First slide

Theory of equations

Question

The quadratic equation p(x) = 0 with real coefficients has purely imaginary roots. Then the equation p(p(x)) = 0 has

Moderate

A

only purely imaginary roots
B

all real roots
C

two real and two purely imaginary roots
D

neither real nor purely imaginary roots

Solution

Since p(x) = 0 has purely imaginary roots,
p(x) = ax² + c, where a and c have same sign.
Also, p(p(x)) = 0
$\Rightarrow$ p(x) is purely imaginary
$\Rightarrow$ ax² + c is purely imaginary
Hence, x cannot be either purely real or purely imaginary.

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