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1993 AMC 12 #20 - Complex Equation

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January 2, 2012
quadraticscomplex numbersalgebraquadratic formulaAMC

Problem Statement

Consider the equation 10z2−3iz−k=010z^2-3iz-k=0, where zz is a complex variable and i2=−1i^2=-1. Which of the following statements is true?
<spanclass=′latex−bold′>(A)</span> For all positive real numbers k, both roots are pure imaginary.<spanclass=′latex−bold′>(B)</span> For all negative real numbers k, both roots are pure imaginary.<spanclass=′latex−bold′>(C)</span> For all pure imaginary numbers k, both roots are real and rational.<spanclass=′latex−bold′>(D)</span> For all pure imaginary numbers k, both roots are real and irrational.<spanclass=′latex−bold′>(E)</span> For all complex numbers k, neither root is real. <span class='latex-bold'>(A)</span>\ \text{For all positive real numbers}\ k,\ \text{both roots are pure imaginary.} \\ \qquad<span class='latex-bold'>(B)</span>\ \text{For all negative real numbers}\ k,\ \text{both roots are pure imaginary.} \\ \qquad<span class='latex-bold'>(C)</span>\ \text{For all pure imaginary numbers}\ k,\ \text{both roots are real and rational.} \\ \qquad<span class='latex-bold'>(D)</span>\ \text{For all pure imaginary numbers}\ k,\ \text{both roots are real and irrational.} \\ \qquad<span class='latex-bold'>(E)</span>\ \text{For all complex numbers}\ k,\ \text{neither root is real.}
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