CAT 2020 Slot 1 QA Question 20: If $f(5 + x) = f(5 - x)$ for every real $x$, and $f(x) = 0$ has four distinct real roots, then the sum of these roots is

Question

If $f(5 + x) = f(5 - x)$ for every real $x$, and $f(x) = 0$ has four distinct real roots, then the sum of these roots is

Accepted Answer

The condition $f(5 + x) = f(5 - x)$ means the function is symmetric about $x = 5$.

If $r$ is a root, then $f(r) = 0$. Write $r$ as $5 + (r - 5)$. By symmetry, $5 - (r - 5) = 10 - r$ is also a root.

So, roots occur in pairs of the form $r$ and $10 - r$, and each pair has sum 10.

There are four di

If f(5+x)=f(5−x)f(5 + x) = f(5 - x)f(5+x)=f(5−x) for every real xxx, and f(x)=0f(x) = 0f(x)=0 has four distinct real roots, then the sum of these roots is