A) \[2,3\]
B) \[3,2\]
C) \[\frac{3}{2},4\]
D) \[1,6\]
E) \[\frac{3}{2},6\]
Correct Answer: C
Solution :
Given, \[{{(1+ax)}^{n}}=1+6x+\frac{27}{2}{{x}^{2}}+...+{{a}^{n}}{{x}^{n}}\] ...(i) The expansion of\[{{(1+ax)}^{n}}\] is, \[{{(1+ax)}^{n}}=1+n\,ax+\frac{n(n-1)}{2!}{{(ax)}^{2}}+....\] ...(ii) On comparing the coefficient of like powers of\[x\]in Eqs. (i) and (ii), \[na=6\] ...(iii) \[\frac{27}{2}=\frac{n(n-1)}{2}.{{a}^{2}}\] \[\Rightarrow \] \[27=(n-1)(na).a\] \[27=(n-1)a\,6\] [from Eq. (iii)] \[(n-1)a=\frac{9}{2}\] ...(iv) From Eqs. (iii) and (iv), \[\frac{(n-1)6}{n}=\frac{9}{2}\] \[\Rightarrow \] \[\frac{n-1}{n}=\frac{3}{4}\] \[\Rightarrow \] \[4n-4=3n\] \[\Rightarrow \] \[n=4\] From Eq. (iii), \[a=\frac{6}{4}\] \[\Rightarrow \] \[a=3/2\]
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