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
A) $x + 2 y - 2 z = 0$
B) $3 x + 2 y - 2 z = 0$
C) $x - 2 y + z = 0$
D) $5 x + 2 y - 4 z = 0$

Match the statements in Column-I with the values in Column-II.
Column I
A) A line from the origin meets the lines $\frac { x - 2 } { 1 } = \frac { y - 1 } { - 2 } = \frac { z + 1 } { 1 }$ and $\frac { x - \frac { 8 } { 3 } } { 2 } = \frac { y + 3 } { - 1 } = \frac { z - 1 } { 1 }$ at $P$ and $Q$ respectively. If length $\mathrm { PQ } = d$, then $d ^ { 2 }$ is
B) The values of $x$ satisfying $\tan ^ { - 1 } ( x + 3 ) - \tan ^ { - 1 } ( x - 3 ) = \sin ^ { - 1 } \left( \frac { 3 } { 5 } \right)$ are
C) Non-zero vectors $\vec { a } , \vec { b }$ and $\vec { c }$ satisfy $\vec { a } \cdot \vec { b } = 0$, $( \overrightarrow { \mathrm { b } } - \overrightarrow { \mathrm { a } } ) \cdot ( \overrightarrow { \mathrm { b } } + \overrightarrow { \mathrm { c } } ) = 0$ and $2 | \overrightarrow { \mathrm {~b} } + \overrightarrow { \mathrm { c } } | = | \overrightarrow { \mathrm { b } } - \overrightarrow { \mathrm { a } } |$. If $\vec { a } = \mu \vec { b } + 4 \vec { c }$, then the possible values of $\mu$ are
D) Let f be the function on $[ - \pi , \pi ]$ given by $f ( 0 ) = 9$ and $f ( x ) = \sin \left( \frac { 9 x } { 2 } \right) / \sin \left( \frac { x } { 2 } \right)$ for $x \neq 0$. The value of $\frac { 2 } { \pi } \int _ { - \pi } ^ { \pi } f ( x ) d x$ is
Column II p) $-4$ q) $0$ r) $4$ s) $-1$ (or as given in paper) t) $6$

If the distance between the plane $\mathrm { Ax } - 2 \mathrm { y } + \mathrm { z } = \mathrm { d }$ and the plane containing the lines $\frac { x - 1 } { 2 } = \frac { y - 2 } { 3 } = \frac { z - 3 } { 4 }$ and $\frac { x - 2 } { 3 } = \frac { y - 3 } { 4 } = \frac { z - 4 } { 5 }$ is $\sqrt { 6 }$, then $| d |$ is

Perpendiculars are drawn from points on the line $\frac { x + 2 } { 2 } = \frac { y + 1 } { - 1 } = \frac { z } { 3 }$ to the plane $x + y + z = 3$. The feet of perpendiculars lie on the line
(A) $\frac { x } { 5 } = \frac { y - 1 } { 8 } = \frac { z - 2 } { - 13 }$
(B) $\frac { x } { 2 } = \frac { y - 1 } { 3 } = \frac { z - 2 } { - 5 }$
(C) $\frac { x } { 4 } = \frac { y - 1 } { 3 } = \frac { z - 2 } { - 7 }$
(D) $\frac { x } { 2 } = \frac { y - 1 } { - 7 } = \frac { z - 2 } { 5 }$

A line $l$ passing through the origin is perpendicular to the lines $l _ { 1 } : ( 3 + t ) \hat { i } + ( - 1 + 2 t ) \hat { j } + ( 4 + 2 t ) \hat { k } , - \infty < t < \infty$ $l _ { 2 } : ( 3 + 2 s ) \hat { i } + ( 3 + 2 s ) \hat { j } + ( 2 + s ) \hat { k } , - \infty < s < \infty$ Then, the coordinate(s) of the point(s) on $l _ { 2 }$ at a distance of $\sqrt { 17 }$ from the point of intersection of $l$ and $l _ { 1 }$ is (are)
(A) $\left( \frac { 7 } { 3 } , \frac { 7 } { 3 } , \frac { 5 } { 3 } \right)$
(B) $( - 1 , - 1,0 )$
(C) $( 1,1,1 )$
(D) $\left( \frac { 7 } { 9 } , \frac { 7 } { 9 } , \frac { 8 } { 9 } \right)$

