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    Maths-
    General
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    Question

    The value of log subscript e end subscript invisible function application open parentheses 1 plus a x to the power of 2 end exponent plus a to the power of 2 end exponent plus fraction numerator a over denominator x to the power of 2 end exponent end fraction close parentheses is

    1. a open parentheses x to the power of 2 end exponent minus fraction numerator 1 over denominator x to the power of 2 end exponent end fraction close parentheses minus fraction numerator a to the power of 2 end exponent over denominator 2 end fraction open parentheses x to the power of 4 end exponent minus fraction numerator 1 over denominator x to the power of 4 end exponent end fraction close parentheses plus fraction numerator a to the power of 3 end exponent over denominator 3 end fraction open parentheses x to the power of 6 end exponent minus fraction numerator 1 over denominator x to the power of 6 end exponent end fraction close parentheses minus.....    
    2. a open parentheses x to the power of 2 end exponent plus fraction numerator 1 over denominator x to the power of 2 end exponent end fraction close parentheses minus fraction numerator a to the power of 2 end exponent over denominator 2 end fraction open parentheses x to the power of 4 end exponent plus fraction numerator 1 over denominator x to the power of 4 end exponent end fraction close parentheses plus fraction numerator a to the power of 3 end exponent over denominator 3 end fraction open parentheses x to the power of 6 end exponent plus fraction numerator 1 over denominator x to the power of 6 end exponent end fraction close parentheses minus.....    
    3. a open parentheses x to the power of 2 end exponent plus fraction numerator 1 over denominator x to the power of 2 end exponent end fraction close parentheses plus fraction numerator a to the power of 2 end exponent over denominator 2 end fraction open parentheses x to the power of 4 end exponent plus fraction numerator 1 over denominator x to the power of 4 end exponent end fraction close parentheses plus fraction numerator a to the power of 3 end exponent over denominator 3 end fraction open parentheses x to the power of 6 end exponent plus fraction numerator 1 over denominator x to the power of 6 end exponent end fraction close parentheses plus.....    
    4. a open parentheses x to the power of 2 end exponent minus fraction numerator 1 over denominator x to the power of 2 end exponent end fraction close parentheses plus fraction numerator a to the power of 2 end exponent over denominator 2 end fraction open parentheses x to the power of 4 end exponent minus fraction numerator 1 over denominator x to the power of 4 end exponent end fraction close parentheses plus fraction numerator a to the power of 3 end exponent over denominator 3 end fraction open parentheses x to the power of 6 end exponent minus fraction numerator 1 over denominator x to the power of 6 end exponent end fraction close parentheses plus.....    

    The correct answer is: a open parentheses x to the power of 2 end exponent plus fraction numerator 1 over denominator x to the power of 2 end exponent end fraction close parentheses minus fraction numerator a to the power of 2 end exponent over denominator 2 end fraction open parentheses x to the power of 4 end exponent plus fraction numerator 1 over denominator x to the power of 4 end exponent end fraction close parentheses plus fraction numerator a to the power of 3 end exponent over denominator 3 end fraction open parentheses x to the power of 6 end exponent plus fraction numerator 1 over denominator x to the power of 6 end exponent end fraction close parentheses minus.....

      log subscript e end subscript invisible function application open square brackets 1 plus a x to the power of 2 end exponent plus a to the power of 2 end exponent plus fraction numerator a over denominator x to the power of 2 end exponent end fraction close square brackets
      equals log subscript e end subscript invisible function application left parenthesis 1 plus a x to the power of 2 end exponent right parenthesis open parentheses 1 plus fraction numerator a over denominator x to the power of 2 end exponent end fraction close parentheses
      equals log subscript e end subscript invisible function application left parenthesis 1 plus a x to the power of 2 end exponent right parenthesis plus log subscript e end subscript invisible function application open parentheses 1 plus fraction numerator a over denominator x to the power of 2 end exponent end fraction close parentheses
      = open square brackets a x to the power of 2 end exponent minus fraction numerator 1 over denominator 2 end fraction a to the power of 2 end exponent x to the power of 4 end exponent plus fraction numerator 1 over denominator 3 end fraction a to the power of 3 end exponent x to the power of 6 end exponent minus........... close square brackets plus open square brackets fraction numerator a over denominator x to the power of 2 end exponent end fraction minus fraction numerator 1 over denominator 2 end fraction a to the power of 2 end exponent open parentheses fraction numerator 1 over denominator x to the power of 4 end exponent end fraction close parentheses plus fraction numerator 1 over denominator 3 end fraction a to the power of 3 end exponent open parentheses fraction numerator 1 over denominator x to the power of 6 end exponent end fraction close parentheses minus..... close square brackets
      equals a open parentheses x to the power of 2 end exponent plus fraction numerator 1 over denominator x to the power of 2 end exponent end fraction close parentheses minus fraction numerator 1 over denominator 2 end fraction. a to the power of 2 end exponent open parentheses x to the power of 4 end exponent plus fraction numerator 1 over denominator x to the power of 4 end exponent end fraction close parentheses plus fraction numerator 1 over denominator 3 end fraction a to the power of 3 end exponent open parentheses x to the power of 6 end exponent plus fraction numerator 1 over denominator x to the power of 6 end exponent end fraction close parentheses minus...

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      open vertical bar table row cell a to the power of 2 end exponent plus x to the power of 2 end exponent end cell cell a b end cell cell c a end cell row cell a b end cell cell b to the power of 2 end exponent plus x to the power of 2 end exponent end cell cell b c end cell row cell c a end cell cell b c end cell cell c to the power of 2 end exponent plus x to the power of 2 end exponent end cell end table close vertical bar is divisor of

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