| Exam Board | OCR MEI |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2019 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Transformations of trigonometric graphs |
| Difficulty | Standard +0.3 Part (a) is a standard harmonic form question requiring routine application of R cos(x+α) = R cos α cos x - R sin α sin x, comparing coefficients to find R = √53 and tan α = 2/7. Part (b) requires recognizing that 1/(7cos x - 2sin x) = 1/(R cos(x+α)) = sec(x+α)/R, leading to a horizontal translation and vertical stretch—straightforward once the connection is made but slightly beyond pure recall. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(7\cos x - 2\sin x = R\cos(x + \alpha)\) | ||
| \(\Rightarrow 7 = R\cos\alpha,\ 2 = R\sin\alpha\) | M1 | Forming two equations soi; allow sign errors |
| \(R = \sqrt{7^2 + 2^2} = \sqrt{53}\) | B1 | Allow even if from equations with sin/cos interchange; must be exact |
| \(\alpha = \arctan\frac{2}{7} = 0.278...\) | M1, A1 [4] | FT their equations; cao (3sf); must be in radians for A mark. Allow M1M1 for \(\alpha = \arctan\left(\pm\frac{2}{7}\right)\); \(\alpha = 15.9°\) gets M1M1A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y = \frac{1}{7\cos x - 2\sin x} = \frac{1}{\sqrt{53}}\sec(x + \alpha)\) | ||
| Stretch scale factor \(\frac{1}{\sqrt{53}}\) in the \(y\)-direction | B1 | Stretch in the \(y\)-direction; for correct scale factor FT their \(R\); do not allow enlargement instead of stretch |
| Translation \(\begin{pmatrix}-0.278\\0\end{pmatrix}\) | B1 [3] | FT their value for \(\alpha\); allow for translation to the left by \(0.278\); do not allow "shift" or "slide" |
## Question 10:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $7\cos x - 2\sin x = R\cos(x + \alpha)$ | | |
| $\Rightarrow 7 = R\cos\alpha,\ 2 = R\sin\alpha$ | M1 | Forming two equations soi; allow sign errors |
| $R = \sqrt{7^2 + 2^2} = \sqrt{53}$ | B1 | Allow even if from equations with sin/cos interchange; must be exact |
| $\alpha = \arctan\frac{2}{7} = 0.278...$ | M1, A1 [4] | FT their equations; cao (3sf); must be in radians for A mark. Allow M1M1 for $\alpha = \arctan\left(\pm\frac{2}{7}\right)$; $\alpha = 15.9°$ gets M1M1A0 |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y = \frac{1}{7\cos x - 2\sin x} = \frac{1}{\sqrt{53}}\sec(x + \alpha)$ | | |
| Stretch scale factor $\frac{1}{\sqrt{53}}$ in the $y$-direction | B1 | Stretch in the $y$-direction; for correct scale factor FT their $R$; do not allow enlargement instead of stretch |
| Translation $\begin{pmatrix}-0.278\\0\end{pmatrix}$ | B1 [3] | FT their value for $\alpha$; allow for translation to the left by $0.278$; do not allow "shift" or "slide" |
---10
\begin{enumerate}[label=(\alph*)]
\item Express $7 \cos x - 2 \sin x$ in the form $R \cos ( x + \alpha )$ where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$, giving the exact value of $R$ and the value of $\alpha$ correct to 3 significant figures.
\item Give details of a sequence of two transformations which maps the curve $y = \sec x$ onto the curve $y = \frac { 1 } { 7 \cos x - 2 \sin x }$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 1 2019 Q10 [7]}}