General term of Binomial expansion $${\left( {{5 \over 2}{x^3} - {1 \over {5{x^2}}}} \right)^{11}}$$ is

$${T_{r + 1}} = {}^{11}{C_r}\,.\,{\left( {{5 \over 2}{x^3}} \right)^{11 - r}}\,.\,{\left( { - {1 \over {5{x^2}}}} \right)^r}$$

$$ = {}^{11}{C_r}\,.\,{\left( {{5 \over 2}} \right)^{11 - r}}\,.\,{\left( { - {1 \over 5}} \right)^r}\,.\,{x^{33 - 5r}}$$

In the term,

$$\left( {1 - {x^2} + 3{x^3}} \right){\left( {{5 \over 2}{x^3} - {1 \over {5{x^2}}}} \right)^{11}}$$

Term independent of x is when

(1) $$33 - 5r = 0$$

$$ \Rightarrow r = {{33} \over 5}\,\, \notin $$ integer

(2) $$33 - 5r = -2$$

$$ \Rightarrow 5r = 35$$

$$ \Rightarrow r = 7\,\, \in $$ integer

(3) $$33 - 5r = - 3$$

$$ \Rightarrow 5r = 36$$

$$ \Rightarrow r = {{36} \over 5}\,\, \notin $$ integer

$$\therefore$$ Only for r = 7 independent of x term possible.

$$\therefore$$ Independent of x term

$$ = - \left( {{}^{11}{C_7}{{\left( {{5 \over 2}} \right)}^4}\,.\,{{\left( { - {1 \over 5}} \right)}^7}} \right)$$

$$ = - \left( {{{11\,.\,10\,.\,9\,.\,8} \over {4\,.\,3\,.\,2\,.\,1}}\,.\,{{{5^4}} \over {{2^4}}}\,.\, - {1 \over {{5^7}}}} \right)$$

$$ = {{11\,.\,10\,.\,3} \over {{2^4}\,.\,{5^3}}}$$

$$ = {{11\,.\,3} \over {{2^3}\,.\,{5^2}}}$$

$$ = {{33} \over {200}}$$