Consider the lines $L _ { 1 } : \frac { x - 1 } { 2 } = \frac { y } { - 1 } = \frac { z + 3 } { 1 } , L _ { 2 } : \frac { x - 4 } { 1 } = \frac { y + 3 } { 1 } = \frac { z + 3 } { 2 }$ and the planes $P _ { 1 } : 7 x + y + 2 z = 3 , P _ { 2 } : 3 x + 5 y - 6 z = 4$. Let $a x + b y + c z = d$ be the equation of the plane passing through the point of intersection of lines $L _ { 1 }$ and $L _ { 2 }$, and perpendicular to planes $P _ { 1 }$ and $P _ { 2 }$.
Match List I with List II and select the correct answer using the code given below the lists:
List I

• [P.] $a =$
• [Q.] $b =$
• [R.] $c =$
• [S.] $d =$

List II

1. $13$
2. $-3$
3. $1$
4. $-2$

Codes:

	P	Q	R	S
(A)	3	2	4	1
(B)	1	3	4	2
(C)	3	2	1	4
(D)	2	4	1	3

In $\mathbb { R } ^ { 3 }$, consider the planes $P _ { 1 } : y = 0$ and $P _ { 2 } : x + z = 1$. Let $P _ { 3 }$ be a plane, different from $P _ { 1 }$ and $P _ { 2 }$, which passes through the intersection of $P _ { 1 }$ and $P _ { 2 }$. If the distance of the point $( 0,1,0 )$ from $P _ { 3 }$ is 1 and the distance of a point $( \alpha , \beta , \gamma )$ from $P _ { 3 }$ is 2, then which of the following relations is (are) true?
(A) $2 \alpha + \beta + 2 \gamma + 2 = 0$
(B) $2 \alpha - \beta + 2 \gamma + 4 = 0$
(C) $2 \alpha + \beta - 2 \gamma - 10 = 0$
(D) $2 \alpha - \beta + 2 \gamma - 8 = 0$

In $\mathbb { R } ^ { 3 }$, let $L$ be a straight line passing through the origin. Suppose that all the points on $L$ are at a constant distance from the two planes $P _ { 1 } : x + 2 y - z + 1 = 0$ and $P _ { 2 } : 2 x - y + z - 1 = 0$. Let $M$ be the locus of the feet of the perpendiculars drawn from the points on $L$ to the plane $P _ { 1 }$. Which of the following points lie(s) on $M$?
(A) $\left( 0 , - \frac { 5 } { 6 } , - \frac { 2 } { 3 } \right)$
(B) $\left( - \frac { 1 } { 6 } , - \frac { 1 } { 3 } , \frac { 1 } { 6 } \right)$
(C) $\left( - \frac { 5 } { 6 } , 0 , \frac { 1 } { 6 } \right)$
(D) $\left( - \frac { 1 } { 3 } , 0 , \frac { 2 } { 3 } \right)$

Let $P$ be a point in the first octant, whose image $Q$ in the plane $x + y = 3$ (that is, the line segment $P Q$ is perpendicular to the plane $x + y = 3$ and the mid-point of $P Q$ lies in the plane $x + y = 3$ ) lies on the $z$-axis. Let the distance of $P$ from the $x$-axis be 5 . If $R$ is the image of $P$ in the $x y$-plane, then the length of $P R$ is $\_\_\_\_$ .

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) $\frac { x } { 1 } = \frac { y - 4 } { - 2 } = \frac { z } { 3 }$

Let $S$ be the reflection of a point $Q$ with respect to the plane given by
$$\vec { r } = - ( t + p ) \hat { \imath } + t \hat { \jmath } + ( 1 + p ) \hat { k }$$
where $t , p$ are real parameters and $\hat { \imath } , \hat { \jmath } , \hat { k }$ are the unit vectors along the three positive coordinate axes. If the position vectors of $Q$ and $S$ are $10 \hat { \imath } + 15 \hat { \jmath } + 20 \hat { k }$ and $\alpha \hat { \imath } + \beta \hat { \jmath } + \gamma \hat { k }$ respectively, then which of the following is/are TRUE ?
(A) $3 ( \alpha + \beta ) = - 101$
(B) $3 ( \beta + \gamma ) = - 71$
(C) $3 ( \gamma + \alpha ) = - 86$
(D) $3 ( \alpha + \beta + \gamma ) = - 121$

