Difference between revisions of "1993 AHSME Problems/Problem 20"
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== Solution == | == Solution == | ||
− | <math>\fbox{B}</math> | + | |
+ | Let <math>r_1</math> and <math>r_2</math> denote the roots of the polynomial. Then <math>r_1 + r_2 = 3i</math> is pure imaginary, so <math>r_1</math> and <math>r_2</math> have offsetting real parts. Write <math>r_1 = a + bi</math> and <math>r_2 = -a + ci</math>. | ||
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+ | Now <math>-k = r_1 r_2 = -a^2 -bc + a(c-b)i</math>. In the case that <math>k</math> is real, then <math>a(c-b)=0</math> so either <math>a=0</math> or that <math>b=c</math>. In the first case, the roots are pure imaginary and in the second case we have <math>k = a^2+b^2</math>, a positive number. | ||
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+ | We can therefore conclude that if <math>k</math> is real and negative, it must be the first case and the roots are pure imaginary. | ||
+ | |||
+ | It's possible to rule out the other cases by reasoning through the cases, but this is enough to show that <math>\fbox{B}</math> is true. | ||
== See also == | == See also == |
Latest revision as of 22:28, 27 May 2021
Problem
Consider the equation , where is a complex variable and . Which of the following statements is true?
Solution
Let and denote the roots of the polynomial. Then is pure imaginary, so and have offsetting real parts. Write and .
Now . In the case that is real, then so either or that . In the first case, the roots are pure imaginary and in the second case we have , a positive number.
We can therefore conclude that if is real and negative, it must be the first case and the roots are pure imaginary.
It's possible to rule out the other cases by reasoning through the cases, but this is enough to show that is true.
See also
1993 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.