the distance of $N$ from the plane $2x - 2y + z + 5 = 0$ is $\_\_\_\_$.

Let $S$ be the set of all values of $\lambda$, for which the shortest distance between the lines $\frac { x - \lambda } { 0 } = \frac { y - 3 } { 4 } = \frac { z + 6 } { 1 }$ and $\frac { x + \lambda } { 3 } = \frac { y } { - 4 } = \frac { z - 6 } { 0 }$ is 13. Then $8 \left| \sum _ { \lambda \in S } \lambda \right|$ is equal to
(1) 306
(2) 304
(3) 308
(4) 302

The shortest distance between the lines $\frac{x-5}{1} = \frac{y-2}{2} = \frac{z-4}{-3}$ and $\frac{x+3}{1} = \frac{y+5}{4} = \frac{z-1}{-5}$ is
(1) $7\sqrt{3}$
(2) $5\sqrt{3}$
(3) $6\sqrt{3}$
(4) $4\sqrt{3}$