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The quadratic equation p(x) = 0 with real coefficients has purely imaginary roots. Then the equation p(p(x)) = 0 has (JEE Adv. 2014)
  • a)
    one purely imaginary root
  • b)
    all real roots
  • c)
    two real and two purely imaginary roots
  • d)
    neither real nor purely imaginary roots
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
The quadratic equation p(x) = 0 with real coefficients has purely imag...
Quadratic equation with real coefficients and purely imaginary roots can be considered as
p(x) = x2 + a = 0 where a > 0 and a ∈R
The p[ p(x)] = 0 ⇒ (x2 + a)2 + a = 0
⇒ x4 + 2ax2 + (a2 + a) = 0
where a, b≠0
∴ p[p(x)] = 0 has complex roots which are neither purely real nor purely imaginary.
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Most Upvoted Answer
The quadratic equation p(x) = 0 with real coefficients has purely imag...
Given Information:

The quadratic equation p(x) = 0 with real coefficients has purely imaginary roots.

To Find:

The nature of the roots of the equation p(p(x)) = 0.

Solution:

Understanding the Given Information:

The given quadratic equation p(x) = 0 has purely imaginary roots. This means that the solutions of the equation are complex numbers of the form ai, where a is a non-zero real number and i is the imaginary unit.

Understanding the Equation p(p(x)) = 0:

We need to find the nature of the roots of the equation p(p(x)) = 0. To do this, we substitute p(x) into the equation:

p(p(x)) = 0

Replacing p(x) with its roots, which are purely imaginary:

p(ai) = 0

Now, we need to find the value of p(ai).


Using the Given Information:

Let's assume the quadratic equation p(x) = 0 to be:

p(x) = ax^2 + bx + c

Since the roots of p(x) = 0 are purely imaginary, they can be written as:

x = ai and x = -ai

Substituting these values in the equation p(x) = 0:

p(ai) = a(ai)^2 + b(ai) + c = 0

p(-ai) = a(-ai)^2 + b(-ai) + c = 0


Using the Fact that p(x) has Real Coefficients:

Since p(x) has real coefficients, the imaginary terms in the above equations should cancel out, resulting in a real number:

a(-a)(i)^2 + b(-a)i + c = 0

a(-a)(-1) + b(-a)i + c = 0

a^2 + c - abi = 0

This implies that the coefficient of the imaginary term should be zero:

abi = 0

Since a is non-zero, this equation implies that b = 0.


Substituting the Value of b = 0:

Substituting b = 0 in the equation p(ai) = a^2 + c - abi:

p(ai) = a^2 + c = 0

This means that the value of c is equal to -a^2.

So, the quadratic equation p(x) = 0 can be written as:

p(x) = ax^2 - a^2 = a(x - a)(x + a) = 0


Understanding the Nature of the Roots of p(p(x)) = 0:

Now, let's analyze the equation p(p(x)) = 0. Substituting p(x) = a(x - a)(x + a) into the equation:

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The quadratic equation p(x) = 0 with real coefficients has purely imaginary roots. Then the equation p(p(x)) = 0 has (JEE Adv. 2014)a)one purely imaginary rootb)all real rootsc)two real and two purely imaginary rootsd)neither real nor purely imaginary rootsCorrect answer is option 'D'. Can you explain this answer?
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