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Term independent of x in (a x^p + b/x^q)^n, or a constant from it

This type of question, asked 6 times: 2021 Paper 22, 1 Feb 2023, 8 Apr 2023, 5 Apr 2024, 8 Apr 2024, 4 Apr 2026 S2. All 6 questions below, oldest first.

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  1. Question 1

    2021 Paper 22

    89
    If (3644)k\left( \frac{3^{6}}{4^{4}} \right)k is the term, independent of xx, in the binomial expansion of (x4−12x2)12\left( \frac{x}{4}-\frac{12}{x^{2}} \right)^{12}, then kk is equal to

  2. Question 2

    1 Feb 2023

    172
    If the term without xx in the expansion of (x23+ax3)22\left( x^{\frac{2}{3}}+\frac{a}{x^{3}} \right)^{22} is 7315, then ∣a∣|a| is equal to

  3. Question 3

    8 Apr 2023

    82
    Let [t] denotes the greatest integer ≤t\leq t. If the constant term in the expansion of (3x2−12x5)7\left( 3x^{2}-\frac{1}{2x^{5}} \right)^{7} is α\alpha, then [α]\lbrack\alpha\rbrack is equal to

  4. Question 4

    5 Apr 2024

    160
    If the constant term in the expansion of (35x+2x53)12,x≠0\left( \frac{\sqrt[5]{3}}{x}+\frac{2x}{\sqrt[3]{5}} \right)^{12},x \neq 0, is α×28×35\alpha \times2^{8}\times\sqrt[5]{3}, then 25α25\alpha is equal to :

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  5. Question 5

    8 Apr 2024

    170
    If the term independent of xx in the expansion of (ax 2+12x3)10\left( \sqrt{ax}\ ^{2}+\frac{1}{2x^{3}} \right)^{10} is 105, then a2a^{2} is equal to :

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  6. Question 6

    4 Apr 2026 S2

    59
    In the expansion of (9x−13x)18,x>0\left( 9x-\frac{1}{3\sqrt{x}} \right)^{18},x> 0, if the term independent of x is (221)k(221)k. then k is equal to:

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