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Complex numbers: OCR MEI A‑Level Further Maths questions

Every OCR MEI A-Level Further Maths exam question on CP.2 Complex numbers, 2018–2025, with its mark scheme and examiner’s report.

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847 questions

  1. Q1

    June 2025 · Core Pure

    4 marksCP.2 Complex numbers
    The whole paper ↗
    June 2025 Core Pure, question 1: The complex number z satisfies the equation z 2 i z * 1 4 i 0 . + + - = You are given that z x i y , where x and y are real numbers. = + Determine the values of

  2. Q9

    June 2025 · Core Pure

    10 marksCP.2 Complex numbers
    The whole paper ↗
    June 2025 Core Pure, question 9: The figure below shows an Argand diagram with a regular pentagon ABCDE. The point A represents the real number 1. The point B represents the complex number w. I

  3. Q12

    June 2025 · Core Pure

    8 marksCP.4 Further algebra and functionsCP.2 Complex numbers
    The whole paper ↗
    June 2025 Core Pure, question 12: In this question you must show detailed reasoning. are a, 2 4 3 2 The roots of the equation z z cz dz 18 0 a , b and - . - + + + = b Determine, in any order, th

  4. Q15

    June 2025 · Core Pure

    9 marksCP.2 Complex numbers
    The whole paper ↗
    June 2025 Core Pure, question 15: ^ ^h h 4 i 4 i 15 (a) Show that i i a b , where a and b are integers to be determined. [2] 3 e 3 e - cos 4 - - = + i The infinite series C and S are defined as

  5. Q5

    June 2025 · AS Core Pure

    10 marksCP.2 Complex numbers
    The whole paper ↗
    June 2025 AS Core Pure, question 5: In this question you must show detailed reasoning. + ^ h . The complex number w is given by w – 4 2 4 2 i = (a) (i) Find w . [2] (ii) Find arg w . ( ) [2] The c

  6. Q7

    June 2025 · AS Core Pure

    11 marksCP.2 Complex numbers
    The whole paper ↗
    June 2025 AS Core Pure, question 7: The region R of the Argand diagram consists of the set of points representing complex numbers z which satisfy the following inequalities. 1 Im z ( ) 0 arg z ( 2

  7. Q8

    June 2025 · AS Core Pure

    8 marksCP.2 Complex numbersCP.4 Further algebra and functions
    The whole paper ↗
    June 2025 AS Core Pure, question 8: 2 8 The three distinct roots of the equation z 4 z pz q 0 , where p and q are real, are drawn on - + + = an Argand diagram. The three points which represent the

  8. Q2

    June 2024 · Core Pure

    7 marksCP.2 Complex numbers
    The whole paper ↗
    June 2024 Core Pure, question 2: Two complex numbers are given by u =- + and v 1 i =- - . 2 i (a) (i) Find u - in the form a v bi , where a and b are real. [1] + (ii) In this question you must

  9. Q6

    June 2024 · Core Pure

    6 marksCP.2 Complex numbers
    The whole paper ↗
    June 2024 Core Pure, question 6: On separate Argand diagrams, sketch the set of points represented by each of the following. (a) z 1 2 i G 4 . [3] - - 1 (b) arg z ( i ) . [3] + = r 3

  10. Q13

    June 2024 · Core Pure

    10 marksCP.2 Complex numbers
    The whole paper ↗
    June 2024 Core Pure, question 13: 1 i 13 The complex number z is defined as z e i where 0 r. = 1 1 i 3 2 On an Argand diagram, the point O represents the complex number 0, and the points P1, P2,

  11. Q1

    June 2024 · AS Core Pure

    4 marksCP.2 Complex numbers
    The whole paper ↗
    June 2024 AS Core Pure, question 1: + 1 The quadratic equation x ax b 0 , where a and b are real constants, has a root 2 3 i. + = - (a) Write down the other root. [1] (b) Hence or otherwise determ

  12. Q8

    June 2024 · AS Core Pure

    9 marksCP.2 Complex numbers
    The whole paper ↗
    June 2024 AS Core Pure, question 8: In an Argand diagram, the point P representing the complex number w lies on the locus defined by ^ h % /. You are given that e 3 r z : arg ( z 7 ) R w 1 . - = =

  13. Q1

    June 2023 · Core Pure

    7 marksCP.2 Complex numbers
    The whole paper ↗
    June 2023 Core Pure, question 1: (a) The complex number a bi is denoted by z. + (i) Write down z*. [1] (ii) Find Re(iz). [2] 5 i 3 + (b) The complex number w is given by w . = 2 i 3 - (i) In th

