A2 June 2022 Paper 1 Q3
3 In this question you must show detailed reasoning.
Solve the equation \(3\cosh x = 2\sinh^2 x\), giving your solutions in exact logarithmic form. [6]
| Scheme | Marks | AO |
|---|---|---|
| DR \(3\cosh x = 2\sinh^2 x = 2\left(\cosh^2 x - 1\right)\) | M1 | 3.1a |
| \(\Rightarrow 2\cosh^2 x - 3\cosh x - 2 = 0\) | A1 | 1.1 |
| \(\Rightarrow (2\cosh x + 1)(\cosh x - 2) = 0\) | M1 | 1.1 |
| \(\Rightarrow \cosh x = -\dfrac{1}{2}\) or 2 | A1 | 1.1 |
| \(\Rightarrow x = \ln\left(2 + \sqrt{3}\right)\) | A1 | 1.1 |
| or \(-\ln\left(2 + \sqrt{3}\right)\) | A1 | 1.1 |
| [6] |
Notes
M1: \(\sinh^2 x = \cosh^2 x - 1\) used
M1: Solve their quadratic
A1: Don’t need to see \(-\frac{1}{2}\) if factorisation or quadratic formula shown
A1: or \(\ln\left(2 - \sqrt{3}\right)\)