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 $\_\_\_\_$ .

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 $\overrightarrow { O P } = \frac { \alpha - 1 } { \alpha } \hat { i } + \hat { j } + \hat { k } , \overrightarrow { O Q } = \hat { i } + \frac { \beta - 1 } { \beta } \hat { j } + \hat { k }$ 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 $\vec { p } = 2 \hat { i } + \hat { j } + 3 \hat { k }$ 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 $\overline { \mathrm { a } } = \hat { \mathrm { i } } + \hat { \mathrm { j } } + \hat { \mathrm { k } } , \overline { \mathrm { b } } = \hat { \mathrm { i } } - \hat { \mathrm { j } } + 2 \hat { \mathrm { k } }$ 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) $\hat{i} + 3\hat{j} - 3\hat{k}$

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) $\frac{5\pi}{6}$

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

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 $\vec{a} = \hat{\mathrm{i}} + \hat{\mathrm{j}} + \sqrt{2}\hat{\mathrm{k}},\, \vec{b} = b_1\hat{\mathrm{i}} + b_2\hat{\mathrm{j}} + \sqrt{2}\hat{\mathrm{k}}$ and $\vec{c} = 5\hat{\mathrm{i}} + \hat{\mathrm{j}} + \sqrt{2}\hat{\mathrm{k}}$ be three vectors such that the projection vector of $\vec{b}$ on $\vec{a}$ is $|\vec{a}|$. If $\vec{a} + \vec{b}$ is perpendicular to $\vec{c}$, then $|\vec{b}|$ is equal to:
(1) $\sqrt{22}$
(2) $\sqrt{32}$
(3) 6
(4) 4

Let $\vec { a } , \vec { b }$ and $\vec { c }$ be three unit vectors, out of which vectors $\vec { b }$ and $\vec { c }$ are non-parallel. If $\alpha$ and $\beta$ are the angles which vector $\vec { a }$ makes with vectors $\vec { b }$ and $\vec { c }$ respectively and $\vec { a } \times ( \vec { b } \times \vec { c } ) = \frac { 1 } { 2 } \vec { b }$, then $| \alpha - \beta |$ is equal to :
(1) $90 ^ { \circ }$
(2) $60 ^ { \circ }$
(3) $45 ^ { \circ }$
(4) $30 ^ { \circ }$

Let $y = y(x)$ be the solution of the differential equation, $\left(x^2 + 1\right)^2 \frac{dy}{dx} + 2x\left(x^2 + 1\right)y = 1$ such that $y(0) = 0$. If $\sqrt{a}\, y(1) = \frac{\pi}{32}$, then the value of $a$ is
(1) $\frac{1}{16}$
(2) $\frac{1}{2}$
(3) $\frac{1}{4}$
(4) $1$

Let $\vec { \alpha } = 3 \hat { i } + \hat { j }$ and $\vec { \beta } = 2 \hat { i } - \hat { j } + 3 \hat { k }$. If $\vec { \beta } = \overrightarrow { \beta _ { 1 } } - \overrightarrow { \beta _ { 2 } }$, where $\overrightarrow { \beta _ { 1 } }$ is parallel to $\vec { \alpha }$ and $\overrightarrow { \beta _ { 2 } }$ is perpendicular to $\vec { \alpha }$, then $\overrightarrow { \beta _ { 1 } } \times \overrightarrow { \beta _ { 2 } }$ is equal to:
(1) $\frac { 1 } { 2 } ( - 3 \hat { i } + 9 \hat { j } + 5 \widehat { k } )$
(2) $3 \hat { i } - 9 \hat { j } - 5 \widehat { k }$
(3) $- 3 \hat { i } + 9 \hat { j } + 5 \widehat { k }$
(4) $\frac { 1 } { 2 } ( 3 \hat { i } - 9 \hat { j } + 5 \hat { k } )$

A vector $\vec { a } = \alpha \hat { i } + 2 \hat { j } + \beta \hat { k }$ $(\alpha, \beta \in R)$ lies in the plane of the vectors, $\vec { b } = \hat { i } + \hat { j }$ and $\vec { c } = \hat { i } - \hat { j } + 4 \hat { k }$. If $\vec { a }$ bisects the angle between $\vec { b }$ and $\vec { c }$, then
(1) $\vec { a } \cdot \hat { i } + 3 = 0$
(2) $\vec { a } \cdot \hat { i } + 1 = 0$
(3) $\vec { a } \cdot \widehat { k } + 2 = 0$
(4) $\vec { a } \cdot \widehat { k } + 4 = 0$

Let $\vec { a } = \hat { i } + \hat { j } + \hat { k }$ and $\vec { b } = \hat { j } - \hat { k }$. If $\vec { c }$ is a vector such that $\vec { a } \times \vec { c } = \vec { b }$ and $\vec { a } \cdot \vec { c } = 3$, then $\vec { a } \cdot ( \vec { b } \times \vec { c } )$ is equal to

