Let $\alpha = 8 - 14 \mathrm { i } , \mathrm { A } = \left\{ \mathrm { z } \in \mathbb { C } : \frac { \alpha \mathrm { z } - \bar { \alpha } \overline { \mathrm { z } } } { \mathrm { z } ^ { 2 } - ( \overline { \mathrm { z } } ) ^ { 2 } - 112 \mathrm { i } } = 1 \right\}$ and $B = \{ z \in \mathbb { C } : | z + 3 i | = 4 \}$
Then, $\sum _ { z \in A \cap B } ( \operatorname { Re } z - \operatorname { Im } z )$ is equal to $\_\_\_\_$

If for $z = \alpha + i \beta , | z + 2 | = z + 4 ( 1 + i )$, then $\alpha + \beta$ and $\alpha \beta$ are the roots of the equation
(1) $x ^ { 2 } + 3 x - 4 = 0$
(2) $x ^ { 2 } + 7 x + 12 = 0$
(3) $x ^ { 2 } + x - 12 = 0$
(4) $x ^ { 2 } + 2 x - 3 = 0$

If $\sin^{-1}\frac{\alpha}{17} + \cos^{-1}\frac{4}{5} - \tan^{-1}\frac{77}{36} = 0$, $0 < \alpha < 13$, then $\sin^{-1}(\sin\alpha) + \cos^{-1}(\cos\alpha)$ is equal to
(1) $\pi$
(2) 16
(3) 0
(4) $16 - 5\pi$

Let $z_1$ and $z_2$ be two complex numbers such that $z_1 + z_2 = 5$ and $z_1^3 + z_2^3 = 20 + 15i$. Then $|z_1^4 + z_2^4|$ equals
(1) $30\sqrt{3}$
(2) $75\sqrt{2}$
(3) $15\sqrt{15}$
(4) $25\sqrt{3}$

Let $r$ and $\theta$ respectively be the modulus and amplitude of the complex number $z = 2 - i \left( 2 \tan \frac { 5 \pi } { 8 } \right)$, then $( r , \theta )$ is equal to
(1) $\left( 2 \sec \frac { 3 \pi } { 8 } , \frac { 3 \pi } { 8 } \right)$
(2) $\left( 2 \sec \frac { 3 \pi } { 8 } , \frac { 5 \pi } { 8 } \right)$
(3) $\left( 2 \sec \frac { 5 \pi } { 8 } , \frac { 3 \pi } { 8 } \right)$
(4) $\left( 2 \sec \frac { 11 \pi } { 8 } , \frac { 11 \pi } { 8 } \right)$

If $z = x + i y , x y \neq 0$, satisfies the equation $z ^ { 2 } + i \bar { z } = 0$, then $\left| z ^ { 2 } \right|$ is equal to:
(1) 9
(2) 1
(3) 4
(4) $\frac { 1 } { 4 }$

The sum of all possible values of $\theta \in [ - \pi , 2 \pi ]$, for which $\frac { 1 + i \cos \theta } { 1 - 2 i \cos \theta }$ is purely imaginary, is equal (1) $3 \pi$ (2) $2 \pi$ (3) $5 \pi$ (4) $4 \pi$

Let $z_1$ and $z_2$ be two complex number such that $z_1 + z_2 = 5$ and $z_1^3 + z_2^3 = 20 + 15i$. Then $z_1^4 + z_2^4$ equals-
(1) $30\sqrt{3}$
(2) 75
(3) $15\sqrt{15}$
(4) $25\sqrt{3}$

If the set $R = \{ ( a , b ) : a + 5 b = 42 , a , b \in \mathbb { N } \}$ has $m$ elements and $\sum _ { n = 1 } ^ { m } \left( 1 - i ^ { n ! } \right) = x + i y$, where $i = \sqrt { - 1 }$, then the value of $m + x + y$ is
(1) 12
(2) 4
(3) 8
(4) 5

If $\alpha$ satisfies the equation $x ^ { 2 } + x + 1 = 0$ and $( 1 + \alpha ) ^ { 7 } = \mathrm { A } + \mathrm { B } \alpha + \mathrm { C } \alpha ^ { 2 } , \mathrm {~A} , \mathrm {~B} , \mathrm { C } \geq 0$, then $5 ( 3 \mathrm {~A} - 2 \mathrm {~B} - \mathrm { C } )$ is equal to

