Let $S$ be the set of all points $z$ in the complex plane such that $$\left(1 + \frac{1}{z}\right)^4 = 1$$ Then, the points of $S$ are
(A) vertices of a rectangle
(B) vertices of a right-angled triangle
(C) vertices of an equilateral triangle
(D) collinear

Let $z$ be a complex number satisfying $|z|^3 + 2z^2 + 4\bar{z} - 8 = 0$, where $\bar{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.
Which of the following statements is TRUE?
(A) $|z|^2 = 2$
(B) $|z|^2 = 4$
(C) $|z|^2 = 8$
(D) $|z|^2 = 16$

Let $\bar { z }$ denote the complex conjugate of a complex number $z$ and let $i = \sqrt { - 1 }$. In the set of complex numbers, the number of distinct roots of the equation
$$\bar { z } - z ^ { 2 } = i \left( \bar { z } + z ^ { 2 } \right)$$
is $\_\_\_\_$.

Let $z$ be a complex number satisfying $| z | ^ { 3 } + 2 z ^ { 2 } + 4 \bar { z } - 8 = 0$, where $\bar { z }$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.
Match each entry in List-I to the correct entries in List-II.
List-I
(P) $| z | ^ { 2 }$ is equal to
(Q) $| z - \bar { z } | ^ { 2 }$ is equal to
(R) $| z | ^ { 2 } + | z + \bar { z } | ^ { 2 }$ is equal to
(S) $| z + 1 | ^ { 2 }$ is equal to
List-II
(1) 12
(2) 4
(3) 8
(4) 10
(5) 7
The correct option is:
(A) $( P ) \rightarrow ( 1 ) \quad ( Q ) \rightarrow ( 3 ) \quad ( R ) \rightarrow ( 5 ) \quad ( S ) \rightarrow ( 4 )$
(B) $( P ) \rightarrow ( 2 ) \quad ( Q ) \rightarrow ( 1 ) \quad ( R ) \rightarrow ( 3 ) \quad ( S ) \rightarrow ( 5 )$
(C) $( P ) \rightarrow ( 2 ) \quad ( Q ) \rightarrow ( 4 ) \quad ( R ) \rightarrow ( 5 ) \quad ( S ) \rightarrow ( 1 )$
(D) $( P ) \rightarrow ( 2 ) \quad ( Q ) \rightarrow ( 3 ) \quad ( R ) \rightarrow ( 5 ) \quad ( S ) \rightarrow ( 4 )$

If the equation $x ^ { 2 } + b x + 45 = 0 , b \in R$ has conjugate complex roots and they satisfy $| z + 1 | = 2 \sqrt { 10 }$, then
(1) $b ^ { 2 } - b = 30$
(2) $b ^ { 2 } + b = 72$
(3) $b ^ { 2 } - b = 42$
(4) $b ^ { 2 } + b = 12$

The area of the polygon, whose vertices are the non-real roots of the equation $\bar { z } = i z ^ { 2 }$ is
(1) $\frac { 3 \sqrt { 3 } } { 2 }$
(2) $\frac { 3 \sqrt { 3 } } { 4 }$
(3) $\frac { \sqrt { 3 } } { 4 }$
(4) $\frac { \sqrt { 3 } } { 2 }$

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

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) $\frac { 1 } { 4 }$

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) $25\sqrt{3}$