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