The sum of coefficients of even powers of $x$ in ${(x + \sqrt{x^{3} - 1})}^{6} + {(x - \sqrt{x^{3} - 1})}^{6}$

$23$

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$24$

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$18$

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$21$

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Solution

The correct option is B
$24$

Explanation for correct option:
We know that ${(a + b)}^{6} + {(a - b)}^{6} = 2 (a^{6} + 15 a^{4} b^{2} + 15 a^{2} b^{4} + b^{6})$
Expanding the help of binomial theorem we get
${(x + \sqrt{x^{3} - 1})}^{6} + {(x - \sqrt{x^{3} - 1})}^{6}$
$= 2 [x^{6} + 15 x^{4} (x^{3} - 1) + 15 x^{2} {(x^{3} - 1)}^{2} + {(x^{3} - 1)}^{3}] = 2 [x^{6} + 15 x^{7} - 15 x^{4} + 15 x^{8} - 30 x^{5} + 15 x^{2} + x^{9} - 3 x^{6} + 3 x^{3} - 1] [∵ {(a - b)}^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} + b^{3}]$
Term with even power is
$= 2 [x^{6} - 15 x^{4} + 15 x^{8} + 15 x^{2} - 3 x^{6} - 1]$
Therefore the sum of the coefficient is
$= 2 (1 - 15 + 15 + 15 - 3 - 1) = 24$
Hence option (B) is correct.

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