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Further calculus: OCR A‑Level Further Maths questions

Every OCR A-Level Further Maths exam question on CP.5 Further calculus, 2018–2025, with its mark scheme and examiner’s report.

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648 questions

  1. Q3

    June 2025 · Pure Core 1

    6 marksCP.5 Further calculus
    The whole paper ↗
    June 2025 Pure Core 1, question 3: 3 The region R1 is bounded by the curve y , the line x k and the coordinate axes, as = = 2 25 x - shown in Fig. 1. Fig. 1 80 y = y 2 x 25 - R1 x O x=k (a) In th

  2. Q7

    June 2025 · Pure Core 2

    10 marksCP.5 Further calculus
    The whole paper ↗
    June 2025 Pure Core 2, question 7: In this question you must show detailed reasoning. 2 4 x 5 x 17 - + - (a) Express x in partial fractions. [6] 3 2 x 3 x 3 - + - y 3 2 4 x 5 x 17 - + - (b) Hence

  3. Q1

    June 2024 · Pure Core 1

    3 marksCP.5 Further calculus
    The whole paper ↗
    June 2024 Pure Core 1, question 1: d . y 1 2 1 y x [3] Given that sin - ( ) , find x = d

  4. Q5

    June 2024 · Pure Core 1

    5 marksCP.5 Further calculus
    The whole paper ↗
    June 2024 Pure Core 1, question 5: 12 x 5 Express ( using partial fractions. [5] 2 2 x 1 2 )( x 1 ) + +

  5. Q6

    June 2024 · Pure Core 1

    4 marksCP.5 Further calculus
    The whole paper ↗
    June 2024 Pure Core 1, question 6: In this question you must show detailed reasoning. 3 y 18 Determine the exact value of d x . [4] 2 x x 9

  6. Q9

    June 2024 · Pure Core 1

    6 marksCP.4 Further algebra and functionsCP.5 Further calculus
    The whole paper ↗
    June 2024 Pure Core 1, question 9: ^ h up to and including the term in x4. 2 9 (a) Find the Maclaurin series of x [3] ln ( 1 ) + ^ h and 2 y x The diagram below shows parts of the graphs of the c

  7. Q7

    June 2024 · Pure Core 2

    10 marksCP.5 Further calculusCP.8 Hyperbolic functions
    The whole paper ↗
    June 2024 Pure Core 2, question 7: bx + x 7 (a) Express 17 coshx - 15 sinhx in the form e - ( a e c ) where a, b and c are integers to be determined. [3] 1 A function is defined by ( ) f x . The

  8. Q6

    June 2023 · Pure Core 1

    4 marksCP.5 Further calculusCP.8 Hyperbolic functions
    The whole paper ↗
    June 2023 Pure Core 1, question 6: In this question you must show detailed reasoning. The power output, p watts, of a machine at time t hours after it is switched on can be modelled by the equati

  9. 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

  10. Q4

    June 2023 · Pure Core 2

    4 marksCP.5 Further calculus
    The whole paper ↗
    June 2023 Pure Core 2, question 4: In this question you must show detailed reasoning. 1 The region R is bounded by the curve with equation y , the x-axis and the lines = 2 x x 3 3 1 - + 1 with eq

  11. Q1

    June 2022 · Pure Core 1

    6 marksCP.5 Further calculusCP.8 Hyperbolic functions
    The whole paper ↗
    June 2022 Pure Core 1, question 1: In this question you must show detailed reasoning. 41 (a) Show that cosh ( 2 ln 3 ) . [2] = 9 The region R is bounded by the curve with equation y sinh x , the

  12. Q7

    June 2022 · Pure Core 1

    10 marksCP.5 Further calculus
    The whole paper ↗
    June 2022 Pure Core 1, question 7: x 18 A B Cx D + + 7 (a) Determine the values of A, B, C and D such that . [4] / + + 2 2 2 2 x ( x 9 ) x x x 9 + + (b) In this question you must show detailed re

  13. Q6

    November 2021 · Pure Core 1

    3 marksCP.5 Further calculus
    The whole paper ↗
    November 2021 Pure Core 1, question 6: O is the origin of a coordinate system whose units are cm. The points A, B, C and D have coordinates (1, 0), (1, 4), (6, 9) and (0, 9) respectively. ^ h 2 2 The

  14. Q7

    November 2021 · Pure Core 2

    10 marksCP.5 Further calculus
    The whole paper ↗
    November 2021 Pure Core 2, question 7: In this question you must show detailed reasoning. 3 2 x x x Bx C 9 1 + + - + (a) Find the values of A, B and C for which x A . [1] / + 3 2 3 2 x 4 x 4 x x 4 x

  15. Q1

    November 2020 · Pure Core 1

    2 marksCP.5 Further calculus
    The whole paper ↗
    November 2020 Pure Core 1, question 1: 1 Find the mean value of ( )x f x 6 x over the interval [ , ]. 0 3 [2] = +

  16. Q8

    November 2020 · Pure Core 1

    10 marksCP.8 Hyperbolic functionsCP.5 Further calculus
    The whole paper ↗
    November 2020 Pure Core 1, question 8: 8 (a) Using exponentials, show that cosh 2 u 2 sinh u 1 . [2] / + (b) By differentiating both sides of the identity in part (a) with respect to u, show that sin

  17. Q12

    November 2020 · Pure Core 1

    6 marksCP.5 Further calculus
    The whole paper ↗
    November 2020 Pure Core 1, question 12: ^ h < 4 3 r 12 Show that d x ln a b where a, b and c are integers to be determined. [6] = + + c 4 1 x - 0

