Consider the equation `10 z^2-3i z-k=0,w h e r ez` is a following complex variable and `i^2=-1.` Which of the following statements ils true? For real complex numbers `k` , both roots are purely imaginary. For all complex numbers `k` , neither both roots is real. For all purely imaginary numbers `k` , both roots are real and irrational. For real negative numbers `k` , both roots are purely imaginary.

Options

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

Answer

D

Explanation

`10z^(2) - 3iz k = 0`
`rArr z = (3ipm sqrt(-9 + 40 K))/(20)`
Now , D `= - 9 + 40 K`. 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 ture.
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 (3) is false.
If k = 0 ( which is a complex number), then the roots are 0 and `(3//10)i` So (2) is false.