Question

In the quadratic equation $p(x) = 0$ with real coefficients has purely imaginary roots. Then, the equation $p[p(x)] = 0$ has

• A. only purely imaginary roots
• B. all real roots
• C. two real and two purely imaginary roots
• D. neither real nor purely imaginary roots

Answer 1

Answer: (D)

neither real nor purely imaginary roots

Answer 2

$P(x)=a x^{2}+b$ with $a, b$ of same sign. $P(P(x))=a\left(a x^{2}+b\right)^{2}+b$
If $x \in R$ or ix $\in R$
$\Rightarrow x ^{2} \in R$
$\Rightarrow P ( x ) \in R$
$\Rightarrow P ( P ( x )) \neq 0$
Hence real or purely imaginary number can not satisfy $P ( P ( x ))=0$.