If $\mathrm{f}(1)=3, \mathrm{f}^{\prime}(1)=2$, then $\frac{\mathrm{d}}{\mathrm{dx}}\left\{\log \left[\mathrm{f}\left(\mathrm{e}^x+2 x\right)\right]\right\}$ at $x=0$ is

If $\frac{1}{6} \sin \theta, \cos \theta, \tan \theta$ are in G.P., then the general solution of $\theta$ is

Let $f$ be a function which is continuous and differentiable for all $x$. If $\mathrm{f}(1)=1$ and $\mathrm{f}^{\prime}(x) \leq 5$ for all $x$ in $[1,5]$, then the maximum value of $\mathrm{f}(5)$ is

In a triangle ABC with usual notations if, $\cot \frac{A}{2}=\frac{b+c}{a}$, then the triangle $A B C$ is