49. 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 given by
(A) $\hat { i } - 3 \hat { j } + 3 \hat { k }$
(B) $- 3 \hat { i } - 3 \hat { j } - \hat { k }$
(C) $3 \hat { i } - \hat { j } + 3 \hat { k }$
(D) $\quad \hat { i } + 3 \hat { j } - 3 \hat { k }$

ANSWER: C

1. Let $P = \{ \theta : \sin \theta - \cos \theta = \sqrt { 2 } \cos \theta \}$ and $Q = \{ \theta : \sin \theta + \cos \theta = \sqrt { 2 } \sin \theta \}$ be two sets. Then
   (A) $P \subset Q$ and $Q - P \neq \varnothing$
   (B) $Q \not \subset P$
   (C) $P \not \subset Q$
   (D) $P = Q$

ANSWER: D

1. Let the straight line $x = b$ divide the area enclosed by $y = ( 1 - x ) ^ { 2 } , y = 0$, and $x = 0$ into two parts $R _ { 1 } ( 0 \leq x \leq b )$ and $R _ { 2 } ( b \leq x \leq 1 )$ such that $R _ { 1 } - R _ { 2 } = \frac { 1 } { 4 }$. Then $b$ equals
   (A) $\frac { 3 } { 4 }$
   (B) $\frac { 1 } { 2 }$
   (C) $\frac { 1 } { 3 }$
   (D) $\frac { 1 } { 4 }$

ANSWER:B

1. Let $\alpha$ and $\beta$ be the roots of $x ^ { 2 } - 6 x - 2 = 0$, with $\alpha > \beta$. If $a _ { n } = \alpha ^ { n } - \beta ^ { n }$ for $n \geq 1$, then the value of $\frac { a _ { 10 } - 2 a _ { 8 } } { 2 a _ { 9 } }$ is
   (A) 1
   (B) 2
   (C) 3
   (D) 4

ANSWER: C 53. A straight line $L$ through the point $( 3 , - 2 )$ is inclined at an angle $60 ^ { \circ }$ to the line $\sqrt { 3 } x + y = 1$. If $L$ also intersects the $x$-axis, then the equation of $L$ is
(A) $y + \sqrt { 3 } x + 2 - 3 \sqrt { 3 } = 0$
(B) $y - \sqrt { 3 } x + 2 + 3 \sqrt { 3 } = 0$
(C) $\sqrt { 3 } y - x + 3 + 2 \sqrt { 3 } = 0$
(D) $\sqrt { 3 } y + x - 3 + 2 \sqrt { 3 } = 0$
ANSWER:B

SECTION - II (Total Marks : 16)

(Multiple Correct Answers Type)

This section contains 4 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE may be correct. 54. The vector(s) which is/are coplanar with vectors $\hat { i } + \hat { j } + 2 \hat { k }$ and $\hat { i } + 2 \hat { j } + \hat { k }$, and perpendicular to the vector $\hat { i } + \hat { j } + \hat { k }$ is/are
(A) $\hat { j } - \hat { k }$
(B) $- \hat { i } + \hat { j }$
(C) $\hat { i } - \hat { j }$
(D) $- \hat { j } + \hat { k }$

ANSWER: AD

1. Let $M$ and $N$ be two $3 \times 3$ non-singular skew-symmetric matrices such that $M N = N M$. If $P ^ { T }$ denotes the transpose of $P$, then $M ^ { 2 } N ^ { 2 } \left( M ^ { T } N \right) ^ { - 1 } \left( M N ^ { - 1 } \right) ^ { T }$ is equal to
   (A) $M ^ { 2 }$
   (B) $- N ^ { 2 }$
   (C) $- M ^ { 2 }$
   (D) $M N$

ANSWER : MARKS TO ALL

1. Let the eccentricity of the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ be reciprocal to that of the ellipse $x ^ { 2 } + 4 y ^ { 2 } = 4$. If the hyperbola passes through a focus of the ellipse, then
   (A) the equation of the hyperbola is $\frac { x ^ { 2 } } { 3 } - \frac { y ^ { 2 } } { 2 } = 1$
   (B) a focus of the hyperbola is $( 2,0 )$
   (C) the eccentricity of the hyperbola is $\sqrt { \frac { 5 } { 3 } }$
   (D) the equation of the hyperbola is $x ^ { 2 } - 3 y ^ { 2 } = 3$

