The least value of $| z |$ where $z$ is complex number which satisfies the inequality $e ^ { \left( \frac { ( | z | + 3 ) ( | z | - 1 ) } { | | z | + 1 | } \log _ { \mathrm { e } } 2 \right) } \geq \log _ { \sqrt { 2 } } | 5 \sqrt { 7 } + 9 i | , i = \sqrt { - 1 }$, is equal to :
(1) 3
(2) $\sqrt { 5 }$
(3) 2
(4) 8

If the real part of the complex number $( 1 - \cos \theta + 2i \sin \theta ) ^ { - 1 }$ is $\frac { 1 } { 5 }$ for $\theta \in ( 0 , \pi )$, then the value of the integral $\int _ { 0 } ^ { \theta } \sin x \mathrm {~d} x$ is equal to:
(1) 1
(2) 2
(3) - 1
(4) 0

If $\alpha , \beta \in R$ are such that $1 - 2 i$ (here $i ^ { 2 } = - 1$ ) is a root of $z ^ { 2 } + \alpha z + \beta = 0$, then ( $\alpha - \beta$ ) is equal to:
(1) - 7
(2) 7
(3) - 3
(4) 3

The value of $\tan \left( 2 \tan ^ { - 1 } \left( \frac { 3 } { 5 } \right) + \sin ^ { - 1 } \left( \frac { 5 } { 13 } \right) \right)$ is equal to:
(1) $\frac { - 181 } { 69 }$
(2) $\frac { 220 } { 21 }$
(3) $\frac { - 291 } { 76 }$
(4) $\frac { 151 } { 63 }$

Let $z$ and $w$ be two complex numbers such that $w = z \bar { z } - 2 z + 2 , \left| \frac { z + i } { z - 3 i } \right| = 1$ and $\operatorname { Re } ( w )$ has minimum value. Then, the minimum value of $n \in N$ for which $w ^ { n }$ is real, is equal to $\_\_\_\_$.

Let $\alpha$ and $\beta$ be the roots of the equation $x ^ { 2 } + ( 2 i - 1 ) = 0$. Then, the value of $\left| \alpha ^ { 8 } + \beta ^ { 8 } \right|$ is equal to
(1) 50
(2) 250
(3) 1250
(4) 1550

If $z = 2 + 3 i$, then $z ^ { 5 } + ( \bar { z } ) ^ { 5 }$ is equal to:
(1) 244
(2) 224
(3) 245
(4) 265

If $z \neq 0$ be a complex number such that $z - \frac{1}{z} = 2$, then the maximum value of $|z|$ is
(1) $\sqrt{2}$
(2) 1
(3) $\sqrt{2} - 1$
(4) $\sqrt{2} + 1$

Let $O$ be the origin and $A$ be the point $z _ { 1 } = 1 + 2i$. If $B$ is the point $z _ { 2 } , \operatorname { Re } \left( z _ { 2 } \right) < 0$, such that $OAB$ is a right angled isosceles triangle with $OB$ as hypotenuse, then which of the following is NOT true?
(1) $\arg z _ { 2 } = \pi - \tan ^ { - 1 } 3$
(2) $\arg \left( z _ { 1 } - 2 z _ { 2 } \right) = - \tan ^ { - 1 } \frac { 4 } { 3 }$
(3) $\left| z _ { 2 } \right| = \sqrt { 10 }$
(4) $\left| 2 z _ { 1 } - z _ { 2 } \right| = 5$

Let the minimum value $v _ { 0 }$ of $v = | z | ^ { 2 } + | z - 3 | ^ { 2 } + | z - 6 i | ^ { 2 } , z \in \mathbb { C }$ is attained at $z = z _ { 0 }$. Then $\left| 2 z _ { 0 } ^ { 2 } - \bar { z } _ { 0 } ^ { 3 } + 3 \right| ^ { 2 } + v _ { 0 } ^ { 2 }$ is equal to
(1) 1000
(2) 1024
(3) 1105
(4) 1196

If $z = x + i y$ satisfies $|z - 2| = 0$ and $|z - i| - |z + 5i| = 0$, then
(1) $x + 2 y - 4 = 0$
(2) $x ^ { 2 } + y - 4 = 0$
(3) $x + 2 y + 4 = 0$
(4) $x ^ { 2 } - y + 3 = 0$

Let for some real numbers $\alpha$ and $\beta , a = \alpha - i \beta$. If the system of equations $4 i x + ( 1 + i ) y = 0$ and $8 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right) x + \bar { a } y = 0$ has more than one solution then $\frac { \alpha } { \beta }$ is equal to
(1) $2 - \sqrt { 3 }$
(2) $2 + \sqrt { 3 }$
(3) $- 2 + \sqrt { 3 }$
(4) $- 2 - \sqrt { 3 }$

Let $S$ be the set of all $( \alpha , \beta ) , \pi < \alpha , \beta < 2 \pi$, for which the complex number $\frac { 1 - i \sin \alpha } { 1 + 2 i \sin \alpha }$ is purely imaginary and $\frac { 1 + i \cos \beta } { 1 - 2 i \cos \beta }$ is purely real. Let $Z _ { \alpha \beta } = \sin 2 \alpha + i \cos 2 \beta , ( \alpha , \beta ) \in S$. Then $\sum _ { ( \alpha , \beta ) \in S } \left( i Z _ { \alpha \beta } + \frac { 1 } { i \bar { Z } _ { \alpha \beta } } \right)$ is equal to
(1) 3
(2) $3 i$
(3) 1
(4) $2 - i$

Let $S = \{ z \in \mathbb { C } : | z - 3 | \leq 1$ and $z ( 4 + 3 i ) + \bar { z } ( 4 - 3 i ) \leq 24 \}$. If $\alpha + i \beta$ is the point in $S$ which is closest to $4 i$, then $25 ( \alpha + \beta )$ is equal to $\_\_\_\_$.

