Let $L _ { 1 }$ and $L _ { 2 }$ denote the lines $$\begin{aligned} & \vec { r } = \hat { i } + \lambda ( - \hat { i } + 2 \hat { j } + 2 \hat { k } ) , \lambda \in \mathbb { R } \text { and } \\ & \vec { r } = \mu ( 2 \hat { i } - \hat { j } + 2 \hat { k } ) , \mu \in \mathbb { R } \end{aligned}$$ respectively. If $L _ { 3 }$ is a line which is perpendicular to both $L _ { 1 }$ and $L _ { 2 }$ and cuts both of them, then which of the following options describe(s) $L _ { 3 }$?
(A) $\vec { r } = \frac { 2 } { 9 } ( 4 \hat { i } + \hat { j } + \hat { k } ) + t ( 2 \hat { i } + 2 \hat { j } - \hat { k } ) , t \in \mathbb { R }$
(B) $\vec { r } = \frac { 2 } { 9 } ( 2 \hat { i } - \hat { j } + 2 \hat { k } ) + t ( 2 \hat { i } + 2 \hat { j } - \hat { k } ) , t \in \mathbb { R }$
(C) $\vec { r } = \frac { 1 } { 3 } ( 2 \hat { i } + \hat { k } ) + t ( 2 \hat { i } + 2 \hat { j } - \hat { k } ) , t \in \mathbb { R }$
(D) $\vec { r } = t ( 2 \hat { i } + 2 \hat { j } - \hat { k } ) , t \in \mathbb { R }$

Three lines are given by $$\begin{aligned} & \vec { r } = \lambda \hat { i } , \lambda \in \mathbb { R } \\ & \vec { r } = \mu ( \hat { i } + \hat { j } ) , \quad \mu \in \mathbb { R } \text { and } \\ & \vec { r } = v ( \hat { i } + \hat { j } + \hat { k } ) , \quad v \in \mathbb { R } \end{aligned}$$ Let the lines cut the plane $x + y + z = 1$ at the points $A , B$ and $C$ respectively. If the area of the triangle $A B C$ is $\triangle$ then the value of $( 6 \Delta ) ^ { 2 }$ equals

Let $\alpha, \beta, \gamma, \delta$ be real numbers such that $\alpha^{2} + \beta^{2} + \gamma^{2} \neq 0$ and $\alpha + \gamma = 1$. Suppose the point $(3, 2, -1)$ is the mirror image of the point $(1, 0, -1)$ with respect to the plane $\alpha x + \beta y + \gamma z = \delta$. Then which of the following statements is/are TRUE?
(A) $\alpha + \beta = 2$
(B) $\delta - \gamma = 3$
(C) $\delta + \beta = 4$
(D) $\alpha + \beta + \gamma = \delta$

Let $L _ { 1 }$ and $L _ { 2 }$ be the following straight lines.
$$L _ { 1 } : \frac { x - 1 } { 1 } = \frac { y } { - 1 } = \frac { z - 1 } { 3 } \text { and } L _ { 2 } : \frac { x - 1 } { - 3 } = \frac { y } { - 1 } = \frac { z - 1 } { 1 } .$$
Suppose the straight line
$$L : \frac { x - \alpha } { l } = \frac { y - 1 } { m } = \frac { z - \gamma } { - 2 }$$
lies in the plane containing $L _ { 1 }$ and $L _ { 2 }$, and passes through the point of intersection of $L _ { 1 }$ and $L _ { 2 }$. If the line $L$ bisects the acute angle between the lines $L _ { 1 }$ and $L _ { 2 }$, then which of the following statements is/are TRUE?
(A) $\alpha - \gamma = 3$
(B) $l + m = 2$
(C) $\alpha - \gamma = 1$
(D) $l + m = 0$

Let $\vec{u}$, $\vec{v}$ and $\vec{w}$ be vectors in three-dimensional space, where $\vec{u}$ and $\vec{v}$ are unit vectors which are not perpendicular to each other and $$\vec{u} \cdot \vec{w} = 1, \quad \vec{v} \cdot \vec{w} = 1, \quad \vec{w} \cdot \vec{w} = 4.$$ If the volume of the parallelepiped, whose adjacent sides are represented by the vectors $\vec{u}$, $\vec{v}$ and $\vec{w}$, is $\sqrt{2}$, then the value of $|3\vec{u} + 5\vec{v}|$ is ____.