  14. Q5

    June 2023 · Core Pure

    7 marksCP.2 Complex numbers
    The whole paper ↗
    June 2023 Core Pure, question 5: (a) In this question you must show detailed reasoning. , expressed in reii form. Determine the sixth roots of 64 [4] - (b) Represent the roots on an Argand diag

  15. Q12

    June 2023 · Core Pure

    7 marksCP.2 Complex numbers
    The whole paper ↗
    June 2023 Core Pure, question 12: 5i 12 Show that sin a sin 5 b sin 3 c sin , where a, b and c are constants to be determined. [7] = + + i i i

  16. Q13

    June 2023 · Core Pure

    14 marksCP.2 Complex numbers
    The whole paper ↗
    June 2023 Core Pure, question 13: (a) On separate Argand diagrams, show the set of points representing each of the following inequalities. (i) z 5 [3] G (ii) z 2 4 i z 2 6 i [3] + - H - - (b) Sh

  17. Q3

    June 2023 · AS Core Pure

    6 marksCP.4 Further algebra and functionsCP.2 Complex numbers
    The whole paper ↗
    June 2023 AS Core Pure, question 3: In this question you must show detailed reasoning. 3 2 The function f (z) is given by ( ) f z 2 z 7 z 16 z 15 . = - + - 3a k, find the roots of f (z) = 0. By fi

  18. Q5

    June 2023 · AS Core Pure

    4 marksCP.2 Complex numbers
    The whole paper ↗
    June 2023 AS Core Pure, question 5: The Argand diagram below shows the points representing 1 and z, where z 2 . = Im z Re O 1 Mark the points representing the following complex numbers on the copy

  19. Q7

    June 2023 · AS Core Pure

    10 marksCP.2 Complex numbers
    The whole paper ↗
    June 2023 AS Core Pure, question 7: In this question you must show detailed reasoning. The complex number 3 + is denoted by z. i ^ h , express z5 in the form a (a) By expanding i 5 3 bi where a an

