In sailing competitions, athletes can use anemometers to measure wind speed and direction. The measured result is called apparent wind speed in nautical science. The vector corresponding to apparent wind speed is the sum of the vector corresponding to true wind speed and the vector corresponding to ship's wind speed, where the vector corresponding to ship's wind speed has the same magnitude as the vector corresponding to ship's speed but opposite direction. Figure 1 shows the correspondence between part of the wind force levels, names, and wind speeds. A sailor measured the vectors corresponding to apparent wind speed and ship's speed at a certain moment as shown in Figure 2 (the magnitude of wind speed and the magnitude of the vector are the same, unit: m/s). Then the true wind is

Level	Wind Speed	Name
2	$1.1 \sim 3.3$	Light breeze
3	$3.4 \sim 5.4$	Gentle breeze
4	$5.5 \sim 7.9$	Moderate breeze
5	$8.0 \sim 10.1$	Fresh breeze

A. Light breeze
B. Gentle breeze
C. Moderate breeze
D. Fresh breeze

Given plane vectors $\vec{a} = (x, 1)$, $\vec{b} = (x-1, 2x)$. If $\vec{a} \perp (\vec{a} - \vec{b})$, then $|\vec{a}| = $ \_\_\_\_

Let $\lambda$ be a real such that $0 \leqslant \lambda \leqslant 1$. For $t$ in $\{-1, 1\}$, $Y_{t}$ denotes the projection of $X'$ onto $C_{t}$. Show that
$$d(X, C) \leqslant \left\|(1 - \lambda)(Y_{\varepsilon_{n}} + \varepsilon_{n} e_{n}) + \lambda(Y_{-\varepsilon_{n}} - \varepsilon_{n} e_{n}) - X\right\|$$

Let $n$ be an integer such that $n \geqslant 2$. We denote by $E' = \operatorname{Vect}(e_{1}, \ldots, e_{n-1})$ and by $\pi$ the orthogonal projection onto $E'$. We set $X' = \pi \circ X = \sum_{i=1}^{n-1} \varepsilon_{i} e_{i}$. For $t$ in $\{-1, 1\}$ we denote $C_{t} = \pi(C \cap H_{t})$ where $H_t = E' + te_n$. For $t$ in $\{-1, 1\}$, we denote by $Y_{t}$ the projection of $X'$ onto the non-empty closed convex set $C_{t}$. Let $\lambda$ be a real such that $0 \leqslant \lambda \leqslant 1$.
Show that
$$d(X, C) \leqslant \left\|(1 - \lambda)(Y_{\varepsilon_{n}} + \varepsilon_{n} e_{n}) + \lambda(Y_{-\varepsilon_{n}} - \varepsilon_{n} e_{n}) - X\right\|$$

Let $\lambda$ be a real such that $0 \leqslant \lambda \leqslant 1$. For $t$ in $\{-1, 1\}$, $Y_{t}$ denotes the projection of $X'$ onto $C_{t}$. Deduce that
$$d(X, C)^{2} \leqslant 4\lambda^{2} + \left\|(1 - \lambda)(Y_{\varepsilon_{n}} - X') + \lambda(Y_{-\varepsilon_{n}} - X')\right\|^{2}$$
then that
$$d(X, C)^{2} \leqslant 4\lambda^{2} + (1 - \lambda)\|Y_{\varepsilon_{n}} - X'\|^{2} + \lambda\|Y_{-\varepsilon_{n}} - X'\|^{2}$$
Thus, show the inequality
$$d(X, C)^{2} \leqslant 4\lambda^{2} + (1 - \lambda)d(X', C_{\varepsilon_{n}})^{2} + \lambda d(X', C_{-\varepsilon_{n}})^{2}$$

