In rectangular parallelepiped $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$, $A B = B C = 2$, and the angle between $A C _ { 1 }$ and plane $B B _ { 1 } C _ { 1 } C$ is $30 ^ { \circ }$. Then the volume of the rectangular parallelepiped is
A. 8
B. $6 \sqrt { 2 }$
C. $8 \sqrt { 2 }$
D. $8 \sqrt { 3 }$

8. As shown in the figure, point $N$ is the center of square $A B C D$ , $\triangle E C D$ is an equilateral triangle, plane $E C D \perp$ plane $A B C D$ , and $M$ is the midpoint of segment $E D$ . Then [Figure]
A. $B M = E N$ , and lines $B M$ and $E N$ are intersecting lines
B. $B M \neq E N$ , and lines $B M$ and $E N$ are intersecting lines
C. $B M = E N$ , and lines $B M$ and $E N$ are skew lines
D. $B M \neq E N$ , and lines $B M$ and $E N$ are skew lines

8. As shown in the figure, point $N$ is the center of square $A B C D$ , $\triangle E C D$ is an equilateral triangle, plane $E C D \perp$ plane $A B C D$ , and $M$ is the midpoint of segment $E D$ . Then [Figure]
A. $B M = E N$ , and lines $B M$ and $E N$ are intersecting lines
B. $B M \neq E N$ , and lines $B M$ and $E N$ are intersecting lines
C. $B M = E N$ , and lines $B M$ and $E N$ are skew lines
D. $B M \neq E N$ , and lines $B M$ and $E N$ are skew lines

As shown in the figure, in the quadrangular pyramid $P - A B C D$, $P A \perp$ plane $A B C D$, $A D \perp C D$, $A D \| B C$, $P A = A D = C D = 2$, $B C =$ 3. $E$ is the midpoint of $P D$, and point $F$ is on $P C$ such that $\frac { P F } { P C } = \frac { 1 } { 3 }$. (I) Prove that: $C D \perp$ plane $P A D$; (II) Find the cosine of the dihedral angle $F - A E - P$; (III) Let point $G$ be on $P B$ such that $\frac { P G } { P B } = \frac { 2 } { 3 }$. Determine whether line $A G$ lies in plane $A E F$, and explain the reason.

19. (12 points) Figure 1 is a planar figure composed of rectangle $A D E B$ , right triangle $A B C$ , and rhombus $B F G C$ , where $A B = 1 , B E = B F = 2$ , $\angle F B C = 60 ^ { \circ }$ . Fold it along $A B$ and $B C$ so that $B E$ and $B F$ coincide, and connect $D G$ , as shown in Figure 2.
(1) Prove: In Figure 2, points $A , C , G , D$ are coplanar, and plane $A B C \perp$ plane $B C G E$ .
(2) Therefore, from the known condition we have $( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - a ) ^ { 2 } \geq \frac { ( 2 + a ) ^ { 2 } } { 3 }$ , equality holds if and only if $x = \frac { 4 - a } { 3 } , y = \frac { 1 - a } { 3 } , z = \frac { 2 a - 2 } { 3 }$ . Thus the minimum value of $( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - a ) ^ { 2 }$ is $\frac { ( 2 + a ) ^ { 2 } } { 3 }$ .
From the given condition we have $\frac { ( 2 + a ) ^ { 2 } } { 3 } \geq \frac { 1 } { 3 }$ , solving gives $a \leq - 3$ or $a \geq - 1$ .

In a cube $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$ with edge length 10, $P$ is a point on the left face $A D D _ { 1 } A _ { 1 }$. Given that the distance from point $P$ to $A _ { 1 } D _ { 1 }$ is 3 and the distance from point $P$ to $A A _ { 1 }$ is 2, a line through point $P$ parallel to $A _ { 1 } C$ intersects the cube at points $P$ and $Q$. On which face of the cube is point $Q$ located? ( )
A. $A A _ { 1 } B _ { 1 } B$
B. $B B _ { 1 } C _ { 1 } C$
C. $C C _ { 1 } D _ { 1 } D$
D. $A B C D$

19. In the quadrangular pyramid $Q - A B C D$ , the base $A B C D$ is a square with $A D = 2$ , $Q D = Q A = \sqrt { 5 }$ , $Q C = 3$ . [Figure]
(1) Prove: plane $Q A D \perp$ plane $A B C D$ ;
(2) Find the cosine of the dihedral angle $B - Q D - A$ . Answer: (1) See proof below; (2) $\frac { 2 } { 3 }$ .

[Solution]

[Analysis] (1) Let $O$ be the midpoint of $A D$ , and connect $Q O$ and $C O$ . We can prove that $Q O \perp$ plane $A B C D$ , thus obtaining plane $Q A D \perp$ plane $A B C D$ .
(2) In plane $A B C D$ , through $O$ draw $O T \parallel C D$ , intersecting $B C$ at $T$ . Then $O T \perp A D$ . Establish a coordinate system as shown in the figure. After finding the normal vectors of planes $Q A D$ and $B Q D$ , we can find the cosine of the dihedral angle.

