If 1 + x - 2 x 2 6 = 1 + a 1 x + a 2 x 2 + . . . + a 12 x 12 , then the value of a 2 + a 4 + . . . + a 12 is

Question

If 1 + x - 2 x 2 6 = 1 + a 1 x + a 2 x 2 + . . . + a 12 x 12 , then the value of a 2 + a 4 + . . . + a 12 is, 31, 32, 64, 1024

MARKS App · Accepted Answer

Given , ( 1 + x - 2 x 2 ) 6 = 1 + a 1 x + a 2 x 2 + . . . + a 12 x 12 ...(i) On putting x = 1 in Equation (i), we get 1 + 1 - 2 6 = 1 + a 1 + a 2 + … + a 12 ⟹ ( 0 ) 6 = 1 + a 1 + a 2 + . . . + a 12 ....(ii) On putting x = - 1 in Equation (i), we get 1 - 1 - 2 6 = 1 - a 1 + a 2 - a 3 + … + a 12 ⟹ ( - 2 ) 6 = 1 - a 1 + a 2 - a 3 + . . . + a 12 ...(iii) On adding Equations (ii) and (iii) we get - 2 6 = 2 1 + a 2 + a 4 + … + a 12 ⟹ 64 2 - 1 = a 2 + a 4 + … + a 12 ∴ a 2 + a 4 + … + a 12 = 31

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