Let $n$ be an integer such that $n \geqslant 2$. We denote by $E' = \operatorname{Vect}(e_{1}, \ldots, e_{n-1})$ and by $\pi$ the orthogonal projection onto $E'$. We set $X' = \pi \circ X = \sum_{i=1}^{n-1} \varepsilon_{i} e_{i}$. For $t$ in $\{-1, 1\}$ we denote $C_{t} = \pi(C \cap H_{t})$ where $H_t = E' + te_n$. For $t$ in $\{-1, 1\}$, we denote by $Y_{t}$ the projection of $X'$ onto the non-empty closed convex set $C_{t}$. Let $\lambda$ be a real such that $0 \leqslant \lambda \leqslant 1$.
Deduce that
$$d(X, C)^{2} \leqslant 4\lambda^{2} + \left\|(1 - \lambda)(Y_{\varepsilon_{n}} - X') + \lambda(Y_{-\varepsilon_{n}} - X')\right\|^{2}$$
then that
$$d(X, C)^{2} \leqslant 4\lambda^{2} + (1 - \lambda) \|Y_{\varepsilon_{n}} - X'\|^{2} + \lambda \|Y_{-\varepsilon_{n}} - X'\|^{2}$$
Thus, we have shown the inequality
$$d(X, C)^{2} \leqslant 4\lambda^{2} + (1 - \lambda) d(X', C_{\varepsilon_{n}})^{2} + \lambda d(X', C_{-\varepsilon_{n}})^{2}$$

Suppose $ABCD$ is a parallelogram and $P, Q$ are points on the sides $BC$ and $CD$ respectively, such that $PB = \alpha BC$ and $DQ = \beta DC$. If the area of the triangles $ABP$, $ADQ$, $PCQ$ are 15, 15 and 4 respectively, then the area of $APQ$ is
(a) 14
(b) 15
(c) 16
(d) 18.

50. Let $\vec { a } , \vec { b } , \vec { c }$ be unit vectors such that $\vec { a } + \vec { b } + \vec { c } = \overrightarrow { 0 }$. Which one of the following is correct?
(A) $\vec { a } \times \vec { b } = \vec { b } \times \vec { c } = \vec { c } \times \vec { a } = \overrightarrow { 0 }$
(B) $\vec { a } \times \vec { b } = \vec { b } \times \vec { c } = \vec { c } \times \vec { a } \neq \overrightarrow { 0 }$
(C) $\vec { a } \times \vec { b } = \vec { b } \times \vec { c } = \vec { a } \times \vec { c } \neq \overrightarrow { 0 }$
(D) $\vec { a } \times \vec { b } , \vec { b } \times \vec { c } , \vec { c } \times \vec { a }$ are mutually perpendicular
Answer [Figure] [Figure] ◯ ◯
(A)
(B)
(C)
(D) 51. Let $A B C D$ be a quadrilateral with area 18, with side $A B$ parallel to the side $C D$ and $A B = 2 C D$. Let $A D$ be perpendicular to $A B$ and $C D$. If a circle is drawn inside the quadrilateral $A B C D$ touching all the sides, then its radius is
(A) 3
(B) 2
(C) $\frac { 3 } { 2 }$
(D) 1
Answer [Figure] [Figure] [Figure] [Figure]
(A)
(B)
(C)
(D) 52. Let $f ( x ) = \frac { x } { \left( 1 + x ^ { n } \right) ^ { 1 / n } }$ for $n \geq 2$ and $g ( x ) = \underbrace { f \circ f \circ \cdots \circ f ) } _ { f \text { occurs } n \text { times } } ( x )$. Then $\int x ^ { n - 2 } g ( x ) d x$ equals
(A) $\frac { 1 } { n ( n - 1 ) } \left( 1 + n x ^ { n } \right) ^ { 1 - \frac { 1 } { n } } + K$
(B) $\frac { 1 } { n - 1 } \left( 1 + n x ^ { n } \right) ^ { 1 - \frac { 1 } { n } } + K$
(C) $\frac { 1 } { n ( n + 1 ) } \left( 1 + n x ^ { n } \right) ^ { 1 + \frac { 1 } { n } } + K$
(D) $\frac { 1 } { n + 1 } \left( 1 + n x ^ { n } \right) ^ { 1 + \frac { 1 } { n } } + K$ Answer [Figure] ◯ ◯
(A)
(B)
(C)
(D) 53. The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is
(A) 360
(B) 192
(C) 96
(D) 48
Answer [Figure]
(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) 54. Consider the planes $3 x - 6 y - 2 z = 15$ and $2 x + y - 2 z = 5$.
STATEMENT-1 : The parametric equations of the line of intersection of the given planes are $x = 3 + 14 t , y = 1 + 2 t , z = 15 t$.

because

STATEMENT-2 : The vector $14 \hat { i } + 2 \hat { j } + 15 \hat { k }$ is parallel to the line of intersection of given planes.
(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

