Hyperbolic functions

Sample questions

Original practice questions written in the style of the exam.

1. Question 15 marks

   Solve the equation $3\cosh x - \sinh x = 3$ giving your answers as exact values.

   Mark scheme

   • M1 Uses exponential definitions: $\dfrac{3(e^x + e^{-x})}{2} - \dfrac{e^x - e^{-x}}{2} = 3$ (1)
   • A1 $e^x + 2e^{-x} = 3$ (1)
   • M1 Multiplies by $e^x$: $e^{2x} - 3e^x + 2 = 0$ (1)
   • A1 $(e^x - 1)(e^x - 2) = 0$ (1)
   • A1 $x = 0$ or $x = \ln 2$ (1)
2. Question 24 marks

   Starting from the definition of $\tanh y$ in terms of exponentials, show that $\operatorname{artanh} x = \frac{1}{2}\ln\left(\frac{1 + x}{1 - x}\right), \qquad -1 < x < 1$