Question

_____________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 :

• A. $\alpha+\beta=-30$
• B. $\alpha-\beta=-132$
• C. $\alpha+\beta=60$
• D. $\alpha-\beta=60$

Answer 1

The Correct Option is B

The correct answer is B:\(\alpha-\beta=-132\)
Given that;
\(\alpha,\beta\) are coefficient of \(x^4 and\space x^2\),and;
\(x+\sqrt{(x^2-1)^6}+x-\sqrt{(x^2-1)^2}\)
\(2\left[^{6}C_{0}.x^{6}+^{6}C_{2}x^{4}\left(x^{2}-1\right)+^{6}C_{4}x^{2}\left(x^{2}-1\right)^{2}+^{6}C_{6}\left(x^{2}-1\right)^{3}\right]\)
\(=2[x^6+15(x^6-x^4)+15x^2(x^4-2x^2+1)+(-1+3x^2-3x^2+x^6)]\)
\(=64x^6-96x^4+36x^2-2\)
as \(\alpha,\beta\) are the coefficient of \(x^4,x^2\)
\(\therefore \alpha = -96\,\&\,\beta = 36\)
\(\therefore \alpha - \beta = -132\)