Question

For a non-zero complex number $z$, let arg(z) denote the principal argument with $- \pi < arg(z) \leq \pi$. Then, which of the following statement(s) is (are) FALSE?

• A. $arg(-1 - i)$ = $\frac{\pi}{4}$, where $i = \sqrt{-1}$
• B. The function $f : \mathbb R \to (- \pi , \pi ]$, defined by $f (t) $ = arg $(-1 + it )$ for all $t \in \mathbb R$ , is continuous at all points of $\mathbb R$, where $ i = \sqrt{-1}$
• C. For any two non-zero complex numbers $z_1$ and $z_2, arg \left(\frac{z_1}{z_2}\right) - arg (z_1) + arg (z_2)$ is an integer multiple of $2 \pi $
• D. For any three given distinct complex numbers $z_1, z_2$ and $z_3$, the locus of the point $z$ satisfying the condition $arg \left( \frac{(z - z_1)(z_2 - z_2)}{(z - z_3)(z_2 - z_1)}\right) = \pi $, lies on a straight line

Answer 1

Answer: (A), (B), (D)

a) $\arg (-1-i)=-\frac{3 \pi}{4}$
b) $f(t)=\arg (-1+i t)=\begin{cases}\pi-\tan ^{-1}(t), & t \geq 0 \\ -\pi+\tan ^{-1} & t < 0\end{cases}$
Discontinuous at $t = 0$
$\arg \left(\frac{z_{1}}{z_{2}}\right)-\arg \left(z_{1}\right)+\arg \left(z_{2}\right)$
$=\arg z_{1}-\arg \left(z_{2}\right)+2 n \pi-\arg \left(z_{1}\right)+\arg \left(z_{2}\right)=2 n \pi$
d) $\arg \left(\frac{\left(z-z_{1}\right)\left(z_{2}-z_{3}\right)}{\left(z-z_{3}\right)\left(z_{2}-z_{1}\right)}\right)=\pi$
$\Rightarrow \frac{\left(z-z_{1}\right)\left(z_{2}-z_{3}\right)}{\left(z-z_{3}\right)\left(z_{2}-z_{1}\right)}$ is real
$\Rightarrow z, z_{1}, z_{2}, z_{3}$ are concyclic