Match the statements/expressions given in Column I with the values given in Column II.
Column I
(A) Root(s) of the equation $$2\sin^{2}\theta+\sin^{2}2\theta=2$$ (B) Points of discontinuity of the function $$f(x)=\left[\frac{6x}{\pi}\right]\cos\left[\frac{3x}{\pi}\right],$$ where $[y]$ denotes the largest integer less than or equal to $y$
(C) Volume of the parallelopiped with its edges represented by the vectors $$\hat{i}+\hat{j},\quad\hat{i}+2\hat{j}\text{ and }\hat{i}+\hat{j}+\pi\hat{k}$$ (D) Angle between vectors $\vec{a}$ and $\vec{b}$ where $\vec{a},\vec{b}$ and $\vec{c}$ are unit vectors satisfying $$\vec{a}+\vec{b}+\sqrt{3}\vec{c}=\overrightarrow{0}$$
Column II
(p) $\frac{\pi}{6}$
(q) $\frac{\pi}{4}$
(r) $\frac{\pi}{3}$
(s) $\frac{\pi}{2}$
(t) $\pi$

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

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 $\_\_\_\_$ .

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 $\vec { x } = 3 \hat { i } - 6 \hat { j } - \widehat { k } , \vec { y } = \hat { i } + 4 \hat { j } - 3 \widehat { k }$ 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 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 $\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k}$ 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 $\vec{a} \times \vec{b}$ is $30^\circ$. Then $\vec{a} \cdot \vec{c}$ is equal to:
(1) $\frac{1}{8}$
(2) $25$
(3) $2$
(4) $5$

The magnitude of the projection of the vector $2\hat{i} + 3\hat{j} + \hat{k}$ on the vector perpendicular to the plane containing the vectors $\hat{i} + \hat{j} + \hat{k}$ and $\hat{i} + 2\hat{j} + 3\hat{k}$, is:
(1) $3\sqrt{6}$
(2) $\sqrt{\frac{3}{2}}$
(3) $\sqrt{6}$
(4) $\frac{\sqrt{3}}{2}$

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 force $\vec { F } = ( \hat { i } + 2 \hat { j } + 3 \hat { k } ) \mathrm { N }$ acts at a point $( 4 \hat { i } + 3 \hat { j } - \widehat { k } ) \mathrm { m }$. Then the magnitude of torque about the point $( \hat { i } + 2 \hat { j } + \widehat { k } ) \mathrm { m }$ will be $\sqrt { x } \mathrm {~N} - \mathrm { m }$. The value of $x$ is.

Let $\vec { a }$ and $\vec { b }$ be the vectors along the diagonal of a parallelogram having area $2 \sqrt { 2 }$. Let the angle between $\vec { a }$ and $\vec { b }$ be acute. $| \vec { a } | = 1$ and $| \vec { a } \cdot \vec { b } | = | \vec { a } \times \vec { b } |$. If $\vec { c } = 2 \sqrt { 2 } ( \vec { a } \times \vec { b } ) - 2 \vec { b }$, then an angle between $\vec { b }$ and $\vec { c }$ is
(1) $\frac { - \pi } { 4 }$
(2) $\frac { 5 \pi } { 6 }$
(3) $\frac { \pi } { 3 }$
(4) $\frac { 3 \pi } { 4 }$

Let $ABC$ be a triangle such that $\overrightarrow { BC } = \vec { a }$, $\overrightarrow { CA } = \vec { b }$, $\overrightarrow { AB } = \vec { c }$, $|\vec{a}| = 6\sqrt{2}$, $|\vec{b}| = 2\sqrt{3}$ and $\vec{b} \cdot \vec{c} = 12$. Consider the statements: S1: $|\vec{a} \times \vec{b} + \vec{c} \times \vec{b}| - |\vec{c}| = 6(2\sqrt{2} - 1)$ S2: $\angle ABC = \cos^{-1}\sqrt{\frac{2}{3}}$. Then
(1) both $S1$ and $S2$ are true
(2) only $S1$ is true
(3) only $S2$ is true
(4) both $S1$ and $S2$ are false

