The velocity of a body is given by $\mathrm{v}={\mathrm{At}}^2+\mathrm{Bt}+\mathrm{C}$ . If v and t are expressed in SI, what are the units of A, B and C ?

Here $\mathrm{v}={\mathrm{At}}^2+\mathrm{Bt}+\mathrm{C}$
According to principle of Homogeneity,
$\begin{array}{l}\left[\mathrm{V}\right]=\left[{\mathrm{At}}^2\right]=\left[\mathrm{Bt}\right]=\left[\mathrm{C}\right];\left[\mathrm{V}\right]=\left[{\mathrm{At}}^2\right]\\ {\mathrm{LT}}^{-1}=\left[\mathrm{A}\right]\cdot {\mathrm{T}}^2\Rightarrow \left[\mathrm{A}\right]={\mathrm{LT}}^{-3}\end{array}$
So, unit of A is ms^–3
Similarly
$\left[\mathrm{V}\right]=\left[\mathrm{Bt}\right]{\mathrm{LT}}^{-1}=\left[\mathrm{B}\right]\cdot \mathrm{T}\Rightarrow \left[\mathrm{B}\right]={\mathrm{LT}}^{-2}$
So unit of B is ms^–2
And $\left[\mathrm{V}\right]=\left[\mathrm{C}\right]\Rightarrow \left[\mathrm{C}\right]={\mathrm{LT}}^{-1}$
So unit of C is ms^–1