Discussion Forum



1)

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 and nor purely imaginary roots

Answer:

Option D

Explanation:

Plan  If quadratic equation has purely imaginary  roots, then coefficient of x must be  equal to zero

 Let p(x)= ax2+b  with a, b of same sign and $a,b \epsilon R$

 then p(p(x))= a(a x2+b)2 +b

p(x)  has imaginary roots say ix

 Then, also ax2+b $\epsilon$   R

  and          (ax2+b)2  >0

 $\therefore$     $  a(ax^{2}+b)^{2}+b\neq0, \forall x$

 Thus     p[p(x)]  $\neq0, \forall x$