A) 106
B) 107
C) 155
D) 108
Correct Answer: A
Solution :
Let \[a=({{(1+2x+3{{x}^{2}})}^{6}}+{{(1-4{{x}^{2}})}^{6}})\] \[\therefore \]Coefficient of \[{{x}^{2}}\]in the expansion of the product \[{{(2-x)}^{2}}({{(1+2x+3{{x}^{2}})}^{6}}+{{(1-4{{x}^{2}})}^{6}})\] \[=2(Coefficient\,of\,{{x}^{2}}in\,a)-1(Constant\,of\,expansion)\]IN the expansion of \[({{(1+2x+3{{x}^{2}})}^{6}}+{{(1-4{{x}^{2}})}^{6}}),\] Constant = 1 + 1 = 2 Coefficient of \[{{x}^{2}}=\]Coefficient of \[{{x}^{2}}\]in \[({{\,}^{6}}{{C}_{0}}{{(1+2x)}^{6}}{{(3{{x}^{2}})}^{0}})+\]Coefficient of \[{{x}^{2}}\]in \[({{\,}^{6}}{{C}_{1}}{{(1+2x)}^{5}}3)-{{\,}^{6}}{{C}_{1}}(4)\] \[={{\,}^{6}}{{C}_{2}}4+6\times 3-24\] \[=6=+18-24=54\] Then, coefficient of \[{{x}^{2}}\]in \[{{(2-x)}^{2}}({{(1+2x+3{{x}^{2}})}^{6}}+{{(1-4{{x}^{2}})}^{6}})\] \[=2\times 54-1(2)=108-2=106\]
You need to login to perform this action.
You will be redirected in
3 sec
