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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

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B

All real roots

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C

Two real and two purely imaginary roots

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D

Neither real nor purely imaginary roots

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Solution

The correct option is D

Neither real nor purely imaginary roots


If a quadratic equation has purely imaginary roots, then the coefficient of x must be equal to zero.
Let p(x)=ax2+b with a,b of same sign and a,b ϵ R.
Then, p[p(x)]=a(ax2+b)2+b
p(x) has imaginary roots say ix.
Then, ax2+b ϵ R and (ax2+b)2>0
a(ax2+b)2+b0, x ϵ R
Thus, p[p(x)]0, x ϵ R.

Hence, p[p(x)]=0 has complex roots.


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