Single Choice

The sum of the co-efficients of all odd degree terms in the expansion of $$\left(x + \sqrt {x^{3} - 1}\right)^{5} + (x - \sqrt {x^{3} - 1})^{5}, (x > 1)$$ is

A$$1$$
B$$2$$
Correct Answer
C$$-1$$
D$$0$$

Solution

$$(x+\sqrt {x^3-1})^5+(x-\sqrt {x^3-1})^5$$

$$(a+b)^5=a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5 -------(i)$$

$$(a-b)^5=a^5-5a^4b+10a^3b^2-10a^2b^3+5ab^4-b^5 -------(ii)$$

$$(a+b)^5+(a-b)^5=2[a^5+10a^3b^2+5ab^4]$$

$$=2[x^5+10x^3(x^3-1)+5x(x^3-1)^4]$$

$$=2[x^5+10x^6-10x^3+5x(x^6-2x^3+1)]$$

$$=2x^5+20x^6-20x^3+10x^7-20x^4+10x$$

Here all the co-efficient of the above equation are odd terms coefficients of even terms are cancelled out.
So, sum of coefficients $$\Rightarrow2+20-20+10-20+10=2$$

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