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The term independent of $\mathrm{x}$ in the expansion of $\mathrm{\left(1-x^{2}+3 x^{3}\right)\left(\frac{5}{2} x^{3}-\frac{1}{5 x^{2}}\right)^{11}, x \neq 0}$ is:
Option: 1
$\frac{7}{40}$

Option: 2
$\frac{33}{200}$

Option: 3
$\frac{39}{200}$

Option: 4
$\frac{11}{50}$

Answers (1)

best_answer

$\mathrm{\text { General } \operatorname{term} \text { in }\left(\frac{5}{2} x^{3}-\frac{1}{5 x^{2}}\right)^{11}} \\$

$\mathrm{={ }^{11} C_{r}\left(\frac{5}{2}\right)^{11-r}\left(x^{3}\right)^{11-r}\left(-\frac{1}{5}\right)^{r}\left(x^{-2}\right)^{r}} \\$

$\mathrm{={ }^{11} C_{r} \frac{5^{11-2 r}(-1)^{r} }{2^{11-r}}x^{33-3 r-2 r}}$

Constant term in $\mathrm{\left(1-x^{2}+3 x^{3}\right)\left(\frac{5}{2} x^{3}-\frac{1}{5 x^{2}}\right)^{11}}$

$\mathrm{=1\cdot constant\: term\: in \: second\: bracket -1. coefficient\: of\: x^{-2} in \: second \: bracket }\\$

$\mathrm{+3\: coefficient\: of\: x^{-3}in\: second \: bracket}$ .........(i)

For constant term in second bracket $\mathrm{33-5r=0}$

$\mathrm{\Rightarrow r=\frac{33}{5} \Rightarrow \text { no such term }}$

$\mathrm{For \: x^{-2}\: term \: \Rightarrow 33-5 r=-2 \Rightarrow r=7}\\$

$\mathrm{For \: x^{-3}\: term \: \Rightarrow 33-5 r=-3 \Rightarrow r=\frac{36}{5}}$

$\mathrm{\Rightarrow no\: such\: term}$

$\therefore$ Required answer $\mathrm{=-1 \times{ }^{11} C_{7} \frac{5^{-3}}{2^{4}}(-1)}$

$\mathrm{\begin{aligned} &=\frac{{ }^{11} C_{7}}{5^{3} \cdot 2^{4}} \\ & \end{aligned}}$

$\mathrm{=\frac{33}{200}}$

Hence answer is option 2

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