ANSWER: BD 57. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function such that
$$f ( x + y ) = f ( x ) + f ( y ) , \quad \forall x , y \in \mathbb { R } .$$
If $f ( x )$ is differentiable at $x = 0$, then
(A) $f ( x )$ is differentiable only in a finite interval containing zero
(B) $f ( x )$ is continuous $\forall x \in \mathbb { R }$
(C) $f ^ { \prime } ( x )$ is constant $\forall x \in \mathbb { R }$
(D) $f ( x )$ is differentiable except at finitely many points
ANSWER: BC, BCD

SECTION - III (Total Marks : 15)

(Paragraph Type)

This section contains 2 paragraphs. Based upon one of the paragraphs 3 multiple choice questions and based on the other paragraph $\mathbf { 2 }$ multiple choice questions have to be answered. Each of these questions has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Paragraph for Question Nos. 58 to 60

Let $a , b$ and $c$ be three real numbers satisfying
$$\left[ \begin{array} { l l l } a & b & c \end{array} \right] \left[ \begin{array} { l l l } 1 & 9 & 7 \\ 8 & 2 & 7 \\ 7 & 3 & 7 \end{array} \right] = \left[ \begin{array} { l l l } 0 & 0 & 0 \end{array} \right]$$

1. If the point $P ( a , b , c )$, with reference to (E), lies on the plane $2 x + y + z = 1$, then the value of $7 a + b + c$ is
   (A) 0
   (B) 12
   (C) 7
   (D) 6

ANSWER: D

1. Let $\omega$ be a solution of $x ^ { 3 } - 1 = 0$ with $\operatorname { Im } ( \omega ) > 0$. If $a = 2$ with $b$ and $c$ satisfying (E), then the value of

$$\frac { 3 } { \omega ^ { a } } + \frac { 1 } { \omega ^ { b } } + \frac { 3 } { \omega ^ { c } }$$
is equal to
(A) - 2
(B) 2
(C) 3
(D) - 3

ANSWER: A

1. Let $b = 6$, with $a$ and $c$ satisfying (E). If $\alpha$ and $\beta$ are the roots of the quadratic equation $a x ^ { 2 } + b x + c = 0$, then

$$\sum _ { n = 0 } ^ { \infty } \left( \frac { 1 } { \alpha } + \frac { 1 } { \beta } \right) ^ { n }$$
is
(A) 6
(B) 7
(C) $\frac { 6 } { 7 }$
(D) $\infty$

ANSWER: B

Paragraph for Question Nos. 61 and 62

Let $U _ { 1 }$ and $U _ { 2 }$ be two urns such that $U _ { 1 }$ contains 3 white and 2 red balls, and $U _ { 2 }$ contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from $U _ { 1 }$ and put into $U _ { 2 }$. However, if tail appears then 2 balls are drawn at random from $U _ { 1 }$ and put into $U _ { 2 }$. Now 1 ball is drawn at random from $U _ { 2 }$. 61. The probability of the drawn ball from $U _ { 2 }$ being white is
(A) $\frac { 13 } { 30 }$
(B) $\frac { 23 } { 30 }$
(C) $\frac { 19 } { 30 }$
(D) $\frac { 11 } { 30 }$

ANSWER: B

1. Given that the drawn ball from $U _ { 2 }$ is white, the probability that head appeared on the coin is
   (A) $\frac { 17 } { 23 }$
   (B) $\frac { 11 } { 23 }$
   (C) $\frac { 15 } { 23 }$
   (D) $\frac { 12 } { 23 }$

ANSWER: D

SECTION - IV (Total Marks : 28)

(Integer Answer Type)

This section contains $\mathbf { 7 }$ questions. The answer to each of the questions is a single-digit integer, ranging from 0 to 9 . The bubble corresponding to the correct answer is to be darkened in the ORS. 63. Consider the parabola $y ^ { 2 } = 8 x$. Let $\Delta _ { 1 }$ be the area of the triangle formed by the end points of its latus rectum and the point $P \left( \frac { 1 } { 2 } , 2 \right)$ on the parabola, and $\Delta _ { 2 }$ be the area of the triangle formed by drawing tangents at $P$ and at the end points of the latus rectum. Then $\frac { \Delta _ { 1 } } { \Delta _ { 2 } }$ is

