Embibe Experts Solutions for Exercise 4: Assignment

If a complex number $z$ moves such that $|Re(\overline{z}\left)\right|+\left|\mathrm{Im}\right(z\left)\right|=\sqrt{3}$, then the area bounded by all such positions of $z$ is

If both the roots of the quadratic equation $a{x}^2+bx+c=0; a, b, c\in R-\left\{0\right\}$ are nonreal and if ' $\theta$ ' denotes the principle argument of the complex number $Z=a+i c,$ then which of the following options is necessarily true?

Consider two complex numbers $z_1$ and $z_2$ such that $\left|z_1+z_2\right|=\left|z_1\right|-\left|z_2\right|,$ then the complex number $z=\frac{z_1}{z_2}$ will always lie on a

The sum of the real roots of polynomial $72-36z-26z^2+13z^3+2z^4-z^5=0$ will be

If $f\left(x\right)=7x^5+3x^4+a_1{x}^3+a_2{x}^2+a_{3}x+1$ such that for any $z=x+i y$ where $xy\ne 0$ and $x, y\in R, f\left(\overline{z}\right)=\overline{f\left(z\right)}$ then the product of $a_1 a_2 a_3$

$a, b, c$ are real numbers in the polynomial $p\left(z\right)=2z^4+a{z}^3+b{z}^2+cz+5.$ If two roots of the equation $p\left(z\right)=0$ are $2$ and $i(=\sqrt{-1}),$ then the value of $a$ is

Given that the equation $z^2+(a+ib)z+c+id=0$ where $a, b, c, d$ are non-zero, has a real root then

It is given that the equation $|z{|}^2-2iz+2\alpha (\lambda +i)=0$ possesses a solution for all $\alpha \in R,$ then the number of integral value(s) of ' $\lambda$ ' for which it is true is