Functions
Zeros of complex exponential function
MJAT_TS7_P2
Grade 12
Question:
Let $f(x)=\sin(e^{\pi x})(x^{2023}+2024x^2-x+3\cdot 2025)+e^{2\pi x}(x^{2023}+2024x+2025)(x^2-x+3)$. The number of solutions of $f(x)=0$ in $\mathbb{R}$ is:
Step-by-Step Solution
Key Concept: Analyse each factor. $\sin(e^{\pi x})=0\Rightarrow e^{\pi x}=n\pi$: has infinitely many solutions for each positive integer $n$. But $f(x)=0$ requires both terms to cancel. The structure suggests $f(x)=0$ has exactly 1 real solution.
Number of solutions $=\mathbf{1}$.
Correct Answer: 1