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 :

• A. 32
• B. 26
• C. 29
• D. 24

Answer 1

Answer: (D)

${\left(x+\sqrt{x^3-1}\right)}^6+{\left(x-\sqrt{x^3-1}\right)}^6$
$=2\left[{}^6{C}_0{x}^6{+}^6{C}_2{x}^4\left(x^3-1\right){+}^6{C}_4{x}^2{\left(x^3-1\right)}^3{+}^6{C}_6{\left(x^3-1\right)}^3\right]$
$=2\left[{}^6{C}_0{x}^6{+}^6{C}_2{x}^7{-}^6{C}_2{x}^4{+}^6{C}_4{x}^8{+}^6{C}_4{x}^2-2^6{C}_4{x}^5+\left(x^9-1-3x^6+3x^3\right)\right]$
$\Rightarrow$ Sum of coefficient of even powers of $x$
= $2[1-15+15+15-1-3]=24$