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Complex numbers: OCR A‑Level Further Maths questions

Every OCR 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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648 questions

  1. Q8

    June 2025 · Pure Core 1

    7 marksCP.2 Complex numbers
    The whole paper ↗
    June 2025 Pure Core 1, question 8: In this question you must show detailed reasoning. In this question, arguments of complex numbers are in the interval [ 0 2r . , ) (a) Given that z 5 5i , expre

  2. Q10

    June 2025 · Pure Core 1

    11 marksCP.2 Complex numbers
    The whole paper ↗
    June 2025 Pure Core 1, question 10: In this question you must show detailed reasoning. (a) Use de Moivre’s Theorem to show that, if cos5 0 , i ! 5 3 tan 10 tan 5 tan - + i i i tan 5 . [4] i / 4 2

  3. Q2

    June 2025 · Pure Core 2

    4 marksCP.2 Complex numbers
    The whole paper ↗
    June 2025 Pure Core 2, question 2: 2 2 You are given that 7 5 i is one root of the equation x 10 x 18 x 296 0 . - - + + - = 3 2 (a) Write down another complex root of the equation x 10 x 18 x 296

  4. Q5

    June 2025 · Pure Core 2

    6 marksCP.2 Complex numbers
    The whole paper ↗
    June 2025 Pure Core 2, question 5: In this question you must show detailed reasoning. (a) Use an algebraic method to determine the two square roots of 3 ( 4 7 )i . - + Give your answers in the fo

  5. Q1

    June 2025 · AS Pure Core

    8 marksCP.2 Complex numbers
    The whole paper ↗
    June 2025 AS Pure Core, question 1: (a) The complex number z is such that z arg z 7 and ( ) 2 2 . radians. = = Express z in cartesian form. [3] + ^ h . (b) Use an algebraic method to determine the

  6. Q5

    June 2025 · AS Pure Core

    8 marksCP.2 Complex numbers
    The whole paper ↗
    June 2025 AS Pure Core, question 5: " ,. 5 The locus L is defined by L z z , z ( 20 15 i ) 7 = | ! - + G C (a) On the Argand diagram in the Printed Answer Booklet, sketch and label L. [2] (b) Dete

  7. Q9

    June 2025 · AS Pure Core

    8 marksCP.2 Complex numbers
    The whole paper ↗
    June 2025 AS Pure Core, question 9: z 2 9 (a) Prove that if z is any complex number then ( *) z z [1] . = (b) Prove that if w and z are any complex numbers then ( wz w z [2] ) * * * . = The comple

  8. Q2

    June 2024 · Pure Core 1

    8 marksCP.2 Complex numbers
    The whole paper ↗
    June 2024 Pure Core 1, question 2: % /. 1 2 The locus C1 is defined by C z z 0 arg ( i ) = + r | G G 4 1 (a) Indicate by shading on the Argand diagram in the Printed Answer Booklet the region rep

  9. Q2

    June 2024 · Pure Core 2

    6 marksCP.2 Complex numbers
    The whole paper ↗
    June 2024 Pure Core 2, question 2: In this question you must show detailed reasoning. 2 - (a) Solve the equation x 6 x 58 0 . Give your solutions in the form a + bi where a and b are + = real num

  10. Q9

    June 2024 · Pure Core 2

    12 marksCP.2 Complex numbers
    The whole paper ↗
    June 2024 Pure Core 2, question 9: In this question, the argument of a complex number is defined as being in the range [0, 2r). ~ , where k = 0, 1, 2, ..., n - 1, are the n nth roots of unity for

  11. Q2

    June 2024 · AS Pure Core

    4 marksCP.2 Complex numbers
    The whole paper ↗
    June 2024 AS Pure Core, question 2: In this question you must show detailed reasoning. 8 i + in the form a (a) Express 2 bi where a and b are real. [2] + i - 2 - (b) Solve the equation x 4 8 x 5 0

  12. Q4

    June 2024 · AS Pure Core

    6 marksCP.2 Complex numbers
    The whole paper ↗
    June 2024 AS Pure Core, question 4: The Argand diagram shows a circle of radius 3. The centre of the circle is the point which represents the complex number 4 2i . - Im O Re 4 –2 (a) Use set notat

  13. Q9

    June 2024 · AS Pure Core

    8 marksCP.2 Complex numbers
    The whole paper ↗
    June 2024 AS Pure Core, question 9: In this question you must show detailed reasoning. 4 3 2 You are given that a is a real root of the equation x x 3 x 5 x 0 . + + - = You are also given that a 2

