A) \[-\frac{5}{3}\]
B) \[\frac{10}{3}\]
C) \[-\frac{3}{10}\]
D) \[\frac{3}{5}\] The coefficient of\[x\]in the middle term of expansion of\[{{(1+\alpha x)}^{4}}{{=}^{4}}{{C}_{2}}.{{\alpha }^{2}}\] The coefficient of x in the middle term of the expansion of\[{{(1-\alpha x)}^{6}}{{=}^{6}}{{C}_{3}}{{(-\alpha )}^{3}}\] According to question, \[^{4}{{C}_{2}}{{\alpha }^{2}}{{=}^{6}}{{C}_{3}}{{(-\alpha )}^{3}}\] \[\Rightarrow \] \[\frac{4!}{2!2!}{{\alpha }^{2}}=-\frac{6!}{3!3!}{{\alpha }^{3}}\] \[\Rightarrow \] \[6{{\alpha }^{2}}=-20{{\alpha }^{3}}\] \[\Rightarrow \] \[\alpha =-\frac{6}{20}\] \[\Rightarrow \] \[\alpha =-\frac{3}{10}\]
Correct Answer: C
Solution :
The coefficient of\[x\]in the middle term of expansion of\[{{(1+\alpha x)}^{4}}{{=}^{4}}{{C}_{2}}.{{\alpha }^{2}}\] The coefficient of x in the middle term of the expansion of\[{{(1-\alpha x)}^{6}}{{=}^{6}}{{C}_{3}}{{(-\alpha )}^{3}}\] According to question, \[^{4}{{C}_{2}}{{\alpha }^{2}}{{=}^{6}}{{C}_{3}}{{(-\alpha )}^{3}}\] \[\Rightarrow \] \[\frac{4!}{2!2!}{{\alpha }^{2}}=-\frac{6!}{3!3!}{{\alpha }^{3}}\] \[\Rightarrow \] \[6{{\alpha }^{2}}=-20{{\alpha }^{3}}\] \[\Rightarrow \] \[\alpha =-\frac{6}{20}\] \[\Rightarrow \] \[\alpha =-\frac{3}{10}\]
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