Let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } - x + 2 = 0$ with $\operatorname { Im } ( \alpha ) > \operatorname { Im } ( \beta )$. Then $\alpha ^ { 6 } + \alpha ^ { 4 } + \beta ^ { 4 } - 5 \alpha ^ { 2 }$ is equal to

If $\alpha$ and $\beta$ are the roots of the equation $2z^2 - 3z - 2i = 0$, where $i = \sqrt{-1}$, then $16 \cdot \operatorname{Re}\left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}}\right) \cdot \operatorname{Im}\left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}}\right)$ is equal to
(1) 441
(2) 398
(3) 312
(4) 409

The number of complex numbers $z$, satisfying $| z | = 1$ and $\left| \frac { z } { \bar { z } } + \frac { \bar { z } } { z } \right| = 1$, is :
(1) 4
(2) 8
(3) 10
(4) 6

If $\alpha + i \beta$ and $\gamma + i \delta$ are the roots of $x ^ { 2 } - ( 3 - 2 i ) x - ( 2 i - 2 ) = 0 , i = \sqrt { - 1 }$, then $\alpha \gamma + \beta \delta$ is equal to :
(1) $-2$
(2) 6
(3) $-6$
(4) 2

If for some $\alpha, \beta$; $\alpha \leq \beta$, $\alpha + \beta = 8$ and $\sec^2(\tan^{-1}\alpha) + \operatorname{cosec}^2(\cot^{-1}\beta) = 36$, then $\alpha^2 + \beta$ is $\underline{\hspace{2cm}}$.

Let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } - a x - b = 0$ with $\operatorname { Im } ( \alpha ) < \operatorname { Im } ( \beta )$. Let $P _ { n } = \alpha ^ { n } - \beta ^ { n }$. If $\mathrm { P } _ { 3 } = - 5 \sqrt { 7 } i , \mathrm { P } _ { 4 } = - 3 \sqrt { 7 } i , \mathrm { P } _ { 5 } = 11 \sqrt { 7 } i$ and $\mathrm { P } _ { 6 } = 45 \sqrt { 7 } i$, then $\left| \alpha ^ { 4 } + \beta ^ { 4 } \right|$ is equal to

Let integers $\mathrm { a } , \mathrm { b } \in [ - 3,3 ]$ be such that $\mathrm { a } + \mathrm { b } \neq 0$. Then the number of all possible ordered pairs $( \mathrm { a } , \mathrm { b } )$, for which $\left| \frac { z - \mathrm { a } } { z + \mathrm { b } } \right| = 1$ and $\left| \begin{array} { c c c } z + 1 & \omega & \omega ^ { 2 } \\ \omega & z + \omega ^ { 2 } & 1 \\ \omega ^ { 2 } & 1 & z + \omega \end{array} \right| = 1 , z \in \mathrm { C }$, where $\omega$ and $\omega ^ { 2 }$ are the roots of $x ^ { 2 } + x + 1 = 0$, is equal to $\_\_\_\_$ .

Q61. Consider the following two statements : Statement I : For any two non-zero complex numbers $z _ { 1 } , z _ { 2 }$, $\left( \left| z _ { 1 } \right| + \left| z _ { 2 } \right| \right) \left| \frac { z _ { 1 } } { \left| z _ { 1 } \right| } + \frac { z _ { 2 } } { \left| z _ { 2 } \right| } \right| \leq 2 \left( \left| z _ { 1 } \right| + \left| z _ { 2 } \right| \right)$, and Statement II : If $x , y , z$ are three distinct complex numbers and $\mathrm { a } , \mathrm { b } , \mathrm { c }$ are three positive real numbers such that $\frac { \mathrm { a } } { | y - z | } = \frac { \mathrm { b } } { | z - x | } = \frac { \mathrm { c } } { | x - y | }$, then $\frac { \mathrm { a } ^ { 2 } } { y - z } + \frac { \mathrm { b } ^ { 2 } } { z - x } + \frac { \mathrm { c } ^ { 2 } } { x - y } = 1$. Between the above two statements,
(1) Statement I is correct but Statement II is
(2) both Statement I and Statement II are correct. incorrect.
(3) both Statement I and Statement II are incorrect.
(4) Statement I is incorrect but Statement II is correct.

