The expression given below shows the variation of velocity $$(\upsilon)$$ with time $$(t),\upsilon = At^{2}+\frac{Bt}{C+t}$$. The dimension of ABC is :

We need to find the dimensions of the product ABC, where $$v = At^2 + \frac{Bt}{C+t}$$.

Since $$C + t$$ must be dimensionally consistent, $$[C] = [t] = T$$.

From the first term $$At^2$$: $$[A][T^2] = [v] = LT^{-1}$$, so $$[A] = LT^{-3}$$.

From the second term $$\frac{Bt}{C+t}$$: for large $$t$$, this approaches $$B$$, so $$[B] = [v] = LT^{-1}$$.

(More rigorously: $$\frac{[B][T]}{[T]} = LT^{-1}$$, so $$[B] = LT^{-1}$$.)

Calculate $$[ABC]$$:

$$[ABC] = [A][B][C] = (LT^{-3})(LT^{-1})(T) = L^2T^{-3} = M^0L^2T^{-3}$$

The correct answer is Option (4): $$[M^0L^2T^{-3}]$$.