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

The distance of the point $(1, -5, 9)$ from the plane $x - y + z = 5$ measured along the line $x = y = z$ is: (1) $3\sqrt{10}$ (2) $10\sqrt{3}$ (3) $\frac{10}{\sqrt{3}}$ (4) $\frac{20}{3}$

The number of distinct real values of $\lambda$, for which the lines $\frac { x - 1 } { 1 } = \frac { y - 2 } { 2 } = \frac { z + 3 } { \lambda ^ { 2 } }$ and $\frac { x - 3 } { 1 } = \frac { y - 2 } { \lambda ^ { 2 } } = \frac { z - 1 } { 2 }$, are coplanar is
(1) 2
(2) 4
(3) 3
(4) 1

The distance of the point $(1, 3, -7)$ from the plane passing through the point $(1, -1, -1)$, having normal perpendicular to both the lines $$\frac { x - 1 } { 1 } = \frac { y + 2 } { - 2 } = \frac { z - 4 } { 3 } \quad \text{and} \quad \frac { x - 2 } { 2 } = \frac { y + 1 } { - 1 } = \frac { z + 7 } { - 1 }$$ is:
(1) $\frac { 10 } { \sqrt { 74 } }$
(2) $\frac { 20 } { \sqrt { 74 } }$
(3) $\frac { 5 } { \sqrt { 83 } }$
(4) $\frac { 10 } { \sqrt { 83 } }$

If the image of the point $P ( 1 , - 2 , 3 )$ in the plane $2 x + 3 y - 4 z + 22 = 0$ measured parallel to the line $\frac { x } { 1 } = \frac { y } { 4 } = \frac { z } { 5 }$ is $Q$, then $P Q$ is equal to:
(1) $3 \sqrt { 5 }$
(2) $2 \sqrt { 42 }$
(3) $\sqrt { 42 }$
(4) $6 \sqrt { 5 }$

The coordinates of the foot of the perpendicular from the point $( 1 , - 2,1 )$ on the plane containing the lines $\frac { x + 1 } { 6 } = \frac { y - 1 } { 7 } = \frac { z - 3 } { 8 }$ and $\frac { x - 1 } { 3 } = \frac { y - 2 } { 5 } = \frac { z - 3 } { 7 }$, is:
(1) $( 2 , - 4,2 )$
(2) $( 1,1,1 )$
(3) $( 0,0,0 )$
(4) $( - 1,2 , - 1 )$

The line of intersection of the planes $\vec { r } \cdot ( 3 \hat { i } - \hat { j } + \widehat { k } ) = 1$ and $\vec { r } \cdot ( \hat { i } + 4 \hat { j } - 2 \widehat { k } ) = 2$, is,
(1) $\frac { x - \frac { 6 } { 13 } } { 2 } = \frac { y - \frac { 5 } { 13 } } { 7 } = \frac { z } { - 13 }$
(2) $\frac { x - \frac { 4 } { 7 } } { 2 } = \frac { y } { - 7 } = \frac { z + \frac { 5 } { 7 } } { 13 }$
(3) $\frac { x - \frac { 6 } { 13 } } { 2 } = \frac { y - \frac { 5 } { 13 } } { - 7 } = \frac { z } { - 13 }$
(4) $\frac { x - \frac { 4 } { 7 } } { - 2 } = \frac { y } { 7 } = \frac { z - \frac { 5 } { 7 } } { 13 }$

If $L _ { 1 }$ is the line of intersection of the planes $2 x - 2 y + 3 z - 2 = 0 , x - y + z + 1 = 0$ and $L _ { 2 }$ is the line of intersection of the planes $x + 2 y - z - 3 = 0,3 x - y + 2 z - 1 = 0$, then the distance of the origin from the plane, containing the lines $L _ { 1 }$ and $L _ { 2 }$ is
(1) $\frac { 1 } { \sqrt { 2 } }$
(2) $\frac { 1 } { 4 \sqrt { 2 } }$
(3) $\frac { 1 } { 3 \sqrt { 2 } }$
(4) $\frac { 1 } { 2 \sqrt { 2 } }$