Let $\vec { a } = \hat { i } + 2 \hat { j } - \widehat { k } , \vec { b } = \hat { i } - \hat { j }$ and $\vec { c } = \hat { i } - \hat { j } - \hat { k }$ be three given vectors. If $\vec { r }$ is a vector such that $\vec { r } \times \vec { a } = \vec { c } \times \vec { a }$ and $\vec { r } \cdot \vec { b } = 0$, then $\vec { r } \cdot \vec { a }$ is equal to

Let for a triangle $ABC$ $$\begin{aligned} & \overrightarrow { AB } = - 2 \hat { i } + \hat { j } + 3 \hat { k } \\ & \overrightarrow { CB } = \alpha \hat { i } + \beta \hat { j } + \gamma \hat { k } \\ & \overrightarrow { CA } = 4 \hat { i } + 3 \hat { j } + \delta \hat { k } \end{aligned}$$ If $\delta > 0$ and the area of the triangle $ABC$ is $5 \sqrt { 6 }$ then $\overrightarrow { CB } \cdot \overrightarrow { CA }$ is equal to
(1) 60
(2) 54
(3) 108
(4) 120

Let $S$ be the set of all $( \lambda , \mu )$ for which the vectors $\lambda \hat { i } - \hat { j } + \widehat { k } , \hat { i } + 2 \hat { j } + \mu \widehat { k }$ and $3 \hat { i } - 4 \hat { j } + 5 \widehat { k }$, where $\lambda - \mu = 5$, are coplanar, then $\sum _ { ( \lambda , \mu ) \in S } 80 \left( \lambda ^ { 2 } + \mu ^ { 2 } \right)$ is equal to
(1) 2210
(2) 2130
(3) 2290
(4) 2370

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

If the four points, whose position vectors are $3 \hat { i } - 4 \hat { j } + 2 \widehat { k } , \hat { i } + 2 \hat { j } - \widehat { k } , - 2 \hat { i } - \hat { j } + 3 \widehat { k }$ and $5 \hat { i } - 2 \alpha \hat { j } + 4 \widehat { k }$ are coplanar, then $\alpha$ is equal to
(1) $\frac { 73 } { 17 }$
(2) $- \frac { 107 } { 17 }$
(3) $- \frac { 73 } { 17 }$
(4) $\frac { 107 } { 17 }$

Let $\vec { a } = - \hat { i } - \hat { j } + \hat { k } , \vec { a } \cdot \vec { b } = 1$ and $\vec { a } \times \vec { b } = \hat { i } - \hat { j }$. Then $\vec { a } - 6 \vec { b }$ is equal to
(1) $3 ( \hat { i } - \hat { j } - \widehat { k } )$
(2) $3 ( \hat { i } + \hat { j } + \hat { k } )$
(3) $3 ( \hat { i } - \hat { j } + \widehat { k } )$
(4) $3 ( \hat { i } + \hat { j } - \widehat { k } )$

If the vectors $\vec { a } = \lambda \hat { i } + \mu \hat { j } + 4 \widehat { k } , \vec { b } = - 2 \hat { i } + 4 \hat { j } - 2 \widehat { k }$ and $\vec { c } = 2 \hat { i } + 3 \hat { j } + \widehat { k }$ are coplanar and the projection of $\vec { a }$ on the vector $\vec { b }$ is $\sqrt { 54 }$ units, then the sum of all possible values of $\lambda + \mu$ is equal to
(1) 0
(2) 6
(3) 24
(4) 18

Let $\vec { a } , \vec { b }$ and $\vec { c }$ be three non-zero non-coplanar vectors. Let the position vectors of four points $A , B , C$ and $D$ be $\overrightarrow { \mathrm { a } } - \overrightarrow { \mathrm { b } } + \overrightarrow { \mathrm { c } } , \lambda \overrightarrow { \mathrm { a } } - 3 \overrightarrow { \mathrm {~b} } + 4 \overrightarrow { \mathrm { c } } , - \vec { a } + 2 \vec { b } - 3 \vec { c }$ and $2 \vec { a } - 4 \vec { b } + 6 \vec { c }$ respectively. If $\overrightarrow { A B } , \overrightarrow { A C }$ and $\overrightarrow { A D }$ are coplanar, then $\lambda$ is :

Let $\lambda \in \mathbb{R}$, $\vec{a} = \lambda\hat{i} + 2\hat{j} - 3\hat{k}$, $\vec{b} = \hat{i} - \lambda\hat{j} + 2\hat{k}$. If $((\vec{a} + \vec{b}) \times (\vec{a} \times \vec{b})) \times (\vec{a} - \vec{b}) = 8\hat{i} - 40\hat{j} - 24\hat{k}$, then $|\lambda(\vec{a} + \vec{b}) \times (\vec{a} - \vec{b})|^{2}$ is equal to
(1) 140
(2) 132
(3) 144
(4) 136

Let $\vec{a}$ and $\vec{b}$ be two vectors. Let $|\vec{a}| = 1$, $|\vec{b}| = 4$ and $\vec{a} \cdot \vec{b} = 2$. If $\vec{c} = (2\vec{a} \times \vec{b}) - 3\vec{b}$, then the value of $\vec{b} \cdot \vec{c}$ is
(1) $-24$
(2) $-48$
(3) $-84$
(4) $-60$