2. $\vec { a } , \vec { b } , \vec { c } , \vec { d }$ are four distinct vectors satisfying the conditions $\vec { a } \times \vec { b } = \vec { c } \times \vec { d }$ and $\vec { a } \times \vec { c } = \vec { b } \times \vec { d }$, then prove that $\vec { a } \cdot \vec { b } + \vec { c } \cdot \vec { d } \neq \vec { a } \cdot \vec { c } + \vec { b } \cdot \vec { d }$.
Sol. Given that $\vec { a } \times \vec { b } = \vec { c } \times \vec { d }$ and $\vec { a } \times \vec { c } = \vec { b } \times \vec { d }$ $\Rightarrow \vec { a } \times ( \vec { b } - \vec { c } ) = ( \vec { c } - \vec { b } ) \times \vec { d } = \vec { d } \times ( \vec { b } - \vec { c } ) \Rightarrow \vec { a } - \vec { d } \| \vec { b } - \vec { c }$ $\Rightarrow ( \vec { a } - \vec { d } ) \cdot ( \vec { b } - \vec { c } ) \neq 0 \Rightarrow \vec { a } \cdot \vec { b } + \vec { d } \cdot \vec { c } \neq \vec { d } \cdot \vec { b } + \vec { a } \cdot \vec { c }$.

9. A plane is parallel to two lines whose direction ratios are $( 1,0 , - 1 )$ and $( - 1,1,0 )$ and it contains the point $( 1,1,1 )$. If it cuts coordinate axis at $\mathrm { A } , \mathrm { B } , \mathrm { C }$, then find the volume of the tetrahedron OABC .
Sol. Let $( l , m , n )$ be the direction ratios of the normal to the required plane so that $l - n = 0$ and $- l + m = 0$ $\Rightarrow \mathrm { l } = \mathrm { m } = \mathrm { n }$ and hence the equation of the plane containing $( 1,1,1 )$ is $\frac { \mathrm { x } } { 3 } + \frac { \mathrm { y } } { 3 } + \frac { \mathrm { z } } { 3 } = 1$. Its intercepts with the coordinate axes are $\mathrm { A } ( 3,0,0 ) ; \mathrm { B } ( 0,3,0 ) ; \mathrm { C } ( 0,0,3 )$. Hence the volume of OABC $= \frac { 1 } { 6 } \left| \begin{array} { l l l } 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{array} \right| = \frac { 27 } { 6 } = \frac { 9 } { 2 }$ cubic units.