If $A ( 3,1 , -1 ) , B \left( \frac { 5 } { 3 } , \frac { 7 } { 3 } , \frac { 1 } { 3 } \right) , C ( 2,2,1 )$ and $D \left( \frac { 10 } { 3 } , \frac { 2 } { 3 } , \frac { -1 } { 3 } \right)$ are the vertices of a quadrilateral $ABCD$, then its area is
(1) $\frac { 2 \sqrt { 2 } } { 3 }$
(2) $\frac { 5 \sqrt { 2 } } { 3 }$
(3) $2 \sqrt { 2 }$
(4) $\frac { 4 \sqrt { 2 } } { 3 }$

Let $\vec { a } = 4 \hat { i } - \hat { j } + \hat { k } , \vec { b } = 11 \hat { i } - \hat { j } + \hat { k }$ and $\vec { c }$ be a vector such that $( \vec { a } + \vec { b } ) \times \vec { c } = \vec { c } \times ( - 2 \vec { a } + 3 \vec { b } )$. If $( 2 \vec { a } + 3 \vec { b } ) \cdot \vec { c } = 1670$, then $| \vec { c } | ^ { 2 }$ is equal to : (1) 1609 (2) 1618 (3) 1600 (4) 1627

If $\vec { a } = \hat { i } + 2 \hat { j } + \hat { k } , \vec { b } = 3 ( \hat { i } - \hat { j } + \hat { k } )$ and $\overrightarrow { \mathrm { c } }$ be the vector such that $\vec { a } \times \vec { c } = \vec { b }$ and $\vec { a } \cdot \vec { c } = 3$, then $\vec { a } \cdot ( ( \vec { c } \times \vec { b } ) - \vec { b } - \vec { c } )$ is equal to
(1) 32
(2) 24
(3) 20
(4) 36

Let $\vec { a } = \hat { i } + \hat { j } + \hat { k } , \vec { b } = 2 \hat { i } + 4 \hat { j } - 5 \hat { k }$ and $\vec { c } = x \hat { i } + 2 \hat { j } + 3 \hat { k } , x \in \mathbb { R }$. If $\vec { d }$ is the unit vector in the direction of $\vec { b } + \vec { c }$ such that $\vec { a } \cdot \vec { d } = 1$, then $( \vec { a } \times \vec { b } ) \cdot \vec { c }$ is equal to
(1) 11
(2) 3
(3) 9
(4) 6

Let $\mathrm { L } _ { 1 } : \frac { x - 1 } { 2 } = \frac { y - 2 } { 3 } = \frac { z - 3 } { 4 }$ and $\mathrm { L } _ { 2 } : \frac { x - 2 } { 3 } = \frac { y - 4 } { 4 } = \frac { z - 5 } { 5 }$ be two lines. Then which of the following points lies on the line of the shortest distance between $L _ { 1 }$ and $L _ { 2 }$?
(1) $\left( \frac { 14 } { 3 } , - 3 , \frac { 22 } { 3 } \right)$
(2) $\left( - \frac { 5 } { 3 } , - 7,1 \right)$
(3) $\left( 2,3 , \frac { 1 } { 3 } \right)$
(4) $\left( \frac { 8 } { 3 } , - 1 , \frac { 1 } { 3 } \right)$

In coordinate space, let $E$ be the plane passing through the three points $A ( 0 , - 1 , - 1 )$ , $B ( 1 , - 1 , - 2 )$ , $C ( 0,1,0 )$ . Assume $H$ is a point in space satisfying $\overrightarrow { AH } = \frac { 2 } { 3 } \overrightarrow { AB } - \frac { 1 } { 3 } \overrightarrow { AC } + 3 ( \overrightarrow { AB } \times \overrightarrow { AC } )$ . Based on the above, answer the following questions.
(1) Find the volume of tetrahedron $ABCH$ . (4 points) (Note: The volume of a tetrahedron is one-third of the base area times the height)
(2) Let $H ^ { \prime }$ be the symmetric point of point $H$ with respect to plane $E$ . Find the coordinates of $H ^ { \prime }$ . (4 points)
(3) Determine whether the projection of point $H ^ { \prime }$ onto plane $E$ lies inside $\triangle ABC$ . Explain your reasoning. (4 points) (Note: The interior of a triangle does not include the three sides of the triangle)