Question

If $z$ be a complex number satisfying $|Re\left(z\right)|+|Im\left(z\right)|=4$, then $|z|$ cannot be:

• A. $\sqrt{7}$
• B. $\sqrt{\frac{17}{2}}$
• C. $\sqrt{10}$
• D. $\sqrt{8}$

Answer 1

The correct option is A $\sqrt{7}$

$|Re\left(z\right)|+|Im\left(z\right)|=4$
Let $z=x+iy$
$\Rightarrow |x|+|y|=4$
$\therefore z$ lies on the square.
Maximum value of $|z|=4$ when $z=4,-4,4i,-4i$
Minimum value of $|z|=2\sqrt{2}$ when $z=\pm 2+2i,2\pm 2i$
$|z|\in [2\sqrt{2},4]$
$|z|\in [\sqrt{8},\sqrt{16}]$
$|z|\ne \sqrt{7}$