Question

Let $f(x)$ be a quadratic polynomial with leading coefficient 1 such that $f(0) = p, p≠0$ and $f(1)=\frac{1}{3}$. If the equation $f(x) = 0$ and $fofofof(x) = 0$ have a common real root, then $f(–3)$ is equal to.......................

Answer 1

The correct answer is 25
Let f(x)=x2+bx+p
f(1)=31​⇒1+b+p=31​.....(1)
Assume common root be α
f(α)=0&f(f(f(f(α))))=0
⇒f(f(f(0)))=0
⇒f(f(p))=0
⇒f(p2+bp+p)=0
⇒f(p(p+b+1))=0
⇒f(3p​)=0
⇒9p2​+b⋅3p​+p=0
⇒9p​+3b​+1=0
p+3b+9=0.....(2)
From (1) & (2) ⇒p=27​
Now, f(−3)=9−3b+p
=9−(−p−9)+p
=18+2p=18+2×27​=25