In the orthonormal frame $(A ; \overrightarrow{AB} ; \overrightarrow{AD} ; \overrightarrow{AE})$, $ABCDEFGH$ denotes a cube with side length 1.
a. The dot product $\overrightarrow{AF} \cdot \overrightarrow{BG}$ is equal to 0. b. The dot product $\overrightarrow{AF} \cdot \overrightarrow{BG}$ is equal to $(-1)$. c. The dot product $\overrightarrow{AF} \cdot \overrightarrow{BG}$ is equal to 1. d. The dot product $\overrightarrow{AF} \cdot \overrightarrow{BG}$ is equal to 2.

In space, with respect to an orthonormal coordinate system, we consider the points $\mathrm { A } ( 1 ; - 1 ; - 1 )$, $\mathrm { B } ( 1 ; 1 ; 1 ) , \mathrm { C } ( 0 ; 3 ; 1 )$ and the plane $\mathscr { P }$ with equation $2 x + y - z + 5 = 0$.
A measure of the angle $\widehat { \mathrm { BAC } }$ rounded to the nearest tenth of a degree is equal to : a. $22,2 ^ { \circ }$ b. $0,4 ^ { \circ }$ c. $67,8 ^ { \circ }$ d. $1,2 ^ { \circ }$

In coordinate space, let B be the foot of the perpendicular from the point $\mathrm { A } ( 3,6,0 )$ to the plane $\sqrt { 3 } y - z = 0$. Find the value of $\overrightarrow { \mathrm { OA } } \cdot \overrightarrow { \mathrm { OB } }$. (Here, O is the origin.) [4 points]

In coordinate space, the figure $S$ is formed by the intersection of the sphere $( x - 1 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - 1 ) ^ { 2 } = 9$ with center C and the plane $x + y + z = 6$. For two points $\mathrm { P } , \mathrm { Q }$ on figure $S$, what is the minimum value of the dot product $\overrightarrow { \mathrm { CP } } \cdot \overrightarrow { \mathrm { CQ } }$ of the two vectors $\overrightarrow { \mathrm { CP } } , \overrightarrow { \mathrm { CQ } }$? [4 points]
(1) - 3
(2) - 2
(3) - 1
(4) 1
(5) 2

In coordinate space, there are two points $\mathrm { P } , \mathrm { Q }$ moving on the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$. Let $\mathrm { P } _ { 1 } , \mathrm { Q } _ { 1 }$ be the feet of the perpendiculars from points P and Q to the plane $y = 4$ respectively, and let $\mathrm { P } _ { 2 } , \mathrm { Q } _ { 2 }$ be the feet of the perpendiculars to the plane $y + \sqrt { 3 } z + 8 = 0$ respectively. Find the maximum value of $2 | \overrightarrow { \mathrm { PQ } } | ^ { 2 } - \left| \overrightarrow { \mathrm { P } _ { 1 } \mathrm { Q } _ { 1 } } \right| ^ { 2 } - \left| \overrightarrow { \mathrm { P } _ { 2 } \mathrm { Q } _ { 2 } } \right| ^ { 2 }$. [4 points]

For two points $\mathrm { A } ( 2 , \sqrt { 2 } , \sqrt { 3 } )$ and $\mathrm { B } ( 1 , - \sqrt { 2 } , 2 \sqrt { 3 } )$ in coordinate space, point P satisfies the following conditions. (가) $| \overrightarrow { \mathrm { AP } } | = 1$ (나) The angle between $\overrightarrow { \mathrm { AP } }$ and $\overrightarrow { \mathrm { AB } }$ is $\frac { \pi } { 6 }$.
For point Q on a sphere centered at the origin with radius 1, the maximum value of $\overrightarrow { \mathrm { AP } } \cdot \overrightarrow { \mathrm { AQ } }$ is $a + b \sqrt { 33 }$. Find the value of $16 \left( a ^ { 2 } + b ^ { 2 } \right)$. (Here, $a$ and $b$ are rational numbers.) [4 points]

In coordinate space, let $\vec { a } , \vec { b } , \vec { c }$ be the position vectors of three points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ with respect to the origin. The dot products between these vectors are shown in the following table.

$\cdot$	$\vec { a }$	$\vec { b }$	$\vec { c }$
$\vec { a }$	2	1	$- \sqrt { 2 }$
$\vec { b }$	1	2	0
$\vec { c }$	$- \sqrt { 2 }$	0	2

For example, $\vec { a } \cdot \vec { c } = - \sqrt { 2 }$. Which of the following correctly shows the order of the distances between the three points $\mathrm { A } , \mathrm { B } , \mathrm { C }$? [4 points]
(1) $\overline { \mathrm { AB } } < \overline { \mathrm { AC } } < \overline { \mathrm { BC } }$
(2) $\overline { \mathrm { AB } } < \overline { \mathrm { BC } } < \overline { \mathrm { AC } }$
(3) $\overline { \mathrm { AC } } < \overline { \mathrm { AB } } < \overline { \mathrm { BC } }$
(4) $\overline { \mathrm { BC } } < \overline { \mathrm { AB } } < \overline { \mathrm { AC } }$
(5) $\overline { \mathrm { BC } } < \overline { \mathrm { AC } } < \overline { \mathrm { AB } }$

In a regular tetrahedron ABCD with edge length 4, let O be the centroid of triangle ABC and P be the midpoint of segment AD. For a point Q on face BCD of the regular tetrahedron ABCD, when the two vectors $\overrightarrow { \mathrm { OQ } }$ and $\overrightarrow { \mathrm { OP } }$ are perpendicular to each other, the maximum value of $| \overrightarrow { \mathrm { PQ } } |$ is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [4 points]

