2.5-2.7 Modulus-argument form and loci Quiz: Pearson Edexcel Further Maths, Unit 2

The 20 questions

1. Which range is the standard principal argument of a non-zero complex number taken to lie in?

   • -pi/2 < arg z <= pi/2
   • 0 <= arg z < 2pi
   • -pi < arg z <= pi
   • 0 < arg z <= pi/2

2. Which expression shows the modulus-argument form of a complex number z?

   • r(sin θ + i cos θ)
   • r(tan θ + i sin θ)
   • r(cos θ + i sin θ)
   • r(cos θ - i sin θ)

3. What is the modulus of 2 - 2i?

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

4. What is the principal argument of 1 + i?

   • pi/2
   • -pi/4
   • 3pi/4
   • pi/4

5. Which result gives the argument of a product z1 z2 of non-zero complex numbers?

   • arg z1 x arg z2
   • arg z1 / arg z2
   • arg z1 - arg z2
   • arg z1 + arg z2

6. Which result gives the modulus of a quotient z1/z2 of non-zero complex numbers?

   • |z1|^2 / |z2|
   • |z1| - |z2|
   • |z1| |z2|
   • |z1| / |z2|

7. What locus is represented by the equation |z - a| = r on an Argand diagram?

   • Straight line
   • Ellipse
   • Circle
   • Half-line

8. Which modulus-argument form represents z = 1 + i sqrt(3)?

   • 2(cos(pi/3) - i sin(pi/3))
   • 2(cos(pi/3) + i sin(pi/3))
   • sqrt(3)(cos(pi/3) + i sin(pi/3))
   • 2(cos(pi/6) + i sin(pi/6))

9. Let z1 = 2(cos(pi/6) + i sin(pi/6)) and z2 = 3(cos(pi/3) + i sin(pi/3)). Which is z1 z2 in Cartesian form?

   • 5i
   • 6i
   • 6
   • -6i

10. Let z1 = 2(cos(pi/6) + i sin(pi/6)) and z2 = 3(cos(pi/3) + i sin(pi/3)). Which is z1/z2 in modulus-argument form?

    • (2/3)(cos(pi/2) + i sin(pi/2))
    • (2/3)(cos(-pi/6) + i sin(-pi/6))
    • (3/2)(cos(pi/6) + i sin(pi/6))
    • (2/3)(cos(pi/6) + i sin(pi/6))

11. Express 4(cos(pi/4) + i sin(pi/4)) in Cartesian form.

    • 2 sqrt(2) + 2 sqrt(2) i
    • 4 + 4i
    • 2 sqrt(2) - 2 sqrt(2) i
    • sqrt(2) + sqrt(2) i

12. What are the modulus and principal argument of -1 - i?

    • sqrt(2) and -pi/4
    • sqrt(2) and 3pi/4
    • 2 and -pi/4
    • sqrt(2) and -3pi/4

13. What is the radius of the circle given by |z - 2| = 3 on an Argand diagram?

    • 9
    • 5
    • 2
    • 3

14. What geometrical shape is represented by arg(z - 1) = π/4 on an Argand diagram?

    • Half-line
    • Circle
    • Parabola
    • Full straight line

15. Which inequality defines the region |z| ≤ |z - 2| on an Argand diagram?

    • Re(z) ≤ 1
    • Re(z) ≥ 1
    • Re(z) ≤ 2
    • Im(z) ≤ 1

16. What is (1 + i)^4 in Cartesian form?

    • 4i
    • -4
    • -4i
    • 4

17. Given |z1| = 3, arg z1 = pi/3, |z2| = 2 and arg z2 = -pi/6, what is arg(z1 z2)?

    • pi/2
    • -pi/6
    • pi/6
    • -pi/2

18. What shape does the region π/4 ≤ arg z ≤ π/2 form on an Argand diagram?

    • Sector
    • Straight line
    • Circle
    • Half-plane

19. A complex number has modulus 2 and argument pi/3. What is it in Cartesian form?

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

20. What is (1 + sqrt(3) i)^6 in Cartesian form?

    • 64i
    • 64
    • 32
    • -64

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