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.