  14. Q3

    June 2023 · Pure Core 1

    5 marksCP.2 Complex numbers
    The whole paper ↗
    June 2023 Pure Core 1, question 3: 3 i r . - + 5 i 3 (a) Show that 3e [2] = 6 2 ^ h, giving the roots in the 9 3 3 i - + 5 = (b) Hence determine the exact roots of the equation z 2 form reiiwhere

  15. Q9

    June 2023 · Pure Core 1

    14 marksCP.2 Complex numbersCP.5 Further calculus
    The whole paper ↗
    June 2023 Pure Core 1, question 9: In this question you must show detailed reasoning. (a) Use de Moivre’s theorem to determine constants A, B and C such that 4 sin A cos 4 B cos 2 C . [5] + i + i

  16. Q2

    June 2023 · Pure Core 2

    7 marksCP.2 Complex numbers
    The whole paper ↗
    June 2023 Pure Core 2, question 2: In this question you must show detailed reasoning. (a) Write the complex number 24 7i in modulus-argument form. [3] - + (b) Solve the simultaneous equations giv

  17. Q3

    June 2023 · AS Pure Core

    8 marksCP.2 Complex numbers
    The whole paper ↗
    June 2023 AS Pure Core, question 3: In this question you must show detailed reasoning. In this question the principal argument of a complex number lies in the interval [ , 0 2r . ) Complex numbers

  18. Q8

    June 2023 · AS Pure Core

    9 marksCP.2 Complex numbers
    The whole paper ↗
    June 2023 AS Pure Core, question 8: (a) Solve the equation - . [4] 2 7 i 3 * i ~ + + = ~ (b) Prove algebraically that, for non-zero z, z z * if and only if z is purely imaginary. [2] =- (c) The co

  19. Q3

    June 2022 · Pure Core 1

    11 marksCP.2 Complex numbers
    The whole paper ↗
    June 2022 Pure Core 1, question 3: In this question you must show detailed reasoning. 2 - (a) Find the roots of the equation z 2 2 z 5 0 . [2] + = The loci C1 and C2 are given by z z 2 i and z 2