20. Two planes $\mathrm { P } _ { 1 }$ and $\mathrm { P } _ { 2 }$ pass through origin. Two lines $\mathrm { L } _ { 1 }$ and $\mathrm { L } _ { 2 }$ also passing through origin are such that $\mathrm { L } _ { 1 }$ lies on $\mathrm { P } _ { 1 }$ but not on $\mathrm { P } _ { 2 } , \mathrm {~L} _ { 2 }$ lies on $\mathrm { P } _ { 2 }$ but not on $\mathrm { P } _ { 1 } . \mathrm { A } , \mathrm { B } , \mathrm { C }$ are three points other than origin, then prove that the permutation $\left[ \mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime } , \mathrm { C } ^ { \prime } \right]$ of $[ \mathrm { A } , \mathrm { B } , \mathrm { C } ]$ exists such that
(i). $\quad \mathrm { A }$ lies on $\mathrm { L } _ { 1 } , \mathrm {~B}$ lies on $\mathrm { P } _ { 1 }$ not on $\mathrm { L } _ { 1 } , \mathrm { C }$ does not lie on $\mathrm { P } _ { 1 }$.
(ii). $\quad \mathrm { A } ^ { \prime }$ lies on $\mathrm { L } _ { 2 } , \mathrm {~B} ^ { \prime }$ lies on $\mathrm { P } _ { 2 }$ not on $\mathrm { L } _ { 2 } , \mathrm { C } ^ { \prime }$ does not lie on $\mathrm { P } _ { 2 }$.
Sol. A corresponds to one of $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime } , \mathrm { C } ^ { \prime }$ and
B corresponds to one of the remaining of $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime } , \mathrm { C } ^ { \prime }$ and
C corresponds to third of $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime } , \mathrm { C } ^ { \prime }$. Hence six such permutations are possible eg One of the permutations may $\mathrm { A } \equiv \mathrm { A } ^ { \prime } ; \mathrm { B } \equiv \mathrm { B } ^ { \prime } , \mathrm { C } \equiv \mathrm { C } ^ { \prime }$ From the given conditions:
A lies on $\mathrm { L } _ { 1 }$.
B lies on the line of intersection of $\mathrm { P } _ { 1 }$ and $\mathrm { P } _ { 2 }$ and ' C ' lies on the line $\mathrm { L } _ { 2 }$ on the plane $\mathrm { P } _ { 2 }$. Now, $\mathrm { A } ^ { \prime }$ lies on $\mathrm { L } _ { 2 } \equiv \mathrm { C }$. $\mathrm { B } ^ { \prime }$ lies on the line of intersection of $\mathrm { P } _ { 1 }$ and $\mathrm { P } _ { 2 } \equiv \mathrm {~B}$ $\mathrm { C } ^ { \prime }$ lie on $\mathrm { L } _ { 1 }$ on plane $\mathrm { P } _ { 1 } \equiv \mathrm {~A}$. Hence there exist a particular set [ $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime } , \mathrm { C } ^ { \prime }$ ] which is the permutation of $[ \mathrm { A } , \mathrm { B } , \mathrm { C } ]$ such that both (i) and
(ii) is satisfied. Here $\left[ \mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime } , \mathrm { C } ^ { \prime } \right] \equiv [ \mathrm { CBA } ]$.

9. A variable plane $x / a + y / b + z / c = 1$ at a unit distance from origin cuts the coordinate axes at $A , B$ and $C$. Centroid $( x , y , z )$ satisfies the equation $1 / x ^ { 2 } + 1 / y ^ { 2 } + 1 / z ^ { 2 } = K$. The value of $K$ is :
(a) 9
(b) 3
(c) $1 / 9$
(d) $1 / 3$

25. If $\vec { a } , \vec { b } , \vec { c }$ are three non zero, non coplanar vectors and $\vec { b } _ { 1 } = \vec { b } - \frac { \vec { b } \cdot \vec { a } } { | \vec { a } | ^ { 2 } } \vec { a }$,
$$\begin{aligned} & \vec { b } _ { 2 } = \vec { b } + \frac { \vec { b } \cdot \vec { a } } { | \vec { a } | ^ { 2 } } \vec { a } , \text { And } \vec { c } _ { 1 } = \vec { c } - \frac { \vec { c } \cdot \vec { a } } { | \vec { a } | ^ { 2 } } \vec { a } - \frac { \vec { c } \cdot \vec { b } } { | \vec { b } | ^ { 2 } } \vec { b } , \quad \vec { c } _ { 2 } = \vec { c } - \frac { \vec { c } \cdot \vec { a } } { | \vec { a } | ^ { 2 } } \vec { a } - \frac { \vec { c } \cdot \vec { b } } { \left| \vec { b } _ { 1 } \right| ^ { 2 } } \vec { b } _ { 1 } \\ & \vec { c } _ { 3 } = \vec { c } - \frac { \vec { c } \cdot \vec { a } } { | \vec { a } | ^ { 2 } } \vec { a } - \frac { \vec { c } \cdot \vec { b } } { | \vec { b } | ^ { 2 } } \vec { b } _ { 2 } , \vec { c } _ { 4 } = \vec { a } - \frac { \vec { c } \cdot \vec { a } } { | \vec { a } | ^ { 2 } } \vec { a } \end{aligned}$$
Then which of the following is a set of mutually orthogonal vectors:
(a) $\left( \vec { a } , \vec { b } _ { 1 } , \vec { c } _ { 1 } \right)$
(b) $\left( \vec { a } , \vec { b } _ { 1 } , \vec { c } _ { 2 } \right)$
(c) $\quad \left( \vec { a } , \vec { b } _ { 2 } , \vec { c } _ { 3 } \right)$
(d) $\left( \vec { a } , \vec { b } _ { 2 } , \vec { c } _ { 4 } \right)$

