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  • If  and  be the coefficients of <img alt="x^{4}" src="http://entr

If \alpha and \beta be the coefficients of x^{4} and x^{2} respectively in the expansion of  \left ( x+\sqrt{x^{2}-1} \right )^{6}+\left ( x-\sqrt{x^{2}-1} \right )^{6}, then : 
Option: 1 \alpha +\beta =30
Option: 2 \alpha -\beta =-132
Option: 3 \alpha +\beta =60
Option: 4 \alpha -\beta =60
 

Answers (1)

best_answer

We know

\mathrm{(x+y)^{n}+(x-y)^{n}=2\left[^{n} C_{0}\; x^{n}\; y^{0}+^{n} C_{2} \;x^{n-2}\; y^{2}+^{n} C_{4}\; x^{n-4} \;y^{4}+\ldots .\right]}

 

Now,

\left ( x+\sqrt{x^{2}-1} \right )^{6}+\left ( x-\sqrt{x^{2}-1} \right )^{6}

={2\left[^{6} \mathrm{C}_{0} \mathrm{x}^{6}+^{6} \mathrm{C}_{2} \mathrm{x}^{4}\left(\mathrm{x}^{2}-1\right)+^{6} \mathrm{C}_{4} \mathrm{x}^{2}\left(\mathrm{x}^{2}-1\right)^{2}+^{6} \mathrm{C}_{6}\left(\mathrm{x}^{2}-1\right)^{3}\right]} \\ {\quad=2\left[\mathrm{x}^{6}+15\left(\mathrm{x}^{6}-\mathrm{x}^{4}\right)+15 \mathrm{x}^{2}\left(\mathrm{x}^{4}-2 \mathrm{x}^{2}+1\right)+\left(\mathrm{x}^{6}-1-3 \mathrm{x}^{4}+3 \mathrm{x}^{2}\right)\right]} \\ {\quad=2\left(32 \mathrm{x}^{6}-48 \mathrm{x}^{4}+18 \mathrm{x}^{2}-1\right)} \\ {\quad\alpha=-96 \text { and } \beta=36} \\ {\therefore \quad \alpha-\beta=-132}Correct Option (2)

Posted by

vishal kumar

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