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Easy

Question

Co-efficient of x to the power of r end exponent in the expansion of fraction numerator 1 plus 4 x to the power of 2 end exponent plus x to the power of 4 end exponent over denominator open parentheses 1 minus x close parentheses to the power of 4 end exponent end fraction is

  1. r to the power of 2 end exponent plus 2 r    
  2. r to the power of 3 end exponent plus 3 r    
  3. r plus 3    
  4. none of these.    

The correct answer is: r to the power of 3 end exponent plus 3 r


    fraction numerator 1 plus 4 x to the power of 2 end exponent plus x to the power of 4 end exponent over denominator open parentheses 1 minus x close parentheses to the power of 4 end exponent end fraction equals open parentheses 1 plus 4 x to the power of 2 end exponent plus x to the power of 4 end exponent close parentheses open parentheses 1 minus x close parentheses to the power of negative 4 end exponent
    equals open parentheses 1 plus 4 x to the power of 2 end exponent plus x to the power of 4 end exponent close parentheses open parentheses 1 plus 4 x plus 10 x to the power of 2 end exponent plus... close parentheses equals open parentheses 1 plus 4 x to the power of 2 end exponent plus x to the power of 4 end exponent close parentheses open parentheses 1 plus to the power of 4 end exponent C subscript 1 end subscript x plus to the power of 5 end exponent C subscript 2 end subscript x to the power of 2 end exponent plus... plus to the power of r minus 1 end exponent C subscript r minus 4 end subscript x to the power of r minus 4 end exponent plus to the power of r end exponent C subscript r minus 3 end subscript x to the power of r minus 3 end exponent close plus to the power of r plus 1 end exponent C subscript r minus 2 end subscript x to the power of r minus 2 end exponent plus to the power of r plus 2 end exponent C subscript r minus 1 end subscript x to the power of r minus 1 end exponent x to the power of r minus 1 end exponent plus to the power of r plus 3 end exponent C subscript r end subscript x to the power of r end exponent
    Hence co- eff. of x to the power of r end exponent in fraction numerator 1 plus 4 x to the power of 2 end exponent plus x to the power of 4 end exponent over denominator open parentheses 1 minus x close parentheses to the power of 4 end exponent end fraction = r plus 3 C subscript r end subscript plus 4. to the power of r plus 1 end exponent C subscript r minus 2 end subscript plus to the power of r minus 1 end exponent C subscript r minus 4 end subscript
    equals fraction numerator open parentheses r plus 3 close parentheses open parentheses r plus 2 close parentheses open parentheses r plus 1 close parentheses over denominator 1.2.3 end fraction plus 4. fraction numerator open parentheses r plus 1 close parentheses r open parentheses r minus 1 close parentheses over denominator 1.2.3 end fraction plus fraction numerator open parentheses r minus 1 close parentheses open parentheses r minus 2 close parentheses open parentheses r minus 3 close parentheses over denominator 1.2.3 end fraction
    equals fraction numerator 1 over denominator 6 end fraction open square brackets r to the power of 3 end exponent plus 6 r to the power of 2 end exponent plus 11 r plus 6 plus 4 open parentheses r to the power of 3 end exponent minus r close parentheses plus r to the power of 3 end exponent minus 6 r to the power of 2 end exponent plus 11 r minus 6 close square brackets equals fraction numerator 1 over denominator 6 end fraction open square brackets 6 r to the power of 3 end exponent plus 18 r close square brackets equals r to the power of 3 end exponent plus 3 r

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