12. 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
(A) $4 \hat { i } - \hat { j } + 4 \hat { k }$
(B) $3 \hat { i } + \hat { j } - 3 \hat { k }$
(C) $2 \hat { i } + \hat { j } - 2 \hat { k }$
(D) $4 \hat { i } + \hat { j } - 4 \hat { k }$

Sol. (A)

Vector lying in the plane of $\vec { a }$ and $\vec { b }$ is $\vec { r } = \lambda _ { 1 } \vec { a } + \lambda _ { 2 } \vec { b }$ and its projection on $\vec { c }$ is $\frac { 1 } { \sqrt { 3 } }$ $\Rightarrow \quad \left[ \left( \lambda _ { 1 } + \lambda _ { 2 } \right) \hat { \mathrm { i } } - \left( 2 \lambda _ { 1 } - \lambda _ { 2 } \right) \hat { \mathrm { j } } + \left( \lambda _ { 1 } + \lambda _ { 2 } \right) \hat { \mathrm { k } } \right] \cdot \frac { [ \hat { \mathrm { i } } - \hat { \mathrm { j } } - \hat { \mathrm { k } } ] } { \sqrt { 3 } } = \frac { 1 } { \sqrt { 3 } }$ $\Rightarrow \quad 2 \lambda _ { 1 } - \lambda _ { 2 } = - 1 \Rightarrow \overrightarrow { \mathrm { r } } = \left( 3 \lambda _ { 1 } + 1 \right) \hat { \mathrm { i } } - \hat { \mathrm { j } } + \left( 3 \lambda _ { 1 } + 1 \right) \hat { \mathrm { k } }$ Hence the required vector is $4 \hat { i } - \hat { j } + 4 \hat { k }$.

Alternate:

Vector lying in the plane of $\vec { a }$ and $\vec { b }$ is $\vec { a } + \lambda \vec { b }$, and its projection on $C$ is $\frac { 1 } { \sqrt { 3 } }$. $\Rightarrow \left( ( 1 + \lambda ) \hat { \mathrm { i } } + ( 2 - \lambda ) \hat { \mathrm { j } } + ( 1 + \lambda ) \hat { \mathrm { k } } \cdot \frac { ( \hat { \mathrm { i } } - \hat { \mathrm { j } } - \hat { \mathrm { k } } ) } { \sqrt { 3 } } \right) = \frac { 1 } { \sqrt { 3 } }$ $\Rightarrow \lambda = 3$. Hence the required vector is $4 \hat { i } - \hat { j } + 4 \hat { k }$.

Section - B (May have more than one option correct)

1. The equations of the common tangents to the parabola $y = x ^ { 2 }$ and $y = - ( x - 2 ) ^ { 2 }$ is/are
   (A) $\mathrm { y } = 4 ( \mathrm { x } - 1 )$
   (B) $\mathrm { y } = 0$
   (C) $y = - 4 ( x - 1 )$
   (D) $y = - 30 x - 50$

Sol. (A), (B) Equation of tangent to $x ^ { 2 } = y$ is
$$\mathrm { y } = \mathrm { mx } - \frac { 1 } { 4 } \mathrm {~m} ^ { 2 }$$
Equation of tangent to $( x - 2 ) ^ { 2 } = - y$ is
$$\mathrm { y } = \mathrm { m } ( \mathrm { x } - 2 ) + \frac { 1 } { 4 } \mathrm {~m} ^ { 2 }$$
(1) and (2) are identical. $\Rightarrow \mathrm { m } = 0$ or 4 $\therefore \quad$ Common tangents are $\mathrm { y } = 0$ and $\mathrm { y } = 4 \mathrm { x } - 4$.

