| Exam Board | OCR MEI |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2023 |
| Session | June |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Trig equation from real-world model |
| Difficulty | Moderate -0.3 Part (a) is direct substitution into a given formula. Part (b) requires understanding that cos(6x) completes multiple periods over the domain 0≤x≤40, making it slightly more conceptual than pure calculation, but this is still a straightforward AS-level question testing basic understanding of periodic functions rather than solving equations. |
| Spec | 1.05f Trigonometric function graphs: symmetries and periodicities1.05g Exact trigonometric values: for standard angles |
| Answer | Marks | Guidance |
|---|---|---|
| Substitute \(x = 40\), \(h = 17 + 15\cos240 = 9.5\) m | B1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Maximum when \(6x = 0\) or \(360\) so \(x = 0, 60\) | M1 | Must include reference to either 60 or 240 or a sketch illustrating the \(x\)-direction stretch. Allow if wrong conclusion reached. Do not allow argument based on mechanics principles alone. |
| But the model is only valid for \(0 \leq x \leq 40\), so Tom's argument is invalid. | E1 | Clear argument |
## Question 2:
**(a)**
Substitute $x = 40$, $h = 17 + 15\cos240 = 9.5$ m | B1 | cao
**(b)**
Maximum when $6x = 0$ or $360$ so $x = 0, 60$ | M1 | Must include reference to either 60 or 240 or a sketch illustrating the $x$-direction stretch. Allow if wrong conclusion reached. Do not allow argument based on mechanics principles alone.
But the model is only valid for $0 \leq x \leq 40$, so Tom's argument is invalid. | E1 | Clear argument
---2 The height of the first part of a rollercoaster track is $h \mathrm {~m}$ at a horizontal distance of $x \mathrm {~m}$ from the start. A student models this using the equation $h = 17 + 15 \cos 6 x$, for $0 \leqslant x \leqslant 40$, using the values of $h$ given when their calculator is set to work in degrees.
\begin{enumerate}[label=(\alph*)]
\item Find the height that the student's model predicts when the horizontal distance from the start is 40 m .
\item The student argues that the model predicts that the rollercoaster track will achieve a maximum height of 32 m more than once because the cosine function is periodic.
Comment on the validity of the student's argument.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI AS Paper 1 2023 Q2 [3]}}