Embibe Experts Solutions for Chapter: Complex Numbers, Exercise 1: Tamilnadu Board-2018

If $\omega$ is a cube root of unity then the value of ${\left(1-\omega +{\omega }^2\right)}^4+{\left(1+\omega -{\omega }^2\right)}^4$ is 

The modulus and amplitude of the complex number ${\left[e^{3-i\frac{\pi }{4}}\right]}^3$ are respectively

If the point is represented by the complex number $iz$ is rotated about the origin through an angle $\frac{\pi }{2}$ in the counterclockwise direction then the complex number representing the new position is

$\text{'}P\text{'}$ represents the variable complex number $z$. Find the locus of $P$. If $Re\left[\frac{z-1}{z+i}\right]=1$.

In the multiplicative group of the cube root of unity, the order of ${\omega }^2$ is : [$\omega$ is a complex cube root of unity]

 If $-x-iy$ lies in the first quadrant, then $-ix+y$ lies in the:

If $z_1=1+2i, z_2=1-3i$ and $z_3=2+4i$ then, the points on the Argand diagram representing $z_1{z}_2{z}_3, 2z_1{z}_2{z}_3, -7z_1{z}_2{z}_3$ are:

If $n$ is a positive integer, prove that ${\left(\frac{1+\sin \theta -i\cos \theta }{1+\sin \theta +i\cos \theta }\right)}^n=\cos n\left(\frac{{\displaystyle \pi }}{{\displaystyle 2}}-\theta \right)-i\sin n\left(\frac{{\displaystyle \pi }}{{\displaystyle 2}}-\theta \right)$.