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 $x$
= $2[1 - 15 + 15 + 15 - 1 - 3 ] = 24$