Question

If $\alpha$ and $\beta$ be the coefficients of $x^4$ and $x^2$ respectively in the expansion of $(x+\sqrt{x^2-1}{)}^6+(x-\sqrt{x^2-1}{)}^6$, then :

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

Answer 1

The correct option is B $\alpha -\beta =-132$

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

$=2[{}^6{C}_0{x}^6+{}^6{C}_2{x}^4(x^2-1)+{}^6{C}_4{x}^2(x^2-1{)}^2+{}^6{C}_6(x^2-1{)}^3]$

$=2[32x^6-48x^4+18x^2-1]$

$\Rightarrow \alpha =-96,\beta =36$
$\Rightarrow \alpha -\beta =-132$