Question

For the equation $10z^2-4iz-m=0$, where $z$ is a complex variable and $i=\sqrt{-1}$, which of the following is true?

• A. Roots are purely imaginary for all negative real values of $m$
• B. Roots are not real for all complex numbers $m$
• C. Roots are purely imaginary for all positive real values of $m$
• D. None of the above

Answer 1

Answer: (A)

Roots are purely imaginary for all negative real values of $m$

We have, $10z^2-4iz-m=0$
$\Rightarrow z=\frac{4i\pm \sqrt{-16+40m}}{20}$
$=\frac{4i\pm 2\sqrt{-4+10m}}{20}$
$=\frac{2i\pm \sqrt{-4+10m}}{10}$
Now, $D=-4+10m$
If $m=1$, then $D=6$. So, option (c) is false. If $m$ is a negative real number, then $D$ is a negative real number. So, option (a) is true.