19. Let $\overrightarrow { \mathrm { A } }$ be vector parallel to line of intersection of planes $\mathrm { P } _ { 1 }$ and $\mathrm { P } _ { 2 }$ through origin. $\mathrm { P } _ { 1 }$ is parallel to the vectors $2 \hat { \mathrm { j } } + 3 \hat { \mathrm { k } }$ and $4 \hat { j } - 3 \hat { k }$ and $P _ { 2 }$ is parallel to $\hat { j } - \hat { k }$ and $3 \hat { i } + 3 \hat { j }$, then the angle between vector $\vec { A }$ and $2 \hat { i } + \hat { j } - 2 \hat { k }$ is
(A) $\frac { \pi } { 2 }$
(B) $\frac { \pi } { 4 }$
(C) $\frac { \pi } { 6 }$
(D) $\frac { 3 \pi } { 4 }$
Sol. (B), (D) Vector AB is parallel to $[ ( 2 \hat { \mathrm { i } } + 3 \hat { \mathrm { k } } ) \times ( 4 ) - 3 \hat { \mathrm { k } } ] \times [ ( \hat { \mathrm { j } } - \hat { \mathrm { k } } ) \times ( 3 \hat { \mathrm { i } } + 3 \hat { \mathrm { j } } ) ] = 54 ( \hat { \mathrm { j } } - \hat { \mathrm { k } } )$ Let $\theta$ is the angle between the vector, then
$$\cos \theta = \pm \left( \frac { 54 + 108 } { 3.54 \sqrt { 2 } } \right) = \pm \frac { 1 } { \sqrt { 2 } }$$
Hence $\theta = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 }$.

The number of distinct real values of $\lambda$ for which the vectors $-\lambda^2\hat{i}+\hat{j}+\hat{k}$, $\hat{i}-\lambda^2\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}-\lambda^2\hat{k}$ are coplanar is
(A) 0
(B) 1
(C) 2
(D) 3

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors $\hat { a } , \hat { b } , \hat { c }$ such that $$\hat { a } \cdot \hat { b } = \hat { b } \cdot \hat { c } = \hat { c } \cdot \hat { a } = \frac { 1 } { 2 }$$ Then, the volume of the parallelopiped is
(A) $\frac { 1 } { \sqrt { 2 } }$
(B) $\frac { 1 } { 2 \sqrt { 2 } }$
(C) $\frac { \sqrt { 3 } } { 2 }$
(D) $\frac { 1 } { \sqrt { 3 } }$

Consider three planes $$\begin{aligned} & P _ { 1 } : x - y + z = 1 \\ & P _ { 2 } : x + y - z = - 1 \\ & P _ { 3 } : x - 3 y + 3 z = 2 . \end{aligned}$$ Let $L _ { 1 } , L _ { 2 } , L _ { 3 }$ be the lines of intersection of the planes $P _ { 2 }$ and $P _ { 3 } , P _ { 3 }$ and $P _ { 1 }$, and $P _ { 1 }$ and $P _ { 2 }$, respectively.
STATEMENT-1: At least two of the lines $L _ { 1 } , L _ { 2 }$ and $L _ { 3 }$ are non-parallel. and STATEMENT-2 : The three planes do not have a common point.
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False
(D) STATEMENT-1 is False, STATEMENT-2 is True

Let $P ( 3,2,6 )$ be a point in space and $Q$ be a point on the line
$$\vec { r } = ( \hat { i } - \hat { j } + 2 \hat { k } ) + \mu ( - 3 \hat { i } + \hat { j } + 5 \hat { k } )$$
Then the value of $\mu$ for which the vector $\overrightarrow { P Q }$ is parallel to the plane $x - 4 y + 3 z = 1$ is
(A) $\frac { 1 } { 4 }$
(B) $- \frac { 1 } { 4 }$
(C) $\frac { 1 } { 8 }$
(D) $- \frac { 1 } { 8 }$

