Embibe Experts Solutions for Chapter: Complex Numbers, Exercise 1: JEE Advanced Paper 1 - 2020

Let ${\theta }_1,{\theta }_2,\dots ,{\theta }_{10}$ be positive valued angles (in radian) such that ${\theta }_1+{\theta }_2+\cdots +{\theta }_{10}=2\pi$. Define the complex numbers $z_1=e^{i{\theta }_1}, z_k=z_{k-1}{e}^{i{\theta }_k}$ for $k=2,3,\dots ,10$, where $i=\sqrt{-1}$. Consider the statements $P$ and $Q$ given below:

$P:\left|z_2-z_1\right|+\left|z_3-z_2\right|+\cdots +\left|z_{10}-z_9\right|+\left|z_1-z_{10}\right|\le 2\pi$

$Q:\left|z_2^2-z_1^2\right|+\left|z_3^2-z_2^2\right|+\cdots +\left|z_{10}^2-z_9^2\right|+\left|z_1^2-z_{10}^2\right|\le 4\pi$

For any complex number $w=c+id$, let $\arg \left(w\right)\in (-\pi ,\pi ]$, where $i=\sqrt{-1}$. Let $\alpha$ and $\beta$ be real numbers such that for all complex numbers $z=x+iy$ satisfying $\arg \left(\frac{z+\alpha }{z+\beta }\right)=\frac{\pi }{4}$, the ordered pair $(x, y)$ lies on the circle $x^2+y^2+5x-3y+4=0$

Then which of the following statements is(are) TRUE?

Let $a, b, x$ and $y$ be real numbers such that $a-b=1$ and $y\ne 0$ . If the complex number $z=x+iy$ satisfies $\mathrm{Im}\left(\frac{az+b}{z+1}\right)=y,$ then which of the following is (are) possible value (s) of $x?$

Let $s, t, r$ be non-zero complex numbers and $L$ be the set of solutions $z=x+iy \left(x, y\in \mathrm{ℝ}, i=\sqrt{-1}\right)$ of the equation $sz+t\overline{z}+r=0$, where $\overline{z}=x-iy.$ Then, which of the following statement(s) is (are) TRUE?

Let $S$ be the set of all complex numbers $z$ satisfying $\left|z^2+z+1\right|=1$. Then which of the following statements is/are TRUE?

Let $S$ be the set of all complex numbers $z$ satisfying $\left|z-2+i\right|\ge \sqrt{5}.$ If the complex number $z_0$ is such that $\frac{1}{\left|z_0-1\right|}$ is the maximum of the set $\left\{\frac{1}{\left|z-1\right|}:z\in S\right\},$ then the principal argument of $\frac{4-z_0-\overline{z_0}}{z_0-\overline{z_0}+2i}$ is

Let $\omega \ne 1$ be a cube root of unity. Then the minimum of the set $\{{\left|a+b\omega +c{\omega }^2\right|}^2:a,b,c$ are distinct non-zero integers $\}$ equals ________

For a non-zero complex number $z$, let $\arg \left(z\right)$ denote the principal argument with $-\pi <\mathrm{a}\mathrm{r}\mathrm{g}\left(z\right)\le \pi$ then, which of the following statement(s) is (are) FALSE?