First slide

Algebra of complex numbers

Question

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

Moderate

A

For real positive numbers 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.

Solution

$10 z^{2} - 3 iz - k = 0$
$\Rightarrow z = \frac{3 i \pm \sqrt{- 9 + 40 k}}{20}$
Now, D = -9 + 40k. If k = 1, then D = 31. So (1) is false.
If k is a negative real number, then D is a negative real number.
So (4) is true.
If k = i, then D = -4 + 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 (2) is false.

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