SATHEE: Theory of Equations 1 Question 18

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

(d) neither real nor purely imaginary roots

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Answer:

Correct Answer: 19. (d)

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, $p [p (x)] = a {(a x^{2} + b)}^{2} + b$

$p (x)$ has imaginary roots say $i x$ .

Then, also $a x^{2} + b \in R$ and ${(a x^{2} + b)}^{2} > 0$

$∴ a {(a x^{2} + b)}^{2} + b \neq 0, \forall x$

Thus,

$p [p (x)] \neq 0, \forall x$

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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

Answer:

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19. In the quadratic equation $p (x) = 0$ with real coefficients has purely imaginary roots. Then, the equation $p [p (x)] = 0$ has