Question

Let $z$ be a complex number such that $z+|z|=3+i$. Then which of the following is/are true?

• A. ${\sin }^3\theta +{\cos }^3\theta =\frac{91}{125}$, where $\theta =\arg z$
• B. $|z|=\frac{3}{5}$
• C. $Re\left(\log z\right)=\log \left(\frac{5}{3}\right)$
• D. $z-\overline{z}=-2i$

Answer 1

Answer: (A), (C)

$Re\left(\log z\right)=\log \left(\frac{5}{3}\right)$

Let $z=x+iy$
Given $z+|z|=3+i$
$\Rightarrow x+iy+\sqrt{x^2+y^2}=3+i$
$\Rightarrow x+\sqrt{x^2+y^2}=3\text{and}y=1$
$\Rightarrow x+\sqrt{x^2+1}=3$
$\Rightarrow x^2+1=(3-x{)}^2\Rightarrow x=\frac{4}{3}$
$\Rightarrow z=\frac{4}{3}+i$
$\Rightarrow |z|=\sqrt{\frac{16}{9}+1}=\frac{5}{3}$

$\therefore \tan \theta =\frac{1}{4/3}=\frac{3}{4}(x>0,y>0,z\text{lies in}{Q}_1)$.
$\Rightarrow \sin \theta =\frac{3}{5},\cos \theta =\frac{4}{5}\Rightarrow {\sin }^3\theta +{\cos }^3\theta =\frac{91}{125}$

We know $\log z=\log |z|+i\cdot \theta \text{, where}\theta =\arg z$
$\therefore Re\left(\log z\right)=\log |z|=\log \left(\frac{5}{3}\right)$
$z-\overline{z}=(\frac{4}{3}+i)-(\frac{4}{3}-i)=2i$