Question

The coefficient of $x^2$ in the expansion of the product $(2–x^2)\left[\right(1+2x+3x^2{)}^6+(1–4x^2{)}^6]$ is

• A. $107$
• B. $108$
• C. $155$
• D. $106$

Answer 1

The correct option is D $106$

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