Multiple Choice

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

Aarg$$(-1-i)=\displaystyle\frac{\pi}{4}$$, where $$i=\sqrt{-1}$$
Correct Answer
BThe function $$f: \mathbb{R}\rightarrow (-\pi , \pi]$$, defined by $$f(t)=arg(-1+it)$$ for all t $$\epsilon\ \mathbb{R}$$, is continuous at all points of $$\mathbb{R}$$, where $$i=\sqrt{-1}$$
Correct Answer
CFor any two non-zero complex numbers $$z_1$$ and $$z_2$$, arg$$\left(\displaystyle\frac{z_1}{z_2}\right)-$$arg $$(z_1)+$$arg$$(z_2)$$ is an integer multiple of $$2\pi$$
DFor any three given distinct complex numbers $$z_1, z_2$$ and $$z_3$$, the locus of the point z satisfying the condition arg $$\left(\displaystyle\frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}\right)=\pi$$, lies on a straight line
Correct Answer

Solution

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

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

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

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

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