If $\vec { a } , \vec { b } , \vec { c }$ and $\vec { d }$ are unit vectors such that
$$( \vec { a } \times \vec { b } ) \cdot ( \vec { c } \times \vec { d } ) = 1$$
and $\quad \vec { a } \cdot \vec { c } = \frac { 1 } { 2 }$,
then

If the distance of the point $\mathrm { P } ( 1 , - 2,1 )$ from the plane $\mathrm { x } + 2 \mathrm { y } - 2 z = \alpha$, where $\alpha > 0$, is 5 , then the foot of the perpendicular from $P$ to the plane is
A) $\left( \frac { 8 } { 3 } , \frac { 4 } { 3 } , - \frac { 7 } { 3 } \right)$
B) $\left( \frac { 4 } { 3 } , - \frac { 4 } { 3 } , \frac { 1 } { 3 } \right)$
C) $\left( \frac { 1 } { 3 } , \frac { 2 } { 3 } , \frac { 10 } { 3 } \right)$
D) $\left( \frac { 2 } { 3 } , - \frac { 1 } { 3 } , \frac { 5 } { 2 } \right)$

Two adjacent sides of a parallelogram ABCD are given by $\overrightarrow { \mathrm { AB } } = 2 \hat { \mathrm { i } } + 10 \hat { \mathrm { j } } + 11 \hat { \mathrm { k } }$ and $\overrightarrow { \mathrm { AD } } = - \hat { \mathrm { i } } + 2 \hat { \mathrm { j } } + 2 \hat { \mathrm { k } }$
The side AD is rotated by an acute angle $\alpha$ in the plane of the parallelogram so that AD becomes $\mathrm { AD } ^ { \prime }$. If $\mathrm { AD } ^ { \prime }$ makes a right angle with the side AB , then the cosine of the angle $\alpha$ is given by
A) $\frac { 8 } { 9 }$
B) $\frac { \sqrt { 17 } } { 9 }$
C) $\frac { 1 } { 9 }$
D) $\frac { 4 \sqrt { 5 } } { 9 }$

Match the statements in Column-I with the values in Column-II.
Column I
A) A line from the origin meets the lines $\frac { x - 2 } { 1 } = \frac { y - 1 } { - 2 } = \frac { z + 1 } { 1 }$ and $\frac { x - \frac { 8 } { 3 } } { 2 } = \frac { y + 3 } { - 1 } = \frac { z - 1 } { 1 }$ at $P$ and $Q$ respectively. If length $\mathrm { PQ } = d$, then $d ^ { 2 }$ is
B) The values of $x$ satisfying $\tan ^ { - 1 } ( x + 3 ) - \tan ^ { - 1 } ( x - 3 ) = \sin ^ { - 1 } \left( \frac { 3 } { 5 } \right)$ are
C) Non-zero vectors $\vec { a } , \vec { b }$ and $\vec { c }$ satisfy $\vec { a } \cdot \vec { b } = 0$, $( \overrightarrow { \mathrm { b } } - \overrightarrow { \mathrm { a } } ) \cdot ( \overrightarrow { \mathrm { b } } + \overrightarrow { \mathrm { c } } ) = 0$ and $2 | \overrightarrow { \mathrm {~b} } + \overrightarrow { \mathrm { c } } | = | \overrightarrow { \mathrm { b } } - \overrightarrow { \mathrm { a } } |$. If $\vec { a } = \mu \vec { b } + 4 \vec { c }$, then the possible values of $\mu$ are
D) Let f be the function on $[ - \pi , \pi ]$ given by $f ( 0 ) = 9$ and $f ( x ) = \sin \left( \frac { 9 x } { 2 } \right) / \sin \left( \frac { x } { 2 } \right)$ for $x \neq 0$. The value of $\frac { 2 } { \pi } \int _ { - \pi } ^ { \pi } f ( x ) d x$ is
Column II p) $-4$ q) $0$ r) $4$ s) $-1$ (or as given in paper) t) $6$

