NIMCET 2023 Question Paper with solutions

${\mathit{e}}^{\mathit{x}}\mathit{s}\mathit{i}\mathit{n}\mathit{x}=1$
$\mathit{f}\left(\mathit{x}\right)=\mathit{s}\mathit{i}\mathit{n}\mathit{x}-{\mathit{e}}^{-\mathit{x}}$
Let it has 2 real roots $\mathit{\alpha },\mathit{\beta }$
${\mathit{f}}^{\prime }\left(\mathit{x}\right)=\cos \mathit{x}+{\mathit{e}}^{-\mathit{x}}=0$
Using Rolle's theorem
There will be at least one real root of its derivative between $\mathit{\alpha }$ and $\mathit{\beta }$
At least one $\mathit{c}\in \left(\mathit{\alpha },\mathit{\beta }\right)$
${\mathit{f}}^{\prime }\left(\mathit{c}\right)=0$
${\mathit{e}}^{-\mathit{c}}+\cos \mathit{c}=0$
$1+{\mathit{e}}^{\mathit{c}}\cos \mathit{c}=0=\mathit{g}\left(\mathit{c}\right)$
$\mathit{g}\left(\mathit{x}\right)={\mathit{e}}^{\mathit{x}}\cos \mathit{x}+1$

At least one real root of $\mathit{g}\left(\mathit{x}\right)$ between two real root of $\mathit{f}\left(\mathit{x}\right)$.