| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Find value where max/min occurs |
| Difficulty | Standard +0.3 This is a standard harmonic form question with routine application of R cos(θ + α) conversion followed by straightforward equation solving and finding max/min values. While multi-part, each step follows a well-practiced procedure taught in C3 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) etc1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain \(R=\sqrt{20}\) or \(R=4.47\) | B1 | |
| Attempt to find value of \(\alpha\) | M1 | Implied by correct value or its complement; allow sin/cos muddles; allow use of radians for M1; condone use of \(\cos\alpha=4\), \(\sin\alpha=2\) here but not for A1 |
| Obtain 26.6 | A1 | Or greater accuracy 26.565…; with no wrong working seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show correct process for finding one answer | M1 | Allowing for case where the answer is negative |
| Obtain 21.3 | A1FT | Or greater accuracy 21.3045…; or anything rounding to 21.3 with no obvious error; following a wrong value of \(\alpha\) but not wrong \(R\) |
| Show correct process for finding second answer | M1 | i.e. attempting fourth quadrant value minus \(\alpha\) value |
| Obtain 286 or 285.6 | A1FT | Or greater accuracy 285.5653…; or anything rounding to 286 with no obvious error; following a wrong value of \(\alpha\) but not wrong \(R\); and no others between \(0°\) and \(360°\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State greatest value is 25 | B1 | allow if \(\alpha\) incorrect |
| Obtain value 63.4 clearly associated with correct greatest value | B1FT | or greater accuracy 63.4349…; following a wrong value of \(\alpha\) |
| State least value is 5 | B1 | allow if \(\alpha\) incorrect |
| Attempt to find \(\theta\) from \(\cos(\theta + \text{their}\,\alpha) = -1\) | M1 | and clearly associated with correct least value |
| Obtain 153 or 153.4 | A1FT | or greater accuracy 153.4349…; following a wrong value of \(\alpha\) |
| [5] |
## Question 8:
### Part (i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain $R=\sqrt{20}$ or $R=4.47$ | B1 | |
| Attempt to find value of $\alpha$ | M1 | Implied by correct value or its complement; allow sin/cos muddles; allow use of radians for M1; condone use of $\cos\alpha=4$, $\sin\alpha=2$ here but not for A1 |
| Obtain 26.6 | A1 | Or greater accuracy 26.565…; with no wrong working seen |
### Part (ii)(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show correct process for finding one answer | M1 | Allowing for case where the answer is negative |
| Obtain 21.3 | A1FT | Or greater accuracy 21.3045…; or anything rounding to 21.3 with no obvious error; following a wrong value of $\alpha$ but not wrong $R$ |
| Show correct process for finding second answer | M1 | i.e. attempting fourth quadrant value minus $\alpha$ value |
| Obtain 286 or 285.6 | A1FT | Or greater accuracy 285.5653…; or anything rounding to 286 with no obvious error; following a wrong value of $\alpha$ but not wrong $R$; and no others between $0°$ and $360°$ |
## Question 8(ii)(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State greatest value is 25 | B1 | allow if $\alpha$ incorrect |
| Obtain value 63.4 clearly associated with correct greatest value | B1FT | or greater accuracy 63.4349…; following a wrong value of $\alpha$ |
| State least value is 5 | B1 | allow if $\alpha$ incorrect |
| Attempt to find $\theta$ from $\cos(\theta + \text{their}\,\alpha) = -1$ | M1 | and clearly associated with correct least value |
| Obtain 153 or 153.4 | A1FT | or greater accuracy 153.4349…; following a wrong value of $\alpha$ |
| **[5]** | | |
---8 (i) Express $4 \cos \theta - 2 \sin \theta$ in the form $R \cos ( \theta + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.\\
(ii) Hence
\begin{enumerate}[label=(\alph*)]
\item solve the equation $4 \cos \theta - 2 \sin \theta = 3$ for $0 ^ { \circ } < \theta < 360 ^ { \circ }$,
\item determine the greatest and least values of
$$25 - ( 4 \cos \theta - 2 \sin \theta ) ^ { 2 }$$
as $\theta$ varies, and, in each case, find the smallest positive value of $\theta$ for which that value occurs.
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2013 Q8 [12]}}