Number Theory T...

If $$i^2= -1$$, then $$1+ i^2+ i^4 +i^6+i^8 +.............to ( 2n +1)$$ terms is equal to

$$i \, \log \left(\dfrac{x - i}{x + i}\right)$$ is equal to

If $$z = -3- i,$$ find $$|z|$$.

If $$a+ ib= \sum_{k=1}^{101} i^k $$, then $$(a, b)$$ equals 

If $$\left| {z - \dfrac{4}{z}} \right| = 2$$ , then the maximum value of$$\left| z \right|$$ is

$$\dfrac{{{{\left( {1 + i} \right)}^3}}}{{2 + i}}$$  is equal to

The modulus of the complex number $$z=\dfrac{1-i}{3-4i}$$ is

The real part of $$(1 - \cos\theta + 2i \sin\theta)^{-1}$$ is:

Which of the following pair of numbers are $$\text{relatively prime}$$ :-

If $$z=\dfrac{1+i}{\sqrt{2}}$$, then the value of $$z^{1929}$$ is