TS EAMCET · Maths · Quadratic Equation

Let $p (x)$ be a quadratic polynomial with real coefficients. If $p (x) = 0$ has only purely imaginary roots, then the zeroes of the polynomial $p (p (x))$ are

A only real numbers
B only purely imaginary numbers
C only rational numbers
D only complex numbers of the form $a + i b$ with $a \neq 0$ and $b \neq 0$

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Answer & Solution

Correct Answer

(D) only complex numbers of the form $a + i b$ with $a \neq 0$ and $b \neq 0$

Step-by-step Solution

Detailed explanation

If the roots of the equation px=0 are purely imaginary then they are of the form iβ. Hence, px=ax2+b with a,b of same sign and a,b∈R. Now, ppx=aax2+b2+b If x∈R or x is purely imaginary, so x2∈R ⇒px∈R ⇒ppx∈0 Hence, real or purely…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Let px be a quadratic polynomial with real coefficients. If px=0 has only purely imaginary roots, then the zeroes of the polynomial ppx are

Step-by-step Solution

Let $p (x)$ be a quadratic polynomial with real coefficients. If $p (x) = 0$ has only purely imaginary roots, then the zeroes of the polynomial $p (p (x))$ are