B8-B11 De Moivre's theorem, exponential form and roots of unity Quiz: AQA Further Maths, Unit 2

The 20 questions

1. By de Moivre's theorem, what is (cos theta + i sin theta)^n equal to?

   • cos(theta^n) + i sin(theta^n)
   • cos(theta) + i sin(n theta)
   • n cos(theta) + i n sin(theta)
   • cos(n theta) + i sin(n theta)

2. What is e^(i pi)?

   • -i
   • 1
   • -1
   • i

3. What are the solutions of z^n = 1 called?

   • The integers n
   • The n-th roots of unity
   • The n-th roots of minus one
   • The zeros of the origin

4. How many distinct 5th roots of unity are there?

   • 6
   • 4
   • 10
   • 5

5. Write cos 3 theta in terms of cos theta.

   • 4 cos^3(theta) + 3 cos(theta)
   • 4 cos^3(theta) - 3 cos(theta)
   • cos^3(theta) - 3 cos(theta)
   • 3 cos^3(theta) - 4 cos(theta)

6. Write sin 3 theta in terms of sin theta.

   • 3 sin(theta) + 4 sin^3(theta)
   • sin^3(theta) - 3 sin(theta)
   • 3 sin(theta) - 4 sin^3(theta)
   • 4 sin(theta) - 3 sin^3(theta)

7. What is the modulus of each cube root of 8?

   • 2
   • 4
   • 8
   • 1/2

8. For w = e^(2 pi i/3), what is 1 + w + w^2?

   • 1
   • 0
   • 3
   • -1

9. Express e^(i pi/3) in the form a + bi.

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

10. Which fourth root of -16 lies in the first quadrant?

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

11. What is the product of all four fourth roots of unity?

    • i
    • -1
    • 1
    • 0

12. What is (1 + i)^6?

    • 8
    • -8i
    • 8i
    • -8

13. What is the modulus of (2 cis(pi/5))^10?

    • 1024
    • 512
    • 20
    • 2048

14. For w = e^(2 pi i/5), what is 1 + w + w^2 + w^3 + w^4?

    • 5
    • 1
    • -1
    • 0

15. Using (cos theta + i sin theta)^2, which expression gives cos^2 theta in terms of cos 2 theta?

    • (1 - cos 2 theta)/2
    • 1 + cos 2 theta
    • (1 + cos 2 theta)/2
    • (cos 2 theta - 1)/2

16. Which is a real cube root of -8?

    • 2 cis(-pi/3)
    • 2 cis(pi/3)
    • 2
    • -2

17. How many sixth roots of unity have order exactly 6?

    • 2
    • 6
    • 1
    • 3

18. Let w be a primitive cube root of unity. What is (1 + w)^3?

    • 2
    • 1
    • w
    • -1

19. Which of these is a cube root of i?

    • cis(pi/2)
    • cis(pi/6)
    • cis(pi/3)
    • cis(-pi/6)

20. What is the real part of (1 + i sqrt(3))^5?

    • -32
    • 32
    • 16
    • -16

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