Hyperbolic functions — AS & A Level Further Mathematics (9231) questions by topic
Hyperbolic identities and their exponential definitions anchor this topic: sketching y = tanh x and stating its asymptote, deriving a double-angle identity linking sinh squared and cosh from first principles, and establishing the Pythagorean-style relationship between tanh and sech.
Show, find and sketch are the top command words across 18 questions from 2023 to 2025 papers, worth from 4 to 15 marks, where starting explicitly from the exponential definitions, as instructed, is what the mark scheme rewards. With no examiner-report extracts logged for this topic, submitting each identity proof for instant marking is the way to confirm every algebraic step from exponential definition to stated identity is shown, not assumed.
May–June 2025, Paper 21
Starting from the definitions of tanh and sech in terms of exponentials, prove that 1 - tanh^2(u) = sech^2(u).
Answer this question and get it marked →May–June 2025, Paper 22
Starting from the definitions of tanh and sech in terms of exponentials, prove that 1 - tanh^2(u) = sech^2(u).
Answer this question and get it marked →May–June 2025, Paper 23
Starting from the definitions of tanh and sech in terms of exponentials, prove that tanh^2(t) + sech^2(t) = 1.
Answer this question and get it marked →The curve C has equation 9y^2 - 3arcsinh(xy) = 1 - 3ln(3). Show that, at the point (4, 1/3) on C, dy/dx = -1/2.
Answer this question and get it marked →May–June 2025, Paper 24
Find the exact value of integral from 1 to 5/2 of 1/sqrt(x^2 - 2x + 5) dx, giving your answer in logarithmic form.
Answer this question and get it marked →The curve C has equation y = tanh x for x = 0. Sketch C and state the equation of the asymptote.
Answer this question and get it marked →May–June 2024, Paper 21
Show that (cosh(x) + sinh(x))^(1/2) = e^((1/2)x).
Answer this question and get it marked →May–June 2024, Paper 22
Show that (cosh(x) + sinh(x))^(1/2) = e^((1/2)x).
Answer this question and get it marked →May–June 2024, Paper 23
Show that d/dx[ (x/2) sqrt(x^2 - 9) - (9/2) cosh^(-1)(x/3) ] = sqrt(x^2 - 9).
Answer this question and get it marked →October–November 2024, Paper 22
Find the value of the integral from 6 to 7 of 1/sqrt((x - 5)^2 - 1) dx, giving your answer in the form ln(a + sqrt(b)), where a and b are integers to be determined.
Answer this question and get it marked →Show that d/dx(ln(tanh x)) = 2 cosech(2x).
Answer this question and get it marked →May–June 2023, Paper 21
Starting from the definitions of sech and tanh in terms of exponentials, prove that 1 - sech^2(t) = tanh^2(t).
Answer this question and get it marked →May–June 2023, Paper 22
Starting from the definitions of sech and tanh in terms of exponentials, prove that 1 - sech^2(t) = tanh^2(t).
Answer this question and get it marked →May–June 2023, Paper 23
Starting from the definitions of cosh and sinh in terms of exponentials, prove that 2 cosh^2(x) = cosh(2x) + 1.
Answer this question and get it marked →October–November 2023, Paper 21
Starting from the definitions of cosh and sinh in terms of exponentials, prove that sinh(2x) = 2 sinh(x) cosh(x).
Answer this question and get it marked →October–November 2023, Paper 22
The diagram shows part of the curve y = x sech^2(x) and its maximum point M. Show that, at M, 2x tanh x - 1 = 0 and verify that this equation has a root between 0.7 and 0.8.
Answer this question and get it marked →Starting from the definitions of cosh and sinh in terms of exponentials, prove that 2 sinh^2 A = cosh 2A - 1.
Answer this question and get it marked →October–November 2023, Paper 23
Starting from the definitions of cosh and sinh in terms of exponentials, prove that sinh(2x) = 2 sinh(x) cosh(x).
Answer this question and get it marked →Other AS & A Level Further Mathematics topics
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