Theory of Equations 1 Question 18
19. In the quadratic equation $p(x)=0$ with real coefficients has purely imaginary roots. Then, the equation $p[p(x)]=0$ has
(2014 Adv.)
(a) only purely imaginary roots
(b) all real roots
(c) two real and two purely imaginary roots
(d) neither real nor purely imaginary roots
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Solution:
- If quadratic equation has purely imaginary roots, then coefficient of $x$ must be equal to zero.
Let $p(x)=a x^{2}+b$ with $a, b$ of same sign and $a, b \in R$.
Then, $\quad p[p(x)]=a\left(a x^{2}+b\right)^{2}+b$
$p(x)$ has imaginary roots say $i x$.
Then, also $a x^{2}+b \in R$ and $\left(a x^{2}+b\right)^{2}>0$
$\therefore \quad a\left(a x^{2}+b\right)^{2}+b \neq 0, \forall x$
Thus,
$$ p[p(x)] \neq 0, \forall x $$
