Question

If the coefficient of x⁷ $open square brackets ax squared space plus open parentheses 1 over bx close parentheses close square brackets to the power of 11$ in equals the coefficient of x^-7 in $open square brackets ax squared space minus open parentheses 1 over bx close parentheses close square brackets to the power of 11$ then a and b satisfy the relation

a – b = 1

a + b = 1

a/b =1

ab =1

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Solution

ab =1

straight T subscript straight r plus 1 end subscript space in space the space expansion space space open square brackets ax squared space plus space 1 over bx close square brackets to the power of 11 space equals space to the power of 11 straight C subscript straight r space left parenthesis ax squared right parenthesis to the power of 11 minus straight r end exponent space open parentheses 1 over bx close parentheses to the power of straight r space equals space to the power of 11 straight C subscript straight r space left parenthesis straight a right parenthesis to the power of 11 minus straight r end exponent space left parenthesis straight b right parenthesis to the power of negative straight r end exponent space left parenthesis straight x right parenthesis to the power of 22 minus 2 straight r minus straight r end exponent rightwards double arrow space 22 minus 3 straight r space equals space 7 straight r space equals 5 therefore comma space coefficient space of space straight x to the power of 7 space equals space to the power of 11 straight C subscript 5 space left parenthesis straight a right parenthesis to the power of 6 space left parenthesis straight b right parenthesis to the power of negative 5 end exponent...... space left parenthesis 1 right parenthesis Again space straight T subscript straight r plus 1 end subscript space in space the space exapansion space open square brackets ax space space minus space 1 over bx squared close square brackets to the power of 11 space equals space to the power of 11 straight C subscript straight r space left parenthesis ax right parenthesis to the power of 11 minus straight r end exponent space open parentheses negative 1 over bx squared close parentheses to the power of straight r space equals space to the power of 11 straight C subscript straight r straight a to the power of 11 minus straight r end exponent space left parenthesis negative 1 right parenthesis to the power of straight r space straight x space left parenthesis straight b right parenthesis to the power of negative straight r end exponent space left parenthesis straight x right parenthesis to the power of negative 2 straight r end exponent space left parenthesis straight x right parenthesis to the power of 11 minus straight r end exponent Now comma space 11 space minus 3 straight r space equals space 7 rightwards double arrow space 3 straight r space equals 18 rightwards double arrow straight r space equals 6 therefore comma space coefficient space of space straight x to the power of negative 7 end exponent space equals space to the power of 11 straight C subscript 6 straight a to the power of 5 space straight x space 1 space straight x space left parenthesis straight b right parenthesis to the power of negative 6 end exponent rightwards double arrow to the power of 11 straight C subscript 5 space left parenthesis straight a right parenthesis to the power of 6 left parenthesis straight b right parenthesis to the power of negative 5 end exponent space equals space to the power of 11 straight C subscript 6 straight a to the power of 5 space straight x space left parenthesis straight b right parenthesis to the power of negative 6 end exponent rightwards double arrow ab space equals space 1

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