Q55 / 90 · KVPY 2012 (SB/SX) · Mathematics

The coefficient of $x^{2012}$ in $\frac{1+x}{\left(1+x^{2}\right)(1-x)}$ is -

  • A
    2010
  • B
    2011
  • C
    2012
  • Correct answer
    D
    None of these

Solution

Coeff. Of $x^{2012}$ $\frac{(1+x)^{2}}{\left(1+x^{2}\right)\left(1-x^{2}\right)}$ $=(1+x)^{2}\left(1-x^{4}\right)^{-1}$ $=\left(1+2 x+x^{2}\right)\left(1-x^{4}\right)^{-1}$ Coeff. Of $x^{2012}+2$ Coeff of $x^{2011}+$ Coeff of $x^{2010}$ in the expansion of $\left(1-x^{4}\right)^{-1}$ $x^{2011}$ and $x^{2010}$ not possible in $\left(1-x^{4}\right)^{-1}$ $=$ only coeff. Of $x^{2012}$ in the expassion of $\left(1-x^{4}\right)^{-1}$ ${ }^{1+503-1} C_{503}=1$

Hence, the correct answer is option: 'D'.

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