13. In the Cartesian coordinate plane, the hyperbola $\Gamma$ is centered at the origin with one focus at $( \sqrt { 5 } , 0 )$ . $\overrightarrow { e _ { 1 } } = ( 2,1 )$ and $\overrightarrow { e _ { 2 } } = ( 2 , - 1 )$ are direction vectors of the two asymptotes respectively. For any point $P$ on the hyperbola $\Gamma$ , if $\overrightarrow { O P } = a \overrightarrow { e _ { 1 } } + b \overrightarrow { e _ { 2 } } ( a , b \in \mathbf { R } )$ , then an equation satisfied by $a$ and $b$ is $\_\_\_\_$.

22. (Total Score: 16 points) Subproblem 1: 3 points, Subproblem 2: 5 points, Subproblem 3: 8 points.
If real numbers $x , y , m$ satisfy $| x - m | < | y - m |$ , then $x$ is said to be closer to $m$ than $y$ is.
(1) If $x ^ { 2 } - 1$ is closer to 3 than to 0, find the range of $x$;
(2) For any two distinct positive numbers $a , b$ , prove that $a ^ { 2 } b + a b ^ { 2 }$ is closer to $2 a b \sqrt { a b }$ than $a ^ { 3 } + b ^ { 3 }$ is;
(3) Given that the domain of function $f ( x )$ is $D = \{ x \mid x \neq k \pi , k \in \mathbf { Z } , x \in \mathbf { R } \}$ . For any $x \in D$ , $f ( x )$ equals whichever of $1 + \sin x$ and $1 - \sin x$ is closer to 0. Write the analytical expression for $f ( x )$ and indicate its parity, minimum positive period, minimum value, and monotonicity (proofs of conclusions are not required).

13. As shown in the figure, the line $x = 2$ intersects the asymptotes of the hyperbola $\Gamma : \frac { x ^ { 2 } } { 4 } - y ^ { 2 } = 1$ at points $E _ { 1 }$ and $E _ { 2 }$. Let $\overrightarrow { O E _ { 1 } } = \overrightarrow { e _ { 1 } }$ and $\overrightarrow { O E _ { 2 } } = \overrightarrow { e _ { 2 } }$. For any point $P$ on the hyperbola $\Gamma$, if $\overrightarrow { O P } = a \overrightarrow { e _ { 1 } } + b \overrightarrow { e _ { 2 } }$ ($a , b \in \mathbb{R}$), then $a$ and $b$ satisfy the equation $\_\_\_\_$ $4 a b = 1$.
Analysis: $E _ { 1 } ( 2,1 )$, $E _ { 2 } ( 2 , - 1 )$
$$\overrightarrow { O P } = a \overrightarrow { e _ { 1 } } + b \overrightarrow { e _ { 2 } } = ( 2 a + 2 b , a - b ), \text{ point } P \text{ is on the hyperbola}$$
$\therefore \frac { ( 2 a + 2 b ) ^ { 2 } } { 4 } - ( a - b ) ^ { 2 } = 1$, which simplifies to $4 a b = 1$ [Figure]

133- Three points $A(2,1,\circ)$, $B(3,-1,2)$, $C(-1,1,3)$ are vertices of a triangle. What is $\cos A$?
(1) $\dfrac{\sqrt{2}}{6}$ (2) $\dfrac{\sqrt{2}}{4}$ (3) $\dfrac{\sqrt{2}}{6}$ (4) $\dfrac{\sqrt{3}}{4}$

136- For which value of $m$, the three vectors $\vec{a} = (-1,2,3)$, $\vec{b} = (2,\circ,1)$, $\vec{c} = (-4,m,5)$ are coplanar?
(1) $-2$ (2) $2$ (3) $3$ (4) $4$

If \overrightarrow { \mathrm { u } } , \mathrm { v } ^ { \rightarrow } , \mathrm { w } ^ { \rightarrow } are three noncoplanar unit vectors and $\alpha , \beta , \gamma$ are the angles between $\mathrm { u } \rightarrow$ are and $\mathrm { v } ^ { \rightarrow } , \mathrm { v } ^ { \rightarrow } are and $\overrightarrow { \mathrm { w } } , \mathrm { w } ^ { \rightarrow } and $\overrightarrow { \mathrm { u } }$ are respectively and $\overrightarrow { \mathrm { x } } , \mathrm { y } ^ { \rightarrow } , \mathrm { z }$ are unit vectors along the bisectors of the angles $\mathrm { a } , \mathrm { b } , \mathrm { g }$ respectively. Prove that $$\left[ \begin{array} { l l l } \vec { x } \times \vec { y } & \vec { y } \times \vec { z } & \vec { z } \times \vec { x } \end{array} \right] = \frac { 1 } { 16 } \left[ \begin{array} { l l l } \vec { u } & \vec { v } & \vec { w } \end{array} \right] ^ { 2 } \sec ^ { 2 } \frac { \alpha } { 2 } \sec ^ { 2 } \frac { \beta } { 2 } \sec ^ { 2 } \frac { \gamma } { 2 }$$

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

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) $\frac { 7 \hat { i } - 7 \hat { j } - \hat { k } } { \sqrt { 99 } }$

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 $\overrightarrow { \mathrm { a } }$ 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

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

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

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 $O$ 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] $\sin ( Q + R )$