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

The 20 questions

1. According to De Moivre's theorem, what does (cos θ + i sin θ)^n equal?

   • cos(nθ) + i sin(nθ)
   • n(cos θ + i sin θ)
   • cosⁿ θ + i sinⁿ θ
   • cos(nθ) - i sin(nθ)

2. Which expression is equal to e^(i θ) according to Euler's formula?

   • cos θ - i sin θ
   • cos θ + i sin θ
   • sin θ + i cos θ
   • cos θ + sin θ

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. If z = e^(i θ), what is z + 1/z simplified?

   • 2i sin θ
   • 2i cos θ
   • 2 cos θ
   • 2 sin θ

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. What is cos 3θ expressed in terms of cos θ?

    • 4 cos³ θ - 3 cos θ
    • 3 cos³ θ - 4 cos θ
    • 4 cos³ θ + 3 cos θ
    • 3 cos θ - 4 cos³ θ

12. What is sin 3θ expressed in terms of sin θ?

    • 3 sin θ - 4 sin³ θ
    • 4 sin³ θ - 3 sin θ
    • 4 sin θ - 3 sin³ θ
    • 3 sin θ + 4 sin³ θ

13. What does 8 cos⁴ θ equal in terms of multiple angles?

    • cos 4θ - 4 cos 2θ + 3
    • cos 4θ + 2 cos 2θ + 3
    • cos 4θ + 4 cos 2θ + 3
    • cos 4θ + 4 cos 2θ - 3

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. What does 8 sin⁴ θ equal in terms of multiple angles?

    • cos 4θ - 2 cos 2θ + 3
    • cos 4θ - 4 cos 2θ + 3
    • cos 4θ - 4 cos 2θ - 3
    • cos 4θ + 4 cos 2θ + 3

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 θ + i sin θ)² + (cos θ - i sin θ)² in simplest form?

    • 0
    • 2 sin 2θ
    • 2i sin 2θ
    • 2 cos 2θ

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

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

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