Help Needed

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 :)

#Algebra

Easy Math Editor

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Comments

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

(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 - 4 years, 4 months ago

thank you bro very much , now i think i get it :) , you are awesome :) thnx again .

A Former Brilliant Member - 4 years, 4 months ago

No problem Bro Just tag me

Rishabh Deep Singh - 4 years, 4 months ago

@Prakhar Bindal @Aniket Sanghi @Rishabh Deep Singh

A Former Brilliant Member - 4 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$