B4-B7 Argand diagrams, modulus-argument form and loci Quiz: AQA Further Maths, Unit 2

The 20 questions

1. On an Argand diagram, which point represents z = -3 + 4i?

   • The point (3, -4)
   • The point (-4, 3)
   • The point (4, -3)
   • The point (-3, 4)

2. What is the modulus of z = 3 + 4i?

   • 5
   • 1
   • 7
   • 25

3. What is the principal argument of z = -1 + i?

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

4. Which locus describes |z - 2| = 3 on the Argand diagram?|A circle with centre (2, 0) and radius 3|A circle with centre (-2, 0) and radius 3

   • z - a
   • A circle with centre (2, 0) and radius 9
   • The modulus
   • A line through (2, 0)

5. Which locus is arg(z - 1) = pi/4?

   • A circle centred at (1, 0)
   • A half-line from (1, 0) at angle pi/4
   • A full line through (1, 0) at angle -pi/4
   • A half-line from (-1, 0) at angle pi/4

6. Convert z = 1 - i sqrt(3) to modulus-argument form.

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

7. Given z1 = 2 cis(pi/6) and z2 = 3 cis(pi/3), what is z1 z2?

   • 5 cis(pi/2)
   • 6 cis(pi/2)
   • 1.5 cis(pi/2)
   • 6 cis(pi/6)

8. Given z1 = 8 cis(5pi/6) and z2 = 2 cis(pi/3), what is z1/z2?

   • 4 cis(pi/2)
   • 4 cis(5pi/6)
   • 16 cis(pi/2)
   • 4 cis(pi/3)

9. Which locus is described by |z| = |z - 3i|?

   • The circle centred at 3i with radius 3
   • The line x = 3/2
   • The line y = 3
   • The line y = 3/2

10. Find z with arg z = pi/3 and |z - 2| = 2.

    • z = 1 + sqrt(3) i
    • z = 1 + i
    • z = 2 + 2 sqrt(3) i
    • z = sqrt(3) + i

11. Write z = 4 - 4i in modulus-argument form.

    • 4 sqrt(2) cis(pi/4)
    • 8 cis(-pi/4)
    • 4 cis(-pi/4)
    • 4 sqrt(2) cis(-pi/4)

12. What is (1 + i)^8?

    • 16i
    • -16
    • 4
    • 16

13. What is the locus described by arg(z + 1 - i) = 0?

    • A half-line from (-1, 1) extending to the right, excluding the start point
    • A half-line from (1, -1) extending to the right, starting at the point (1, -1)
    • A full horizontal line through (-1, 1), with no start point or direction
    • A half-line from (-1, 1) extending upward, starting at the point (-1, 1)

14. For z on the locus |z - 4| = 2, what is the least possible value of |z|?

    • 6
    • 4
    • sqrt(20)
    • 2

15. For z on |z - 3 - 4i| = 2, what is the least possible value of |z|?

    • 5
    • 3
    • 7
    • 1

16. What is arg((1 + i)/(1 - i))?

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

17. What is the modulus of (3 + 4i)/(1 - 2i)?

    • 5
    • sqrt(5)
    • sqrt(7)
    • 1

18. Find z with arg z = pi/3 and |z - 4| = 4.

    • 2 + sqrt(3) i
    • 4 + 4 sqrt(3) i
    • 2 + 2 sqrt(3) i
    • 1 + sqrt(3) i

19. Which set is described by |z - 1| = |z + 1|?

    • The real axis, the line Im z = 0
    • The line x = 1
    • The circle centred at 0 with radius 1
    • The imaginary axis, the line Re z = 0

20. What is the argument of (1 - i)^4?

    • -pi/2
    • 0
    • pi
    • pi/2

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