$\underline{\mathbf{Answer:A}}$
$\underline{\mathbf{Solution:}}$
$\mathrm{e^a+e^b = 10,\;\;a \in R\;\;\;[Given]} \tag{1}$
Let $\mathrm{x = e^a}$, and $\mathrm {y = e^b}$
Then,
$\mathrm{\mathbf{max} (e^a+e^b+e^ae^b + 1) = \mathbf{max}(x+y+xy+1) = x+y+\mathbf{max}(xy) +1}$
Now, $\mathbf{xy}$ is maximum only when: $\mathrm{x = 5,\;y = 5\tag{2}}$
$\therefore \text{Maximum Value} = \underbrace{10}_\text{from (1)} + \underbrace{25}_\text{from(2)} + 1 = 36$
$\therefore \mathbf A$ is the correct option.