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

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For real positive numbers k, both roots are purely imaginary.

For all complex numbers k, neither root is real.

For all purely imaginary numbers k, both roots are real and irrational.

For real negative numbers k, both roots are purely imaginary.

answer is D.

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Detailed Solution

10z2−3iz−k=0⇒ z=3i±−9+40k20Now, 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 15+25i and −-15-110i. So (3) is false.If k = 0 (which is a complex number), then the roots are 0 and 310i So (2) is false.

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