Question

The sum of the co-efficients of all odd degree terms in the expansion
${\left(x+\sqrt{x^3-1}\right)}^5+{\left(x-\sqrt{x^3-1}\right)}^5,(x>1)$ is

• A. -1
• B. 0
• C. 1
• D. 2

Answer 1

Answer: (D)

${\left(x+\sqrt{x^3-1}\right)}^5+{\left(x-\sqrt{x^3-1}\right)}^5$
$=2\left[{}^5{C}_0{x}^5+{}^5{C}_2{x}^3\left(x^3-1\right)+{}^5{C}_{4}x{\left(x^3-1\right)}^2\right]$
$=2\left[x^5+10\left(x^6-x^3\right)+5x\left(x^6-2x^3+1\right)\right]$
$=2\left[x^5+10x^6-10x^3+5x^7-10x^4+5x\right]$
$=2\left[5x^7+10x^6+x^5-10x^4-10x^3+5x\right]$
Sum of odd degree terms coefficients
$=2(5+1-10+5)$
$=2$