Let the position vectors of the points $P , Q , R$ and $S$ be $\vec { a } = \hat { i } + 2 \hat { j } - 5 \hat { k } , \vec { b } = 3 \hat { i } + 6 \hat { j } + 3 \hat { k }$, $\vec { c } = \frac { 17 } { 5 } \hat { i } + \frac { 16 } { 5 } \hat { j } + 7 \hat { k }$ and $\vec { d } = 2 \hat { i } + \hat { j } + \hat { k }$, respectively. Then which of the following statements is true?
(A) The points $P , Q , R$ and $S$ are NOT coplanar
(B) $\frac { \vec { b } + 2 \vec { d } } { 3 }$ is the position vector of a point which divides $P R$ internally in the ratio $5 : 4$
(C) $\frac { \vec { b } + 2 \vec { d } } { 3 }$ is the position vector of a point which divides $P R$ externally in the ratio $5 : 4$
(D) The square of the magnitude of the vector $\vec { b } \times \vec { d }$ is 95

Let $Q$ be the cube with the set of vertices $\left\{ \left( x _ { 1 } , x _ { 2 } , x _ { 3 } \right) \in \mathbb { R } ^ { 3 } : x _ { 1 } , x _ { 2 } , x _ { 3 } \in \{ 0,1 \} \right\}$. 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 $d \left( \ell _ { 1 } , \ell _ { 2 } \right)$ denote the shortest distance between them. Then the maximum value of $d \left( \ell _ { 1 } , \ell _ { 2 } \right)$, as $\ell _ { 1 }$ varies over $F$ and $\ell _ { 2 }$ varies over $S$, is
(A) $\frac { 1 } { \sqrt { 6 } }$
(B) $\frac { 1 } { \sqrt { 8 } }$
(C) $\frac { 1 } { \sqrt { 3 } }$
(D) $\frac { 1 } { \sqrt { 12 } }$

A straight line drawn from the point $P ( 1,3,2 )$, parallel to the line $\frac { x - 2 } { 1 } = \frac { y - 4 } { 2 } = \frac { z - 6 } { 1 }$, intersects the plane $L _ { 1 } : x - y + 3 z = 6$ at the point $Q$. Another straight line which passes through $Q$ and is perpendicular to the plane $L _ { 1 }$ intersects the plane $L _ { 2 } : 2 x - y + z = - 4$ at the point $R$. Then which of the following statements is (are) TRUE?
(A) The length of the line segment $PQ$ is $\sqrt { 6 }$
(B) The coordinates of $R$ are $( 1,6,3 )$
(C) The centroid of the triangle $PQR$ is $\left( \frac { 4 } { 3 } , \frac { 14 } { 3 } , \frac { 5 } { 3 } \right)$
(D) The perimeter of the triangle $PQR$ is $\sqrt { 2 } + \sqrt { 6 } + \sqrt { 11 }$

Let $\vec { p } = 2 \hat { i } + \hat { j } + 3 \hat { k }$ and $\vec { q } = \hat { i } - \hat { j } + \hat { k }$. If for some real numbers $\alpha , \beta$, and $\gamma$, we have
$$15 \hat { i } + 10 \hat { j } + 6 \hat { k } = \alpha ( 2 \vec { p } + \vec { q } ) + \beta ( \vec { p } - 2 \vec { q } ) + \gamma ( \vec { p } \times \vec { q } )$$
then the value of $\gamma$ is $\_\_\_\_$ .

Let $\overrightarrow { O P } = \frac { \alpha - 1 } { \alpha } \hat { i } + \hat { j } + \hat { k } , \overrightarrow { O Q } = \hat { i } + \frac { \beta - 1 } { \beta } \hat { j } + \hat { k }$ and $\overrightarrow { O R } = \hat { i } + \hat { j } + \frac { 1 } { 2 } \hat { k }$ be three vectors, where $\alpha , \beta \in \mathbb { R } - \{ 0 \}$ and $O$ denotes the origin. If $( \overrightarrow { O P } \times \overrightarrow { O Q } ) \cdot \overrightarrow { O R } = 0$ and the point $( \alpha , \beta , 2 )$ lies on the plane $3 x + 3 y - z + l = 0$, then the value of $l$ is $\_\_\_\_$ .

Let $\overline { \mathrm { a } } = \hat { \mathrm { i } } + \hat { \mathrm { j } } + \hat { \mathrm { k } } , \overline { \mathrm { b } } = \hat { \mathrm { i } } - \hat { \mathrm { j } } + 2 \hat { \mathrm { k } }$ and $\overline { \mathrm { c } } = \mathrm { x } \hat { \mathrm { i } } + ( \mathrm { x } - 2 ) \hat { \mathrm { j } } - \hat { \mathrm { k } }$. If the vector $\overline { \mathrm { c } }$ lies in the plane of $\bar { a }$ and $\bar { b }$, then $x$ equals
(1) 0
(2) 1
(3) - 4
(4) - 2

Let $L$ be the line of intersection of the planes $2 x + 3 y + z = 1$ and $x + 3 y + 2 z = 2$. If $L$ makes an angles $\alpha$ with the positive $x$-axis, then $\cos \alpha$ equals
(1) $\frac { 1 } { \sqrt { 3 } }$
(2) $\frac { 1 } { 2 }$
(3) 1
(4) $\frac { 1 } { \sqrt { 2 } }$

If a line makes an angle of $\frac { \pi } { 4 }$ with the positive directions of each of $x$-axis and $y$-axis, then the angle that the line makes with the positive direction of the $z$-axis is
(1) $\frac { \pi } { 6 }$
(2) $\frac { \pi } { 3 }$
(3) $\frac { \pi } { 4 }$
(4) $\frac { \pi } { 2 }$