If $\overrightarrow { \mathrm { a } }$ and $\overrightarrow { \mathrm { b } }$ are vectors in space given by $\overrightarrow { \mathrm { a } } = \frac { \hat { \mathrm { i } } - 2 \hat { \mathrm { j } } } { \sqrt { 5 } }$ and $\overrightarrow { \mathrm { b } } = \frac { 2 \hat { \mathrm { i } } + \hat { \mathrm { j } } + 3 \hat { \mathrm { k } } } { \sqrt { 14 } }$, then the value of $( 2 \vec { a } + \vec { b } ) \cdot [ ( \vec { a } \times \vec { b } ) \times ( \vec { a } - 2 \vec { b } ) ]$ is

Consider the set of eight vectors $V = \{ \mathrm { a } \hat { i } + \mathrm { b } \hat { j } + \mathrm { c } \hat { k } : a , b , c \in \{ - 1,1 \} \}$. Three noncoplanar vectors can be chosen from $V$ in $2 ^ { p }$ ways. Then $p$ is

From a point $P(\lambda, \lambda, \lambda)$, perpendiculars $PQ$ and $PR$ are drawn respectively on the lines $y = x, z = 1$ and $y = -x, z = -1$. If $P$ is such that $\angle QPR$ is a right angle, then the possible value(s) of $\lambda$ is(are)
(A) $\sqrt{2}$
(B) $1$
(C) $-1$
(D) $-\sqrt{2}$

Let $\vec{x}, \vec{y}$ and $\vec{z}$ be three vectors each of magnitude $\sqrt{2}$ and the angle between each pair of them is $\frac{\pi}{3}$. If $\vec{a}$ is a nonzero vector perpendicular to $\vec{x}$ and $\vec{y} \times \vec{z}$ and $\vec{b}$ is a nonzero vector perpendicular to $\vec{y}$ and $\vec{z} \times \vec{x}$, then
(A) $\vec{b} = (\vec{b} \cdot \vec{z})(\vec{z} - \vec{x})$
(B) $\vec{a} = (\vec{a} \cdot \vec{y})(\vec{y} - \vec{z})$
(C) $\vec{a} \cdot \vec{b} = -(\vec{a} \cdot \vec{y})(\vec{b} \cdot \vec{z})$
(D) $\vec{a} = (\vec{a} \cdot \vec{y})(\vec{z} - \vec{y})$

Suppose that $\vec { p } , \vec { q }$ and $\vec { r }$ are three non-coplanar vectors in $\mathbb { R } ^ { 3 }$. Let the components of a vector $\vec { s }$ along $\vec { p } , \vec { q }$ and $\vec { r }$ be 4,3 and 5 , respectively. If the components of this vector $\vec { s }$ along $( - \vec { p } + \vec { q } + \vec { r } ) , ( \vec { p } - \vec { q } + \vec { r } )$ and $( - \vec { p } - \vec { q } + \vec { r } )$ are $x , y$ and $z$, respectively, then the value of $2 x + y + z$ is

In $\mathbb { R } ^ { 3 }$, consider the planes $P _ { 1 } : y = 0$ and $P _ { 2 } : x + z = 1$. Let $P _ { 3 }$ be a plane, different from $P _ { 1 }$ and $P _ { 2 }$, which passes through the intersection of $P _ { 1 }$ and $P _ { 2 }$. If the distance of the point $( 0,1,0 )$ from $P _ { 3 }$ is 1 and the distance of a point $( \alpha , \beta , \gamma )$ from $P _ { 3 }$ is 2, then which of the following relations is (are) true?
(A) $2 \alpha + \beta + 2 \gamma + 2 = 0$
(B) $2 \alpha - \beta + 2 \gamma + 4 = 0$
(C) $2 \alpha + \beta - 2 \gamma - 10 = 0$
(D) $2 \alpha - \beta + 2 \gamma - 8 = 0$

