For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix $$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$ and $F(t)$ denotes the limit of $F_n(t) = I_3 + \sum_{k=1}^n \frac{t^k \mathcal{M}^k}{k!}$ as defined in question 9.
Show that for all $t \in \mathbb{R}$ and $X, Y$ vectors of $\mathbb{R}^{3}$, we have $F(t)X \cdot Y = X \cdot F(-t)Y$. Deduce that $F(t)(X \wedge Y) = (F(t)X) \wedge (F(t)Y)$.

133- If $\mathbf{a} = \mathbf{i} - 2\mathbf{j}$, $\mathbf{b} = 3\mathbf{j} + 2\mathbf{k}$, and $\mathbf{c} = 4\mathbf{i} + \mathbf{j} - 2\mathbf{k}$, then the image of vector $(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}$ on the $x$-axis is which of the following?
(1) $1$ (2) $2$ (3) $3$ (4) $4$

134- Two vectors $\mathbf{a} = (1,-2,3)$ and $\mathbf{b} = (2,1,-1)$ are given. The volume of the parallelepiped built on vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{a} \times \mathbf{b}$ is:
(1) $54$ (2) $72$ (3) $75$ (4) $80$

134- If $\mathbf{a} = (2, -3, 1)$ and $\mathbf{b} = (1, 2, -4)$, the volume of the parallelepiped constructed on vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{a} \times \mathbf{b}$ is which of the following?
(1) $225$ (2) $220$ (3) $245$ (4) $250$

141. Vector $\vec{a} = (-1, \alpha, 1)$ makes a $45°$ angle with the $z$-axis in space. If $\vec{b} = \left(-\dfrac{4}{3}, \dfrac{2}{3}, 2\right)$ and the angle of vector $\vec{a} \times \vec{b}$ with the $z$-axis is $\theta$, then $\cos\theta$ is which of the following?
(1) $-\dfrac{\sqrt{3}}{3}$ (2) $-\dfrac{\sqrt{3}}{4}$ (3) $\dfrac{\sqrt{3}}{4}$ (4) $\dfrac{\sqrt{3}}{2}$

27. Let the vectors $a , b , c$ and $d$ be such that $( a \times b ) \times ( c \times d ) = 0$. Let P1 and P2be planes determined by the pairs of vectors $a , b$ and $c , d$ respectively, then the angle between $P 1$ and P 2 is :
(A) 0
(B) $\mathrm { p } / 4$
(C) $\mathrm { p } / 3$
(D) $\mathrm { p } / 2$

8. Let V be the volume of the parallelepiped formed by the vectors
$$\begin{aligned} & \vec { a } = a _ { 1 } \hat { \imath } + a _ { 2 } \hat { \jmath } + a _ { 3 } \hat { k } \\ & \vec { b } = b _ { 1 } \hat { \imath } + b _ { 2 } \hat { \jmath } + b _ { 3 } \hat { k } \\ & \vec { c } = c _ { 1 } \hat { \imath } + c _ { 2 } \hat { \jmath } + c _ { 3 } \hat { k } \end{aligned}$$
$r$, bsi,fą, where $\mathrm { r } = 1,2,3$ are non-negative real numbers and $\sum \mathrm { r } = 13 ( \mathrm { ar } + \mathrm { br } + \mathrm { cr } ) = 3 \mathrm {~L}$. Show that $\mathrm { V } < \mathrm { L } ^ { 3 }$.

