Amit M Agarwal Solutions for Chapter: Complex Numbers, Exercise 6: Entrances Gallery

If ${\left|z_1+z_2\right|}^2={\left|z_1\right|}^2+{\left|z_2\right|}^2,$ then $\frac{z_1}{z_2}$ is

The value of $\left|\frac{1+i\sqrt{3}}{{\left(1+\frac{1}{i+1}\right)}^2}\right|$ is

If $z_1$ and $z_2$ are two non-zero complex numbers such that $\left|z_1+z_2\right|=\left|z_1\right|+\left|z_2\right|$, then $\arg \left(\frac{z_1}{z_2}\right)$ is

If $z_1=\sqrt{2}\left(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}\right)$ and $z_2=\sqrt{3}\left(\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}\right),$ then $\left|z_1{z}_2\right|$ is

If $z=r\left(\cos \theta +i\sin \theta \right),$ then the value of $\frac{z}{\overline{z}}+\frac{\overline{z}}{z}$ is

If $z=\frac{4}{1+i},$ then $\overline{z}$ is (where, $\overline{z}$ is complex conjugate of $z$)

If $-\pi <\arg \left(z\right)<-\frac{\pi }{2},$ then $\arg \left(\overline{z}\right)-\arg \left(-\overline{z}\right)$ is

The value of $\frac{\cos 30°+i\sin 30°}{\cos 60°-i\sin 60°}$ is equal to