    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+b≠0,∀ x ϵ R Thus, p[p(x)]≠0,∀ x ϵ R. Hence, p[p(x)]=0 has complex roots.  Suggest Corrections  0      Similar questions  Explore more