If the lines $\frac{x-1}{2} = \frac{y+1}{3} = \frac{z-1}{4}$ and $\frac{x-3}{1} = \frac{y-k}{2} = \frac{z}{1}$ intersect, then $k$ is equal to
(1) $\frac{2}{9}$
(2) $\frac{9}{2}$
(3) 0
(4) $-1$

Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = \hat{i} - \hat{j} + \hat{k}$ and $\vec{c} = \hat{i} - \hat{j} - \hat{k}$ be three vectors. A vector $\vec{v}$ in the plane of $\vec{a}$ and $\vec{b}$, whose projection on $\vec{c}$ is $\frac{1}{\sqrt{3}}$, is
(1) $\hat{i} - 3\hat{j} + 3\hat{k}$
(2) $-3\hat{i} - 3\hat{j} - \hat{k}$
(3) $3\hat{i} - \hat{j} + 3\hat{k}$
(4) $\hat{i} + 3\hat{j} - 3\hat{k}$

If the lines $\frac{x-2}{1} = \frac{y-3}{1} = \frac{z-4}{-k}$ and $\frac{x-1}{k} = \frac{y-4}{2} = \frac{z-5}{1}$ are coplanar, then $k$ can be
(1) $-1$ or $-3$
(2) $-1$ or $3$
(3) $1$ or $-1$
(4) $0$ or $-3$

If two lines $L _ { 1 }$ and $L _ { 2 }$ in space, are defined by
$$\begin{gathered} L _ { 1 } = \{ x = \sqrt { \lambda } y + ( \sqrt { \lambda } - 1 ) , \\ z = ( \sqrt { \lambda } - 1 ) y + \sqrt { \lambda } \} \text { and } \\ L _ { 2 } = \{ x = \sqrt { \mu } y + ( 1 - \sqrt { \mu } ) , \\ z = ( 1 - \sqrt { \mu } ) y + \sqrt { \mu } \} \end{gathered}$$
then $L _ { 1 }$ is perpendicular to $L _ { 2 }$, for all nonnegative reals $\lambda$ and $\mu$, such that :
(1) $\sqrt { \lambda } + \sqrt { \mu } = 1$
(2) $\lambda \neq \mu$
(3) $\lambda + \mu = 0$
(4) $\lambda = \mu$

If the lines $\frac{x-2}{1} = \frac{y-3}{1} = \frac{z-4}{-k}$ and $\frac{x-1}{k} = \frac{y-4}{2} = \frac{z-5}{1}$ are coplanar, then $k$ can have
(1) exactly two values.
(2) exactly three values.
(3) any value.
(4) exactly one value.

Distance between two parallel planes $2x + y + 2z = 8$ and $4x + 2y + 4z + 5 = 0$ is
(1) $\frac{7}{2}$
(2) $\frac{9}{2}$
(3) $\frac{3}{2}$
(4) $\frac{5}{2}$

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) $2\sqrt{14}$
(2) $8$
(3) $3\sqrt{21}$
(4) $13$

$ABC$ is a triangle in a plane with vertices $A ( 2,3,5 ) , B ( - 1,3,2 )$ and $C ( \lambda , 5 , \mu )$. If the median through $A$ is equally inclined to the coordinate axes, then the value of $\left( \lambda ^ { 3 } + \mu ^ { 3 } + 5 \right)$ is
(1) 1130
(2) 1348
(3) 1077
(4) 676

Let $ABC$ be a triangle whose circumcentre is at $P$. If the position vectors $A , B , C$ and $P$ are $\vec{a}, \vec{b}, \vec{c}$ and $\frac { \vec { a } + \vec { b } + \vec { c } } { 4 }$ respectively, then the position vector of the orthocentre of this triangle, is :
(1) $- \left( \frac { \vec { a } + \vec { b } + \vec { c } } { 2 } \right)$
(2) $\vec { a } + \vec { b } + \vec { c }$
(3) $\frac { ( \vec { a } + \vec { b } + \vec { c } ) } { 2 }$
(4) $\overrightarrow { 0 }$

Given, $\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k}$ and $\vec{b} = \hat{i} + \hat{j}$. Let $\vec{c}$ be a vector such that $|\vec{c} - \vec{a}| = 3$, $|(\vec{a} \times \vec{b}) \times \vec{c}| = 3$ and the angle between $\vec{c}$ and $\vec{a} \times \vec{b}$ be $30^\circ$. Then $\vec{a} \cdot \vec{c}$ is equal to:
(1) $\dfrac{25}{8}$
(2) $2$
(3) $5$
(4) $\dfrac{1}{8}$

Let $\vec { u } = \hat { i } + \hat { j }$, $\vec { v } = \hat { i } - \hat { j }$ and $\vec { w } = \hat { i } + 2 \hat { j } + 3 \hat { k }$. If $\hat { n }$ is a unit vector such that $\vec { u } \cdot \hat { n } = 0$ and $\vec { v } \cdot \hat { n } = 0$, then $| \vec { w } \cdot \hat { n } |$ is equal to:
(1) 0
(2) 1
(3) 2
(4) 3

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