Question

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

• A. arg$(-1-i)=\frac{\pi }{4}$, where $i=\sqrt{-1}$
• B. The function $f:R\to (-\pi ,\pi ]$, defined by $f\left(t\right)=arg(-1+it)$ for all t $ϵR$, is continuous at all points of $R$, 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 $\left(z_1\right)+$arg$\left(z_2\right)$ 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 $\left(\frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}\right)=\pi$, lies on a straight line

Answer 1

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

arg$(-1-i)=\frac{\pi }{4}$, where $i=\sqrt{-1}$
The function $f:R\to (-\pi ,\pi ]$, defined by $f\left(t\right)=arg(-1+it)$ for all t $ϵR$, is continuous at all points of $R$, where $i=\sqrt{-1}$
For any three given distinct complex numbers $z_1,z_2$ and $z_3$, the locus of the point z satisfying the condition arg $\left(\frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}\right)=\pi$, lies on a straight line

(A) $arg(-1-i)=-\frac{3\pi }{4}$,
(B) $f\left(t\right)=arg(-1+it)=\{\begin{array}{cc}\pi +{\tan }^{-1}\left(t\right),& t\ge 0\\ -\pi +{\tan }^{-1}\left(t\right),& t<0\end{array}$
Discontinuous at $t=0$.
(C) $arg\left(\frac{z_1}{z_2}\right)-arg\left(z_1\right)+arg\left(z_2\right)$ $=arg\left(z_1\right)-arg\left(z_2\right)+2n\pi -arg\left(z_1\right)+arg\left(z_2\right)=2n\pi$.

(D) $arg\left(\frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}\right)=\pi$

$\Rightarrow \frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}$ is real.

$\Rightarrow z,z_1,z_2,z_3$ are concyclic.