CAIE A-Level Further Math 2.1 Hyperbolic functions Question Bank

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

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 functions 2.1.2—Hyperbolic functions 2.1.3—Hyperbolic functions 2.1.4—Hyperbolic functions

[Maximum number: 5]

(a)

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

[ 2 ]

Figure for Answer to Question (a) — CAIE A-Level Further Math A2

B1

Correct shape and position, not too truncated.

$x=0, \quad y=1$

B1

States equations of asymptotes.

2

(b)

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

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

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

[ 3 ]

$\operatorname{coth} x=\frac{\mathrm{e}^{x}+\mathrm{e}^{-x}}{\mathrm{e}^{x}-\mathrm{e}^{-x}} \quad \operatorname{cosech} x=\frac{2}{\mathrm{e}^{x}-\mathrm{e}^{-x}}$

B1

$\left(\frac{\mathrm{e}^{x}+\mathrm{e}^{-x}}{\mathrm{e}^{x}-\mathrm{e}^{-x}}\right)^{2}-\frac{4}{\left(\mathrm{e}^{x}-\mathrm{e}^{-x}\right)^{2}}=\frac{\mathrm{e}^{2 x}+\mathrm{e}^{-2 x}-2}{\left(\mathrm{e}^{x}-\mathrm{e}^{-x}\right)^{2}}=1$

M1 A1

Writes over common denominator, AG.

3

[Maximum number: 3]

Starting from the definition of tanh in terms of exponentials, prove that $\tanh ^{-1} x=\frac{1}{2} \ln \left(\frac{1+x}{1-x}\right)$ . [3]

$x=\frac{\mathrm{e}^{u}-\mathrm{e}^{-u}}{\mathrm{e}^{u}+\mathrm{e}^{-u}}$

1

М1

Write in exponential form

$\mathrm{e}^{2 u}(1-x)=1+x$

1

M1

Rearrange

$u=\tanh ^{-1} x=\frac{1}{2} \ln \left(\frac{1+x}{1-x}\right)$

1

A1

AG CAO

3

All question bank results loaded

Covered syllabus points

2.1 Hyperbolic functions question 1

Question (a)

Question (b)

2.1 Hyperbolic functions question 2