| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Year | 2021 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Solve using substitution u = cosh x or u = sinh x |
| Difficulty | Standard +0.8 This is a two-part question requiring polynomial factorization followed by solving a hyperbolic equation via substitution. Part (a) is routine (factor out (x-2) and solve quadratic), but part (b) requires recognizing the substitution u = cosh θ, then using inverse hyperbolic functions and considering the range restriction cosh θ ≥ 1. The multi-step nature and the need to handle the domain constraint elevates this above average difficulty for Further Maths. |
| Spec | 4.05b Transform equations: substitution for new roots4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07d Differentiate/integrate: hyperbolic functions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x^3 - 2x^2 - x + 2 = (x-2)(x^2-1) = (x-2)(x-1)(x+1)\) | B1 | Obtains the other two roots |
| \(x = 1\) and \(x = -1\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\cosh\theta = 2,\ \cosh\theta = 1,\ \cosh\theta = -1\) | M1 | Obtains at least one correct value for \(\cosh\theta\), or uses part (a) answer to write a value for \(\cosh\theta\) |
| Reject \(\cosh\theta = -1\) since \(\cosh\theta \geq 1\) | B1ft | Rejects \(\cosh\theta = -1\); follow through their part (a) \(< 1\); PI by only considering values for which \(\cosh\theta \geq 1\) |
| \(\theta = \pm\ln(2+\sqrt{3}),\ \theta = 0\) | A1ft | Correctly finds at least one non-zero root; follow through any of their answers to part (a) if greater than 1 |
| Three correct values of \(\theta\), no incorrect answers; accept \(\theta = \pm\cosh^{-1}(2)\) | A1 | Obtains the three correct values with no incorrect answers |
## Question 12(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^3 - 2x^2 - x + 2 = (x-2)(x^2-1) = (x-2)(x-1)(x+1)$ | B1 | Obtains the other two roots |
| $x = 1$ and $x = -1$ | | |
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## Question 12(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\cosh\theta = 2,\ \cosh\theta = 1,\ \cosh\theta = -1$ | M1 | Obtains at least one correct value for $\cosh\theta$, or uses part (a) answer to write a value for $\cosh\theta$ |
| Reject $\cosh\theta = -1$ since $\cosh\theta \geq 1$ | B1ft | Rejects $\cosh\theta = -1$; follow through their part (a) $< 1$; PI by only considering values for which $\cosh\theta \geq 1$ |
| $\theta = \pm\ln(2+\sqrt{3}),\ \theta = 0$ | A1ft | Correctly finds at least one non-zero root; follow through any of their answers to part (a) if greater than 1 |
| Three correct values of $\theta$, no incorrect answers; accept $\theta = \pm\cosh^{-1}(2)$ | A1 | Obtains the three correct values with no incorrect answers |
---12 The equation $x ^ { 3 } - 2 x ^ { 2 } - x + 2 = 0$ has three roots. One of the roots is 2
12
\begin{enumerate}[label=(\alph*)]
\item Find the other two roots of the equation.
12
\item Hence, or otherwise, solve
$$\cosh ^ { 3 } \theta - 2 \cosh ^ { 2 } \theta - \cosh \theta + 2 = 0$$
giving your answers in an exact form.
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 2021 Q12 [5]}}