Question

If $(1 + x - 2x^2)^6 = 1 + a_1x + a_2x^2 + ... + a_{12}x^{12}$, then the expression $a_2 + a_4 + a_6+ .... + a_{12}$ has the value

• A. $32$
• B. $63$
• C. $64$
• D. $31$

Answer 1

Answer: (D)

$\left(1 + x - 2x^{2}\right)^{6} = 1 + a_{1}x + a_{2}x^{2} + ... + a_{12}x^{12}$
On putting $x = 1$ and $x = -1$ in the given expansion and adding the results, we get
$64 = 2\left(1 +a_{2} + a_{4 }+ ....+ a_{12}\right)$
$\Rightarrow a_{2} + a_{4} + a_{6} + .... + a_{12} = 31$