Complex Numbers...

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

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

A particle starts from a point $$z_0= I + i$$, where $$i
=\sqrt{-1}$$ It moves horizontally away from origin by $$2$$ units and then
vertically away from origin by $$3$$ units to reach a point$$ z_1$$. From $$z_1$$
particle moves $$\sqrt{5}$$ units in the direction of $$2\hat i + \hat j$$ and
then it moves through an angle of $$\cos e{c^{ - 1}}\sqrt 2 $$ in anticlockwise
direction of a circle with centre at origin to reach a point $$z_2$$ . The arg $$z_2$$ is given by

Find the nature of the roots for the equation $$n^{2}+2\sqrt{2} n+1=0$$

For the equation $$|x|^2+|x|-6=0$$, the roots are 

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

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

The equation $$4\sin^2x+4\sin x+a^2-3=0$$ possesses a solution if 'a' belongs to the interval.