Question

The quadratic equation `p(x)=0` with real coefficients has purely imaginary roots. Then the equation `p(p(x))=0` has only purely imaginary roots at real roots two real and purely imaginary roots neither real nor purely imaginary roots
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

Correct Answer - D
If quadratric equation has purely imaginary roots, then coefficient of x must be equal to zero.
Let `p(x)=ax^(2)+b` with a, b of same sing and `a, b in R.` Then `p[p(x)=a(ax^(2)+b)^(2)+b`
P(x) has imaginary roots say ix.
Then , also `ax^(2)+b in Rand (ax^(2)+b)^(2)gt0`
`a (ax^(2)+b)^(2)+b ne0, AAx`
Thus, `p[p(x)] ne0, AAx`