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 $\_\_\_\_$ .

Consider the cube in the first octant with sides $O P , O Q$ and $O R$ of length 1 , along the $x$-axis, $y$-axis and $z$-axis, respectively, where $O ( 0,0,0 )$ is the origin. Let $S \left( \frac { 1 } { 2 } , \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$ be the centre of the cube and $T$ be the vertex of the cube opposite to the origin $O$ such that $S$ lies on the diagonal $O T$. If $\vec { p } = \overrightarrow { S P } , \vec { q } = \overrightarrow { S Q } , \vec { r } = \overrightarrow { S R }$ and $\vec { t } = \overrightarrow { S T }$, then the value of $| ( \vec { p } \times \vec { q } ) \times ( \vec { r } \times \vec { t } ) |$ is $\_\_\_\_$ .

Three lines $$\begin{array}{ll} L_1: & \vec{r} = \lambda\hat{i}, \lambda \in \mathbb{R}, \\ L_2: & \vec{r} = \hat{k} + \mu\hat{j}, \mu \in \mathbb{R} \text{ and} \\ L_3: & \vec{r} = \hat{i} + \hat{j} + v\hat{k}, v \in \mathbb{R} \end{array}$$ are given. For which point(s) $Q$ on $L_2$ can we find a point $P$ on $L_1$ and a point $R$ on $L_3$ so that $P$, $Q$ and $R$ are collinear?
(A) $\hat{k} - \frac{1}{2}\hat{j}$
(B) $\hat{k}$
(C) $\hat{k} + \frac{1}{2}\hat{j}$
(D) $\hat{k} + \hat{j}$

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 $O$ be the origin and $\overrightarrow { O A } = 2 \hat { \mathrm { i } } + 2 \hat { \mathrm { j } } + \hat { \mathrm { k } } , \quad \overrightarrow { O B } = \hat { \mathrm { i } } - 2 \hat { \mathrm { j } } + 2 \hat { \mathrm { k } }$ and $\overrightarrow { O C } = \frac { 1 } { 2 } ( \overrightarrow { O B } - \lambda \overrightarrow { O A } )$ for some $\lambda > 0$. If $| \overrightarrow { O B } \times \overrightarrow { O C } | = \frac { 9 } { 2 }$, then which of the following statements is (are) TRUE ?
(A) Projection of $\overrightarrow { O C }$ on $\overrightarrow { O A }$ is $- \frac { 3 } { 2 }$
(B) Area of the triangle $O A B$ is $\frac { 9 } { 2 }$
(C) Area of the triangle $A B C$ is $\frac { 9 } { 2 }$
(D) The acute angle between the diagonals of the parallelogram with adjacent sides $\overrightarrow { O A }$ and $\overrightarrow { O C }$ is $\frac { \pi } { 3 }$

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 $\hat { \imath } , \hat { \jmath }$ and $\hat { k }$ be the unit vectors along the three positive coordinate axes. Let
$$\begin{aligned} & \vec { a } = 3 \hat { \imath } + \hat { \jmath } - \hat { k } , \\ & \vec { b } = \hat { \imath } + b _ { 2 } \hat { \jmath } + b _ { 3 } \hat { k } , \quad b _ { 2 } , b _ { 3 } \in \mathbb { R } , \\ & \vec { c } = c _ { 1 } \hat { \imath } + c _ { 2 } \hat { \jmath } + c _ { 3 } \hat { k } , \quad c _ { 1 } , c _ { 2 } , c _ { 3 } \in \mathbb { R } \end{aligned}$$
be three vectors such that $b _ { 2 } b _ { 3 } > 0 , \vec { a } \cdot \vec { b } = 0$ and
$$\left( \begin{array} { r c r } 0 & - c _ { 3 } & c _ { 2 } \\ c _ { 3 } & 0 & - c _ { 1 } \\ - c _ { 2 } & c _ { 1 } & 0 \end{array} \right) \left( \begin{array} { l } 1 \\ b _ { 2 } \\ b _ { 3 } \end{array} \right) = \left( \begin{array} { r } 3 - c _ { 1 } \\ 1 - c _ { 2 } \\ - 1 - c _ { 3 } \end{array} \right)$$
Then, which of the following is/are TRUE ?
(A) $\vec { a } \cdot \vec { c } = 0$
(B) $\vec { b } \cdot \vec { c } = 0$
(C) $| \vec { b } | > \sqrt { 10 }$
(D) $| \vec { c } | \leq \sqrt { 11 }$

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) $\frac { 1 } { \sqrt { 12 } }$

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

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 $\mathbb { R } ^ { 3 }$ denote the three-dimensional space. Take two points $P = ( 1,2,3 )$ and $Q = ( 4,2,7 )$. Let $\operatorname { dist } ( X , Y )$ denote the distance between two points $X$ and $Y$ in $\mathbb { R } ^ { 3 }$. Let
$$\begin{gathered} S = \left\{ X \in \mathbb { R } ^ { 3 } : ( \operatorname { dist } ( X , P ) ) ^ { 2 } - ( \operatorname { dist } ( X , Q ) ) ^ { 2 } = 50 \right\} \text { and } \\ T = \left\{ Y \in \mathbb { R } ^ { 3 } : ( \operatorname { dist } ( Y , Q ) ) ^ { 2 } - ( \operatorname { dist } ( Y , P ) ) ^ { 2 } = 50 \right\} \end{gathered}$$
Then which of the following statements is (are) TRUE?
(A) There is a triangle whose area is 1 and all of whose vertices are from $S$.
(B) There are two distinct points $L$ and $M$ in $T$ such that each point on the line segment $L M$ is also in $T$.
(C) There are infinitely many rectangles of perimeter 48, two of whose vertices are from $S$ and the other two vertices are from $T$.
(D) There is a square of perimeter 48, two of whose vertices are from $S$ and the other two vertices are from $T$.

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 $L _ { 1 }$ 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 $\cos ^ { - 1 } \left( \frac { 1 } { 2 \sqrt { 3 } } \right)$

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 $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = \hat{i} - \hat{j} + \hat{k}$ 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 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 vectors $\overrightarrow{AB} = 3\hat{i} + 4\hat{k}$ and $\overrightarrow{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}$ are the sides of a triangle $ABC$, then the length of the median through $A$ is:
(1) $\sqrt{33}$
(2) $\sqrt{45}$
(3) $\sqrt{18}$
(4) $\sqrt{72}$

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) $13$

Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three unit vectors such that $\vec{a} \times (\vec{b} \times \vec{c}) = \frac{\sqrt{3}}{2}(\vec{b}+\vec{c})$. If $\vec{b}$ is not parallel to $\vec{c}$, then the angle between $\vec{a}$ and $\vec{b}$ is: (1) $\frac{3\pi}{4}$ (2) $\frac{\pi}{2}$ (3) $\frac{2\pi}{3}$ (4) $\frac{5\pi}{6}$