JEE Main 2022 · 25 Jul · Shift 2 · Mathematics Q22

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MathematicsInteger+4 / -1Question 22 of 90

Let $f(x)$ be a quadratic polynomial with leading coefficient 1 such that $f(0)=p$ , $p\ne0$ , and $f(1)=\dfrac13$ . If the equations $f(x)=0$ and $f\circ f\circ f\circ f(x)=0$ have a common real root, then $f(-3)$ is equal to ______.

Your answer

Answer and worked solution

Official answer: 25

Common root α. $f(\alpha)=0$ makes $f^{4}(\alpha)=f(f(f(0)))=f(f(p))$ , so $f(p)$ must be a root. With $b=-\frac23-p$ : $f(p)=\frac p3$ .

Condition $f\left(\frac p3\right)=0$ : $p+3b+9=0$ , so $p=\frac72$ , $b=-\frac{25}6$ .

Value $f(-3)=9+12.5+3.5=25$ .

Answer: 25

Topics:Roots and coefficients Composition and inverse functions

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JEE Main 2022 · 25 Jul · Shift 2, Mathematics question 22