Let t denote the greatest integer ≤ t . if the constant term in the expansion of 3 x 2 - 1 2 x 5 7 is…

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

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

$\Rightarrow T_{r+1}={}^{7}C_r{\left(3x^2\right)}^{7-r}{\left(\frac{-1}{2x^5}\right)}^r$

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

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

$\Rightarrow r=2$

Coefficient of $x^0$ is ${}^{7}C_2{\left(-1\right)}^2\times \left(\frac{3^{7-2}}{2^2}\right)$

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

$\Rightarrow \alpha =\frac{5103}{4}$

$\Rightarrow \left[\alpha \right]=1275$

Hence this is the required answer.