JEE Advanced 2024 Paper 2, Mathematics Section 3 Q3: Monotonocity

Factor:

$f(x)=\frac{x^{2023}+2024x+2025}{x^2-x+3}\cdot \left(\frac{\sin x+2}{e^{\pi x}}\right).$

The denominator $x^2-x+3$ has discriminant $1-12<0$, so it is positive for all real $x$. The factor $\sin x+2\ge 1>0$, and $e^{\pi x}>0$. So $f(x)=0$ iff

$g(x):=x^{2023}+2024x+2025=0.$

Compute the derivative: $g^{\prime }(x)=2023x^{2022}+2024>0$ for every real $x$ (even power keeps the leading term non-negative, and $+2024$ keeps it strictly positive). Hence $g$ is strictly increasing on $\mathbb{R}$ and crosses zero exactly once.

$\therefore$ The number of solutions is $1$.