Question

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

• A. For real positive numbers k, both roots are purely imaginary.
• B. For all purely imaginary 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.

$10z^2-3iz-k=0$
$⟹z=\frac{3i\pm \sqrt{-9+40k}}{20}$
Now, D = $-$ 9 + 40k.
If k = 1, then D = 31. So (a) is false If k is a negative real number, then D is a negative real number. So (d) is true.
If k = i, then D = 9 + 40i = 16 + 40i - 25 = ${(4+5i)}^2$, and the roots are (1/5) + (2/5)I and - (1/5 -1/10)I. So (c) Is false.
If k = 0 (which is a complex number), then the roots are 0 and (3/10)i. so (b) is false.