Lesson 2.3-2.4

2.3-2.4 Conjugate roots and Argand diagrams Quiz: Pearson Edexcel Further Maths, Unit 2

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

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Lesson 2.3-2.4, Conjugate roots and Argand diagrams: 20 multiple choice questions for the Pearson Edexcel Further Maths (9FM0), Unit 2: Complex numbers, written with Revision Ninja.

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The 20 questions

  1. For z = x + iy, what is the complex conjugate z*?

    • -x + iy
    • x + iy
    • -x - iy
    • x - iy

  2. If z1 is a root of a polynomial equation with real coefficients, which other number is also a root?

    • The reciprocal 1/z1
    • The negative -z1
    • The conjugate z1*
    • The modulus |z1|

  3. On an Argand diagram, what is the geometrical effect of taking the conjugate z* of z?

    • Translation of z by one unit to the right
    • Rotation of z through 90 degrees about the origin
    • Reflection of z in the real axis
    • Reflection of z in the imaginary axis

  4. What is z + z* for a complex number z = x + iy?

    • 2iy
    • 2x
    • x^2 + y^2
    • 0

  5. What is z z* for a complex number z = x + iy?

    • (x + y)^2
    • 2xy
    • x^2 - y^2
    • x^2 + y^2

  6. On an Argand diagram, the sum z1 + z2 is represented by which feature of the parallelogram formed by the points 0, z1 and z2?

    • The point reflected through the origin from z1
    • The midpoint of the segment joining z1 and z2
    • The side joining z1 to z2
    • The diagonal from the origin to the fourth vertex

  7. On an Argand diagram, how are the points z and -z related?

    • They are symmetric about the origin
    • They are reflections in the real axis
    • They are the same point
    • They are reflections in the imaginary axis

  8. For z = 1 + 2i and w = 3 - i, what is z + w?

    • 4 + 3i
    • 4 + i
    • 2 + 3i
    • 4 - i

  9. For z = 1 + 2i and w = 3 - i, what is |z - w|?

    • 5
    • sqrt(5)
    • 13
    • sqrt(13)

  10. For z = 3 + 4i, what is |z - z*|?

    • 0
    • 8
    • 6
    • 4

  11. A real cubic has root 1 + i. Given the cubic x^3 - 3x^2 + 4x - 2 = 0, what is its real root?

    • -1
    • 2
    • 0
    • 1

  12. A complex number z satisfies z + z* = 6 and z z* = 10. Which values can z take?

    • 3 + sqrt(10)i or 3 - sqrt(10)i
    • 6 + 10i or 6 - 10i
    • 1 + 3i or 1 - 3i
    • 3 + i or 3 - i

  13. A real quartic has roots 1 + i and 2i. Which two roots must also be present?

    • 1 + i and -2i
    • 1 - i and 2i
    • 1 - i and -2i
    • -1 - i and -2i

  14. What is the constant term of the real cubic whose roots are 2 + i, 2 - i and 1?

    • -5
    • -1
    • 5
    • 9

  15. Which locus describes all points z with |z - 1| = |z + i| on an Argand diagram?

    • The line y = x
    • The circle with centre 1 and radius 1
    • The line y = x - 1
    • The line y = -x

  16. The cubic z^3 + z^2 + 4z + 4 = 0 has root 2i. Which of the roots is real?

    • 1
    • 2
    • -1
    • -2

  17. Which statement correctly justifies that a real polynomial's non-real root z has conjugate z* also a root?

    • Conjugating f(z) = 0 term by term, and using that the coefficients are real, gives f(z*) = 0
    • Adding f(z) and f(z*) gives zero for every complex number z
    • Multiplying z by z* gives the other root in every case
    • Dividing f(z) by z gives f(z*) directly for any polynomial

  18. Which real quadratic has the complex number 2 + i as a root?

    • x^2 - 4x + 5
    • x^2 - 4x - 3
    • x^2 + 4x + 5
    • x^2 - 2x + 5

  19. What is the reflection of the point 3 - 2i in the imaginary axis?

    • 2 + 3i
    • -3 + 2i
    • -3 - 2i
    • 3 + 2i

  20. A real quadratic has roots 2 + 3i and 2 - 3i. Which statement about its discriminant is correct?

    • The discriminant is undefined
    • The discriminant is negative
    • The discriminant is positive
    • The discriminant is zero

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