ANSWER:2

1. Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots , a _ { 100 }$ be an arithmetic progression with $a _ { 1 } = 3$ and $S _ { p } = \sum _ { i = 1 } ^ { p } a _ { i } , 1 \leq p \leq 100$. For any integer $n$ with $1 \leq n \leq 20$, let $m = 5 n$. If $\frac { S _ { m } } { S _ { n } }$ does not depend on $n$, then $a _ { 2 }$ is ANSWER : 3, 9, 3 \& 9 BOTH
2. The positive integer value of $n > 3$ satisfying the equation

$$\frac { 1 } { \sin \left( \frac { \pi } { n } \right) } = \frac { 1 } { \sin \left( \frac { 2 \pi } { n } \right) } + \frac { 1 } { \sin \left( \frac { 3 \pi } { n } \right) }$$
is

ANSWER: 7

1. Let $f : [ 1 , \infty ) \rightarrow [ 2 , \infty )$ be a differentiable function such that $f ( 1 ) = 2$. If

$$6 \int _ { 1 } ^ { x } f ( t ) d t = 3 x f ( x ) - x ^ { 3 }$$
for all $x \geq 1$, then the value of $f ( 2 )$ is

ANSWER : MARKS TO ALL

1. If $z$ is any complex number satisfying $| z - 3 - 2 i | \leq 2$, then the minimum value of $| 2 z - 6 + 5 i |$ is

ANSWER: 5

1. The minimum value of the sum of real numbers $a ^ { - 5 } , a ^ { - 4 } , 3 a ^ { - 3 } , 1 , a ^ { 8 }$ and $a ^ { 10 }$ with $a > 0$ is

ANSWER: 8

1. Let $f ( \theta ) = \sin \left( \tan ^ { - 1 } \left( \frac { \sin \theta } { \sqrt { \cos 2 \theta } } \right) \right)$, where $- \frac { \pi } { 4 } < \theta < \frac { \pi } { 4 }$. Then the value of

$$\frac { d } { d ( \tan \theta ) } ( f ( \theta ) )$$
is

ANSWER: 1

Let $\vec{a}, \vec{b}$, and $\vec{c}$ be three non-coplanar unit vectors such that the angle between every pair of them is $\frac{\pi}{3}$. If $\vec{a} \times \vec{b} + \vec{b} \times \vec{c} = p\vec{a} + q\vec{b} + r\vec{c}$, where $p$, $q$ and $r$ are scalars, then the value of $\frac{p^2 + 2q^2 + r^2}{q^2}$ is

Let $\triangle P Q R$ be a triangle. Let $\vec { a } = \overrightarrow { Q R } , \vec { b } = \overrightarrow { R P }$ and $\vec { c } = \overrightarrow { P Q }$. If $| \vec { a } | = 12 , | \vec { b } | = 4 \sqrt { 3 }$ and $\vec { b } \cdot \vec { c } = 24$, then which of the following is (are) true?
(A) $\frac { | \vec { c } | ^ { 2 } } { 2 } - | \vec { a } | = 12$
(B) $\frac { | \vec { c } | ^ { 2 } } { 2 } + | \vec { a } | = 30$
(C) $| \vec { a } \times \vec { b } + \vec { c } \times \vec { a } | = 48 \sqrt { 3 }$
(D) $\vec { a } \cdot \vec { b } = - 72$

Column I
(A) In $\mathbb { R } ^ { 2 }$, if the magnitude of the projection vector of the vector $\alpha \hat { i } + \beta \hat { j }$ on $\sqrt { 3 } \hat { i } + \hat { j }$ is $\sqrt { 3 }$ and if $\alpha = 2 + \sqrt { 3 } \beta$, then possible value(s) of $| \alpha |$ is (are)
(B) Let $a$ and $b$ be real numbers such that the function $$f ( x ) = \left\{ \begin{array} { c c } - 3 a x ^ { 2 } - 2 , & x < 1 \\ b x + a ^ { 2 } , & x \geq 1 \end{array} \right.$$ is differentiable for all $x \in \mathbb { R }$. Then possible value(s) of $a$ is (are)
(C) Let $\omega \neq 1$ be a complex cube root of unity. If $\left( 3 - 3 \omega + 2 \omega ^ { 2 } \right) ^ { 4 n + 3 } + \left( 2 + 3 \omega - 3 \omega ^ { 2 } \right) ^ { 4 n + 3 } + \left( - 3 + 2 \omega + 3 \omega ^ { 2 } \right) ^ { 4 n + 3 } = 0$, then possible value(s) of $n$ is (are)
(D) Let the harmonic mean of two positive real numbers $a$ and $b$ be 4. If $q$ is a positive real number such that $a , 5 , q , b$ is an arithmetic progression, then the value(s) of $| q - a |$ is (are) Column II (P) 1 (Q) 2 (R) 3 (S) 4 (T) 5

