2.10-2.11 The n distinct nth roots and complex roots of polynomials Quiz: Pearson Edexcel Further Maths, Unit 2

The 20 questions

1. How many distinct nth roots does a non-zero complex number have?

   • 2
   • n^2
   • n - 1
   • n

2. On an Argand diagram, what shape do the n distinct nth roots of a non-zero complex number form?

   • A circle of radius n
   • A regular n-gon
   • An irregular polygon with n vertices
   • A straight line

3. For an integer n > 1, what is the sum of all the nth roots of unity?

   • n
   • 0
   • 1
   • -1

4. What are the four roots of z^4 = 1?

   • 1, i, -1, -i
   • 2, 2i, -2, -2i
   • 0, 1, i, -1
   • 1 and -1 only

5. What are the three cube roots of 8?

   • -2, 1 + sqrt(3) i, 1 - sqrt(3) i
   • 2, -1 + sqrt(3) i, -1 - sqrt(3) i
   • 2, 2i, -2
   • 2, 1 + sqrt(3) i, 1 - sqrt(3) i

6. Every root of z^5 = 32 has the same modulus. What is that modulus?

   • sqrt(2)
   • 2
   • 32
   • 32/5

7. The 7th roots of unity are equally spaced in argument. What is the angular spacing between adjacent roots?

   • 2 pi / 7
   • pi / 7
   • 7 pi / 2
   • pi / 14

8. Which of the following is a non-real cube root of unity?

   • -1 + (sqrt(3)/2) i
   • 1/2 + (sqrt(3)/2) i
   • -1/2 + (sqrt(3)/2) i
   • -1/2 + i

9. What is the product of the roots of z^3 - 1 = 0?

   • 1
   • -1
   • 0
   • 3

10. The vertices of a square are the 4th roots of unity 1, i, -1 and -i. What is its area?

    • sqrt(2)
    • 2
    • 4
    • 1

11. A regular hexagon has its vertices at the 6th roots of unity. What is its side length?

    • sqrt(3)
    • 1/2
    • 2
    • 1

12. Which pair of numbers are the solutions of z^2 = -4?

    • 4i and -4i
    • i and -i
    • 2 and -2
    • 2i and -2i

13. Which of the following is a root of z^4 = -1?

    • (1 - i)/2
    • i
    • (1 + i)/sqrt(2)
    • 1 + i

14. Which monic cubic polynomial has the three roots 2, -1 + sqrt(3) i and -1 - sqrt(3) i?

    • z^3 - 4
    • z^3 - 8
    • z^3 + 8
    • z^3 - 8z

15. Solve z^2 - 2z + 4 = 0 over the complex numbers.

    • z = 1 + sqrt(3) i or z = 1 - sqrt(3) i
    • z = 1 + 2i or z = 1 - 2i
    • z = 2 + sqrt(3) i or z = 2 - sqrt(3) i
    • z = -1 + sqrt(3) i or z = -1 - sqrt(3) i

16. What are the three distinct cube roots of -1?

    • 1, -1/2 + (sqrt(3)/2) i, -1/2 - (sqrt(3)/2) i
    • -1, 1/2 + (sqrt(3)/2) i, 1/2 - (sqrt(3)/2) i
    • -1, -1/2 + (sqrt(3)/2) i, -1/2 - (sqrt(3)/2) i
    • 1, 1/2 + (sqrt(3)/2) i, 1/2 - (sqrt(3)/2) i

17. Which set gives the vertices of a regular pentagon centred at the origin with one vertex at 1?

    • e^(pi i k / 5) for k = 0, 1, 2, 3, 4
    • e^(2 pi i k / 5) for k = 0, 1, 2, 3, 4
    • e^(2 pi i k / 4) for k = 0, 1, 2, 3
    • 5 e^(2 pi i k / 5) for k = 0, 1, 2, 3, 4

18. The vertices of a regular hexagon are the roots of z^6 = 64. What is the circumradius?

    • 8
    • 6
    • 2
    • 64

19. If omega = e^(2 pi i / 3), what is omega^100?

    • e^(4 pi i / 3)
    • -1
    • 1
    • e^(2 pi i / 3)

20. Which of the following is a cube root of i?

    • (sqrt(3) - i)/2
    • (1 + sqrt(3) i)/2
    • (sqrt(3) + i)/2
    • (1 + i)/sqrt(2)

Related quizzes

• Quadratic equations and complex arithmetic Quiz · 2.1-2.2 · 20 questions
• Conjugate roots and Argand diagrams Quiz · 2.3-2.4 · 20 questions
• Modulus-argument form and loci Quiz · 2.5-2.7 · 20 questions
• De Moivre's theorem and roots of unity Quiz · 2.8-2.9 · 20 questions
• Proof by mathematical induction Quiz · 1.1 · 20 questions
• Matrix arithmetic and inverses Quiz · 3.1-3.2 · 20 questions
• Expectation of discrete random variables Quiz · 3B.1.1 · 20 questions
• The Poisson distribution Quiz · 3B.2.1 · 20 questions
• Geometric and negative binomial models Quiz · 3B.3.1 · 20 questions
• Hypothesis tests for the Poisson distribution Quiz · 3B.4.1 · 20 questions