  20. Q9

    June 2022 · Pure Core 1

    5 marksCP.2 Complex numbers
    The whole paper ↗
    June 2022 Pure Core 1, question 9: The cube roots of unity are represented on the Argand diagram below by the points A, B and C. Im(z) B L O M A Re(z) N C The points L, M and N are the midpoints
All 47 questions on Complex numbers, as a list
  1. June 2025, Pure Core 1, Q8 (7 marks): In this question you must show detailed reasoning. In this question, arguments of complex numbers are in the interval [ 0 2r . , ) (a) Given that z 5 5i , expre
  2. June 2025, Pure Core 1, Q10 (11 marks): In this question you must show detailed reasoning. (a) Use de Moivre’s Theorem to show that, if cos5 0 , i ! 5 3 tan 10 tan 5 tan - + i i i tan 5 . [4] i / 4 2
  3. June 2025, Pure Core 2, Q2 (4 marks): 2 2 You are given that 7 5 i is one root of the equation x 10 x 18 x 296 0 . - - + + - = 3 2 (a) Write down another complex root of the equation x 10 x 18 x 296
  4. June 2025, Pure Core 2, Q5 (6 marks): In this question you must show detailed reasoning. (a) Use an algebraic method to determine the two square roots of 3 ( 4 7 )i . - + Give your answers in the fo
  5. June 2025, AS Pure Core, Q1 (8 marks): (a) The complex number z is such that z arg z 7 and ( ) 2 2 . radians. = = Express z in cartesian form. [3] + ^ h . (b) Use an algebraic method to determine the
  6. June 2025, AS Pure Core, Q5 (8 marks): " ,. 5 The locus L is defined by L z z , z ( 20 15 i ) 7 = | ! - + G C (a) On the Argand diagram in the Printed Answer Booklet, sketch and label L. [2] (b) Dete
  7. June 2025, AS Pure Core, Q9 (8 marks): z 2 9 (a) Prove that if z is any complex number then ( *) z z [1] . = (b) Prove that if w and z are any complex numbers then ( wz w z [2] ) * * * . = The comple
  8. June 2024, Pure Core 1, Q2 (8 marks): % /. 1 2 The locus C1 is defined by C z z 0 arg ( i ) = + r | G G 4 1 (a) Indicate by shading on the Argand diagram in the Printed Answer Booklet the region rep
  9. June 2024, Pure Core 2, Q2 (6 marks): In this question you must show detailed reasoning. 2 - (a) Solve the equation x 6 x 58 0 . Give your solutions in the form a + bi where a and b are + = real num
  10. June 2024, Pure Core 2, Q9 (12 marks): In this question, the argument of a complex number is defined as being in the range [0, 2r). ~ , where k = 0, 1, 2, ..., n - 1, are the n nth roots of unity for
  11. June 2024, AS Pure Core, Q2 (4 marks): In this question you must show detailed reasoning. 8 i + in the form a (a) Express 2 bi where a and b are real. [2] + i - 2 - (b) Solve the equation x 4 8 x 5 0
  12. June 2024, AS Pure Core, Q4 (6 marks): The Argand diagram shows a circle of radius 3. The centre of the circle is the point which represents the complex number 4 2i . - Im O Re 4 –2 (a) Use set notat
  13. June 2024, AS Pure Core, Q9 (8 marks): In this question you must show detailed reasoning. 4 3 2 You are given that a is a real root of the equation x x 3 x 5 x 0 . + + - = You are also given that a 2
  14. June 2023, Pure Core 1, Q3 (5 marks): 3 i r . - + 5 i 3 (a) Show that 3e [2] = 6 2 ^ h, giving the roots in the 9 3 3 i - + 5 = (b) Hence determine the exact roots of the equation z 2 form reiiwhere
  15. June 2023, Pure Core 1, Q9 (14 marks): In this question you must show detailed reasoning. (a) Use de Moivre’s theorem to determine constants A, B and C such that 4 sin A cos 4 B cos 2 C . [5] + i + i
  16. June 2023, Pure Core 2, Q2 (7 marks): In this question you must show detailed reasoning. (a) Write the complex number 24 7i in modulus-argument form. [3] - + (b) Solve the simultaneous equations giv
  17. June 2023, AS Pure Core, Q3 (8 marks): In this question you must show detailed reasoning. In this question the principal argument of a complex number lies in the interval [ , 0 2r . ) Complex numbers
  18. June 2023, AS Pure Core, Q8 (9 marks): (a) Solve the equation - . [4] 2 7 i 3 * i ~ + + = ~ (b) Prove algebraically that, for non-zero z, z z * if and only if z is purely imaginary. [2] =- (c) The co
  19. June 2022, Pure Core 1, Q3 (11 marks): In this question you must show detailed reasoning. 2 - (a) Find the roots of the equation z 2 2 z 5 0 . [2] + = The loci C1 and C2 are given by z z 2 i and z 2
  20. June 2022, Pure Core 1, Q9 (5 marks): The cube roots of unity are represented on the Argand diagram below by the points A, B and C. Im(z) B L O M A Re(z) N C The points L, M and N are the midpoints
  21. June 2022, Pure Core 2, Q9 (9 marks): In this question you must show detailed reasoning. ^ h 4 i i i 4 4 (a) Show that Re e i ( e i e - i ) a cos 4 cos , where a is an integer to be determined. [3]
  22. June 2022, AS Pure Core, Q5 (7 marks): In this question you must show detailed reasoning. (a) Use an algebraic method to find the square roots of -16 + 30i. [5] 99 5i - + (b) By finding the cube of o
