plumiCalculus with hyperbolic functions: OCR A‑Level Further Maths questions
OCR A-Level Further Maths past-paper questions on 8.3 Calculus with hyperbolic functions, 2018–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.
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All 10 questions on Calculus with hyperbolic functions, as a list
- June 2025, Pure Core 2, Q8 (12 marks): A function f (x) is defined by f (x) = xsinh 2x. 2 n 0 d f d f n (a) Prove by induction that 4 ( x sinh 2 x n cosh 2 x ) for n 0 where d is defined as H = + 2 n
- 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 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, 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 1, Q6 (6 marks): Let y x cosh x . = 2 n 1 d - y Prove by induction that, for all integers n H , 1 x sinh x ( 2 n 1 ) cosh x . [6] = + - 2 n 1 d x -
- 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 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
- 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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