| Exam Board | OCR MEI |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2018 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Range of simple harmonic function |
| Difficulty | Moderate -0.3 This is a standard harmonic form question requiring routine application of the R cos(x + α) formula using R = √(7² + 24²) = 25 and tan α = 24/7. Part (ii) simply requires recognizing that the range is [12-25, 12+25] = [-13, 37]. While it involves multiple steps, it's a textbook exercise with no novel insight required, making it slightly easier than average. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(R=25\) | B1 (AO1.1) | |
| \(\tan^{-1}\!\left(\frac{24}{7}\right)\) or \(\sin^{-1}\!\left(\frac{24}{25}\right)\) or \(\cos^{-1}\!\left(\frac{7}{25}\right)\) | M1 (AO1.1) | 73.739795° rounded to 2 or more sf may imply M1A0 |
| \(25\cos(x+1.29)\) | A1 (AO1.1) | \(\alpha=1.28700221759\) rounded to 2 or more sf; allow A1 for \(\alpha\) found to 2 or more sf |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(12\pm \textit{their } 25\) | M1 (AO3.1a) | Or one of \(-13\) and \(37\) identified |
| \(-13\leq f(x)\leq 37\) | A1 (AO1.1) | Allow e.g. from \(-13\) to \(37\) inclusive; A0 if inequality is strict |
| [2] |
## Question 6(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $R=25$ | B1 (AO1.1) | |
| $\tan^{-1}\!\left(\frac{24}{7}\right)$ or $\sin^{-1}\!\left(\frac{24}{25}\right)$ or $\cos^{-1}\!\left(\frac{7}{25}\right)$ | M1 (AO1.1) | 73.739795° rounded to 2 or more sf may imply M1A0 |
| $25\cos(x+1.29)$ | A1 (AO1.1) | $\alpha=1.28700221759$ rounded to 2 or more sf; allow A1 for $\alpha$ found to 2 or more sf |
| **[3]** | | |
## Question 6(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $12\pm \textit{their } 25$ | M1 (AO3.1a) | Or one of $-13$ and $37$ identified |
| $-13\leq f(x)\leq 37$ | A1 (AO1.1) | Allow e.g. from $-13$ to $37$ inclusive; A0 if inequality is strict |
| **[2]** | | |
---6 (i) Express $7 \cos x - 24 \sin x$ in the form $R \cos ( x + \alpha )$, where $0 < \alpha < \frac { \pi } { 2 }$.\\
(ii) Write down the range of the function
$$f ( x ) = 12 + 7 \cos x - 24 \sin x , \quad 0 \leqslant x \leqslant 2 \pi .$$
\hfill \mbox{\textit{OCR MEI Paper 2 2018 Q6 [5]}}