| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2016 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Deduce related equation solution |
| Difficulty | Standard +0.3 This is a standard harmonic form question with routine steps: (a) convert to R cos(θ-α) using standard formulas, (b) solve the resulting equation, (c) deduce a related solution using symmetry. All techniques are textbook exercises with no novel insight required. Part (c) adds slight complexity but follows directly from recognizing the sign change pattern. Slightly easier than average due to being highly procedural. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc |
| HJUV SIHI NI III HM ION OC | VI4V SIHI NI JIHM ION OC | VEXV SIHI NI JIIHM IONOO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(R = \sqrt{34}\) | B1 | Must be exact; ignore decimal value (5.83...) |
| \(\tan\alpha = \pm\frac{5}{3}\), \(\tan\alpha = \pm\frac{3}{5}\) leading to \(\alpha = ...\) (Allow \(\cos\alpha = \pm\frac{5}{\sqrt{34}}\) or \(\pm\frac{3}{\sqrt{34}}\), \(\sin\alpha = \pm\frac{5}{\sqrt{34}}\) or \(\pm\frac{3}{\sqrt{34}}\)) where \(\sqrt{34}\) is their \(R\) | M1 | |
| \(\alpha = 59.04°\) | A1 | awrt 59.04° |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\sqrt{34}\cos(\theta - 59.04) = 2 \Rightarrow \cos(\theta - 59.04) = \frac{2}{\sqrt{34}}\) (0.343); attempts to use part (a), proceeds to \(\cos(\theta \pm \text{"59.04"}) = K\), \( | K | \leq 1\) |
| \(\theta_1 - 59.04 = 69.94 \Rightarrow \theta_1 =\) awrt \(129.0°\) | A1 | |
| \(\theta_2 \pm 59.04 = 360 - \text{'69.94'} \Rightarrow \theta_2 = ...\); correct attempt at second solution in range | dM1 | Dependent on previous M; usually \(\theta - \text{their } 59.04 = 360 - \text{their '69.94'}\) |
| \(\theta_2 = 349.1°\) | A1 | awrt 349.1° |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\theta + \text{their } 59.04 = \cos^{-1}\!\left(\frac{2}{\text{their}\sqrt{34}}\right) \Rightarrow \theta = ...\) (Allow \(\theta - \text{their } 59.04 = \cos^{-1}\!\left(\frac{2}{\text{their}\sqrt{34}}\right) \Rightarrow \theta = ...\) if they have \(\theta +\) .. in (b)) | M1 | Evidence of use of parts (a) and (b) to obtain \(\theta\); may be implied by use of answers to part (b) |
| \(\theta = 10.9°\) | A1 | awrt 10.9 |
## Question 1:
### Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $R = \sqrt{34}$ | B1 | Must be exact; ignore decimal value (5.83...) |
| $\tan\alpha = \pm\frac{5}{3}$, $\tan\alpha = \pm\frac{3}{5}$ leading to $\alpha = ...$ (Allow $\cos\alpha = \pm\frac{5}{\sqrt{34}}$ or $\pm\frac{3}{\sqrt{34}}$, $\sin\alpha = \pm\frac{5}{\sqrt{34}}$ or $\pm\frac{3}{\sqrt{34}}$) where $\sqrt{34}$ is their $R$ | M1 | |
| $\alpha = 59.04°$ | A1 | awrt 59.04° |
**(3 marks)**
### Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sqrt{34}\cos(\theta - 59.04) = 2 \Rightarrow \cos(\theta - 59.04) = \frac{2}{\sqrt{34}}$ (0.343); attempts to use part (a), proceeds to $\cos(\theta \pm \text{"59.04"}) = K$, $|K| \leq 1$ | M1 | May be implied by $\theta - \text{"59.04"} = 69.94...°$ or $\theta - \text{"59.04"} = \cos^{-1}\!\left(\frac{2}{\text{their}\sqrt{34}}\right)$ |
| $\theta_1 - 59.04 = 69.94 \Rightarrow \theta_1 =$ awrt $129.0°$ | A1 | |
| $\theta_2 \pm 59.04 = 360 - \text{'69.94'} \Rightarrow \theta_2 = ...$; correct attempt at second solution in range | dM1 | Dependent on previous M; usually $\theta - \text{their } 59.04 = 360 - \text{their '69.94'}$ |
| $\theta_2 = 349.1°$ | A1 | awrt 349.1° |
Extra answers in range: deduct final A mark.
**(4 marks)**
### Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\theta + \text{their } 59.04 = \cos^{-1}\!\left(\frac{2}{\text{their}\sqrt{34}}\right) \Rightarrow \theta = ...$ (Allow $\theta - \text{their } 59.04 = \cos^{-1}\!\left(\frac{2}{\text{their}\sqrt{34}}\right) \Rightarrow \theta = ...$ if they have $\theta +$ .. in (b)) | M1 | Evidence of use of parts (a) and (b) to obtain $\theta$; may be implied by use of answers to part (b) |
| $\theta = 10.9°$ | A1 | awrt 10.9 |
**(2 marks)**
---\begin{enumerate}
\item (a) Express $3 \cos \theta + 5 \sin \theta$ in the form $R \cos ( \theta - \alpha )$, where $R$ and $\alpha$ are constants, $R > 0$ and $0 < \alpha < 90 ^ { \circ }$. Give the exact value of $R$ and give the value of $\alpha$ to 2 decimal places.\\
(b) Hence solve, for $0 \leqslant \theta < 360 ^ { \circ }$, the equation
\end{enumerate}
$$3 \cos \theta + 5 \sin \theta = 2$$
Give your answers to one decimal place.\\
(c) Use your solutions to parts (a) and (b) to deduce the smallest positive value of $\theta$ for which
$$3 \cos \theta - 5 \sin \theta = 2$$
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HJUV SIHI NI III HM ION OC & VI4V SIHI NI JIHM ION OC & VEXV SIHI NI JIIHM IONOO \\
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\hfill \mbox{\textit{Edexcel C34 2016 Q1 [9]}}