Number Theory T...

The expression $$\dfrac {(1 + i)^{n}}{(1 - i)^{n - 2}}$$ equals.

Given : $$u = 1+i \sqrt{3}$$ and $$v = \sqrt{3} + i$$

Calculate $$\dfrac{u^3 }{ v^4}$$

If $$z_1$$ and $$z_2$$ are complex numbers with $$|z_1|=|z_2|$$, then which of the following is/are correct?
1. $$z_1=z_2$$
2. Real part of $$z_1 =$$ Real part of $$z_2$$
3. Imaginary part of $$z_1 =$$ Imaginary part of $$z_2$$
Select the correct answer using the statements given below :

$$p + iq = (2 - 3i) (4 + 2i)$$ then $$q$$ is

Two complex numbers are represented by ordered pairs $$z_1: (a,0)\ \&\  z_2: (c,d)$$, which of the following is correct simplification for $$z_1\times z_2=$$?

If $${ x }^{ 2 }+{ y }^{ 2 }=1$$ then value of $$\dfrac { 1+x+iy }{ 1+x-iy } $$ is

The value of $$ \sum _{ k=0 }^{ n }{ (i^k + i^{k+1} ) } , $$ where $$ i^2 = -1 ,$$ is equal to :

If $$f\left( z \right) =\dfrac { 1-{ z }^{ 3 } }{ 1-z } $$, where $$z=x+iy$$ with $$z\neq 1$$, then $$Re\overline { \left\{ f\left( z \right)  \right\}  } =0$$ reduces to

If $$z_{1}$$ and $$z_{2}$$ be complex numbers such that $$z_{1} + i(\overline {z_{2}}) = 0$$ and $$arg (\overline {z_{1}}z_{2}) = \dfrac {\pi}{3}$$. Then, $$arg (\overline {z_{1}})$$ is equal to

If $$\left( \dfrac{1 + i}{1 - i} \right)^m = 1$$, then the least positive integral value of m is