Question

Consider the equation 10z² – 3iz – k = 0, where z is a complex variable and i² = –1.Which of the following statements is True?

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

Answer 1

Answer: (B)

For real negative numbers k, both roots are pure imaginary

Answer 2

Use the quadratic formula to obtain

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

which has the discriminant, D = –9 + 40k.

If k = 1, then D = 31, so (1) is false.

If k = i, then D = –9 + 40i = 16 + 40i – 25 = (4 + 5i)²,

and the roots are $\frac{1}{5} + \frac{2}{5}i$ and $- \frac{1}{5} - \frac{1}{10}i$, So (3) is false.

If k = 0(which is a complex number), then the roots are 0 and $\frac{3}{10}i$, so (4) is false.