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 is the modulus-argument form of a complex number with modulus r and argument theta?

   • r + i theta
   • r sin theta + i r cos theta
   • r(cos theta + i sin theta)
   • r(cos theta - i sin theta) with theta negated

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. Which locus is described by |z - a| = r, where a is a fixed complex number and r is positive?

   • A straight line through a
   • A disc of radius a centred at r
   • A circle with centre a and radius r
   • A half-line from a at angle r

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. Which locus is described by |z - 2| = 3 on an Argand diagram?

    • Circle with centre 2 and radius 9
    • Circle with centre 3 and radius 2
    • Circle with centre -2 and radius 3
    • Circle with centre 2 and radius 3

14. Which locus is described by arg(z - 1) = pi/4?

    • A half-line from 1 at angle pi/4 to the positive real direction, excluding the point 1
    • A circle centred at 1 with radius pi/4
    • A half-line from 0 at angle 1 radian
    • A line through 0 with gradient 1

15. Which region is described by |z| <= |z - 2| on an Argand diagram?

    • Re(z) <= 1, a closed half-plane
    • Re(z) >= 1, a closed half-plane
    • |z| < 2, the interior of a disc
    • Im(z) <= 1, a closed half-plane

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. Which region is described by pi/4 <= arg z <= pi/2 on an Argand diagram?

    • A sector bounded by the half-lines at angles pi/4 and pi/2 from the origin, including its boundary
    • The strip between Im(z) = 1 and Im(z) = 2
    • The half-plane Re(z) >= 0
    • The disc |z| <= pi/2

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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