Question

The sum of the co-efficient of all odd degree terms in the expansion of ${(x+\sqrt{x^3-1})}^5+{(x-\sqrt{x^3-1})}^5$ $a<n$ is

• A. $0$
• B. $1$
• C. $2$
• D. $-1$

Answer 1

Answer: (C)

$2$

Let $y=\sqrt{x^3-1}$

$(x+y{)}^5+(x-y{)}^5$

$=x^5+5C_1{x}^{4}y+5C_2{x}^3{y}^2+5C_3{x}^2{y}^3+5C_{4}x{y}^4+5C_5{y}^5+x^5-5C_1{x}^{4}y+5C_2{x}^3{y}^2-5C_3{x}^2{y}^3+5C_{4}x{y}^4-5C_5{y}^5$

$=2(x^5+10x^3{y}^2+5x{y}^4)$

$=2(x^5+10x^3(x^3-1)+5x(x^3-1{)}^2)$

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

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

Sum of coefficient of off powers of x.

$2[5+1-10+5]=2\left(1\right)=2$.