Circle $\odot O$ has radius 1. Line $PA$ is tangent to $\odot O$ at point $A$. Line $PB$ intersects $\odot O$ at points $B$ and $C$. $D$ is the midpoint of $BC$. If $| P O | = \sqrt { 2 }$, then the maximum value of $\overrightarrow { P A } \cdot \overrightarrow { P D }$ is
A. $\frac { 1 + \sqrt { 2 } } { 2 }$
B. $\frac { 1 + 2 \sqrt { 2 } } { 2 }$
C. $1 + \sqrt { 2 }$
D. $2 + \sqrt { 2 }$

Given vectors $\vec { a } , \vec { b }$ satisfying $| \vec { a } | = 1 , | \vec { a } + 2 \vec { b } | = 2$, and $( \vec { b } - 2 \vec { a } ) \perp \vec { b }$, then $| \vec { b } | =$
A. $\frac { 1 } { 2 }$
B. $\frac { \sqrt { 2 } } { 2 }$
C. $\frac { \sqrt { 3 } } { 2 }$
D. 1

Given vectors $\boldsymbol { a } = ( 0,1 ) , \boldsymbol { b } = ( 2 , x )$ , if $\boldsymbol { b } \perp ( \boldsymbol { b } - 4 \boldsymbol { a } )$ , then $x =$
A. $- 2$
B. $- 1$
C. $1$
D. $2$

Given vectors $\boldsymbol { a } , \boldsymbol { b }$, then ``$( \boldsymbol { a } + \boldsymbol { b } ) ( \boldsymbol { a } - \boldsymbol { b } ) = 0$'' is ``$\boldsymbol { a } = \boldsymbol { b }$ or $\boldsymbol { a } = - \boldsymbol { b }$'' a \_\_\_\_ condition.

In sailing competitions, athletes can use an anemometer 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 velocity but opposite direction. Figure 1 shows the correspondence between part of the wind force levels, names, and wind speeds. An athlete measured the vector corresponding to apparent wind speed and the vector corresponding to ship's velocity as shown in Figure 2 (the magnitude of wind speed and the magnitude of the vector are the same, unit: $\mathrm{m/s}$). Then the true wind is

Level	Wind Speed $\mathrm{m/s}$	Name
2	$1.1 \sim 3.3$	Light Breeze
3	$3.4 \sim 5.4$	Gentle Breeze
4	$5.5 \sim 7.9$	Moderate Wind
5	$8.0 \sim 10.1$	Fresh Wind

A. Light Breeze
B. Gentle Breeze
C. Moderate Wind
D. Fresh Wind

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}| = $ \_\_\_\_

Calculate the coordinates of $T$.

133. Given $\mathbf{a} = (3, m, 5)$ and $\mathbf{b} = (3-m, 7, \circ)$. For a value of $m$, the two vectors $\mathbf{a}+\mathbf{b}$ and $\mathbf{a}-\mathbf{b}$ are perpendicular to each other. What is the angle between vectors $\mathbf{a}$ and $\mathbf{b}$ in this case?
(1) $30$ (2) $45$ (3) $60$ (4) $90$
%% Page 7 Mathematics 120-C Page 6

142- Three vectors $\vec{a} = (1,1,0)$, $\vec{b} = (-1,2,0)$, and $\vec{c}$ are non-coplanar, and $\vec{h} = (x,y,4)$ is the altitude vector of the parallelepiped formed by these three vectors. If $\vec{a} \cdot \vec{c} = 1$ and $\vec{b} \cdot \vec{c} = 5$, what is the magnitude of vector $\vec{c}$?
(4) $\sqrt{21}$ (3) $\sqrt{19}$ (2) $4$ (1) $5$

35. Suppose $\vec{a}$ and $\vec{b}$ are non-zero vectors whose dot product is $-\dfrac{3}{5}$ times the product of their magnitudes. What is the area of the triangle formed by the vectors $\left(\dfrac{3\vec{a}}{|\vec{a}|}+\dfrac{2\vec{b}}{|\vec{b}|}\right)$ and $\left(\dfrac{\vec{a}}{|\vec{a}|}-\dfrac{2\vec{b}}{|\vec{b}|}\right)$?
(1) $6/4$ (2) $4/8$ (3) $3/2$ (4) $1/6$

Consider a triangle $ABC$. The sides $AB$ and $AC$ are extended to points $D$ and $E$, respectively, such that $AD = 3AB$ and $AE = 3AC$. Then one diagonal of $BDEC$ divides the other diagonal in the ratio
(A) $1 : 3$
(B) $1 : \sqrt{3}$
(C) $1 : 2$
(D) $1 : \sqrt{2}$.

