PYQ Vault
EbooksTutor

Sums of (w^k + 1/w^k) raised to a power, over k, for a root of unity w

This type of question, asked 4 times: 2021 Paper 18, 26 Jun 2022, 4 Apr 2025, 21 Jan 2026 S1. All 4 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 18

    81
    Let z=1−i32,i=−1z =\frac{1 -i\sqrt{3}}{2},i=\sqrt{- 1}. Then the value of 21+(z+1z)3+(z2+1z2)3+(z3+1z3)3+…+(z21+1z21)321 +\left( z +\frac{1}{z} \right)^{3}+\left( z^{2}+\frac{1}{z^{2}} \right)^{3}+\left( z^{3}+\frac{1}{z^{3}} \right)^{3}+ \ldots +\left( z^{21}+\frac{1}{z^{21}} \right)^{3} is

  2. Question 2

    26 Jun 2022

    180
    If z2+z+1=0,z∈Cz^{2}+ z + 1 = 0,z \in\mathbb{C}, then ∣∑n=115(Zn+(−1)n1Zn)2∣\left| \sum_{n = 1}^{15} \left( Z^{n}+ ( - 1)^{n}\frac{1}{Z^{n}} \right)^{2} \right| is equal to

  3. Question 3

    4 Apr 2025

    146
    If α\alpha is a root of the equation x2+x+1=0x^{2} + x + 1 = 0 and ∑k=1n(αk+1αk)2=20\sum_{k = 1}^{n} \left( \alpha^{k} + \frac{1}{\alpha^{k}} \right)^{2} = 20, then nn is equal to .

  4. Question 4

    21 Jan 2026 S1

    67
    If x2+x+1=0x^{2}+ x + 1 = 0, then the value of (x+1x)4+(x2+1x2)4+(x3+1x3)4+…+(x25+1x25)4\left( x +\frac{1}{x} \right)^{4}+\left( x^{2}+\frac{1}{x^{2}} \right)^{4}+\left( x^{3}+\frac{1}{x^{3}} \right)^{4}+ \ldots +\left( x^{25}+\frac{1}{x^{25}} \right)^{4} is :

    Tap an option to check your answer.