Consider the lines
$$\begin{aligned} & L _ { 1 } : \frac { x + 1 } { 3 } = \frac { y + 2 } { 1 } = \frac { z + 1 } { 2 } \\ & L _ { 2 } : \frac { x - 2 } { 1 } = \frac { y + 2 } { 2 } = \frac { z - 3 } { 3 } \end{aligned}$$
The unit vector perpendicular to both $L _ { 1 }$ and $L _ { 2 }$ is
(A) $\frac { - \hat { i } + 7 \hat { j } + 7 \hat { k } } { \sqrt { 99 } }$
(B) $\frac { - \hat { i } - 7 \hat { j } + 5 \hat { k } } { 5 \sqrt { 3 } }$
(C) $\frac { - \hat { i } + 7 \hat { j } + 5 \hat { k } } { 5 \sqrt { 3 } }$
(D) $\frac { 7 \hat { i } - 7 \hat { j } - \hat { k } } { \sqrt { 99 } }$

Consider the lines
$$\begin{aligned} & L _ { 1 } : \frac { x + 1 } { 3 } = \frac { y + 2 } { 1 } = \frac { z + 1 } { 2 } \\ & L _ { 2 } : \frac { x - 2 } { 1 } = \frac { y + 2 } { 2 } = \frac { z - 3 } { 3 } \end{aligned}$$
The shortest distance between $L _ { 1 }$ and $L _ { 2 }$ is
(A) 0
(B) $\frac { 17 } { \sqrt { 3 } }$
(C) $\frac { 41 } { 5 \sqrt { 3 } }$
(D) $\frac { 17 } { 5 \sqrt { 3 } }$

Consider the lines
$$\begin{aligned} & L _ { 1 } : \frac { x + 1 } { 3 } = \frac { y + 2 } { 1 } = \frac { z + 1 } { 2 } \\ & L _ { 2 } : \frac { x - 2 } { 1 } = \frac { y + 2 } { 2 } = \frac { z - 3 } { 3 } \end{aligned}$$
The distance of the point $( 1,1,1 )$ from the plane passing through the point $( - 1 , - 2 , - 1 )$ and whose normal is perpendicular to both the lines $L _ { 1 }$ and $L _ { 2 }$ is
(A) $\frac { 2 } { \sqrt { 75 } }$
(B) $\frac { 7 } { \sqrt { 75 } }$
(C) $\frac { 13 } { \sqrt { 75 } }$
(D) $\frac { 23 } { \sqrt { 75 } }$

Let $\overrightarrow { P R } = 3 \hat { i } + \hat { j } - 2 \hat { k }$ and $\overrightarrow { S Q } = \hat { i } - 3 \hat { j } - 4 \hat { k }$ determine diagonals of a parallelogram $P Q R S$ and $\overrightarrow { P T } = \hat { i } + 2 \hat { j } + 3 \hat { k }$ be another vector. Then the volume of the parallelepiped determined by the vectors $\overrightarrow { P T } , \overrightarrow { P Q }$ and $\overrightarrow { P S }$ is
(A) 5
(B) 20
(C) 10
(D) 30

Match List I with List II and select the correct answer using the code given below the lists:
List I

• [P.] Volume of parallelepiped determined by vectors $\vec { a } , \vec { b }$ and $\vec { c }$ is 2. Then the volume of the parallelepiped determined by vectors $2 ( \vec { a } \times \vec { b } ) , 3 ( \vec { b } \times \vec { c } )$ and $( \vec { c } \times \vec { a } )$ is
• [Q.] Volume of parallelepiped determined by vectors $\vec { a } , \vec { b }$ and $\vec { c }$ is 5. Then the volume of the parallelepiped determined by vectors $3 ( \vec { a } + \vec { b } ) , ( \vec { b } + \vec { c } )$ and $2 ( \vec { c } + \vec { a } )$ is
• [R.] Area of a triangle with adjacent sides determined by vectors $\vec { a }$ and $\vec { b }$ is 20. Then the area of the triangle with adjacent sides determined by vectors $( 2 \vec { a } + 3 \vec { b } )$ and $( \vec { a } - \vec { b } )$ is
• [S.] Area of a parallelogram with adjacent sides determined by vectors $\vec { a }$ and $\vec { b }$ is 30. Then the area of the parallelogram with adjacent sides determined by vectors $( \vec { a } + \vec { b } )$ and $\vec { a }$ is

List II

1. $100$
2. $30$
3. (values as given in the list)

