plumiHyperbolic functions: OCR A‑Level Further Maths questions
Every OCR A-Level Further Maths exam question on CP.8 Hyperbolic functions, 2018–2025, with its mark scheme and examiner’s report.
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![June 2025 Pure Core 1, question 2: - b y d in terms of x. l, find x 1 1 2 (a) Given that y cosh x [1] = d 3 - b l at the point where x 1 1 (b) Determine an equation of the normal to the curve y c](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y54001-jun25-02-qp.webp?v=3)
![June 2025 Pure Core 2, question 10: u 10 (a) Find e - as a percentage of sinh u for the following values of u. 2 • u = 2 • u = 5 [1] (b) Find, as a percentage of sinh u, the difference between sin](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y54101-jun25-10-qp.webp?v=3)
![June 2024 Pure Core 1, question 7: - , show that (a) By using the definitions of cosh u and sinh u in terms of eu and e u 7 sinh 2 u 2 sinh cosh u u . [2] / The equation of a curve, C, is y 16 co](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y54001-jun24-07-qp.webp?v=3)
![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](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y54101-jun24-07-qp.webp?v=3)
![June 2023 Pure Core 2, question 3: ^ h d 1 1 3 (a) Show that sinh - u . [2] = u d 2 u 1 + 1 (b) Find the equation of the normal to the graph of y sinh - 2 x at the point where x 6 . = = Give your](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y54101-jun23-03-qp.webp?v=3)
![June 2023 Pure Core 2, question 5: In this question you must show detailed reasoning. (a) Using the definitions of sinh x and cosh x in terms of exponentials, show that x x x [2] sinh 2 2 sinh co](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y54101-jun23-05-qp.webp?v=3)
![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](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y54001-jun22-01-qp.webp?v=3)
![June 2022 Pure Core 2, question 5: (a) By using the exponential definitions of sinh x and cosh x, prove the identity 2 2 cosh 2 x cosh x sinh x . [2] / + (b) Hence find an expression for cosh 2x](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y54101-jun22-05-qp.webp?v=3)
![November 2021 Pure Core 1, question 8: You are given that ( ) f x 4 sinh x 3 cosh x . = + (a) Show that the curve y f ( ) x has no turning points. [3] = (b) Determine the exact solution of the equati](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y54001-nov21-08-qp.webp?v=3)
![November 2021 Pure Core 2, question 5: In this question you must show detailed reasoning. 2 (a) Using the definition of cosh x in terms of exponentials, show that cosh 2 x 2 cosh x - . [2] 1 / (b) So](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y54101-nov21-05-qp.webp?v=3)
![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](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y54001-nov20-08-qp.webp?v=3)
![June 2019 Pure Core 1, question 5: The diagram shows part of the curve y 5 cosh x 3 sinh x . = + y x O (a) Solve the equation cosh 5 x 3 sinh x 4 giving your solution in exact form. [4] + = (b) I](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y54001-jun19-05-qp.webp?v=3)
![June 2019 Pure Core 1, question 7: 7 The function sech x is defined by sec h x . = cosh x x 2 e (a) Show that sec h x . [2] = 2 x e 1 + y (b) Using a suitable substitution, find sech x x d . [4]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/y54001-jun19-07-qp.webp?v=3)
All 16 questions on Hyperbolic functions, as a list
- June 2025, Pure Core 1, Q2 (5 marks): - b y d in terms of x. l, find x 1 1 2 (a) Given that y cosh x [1] = d 3 - b l at the point where x 1 1 (b) Determine an equation of the normal to the curve y c
- June 2025, Pure Core 2, Q10 (8 marks): u 10 (a) Find e - as a percentage of sinh u for the following values of u. 2 • u = 2 • u = 5 [1] (b) Find, as a percentage of sinh u, the difference between sin
- June 2024, Pure Core 1, Q7 (9 marks): - , show that (a) By using the definitions of cosh u and sinh u in terms of eu and e u 7 sinh 2 u 2 sinh cosh u u . [2] / The equation of a curve, C, is y 16 co
- 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
- 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
- June 2023, Pure Core 2, Q3 (6 marks): ^ h d 1 1 3 (a) Show that sinh - u . [2] = u d 2 u 1 + 1 (b) Find the equation of the normal to the graph of y sinh - 2 x at the point where x 6 . = = Give your
- June 2023, Pure Core 2, Q5 (7 marks): In this question you must show detailed reasoning. (a) Using the definitions of sinh x and cosh x in terms of exponentials, show that x x x [2] sinh 2 2 sinh co
- June 2023, Pure Core 2, Q10 (7 marks): In this question you must show detailed reasoning. A region, R, of the floor of an art gallery is to be painted for the purposes of an art installation. A suita
- 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
- June 2022, Pure Core 2, Q5 (7 marks): (a) By using the exponential definitions of sinh x and cosh x, prove the identity 2 2 cosh 2 x cosh x sinh x . [2] / + (b) Hence find an expression for cosh 2x
- November 2021, Pure Core 1, Q8 (8 marks): You are given that ( ) f x 4 sinh x 3 cosh x . = + (a) Show that the curve y f ( ) x has no turning points. [3] = (b) Determine the exact solution of the equati
- November 2021, Pure Core 2, Q5 (8 marks): In this question you must show detailed reasoning. 2 (a) Using the definition of cosh x in terms of exponentials, show that cosh 2 x 2 cosh x - . [2] 1 / (b) So
- 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
- November 2020, Pure Core 2, Q9 (11 marks): Two thin poles, OA and BC, are fixed vertically on horizontal ground. A chain is fixed at A and C such that it touches the ground at point D as shown in the dia
- June 2019, Pure Core 1, Q5 (7 marks): The diagram shows part of the curve y 5 cosh x 3 sinh x . = + y x O (a) Solve the equation cosh 5 x 3 sinh x 4 giving your solution in exact form. [4] + = (b) I
- June 2019, Pure Core 1, Q7 (6 marks): 7 The function sech x is defined by sec h x . = cosh x x 2 e (a) Show that sec h x . [2] = 2 x e 1 + y (b) Using a suitable substitution, find sech x x d . [4]
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