Let $O$ be the origin and let $P Q R$ be an arbitrary triangle. The point $S$ is such that
$$\overrightarrow { O P } \cdot \overrightarrow { O Q } + \overrightarrow { O R } \cdot \overrightarrow { O S } = \overrightarrow { O R } \cdot \overrightarrow { O P } + \overrightarrow { O Q } \cdot \overrightarrow { O S } = \overrightarrow { O Q } \cdot \overrightarrow { O R } + \overrightarrow { O P } \cdot \overrightarrow { O S }$$
Then the triangle $P Q R$ has $S$ as its
[A] centroid
[B] circumcentre
[C] incentre
[D] orthocenter

Let $\vec { a }$ and $\vec { b }$ be two unit vectors such that $\vec { a } \cdot \vec { b } = 0$. For some $x , y \in \mathbb { R }$, let $\vec { c } = x \vec { a } + y \vec { b } + ( \vec { a } \times \vec { b } )$. If $| \vec { c } | = 2$ and the vector $\vec { c }$ is inclined at the same angle $\alpha$ to both $\vec { a }$ and $\vec { b }$, then the value of $8 \cos ^ { 2 } \alpha$ is $\_\_\_\_$.

Let $\vec{a} = 2\hat{i} + \hat{j} - \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} + \hat{k}$ be two vectors. Consider a vector $\vec{c} = \alpha\vec{a} + \beta\vec{b}$, $\alpha, \beta \in \mathbb{R}$. If the projection of $\vec{c}$ on the vector $(\vec{a} + \vec{b})$ is $3\sqrt{2}$, then the minimum value of $(\vec{c} - (\vec{a} \times \vec{b})) \cdot \vec{c}$ equals

Let $a$ and $b$ be positive real numbers. Suppose $\overrightarrow{PQ} = a\hat{i} + b\hat{j}$ and $\overrightarrow{PS} = a\hat{i} - b\hat{j}$ are adjacent sides of a parallelogram $PQRS$. Let $\vec{u}$ and $\vec{v}$ be the projection vectors of $\vec{w} = \hat{i} + \hat{j}$ along $\overrightarrow{PQ}$ and $\overrightarrow{PS}$, respectively. If $|\vec{u}| + |\vec{v}| = |\vec{w}|$ and if the area of the parallelogram $PQRS$ is 8, then which of the following statements is/are TRUE?
(A) $a + b = 4$
(B) $a - b = 2$
(C) The length of the diagonal $PR$ of the parallelogram $PQRS$ is 4
(D) $\vec{w}$ is an angle bisector of the vectors $\overrightarrow{PQ}$ and $\overrightarrow{PS}$

In a triangle $P Q R$, let $\vec { a } = \overrightarrow { Q R } , \vec { b } = \overrightarrow { R P }$ and $\vec { c } = \overrightarrow { P Q }$. If
$$| \vec { a } | = 3 , \quad | \vec { b } | = 4 \quad \text { and } \quad \frac { \vec { a } \cdot ( \vec { c } - \vec { b } ) } { \vec { c } \cdot ( \vec { a } - \vec { b } ) } = \frac { | \vec { a } | } { | \vec { a } | + | \vec { b } | }$$
then the value of $| \vec { a } \times \vec { b } | ^ { 2 }$ is $\_\_\_\_$

For any two points $M$ and $N$ in the $XY$-plane, let $\overrightarrow { MN }$ denote the vector from $M$ to $N$, and $\overrightarrow { 0 }$ denote the zero vector. Let $P , Q$ and $R$ be three distinct points in the $XY$-plane. Let $S$ be a point inside the triangle $\triangle PQR$ such that
$$\overrightarrow { SP } + 5 \overrightarrow { SQ } + 6 \overrightarrow { SR } = \overrightarrow { 0 }$$
Let $E$ and $F$ be the mid-points of the sides $PR$ and $QR$, respectively. Then the value of
$$\frac { \text { length of the line segment } EF } { \text { length of the line segment } ES }$$
is $\_\_\_\_$ .

