NIMCET 2023 — Mathematics PYQ

Given $e^x\sin x=1$, we can write it as:

Let $f(x)=e^{-x}-\sin x$.

Let $\alpha$ and $\beta$ be two real roots of $f(x)=0$.

So, $f(\alpha )=0$ and $f(\beta )=0$.

2. Apply Rolle's Theorem:

According to Rolle's Theorem, if $f(x)$ is continuous and differentiable, and $f(\alpha )=f(\beta )$, then there exists at least one value $c\in (\alpha ,\beta )$ such that:

3. Differentiate $f(x)$:

4. Set $f^{\prime }(x)=0$:

Conclusion:

By Rolle's Theorem, there is at least one root of $f^{\prime }(x)=0$ between any two roots of $f(x)=0$. Therefore, the equation $e^x\cos x=-1$ has at least one root between any two roots of $e^x\sin x=1$.

Correct Option: 1