A variable plane passes through a fixed point $( 3,2,1 )$ and meets $x , y$ and $z$-axes at $A , B \& C$ respectively. A plane is drawn parallel to the $yz$-plane through $A$, a second plane is drawn parallel to the $zx$-plane through $B$ and a third plane is drawn parallel to the $xy$-plane through $C$. Then the locus of the point of intersection of these three planes, is
(1) $\frac { 3 } { x } + \frac { 2 } { y } + \frac { 1 } { z } = 1$
(2) $\frac { 1 } { x } + \frac { 1 } { y } + \frac { 1 } { z } = \frac { 11 } { 6 }$
(3) $x + y + z = 6$
(4) $\frac { x } { 3 } + \frac { y } { 2 } + \frac { z } { 1 } = 1$

A variable plane passes through a fixed point ( $3,2,1$ ) and meets $x , y$ and $z$ axes at $A , B$ and $C$ respectively. A plane is drawn parallel to $y z$ - plane through $A$, a second plane is drawn parallel $z x$ plane through $B$ and a third plane is drawn parallel to $x y$ - plane through $C$. Then the locus of the point of intersection of these three planes, is
(1) $( x + y + z = 6 )$
(2) $\frac { x } { 3 } + \frac { y } { 2 } + \frac { z } { 1 } = 1$
(3) $\frac { 3 } { x } + \frac { 2 } { y } + \frac { 1 } { z } = 1$
(4) $\frac { 1 } { x } + \frac { 1 } { y } + \frac { 1 } { z } = \frac { 11 } { 6 }$

An angle between the plane $x + y + z = 5$ and the line of intersection of the planes, $3 x + 4 y + z - 1 = 0$ and $5 x + 8 y + 2 z + 14 = 0$ is
(1) $\cos ^ { - 1 } \left( \sqrt { \frac { 3 } { 17 } } \right)$
(2) $\cos ^ { - 1 } \left( \frac { 3 } { \sqrt { 17 } } \right)$
(3) $\sin ^ { - 1 } \left( \frac { 3 } { \sqrt { 17 } } \right)$
(4) $\sin ^ { - 1 } \left( \sqrt { \frac { 3 } { 17 } } \right)$

An angle between the plane, $x + y + z = 5$ and the line of intersection of the planes, $3 x + 4 y + z - 1 = 0$ and $5 x + 8 y + 2 z + 14 = 0$, is
(1) $\cos ^ { - 1 } \left( \frac { 3 } { \sqrt { 17 } } \right)$
(2) $\cos ^ { - 1 } \left( \sqrt { \frac { 3 } { 17 } } \right)$
(3) $\sin ^ { - 1 } \left( \frac { 3 } { \sqrt { 17 } } \right)$
(4) $\sin ^ { - 1 } \left( \sqrt { \frac { 3 } { 17 } } \right)$

If the lines $x = ay + b,\, z = cy + d$ and $x = a'z + b',\, y = c'z + d'$ are perpendicular, then
(1) $cc' + a + a' = 0$
(2) $aa' + c + c' = 0$
(3) $bb' + cc' + 1 = 0$
(4) $ab' + bc' + 1 = 0$

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) $\frac { 1 } { 8 }$

The vector equation of the plane through the line of intersection of the planes $x + y + z = 1$ and $2 x + 3 y + 4 z = 5$ which is perpendicular to the plane $x - y + z = 0$ is
(1) $\vec { r } \times ( \hat { i } + \hat { k } ) + 2 = 0$
(2) $\vec { r } \cdot ( \hat { i } - \hat { k } ) - 2 = 0$
(3) $\vec { r } \times ( \hat { i } - \hat { k } ) + 2 = 0$
(4) $\vec { r } \cdot ( \hat { i } - \hat { k } ) + 2 = 0$

