The velocity v of a particle at time t is given by the equation v =...

According to the principle of homogeneity of dimensions, only quantities with the same dimensions can be added or subtracted.

In the denominator $(t + C)$, the constant $C$ is added to time $t$. Therefore, $[C] = [t] = [T]$.

Each term in the equation must have the same dimensions as velocity $v$, which is $[L T^{-1}]$.

For the term $D$: $[D] = [v] = [L T^{-1}]$.

For the term $At$: $[A][t] = [v] \Rightarrow [A][T] = [L T^{-1}] \Rightarrow [A] = [L T^{-2}]$.

For the term $\dfrac{B}{t + C}$: $\dfrac{[B]}{[T]} = [L T^{-1}] \Rightarrow [B] = [L]$.

Thus, the correct matching is (A)-(IV), (B)-(I), (C)-(II), (D)-(III).

Answer: (A)-(IV), (B)-(I), (C)-(II), (D)-(III)