Let  \(z \in C\).

Given that  \(|z|=1\)  and  \(z \neq 1, \operatorname{Re}\left(\dfrac{1}{1-z}\right)\) is

1. \(-\dfrac{1}{2}\)
2. \(0\)
3. \(\dfrac{1}{2}\)
4. \(\dfrac{\sqrt{3}}{2}\)

Let  \(z=3+k i\)  where  \(k \in R\).

A value of \(k\) that makes  \(z^2+4 i z+3\)  purely imaginary is

1. \(-2\)
2. \(-1\)
3. \(1\)
4. \(2\)

Given the complex number \(z=a+b i\), where \(a \in R \backslash\{0\}\) and \(b \in R, \dfrac{4 z \bar{z}}{(z+\bar{z})^2}\) is equivalent to

1. \(1+\left(\dfrac{\operatorname{Im}(z)}{\operatorname{Re}(z)}\right)^2\)
2. \(4\bigl[\operatorname{Re}(z) \times \operatorname{Im}(z)\bigr]\)
3. \(4\Bigl([\operatorname{Re}(z)]^2+[\operatorname{Im}(z)]^2\Bigr)\)
4. \(4\Bigl[1+(\operatorname{Re}(z)+\operatorname{Im}(z))^2\Bigr]\)
5. \(\dfrac{2 \times \operatorname{Im}(z)}{[\operatorname{Re}(z)]^2}\)

Which one of the following statements is false for  `z_1, z_2 ∈ C`?

1. `z^(−1) = (barz)/(|z|^2), z != 0`
2. `|z_1 + z_2| > |z_1| + |z_2|`
3. `(z_1)/(z_2) = (z_1barz_2)/(|z_2|^2), z_2 != 0`
4. `|z_1z_2| = |z_1||z_2|`
5. `|(z_1)/(z_2)| = (|z_1|)/(|z_2|), z_2 != 0`

If  `z = a + bi`, where both `a` and `b` are non-zero real numbers and  `z in C`, which of the following does not represent a real number?

1. `z + barz`
2. `|\ z\ |`
3. `zbarz`
4. `z^2-2abi`
5. `(z-bar z) (z + bar z)`

Which one of the following relations has a graph that passes through the point  `1 + 2i`  in the complex plane?

1. `zbarz = sqrt5`
2. `text(Arg)(z) = pi/3`
3. `| z - 1 | = | z - 2i |`
4. `text(Re)(z) = 2text(Im)(z)`
5. `z + barz = 2`