Answer

[Figure]
(A)
(B)
(C)
(D) 55. STATEMENT-1 : The curve $y = \frac { - x ^ { 2 } } { 2 } + x + 1$ is symmetric with respect to the line $x = 1$. because STATEMENT-2 : A parabola is symmetric about its axis.
(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 Answer
(A) [Figure] [Figure] [Figure]
(A)
(B)
(C)
(D) 56. Let $f ( x ) = 2 + \cos x$ for all real $x$.
STATEMENT-1 : For each real $t$, there exists a point $c$ in $[ t , t + \pi ]$ such that $f ^ { \prime } ( c ) = 0$. because STATEMENT-2 : $f ( t ) = f ( t + 2 \pi )$ for each real $t$.
(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
Answer
[Figure]
(A)
[Figure]
(B)
[Figure]
(C)
◯
(D) 57. Lines $L _ { 1 } : y - x = 0$ and $L _ { 2 } : 2 x + y = 0$ intersect the line $L _ { 3 } : y + 2 = 0$ at $P$ and $Q$, respectively. The bisector of the acute angle between $L _ { 1 }$ and $L _ { 2 }$ intersects $L _ { 3 }$ at $R$.
STATEMENT-1 : The ratio $P R : R Q$ equals $2 \sqrt { 2 } : \sqrt { 5 }$.

because

STATEMENT-2 : In any triangle, bisector of an angle divides the triangle into two similar triangles.
(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 Answer
[Figure]
(A)
[Figure]
(B)
(B)
[Figure]
(D)

1. Which one of the following statements is correct?
   (A) $G _ { 1 } > G _ { 2 } > G _ { 3 } > \cdots$
   (B) $G _ { 1 } < G _ { 2 } < G _ { 3 } < \cdots$
   (C) $G _ { 1 } = G _ { 2 } = G _ { 3 } = \cdots$
   (D) $G _ { 1 } < G _ { 3 } < G _ { 5 } < \cdots$ and $G _ { 2 } > G _ { 4 } > G _ { 6 } > \cdots$ Answer ◯
   (A) [Figure]
   (B) [Figure]
   (C) ◯
   (D)
2. Which one of the following statements is correct?
   (A) $A _ { 1 } > A _ { 2 } > A _ { 3 } > \cdots$
   (B) $A _ { 1 } < A _ { 2 } < A _ { 3 } < \cdots$
   (C) $A _ { 1 } > A _ { 3 } > A _ { 5 } > \cdots$ and $A _ { 2 } < A _ { 4 } < A _ { 6 } < \cdots$
   (D) $A _ { 1 } < A _ { 3 } < A _ { 5 } < \cdots$ and $A _ { 2 } > A _ { 4 } > A _ { 6 } > \cdots$

Answer

[Figure] ◯ ◯
(A)
(B)
(C)
(D) 60. Which one of the following statements is correct?
(A) $H _ { 1 } > H _ { 2 } > H _ { 3 } > \cdots$
(B) $H _ { 1 } < H _ { 2 } < H _ { 3 } < \cdots$
(C) $H _ { 1 } > H _ { 3 } > H _ { 5 } > \cdots$ and $H _ { 2 } < H _ { 4 } < H _ { 6 } < \cdots$
(D) $H _ { 1 } < H _ { 3 } < H _ { 5 } < \cdots$ and $H _ { 2 } > H _ { 4 } > H _ { 6 } > \cdots$

M61-63: Paragraph for Question Nos. 61 to 63

If a continuous function $f$ defined on the real line $\mathbf { R }$, assumes positive and negative values in $\mathbf { R }$ then the equation $f ( x ) = 0$ has a root in $\mathbf { R }$. For example, if it is known that a continuous function $f$ on $\mathbf { R }$ is positive at some point and its minimum value is negative then the equation $f ( x ) = 0$ has a root in $\mathbf { R }$. Consider $f ( x ) = k e ^ { x } - x$ for all real $x$ where $k$ is a real constant. Answer ◯
(A) [Figure]
(B) O
(C) [Figure]
(D) 61. The line $y = x$ meets $y = k e ^ { x }$ for $k \leq 0$ at
(A) no point
(B) one point
(C) two points
(D) more than two points
Answer
[Figure]
(A)
[Figure]
(B)
[Figure]
(C)
[Figure]
(D)

1. The positive value of $k$ for which $k e ^ { x } - x = 0$ has only one root is
   (A) $\frac { 1 } { e }$
   (B) 1
   (C) $e$
   (D) $\log _ { e } 2$

