AS June 2021 Paper 1 Q12
12 The equation \(x^3 - 2x^2 - x + 2 = 0\) has three roots. One of the roots is 2
(a) Find the other two roots of the equation. [1 mark]
(b) Hence, or otherwise, solve\[\cosh^3\theta - 2\cosh^2\theta - \cosh\theta + 2 = 0\]
giving your answers in an exact form. [4 marks]
| Scheme | Marks | AO |
|---|---|---|
| Obtains the other two roots. | B1 | 1.1b |
| (1) |
Typical solution
\[\begin{aligned}x^3 - 2x^2 - x + 2 &= (x - 2)(x^2 - 1) \\ &= (x - 2)(x - 1)(x + 1)\end{aligned}\]\[x = 1 \quad \text{and} \quad x = -1\]| Scheme | Marks | AO |
|---|---|---|
| Obtains at least one correct value for \(\cosh\theta\) Or uses their answer to part (a) to write down a value for \(\cosh\theta\) | M1 | 1.1a |
| Rejects \(\cosh\theta = -1\) Follow through their part (a) \(\lt 1\) PI by only considering values for which \(\cosh\theta \geqslant 1\) | B1ft | 1.1b |
| Correctly finds at least one non-zero root of the equation. Follow through any of their answers to part (a) if greater than 1. | A1ft | 1.1b |
| Obtains the three correct values of \(\theta\) with no incorrect answers. Accept \(\theta = \pm\cosh^{-1}(2)\) | A1 | 1.1b |
| (4) | ||
| (5 marks) |
Typical solution
\[\cosh\theta = 2, \quad \cosh\theta = 1, \quad \cosh\theta = -1\]but \(\cosh\theta \geqslant 1 \quad \therefore \cosh\theta \neq -1\)
\[\theta = \pm\ln\big(2 + \sqrt{3}\big), \quad \theta = 0\]