Q33 / 90 · JEE Main 2018 (16 Apr Online) · Mathematics

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

Solution

We have,

{(1 + 2 x + 3 x^{2})}^{6} = \sum_{r = 0}^{6}^{6} C_{r} {(2 x + 3 x^{2})}^{r}

=^{6} C_{0} +^{6} C_{1} (2 x + 3 x^{2}) +^{6} C_{2} {(2 x + 3 x^{2})}^{2} + \dots +^{6} C_{6} {(2 x + 3 x^{2})}^{6}

∴

Coefficient of

x^{2} = 18 + 60 = 78

Again, coefficient of

x^{2}

{(1 - 4 x^{2})}^{6} = - 24

Constant term in

({(1 + 2 x + 3 x^{2})}^{6} + {(1 - 4 x^{2})}^{6})

1 + 1 = 2

.
Thus, the required coefficient of

x^{2} = 2

(coefficient of

x^{2}

) in

({(1 + 2 x + 3 x^{2})}^{6} + {(1 - 4 x^{2})}^{6}) -

constant term in

({(1 + 2 x + 3 x^{2})}^{6} + {(1 - 4 x^{2})}^{6})

= 2 (78 - 24) - 2

= 106

Hence, the correct answer is option: 'D'.