The velocity v of a particle is given in terms of time t by the equation $\mathrm{v}=\mathrm{at}+\frac{\mathrm{b}}{\mathrm{t}+\mathrm{c}}$. The dimensions of a, b and c are :

$\mathrm{\upsilon }=\mathrm{at}+\frac{\mathrm{b}}{\mathrm{t}+\mathrm{c}}$

By principle of homogeneity, 

$\dim \left(\mathrm{at}\right)=\dim \left(\mathrm{\upsilon }\right)\Rightarrow \mathrm{a}=\frac{{\mathrm{LT}}^{-1}}{\mathrm{T}}=\left[{\mathrm{LT}}^{-2}\right]$

Also dim(c)  = dim(t) = [T]

$\text{ Now }\frac{\mathrm{b}}{\text{ time }}=\text{ Velocity }$

$\therefore \mathrm{b}=\text{ velocity }\times \text{ time }={\mathrm{LT}}^{-1}\times \mathrm{T}=\left[\mathrm{L}\right]$

So correct choice is (3).