If (1 + x - 2x2)6 = 1 + a1x + a2x2 + ... + a12x12, then the expression a2 + a4 + a6+ .... + a12 has the value

Question

If (1 + x - 2x2)6 = 1 + a1x + a2x2 + ... + a12x12, then the expression a2 + a4 + a6+ .... + a12 has the value (A) 32 (B) 63 (C) 64 (D) 31

Tardigrade Admin · Accepted Answer

(1 + x - 2x2)6 = 1 + a1x + a2x2 + ... + a12x12 On putting x = 1 and x = -1 in the given expansion and adding the results, we get 64 = 2(1 +a2 + a4 + ....+ a12) ⇒ a2 + a4 + a6 + .... + a12 = 31

Q. If $(1 + x - 2 x^{2})^{6} = 1 + a_{1} x + a_{2} x^{2} + ... + a_{12} x^{12}$ , then the expression $a_{2} + a_{4} + a_{6} + .... + a_{12}$ has the value

$32$

$63$

$64$

$31$

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

Q. If (1+x−2x2)6=1+a1​x+a2​x2+...+a12​x12, then the expression a2​+a4​+a6​+....+a12​ has the value

32

63

64

31