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 numbers $k$ , both roots are purely imaginary.
• B. For all complex numbers $k$ , neither root is real.
• C. For all complex 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.

Applying quadratic formula
$z=\frac{3i\pm \sqrt{-9+40k}}{20}$
$=\frac{3i\pm \sqrt{40k-9}}{20}$
Now
For $40k-9<0$
Both the roots are purely imaginary.
Therefore
For $k<\frac{9}{40}$
The roots are imaginary.
Therefore, for all negative values of k, both the roots are purely imaginary.
Hence, option 'D' is correct.