For continuity at $x=3$: $k\sqrt{4}=3m+2\Rightarrow 2k=3m+2$ ...(i)

For differentiability at $x=3$: Left derivative $=\frac{k}{2\sqrt{x+1}}\big|_{x=3}=\frac{k}{4}$; Right derivative $=m$.

So $\frac{k}{4}=m$ ...(ii)

From (ii): $k=4m$. Substitute in (i): $8m=3m+2\Rightarrow5m=2\Rightarrow m=\frac{2}{5}$, $k=\frac{8}{5}$.

$k+m=\frac{8}{5}+\frac{2}{5}=\frac{10}{5}=\mathbf{2}$

So correct answer is $k+m=2$, which is option A (index 0). But official JEE answer is $\frac{10}{3}$... Re-checking: $5m=2 \Rightarrow m=2/5$. $k=4(2/5)=8/5$. $k+m = 8/5+2/5=2$. Answer is $\mathbf{2}$.