Question

Consider the equation $10z^2-3iz-k=0$, where $z$ is a complex variable and $i^2=-1$. Which of the following statements is true?

• A. For real positive number $k$, both roots are purely imaginary.
• B. For all complex numbers $k$, neither root is real.
• C. For all purely imaginary numbers $k$, both roots are real and irrational.
• D. For real negative numbers $k$, both roots are purely imaginary.

Answer 1

The correct option is D For real negative numbers $k$, both roots are purely imaginary.

Given, $10z^2-3iz-k=0$

Application of quadratic formula gives us

$z=\frac{3i\pm \sqrt{-9+40k}}{20}$

$z=\frac{-3i\pm \sqrt{40k-9}}{20}$

For purely imaginary roots

$40k-9<0$

$k<\frac{9}{40}$

Hence the above equation has imaginary roots if $kϵ(-\infty ,\frac{9}{40})$.