If (1 + x - 2x 2 ) 6 = 1 + a 1 x + a 2 x 2 + ... + a 12 x 12 , then a 2 + a 4 + ... a 12 is?

Let \(P(x) = (1 + x - 2x 2) 6\). \(P(1) = (1 + 1 - 2) 6 = 0\), which is the sum of all coefficients. \(P(-1) = (1 - 1 - 2) 6 = 64\), which is the even-power coefficients minus the odd-power ones. Sum of even-power coefficients (including \(a 0 = 1\)) = \(P(1) + P(-1)/2 = 32\). \(a 2 + a 4 + + a 12 = 32 - 1 = 31\) Answer: (C) 31

If (1 + x - 2x2)6&nb... - Solved

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Question 13

Mathematics Limit of Functions Hard

If (1 + x - 2x2)6 = 1 + a1x + a2x2 + ... + a12x12, then a2 + a4 + ... a12 is?

(A) 29

(B) 30

(C) 31

(D) 32

✓ Correct Answer: Option C Verified Solution

📘 Step-by-Step Solution & Detailed Explanation:

Let \(P(x) = (1 + x - 2x^2)^6\).

\(P(1) = (1 + 1 - 2)^6 = 0\), which is the sum of all coefficients.
\(P(-1) = (1 - 1 - 2)^6 = 64\), which is the even-power coefficients minus the odd-power ones.

Sum of even-power coefficients (including \(a_0 = 1\)) = \(\frac{P(1) + P(-1)}{2} = 32\).

\(a_2 + a_4 + \dots + a_{12} = 32 - 1 = 31\)

Answer: (C) 31

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