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(B) The expression is $(1-x^{2}+3x^{3})(\frac{5}{2}x^{3}-\frac{1}{5x^{2}})^{11}$.The general term of $(\frac{5}{2}x^{3}-\frac{1}{5x^{2}})^{11}$ is giv

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$\frac{33}{200}$

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(B) The expression is $(1-x^{2}+3x^{3})(\frac{5}{2}x^{3}-\frac{1}{5x^{2}})^{11}$.The general term of $(\frac{5}{2}x^{3}-\frac{1}{5x^{2}})^{11}$ is given by $T_{r+1} = {}^{11}C_{r}(\frac{5}{2}x^{3})^{11-r}(-\frac{1}{5x^{2}})^{r}$.Simplifying this,we get $T_{r+1} = {}^{11}C_{r}(\frac{5}{2})^{11-r}(-\frac{1}{5})^{r}x^{33-3r-2r} = {}^{11}C_{r}(\frac{5}{2})^{11-r}(-\frac{1}{5})^{r}x^{33-5r}$.To find the term independent of $x$ in the product $(1-x^{2}+3x^{3})(\frac{5}{2}x^{3}-\frac{1}{5x^{2}})^{11}$,we need:$1$. $1 	imes$ coefficient of $x^{0}$ in the expansion.$2$. $-x^{2} 	imes$ coefficient of $x^{-2}$ in the expansion.$3$. $3x^{3} 	imes$ coefficient of $x^{-3}$ in the expansion.For $x^{0}$: $33-5r = 0 \Rightarrow r = 6.6$ (not an integer).For $x^{-2}$: $33-5r = -2$ $\Rightarrow 5r = 35$ $\Rightarrow r = 7$.For $x^{-3}$: $33-5r = -3 \Rightarrow 5r = 36$ (not an integer).Thus,only the second case contributes to the constant term:$-1 	imes [{}^{11}C_{7}(\frac{5}{2})^{11-7}(-\frac{1}{5})^{7}] = -{}^{11}C_{4}(\frac{5}{2})^{4}(-\frac{1}{5})^{7}$.Calculating this: $-330 	imes \frac{625}{16} 	imes (-\frac{1}{78125}) = 330 	imes \frac{625}{16 	imes 78125} = 330 	imes \frac{1}{16 	imes 125} = \frac{330}{2000} = \frac{33}{200}$.

The term independent of $x$ in the expression of $(1-x^{2}+3x^{3})(\frac{5}{2}x^{3}-\frac{1}{5x^{2}})^{11}, x \neq 0$ is

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