QJEEM25Mac2c3: JEE Main 2025 Mathematics - Class 11 JEEM25Mac2c3 Solution: Let be a polynomial of degree 2 , satisfying . If , then the…

as $f(x)$ is a polynomial of degree two let it be

$f(x)=a x^2+b x+c \quad(a \neq 0)$

on satisfying given conditions we get

$C=1 \& a= \pm 1$

hence $f(x)=1 \pm x^2$

also range $\in(-\infty, 1]$ hence

$\begin{gathered}f(x)=1-x^2 \\\text { now } f(k)=-2 k\end{gathered}$

$1-\mathrm{k}^2=-2 \mathrm{k} \rightarrow \mathrm{k}^2-2 \mathrm{k}-1=0$

let roots of this equation be $\alpha \& \beta$ then $\alpha^2+\beta^2=(\alpha+\beta)^2-2 \alpha \beta$

$=4-2(-1)=6$