1993 AMC 12 Problem 20

Question

Consider the equation

10z^2-3iz-k=0,

where

z

is a complex variable and

i^2=-1.

Which of the following statements is true?

For all positive real numbers

k,

both roots are pure imaginary.

For all negative real numbers

k,

both roots are pure imaginary.

For all pure imaginary numbers

k,

both roots are real and rational.

For all pure imaginary numbers

k,

both roots are real and irrational.

For all complex numbers

k,

neither root is real.

Apply the quadratic formula and simplify the discriminant

For negative real

k,

determine the type of

\sqrt{-9+40k}

The roots are

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

If

k

is negative and real, then

-9+40k

is negative real, so its square roots are pure imaginary. Both resulting values of

z

are therefore pure imaginary. The other universal claims fail, for example by taking

k=1,

i,

or

0

as appropriate. Thus the correct answer is B.

Problem 20 in other years