The coefficient of the term independent of x in the expansion of $\left(1+\mathrm{x}+2{\mathrm{x}}^3\right){\left(\frac{3}{2}{\mathrm{x}}^2-\frac{1}{3\mathrm{x}}\right)}^9$is 

detailed solution

Correct option is C

The general term in the expansion of 32x2−13x9 is Tr+1=9Cr32x29−r−13xr=9Cr329−r−13rx18−3r    ------(i)Now, the coefficients of the terms x0,x−1 and x−3 in32x2−13x9isFor xo, 18-3r =0 ⇒+ r=6For x-1 there exists no integer value of rFor x-3,18 -3r = -3 ⇒ r=7Now, the coefficient of the term independent of x in the expansion of 1+x+2x332x2−13x9=1⋅9C6(−1)6329−6136+0 +2⋅9C7(−1)7329−7137=9⋅8⋅71⋅2⋅3⋅3323⋅136+2⋅9⋅81⋅2(−1)3222⋅137=718−227=1754