Let $P$ be the plane $\sqrt { 3 } x + 2 y + 3 z = 16$ and let $S = \left\{ \alpha \hat { i } + \beta \hat { j } + \gamma \hat { k } : \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 1 \right.$ and the distance of $( \alpha , \beta , \gamma )$ from the plane $P$ is $\left. \frac { 7 } { 2 } \right\}$. 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 $\ell _ { 1 }$ and $\ell _ { 2 }$ be the lines $\vec { r } _ { 1 } = \lambda ( \hat { i } + \hat { j } + \hat { k } )$ and $\vec { r } _ { 2 } = ( \hat { j } - \hat { k } ) + \mu ( \hat { i } + \hat { k } )$, 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 $d \left( H _ { 0 } \right)$ 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.
List-I
(P) The value of $d \left( H _ { 0 } \right)$ is
(Q) The distance of the point $( 0,1,2 )$ from $H _ { 0 }$ is
(R) The distance of origin from $H _ { 0 }$ is
(S) The distance of origin from the point of intersection of planes $y = z , x = 1$ and $H _ { 0 }$ is
List-II
(1) $\sqrt { 3 }$
(2) $\frac { 1 } { \sqrt { 3 } }$
(3) 0
(4) $\sqrt { 2 }$
(5) $\frac { 1 } { \sqrt { 2 } }$
The correct option is:
(A) $( P ) \rightarrow ( 2 )$ $( Q ) \rightarrow ( 4 )$ $( R ) \rightarrow ( 5 )$ $( S ) \rightarrow ( 1 )$
(B) $( P ) \rightarrow ( 5 )$ $( Q ) \rightarrow ( 4 )$ $( R ) \rightarrow ( 3 )$ $( S ) \rightarrow ( 1 )$
(C) $( P ) \rightarrow ( 2 )$ $( Q ) \rightarrow ( 1 )$ $( R ) \rightarrow ( 3 )$ $( S ) \rightarrow ( 2 )$
(D) $( P ) \rightarrow ( 5 )$ $( Q ) \rightarrow ( 1 )$ $( R ) \rightarrow ( 4 )$ $( S ) \rightarrow ( 2 )$

Let $\gamma \in \mathbb { R }$ be such that the lines $L _ { 1 } : \frac { x + 11 } { 1 } = \frac { y + 21 } { 2 } = \frac { z + 29 } { 3 }$ and $L _ { 2 } : \frac { x + 16 } { 3 } = \frac { y + 11 } { 2 } = \frac { z + 4 } { \gamma }$ 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.
List-I
(P) $\gamma$ equals
(Q) A possible choice for $\hat { n }$ is
(R) $\overrightarrow { O R _ { 1 } }$ equals
(S) A possible value of $\overrightarrow { O R _ { 1 } } \cdot \hat { n }$ is
List-II
(1) $- \hat { i } - \hat { j } + \hat { k }$
(2) $\sqrt { \frac { 3 } { 2 } }$
(3) 1
(4) $\frac { 1 } { \sqrt { 6 } } \hat { i } - \frac { 2 } { \sqrt { 6 } } \hat { j } + \frac { 1 } { \sqrt { 6 } } \hat { k }$
(5) $\sqrt { \frac { 2 } { 3 } }$
The correct option is
(A) $(\mathrm{P}) \rightarrow (3)$, $(\mathrm{Q}) \rightarrow (4)$, $(\mathrm{R}) \rightarrow (1)$, $(\mathrm{S}) \rightarrow (2)$
(B) $(\mathrm{P}) \rightarrow (5)$, $(\mathrm{Q}) \rightarrow (4)$, $(\mathrm{R}) \rightarrow (1)$, $(\mathrm{S}) \rightarrow (2)$
(C) $(\mathrm{P}) \rightarrow (3)$, $(\mathrm{Q}) \rightarrow (4)$, $(\mathrm{R}) \rightarrow (1)$, $(\mathrm{S}) \rightarrow (5)$
(D) $(\mathrm{P}) \rightarrow (3)$, $(\mathrm{Q}) \rightarrow (1)$, $(\mathrm{R}) \rightarrow (4)$, $(\mathrm{S}) \rightarrow (5)$

Let $L _ { 1 }$ be the line of intersection of the planes given by the equations
$$2 x + 3 y + z = 4 \text { and } x + 2 y + z = 5 .$$
Let $L _ { 2 }$ be the line passing through the point $P ( 2 , - 1,3 )$ and parallel to $L _ { 1 }$. Let $M$ denote the plane given by the equation
$$2 x + y - 2 z = 6$$
Suppose that the line $L _ { 2 }$ meets the plane $M$ at the point $Q$. Let $R$ be the foot of the perpendicular drawn from $P$ to the plane $M$.
Then which of the following statements is (are) TRUE?

