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High powers of the cube roots of unity

This type of question, asked 5 times: 2021 Paper 25, 29 Jun 2022, 27 Jan 2024, 30 Jan 2024, 22 Jan 2026 S1. All 5 questions below, oldest first.

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  1. Question 1

    2021 Paper 25

    89
    The sum of 162th 162^{\text{th~}} power of the roots of the equation x3−2x2+2x−1=0x^{3}- 2x^{2}+ 2x- 1 = 0 is

  2. Question 2

    29 Jun 2022

    151
    Let α\alpha be a root of the equation 1+x2+x4=01 +x^{2}+x^{4}= 0. Then the value of α1011+α2022−α3033\alpha^{1011}+\alpha^{2022}-\alpha^{3033} is equal to:

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  3. Question 3

    27 Jan 2024

    90
    If α\alpha satisfies the equation x2+x+1=0x^{2}+ x + 1 = 0 and (1+α)7=A+Bα+Cα2, A, B,C≥0(1 + \alpha)^{7}= A + B\alpha +C\alpha^{2},\text{ }A,\text{ }B,C \geq 0, then 5(3 A−2 B−C)5(3\text{ }A - 2\text{ }B - C) is equal to

  4. Question 4

    30 Jan 2024

    166
    If zz is a complex number, then the number of common roots of the equation z1985+z100+1=0z^{1985}+z^{100}+ 1 = 0 and z3+2z2+2z+1=0z^{3}+ 2z^{2}+ 2z+ 1 = 0, is equal to :

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  5. Question 5

    22 Jan 2026 S1

    71
    Let α=−1+i32\alpha = \frac{- 1 + i\sqrt{3}}{2} and β=−1−i32,i=−1\beta = \frac{- 1 - i\sqrt{3}}{2},i = \sqrt{- 1}. If (7−7α+9β)20+(9+7α−7β)20+(−7+9α+7β)20+(14+7α+7β)20=m10(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + ( - 7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}, then m is ____\_\_\_\_ .