Codes:

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

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 $\vec { w } = \hat { \imath } + \hat { \jmath } - 2 \hat { k }$, and $\vec { u }$ and $\vec { v }$ be two vectors, such that $\vec { u } \times \vec { v } = \vec { w }$ and $\vec { v } \times \vec { w } = \vec { u }$. Let $\alpha , \beta , \gamma$, and $t$ be real numbers such that $\vec { u } = \alpha \hat { \imath } + \beta \hat { \jmath } + \gamma \hat { k } , \quad - t \alpha + \beta + \gamma = 0 , \quad \alpha - t \beta + \gamma = 0 , \quad$ and $\alpha + \beta - t \gamma = 0$.
Match each entry in List-I to the correct entry in List-II and choose the correct option.

List-I

(P) $| \vec { v } | ^ { 2 }$ is equal to (Q) If $\alpha = \sqrt { 3 }$, then $\gamma ^ { 2 }$ is equal to (R) If $\alpha = \sqrt { 3 }$, then $( \beta + \gamma ) ^ { 2 }$ is equal to (S) If $\alpha = \sqrt { 2 }$, then $t + 3$ is equal to

List-II

(1) 0
(2) 1
(3) 2
(4) 3
(5) 5

(A)	$( \mathrm { P } ) \rightarrow ( 2 )$	$( \mathrm { Q } ) \rightarrow ( 1 )$	$( \mathrm { R } ) \rightarrow ( 4 )$	$( \mathrm { S } ) \rightarrow ( 5 )$
(B)	$( \mathrm { P } ) \rightarrow ( 2 )$	$( \mathrm { Q } ) \rightarrow ( 4 )$	$( \mathrm { R } ) \rightarrow ( 3 )$	$( \mathrm { S } ) \rightarrow ( 5 )$
(C)	$( \mathrm { P } ) \rightarrow ( 2 )$	$( \mathrm { Q } ) \rightarrow ( 1 )$	$( \mathrm { R } ) \rightarrow ( 4 )$	$( \mathrm { S } ) \rightarrow ( 3 )$
(D)	$( \mathrm { P } ) \rightarrow ( 5 )$	$( \mathrm { Q } ) \rightarrow ( 4 )$	$( \mathrm { R } ) \rightarrow ( 1 )$	$( \mathrm { S } ) \rightarrow ( 3 )$

If $\vec{a} = \frac{1}{\sqrt{10}}(3\hat{i}+\hat{k})$ and $\vec{b} = \frac{1}{7}(2\hat{i}+3\hat{j}-6\hat{k})$, then the value of $(2\vec{a}-\vec{b})\cdot[(\vec{a}\times\vec{b})\times(\vec{a}+2\vec{b})]$ is
(1) $-5$
(2) $-3$
(3) 5
(4) 3

If $[ \vec { a } \times \vec { b } \quad \vec { b } \times \vec { c } \quad \vec { c } \times \vec { a } ] = \lambda \left[ \vec { a } \quad \vec { b } \quad \vec { c } \right] ^ { 2 }$ then $\lambda$ is equal to
(1) 0
(2) 1
(3) 2
(4) 3

If $\vec { x } = 3 \hat { i } - 6 \hat { j } - \widehat { k } , \vec { y } = \hat { i } + 4 \hat { j } - 3 \widehat { k }$ and $\vec { z } = 3 \hat { i } - 4 \hat { j } - 12 \widehat { k }$, then the magnitude of the projection of $\vec { x } \times \vec { y }$ on $\vec { z }$ is
(1) 14
(2) 12
(3) 15
(4) 10

Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three non-zero vectors such that no two of them are collinear and $(\vec{a} \times \vec{b}) \times \vec{c} = \frac{1}{3}|\vec{b}||\vec{c}|\vec{a}$. If $\theta$ is the angle between vectors $\vec{b}$ and $\vec{c}$, then a value of $\sin\theta$ is:
(1) $\frac{2\sqrt{2}}{3}$
(2) $-\frac{\sqrt{2}}{3}$
(3) $\frac{2}{3}$
(4) $-\frac{2\sqrt{3}}{3}$

