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Manipal MET2016MathematicsBinomial Theorem

The coefficient of the term independent of x in the expansion of ( x+1 x^ 2 / 3 -x^ 1 / 3 +1 - x-1 x-x^ 1 / 2 )¹⁰ is

Options

  1. A210
  2. B105
  3. C70
  4. D112

Correct answer

A. 210

Step-by-step solution

x+1 x^ 2 / 3 -x^ 1 / 3 +1 - x-1 x-x^ 1 / 2 aligned & = (x^ 1 / 3 )^3+1^3 x^ 2 / 3 -x^ 1 / 3 +1 - x-1 x^ 1 / 2 (x^ 1 / 2 -1 ) & = (x^ 1 / 3 +1 ) (x^ 2 / 3 -x^ 1 / 3 +1 ) x^ 2 / 3 -x^ 1 / 3 +1 - x^ 1 / 2 +1 x^ 1 / 2 aligned aligned & =x^ 1 / 3 +1-1-x^ -1 / 2 & =x^ 1 / 3 -x^ -1 / 2 aligned ( x+1 x^ 2 / 3 -x^ 1 / 3 +1 - x-1 x-x^ 1 / 2 )¹⁰= (x^ 1 / 3 -x^ -1 / 2 )¹⁰ aligned & T_ r+1 for (x^ 1 / 3 -x^ -1 / 2 ) is & ¹⁰ C_r (x^ 1 / 3 )^ 10-r (-1)^r (x^ -1 / 2 )^r . aligned For term independent of x , 10-r 3 - r 2 =0 array r

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