| Exam Board | OCR MEI |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2019 |
| Session | June |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Calculate intersection coordinates algebraically |
| Difficulty | Standard +0.3 This is a straightforward intersection problem requiring students to solve 4sin(x) = 3cos(x), which reduces to tan(x) = 3/4. The solution involves basic algebraic manipulation and calculator use for inverse tan, slightly easier than average due to its routine nature and clear method. |
| Spec | 1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(4\sin x° = 3\cos x°\) | M1 | 1.1a – Attempt to solve simultaneously |
| \(\tan x° = \frac{3}{4}\) | M1 | 1.1a – Using trig identity to form equation in \(\tan x\) |
| \(x° = 36.9°,\ 216.9°\) | A1 [3] | 1.1 – Both answers required with no extras in interval \(0° \leq x° \leq 360°\). Answers must be given to 1 dp. ISW answers outside interval. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(4\sin x° = 3\cos x°\) | M1 | 1.1a – Attempt to solve simultaneously |
| \(\Rightarrow 16\sin^2 x° = 9\cos^2 x°\) | ||
| \(\sin^2 x° = \frac{9}{25}\) or \(\cos^2 x° = \frac{16}{25}\) | M1 | 1.1a – Using \(\sin^2 x° + \cos^2 x° = 1\) to form equation in \(\sin^2 x°\) or \(\cos^2 x°\) |
| \(x° = 36.9°,\ 143.1°,\ 216.9°,\ 323.9°\) | ||
| Check solutions satisfy original equation; give \(x° = 36.9°,\ 216.9°\) | A1 [3] | 1.1 – Both answers required with no extras in interval \(0° \leq x° \leq 360°\). ISW answers outside interval. |
## Question 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $4\sin x° = 3\cos x°$ | M1 | 1.1a – Attempt to solve simultaneously |
| $\tan x° = \frac{3}{4}$ | M1 | 1.1a – Using trig identity to form equation in $\tan x$ |
| $x° = 36.9°,\ 216.9°$ | A1 [3] | 1.1 – Both answers required with no extras in interval $0° \leq x° \leq 360°$. Answers must be given to 1 dp. ISW answers outside interval. |
**Alternative method:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $4\sin x° = 3\cos x°$ | M1 | 1.1a – Attempt to solve simultaneously |
| $\Rightarrow 16\sin^2 x° = 9\cos^2 x°$ | | |
| $\sin^2 x° = \frac{9}{25}$ or $\cos^2 x° = \frac{16}{25}$ | M1 | 1.1a – Using $\sin^2 x° + \cos^2 x° = 1$ to form equation in $\sin^2 x°$ or $\cos^2 x°$ |
| $x° = 36.9°,\ 143.1°,\ 216.9°,\ 323.9°$ | | |
| Check solutions satisfy original equation; give $x° = 36.9°,\ 216.9°$ | A1 [3] | 1.1 – Both answers required with no extras in interval $0° \leq x° \leq 360°$. ISW answers outside interval. |
---2 In this question you must show detailed reasoning.
Fig. 2 shows the graphs of $y = 4 \sin x ^ { \circ }$ and $y = 3 \cos x ^ { \circ }$ for $0 \leqslant x \leqslant 360$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{0b1c272a-f0f4-4931-be89-5d045804a7af-3_549_768_813_258}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{center}
\end{figure}
Find the $x$-coordinates of the two points of intersection, giving your answers correct to 1 decimal place.
\hfill \mbox{\textit{OCR MEI AS Paper 1 2019 Q2 [3]}}