Consider the vectors
$$\vec { x } = \hat { \imath } + 2 \hat { \jmath } + 3 \hat { k } , \quad \vec { y } = 2 \hat { \imath } + 3 \hat { \jmath } + \hat { k } , \quad \text { and } \quad \vec { z } = 3 \hat { \imath } + \hat { \jmath } + 2 \hat { k }$$
For two distinct positive real numbers $\alpha$ and $\beta$, define
$$\vec { X } = \alpha \vec { x } + \beta \vec { y } - \vec { z } , \quad \vec { Y } = \alpha \vec { y } + \beta \vec { z } - \vec { x } , \quad \text { and } \quad \vec { Z } = \alpha \vec { z } + \beta \vec { x } - \vec { y }$$
If the vectors $\vec { X } , \vec { Y }$, and $\vec { Z }$ lie in a plane, then the value of $\alpha + \beta - 3$ is $\_\_\_\_$.

If $\hat { u }$ and $\hat { v }$ are unit vectors and $\theta$ is the acute angle between them, then $2 \hat { u } \times 3 \hat { v }$ is a unit vector for
(1) exactly two values of $\theta$
(2) more than two values of $\theta$
(3) no value of $\theta$
(4) exactly one value of $\theta$

Let $\hat{a}$ and $\hat{b}$ be two unit vectors. If the vectors $\vec{c} = \hat{a} + 2\hat{b}$ and $\vec{d} = 5\hat{a} - 4\hat{b}$ are perpendicular to each other, then the angle between $\hat{a}$ and $\hat{b}$ is
(1) $\frac{\pi}{6}$
(2) $\frac{\pi}{2}$
(3) $\frac{\pi}{3}$
(4) $\frac{\pi}{4}$

If $| \vec { a } | = 2 , | \vec { b } | = 3$ and $| \overrightarrow { 2 a } - \vec { b } | = 5$, then $| \overrightarrow { 2 a } + \vec { b } |$ equals:
(1) 5
(2) 7
(3) 17
(4) 1

If $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors satisfying $|\vec{a} - \vec{b}|^2 + |\vec{b} - \vec{c}|^2 + |\vec{c} - \vec{a}|^2 = 9$, then $|2\vec{a} + 5\vec{b} + 5\vec{c}|$ is:
(1) $3$
(2) $\sqrt{10}$
(3) $2$
(4) $\sqrt{5}$

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) $5$

Two vectors $\vec { A }$ and $\vec { B }$ have equal magnitudes. The magnitude of $( \vec { A } + \vec { B } )$ is ' $n$ ' times the magnitude of $( \vec { A } - \vec { B } )$. The angle between $\vec { A }$ and $\vec { B }$ is:
(1) $\cos ^ { - 1 } \left[ \frac { n ^ { 2 } - 1 } { n ^ { 2 } + 1 } \right]$
(2) $\sin ^ { - 1 } \left[ \frac { n - 1 } { n + 1 } \right]$
(3) $\cos ^ { - 1 } \left[ \frac { n - 1 } { n + 1 } \right]$
(4) $\sin ^ { - 1 } \left[ \frac { n ^ { 2 } - 1 } { n ^ { 2 } + 1 } \right]$

The sum of two forces $\overrightarrow { \mathrm { P } }$ and $\overrightarrow { \mathrm { Q } }$ is $\overrightarrow { \mathrm { R } }$ such that $| \overrightarrow { \mathrm { R } } | = | \overrightarrow { \mathrm { P } } |$. Find the angle between resultant of $2 \overrightarrow { \mathrm { P } }$ and $\overrightarrow { \mathrm { Q } }$ and $\overrightarrow { \mathrm { Q } }$.

