Let [t] denote the greatest integer ≤ t . if the constant term in...

The given expansion is ${\left(3{x}^{2} - \dfrac{1}{2{x}^{5}}\right)}^{7}$.

The general term in the binomial expansion of ${\left(x + a\right)}^{n}$ is given by ${T}_{r + 1} = Cr n {x}^{n - r}{a}^{r}$.

$⇒ {T}_{r + 1} = Cr 7 {\left(3{x}^{2}\right)}^{7 - r}{\left(\dfrac{- 1}{2{x}^{5}}\right)}^{r}$

$= Cr 7 {\left(- 1\right)}^{r}\left(\dfrac{{3}^{7 - r}}{{2}^{r}}\right){x}^{14 - 2r - 5r}$

Now for term independent of $x$$⇒ 14 - 2r - 5r = 0$

$⇒ r = 2$

Coefficient of ${x}^{0}$ is $C2 7 {\left(- 1\right)}^{2} \times \left(\dfrac{{3}^{7 - 2}}{{2}^{2}}\right)$

$= \dfrac{7 \times 6}{2} \times \dfrac{{3}^{5}}{{2}^{2}}$.

$⇒ \alpha = \dfrac{5103}{4}$

$⇒ \left[\alpha\right] = 1275$

Hence this is the required answer.