  18. Q10

    November 2020 · Pure Core 2

    10 marksCP.4 Further algebra and functionsCP.5 Further calculus
    The whole paper ↗
    November 2020 Pure Core 2, question 10: 10 Let ( ) f x sin - ( ) x . = (a) (i) Determine f ll ( )x . [2] (ii) Determine the first two non-zero terms of the Maclaurin expansion for ( )x f . [3] (iii) B

  19. Q6

    June 2019 · Pure Core 1

    6 marksCP.5 Further calculus
    The whole paper ↗
    June 2019 Pure Core 1, question 6: tan 1 6 You are given that y - 2 x . = d . y (a) Find d [2] x x 1 y (b) Show that d x k where k is a number to be determined in exact form. [4] 2 ^ h = r 2 2 x

  20. Q3

    June 2019 · Pure Core 2

    5 marksCP.5 Further calculus
    The whole paper ↗
    June 2019 Pure Core 2, question 3: In this question you must show detailed reasoning. 3 y [5] Show that . 3 x x ( 1 ) - d 1 - = 2 5
All 20 questions on Further calculus, as a list
  1. June 2025, Pure Core 1, Q3 (6 marks): 3 The region R1 is bounded by the curve y , the line x k and the coordinate axes, as = = 2 25 x - shown in Fig. 1. Fig. 1 80 y = y 2 x 25 - R1 x O x=k (a) In th
  2. June 2025, Pure Core 2, Q7 (10 marks): In this question you must show detailed reasoning. 2 4 x 5 x 17 - + - (a) Express x in partial fractions. [6] 3 2 x 3 x 3 - + - y 3 2 4 x 5 x 17 - + - (b) Hence
  3. June 2024, Pure Core 1, Q1 (3 marks): d . y 1 2 1 y x [3] Given that sin - ( ) , find x = d
  4. June 2024, Pure Core 1, Q5 (5 marks): 12 x 5 Express ( using partial fractions. [5] 2 2 x 1 2 )( x 1 ) + +
  5. June 2024, Pure Core 1, Q6 (4 marks): In this question you must show detailed reasoning. 3 y 18 Determine the exact value of d x . [4] 2 x x 9
  6. June 2024, Pure Core 1, Q9 (6 marks): ^ h up to and including the term in x4. 2 9 (a) Find the Maclaurin series of x [3] ln ( 1 ) + ^ h and 2 y x The diagram below shows parts of the graphs of the c
  7. June 2024, Pure Core 2, Q7 (10 marks): bx + x 7 (a) Express 17 coshx - 15 sinhx in the form e - ( a e c ) where a, b and c are integers to be determined. [3] 1 A function is defined by ( ) f x . The
  8. June 2023, Pure Core 1, Q6 (4 marks): In this question you must show detailed reasoning. The power output, p watts, of a machine at time t hours after it is switched on can be modelled by the equati
  9. 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
  10. June 2023, Pure Core 2, Q4 (4 marks): In this question you must show detailed reasoning. 1 The region R is bounded by the curve with equation y , the x-axis and the lines = 2 x x 3 3 1 - + 1 with eq
  11. June 2022, Pure Core 1, Q1 (6 marks): In this question you must show detailed reasoning. 41 (a) Show that cosh ( 2 ln 3 ) . [2] = 9 The region R is bounded by the curve with equation y sinh x , the
  12. June 2022, Pure Core 1, Q7 (10 marks): x 18 A B Cx D + + 7 (a) Determine the values of A, B, C and D such that . [4] / + + 2 2 2 2 x ( x 9 ) x x x 9 + + (b) In this question you must show detailed re
  13. November 2021, Pure Core 1, Q6 (3 marks): O is the origin of a coordinate system whose units are cm. The points A, B, C and D have coordinates (1, 0), (1, 4), (6, 9) and (0, 9) respectively. ^ h 2 2 The
  14. November 2021, Pure Core 2, Q7 (10 marks): In this question you must show detailed reasoning. 3 2 x x x Bx C 9 1 + + - + (a) Find the values of A, B and C for which x A . [1] / + 3 2 3 2 x 4 x 4 x x 4 x
  15. November 2020, Pure Core 1, Q1 (2 marks): 1 Find the mean value of ( )x f x 6 x over the interval [ , ]. 0 3 [2] = +
  16. November 2020, Pure Core 1, Q8 (10 marks): 8 (a) Using exponentials, show that cosh 2 u 2 sinh u 1 . [2] / + (b) By differentiating both sides of the identity in part (a) with respect to u, show that sin
  17. November 2020, Pure Core 1, Q12 (6 marks): ^ h < 4 3 r 12 Show that d x ln a b where a, b and c are integers to be determined. [6] = + + c 4 1 x - 0
  18. November 2020, Pure Core 2, Q10 (10 marks): 10 Let ( ) f x sin - ( ) x . = (a) (i) Determine f ll ( )x . [2] (ii) Determine the first two non-zero terms of the Maclaurin expansion for ( )x f . [3] (iii) B
  19. June 2019, Pure Core 1, Q6 (6 marks): tan 1 6 You are given that y - 2 x . = d . y (a) Find d [2] x x 1 y (b) Show that d x k where k is a number to be determined in exact form. [4] 2 ^ h = r 2 2 x
  20. June 2019, Pure Core 2, Q3 (5 marks): In this question you must show detailed reasoning. 3 y [5] Show that . 3 x x ( 1 ) - d 1 - = 2 5

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