4.02o-s Loci, de Moivre's theorem and roots of unity Quiz: OCR Further Maths, Unit 1

The 20 questions

1. What does de Moivre's theorem state for (cos θ + i sin θ)^n?

   • n cos(θ) + i n sin(θ)
   • cos(nθ) - i sin(nθ)
   • cos^n(θ) + i sin^n(θ)
   • cos(nθ) + i sin(nθ)

2. Which expression is equivalent to z + 1/z where z = e^(iθ)?

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

3. What is the simplified value of z^n - 1/z^n where z = cos θ + i sin θ?

   • 2 sin(nθ)
   • 2i cos(nθ)
   • 2 cos(nθ)
   • 2i sin(nθ)

4. What shape is formed on an Argand diagram by the locus |z - 3 - 4i| = 5?

   • Circle
   • Half-line
   • Perpendicular bisector
   • Ellipse

5. What is the radius of the circle given by the locus equation |z - 2 + i| = 9?

   • 9
   • 81
   • 3
   • 2

6. What is the centre of the circle represented by the locus |z + 4 - 3i| = 7?

   • 4 - 3i
   • -4 - 3i
   • -4 + 3i
   • 4 + 3i

7. What geometric locus is represented by the equation |z - 2| = |z - 6|?

   • Circle
   • Perpendicular bisector
   • Parabola
   • Half-line

8. Which Cartesian line equation corresponds to the locus |z - 1| = |z - 5|?

   • y = 3
   • x = 2
   • x = 4
   • x = 3

9. What geometric shape is represented by the locus equation arg(z - 2i) = π/4?

   • Perpendicular bisector
   • Circle
   • Full line
   • Half-line

10. What is the sum of all n distinct nth roots of unity for n > 1?

    • 0
    • i
    • 1
    • n

11. What polygon is formed by plotting the nth roots of unity on an Argand diagram?

    • Rectangle
    • Square
    • Regular n-gon
    • Irregular polygon

12. How many cube roots of unity exist in the complex plane?

    • 1
    • 3
    • 2
    • 6

13. If ω is a non-real cube root of unity, what is the value of 1 + ω + ω^2?

    • 3
    • -1
    • 1
    • 0

14. What is the value of ω^3 if ω is a complex cube root of unity?

    • 1
    • -1
    • 0
    • i

15. What region in the Argand diagram is defined by the inequality |z - 3i| ≤ 2?

    • Circumference only
    • Interior only
    • Interior and boundary
    • Exterior only

16. Using de Moivre's theorem, what is (cos(π/6) + i sin(π/6))^3 equal to?

    • -i
    • 1
    • -1
    • i

17. What is the product of all 4 fourth roots of unity?

    • 1
    • 0
    • i
    • -1

18. What is the radius of the circle given by |2z - 6 + 8i| = 10?

    • 5
    • 20
    • 2.5
    • 10

19. What locus is represented by arg((z - 2)/(z - 4)) = π/2?

    • Full circle
    • Half-line
    • Straight line
    • Semicircle

20. Express cos(3θ) in terms of cos θ using de Moivre's theorem.

    • 4 cos^3 θ + 3 cos θ
    • 3 cos θ - 4 cos^3 θ
    • 4 cos^3 θ - 3 cos θ
    • 3 cos^3 θ - 4 cos θ

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