Question

The sum of the co-efficients of all even degree terms in $x$ in the expansion of ${\left(x+\sqrt{x^3-1}\right)}^6+{\left(x-\sqrt{x^3-1}\right)}^6,\left(x>1\right)$ is equal to :

Answer 1

Answer: 24

${\left(x+\sqrt{x^3-1}\right)}^6+{\left(x-\sqrt{x^3-1}\right)}^6$
$=2[{}^6{C}_0{x}^6+{}^6{C}_2{x}^4\left(x^3-1\right)+{}^6{C}_4{x}^2{\left(x^3-1\right)}^2$
$+{}^6{C}_6{\left(x^3-1\right)}^3]$
$=2[x^6+15x^7-15x^4+15x^8-30x^5+15x^2$
$+x^9-3x^6+3x^3-1]$
Hence, the sum of coefficients of even powers of
$x=2[1-15+15+15-3-1]=24$