In $\mathbb { R } ^ { 3 }$, let $L$ be a straight line passing through the origin. Suppose that all the points on $L$ are at a constant distance from the two planes $P _ { 1 } : x + 2 y - z + 1 = 0$ and $P _ { 2 } : 2 x - y + z - 1 = 0$. Let $M$ be the locus of the feet of the perpendiculars drawn from the points on $L$ to the plane $P _ { 1 }$. Which of the following points lie(s) on $M$?
(A) $\left( 0 , - \frac { 5 } { 6 } , - \frac { 2 } { 3 } \right)$
(B) $\left( - \frac { 1 } { 6 } , - \frac { 1 } { 3 } , \frac { 1 } { 6 } \right)$
(C) $\left( - \frac { 5 } { 6 } , 0 , \frac { 1 } { 6 } \right)$
(D) $\left( - \frac { 1 } { 3 } , 0 , \frac { 2 } { 3 } \right)$

Consider a pyramid $OPQRS$ located in the first octant $(x \geq 0, y \geq 0, z \geq 0)$ with $O$ as origin, and $OP$ and $OR$ along the $x$-axis and the $y$-axis, respectively. The base $OPQR$ of the pyramid is a square with $OP = 3$. The point $S$ is directly above the mid-point $T$ of diagonal $OQ$ such that $TS = 3$. Then
(A) the acute angle between $OQ$ and $OS$ is $\frac{\pi}{3}$
(B) the equation of the plane containing the triangle $OQS$ is $x - y = 0$
(C) the length of the perpendicular from $P$ to the plane containing the triangle $OQS$ is $\frac{3}{\sqrt{2}}$
(D) the perpendicular distance from $O$ to the straight line containing $RS$ is $\sqrt{\frac{15}{2}}$

Let $\hat { u } = u _ { 1 } \hat { i } + u _ { 2 } \hat { j } + u _ { 3 } \hat { k }$ be a unit vector in $\mathbb { R } ^ { 3 }$ and $\hat { w } = \frac { 1 } { \sqrt { 6 } } ( \hat { i } + \hat { j } + 2 \hat { k } )$. Given that there exists a vector $\vec { v }$ in $\mathbb { R } ^ { 3 }$ such that $| \hat { u } \times \vec { v } | = 1$ and $\hat { w } \cdot ( \hat { u } \times \vec { v } ) = 1$. Which of the following statement(s) is(are) correct?
(A) There is exactly one choice for such $\vec { v }$
(B) There are infinitely many choices for such $\vec { v }$
(C) If $\hat { u }$ lies in the $x y$-plane then $\left| u _ { 1 } \right| = \left| u _ { 2 } \right|$
(D) If $\hat { u }$ lies in the $x z$-plane then $2 \left| u _ { 1 } \right| = \left| u _ { 3 } \right|$

Let $P _ { 1 } : 2 x + y - z = 3$ and $P _ { 2 } : x + 2 y + z = 2$ be two planes. Then, which of the following statement(s) is (are) TRUE?
(A) The line of intersection of $P _ { 1 }$ and $P _ { 2 }$ has direction ratios $1,2 , - 1$
(B) The line $$\frac { 3 x - 4 } { 9 } = \frac { 1 - 3 y } { 9 } = \frac { z } { 3 }$$ is perpendicular to the line of intersection of $P _ { 1 }$ and $P _ { 2 }$
(C) The acute angle between $P _ { 1 }$ and $P _ { 2 }$ is $60 ^ { \circ }$
(D) If $P _ { 3 }$ is the plane passing through the point $( 4,2 , - 2 )$ and perpendicular to the line of intersection of $P _ { 1 }$ and $P _ { 2 }$, then the distance of the point $( 2,1,1 )$ from the plane $P _ { 3 }$ is $\frac { 2 } { \sqrt { 3 } }$