9. If $\vec { a } = \hat { i } + \hat { j } + \hat { k } , \vec { b } = 4 \hat { i } + 3 \hat { j } + 4 \hat { k }$ and $\vec { c } = \hat { i } + \alpha \hat { j } + \beta \hat { k }$ are linearly dependent vectors and $| \vec { c } | = \sqrt { 3 }$, then:
(A) $\alpha = 1 , \beta = - 1$
(B) $\alpha = 1 , \beta = \pm 1$
(C) $\alpha = - 1 , \beta = \pm 1$
(D) $\alpha = \pm 1 , \beta = 1$

10. Let $\vec { a } = 2 \hat { \imath } + \hat { \jmath } - 2 \hat { k }$ and $\hat { b } = \hat { \imath } + \hat { \jmath }$. If $\hat { c }$ is a vector such that $\vec { a } \cdot \vec { c } = | \vec { c } - \vec { a } | = 2 \sqrt { 2 }$ and the angle between $( \vec { a } \times \vec { b } )$ and $\vec { c }$ is $30 ^ { \circ }$, then $| ( \vec { a } \times \vec { b } ) \times \vec { c } | =$
(A) $\frac { 3 } { 2 }$
(B) $\frac { 3 } { 2 }$
(C) 2
(D) 3

27. Let $\vec { a }$ and $\vec { b }$ be two non-collinear unit vectors. If $\vec { u } = \vec { a } - ( \vec { a } , \vec { b } ) \vec { b }$ and $\vec { v } = \vec { a } \times \vec { b }$, then $| \vec { v } |$ is:
(A) $| \vec { u } |$
(B) $\quad | \vec { u } | + | \vec { u } \cdot \vec { a } |$
(C) $| \vec { u } | + | \vec { u } , \vec { b } |$
(D) $| \vec { u } | + \vec { u } \cdot ( \vec { a } + \vec { b } )$

24. If the circles $x 2 + y 2 + 2 x + 2 k y + 6 = 0$ and $x 2 + y 2 + 2 k y + k = 0$ intersect orthogonally, then $k$ is :
(A) 2 or $- 3 / 2$
(B) - 2 or $3 / 2$
(C) 2 or $3 / 2$
(D) $( 2 , + \infty )$

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10. Show, by vector methods, that the angular bisectors of a triangle are concurrent and find an expression for the position vector of the point of concurrency in terms of the position vectors of the vertices.

15. (a) Find 3 -dimensional vectors $\overrightarrow { v _ { 1 } } , \overrightarrow { v _ { 2 } } , \overrightarrow { v _ { 3 } }$ satisfying
$$\overrightarrow { v _ { 1 } } \cdot \overrightarrow { v _ { 1 } } = 4 , \overrightarrow { v _ { 1 } } \cdot \overrightarrow { v _ { 2 } } = - 2 , \overrightarrow { v _ { 1 } } \cdot \overrightarrow { v _ { 3 } } = 6 , \overrightarrow { v _ { 2 } } \cdot \overrightarrow { v _ { 2 } } = 2 , \overrightarrow { v _ { 2 } } \cdot \overrightarrow { v _ { 3 } } = - 5 , \overrightarrow { v _ { 3 } } \cdot \overrightarrow { v _ { 3 } } = 29 .$$
(b) Let $\vec { A } ( t ) = f _ { 1 } ( t ) \hat { i } + f _ { 2 } ( t ) \hat { j }$ and $B ( t ) = g _ { 1 } ( t ) \hat { i } + g _ { 2 } ( t ) \hat { j } , t \in [ 0,1 ]$, where $f _ { 1 \text {, } } \mathrm { f } _ { 2 } , \mathrm {~g} _ { 1 } , \mathrm {~g} _ { 2 }$ are continuous functions. If $\vec { A } ( \mathrm { t } )$ and $\vec { B } ( \mathrm { t } )$ are non-zero vectors for all t and $\vec { A } ( 0 ) = 2 \hat { \imath } + 3 \hat { \jmath } , \vec { A } ( 1 ) = 6 \hat { \imath } + 2 \hat { \jmath } , \vec { B } ( 0 ) = 3 \hat { \imath } + 2 \hat { \jmath }$ and $\vec { B } ( 1 ) = 2 \hat { \imath } + 6 \hat { j }$. The show that $\vec { A } ( t )$ and $\vec { B } ( t )$ are parallel for some $t$.

26. If If $\vec { a } , \vec { b }$, and $\vec { c }$ are unit vectors, then
$$| \vec { a } - \vec { b } | ^ { 2 } + | \vec { b } - \vec { c } | ^ { 2 } + | \vec { c } - \vec { a } | ^ { 2 }$$
does not exceed :
(A) 4
(B) 9
(C) 8
(D) 6

9. Incident ray is along the unit vector v and the reflected ray is along the unit vector w . The normal is along unit vector a outwards. Express vector $w$ in terms of vector $a$ and $v$.

45. Let $O ( 0,0 ) , P ( 3,4 ) , Q ( 6,0 )$ be the vertices of the triangle $O P Q$. The point $R$ inside the triangle $O P Q$ is such that the triangles $O P R , P Q R , O Q R$ are of equal area. The coordinates of $R$ are
(A) $\left( \frac { 4 } { 3 } , 3 \right)$
(B) $\left( 3 , \frac { 2 } { 3 } \right)$
(C) $\left( 3 , \frac { 4 } { 3 } \right)$
(D) $\left( \frac { 4 } { 3 } , \frac { 2 } { 3 } \right)$
Answer ◯ [Figure] [Figure] [Figure]
(A)
(B)
(C)
(D)

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 $\vec{a}, \vec{b}, \vec{c}$ be unit vectors such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$. Which one of the following is correct?
(A) $\vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{c} \times \vec{a} = \vec{0}$
(B) $\vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{c} \times \vec{a} \neq \vec{0}$
(C) $\vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{a} \times \vec{c} \neq \vec{0}$
(D) $\vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a}$ are mutually perpendicular

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