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. Two real and two purely imaginary roots
• B. Neither real nor purely imaginary roots
• C. Only purely imaginary roots
• D. All real roots

Answer 1

The correct option is B Neither real nor purely imaginary roots

Let $p\left(x\right)=x^2+a(a>0)$ (roots are purely imaginary)
$p\left(p\right(x\left)\right)=(x^2+a{)}^2+a(a\in R)$
$x^4+2a\left(x^2\right)+a^2+a=0$
$\Rightarrow x^2=\frac{-2a\pm \sqrt{4a^2-4a^2-4a}}{2}=-a\pm \sqrt{ai}$
$x=\pm \sqrt{a\pm \sqrt{ai}}=x_1+i{y}_1$