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I was doing problems when I encountered this: IIT-JEE adv. 201 paper-II problem no. 50 (maths), I think the answer should be a) but answer given is d) , share your thoughts and help me please. :)

paper here :)

Note by Shubham Dhull
1 month, 2 weeks ago

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Q: The The quadratic equation \(f\left( x \right) =0\) with real coefficients has purely imaginary roots. Then the equation \(f\left( f\left( x \right) \right) =0\) has

(A) only purely imaginary roots

(B) all real roots

(C) two real and two purely imaginary roots

(D) neither real nor purely imaginary roots

Answer Let \(f\left( x \right) =a{ x }^{ 2 }+c\) Where \(a\) and \(c\) are Real and \(\alpha i\) and \(-\alpha i\) be the roots of the given equation.

we can write from above equation that \(f\left( \alpha i \right) =f\left( -\alpha i \right) =0\)

We need to find the roots of \(f\left( f\left( x \right) \right) =0\)

For \(f\left( f\left( x \right) \right) =0\), \(f\left( x \right) \) Should be Purely Imaginary which gives us

\(f\left( x \right) =\alpha i=a{ x }^{ 2 }+c\) \(--------(1)\)

Case 1 : x is Purely Real

Now since a and c are Real then x cannot be real to satisfy the Equation \((1)\) => No real Roots.

Case 2 : x is Purely Imaginary

since x is purely imaginary \({ x }^{ 2 }\) is Purely Real which gives us No Imaginary Roots.

Hence the answer is \((D)\) Rishabh Deep Singh · 1 month, 2 weeks ago

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@Rishabh Deep Singh thank you bro very much , now i think i get it :) , you are awesome :) thnx again . Shubham Dhull · 1 month, 2 weeks ago

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@Shubham Dhull No problem Bro Just tag me Rishabh Deep Singh · 1 month, 2 weeks ago

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