| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Transformations of trigonometric graphs |
| Difficulty | Standard +0.8 This question requires converting a sum of cosines to harmonic form (R sin(θ+α)), which involves compound angle expansion and algebraic manipulation beyond routine exercises. Part (ii)(b) adds complexity by requiring substitution and solving with a fractional angle argument. The multi-step nature and need to connect transformations conceptually makes this moderately challenging, though still within standard C3 scope. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Simplify to obtain \(\frac{11}{2}\cos\theta + \frac{5\sqrt{3}}{2}\sin\theta\) | B1 | Or equiv with two terms perhaps with \(\sin 60\) retained; accept decimal values |
| Attempt correct process to find \(R\) | M1 | For expression of form \(a\cos\theta + b\sin\theta\); obtained after initial simplification |
| Attempt correct process to find \(\alpha\) | M1 | For expression of form \(a\cos\theta + b\sin\theta\); condone \(\sin\alpha = \frac{11}{2}\), \(\cos\alpha = \frac{5}{2}\sqrt{3}\); obtained after initial simplification |
| Obtain \(7\sin(\theta + 51.8)\) | A1 | Or greater accuracy \(51.786\ldots\) |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State stretch and translation in either order | M1 | Or equiv but using correct terminology, not move, squash, \(\ldots\) SC: if M0 but one transformation completely correct, award B1 for 1/3 |
| State stretch parallel to \(y\)-axis with factor \(\frac{1}{7}\) | A1ft | Following their \(R\) and clearly indicating correct direction |
| State translation parallel to \(\theta\)-axis or \(x\)-axis by \(51.8\) in positive direction or state translation by vector \(\begin{pmatrix}51.8\\0\end{pmatrix}\) | A1ft | Following their \(\alpha\) and clearly indicating correct direction; or equiv such as \(308.2\) parallel to \(x\)-axis in negative direction |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State left-hand side (their \(R\))\(\sin(\frac{1}{3}\beta+\gamma)\) where \(\gamma \neq \pm\)(their \(\alpha\)), \(\gamma \neq \pm 40\), \(\gamma \neq \pm 20\) | M1 | Or equiv such as stating \(\theta = \frac{1}{3}\beta + 20\) (and, in this case, allowing A1ft provided value of \(\frac{1}{3}\beta\) attempted later) |
| Obtain (their \(R\))\(\sin(\frac{1}{3}\beta +\) their \(\alpha + 20) = 3\) | A1ft | |
| Attempt correct process to find any value of \(\frac{1}{3}\beta\) | M1 | For equation of form \(\sin(\frac{1}{3}\beta+\gamma)=k\) where \( |
| Attempt complete process to find positive value of \(\beta\) | M1 | Including choosing second quadrant value of their \(\sin^{-1}\frac{3}{7}\) |
| Obtain \(248\) or \(249\) or \(248.5\) | A1 | Or greater accuracy \(248.508\ldots\) |
| [5] |
# Question 9:
## Part (i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Simplify to obtain $\frac{11}{2}\cos\theta + \frac{5\sqrt{3}}{2}\sin\theta$ | B1 | Or equiv with two terms perhaps with $\sin 60$ retained; accept decimal values |
| Attempt correct process to find $R$ | M1 | For expression of form $a\cos\theta + b\sin\theta$; obtained after initial simplification |
| Attempt correct process to find $\alpha$ | M1 | For expression of form $a\cos\theta + b\sin\theta$; condone $\sin\alpha = \frac{11}{2}$, $\cos\alpha = \frac{5}{2}\sqrt{3}$; obtained after initial simplification |
| Obtain $7\sin(\theta + 51.8)$ | A1 | Or greater accuracy $51.786\ldots$ |
| | **[4]** | |
## Part (ii)(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State stretch and translation in either order | M1 | Or equiv but using correct terminology, not move, squash, $\ldots$ SC: if M0 but one transformation completely correct, award B1 for 1/3 |
| State stretch parallel to $y$-axis with factor $\frac{1}{7}$ | A1ft | Following their $R$ and clearly indicating correct direction |
| State translation parallel to $\theta$-axis or $x$-axis by $51.8$ in positive direction or state translation by vector $\begin{pmatrix}51.8\\0\end{pmatrix}$ | A1ft | Following their $\alpha$ and clearly indicating correct direction; or equiv such as $308.2$ parallel to $x$-axis in negative direction |
| | **[3]** | |
## Part (ii)(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State left-hand side (their $R$)$\sin(\frac{1}{3}\beta+\gamma)$ where $\gamma \neq \pm$(their $\alpha$), $\gamma \neq \pm 40$, $\gamma \neq \pm 20$ | M1 | Or equiv such as stating $\theta = \frac{1}{3}\beta + 20$ (and, in this case, allowing A1ft provided value of $\frac{1}{3}\beta$ attempted later) |
| Obtain (their $R$)$\sin(\frac{1}{3}\beta +$ their $\alpha + 20) = 3$ | A1ft | |
| Attempt correct process to find any value of $\frac{1}{3}\beta$ | M1 | For equation of form $\sin(\frac{1}{3}\beta+\gamma)=k$ where $|k|<1$, $k\neq 0$ |
| Attempt complete process to find positive value of $\beta$ | M1 | Including choosing second quadrant value of their $\sin^{-1}\frac{3}{7}$ |
| Obtain $248$ or $249$ or $248.5$ | A1 | Or greater accuracy $248.508\ldots$ |
| | **[5]** | |9 (i) Express $5 \cos \left( \theta - 60 ^ { \circ } \right) + 3 \cos \theta$ in the form $R \sin ( \theta + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.\\
(ii) Hence
\begin{enumerate}[label=(\alph*)]
\item give details of the transformations needed to transform the curve $y = 5 \cos \left( \theta - 60 ^ { \circ } \right) + 3 \cos \theta$ to the curve $y = \sin \theta$,
\item find the smallest positive value of $\beta$ satisfying the equation
$$5 \cos \left( \frac { 1 } { 3 } \beta - 40 ^ { \circ } \right) + 3 \cos \left( \frac { 1 } { 3 } \beta + 20 ^ { \circ } \right) = 3 .$$
\section*{END OF QUESTION PAPER}
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2014 Q9 [12]}}