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CAIE A-Level Further Math A2 2.1 Hyperbolic Functions Questions

Practise hyperbolic definitions, identities, inverse functions and graphs, with work linking exponentials, asymptotes and calculus.

Syllabus
2028–2030
Course
Further Mathematics 9231
Level
A2

Exam points

  • prove tanh and sech identities by writing each function in exponential form
  • sketch coth or related curves and state asymptotes from the graph
  • differentiate inverse hyperbolic functions using implicit differentiation and identities

Covered syllabus points

2.1.1—Hyperbolic functions2.1.2—Hyperbolic functions2.1.3—Hyperbolic functions2.1.4—Hyperbolic functions

Question 1

[Maximum number: 5]

Question (a)

(a)

Sketch the graph of y=coth⁡xy=\operatorname{coth} x for x>0 and state the equations of the asymptotes.

[ 2 ]

Question (b)

(b)

Starting from the definitions of coth and cosech in terms of exponentials, prove that

coth⁡2x−cosech⁡2x=1.\operatorname{coth}^{2} x-\operatorname{cosech}^{2} x=1 .

The curve C has equation y=ln⁡coth⁡(12x)y=\ln \operatorname{coth}\left(\frac{1}{2} x\right) for x>0.

[ 3 ]

Question 2

[Maximum number: 3]

Starting from the definitions of tanh and sech in terms of exponentials, prove that

tanh⁡2t+sech⁡2t=1.\tanh ^{2} t+\operatorname{sech}^{2} t=1 .

Question 3

[Maximum number: 6]

The curves C1:y=cosh⁡xC_{1}: y=\cosh x and C2:y=sinh⁡2xC_{2}: y=\sinh 2 x intersect at the point where x=a.

Question (a)

(a)

Find the exact value of a, giving your answer in logarithmic form.

[ 4 ]

Question (b)

(b)

Sketch C1C_{1} and C2C_{2} on the same diagram.

[ 2 ]
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