A) \[\alpha -\beta =60\]
B) \[\alpha +\beta =60\]
C) \[\alpha -\beta =-132\]
D) \[\alpha +\beta =-30\]
Correct Answer: C
Solution :
\[\because \,\,\,\,\,{{\left( x+\sqrt{{{x}^{2}}-1} \right)}^{6}}+{{\left( x-\sqrt{{{x}^{2}}-1} \right)}^{6}}=2\] \[\left[ ^{6}{{C}_{0}}{{x}^{6}}+{}^{6}{{C}_{2}}{{x}^{4}}\left( {{x}^{2}}-1 \right)+{}^{6}{{C}_{4}}{{x}^{2}}{{\left( {{x}^{2}}-1 \right)}^{2}}+{}^{6}{{C}_{6}}{{\left( {{x}^{2}}-1 \right)}^{3}} \right]\] \[=2\left[ 32{{x}^{6}}-48{{x}^{4}}+18{{x}^{2}}-1 \right]\] \[\therefore \]\[\alpha \] = coefficient of \[{{x}^{4}}=-96\] \[\beta \] = coefficient of \[{{x}^{2}}\text{= }36\] \[\Rightarrow \] \[\alpha -\beta =-96-36=-132\]
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