plumiHyperbolic functions: AQA A‑Level Further Maths questions
Every AQA A-Level Further Maths exam question on H Hyperbolic functions, 2019–2025, with its mark scheme and examiner’s report.
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All 38 questions on Hyperbolic functions, 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, Paper 2, Q12 (5 marks): Find the value of d + + ∫ 5 2 1 1 6 5 x x x Give your answer in the form ( ) + + + ln 8 3 5 15 a b c where a, b and c are integers. [5 marks]
- June 2025, AS Paper 1, Q7 (4 marks): (a) Geraldine wants to show that tanh–1 x = 1 2 In ( 1 + x 1 – x) She writes her steps as follows: Let ⇐ ⇐ ⇐ ⇐ ⇐ ⇐ ∴ y = tanh–1 x tanh y = x e y + e–y e y – e–y
- 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 1, Q14 (7 marks): Solve the differential equation dy dx + ytanh x = sinh3 x given that y = 3 when x = ln2 Give your answer in an exact form. [7 marks]
- June 2024, Paper 1, Q17 (7 marks): By making a suitable substitution, show that ∫ 1 –2√ x2 + 6x + 8 dx = 2√ 15 – 1 2 cosh–1 (4) [7 marks]
- 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, Paper 2, Q1 (1 mark): Given that y = sinx + sinhx , find d2y dx2 + y Circle your answer. [1 mark] 2 sin x −2 sin x 2 sinhx −2 sinhx
- June 2023, Paper 2, Q4 (1 mark): It is given that f (x) = cosh−1 (x − 3) Which of the sets listed below is the greatest possible domain of the function f ? Circle your answer. [1 mark] fx : x ×
- 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, Q3 (1 mark): Given that y = sechx , find dy dx Tick (3) one box. [1 mark] sech x tanhx − sech x tanhx cosech x cothx − cosech x cothx
- 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, Paper 1, Q14 (12 marks): The hyperbola H has equation y2 − x2 = 16 The circle C has equation x2 + y2 = 32 The diagram below shows part of the graph of H and part of the graph of C. x y
- November 2021, Paper 2, Q12 (12 marks): The integral Sn is defined by Sn = (a 0 xn sinhx dx (n ≤ 0) 12 (a) Show that for n ≤ 2 Sn = n(n − 1)Sn−2 + an cosh a − nan−1 sinha [7 marks] 12 (b) Hence show t
- 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, Q12 (8 marks): (a) Use the definition of the cosh function to prove that cosh−1 x a = ln x + x2 − a2 p a ! for a > 0 [6 marks] 12 (b) The formulae booklet gives the integral o
- 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, Q1 (1 mark): Which one of these functions has the set fx : jxj < 1g as its greatest possible domain? Circle your answer. [1 mark] cosh x cosh−1 x tanhx tanh−1 x
- 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, Paper 2, Q5 (4 marks): A curve has equation y = coshx Show that the arc length of the curve from x = a to x = b , where 0 < a < b , is equal to sinh b − sinha [4 marks]
- 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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