# 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 :   Option 1) $32$ Option 2) $26$ Option 3) $24$   Option 4) $29$

$(x+\sqrt{x^{3}-1})^{6}+(x-\sqrt{x^{3}-1})^{6}\: \: \: \: \: x>1$

$=2\left ( ^{6}C_{o}\; x^{6}+ ^{6}C_{2}\; x^{2}(x^{3}-1)+^{6}C_{4}x^{A}(x^{3}-1)^{2}+^{6}C_{6}(x^{3}-1)^{3}\right )$

$\therefore 2\left ( ^{6}C_{o}\; x^{6}+ ^{6}C_{2}(x^{5}-x^{2})+^{6}C_{4}x^{4}(x^{6}-2x^{3}+1)^{2}+^{6}C_{6}(x^{9}-3x^{6}+3x^{3}-1)\right )$

$\therefore$ Sum of the coefficient of even powers

$2(^{6}C_{o}-^{6}C_{2}+^{6}C_{4}+C_{4}-3-^{6}C_{6}-^{6}C_{6})$

$=2\left ( 1-15+15+15-3-1 \right )=24$

Option 1)

$32$

Option 2)

$26$

Option 3)

$24$

Option 4)

$29$

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