Q62. Let $z$ be a complex number such that $| z + 2 | = 1$ and $\operatorname { Im } \left( \frac { z + 1 } { z + 2 } \right) = \frac { 1 } { 5 }$. Then the value of $| \operatorname { Re } ( \overline { z + 2 } ) |$ is
(1) $\frac { 2 \sqrt { 6 } } { 5 }$
(2) $\frac { 24 } { 5 }$
(3) $\frac { 1 + \sqrt { 6 } } { 5 }$
(4) $\frac { \sqrt { 6 } } { 5 }$

Q81. The sum of the square of the modulus of the elements in the set $\{ z = \mathrm { a } + \mathrm { ib } : \mathrm { a } , \mathrm { b } \in \mathbf { Z } , z \in \mathbf { C } , | z - 1 | \leq 1 , | z - 5 | \leq | z - 5 \mathrm { i } | \}$ is

If complex numbers $Z _ { 1 } , Z _ { 2 } , \ldots . Z _ { n }$ satisfy the equation $\mathbf { 4 Z } \mathbf { Z } ^ { \mathbf { 2 } } + \bar { Z } = \mathbf { 0 }$, then $\sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } \left| \mathrm { Z } _ { \mathrm { i } } \right| ^ { 2 }$ is equal to (A) $\frac { 3 } { 64 }$ (B) $\frac { 3 } { 16 }$ (C) $\frac { 19 } { \frac { 17 } { 64 } }$ (D) $\frac { 1 } { 16 }$

let $\mathrm { z } = ( 1 + i ) ( 1 + 2 i ) ( 1 + 3 i )$ and $( 1 + \mathrm { n } i )$, where $i = \sqrt { - 1 }$ if If $| z | ^ { 2 } = 44200$, then $n$ is equal to:

Let $s = \left\{ z \in C : 4 z ^ { 2 } + \bar { z } = 0 \right\}$ Then $16 \sum _ { z \in s } | z | ^ { 2 }$ is equal to

In a complex number plane, consider the complex numbers $z$ such that $z^3$ is a real number.
(1) Let $C$ be the figure formed by the set of complex numbers $z = x + iy$ satisfying the above condition. Since the arguments of the complex numbers $z$ satisfy $$\arg z = \frac{\pi}{\mathbf{M}}k \quad (k : \text{integer}),$$ figure $C$ consists of three straight lines represented in terms of $x$ and $y$ by the equations $$y = \mathbf{N}, \quad y = \sqrt{\mathbf{O}}\,x, \quad y = -\sqrt{\mathbf{O}}\,x.$$
(2) Suppose that on $C$ there exists only one complex number $z$ satisfying $|z - 1 - i| = r$. Then the value of $r$ is $$r = \frac{\sqrt{\mathbf{Q}} - \square\mathbf{R}}{\square}$$ and the value of $z$ is $$z = \frac{\mathbf{T} + \sqrt{\mathbf{U}}}{\square\mathbf{V}}(1 + \sqrt{\mathbf{W}}\,i).$$

Let us consider the solutions to the equation in the complex number $z$
$$z ^ { 4 } = - 324 \quad \cdots (1)$$
and the solutions to the equation in the complex number $z$
$$z ^ { 4 } = t ^ { 4 } \quad \cdots (2)$$
where $t$ is a positive real number.
(1) To find the solutions to (1), let us set
$$z = r ( \cos \theta + i \sin \theta ) \quad ( r > 0,0 < \theta \leqq 2 \pi )$$
Then
$$z ^ { 4 } = r ^ { \mathbf { M } } ( \cos \mathbf { N } \theta + i \sin \mathbf { N } \theta ) .$$
The values of $r$ and $\theta$ such that this expression is equal to $-324$ are
$$\begin{aligned} & r = \mathbf { O } \sqrt { \mathbf { P } } , \\ & \theta = \frac { \mathbf { Q } } { \mathbf { R } } \pi , \frac { \mathbf { S } } { \mathbf { R } } \pi , \frac { \mathbf { T } } { \mathbf { R } } \pi , \frac { \mathbf { U } } { \mathbf { R } } \pi , \end{aligned}$$
where $\mathbf { Q } < \mathbf { S } < \mathbf { T } < \mathbf { U }$.
(2) There are $\mathbf { V }$ solutions to equation (2), and these solutions are dependent on $t$. Now, consider any one of the solutions to (1) and any one of the solutions to (2), and let $d$ be the distance on the complex number plane between these two solutions. Then, over the interval $0 < t \leqq 4$, the minimum value of $d$ is $\mathbf { W }$ and the maximum value is $\sqrt { \mathbf { X Y } }$.