  20. Q8

    June 2022 · Core Pure

    11 marksCP.2 Complex numbers
    The whole paper ↗
    June 2022 Core Pure, question 8: " , where k % / and 3 r 8 Two sets of complex numbers are given by z : arg ( z 10 ) z : z 3 6i k , - = - - = 4 is a positive constant. In an Argand diagram, one
All 48 questions on Complex numbers, as a list
  1. June 2025, Core Pure, Q1 (4 marks): The complex number z satisfies the equation z 2 i z * 1 4 i 0 . + + - = You are given that z x i y , where x and y are real numbers. = + Determine the values of
  2. June 2025, Core Pure, Q9 (10 marks): The figure below shows an Argand diagram with a regular pentagon ABCDE. The point A represents the real number 1. The point B represents the complex number w. I
  3. June 2025, Core Pure, Q12 (8 marks): In this question you must show detailed reasoning. are a, 2 4 3 2 The roots of the equation z z cz dz 18 0 a , b and - . - + + + = b Determine, in any order, th
  4. June 2025, Core Pure, Q15 (9 marks): ^ ^h h 4 i 4 i 15 (a) Show that i i a b , where a and b are integers to be determined. [2] 3 e 3 e - cos 4 - - = + i The infinite series C and S are defined as
  5. June 2025, AS Core Pure, Q5 (10 marks): In this question you must show detailed reasoning. + ^ h . The complex number w is given by w – 4 2 4 2 i = (a) (i) Find w . [2] (ii) Find arg w . ( ) [2] The c
  6. June 2025, AS Core Pure, Q7 (11 marks): The region R of the Argand diagram consists of the set of points representing complex numbers z which satisfy the following inequalities. 1 Im z ( ) 0 arg z ( 2
  7. June 2025, AS Core Pure, Q8 (8 marks): 2 8 The three distinct roots of the equation z 4 z pz q 0 , where p and q are real, are drawn on - + + = an Argand diagram. The three points which represent the
  8. June 2024, Core Pure, Q2 (7 marks): Two complex numbers are given by u =- + and v 1 i =- - . 2 i (a) (i) Find u - in the form a v bi , where a and b are real. [1] + (ii) In this question you must
  9. June 2024, Core Pure, Q6 (6 marks): On separate Argand diagrams, sketch the set of points represented by each of the following. (a) z 1 2 i G 4 . [3] - - 1 (b) arg z ( i ) . [3] + = r 3
  10. June 2024, Core Pure, Q13 (10 marks): 1 i 13 The complex number z is defined as z e i where 0 r. = 1 1 i 3 2 On an Argand diagram, the point O represents the complex number 0, and the points P1, P2,
  11. June 2024, AS Core Pure, Q1 (4 marks): + 1 The quadratic equation x ax b 0 , where a and b are real constants, has a root 2 3 i. + = - (a) Write down the other root. [1] (b) Hence or otherwise determ
  12. June 2024, AS Core Pure, Q8 (9 marks): In an Argand diagram, the point P representing the complex number w lies on the locus defined by ^ h % /. You are given that e 3 r z : arg ( z 7 ) R w 1 . - = =
  13. June 2023, Core Pure, Q1 (7 marks): (a) The complex number a bi is denoted by z. + (i) Write down z*. [1] (ii) Find Re(iz). [2] 5 i 3 + (b) The complex number w is given by w . = 2 i 3 - (i) In th
  14. June 2023, Core Pure, Q5 (7 marks): (a) In this question you must show detailed reasoning. , expressed in reii form. Determine the sixth roots of 64 [4] - (b) Represent the roots on an Argand diag
  15. June 2023, Core Pure, Q12 (7 marks): 5i 12 Show that sin a sin 5 b sin 3 c sin , where a, b and c are constants to be determined. [7] = + + i i i
  16. June 2023, Core Pure, Q13 (14 marks): (a) On separate Argand diagrams, show the set of points representing each of the following inequalities. (i) z 5 [3] G (ii) z 2 4 i z 2 6 i [3] + - H - - (b) Sh
  17. June 2023, AS Core Pure, Q3 (6 marks): In this question you must show detailed reasoning. 3 2 The function f (z) is given by ( ) f z 2 z 7 z 16 z 15 . = - + - 3a k, find the roots of f (z) = 0. By fi
  18. June 2023, AS Core Pure, Q5 (4 marks): The Argand diagram below shows the points representing 1 and z, where z 2 . = Im z Re O 1 Mark the points representing the following complex numbers on the copy
  19. June 2023, AS Core Pure, Q7 (10 marks): In this question you must show detailed reasoning. The complex number 3 + is denoted by z. i ^ h , express z5 in the form a (a) By expanding i 5 3 bi where a an
  20. June 2022, Core Pure, Q8 (11 marks): " , where k % / and 3 r 8 Two sets of complex numbers are given by z : arg ( z 10 ) z : z 3 6i k , - = - - = 4 is a positive constant. In an Argand diagram, one
  21. June 2022, Core Pure, Q10 (10 marks): The equation 4 3 2 4 x 16 x ax bx 6 0 , + + + + = where a and b are real, has roots a, 2 a , b and 3b. (a) Given that ,0 determine all 4 roots. [6] 1 b (b) Dete
  22. June 2022, Core Pure, Q11 (8 marks): An Argand diagram with the point A representing a complex number z1 is shown below. Im z1 A Re O 2 r and z e 4 r respectively. i i The complex numbers z2 and z3
  23. June 2022, Core Pure, Q14 (8 marks): i 2 i 14 (a) Find ( [2] 3 e i )( 3 e - i ) in terms of cos2i . - - (b) Hence show that the sum of the infinite series 1 1 1 sin sin 3 sin 5 sin 7 … + + + + i i