(A)	The length of the line segment $PQ$ is $9 \sqrt { 3 }$
(B)	The length of the line segment $QR$ is 15
(C)	The area of $\triangle PQR$ is $\frac { 3 } { 2 } \sqrt { 234 }$
(D)	The acute angle between the line segments $PQ$ and $PR$ is $\cos ^ { - 1 } \left( \frac { 1 } { 2 \sqrt { 3 } } \right)$

The image of the line $\frac { x - 1 } { 3 } = \frac { y - 3 } { 1 } = \frac { z - 4 } { - 5 }$ in the plane $2 x - y + z + 3 = 0$ is the line
(1) $\frac { x - 3 } { 3 } = \frac { y + 5 } { 1 } = \frac { z - 2 } { - 5 }$
(2) $\frac { x - 3 } { - 3 } = \frac { y + 5 } { - 1 } = \frac { z - 2 } { 5 }$
(3) $\frac { x + 3 } { 3 } = \frac { y - 5 } { 1 } = \frac { z - 2 } { - 5 }$
(4) $\frac { x + 3 } { - 3 } = \frac { y - 5 } { - 1 } = \frac { z + 2 } { 5 }$

Equation of the plane which passes through the point of intersection of lines $\frac { x - 1 } { 3 } = \frac { y - 2 } { 1 } = \frac { z - 3 } { 2 }$ and $\frac { x - 3 } { 1 } = \frac { y - 1 } { 2 } = \frac { z - 2 } { 3 }$ and has the largest distance from the origin is:
(1) $4 x + 3 y + 5 z = 50$
(2) $3 x + 4 y + 5 z = 49$
(3) $5 x + 4 y + 3 z = 57$
(4) $7 x + 2 y + 4 z = 54$

If the angle between the line $2 ( x + 1 ) = y = z + 4$ and the plane $2 x - y + \sqrt { \lambda } z + 4 = 0$ is $\frac { \pi } { 6 }$, then the value of $\lambda$ is
(1) $\frac { 45 } { 7 }$
(2) $\frac { 135 } { 11 }$
(3) $\frac { 135 } { 7 }$
(4) $\frac { 45 } { 11 }$

The angle between the lines whose direction cosines satisfy the equations $l + m + n = 0$ and $l ^ { 2 } = m ^ { 2 } + n ^ { 2 }$ is
(1) $\frac { \pi } { 6 }$
(2) $\frac { \pi } { 2 }$
(3) $\frac { \pi } { 3 }$
(4) $\frac { \pi } { 4 }$

A line in the 3-dimensional space makes an angle $\theta \left( 0 < \theta \leq \frac { \pi } { 2 } \right)$ with both the $X$ and $Y$-axes. Then, the set of all values of $\theta$ is in the interval:
(1) $\left( \frac { \pi } { 3 } , \frac { \pi } { 2 } \right]$
(2) $\left( 0 , \frac { \pi } { 4 } \right]$
(3) $\left[ \frac { \pi } { 4 } , \frac { \pi } { 2 } \right]$
(4) $\left[ \frac { \pi } { 6 } , \frac { \pi } { 3 } \right]$

Equation of the line of the shortest distance between the lines $\frac { x } { 1 } = \frac { y } { - 1 } = \frac { z } { 1 }$ and $\frac { x - 1 } { 0 } = \frac { y + 1 } { - 2 } = \frac { z } { 1 }$ is
(1) $\frac { x } { - 2 } = \frac { y } { 1 } = \frac { z } { 2 }$
(2) $\frac { x } { 1 } = \frac { y } { - 1 } = \frac { z } { - 2 }$
(3) $\frac { x - 1 } { 1 } = \frac { y + 1 } { - 1 } = \frac { z } { - 2 }$
(4) $\frac { x - 1 } { 1 } = \frac { y + 1 } { - 1 } = \frac { z } { 1 }$

The equation of the plane containing the line $2x - 5y + z = 3$; $x + y + 4z = 5$, and parallel to the plane $x + 3y + 6z = 1$, is:
(1) $2x + 6y + 12z = 13$
(2) $x + 3y + 6z = -7$
(3) $x + 3y + 6z = 7$
(4) $2x + 6y + 12z = -13$

The distance of the point $( 1,0,2 )$ from the point of intersection of the line $\frac { x - 2 } { 3 } = \frac { y + 1 } { 4 } = \frac { z - 2 } { 12 }$ and the plane $x - y + z = 16$, is
(1) 13
(2) $2 \sqrt { 14 }$
(3) 8
(4) $3 \sqrt { 21 }$

The equation of the plane containing the line of intersection of $2 x - 5 y + z = 3 ; x + y + 4 z = 5$, and parallel to the plane, $x + 3 y + 6 z = 1$, is
(1) $2 x + 6 y + 12 z = - 13$
(2) $2 x + 6 y + 12 z = 13$
(3) $x + 3 y + 6 z = - 7$
(4) $x + 3 y + 6 z = 7$

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) $\frac{20}{3}$