Answer [Figure] [Figure] [Figure]
(A)
(B)
(C)
(D) 63. For $k > 0$, the set of all values of $k$ for which $k e ^ { x } - x = 0$ has two distinct roots is
(A) $\left( 0 , \frac { 1 } { e } \right)$
(B) $\left( \frac { 1 } { e } , 1 \right)$
(C) $\left( \frac { 1 } { e } , \infty \right)$
(D) $( 0,1 )$
Answer ◯ ◯ ◯
(A)
(B)
(C)
(D) 64. Let $f ( x ) = \frac { x ^ { 2 } - 6 x + 5 } { x ^ { 2 } - 5 x + 6 }$.
Match the expressions/statements in Column I with expressions/statements in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.

Column I

(A) If $- 1 < x < 1$, then $f ( x )$ satisfies
(B) If $1 < x < 2$, then $f ( x )$ satisfies
(C) If $3 < x < 5$, then $f ( x )$ satisfies
(D) If $x > 5$, then $f ( x )$ satisfies

Column II

(p) $0 < f ( x ) < 1$
(q) $f ( x ) < 0$
(r) $f ( x ) > 0$
(s) $f ( x ) < 1$
Answer [Figure] 65. Let $( x , y )$ be such that
$$\sin ^ { - 1 } ( a x ) + \cos ^ { - 1 } ( y ) + \cos ^ { - 1 } ( b x y ) = \frac { \pi } { 2 }$$
Match the statements in Column I with statements in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.

Column I

(A) If $a = 1$ and $b = 0$, then $( x , y )$
(B) If $a = 1$ and $b = 1$, then $( x , y )$
(C) If $a = 1$ and $b = 2$, then $( x , y )$
(D) If $a = 2$ and $b = 2$, then $( x , y )$
(p) lies on the circle $x ^ { 2 } + y ^ { 2 } = 1$
(q) lies on $\left( x ^ { 2 } - 1 \right) \left( y ^ { 2 } - 1 \right) = 0$
(r) lies on $y = x$
(s) lies on $\left( 4 x ^ { 2 } - 1 \right) \left( y ^ { 2 } - 1 \right) = 0$

Column II

1. Match the statements in Column I with the properties in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.

Column I

(A) Two intersecting circles
(B) Two mutually external circles
(C) Two circles, one strictly inside the other
(D) Two branches of a hyperbola

Column II

(p) have a common tangent
(q) have a common normal
(r) do not have a common tangent
(s) do not have a common normal
Answer [Figure]

Let two non-collinear unit vectors $\hat { a }$ and $\hat { b }$ form an acute angle. A point $P$ moves so that at any time $t$ the position vector $\overrightarrow { O P }$ (where $O$ is the origin) is given by $\hat { a } \cos t + \hat { b } \sin t$. When $P$ is farthest from origin $O$, let $M$ be the length of $\overrightarrow { O P }$ and $\hat { u }$ be the unit vector along $\overrightarrow { O P }$. Then,
(A) $\hat { u } = \frac { \hat { a } + \hat { b } } { | \hat { a } + \hat { b } | }$ and $M = ( 1 + \hat { a } \cdot \hat { b } ) ^ { \frac { 1 } { 2 } }$
(B) $\hat { u } = \frac { \hat { a } - \hat { b } } { | \hat { a } - \hat { b } | }$ and $M = ( 1 + \hat { a } \cdot \hat { b } ) ^ { \frac { 1 } { 2 } }$
(C) $\hat { u } = \frac { \hat { a } + \hat { b } } { | \hat { a } + \hat { b } | }$ and $M = ( 1 + 2 \hat { a } \cdot \hat { b } ) ^ { \frac { 1 } { 2 } }$
(D) $\hat { u } = \frac { \hat { a } - \hat { b } } { | \hat { a } - \hat { b } | }$ and $M = ( 1 + 2 \hat { a } \cdot \hat { b } ) ^ { \frac { 1 } { 2 } }$

Let $P , Q , R$ and $S$ be the points on the plane with position vectors $- 2 \hat { i } - \hat { j } , 4 \hat { i } , 3 \hat { i } + 3 \hat { j }$ and $- 3 \hat { i } + 2 \hat { j }$ respectively. The quadrilateral $P Q R S$ must be a
A) parallelogram, which is neither a rhombus nor a rectangle
B) square
C) rectangle, but not a square
D) rhombus, but not a square

