QJEEM24Ma576c: JEE Main 2024 Mathematics - Class 12 JEEM24Ma576c Solution: If α = lim x → 0 + ( e tan ⁡ x − e x tan ⁡ x − x ) and β = lim…

$\begin{aligned}\alpha & =\lim _{x \rightarrow 0^{+}} \frac{e^{\sqrt{\tan x}}-e^{\sqrt{x}}}{(\sqrt{\tan x}-\sqrt{x})} \\& =\lim _{x \rightarrow 0} \frac{e^{\sqrt{x}}\left(e^{\sqrt{\tan x}-\sqrt{x}}-1\right)}{(\sqrt{\tan x}-\sqrt{x})}=1 \\\beta & =\lim _{x \rightarrow 0}(1+\sin x)^{\frac{1}{2} \cot x}=\lim _{x \rightarrow 0} e^{(\sin x)\left(\frac{1}{2} \cot x\right)} \\& =\lim _{x \rightarrow 0} e^{\frac{1}{2} \cos x}=e^{1 / 2}\end{aligned}$

$\begin{aligned}& \text { Product of roots }=\sqrt{e}=\frac{-\sqrt{e}}{a} \Rightarrow a=-1 \\& \text { Sum of roots }=\frac{-b}{a}=1+\sqrt{e} \\& \qquad=b=\sqrt{e}+1 \\& \Rightarrow 12 \ln (a+b)=12 \ln (\sqrt{e}+1-1)=12 \ln \left(e^{1 / 2}\right)=6\end{aligned}$