Question

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

The correct option is D Neither real nor purely imaginary roots

If the roots are purely imaginary then they are of the form $\pm i\beta$Hence,$p\left(x\right)=a{x}^2+b$ with $a,b$ of same sign.

$p\left(p\right(x\left)\right)=a(a{x}^2+b{)}^2+b$

If $x$$\in$ $R$ or $x$ is purely imaginary

$\Rightarrow x^2\in R$

$\Rightarrow p\left(x\right)\in R$

$\Rightarrow p\left(p\right(x\left)\right)\ne 0$

Hence, real or purely imaginary number can not satisfy $p\left(p\right(x\left)\right)=0$.