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Coefficients of x and x^2 in (1+x)^p (1-x)^q are given: find p and q

The same question, asked 4 times: 2021 Paper 12*, 26 Jul 2022*, 10 Apr 2023*, 23 Jan 2025. Every wording below, oldest first.

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  1. Question 1

    2021 Paper 12

    39
    For the natural numbers mm, nn, if (1−y)m(1+y)n=1+a1y+a2y2+….+am+nym+n(1 -y)^{m}(1 +y)^{n}= 1 +a_{1}y+a_{2}y^{2}+ \ldots. +a_{m+n}y^{m+n} and a1=a2=10a_{1}=a_{2}= 10, then the value of (m+n)(m+n) is equal to:

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  2. Question 2

    26 Jul 2022

    85
    If the coefficients of xx and x2x^{2} in the expansion of (1+x)p(1−x)q, p,q≤15(1 + x)^{p}(1 - x)^{q},\ p,q \leq 15, are -3 and -5 respectively, then the coefficient of x3x^{3} is equal to

  3. Question 3

    10 Apr 2023

    166
    If the coefficients of xx and x2x^{2} in (1+x)p(1−x)q(1 + x)^{p}(1 - x)^{q} are 4 and -5 respectively, then 2p+3q2p + 3q is equal to

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  4. Question 4

    23 Jan 2025

    126
    If in the expansion of (1+x)p(1−x)q(1 + x)^{p}(1 - x)^{q}, the coefficients of xx and x2x^{2} are 1 and -2 , respectively, then p2+q2p^{2}+q^{2} is equal to :

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