[Detailed Solution]

[Figure]
(1) Let $O$ be the midpoint of $A D$ , and connect $Q O$ and $C O$ . Since $Q A = Q D$ and $O A = O D$ , we have $Q O \perp A D$ . Since $A D = 2$ and $Q A = \sqrt { 5 }$ , we have $Q O = \sqrt { 5 - 1 } = 2$ . In square $A B C D$ , since $A D = 2$ , we have $D O = 1$ , thus $C O = \sqrt { 1 + 4 } = \sqrt { 5 }$ . Since $Q C = 3$ , we have $Q C ^ { 2 } = 9 = 4 + 5 = Q O ^ { 2 } + O C ^ { 2 }$ , so $\triangle Q O C$ is a right triangle with $Q O \perp O C$ . Since $O C \cap A D = O$ , we have $Q O \perp$ plane $A B C D$ . Since $Q O \subset$ plane $Q A D$ , we have plane $Q A D \perp$ plane $A B C D$ .
(2) In plane $A B C D$ , through $O$ draw $O T \parallel C D$ , intersecting $B C$ at $T$ . Then $O T \perp A D$ . Combined with $Q O \perp$ plane $A B C D$ from part (1), we can establish a coordinate system as shown in the figure. [Figure]
Then $D ( 0,1,0 ) , Q ( 0,0,2 ) , B ( 2 , -1,0 )$ , so $\overrightarrow { B Q } = ( -2,1,2 ) , \overrightarrow { B D } = ( -2,2,0 )$ . Let the normal vector of plane $Q B D$ be $\vec { n } = ( x , y , z )$ . Then $\left\{ \begin{array} { l } \vec { n } \cdot \overrightarrow { B Q } = 0 \\ \vec { n } \cdot \overrightarrow { B D } = 0 \end{array} \right.$ , i.e., $\left\{ \begin{array} { l } - 2 x + y + 2 z = 0 \\ - 2 x + 2 y = 0 \end{array} \right.$ . Taking $x = 1$ , we get $y = 1 , z = \frac { 1 } { 2 }$ , Thus $\vec { n } = \left( 1,1 , \frac { 1 } { 2 } \right)$ . The normal vector of plane $Q A D$ is $\vec { m } = ( 1,0,0 )$ . Therefore $\cos \langle \vec { m } , \vec { n } \rangle = \frac { |\vec{m} \cdot \vec{n}| } { |\vec{m}| \cdot |\vec{n}| } = \frac

In the right triangular prism $ABC - A_1B_1C_1$, let $D$ be the midpoint of $BC$. Then
A. $AD \perp A_1C$
B. $BC \perp$ plane $AA_1D$
C. $CC_1 \parallel$ plane $AA_1D$
D. $AD \parallel A_1B_1$

A regular tetrahedron has all its vertices on a sphere of radius $R$. Then the length of each edge of the tetrahedron is
(a) $( \sqrt{2} / \sqrt{3} ) R$
(b) $( \sqrt{3} / 2 ) R$
(c) $( 4 / 3 ) R$
(d) $( 2 \sqrt{2} / \sqrt{3} ) R$

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) $\frac { 17 } { 5 \sqrt { 3 } }$

A line with positive direction cosines passes through the point $P(2,-1,2)$ and makes equal angles with the coordinate axes. The line meets the plane $$2x+y+z=9$$ at point $Q$. The length of the line segment $PQ$ equals
(A) 1
(B) $\sqrt{2}$
(C) $\sqrt{3}$
(D) 2

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) $\frac { 4 \sqrt { 5 } } { 9 }$

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

Two lines $L _ { 1 } : x = 5 , \frac { y } { 3 - \alpha } = \frac { z } { - 2 }$ and $L _ { 2 } : x = \alpha , \frac { y } { - 1 } = \frac { z } { 2 - \alpha }$ are coplanar. Then $\alpha$ can take value(s)
(A) 1
(B) 2
(C) 3
(D) 4

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) $-\sqrt{2}$

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 $P$ be the image of the point $( 3,1,7 )$ with respect to the plane $x - y + z = 3$. Then the equation of the plane passing through $P$ and containing the straight line $\frac { x } { 1 } = \frac { y } { 2 } = \frac { z } { 1 }$ is
(A) $x + y - 3 z = 0$
(B) $3 x + z = 0$
(C) $x - 4 y + 7 z = 0$
(D) $2 x - y = 0$

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 $2 \left| u _ { 1 } \right| = \left| u _ { 3 } \right|$

The equation of the plane passing through the point $( 1,1,1 )$ and perpendicular to the planes $2 x + y - 2 z = 5$ and $3 x - 6 y - 2 z = 7$, is
[A] $14 x + 2 y - 15 z = 1$
[B] $14 x - 2 y + 15 z = 27$
[C] $14 x + 2 y + 15 z = 31$
[D] $- 14 x + 2 y + 15 z = 3$

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 $O$ be the origin, and $\overrightarrow { O X } , \overrightarrow { O Y } , \overrightarrow { O Z }$ be three unit vectors in the directions of the sides $\overrightarrow { Q R } , \overrightarrow { R P }$, $\overrightarrow { P Q }$, respectively, of a triangle $P Q R$.
$| \overrightarrow { O X } \times \overrightarrow { O Y } | =$
[A] $\sin ( P + Q )$
[B] $\sin 2 R$
[C] $\sin ( P + R )$
[D] $\sin ( Q + R )$

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 $\frac { 2 } { \sqrt { 3 } }$

Three lines $$\begin{array}{ll} L_1: & \vec{r} = \lambda\hat{i}, \lambda \in \mathbb{R}, \\ L_2: & \vec{r} = \hat{k} + \mu\hat{j}, \mu \in \mathbb{R} \text{ and} \\ L_3: & \vec{r} = \hat{i} + \hat{j} + v\hat{k}, v \in \mathbb{R} \end{array}$$ are given. For which point(s) $Q$ on $L_2$ can we find a point $P$ on $L_1$ and a point $R$ on $L_3$ so that $P$, $Q$ and $R$ are collinear?
(A) $\hat{k} - \frac{1}{2}\hat{j}$
(B) $\hat{k}$
(C) $\hat{k} + \frac{1}{2}\hat{j}$
(D) $\hat{k} + \hat{j}$