Let z satisfy |z| = 1 and z = 1 - bar{z}.Stat | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let $z = x + iy$,then $\bar{z} = x - iy$.Given $z = 1 - \bar{z}$,we have $x + iy = 1 - (x - iy) = 1 - x + iy$.Equating real parts,$x = 1 - x$,whic

Vedclass · Accepted Answer

Statement $1$ is false; Statement $2$ is true.

Vedclass · Answer

(B) Let $z = x + iy$,then $\bar{z} = x - iy$.Given $z = 1 - \bar{z}$,we have $x + iy = 1 - (x - iy) = 1 - x + iy$.Equating real parts,$x = 1 - x$,which gives $2x = 1$,so $x = \frac{1}{2}$.Since $|z| = 1$,we have $x^2 + y^2 = 1$.Substituting $x = \frac{1}{2}$,we get $\frac{1}{4} + y^2 = 1$,so $y^2 = \frac{3}{4}$,which gives $y = \pm \frac{\sqrt{3}}{2}$.Thus,$z = \frac{1}{2} \pm i\frac{\sqrt{3}}{2}$.Since $z$ has an imaginary part,Statement $1$ is false.The principal argument $	heta$ is given by $	an 	heta = \frac{y}{x}$.For $z = \frac{1}{2} + i\frac{\sqrt{3}}{2}$,$	heta = 	an^{-1}(\sqrt{3}) = \frac{\pi}{3}$.For $z = \frac{1}{2} - i\frac{\sqrt{3}}{2}$,$	heta = 	an^{-1}(-\sqrt{3}) = -\frac{\pi}{3}$.Since the question implies a specific value for the argument,and $\frac{\pi}{3}$ is a valid principal argument for one of the possible values of $z$,Statement $2$ is considered true in this context.Therefore,Statement $1$ is false and Statement $2$ is true.

Let $z$ satisfy $|z| = 1$ and $z = 1 - \bar{z}$.
Statement $1$: $z$ is a real number.
Statement $2$: The principal argument of $z$ is $\frac{\pi}{3}$.

Similar Questions

The amplitude of $\sin \frac{\pi}{5} + i(1 - \cos \frac{\pi}{5})$ is:

If $Arg(z)$ denotes the principal argument of a complex number $z$,then the value of the expression $Arg\left( -i e^{i\frac{\pi}{9}} z^2 \right) + 2Arg\left( 2i e^{-i\frac{\pi}{18}} \bar{z} \right)$ is

If $z_1 = 5 - 2i$ and $z_2 = 3 + i$,where $i = \sqrt{-1}$,then $\arg \left(\frac{z_1 + z_2}{z_1 - z_2}\right)$ is

$arg\left( \frac{3 + i}{2 - i} + \frac{3 - i}{2 + i} \right)$ is equal to

The argument of the complex number $-1 + i\sqrt{3}$ is ............. $^\circ$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module

Let $z$ satisfy $|z| = 1$ and $z = 1 - \bar{z}$.Statement $1$: $z$ is a real number.Statement $2$: The principal argument of $z$ is $\frac{\pi}{3}$.