| Exam Board | OCR |
|---|---|
| Module | PURE |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Trig equation from real-world model |
| Difficulty | Standard +0.3 This is a straightforward application of trigonometric modeling with standard techniques: reading amplitude/vertical shift from the equation (part a), using the period formula to find k (part b), and solving a basic trig inequality (part c). All steps are routine for A-level students who have covered trig equations, making it slightly easier than average. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.05f Trigonometric function graphs: symmetries and periodicities1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(29\) (m) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(1\) (m) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(15 - 14\cos(150k) = 1 \Rightarrow \cos(150k) = 1\) | M1 | Setting given expression equal to (a)(ii) and re-arranging to get \(\cos(150k) = \ldots\) or for stating \(150k = 360\). Must substitute/use \(t=150\). May use \(t=75\) with \(h=29\), \(150k = 2\pi\) (using radians) M1. |
| \(150k = 360 \Rightarrow k = 2.4\) | A1 | The correct answer only implies the M mark so "\(k = 2.4\)" without working is M1A1. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(15 - 14\cos(kt) = 20 \Rightarrow \cos(kt) = -\frac{5}{14}\) | M1* | Setting given expression equal to 20 to obtain equation of the form \(\cos(kt) = k_1\). Need \(-1 \leqslant k_1 \leqslant 1\). Could use inequalities. |
| \(t = 46.2186\ldots, 103.7813\ldots\) | M1dep* | Obtaining at least one value of \(t\) correctly from their equation above. \(2.4t = 110.9248\ldots, 249.0751\ldots\) |
| \(103.7813\ldots - 46.2186\ldots\) or \(150 - (2 \times 46.2186\ldots)\) o.e. | M1 | Subtracting their two positive values of \(t\) (both obtained correctly from their equation above) or \(150 - (2 \times \text{their } t)\). Dependent on previous two M marks. |
| Therefore above 20 m for \(57.6\) (s) | A1 | Must be to 1 decimal place. \(57.562639\ldots\) A0 if using radians. |
## Question 3:
### Part (a)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $29$ (m) | B1 | |
### Part (a)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $1$ (m) | B1 | |
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $15 - 14\cos(150k) = 1 \Rightarrow \cos(150k) = 1$ | M1 | Setting given expression equal to **(a)(ii)** and re-arranging to get $\cos(150k) = \ldots$ **or** for stating $150k = 360$. Must substitute/use $t=150$. May use $t=75$ with $h=29$, $150k = 2\pi$ (using radians) **M1**. |
| $150k = 360 \Rightarrow k = 2.4$ | A1 | The correct answer only implies the M mark so "$k = 2.4$" without working is **M1A1**. |
### Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $15 - 14\cos(kt) = 20 \Rightarrow \cos(kt) = -\frac{5}{14}$ | M1* | Setting given expression equal to 20 to obtain equation of the form $\cos(kt) = k_1$. Need $-1 \leqslant k_1 \leqslant 1$. Could use inequalities. |
| $t = 46.2186\ldots, 103.7813\ldots$ | M1dep* | Obtaining at least one value of $t$ correctly from their equation above. $2.4t = 110.9248\ldots, 249.0751\ldots$ |
| $103.7813\ldots - 46.2186\ldots$ or $150 - (2 \times 46.2186\ldots)$ o.e. | M1 | Subtracting their two positive values of $t$ (**both** obtained correctly from their equation above) or $150 - (2 \times \text{their } t)$. **Dependent on previous two M marks**. |
| Therefore above 20 m for $57.6$ (s) | A1 | Must be to 1 decimal place. $57.562639\ldots$ A0 if using radians. |3 A Ferris wheel at a fairground rotates in a vertical plane. The height above the ground of a seat on the wheel is $h$ metres at time $t$ seconds after the seat is at its lowest point.
The height is given by the equation $h = 15 - 14 \cos ( k t ) ^ { \circ }$, where $k$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Write down the greatest height of a seat above the ground.
\item Write down the least height of a seat above the ground.
\end{enumerate}\item Given that a seat first returns to its lowest point after 150 seconds, calculate the value of $k$.
\item Determine for how long a seat is 20 metres or more above the ground during one complete revolution. Give your answer correct to the nearest tenth of a second.
\end{enumerate}
\hfill \mbox{\textit{OCR PURE Q3 [8]}}