Let $S = \left\{ z \in \mathbb { C } : z ^ { 2 } + \bar { z } = 0 \right\}$. Then $\sum _ { z \in S } ( \operatorname { Re } ( z ) + \operatorname { Im } ( z ) )$ is equal to $\_\_\_\_$ .

Let $\mathrm { z } = \mathrm { a } + i b , \mathrm { b } \neq 0$ be complex numbers satisfying $\mathrm { z } ^ { 2 } = \overline { \mathrm { z } } \cdot 2 ^ { 1 - | z | }$. Then the least value of $n \in N$, such that $z ^ { n } = ( z + 1 ) ^ { n }$, is equal to $\_\_\_\_$.

Let $p , \quad q \in \mathbb { R }$ and $( 1 - \sqrt { 3 } i ) ^ { 200 } = 2 ^ { 199 } ( p + i q ) , i = \sqrt { - 1 }$. Then, $p + q + q ^ { 2 }$ and $p - q + q ^ { 2 }$ are roots of the equation.
(1) $x ^ { 2 } + 4 x - 1 = 0$
(2) $x ^ { 2 } - 4 x + 1 = 0$
(3) $x ^ { 2 } + 4 x + 1 = 0$
(4) $x ^ { 2 } - 4 x - 1 = 0$

Let $a \neq b$ be two non-zero real numbers. Then the number of elements in the set $X = \left\{ z \in C : \operatorname{Re}\left(az^{2} + bz\right) = a$ and $\operatorname{Re}\left(bz^{2} + az\right) = b\right\}$ is equal to
(1) 0
(2) 1
(3) 3
(4) 2

Let the complex number $z = x + iy$ be such that $\frac{2z - 3i}{2z + i}$ is purely imaginary. If $x + y^2 = 0$, then $y^4 + y^2 - y$ is equal to
(1) $\frac{2}{3}$
(2) $\frac{3}{2}$
(3) $\frac{3}{4}$
(4) $\frac{1}{3}$

The value of $\left( \frac { 1 + \sin \frac { 2 \pi } { 9 } + i \cos \frac { 2 \pi } { 9 } } { 1 + \sin \frac { 2 \pi } { 9 } - i \cos \frac { 2 \pi } { 9 } } \right) ^ { 3 }$ is
(1) $\frac { - 1 } { 2 } ( 1 - i \sqrt { 3 } )$
(2) $\frac { 1 } { 2 } ( 1 - i \sqrt { 3 } )$
(3) $\frac { - 1 } { 2 } ( \sqrt { 3 } - i )$
(4) $\frac { 1 } { 2 } ( \sqrt { 3 } + i )$

The complex number $z = \frac{i-1}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}}$ is equal to:
(1) $\sqrt{2}i\left(\cos\frac{5\pi}{12} - i\sin\frac{5\pi}{12}\right)$
(2) $\cos\frac{\pi}{12} - i\sin\frac{\pi}{12}$
(3) $\sqrt{2}\left(\cos\frac{\pi}{12} + i\sin\frac{\pi}{12}\right)$
(4) $\sqrt{2}\left(\cos\frac{5\pi}{12} + i\sin\frac{5\pi}{12}\right)$

For $\alpha, \beta, z \in \mathbb{C}$ and $\lambda > 1$, if $\sqrt{\lambda - 1}$ is the radius of the circle $|z - \alpha|^{2} + |z - \beta|^{2} = 2\lambda$, then $|\alpha - \beta|$ is equal to $\_\_\_\_$.

Let $A = \left\{ \theta \in ( 0,2 \pi ) : \frac { 1 + 2 i \sin \theta } { 1 - i \sin \theta } \right.$ is purely imaginary $\}$ Then the sum of the elements in $A$ is
(1) $4 \pi$
(2) $3 \pi$
(3) $\pi$
(4) $2 \pi$

Let $S = \{ z \in \mathbb { C } : \bar { z } = i z ^ { 2 } + \operatorname { Re } ( \bar { z } ) \}$. Then $\sum _ { z \in S } | z | ^ { 2 }$ is equal to
(1) $\frac { 5 } { 2 }$
(2) 4
(3) $\frac { 7 } { 2 }$
(4) 3

For two non-zero complex number $z _ { 1 }$ and $z _ { 2 }$, if $\operatorname { Re } \left( z _ { 1 } z _ { 2 } \right) = 0$ and $\operatorname { Re } \left( z _ { 1 } + z _ { 2 } \right) = 0$, then which of the following are possible?
(A) $\operatorname { Im } \left( \mathrm { z } _ { 1 } \right) > 0$ and $\operatorname { Im } \left( \mathrm { z } _ { 2 } \right) > 0$
(B) $\operatorname { Im } \left( \mathrm { z } _ { 1 } \right) < 0$ and $\operatorname { Im } \left( \mathrm { z } _ { 2 } \right) > 0$
(C) $\operatorname { Im } \left( \mathrm { z } _ { 1 } \right) > 0$ and $\operatorname { Im } \left( \mathrm { z } _ { 2 } \right) < 0$
(D) $\operatorname { Im } \left( \mathrm { z } _ { 1 } \right) < 0$ and $\operatorname { Im } \left( \mathrm { z } _ { 2 } \right) < 0$ Choose the correct answer from the options given below:
(1) B and D
(2) B and C
(3) A and B
(4) A and C