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Coefficient of x^k in one binomial equals coefficient of x^-k in a related binomial: find a relation between a and b

The same question, asked 5 times: 2021 Paper 16*, 29 Jan 2023*, 30 Jan 2023*, 6 Apr 2023*, 10 Apr 2023. Every wording below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 16

    43
    If Coefficient of x7x^{7} in (x2+1bx)11\left( x^{2}+\frac{1}{bx} \right)^{11} and x−7x^{- 7} in [x−1bx2]11,b≠0\left\lbrack x-\frac{1}{bx^{2}} \right\rbrack^{11},b\neq 0, are equal then find bb.

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  2. Question 2

    29 Jan 2023

    87
    If the co-efficient of x9x^{9} in (αx3+1βx)11\left( \alpha x^{3}+\frac{1}{\beta x} \right)^{11} and the co-efficient of x−9x^{- 9} in (αx−1βx3)11\left( \alpha x -\frac{1}{\beta x^{3}} \right)^{11} are equal, then (αβ)2(\alpha\beta)^{2} is equal to

  3. Question 3

    30 Jan 2023

    79
    If the coefficient of x15x^{15} in the expansion of (ax3+1bx1/3)15\left( ax^{3}+\frac{1}{bx^{1/3}} \right)^{15} is equal to the coefficient of x−15x^{- 15} in the expansion of (ax1/3−1bx3)15\left( ax^{1/3}-\frac{1}{bx^{3}} \right)^{15}, where aa and bb are positive real numbers, then for each such ordered pair (a,b)(a,b) :

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  4. Question 4

    6 Apr 2023

    165
    If the coefficients of x7x^{7} in (ax2+12bx)11\left( ax^{2}+\frac{1}{2bx} \right)^{11} and x−7x^{- 7} in (ax−13bx2)11\left( ax -\frac{1}{3bx^{2}} \right)^{11} are equal, then

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  5. Question 5

    10 Apr 2023

    75
    If the coefficient of x7x^{7} in (ax−1bx2)13\left( ax -\frac{1}{bx^{2}} \right)^{13} and the coefficient of x−5x^{- 5} in (ax+1bx2)13\left( ax +\frac{1}{bx^{2}} \right)^{13} are equal, then a4b4a^{4}b^{4} is equal to:

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