Let $\vec { a } , \vec { b }$ and $\vec { c }$ be three non-zero vectors such that no two of them are collinear and $( \vec { a } \times \vec { b } ) \times \vec { c } = \frac { 1 } { 3 } | \vec { b } | | \vec { c } | \vec { a }$. If $\theta$ is the angle between vectors $\vec { b }$ and $\vec { c }$, then a value of $\sin \theta$ is
(1) $\frac { - 2 \sqrt { 3 } } { 3 }$
(2) $\frac { 2 \sqrt { 2 } } { 3 }$
(3) $\frac { - \sqrt { 2 } } { 3 }$
(4) $\frac { 2 } { 3 }$

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) $\dfrac{1}{8}$

If $\vec { a } , \vec { b }$, and $\overrightarrow { \mathrm { c } }$ are unit vectors such that $\vec { a } + 2 \vec { b } + 2 \overrightarrow { \mathbf { c } } = \overrightarrow { 0 }$, then $| \vec { a } \times \overrightarrow { \mathbf { c } } |$ is equal to
(1) $\frac { 1 } { 4 }$
(2) $\frac { \sqrt { 15 } } { 4 }$
(3) $\frac { 15 } { 16 }$
(4) $\frac { \sqrt { 15 } } { 16 }$

If the volume of a parallelopiped, whose coterminous edges are given by the vectors $\overrightarrow { \mathrm { a } } = \hat { i } + \hat { j } + n \widehat { k }$, $\overrightarrow { \mathrm { b } } = 2 \hat { \mathrm { i } } + 4 \hat { \mathrm { j } } - n \hat { k }$ and, $\overrightarrow { \mathrm { c } } = \hat { \mathrm { i } } + n \hat { j } + 3 \hat { k } ( \mathrm { n } \geq 0 )$ is 158 cubic units, then :
(1) $\overrightarrow { \mathrm { a } } \cdot \overrightarrow { \mathrm { c } } = 17$
(2) $\overrightarrow { \mathrm { b } } \cdot \overrightarrow { \mathrm { c } } = 10$
(3) $n = 7$
(4) $n = 9$

If $\vec { a } \cdot \vec { b } = 1 , \vec { b } \cdot \vec { c } = 2$ and $\vec { c } \cdot \vec { a } = 3$, then the value of $[ \vec { a } \times ( \vec { b } \times \vec { c } ) \quad \vec { b } \times ( \vec { c } \times \vec { a } ) \quad \vec { c } \times ( \vec { b } \times \vec { a } ) ]$ is
(1) 0
(2) $- 6 \vec { a } \cdot ( \vec { b } \times \vec { c } )$
(3) $12 \vec { c } \cdot ( \vec { a } \times \vec { b } )$
(4) $- 12 \vec { b } \cdot ( \vec { c } \times \vec { a } )$

Let $\vec { a } = \alpha \hat { i } + \hat { j } - \hat { k }$ and $\vec { b } = 2 \hat { i } + \hat { j } - \alpha \hat { k } , \alpha > 0$. If the projection of $\vec { a } \times \vec { b }$ on the vector $- \hat { i } + 2 \hat { j } - 2 \hat { k }$ is 30, then $\alpha$ is equal to
(1) $\frac { 15 } { 2 }$
(2) 8
(3) $\frac { 13 } { 2 }$
(4) 7

Let $| \vec { a } | = 2 , | \vec { b } | = 3$ and the angle between the vectors $\vec { a }$ and $\vec { b }$ be $\frac { \pi } { 4 }$. Then $| ( \vec { a } + 2 \vec { b } ) \times ( 2 \vec { a } - 3 \vec { b } ) | ^ { 2 }$ is equal to
(1) 441
(2) 482
(3) 841
(4) 882