The sum of the co-efficients of all even degree terms in x in the expansion ofx+x3−16+x−x3−16,(x>1) is equal to

A
26
B
32
C
29
D
24

Solution:

Even degree terms of given expression

$\begin{array}{l} = [^{6} C_{0} x^{6} +^{6} C_{2} x^{4} {(\sqrt{(x^{3} - 1)})}^{2} +^{6} C_{4} x^{2} {(\sqrt{x^{3} - 1})}^{4} \\ +^{6} C_{6} x^{0} {(\sqrt{x^{3} - 1})}^{6}] + [^{6} C_{0} x^{6} +^{6} C_{2} x^{4} {(- \sqrt{x^{3} - 1})}^{2} \\ +^{6} C_{4} x^{2} {(- \sqrt{x^{3} - 1})}^{4} +^{6} C_{6} x^{0} {(- \sqrt{x^{3} - 1})}^{6}] \\ = 2 [^{6} C_{0} x^{6} +^{6} C_{2} (x^{7} - x^{4}) +^{6} C_{4} (x^{8} + x^{2} - 2 x^{5}) +^{6} C_{6} (x^{9} - 1 - 3 x^{6} + 3 x^{3})] \end{array}$

$∴$ Required sum of coefficients

$\begin{array}{l} = 2 [^{6} C_{0} -^{6} C_{2} + 2 \times^{6} C_{4} -^{6} C_{6} - 3 \times^{6} C_{6}] \\ = 2 [1 - 15 + 30 - 1 - 3] = 24 \end{array}$

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

Solution:

Related content