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

The correct option is D neither real nor purely imaginary roots

If quadratic equation has purely imaginary roots, then coefficient of x must be equal to zero.
$\therefore p\left(x\right)=a{x}^2+c$ with a,c having same sign and a,c $\in$ R.
Then, $p\left[p\right(x\left)\right]=a(a{x}^2+c{)}^2+c\therefore p[p\left(x\right)]=a^3{x}^4+2a^{2}c{x}^2+(a{c}^2+c)$
Considering this as a quadratic in $x^2$, and solving it we get:
$x^2=\frac{-2a^{2}c\pm \sqrt{(2a^{2}c{)}^2-4(a^3\left)\right(a{c}^2+c)}}{2a^3}{x}^2=\frac{-c}{a}\pm \frac{i\sqrt{ac}}{a^2}$[$\therefore$ a,c have the same sign]
$\therefore$ $x^2$has both real and imaginary parts $\Rightarrow$ x too has both real and imaginary parts.