2.8-2.9 De Moivre's theorem and roots of unity Quiz: Pearson Edexcel Further Maths, Unit 2

The 20 questions

1. Which statement is de Moivre's theorem for an integer n?

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

2. In exponential form, which expression equals e^(i theta)?

   • cos theta - i sin theta
   • cos theta + i sin theta
   • sin theta + i cos theta
   • cos theta + sin theta

3. What is the value of e^(i pi)?

   • 1
   • -i
   • i
   • -1

4. Which expression gives cos theta in terms of exponentials?

   • e^(i theta) + e^(-i theta)
   • (e^(i theta) + e^(-i theta)) / 2
   • (e^(i theta) - e^(-i theta)) / 2
   • (e^(i theta) - e^(-i theta)) / (2i)

5. Which expression gives sin theta in terms of exponentials?

   • (e^(i theta) - e^(-i theta)) / 2
   • (e^(i theta) + e^(-i theta)) / 2
   • (e^(i theta) + e^(-i theta)) / (2i)
   • (e^(i theta) - e^(-i theta)) / (2i)

6. What is the modulus of e^(i theta) for real theta?

   • 2
   • 1
   • theta
   • 0

7. For z = e^(i theta), what is z + 1/z?

   • 2 sin theta
   • cos theta + sin theta
   • 2 cos theta
   • 2i sin theta

8. What is the value of (1 + i)^8?

   • 4
   • 8
   • 16
   • -16

9. What is (cos(pi/6) + i sin(pi/6))^6?

   • i
   • -i
   • 1
   • -1

10. Express 1 + i in exponential form r e^(i theta) with r > 0 and theta in (-pi, pi].

    • sqrt(2) e^(i pi/4)
    • sqrt(2) e^(-i pi/4)
    • 2 e^(i pi/4)
    • sqrt(2) e^(i pi/2)

11. Which expression gives cos 3 theta in terms of cos theta?

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

12. Which expression gives sin 3 theta in terms of sin theta?

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

13. Which expression gives cos^4 theta in terms of multiple angles?

    • (cos 4 theta + 4 cos 2 theta + 3) / 16
    • (cos 4 theta - 4 cos 2 theta + 3) / 8
    • (cos 4 theta + 4 cos 2 theta + 3) / 8
    • (cos 4 theta + 4 cos 2 theta) / 8

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

    • -16
    • -32 sqrt(3)
    • 16 sqrt(3)
    • -16 sqrt(3)

15. Using z = e^(i theta), what is z^n + z^(-n) for integer n?

    • 0
    • 2 sin(n theta)
    • 2 cos(n theta)
    • cos(n theta)

16. For real theta, what is |1 + e^(i theta)|?

    • |cos theta|
    • 2
    • 2|sin(theta/2)|
    • 2|cos(theta/2)|

17. Which expression gives sin^4 theta in terms of multiple angles?

    • (cos 4 theta - 4 cos 2 theta) / 8
    • (cos 4 theta - 4 cos 2 theta + 3) / 8
    • (3 - 4 cos 2 theta) / 8
    • (cos 4 theta + 4 cos 2 theta + 3) / 8

18. For positive integer n > 1 and z = e^(2 pi i / n), what is the sum 1 + z + z^2 + ... + z^(n-1)?

    • 0
    • 1
    • n z
    • n

19. What is (cos theta + i sin theta)^2 + (cos theta - i sin theta)^2 in simplest form?

    • cos 2 theta + i sin 2 theta
    • 2
    • 2 sin 2 theta
    • 2 cos 2 theta

20. What is the modulus of (1 + i)^5?

    • 5 sqrt(2)
    • 4
    • 4 sqrt(2)
    • 2 sqrt(2)

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