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) $\vec{a} = (\vec{a} \cdot \vec{y})(\vec{z} - \vec{y})$

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

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

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 $\vec { a }$ 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 $\_\_\_\_$

Let $O$ be the origin and $\overrightarrow { O A } = 2 \hat { \mathrm { i } } + 2 \hat { \mathrm { j } } + \hat { \mathrm { k } } , \quad \overrightarrow { O B } = \hat { \mathrm { i } } - 2 \hat { \mathrm { j } } + 2 \hat { \mathrm { k } }$ and $\overrightarrow { O C } = \frac { 1 } { 2 } ( \overrightarrow { O B } - \lambda \overrightarrow { O A } )$ for some $\lambda > 0$. If $| \overrightarrow { O B } \times \overrightarrow { O C } | = \frac { 9 } { 2 }$, then which of the following statements is (are) TRUE ?
(A) Projection of $\overrightarrow { O C }$ on $\overrightarrow { O A }$ is $- \frac { 3 } { 2 }$
(B) Area of the triangle $O A B$ is $\frac { 9 } { 2 }$
(C) Area of the triangle $A B C$ is $\frac { 9 } { 2 }$
(D) The acute angle between the diagonals of the parallelogram with adjacent sides $\overrightarrow { O A }$ and $\overrightarrow { O C }$ is $\frac { \pi } { 3 }$

Let $\hat { \imath } , \hat { \jmath }$ and $\hat { k }$ be the unit vectors along the three positive coordinate axes. Let
$$\begin{aligned} & \vec { a } = 3 \hat { \imath } + \hat { \jmath } - \hat { k } , \\ & \vec { b } = \hat { \imath } + b _ { 2 } \hat { \jmath } + b _ { 3 } \hat { k } , \quad b _ { 2 } , b _ { 3 } \in \mathbb { R } , \\ & \vec { c } = c _ { 1 } \hat { \imath } + c _ { 2 } \hat { \jmath } + c _ { 3 } \hat { k } , \quad c _ { 1 } , c _ { 2 } , c _ { 3 } \in \mathbb { R } \end{aligned}$$
be three vectors such that $b _ { 2 } b _ { 3 } > 0 , \vec { a } \cdot \vec { b } = 0$ and
$$\left( \begin{array} { r c r } 0 & - c _ { 3 } & c _ { 2 } \\ c _ { 3 } & 0 & - c _ { 1 } \\ - c _ { 2 } & c _ { 1 } & 0 \end{array} \right) \left( \begin{array} { l } 1 \\ b _ { 2 } \\ b _ { 3 } \end{array} \right) = \left( \begin{array} { r } 3 - c _ { 1 } \\ 1 - c _ { 2 } \\ - 1 - c _ { 3 } \end{array} \right)$$
Then, which of the following is/are TRUE ?
(A) $\vec { a } \cdot \vec { c } = 0$
(B) $\vec { b } \cdot \vec { c } = 0$
(C) $| \vec { b } | > \sqrt { 10 }$
(D) $| \vec { c } | \leq \sqrt { 11 }$

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 $\_\_\_\_$ .

A uniform chain of length 2 m is kept on a table such that a length of 60 cm hangs freely from the edge of the table. The total mass of the chain is 4 kg . What is the work done in pulling the entire chain on the table?
(1) 7.2 J
(2) 3.6 J
(3) 120 J
(4) 1200 J

A particle is moving eastwards with a velocity of $5 \mathrm{~m}/\mathrm{s}$ in 10 seconds the velocity changes to $5 \mathrm{~m}/\mathrm{s}$ northwards. The average acceleration in this time is
(1) $\frac{1}{\sqrt{2}} \mathrm{~m}/\mathrm{s}^2$ towards north-east
(2) $\frac{1}{2} \mathrm{~m}/\mathrm{s}^2$ towards north.
(3) zero
(4) $\frac{1}{\sqrt{2}} \mathrm{~m}/\mathrm{s}^2$ towards north-west

The resultant of two forces P N and 3 N is a force of 7 N . If the direction of 3 N force were reversed, the resultant would be $\sqrt { 19 } \mathrm {~N}$. The value of P is
(1) 5 N
(2) 6 N
(3) 3 N
(4) 4 N