G Tewani Solutions for Chapter: Complex Numbers, Exercise 1: DPP

$a, b, c$ are positive numbers in Geometric Progression and the equation $(a+di)x^2+2(b+ei)x+(c+if)=0$ has one real root. Then, $\frac{a}{d}, \frac{b}{e}, \frac{c}{f}$  are in  $(a, b, c, d, e, f\in R \& i=\sqrt{-1})$.

The equation $z^3+iz-1=0, i=\sqrt{-1}$ has

If $a,b$ are complex numbers and one of the roots of the equation $x^2+ax+b=0$ is purely real whereas, the other root is purely imaginary, and ${\mathrm{a}}^2-{\overline{\mathrm{a}}}^2=kb$. Then $k$ is:

If $Z$ is a non-real complex number, then the minimum value of $\frac{\mathit{Im}{Z}^5}{{\mathit{Im}}^{5}Z}$ is

If $z_1, z_2, z_3$ are any three complex numbers on the argand plane, then $z_1\left(Im\left({\overline{z}}_2{z}_3\right)\right)+z_2\left(\mathrm{Im}\left({\overline{z}}_3{z}_1\right)\right)+z_3\left(\mathrm{Im}\left({\overline{z}}_1{z}_2\right)\right)$ is equal to

Complex numbers whose real and imaginary parts $x$ and $y$ are integers and satisfy the equation $3x^2-\left|xy\right|-2y^2+7=0$.

For a complex number $z$, if all the three roots of $a{z}^3+b{z}^2+cz+d=0$ have negative real parts $(a, b, c\in R)$, then 

Suppose three real numbers $a, b, c$ are in $\mathrm{G}.\mathrm{P}.$ and $z=\frac{a+ib}{c-ib}$. Then,