Amit M Agarwal Solutions for Chapter: Complex Numbers, Exercise 2: Work Book Exercise 4.2

If $z_n=\cos \frac{\mathrm{\pi }}{\left(2n+1\right)\left(2n+3\right)}+i\sin \frac{\mathrm{\pi }}{\left(2n+1\right)\left(2n+3\right)}$, then $\underset{n\to \infty }{\lim }\left(z_1·z_2·z_3\dots z_n\right)$ is equal to

If $z_1, z_2$ are two complex numbers and $a, b$ are two real numbers, then ${\left|a{z}_1-b{z}_2\right|}^2+{\left|b{z}_1+a{z}_2\right|}^2$ is equal to

All real numbers $x,$ which satisfy the inequality $\left|1+4i-2^{-x}\right|\le 5,$ where $i=\sqrt{-1},x\in R$ are

Square root of $x^2+\frac{1}{x^2}-\frac{4}{i}\left(x-\frac{1}{x}\right)-6$, where $x\in R$ is equal to

The minimum value of $\left|1+z\right|+\left|1-z\right|$, where $z$ is a complex number, is

If $\arg \left(z+a\right)=\frac{\mathrm{\pi }}{6}$ and $\arg \left(z-a\right)=\frac{2\mathrm{\pi }}{3}, a\in R^{+},$ then

Let $z$ be a complex number satisfying the equation ${\left(z^3+3\right)}^2=-16$, then $\left|z\right|$ has the value equal to

For $z_1=\sqrt[6]{\frac{1-i}{1+i\sqrt{3}}}, z_2=\sqrt[6]{\frac{1-i}{\sqrt{3}+i}}, z_3=\sqrt[6]{\frac{1+i}{\sqrt{3}-i}}$ which of the following holds good?