The length of the perpendicular from the point $(2,-1,4)$ on the straight line $\frac{x+3}{10} = \frac{y-2}{-7} = \frac{z}{1}$ is
(1) greater than 3 but less than 4
(2) greater than 4
(3) less than 2
(4) greater than 2 but less than 3

A plane passing though the points $( 0 , - 1,0 )$ and $( 0,0,1 )$ and making an angle $\frac { \pi } { 4 }$ with the plane $y - z + 5 = 0$, also passes through the point
(1) $( \sqrt { 2 } , - 1,4 )$
(2) $( \sqrt { 2 } , 1,4 )$
(3) $( - \sqrt { 2 } , - 1 , - 4 )$
(4) $( - \sqrt { 2 } , 1 , - 4 )$

The equation of the plane containing the straight line $\frac{x}{2} = \frac{y}{3} = \frac{z}{4}$ and perpendicular to the plane containing the straight lines $\frac{x}{3} = \frac{y}{4} = \frac{z}{2}$ and $\frac{x}{4} = \frac{y}{2} = \frac{z}{3}$ is:
(1) $3x + 2y - 3z = 0$
(2) $x + 2y - 2z = 0$
(3) $x - 2y + z = 0$
(4) $5x + 2y - 4z = 0$

The plane passing through the point $( 4 , - 1,2 )$ and parallel to the lines $\frac { x + 2 } { 3 } = \frac { y - 2 } { - 1 } = \frac { z + 1 } { 2 }$ and $\frac { x - 2 } { 1 } = \frac { y - 3 } { 2 } = \frac { z - 4 } { 3 }$ also passes through the point
(1) $( 1,1 , - 1 )$
(2) $( - 1 , - 1 , - 1 )$
(3) $( - 1 , - 1,1 )$
(4) $( 1,1,1 )$

The equation of a plane containing the line of intersection of the planes $2x - y - 4 = 0$ and $y + 2z - 4 = 0$ and passing through the point $(1,1,0)$ is
(1) $x - 3y - 2z = -2$
(2) $x + 3y + z = 4$
(3) $x - y - z = 0$
(4) $2x - z = 2$

If the line, $\frac { x - 1 } { 2 } = \frac { y + 1 } { 3 } = \frac { z - 2 } { 4 }$ meets the plane, $x + 2 y + 3 z = 15$ at a point $P$, then the distance of $P$ from the origin is,
(1) $2 \sqrt { 5 }$
(2) $\frac { 9 } { 2 }$
(3) $\frac { \sqrt { 5 } } { 2 }$
(4) $\frac { 7 } { 2 }$

Let $P$ be the plane, which contains the line of intersection of the planes, $x + y + z - 6 = 0$ and $2 x + 3 y + z + 5 = 0$ and it is perpendicular to the $x y$-plane. Then the distance of the point $( 0,0,256 )$ from $P$ is equal to:
(1) $205 \sqrt { 5 }$ units
(2) $\frac { 17 } { \sqrt { 5 } }$ units
(3) $\frac { 11 } { \sqrt { 5 } }$ units
(4) $63 \sqrt { 5 }$ units

If for some $\alpha$ and $\beta$ in $R$, the intersection of the following three planes $x + 4 y - 2 z = 1$ $x + 7 y - 5 z = \beta$ $x + 5 y + \alpha z = 5$ is a line in $R ^ { 3 }$, then $\alpha + \beta$ is equal to:
(1) 0
(2) 10
(3) 2
(4) - 10

The foot of the perpendicular drawn from the point $( 4,2,3 )$ to the line joining the points $( 1 , - 2,3 )$ and $( 1,1,0 )$ lies on the plane
(1) $2 x + y - z = 1$
(2) $x - y - 2 z = 1$
(3) $x - 2 y + z = 1$
(4) $x + 2 y - z = 1$