Let $A(1,0)$, $B(6,2)$ and $C \left( \frac { 3 } { 2 } , 6 \right)$ be the vertices of a triangle $ABC$. If $P$ is a point inside the triangle $ABC$ such that the triangles $APC$, $APB$ and $BPC$ have equal areas, then the length of the line segment $PQ$, where $Q$ is the point $\left( - \frac { 7 } { 6 } , - \frac { 1 } { 3 } \right)$, is

Let the vectors $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ be such that $|\overrightarrow{\mathrm{a}}| = 2$, $|\overrightarrow{\mathrm{b}}| = 4$ and $|\overrightarrow{\mathrm{c}}| = 4$. If the projection of $\overrightarrow{\mathrm{b}}$ on $\overrightarrow{\mathrm{a}}$ is equal to the projection of $\overrightarrow{\mathrm{c}}$ on $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ is perpendicular to $\overrightarrow{\mathrm{c}}$, then the value of $|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} - \overrightarrow{\mathrm{c}}|$ is ...

Assertion $A$ : If $A , B , C , D$ are four points on a semi-circular arc with a centre at $O$ such that $| \overrightarrow { A B } | = | \overrightarrow { B C } | = | \overrightarrow { C D } |$. Then, $\overrightarrow { A B } + \overrightarrow { A C } + \overrightarrow { A D } = 4 \overrightarrow { A O } + \overrightarrow { O B } + \overrightarrow { O C }$
Reason $R$ : Polygon law of vector addition yields $\overrightarrow { A B } + \overrightarrow { B C } + \overrightarrow { C D } + \overrightarrow { A D } = 2 \overrightarrow { A O }$
In the light of the above statements, choose the most appropriate answer from the options given below.
(1) $A$ is correct but $R$ is not correct.
(2) $A$ is not correct but $R$ is correct.
(3) Both $A$ and $R$ are correct and $R$ is the correct explanation of $A$.
(4) Both $A$ and $R$ are correct but $R$ is not the correct explanation of $A$.

Three particles $P , Q$ and $R$ are moving along the vectors $\vec { A } = \hat { \mathrm { i } } + \hat { \mathrm { j } } , \overrightarrow { \mathrm { B } } = \hat { \mathrm { j } } + \widehat { \mathrm { k } }$ and $\vec { C } = - \hat { \mathrm { i } } + \hat { \mathrm { j } }$, respectively. They strike on a point and start to move in different directions. Now particle $P$ is moving normal to the plane which contains vector $\vec { A }$ and $\vec { B }$. Similarly particle $Q$ is moving normal to the plane which contains vector $\vec { A }$ and $\vec { C }$. The angle between the direction of motion of $P$ and $Q$ is $\cos ^ { - 1 } \left( \frac { 1 } { \sqrt { x } } \right)$. Then the value of $x$ is $\_\_\_\_$ .

Let a vector $\alpha \hat { \mathrm { i } } + \beta \hat { \mathrm { j } }$ be obtained by rotating the vector $\sqrt { 3 } \hat { \mathrm { i } } + \hat { \mathrm { j } }$ by an angle $45 ^ { \circ }$ about the origin in counterclockwise direction in the first quadrant. Then the area (in sq. units) of triangle having vertices $( \alpha , \beta ) , ( 0 , \beta )$ and $( 0,0 )$ is equal to
(1) $\frac { 1 } { 2 }$
(2) 1
(3) $\frac { 1 } { \sqrt { 2 } }$
(4) $2 \sqrt { 2 }$

In a triangle $ABC$, if $| \overrightarrow { BC } | = 3 , | \overrightarrow { CA } | = 5$ and $| \overrightarrow { BA } | = 7$, then the projection of the vector $\overrightarrow { BA }$ on $\overrightarrow { BC }$ is equal to
(1) $\frac { 19 } { 2 }$
(2) $\frac { 13 } { 2 }$
(3) $\frac { 11 } { 2 }$
(4) $\frac { 15 } { 2 }$

Let $O$ be the origin. Let $\overrightarrow { O P } = x \hat { i } + y \hat { j } - \widehat { k }$ and $\overrightarrow { O Q } = - \hat { i } + 2 \hat { j } + 3 x \hat { k } , x , y \in R , x > 0$, be such that $| \overrightarrow { P Q } | = \sqrt { 20 }$ and the vector $\overrightarrow { O P }$ is perpendicular to $\overrightarrow { O Q }$. If $\overrightarrow { O R } = 3 \hat { i } + \mathrm { z } \hat { j } - 7 \hat { k } , z \in R$, is coplanar with $\overrightarrow { O P }$ and $\overrightarrow { O Q }$, then the value of $x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ is equal to
(1) 7
(2) 9
(3) 2
(4) 1