Question

For a complex number $z$ , if $z^2+\stackrel{−}{z}−z=4i$ and $z$ does not lie in the first quadrant, then (where $i^2=−1$ )

• A. $âˆ\pounds zâˆ\pounds =\sqrt{2}$
• B. $âˆ\pounds zâˆ\pounds =2\sqrt{2}$
• C. $arg\left(z\right)=\frac{−π}{4}$
• D. $arg\left(z\right)=\frac{3Ï€}{4}$

Answer 1

Answer: (A)

Put $z=x+iy$ $⇒x^2−y^2+2ixy−2iy=4i$
$⇒\{\begin{array}{c}{x}^2−y^2=0⇒x=y\text{ or }−y\\ xy−y=2\end{array}$
Case
(i) if $x=y,$ then $x^2−x−2=0$ $⇒x=−1,2$
$⇒\{\begin{array}{c}{z}_1=−1−i\\ z_2=2+2i\text{ (rejected) }\end{array}$
Case (ii) if $x=−y,$ then $x^2−x+2=0$
$⇒x∈ϕ$
Hence, $âˆ\pounds zâˆ\pounds =âˆ\pounds −1−iâˆ\pounds =\sqrt{2}$