Hyperbolic functions and identities: AQA A‑Level Further Maths questions

AQA A-Level Further Maths past-paper questions on H.1 Hyperbolic functions and identities, 2019–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.

All 26 questions on Hyperbolic functions and identities, as a list

June 2025, Paper 1, Q4 (1 mark): Which one of the following statements is always correct? Tick () one box. [1 mark] ( ) = − 2 1 sinh cosh 2 2 1 x x + = 2 2 sech tanh 1 x x + = 2 2 cosech coth
June 2025, Paper 1, Q14 (6 marks): (a) Using the exponential definitions of sinh x and cosh x, prove that ( ) 1 1 1 coth ln 2 1 x x x − +   =   −   [3 marks] 14 (b) Hence, solve the equatio
June 2025, Paper 1, Q18 (15 marks): The diagram shows part of the graph of = 15cosech y x and part of the graph of = + 1 4sinh 2 y x O x y 18 (a) Solve the inequality < + 1 15 cosech 4sinh 2 x x G
June 2025, AS Paper 1, Q12 (6 marks): Use the definitions of sinh x and cosh x in terms of ex and e–x to solve 5 sinh x – 3 cosh x = 6 Give your answer in the form x = In (a + √ b) where a and b are
June 2024, Paper 1, Q9 (8 marks): (a) It is given that _ p = In(r + √ r2 + 1) Starting from the exponential definition of the sinh function, show that sinh p = r [4 marks] 9 (b) Solve the equati
June 2024, Paper 2, Q3 (1 mark): The function g is defined by g(x) = sech x (x ∈ ℝ) Which one of the following is the range of g? Tick () one box. [1 mark] –∞ < g(x) ≤ –1 –1 ≤ g(x) < 0 0 < g(x
June 2024, AS Paper 1, Q1 (1 mark): Express cosh2 x in terms of sinh x Circle your answer. [1 mark] 1 + sinh2 x 1 – sinh2 x sinh2 x – 1 –1 – sinh2 x
June 2024, AS Paper 1, Q6 (4 marks): (a) On the axes below, sketch the graph of y = cosh x Indicate the value of any intercept of the curve with the axes. [2 marks] x y O 6 (b) Solve the equation c
June 2023, Paper 1, Q1 (1 mark): Find the number of solutions of the equation tanhx = cosh x Circle your answer. [1 mark] 0 1 2 3
June 2023, Paper 1, Q12 (6 marks): (a) Starting from the identities for sinh 2x and cosh 2x, prove the identity tanh 2x = 2 tanhx 1 + tanh2 x [2 marks] 12 (b) (i) The function f is defined by f (
June 2023, AS Paper 1, Q1 (1 mark): Which expression below is equivalent to tanh x ? Circle your answer. [1 mark] sinh x cosh x sinh x cosh x cosh x sinh x sinh x + cosh x
June 2023, AS Paper 1, Q6 (6 marks): (a) Find and simplify the first five terms in the Maclaurin series for e2x [2 marks] 6 (b) Hence, or otherwise, write down the first five terms in the Maclaurin
June 2022, Paper 1, Q6 (8 marks): (a) Given that jxj < 1 , prove that tanh−1 x = 1 2 ln 1 + x 1 − x [4 marks] 6 (b) Solve the equation 20 sech2 x − 11 tanhx = 16 Give your answer in logarithmic
June 2022, Paper 2, Q4 (1 mark): Which of the following graphs intersects the graph of y = sinhx at exactly one point? Circle your answer. [1 mark] y = cosech x y = cosh x y = cothx y = sechx f
June 2022, AS Paper 1, Q1 (1 mark): Which of the following exponential expressions is equivalent to 2 sinhx ? Circle your answer. [1 mark] ex ex + e−x ex − e−x e−x
June 2022, AS Paper 1, Q15 (6 marks): The two values of y that satisfy the equation sinh2 y − sinhy − 2 = 0 are y1 and y2 15 (a) Hamzah is asked to find the value of y1 + y2 He writes his answer as
November 2021, Paper 1, Q4 (5 marks): Show that the solutions to the equation 3 tanh2 x − 2 sech x = 2 can be expressed in the form x = × ln (a + b p ) where a and b are integers to be found. You ma
November 2021, AS Paper 1, Q6 (2 marks): Prove the identity cosh2 x − sinh2 x = 1 [2 marks]
November 2021, AS Paper 1, Q12 (5 marks): The equation x3 − 2x2 − x + 2 = 0 has three roots. One of the roots is 2 12 (a) Find the other two roots of the equation. [1 mark] 12 (b) Hence, or otherwise, s
November 2020, Paper 1, Q14 (6 marks): (a) Given that sinh (A + B) = sinh A cosh B + coshA sinhB express sinh (m + 1)x and sinh (m − 1)x in terms of sinhmx, coshmx, sinhx and coshx [1 mark] 14 (b) He
November 2020, AS Paper 1, Q8 (8 marks): (a) Prove that tanh−1 x = 1 2 ln 1 + x 1 − x [5 marks] 8 (b) Prove that the graphs of y = sinh x and y = cosh x do not intersect. [3 marks]
November 2020, AS Paper 1, Q11 (3 marks): Sketch the polar graph of r = sinhy + coshy for 0 × y × 2p [3 marks]
June 2019, Paper 1, Q6 (8 marks): (a) Show that cosh3 x + sinh3 x = 1 4 emx + 3 4 enx where m and n are integers. [3 marks] 6 (b) Hence find cosh6 x − sinh6 x in the form a cosh (kx) + b 8 where
June 2019, AS Paper 1, Q6 (5 marks): (a) On the axes provided, sketch the graph of x = cosh (y + b) where b is a positive constant. [4 marks] x y O 6 (b) Determine the minimum distance between the
June 2019, AS Paper 1, Q9 (7 marks): (a) Saul is solving the equation 2 cosh x + sinh2 x = 1 He writes his steps as follows: 2 coshx + sinh2 x = 1 2 cosh x + 1 − cosh2 x = 1 2 coshx − cosh2 x = 0 c
June 2019, AS Paper 1, Q10 (6 marks): (a) Using the definition of coshx and the Maclaurin series expansion of ex, find the first three non-zero terms in the Maclaurin series expansion of cosh x. [3

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