JEE Mathematics 2024 Question with Solution

Given:

• $f(x)=\sqrt{\lim_{r\to x}\left[\frac{2r^2(f(r^2)-f(x))f(r)}{r^2-x^2}-r^2e^{f(r)/r}\right]}$
• $f(1)=1$

Find: $e^a$ such that $f(a)=0$.

From the provided solution, the limit expression is simplified into the differential equation

Let $y=f(x)$. Then

Now use the substitution $y=vx$, so

Substituting this into the differential equation and following the working given in the source solution, we obtain an integrable relation.

The provided solution then states:

and after integration writes a relation leading to the constant by using $f(1)=1$. Using the initial condition, the source concludes the required value corresponding to $f(a)=0$ gives the final result

Therefore, the required numerical value is $2$.

Note: The solution contains inconsistent intermediate algebra, but both the displayed correct answer and the final conclusion on the page state that the required value is $2$.