Question

The coefficient of $x^2$ in the expansion of the product $(2-x^2)\cdot ((1+2x+3x^2{)}^6+(1-4x^2{)}^6)$ is :

• A. 107
• B. 106
• C. 108
• D. 155

Answer 1

Answer: (B)

coefficient of $x^2=2$ coefficient of $x^2\text{ in }$
$\left({\left(1+2x+3x^2\right)}^6+\left(1-4x^2\right)6\right)$ - constant term
$\left(1+2x+3x^2\right)6\sum _{r=0}^6$
${}^6{C}_r{\left(2x+3x^2\right)}^r$
$={}^6{C}_0+{}^6{C}_1\left(2x+3x^2\right)+{}^6{C}_2{\left(2x+3x^2\right)}^2+\dots$
coefficient of $x^2=2(18+60-24)-2$
$=108-2=106$