  23. June 2022, AS Pure Core, Q7 (7 marks): In this question you must show detailed reasoning. Two loci, C1 and C2, are defined as follows. ' 1 % / and 1 2 C z : arg ( z 2 i ) C z : arg ( z 2 3 2 i ) = +
  24. November 2021, Pure Core 1, Q1 (6 marks): (a) Sketch on a single Argand diagram the loci given by (i) z 1 2 i = , 3 [2] - + (ii) z 1 z 2 . [2] + = - (b) Indicate, by shading, the region of the Argand di
  25. November 2021, Pure Core 1, Q3 (8 marks): 2 3 A function f(z) is defined on all complex numbers z by ( )z f z 3 z kz - where k is a real 5 = - + 2 2 2 constant. The roots of the equation ( )z f 0 are a,
  26. November 2021, Pure Core 1, Q5 (4 marks): Use de Moivre’s theorem to find the constants A, B and C in the identity 5 sin A sin B sin 3 C sin 5 . [4] i / i + i + i
  27. November 2021, Pure Core 2, Q2 (8 marks): In this question you must show detailed reasoning. The complex numbers z1 and z2 are given by z 3 7i and z 2 4i . 1 = - 2 = + (a) Express each of the following
  28. November 2021, AS Pure Core, Q3 (6 marks): In this question you must show detailed reasoning. 4 3 2 The equation x 7 x 2 x 218 x 1428 0 has a root 3 5i . - - + - = - Find the other three roots of this eq
  29. November 2021, AS Pure Core, Q4 (7 marks): " ,. 4 (a) A locus C1 is defined by C z z i z 2 = | + G - 1 (i) Indicate by shading on the Argand diagram in the Printed Answer Booklet the region representing
  30. November 2021, AS Pure Core, Q6 (6 marks): In this question you must show detailed reasoning. 2 - (a) Solve the equation z 2 10 z 25 0 giving your answers in the form a bi . [2] + = + (b) Solve the equat
  31. November 2020, Pure Core 1, Q4 (4 marks): In this question you must show detailed reasoning. (a) Determine the square roots of 25i in the form reii, where 0 1 2 r . [3] G i (b) Illustrate the number 25i
  32. November 2020, Pure Core 1, Q5 (5 marks): J N 3 1 3 i KK 2 OO where ei 5 By expanding z , z i show that , 4 cos 2 cos 6 3 cos 2 [5] . + = = i + i 2 z L P
  33. November 2020, Pure Core 1, Q9 (9 marks): 2 9 You are given that the cubic equation 2 x px qx 3 0 , where p and q are real numbers, has a + + - = complex root 1 i 2 . a = + (a) Write down a second compl
  34. November 2020, Pure Core 2, Q1 (5 marks): In this question you must show detailed reasoning. 2 - Solve the equation z 4 20 z 169 0 . Give your answers in modulus-argument form. [5] + =
  35. November 2020, Pure Core 2, Q8 (9 marks): In this question you must show detailed reasoning. The complex number 4 i 48 is denoted by z. - + (a) Determine the cube roots of z, giving the roots in exponen
  36. November 2020, AS Pure Core, Q1 (6 marks): In this question you must show detailed reasoning. Use an algebraic method to find the square roots of -77 - 36i. [6]
  37. November 2020, AS Pure Core, Q3 (12 marks): In this question you must show detailed reasoning. The complex number 7 - 4i is denoted by z. (a) Giving your answers in the form a + bi, where a and b are rati
  38. November 2020, AS Pure Core, Q8 (8 marks): Two loci, C1 and C2, are defined by % / 2 C z z : z 4 d 36 = = - - 1 ' 1 1 C z : arg ( z 12 d 3 i ) = - - = r 4 2 where d is a real number. (a) Find, in terms o
  39. June 2019, Pure Core 1, Q2 (3 marks): Indicate by shading on an Argand diagram the region " , " , z : z z 4 z z : 3 2 i 2 . [3] G - + - - G
  40. June 2019, Pure Core 1, Q3 (4 marks): In this question you must show detailed reasoning. 3 2 You are given that x 2 5 i is a root of the equation x 2 x 21 x 58 0 . = + - + + = Solve the equation. [4
  41. June 2019, Pure Core 1, Q9 (12 marks): In this question you must show detailed reasoning. 5 = 2 2 You are given the complex number cos i sin and the equation z 1 . ~ = r + r 5 5 (a) Show that ~ is a
  42. June 2019, Pure Core 2, Q7 (7 marks): In an Argand diagram the points representing the numbers 2 + 3i and 1 - i are two adjacent vertices of a square, S. (a) Find the area of S. [3] (b) Find all the
  43. June 2019, Pure Core 2, Q8 (8 marks): In this question you must show detailed reasoning. (a) By writing sini in terms of eii and e ii - show that 1 6 sin ( 10 15 cos 2 6 cos 4 cos 6 ) . [5] = - + -
  44. June 2019, AS Pure Core, Q1 (5 marks): You are given that z 3 4 i . = - (a) Find • z , • arg z , ( ) • z*. [3] On an Argand diagram the complex number w is represented by the point A and w* is repres
  45. June 2019, AS Pure Core, Q4 (14 marks): In this question you must show detailed reasoning. 4 3 2 You are given that ( ) f z 4 z 12 z 41 z 128 z 185 and that 2 + is a root of the equation i = - + - + f
  46. June 2018, AS Pure Core, Q3 (9 marks): In this question you must show detailed reasoning. The complex numbers z1 and z2 are given by z and z a where a is a real number. 2 3i 4i 1 = - 2 = + (i) Expres
  47. June 2018, AS Pure Core, Q5 (10 marks): In this question you must show detailed reasoning. 3i 3 (i) Express ( 2 ) in the form a + ib. [3] + 3 2 (ii) Hence verify that 2 3i is a root of the equation z

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