  24. June 2022, AS Core Pure, Q3 (5 marks): The complex number z satisfies the equation ( z z 5 i ) ( 1 2 i ) *. - = - + Determine z, giving your answer in the form a bi , where a and b are real. [5] +
  25. June 2022, AS Core Pure, Q4 (6 marks): In this question you must show detailed reasoning. where k is a constant, has roots a, 1 3 2 z z kz The equation 2 3 0 , a and b. + + + = [6] Determine the root
  26. June 2022, AS Core Pure, Q5 (5 marks): An Argand diagram is shown below. The circle has centre at the point representing 1 3i , and the + half line intersects the circle at the origin and at the poin
  27. June 2022, AS Core Pure, Q7 (9 marks): On an Argand diagram, the point A represents the complex number z with modulus 2 and 1 1 r The point B represents . argument . z 3 (a) Sketch an Argand diagram
  28. November 2021, Core Pure, Q3 (6 marks): In this question you must show detailed reasoning. a k. 1 1 The complex numbers z1 and z2 are given by z 2 2i and z 2 cos i sin 1 =- + = r + r 2 6 6 (a) Find th
  29. November 2021, Core Pure, Q8 (9 marks): , where p and q are constants, has roots a, -a, b and 1 4 3 2 8 The equation x 4 4 x px qx 9 0 b . - + + - = (a) Determine the exact roots of the equation. [5]
  30. November 2021, Core Pure, Q10 (13 marks): (a) Show on an Argand diagram the points representing the three cube roots of unity. [2] 3 - , expressing them in the form reii, where (b) (i) Find the exact ro
  31. November 2021, Core Pure, Q12 (4 marks): Fig. 12 shows a rhombus OACB in an Argand diagram. The points A and B represent the complex numbers z and w respectively. Im C B A O Re Fig. 12 1 Prove that arg
  32. November 2021, AS Core Pure, Q7 (9 marks): (a) (i) Find the modulus and argument of z1, where z 1 + . i [2] 1 = 1 r (ii) Given that z 2 and arg z ( ) express z2 in a bi form, where a and b are exact , 2
  33. November 2021, AS Core Pure, Q8 (7 marks): In this question you must show detailed reasoning. a . Given that 3 2 The equation x kx 15 x 25 0 has roots a, b and b 2 0 , find, in any order, + + - = a • the
  34. November 2021, AS Core Pure, Q9 (9 marks): (a) On a single Argand diagram, sketch the loci defined by • 3 r arg z ( 2 ) , - = 4 • z z 2 i . [4] = + - (b) In this question you must show detailed reasoning
  35. November 2020, Core Pure, Q6 (4 marks): - 6 The complex number z satisfies the equation z 4 i z * 11 0 . + = Given that Re z ( ) 2 0 , find z in the form a + bi, where a and b are real numbers. [4]
  36. November 2020, Core Pure, Q11 (8 marks): In this question you must show detailed reasoning. In Fig. 11, the points A, B, C, D, E and F represent the complex sixth roots of 64 on an Argand diagram. The
  37. November 2020, Core Pure, Q12 (8 marks): 1 n n 12 (a) Given that z cos isin , express z and z in simplified trigonometric form. = i + i + - n n z z [2] J N J N 3 3 1 1 KK OO KK OO , find constants A an
  38. November 2020, AS Core Pure, Q2 (4 marks): Fig. 2 shows two complex numbers z1 and z2 represented on an Argand diagram. Im z1 z2 Re O Fig. 2 (a) On the copy of Fig. 2 in the Printed Answer Booklet, mark
  39. November 2020, AS Core Pure, Q3 (7 marks): In this question you must show detailed reasoning. 2 - The roots of the equation x 2 x 4 0 are a and b. + = (a) Find a and b in modulus-argument form. [4] 3 (b)
  40. November 2020, AS Core Pure, Q7 (7 marks): 3 2 7 In the quartic equation x 2 20 x ax bx 250 0 , the coefficients a and b are real. One root - + + + = of the equation is 2 + . i Find the other roots. [7]
  41. June 2019, Core Pure, Q8 (8 marks): In this question you must show detailed reasoning. 1 3 2 The roots of the equation x x kx 2 0 are , and b. - + - = a a (a) Evaluate, in exact form, the roots of
  42. June 2019, Core Pure, Q10 (8 marks): In this question you must show detailed reasoning. 3 = (a) You are given that - + is a root of the equation 1 i z a b i , where a and b are real numbers. + Find
  43. June 2019, Core Pure, Q16 (12 marks): i i 16 (a) Show that ( 2 e i )( 2 e i ) 5 4 cos . [3] - - - = - i Series C and S are defined by 1 1 1 1 C cos cos 2 cos 3 cos n , = i + i + i + f + i 2 4 8 n 2
  44. June 2019, AS Core Pure, Q7 (12 marks): (a) Sketch on a single Argand diagram (i) the set of points for which z 1 3 i = , 3 [3] - - 1 r (ii) the set of points for which arg z ( 4 ) . [3] + = 4 1 r (b)
  45. June 2019, AS Core Pure, Q8 (11 marks): In this question you must show detailed reasoning. 4 3 2 You are given that i is a root of the equation z 2 z 3 z az b 0 , where a and b are real - + + + = cons
  46. June 2018, AS Core Pure, Q3 (5 marks): Find real numbers a and b such that ( a b [5] 3 i )( 5 i ) 17 i . - - = -
  47. June 2018, AS Core Pure, Q4 (5 marks): [5] Find a cubic equation with real coefficients, two of whose roots are 2 - i and 3.
  48. June 2018, AS Core Pure, Q9 (9 marks): Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by % / % /. 1 1 z z z z 4 2 arg | G k | r G G r 4 3 Im Q P Re O Fig. 9 (i) Find, in modulu

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