| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2011 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Applied context with trigonometry |
| Difficulty | Standard +0.3 This is a structured multi-part question with clear scaffolding through parts (i)-(v). While it involves 3D geometry visualization and requires addition formulae and algebraic manipulation, each step is guided with results to show. The trigonometry is standard C4 level (sec, tan, addition formulae), and the algebraic work, though somewhat lengthy, follows predictable patterns. Slightly easier than average due to extensive scaffolding. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals3.01c Moment unit: N m |
| Answer | Marks | Guidance |
|---|---|---|
| I don't see any mark scheme content to clean up in your message. You've provided a question number (8) and what appears to be a score range (8 | 7 | 6 |
I don't see any mark scheme content to clean up in your message. You've provided a question number (8) and what appears to be a score range (8 | 7 | 6 | 5), but there are no actual marking criteria, annotations, or guidance notes.
Please provide the full mark scheme content you'd like me to clean up, including:
- The marking points (M1, A1, B1, etc.)
- The criteria or solutions
- Any guidance notes
Then I'll be happy to format it as requested with unicode-to-LaTeX conversions.8 Fig. 8 shows a searchlight, mounted at a point A, 5 metres above level ground. Its beam is in the shape of a cone with axis AC , where C is on the ground. AC is angled at $\alpha$ to the vertical. The beam produces an oval-shaped area of light on the ground, of length DE . The width of the oval at C is GF . Angles DAC, EAC, FAC and GAC are all $\beta$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{f657e167-e6f8-4df2-901b-067c32835877-04_684_872_461_278}
\captionsetup{labelformat=empty}
\caption{Fig. 8}
\end{center}
\end{figure}
In the following, all lengths are in metres.\\
(i) Find AC in terms of $\alpha$, and hence show that $\mathrm { GF } = 10 \sec \alpha \tan \beta$.\\
(ii) Show that $\mathrm { CE } = 5 ( \tan ( \alpha + \beta ) - \tan \alpha )$.
$$\text { Hence show that } \mathrm { CE } = \frac { 5 \tan \beta \sec ^ { 2 } \alpha } { 1 - \tan \alpha \tan \beta } \text {. }$$
Similarly, it can be shown that $\mathrm { CD } = \frac { 5 \tan \beta \sec ^ { 2 } \alpha } { 1 + \tan \alpha \tan \beta }$. [You are not required to derive this result.]\\
You are now given that $\alpha = 45 ^ { \circ }$ and that $\tan \beta = t$.\\
(iii) Find CE and CD in terms of $t$. Hence show that $\mathrm { DE } = \frac { 20 t } { 1 - t ^ { 2 } }$.\\
(iv) Show that $\mathrm { GF } = 10 \sqrt { 2 } t$.
For a certain value of $\beta , \mathrm { DE } = 2 \mathrm { GF }$.\\
(v) Show that $t ^ { 2 } = 1 - \frac { 1 } { \sqrt { 2 } }$.
Hence find this value of $\beta$.
\hfill \mbox{\textit{OCR MEI C4 2011 Q8 [18]}}