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AP EAMCET201920 Apr 2019Evening ShiftMathematicsBinomial TheoremActual

If (1+x)^n=a₀+a₁ x+a₂ x^2+ +a_n x^n and a₀-a₂+a₄-a₆+ =k n 4 , then k=

Options

  1. A2^n
  2. B2^ 2 n
  3. C2^n 2
  4. D2^ n 2

Correct answer

D. 2^ n 2

Step-by-step solution

If (1+x)^n=a₀+a₁ x+a₂ x^2+a₃ x^3+ +a_n x^n Put x=i , we get aligned (1+i)^n= (a₀-a₂+a₄-a₆+ ) & + & +i (a₁-a₃+a₅-a₇+ ) [ 2 ( 4 +i 4 ) ]^n= & (a₀-a₂+a₄-a₆+ ) & +i (a₁-a₃+a₅-a₇+ ) aligned On comparing the real and imaginary parts, we get 2^ n / 2 n 4 =a₀-a₂+a₄-a₆+ and 2^ n / 2 n 4 =a₁-a₃+a₅-a₇+ a₀-a₂+a₄-a₆+ =k n 4 k=2^ n 2 Hence, option (d) is correct.

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