| Exam Board | OCR MEI |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Calculate intersection coordinates algebraically |
| Difficulty | Challenging +1.2 This question requires solving 3sin²θ = 2cosθ algebraically, which involves using sin²θ = 1-cos²θ to form a quadratic in cosθ, then solving and finding all solutions in the given range. While it requires multiple steps (substitution, quadratic formula, inverse trig, finding all solutions), these are standard A-level techniques with no novel insight needed. The algebraic manipulation is moderately challenging but routine for AS level. |
| Spec | 1.05f Trigonometric function graphs: symmetries and periodicities1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Correct cosine curve sketch with correct shape and symmetry | B1 | 1.1a — Correct shape and symmetry for cosine graph |
| Correct maximum and minimum values shown | B1 | 1.1 — Correct maximum and minimum values |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| DR: \(2\cos\theta = 3\sin^2\theta\) | B1 | 1.2 |
| \(2\cos\theta = 3(1 - \cos^2\theta)\) | M1 | 3.1a — Correct use of identity must be seen |
| \(3\cos^2\theta + 2\cos\theta - 3 = 0\) | M1 | 1.1 — Rearranging to zero must be seen, condone one error |
| \(\cos\theta = \frac{-1}{3} + \frac{\sqrt{10}}{3}\) | A1 | 1.1 — Solve quadratic |
| \(\theta = 43.9°,\ 316.1°\) | A1 | 1.1 |
| \(\cos\theta = \frac{-1}{3} - \frac{\sqrt{10}}{3} < -1\) gives no solution | E1 | 2.4 — Or state that graph in part (i) only shows two solutions |
| [6] |
# Question 6:
## Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct cosine curve sketch with correct shape and symmetry | B1 | 1.1a — Correct shape and symmetry for cosine graph |
| Correct maximum and minimum values shown | B1 | 1.1 — Correct maximum and minimum values |
| **[2]** | | |
## Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| DR: $2\cos\theta = 3\sin^2\theta$ | B1 | 1.2 |
| $2\cos\theta = 3(1 - \cos^2\theta)$ | M1 | 3.1a — Correct use of identity **must be seen** |
| $3\cos^2\theta + 2\cos\theta - 3 = 0$ | M1 | 1.1 — Rearranging to zero **must be seen**, condone one error |
| $\cos\theta = \frac{-1}{3} + \frac{\sqrt{10}}{3}$ | A1 | 1.1 — Solve quadratic |
| $\theta = 43.9°,\ 316.1°$ | A1 | 1.1 |
| $\cos\theta = \frac{-1}{3} - \frac{\sqrt{10}}{3} < -1$ gives no solution | E1 | 2.4 — Or state that graph in part (i) only shows two solutions |
| **[6]** | | |
---6
\begin{enumerate}[label=(\alph*)]
\item The graph of $y = 3 \sin ^ { 2 } \theta$ for $0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }$ is shown in Fig. 6.\\
On the copy of Fig. 6 in the Printed Answer Booklet, sketch the graph of $y = 2 \cos \theta$ for $0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{05376a51-e768-4b45-9c18-c98255a4bd70-05_818_1507_571_351}
\captionsetup{labelformat=empty}
\caption{Fig. 6}
\end{center}
\end{figure}
\item In this question you must show detailed reasoning.
Determine the values of $\theta , 0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }$, for which the two graphs cross.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI AS Paper 2 Q6 [8]}}