For a non-zero complex number z , let arg ( z ) denote the principal argument with - π < a r g z…

1. Arg$\left(-1-i\right)=\frac{\pi }{4}, \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} i=\sqrt{-1}$
2. The function $f:\mathbb{R}\to \left(-\pi ,\pi \right]$ defined by $f\left(t\right)=\mathrm{a}\mathrm{r}\mathrm{g}\left(-1+\mathrm{i}\mathrm{t}\right) \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} t\in \mathbb{R}$ , is continuous at all points of $\mathbb{R},\mathbb{ }\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} i=\sqrt{-1}$
3. For any two non-zero complex numbers $z_1 \mathrm{a}\mathrm{n}\mathrm{d} z_2$ , $\mathrm{a}\mathrm{r}\mathrm{g} \left(\frac{z_1}{z_2}\right)-\mathrm{a}\mathrm{r}\mathrm{g}\left(z_1\right)+\mathrm{a}\mathrm{r}\mathrm{g} \left(z_2\right)$ is an integer multiple of $2\pi$
4. For any three given distinct complex numbers $z_1, z_2 \mathrm{a}\mathrm{n}\mathrm{d} z_3$ , the locus of the point $z$ satisfying the condition $\text{a}\text{r}\text{g}\left(\frac{\left(z-z_1\right)\left(z_2-z_3\right)}{\left(z-z_3\right)\left(z_2-z_1\right)}\right)=\pi$ lies on a straight line

Solution

Asked in: JEE Advanced 2018 (Paper 1)

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