AS June 2019 Paper 1 Q9
9
(a) Saul is solving the equation\[2\cosh x + \sinh^2 x = 1\]
He writes his steps as follows:
\[\begin{aligned}2\cosh x + \sinh^2 x &= 1 \\ 2\cosh x + 1 - \cosh^2 x &= 1 \\ 2\cosh x - \cosh^2 x &= 0 \\ \cosh x \neq 0 \;\therefore\; 2 - \cosh x &= 0 \\ \cosh x &= 2 \\ x &= \pm\cosh^{-1}(2)\end{aligned}\]Identify and explain the error in Saul’s method. [2 marks]
(b) Anna is solving the different equation\[\sinh^2(2x) - 2\cosh(2x) = 1\]
and finds the correct answers in the form \(x = \dfrac{1}{p}\cosh^{-1}(q + \sqrt{r})\), where \(p\), \(q\) and \(r\) are integers.
Find the possible values of \(p\), \(q\) and \(r\).
Fully justify your answer. [5 marks]
| Scheme | Marks | AO |
|---|---|---|
| Identifies the 2nd line or \(1 - \cosh^2 x\) as the error. Possibly implied by the explanation. | B1 | 2.3 |
| Explains that \(\sinh^2 x\) should not be replaced with \(1 - \cosh^2 x\) Accept any correct arrangement of the identity \(\cosh^2 x - \sinh^2 x = 1\) | E1 | 2.4 |
Typical solution
In the 2nd line, \(\sinh^2 x\) should not have been replaced with \(1 - \cosh^2 x\)
| Scheme | Marks | AO |
|---|---|---|
| Uses a correct identity to express the equation in terms of one function. e.g. uses \(\cosh^2(2x) - \sinh^2(2x) = 1\) to write the equation in terms of \(\cosh(2x)\) only | B1 | 1.1b |
| Rearranges their equation into a quadratic\(= 0\) or quartic\(= 0\) Must be in terms of just one function. Possibly implied by a correct value for their function. | M1 | 1.1a |
| Finds a correct exact value for their function, e.g. \(\cosh x = \pm\sqrt{1 + \frac{\sqrt{3}}{2}}\) ISW incorrect work following a correct unsimplified answer. | A1 | 1.1b |
| Correctly explains why they reject at least one of their solutions, e.g. \(\cosh 2x \lt 1\) cannot be a solution. Follow through B1M1 only. | E1F | 2.4 |
| Finds both correct sets of values for \(p\), \(q\) and \(r\) and no others. Accept \(\pm\frac{1}{2}\cosh^{-1}(1 + \sqrt{3})\) for full marks. | A1 | 1.1b |
| (7 marks) |
Typical solution
\[\cosh^2(2x) - 1 - 2\cosh(2x) = 1\]\[\cosh^2(2x) - 2\cosh(2x) - 2 = 0\]\[\cosh(2x) = 1 \pm \sqrt{3}\]but \(\cosh(2x) \geqslant 1\) \(\therefore \cosh(2x) = 1 + \sqrt{3}\)
\[x = \pm\frac{1}{2}\cosh^{-1}(1 + \sqrt{3})\]\[p = \pm 2, \quad q = 1, \quad r = 3\]