Standard +0.3 This is a standard C3 harmonic form question with routine application of R sin(θ - α) conversion followed by straightforward max/min finding. Part (a) uses the standard formula R = √(a² + b²) and tan α = b/a. Parts (b) and (c) require recognizing that squaring creates max/min at ±R, then solving simple trigonometric equations within a given range. All steps are algorithmic with no novel insight required, making it slightly easier than average.
9. (a) Express \(2 \sin \theta - 4 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)
Give the value of \(\alpha\) to 3 decimal places.
$$H ( \theta ) = 4 + 5 ( 2 \sin 3 \theta - 4 \cos 3 \theta ) ^ { 2 }$$
Find
(b) (i) the maximum value of \(\mathrm { H } ( \theta )\),
(ii) the smallest value of \(\theta\), for \(0 \leqslant \theta < \pi\), at which this maximum value occurs.
Find
(c) (i) the minimum value of \(\mathrm { H } ( \theta )\),
(ii) the largest value of \(\theta\), for \(0 \leqslant \theta < \pi\), at which this minimum value occurs.
Accept \(\sqrt{20}\) or \(2\sqrt{5}\) or awrt 4.47. Do not accept \(R = \pm\sqrt{20}\). Can be scored in parts (b) or (c) as long as it is clearly \(R\)
Using \(3\theta \pm \text{their '1.107'} = 2\pi \Rightarrow \theta = ..\) Accept \(= n\pi\) where \(n\) is an integer including 0. Degree equivalent: \(3\theta - \text{'63.4°'} = 360° \Rightarrow \theta =\). No mixed units
\(\theta =\) awrt \(2.46\)
A1
awrt 2.46 radians or \(141.1°\). Do not allow multiple solutions for this mark
# Question 9:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $R = \sqrt{20}$ | B1 | Accept $\sqrt{20}$ or $2\sqrt{5}$ or awrt 4.47. Do not accept $R = \pm\sqrt{20}$. Can be scored in parts (b) or (c) as long as it is clearly $R$ |
| $\tan\alpha = \dfrac{4}{2} \Rightarrow \alpha =$ awrt $1.107$ | M1 | For sight of $\tan\alpha = \pm\dfrac{4}{2}$, $\tan\alpha = \pm\dfrac{2}{4}$. Condone $\sin\alpha = 4, \cos\alpha = 2 \Rightarrow \tan\alpha = \dfrac{4}{2}$. If $R$ found first, accept $\sin\alpha = \pm\dfrac{4}{R}$, $\cos\alpha = \pm\dfrac{2}{R}$ |
| $\alpha =$ awrt $1.107$ | A1 | Degrees equivalent $63.4°$ is A0. If all done in degrees, lose just this mark |
## Part (b)(i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $4 + 5R^2 = 104$ | B1ft | Either 104, or if $R$ was incorrect allow numerical value of their $4 + 5R^2$. Allow tolerance of 1 dp on decimal $R$s |
## Part (b)(ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3\theta - \text{'1.107'} = \dfrac{\pi}{2} \Rightarrow \theta =$ awrt $0.89$ | M1 | Using $3\theta \pm \text{their '1.107'} = \dfrac{\pi}{2} \Rightarrow \theta = ..$ Accept $(2n+1)\dfrac{\pi}{2}$. Degree equivalent acceptable: $3\theta - \text{'63.4°'} = 90° \Rightarrow \theta =$. No mixed units |
| $\theta =$ awrt $0.89$ | A1 | awrt 0.89 radians or $51.1°$. Do not allow multiple solutions for this mark |
## Part (c)(i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $4$ | B1 | |
## Part (c)(ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3\theta - \text{'1.107'} = 2\pi \Rightarrow \theta =$ awrt $2.46$ | M1 | Using $3\theta \pm \text{their '1.107'} = 2\pi \Rightarrow \theta = ..$ Accept $= n\pi$ where $n$ is an integer including 0. Degree equivalent: $3\theta - \text{'63.4°'} = 360° \Rightarrow \theta =$. No mixed units |
| $\theta =$ awrt $2.46$ | A1 | awrt 2.46 radians or $141.1°$. Do not allow multiple solutions for this mark |
9. (a) Express $2 \sin \theta - 4 \cos \theta$ in the form $R \sin ( \theta - \alpha )$, where $R$ and $\alpha$ are constants, $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$
Give the value of $\alpha$ to 3 decimal places.
$$H ( \theta ) = 4 + 5 ( 2 \sin 3 \theta - 4 \cos 3 \theta ) ^ { 2 }$$
Find\\
(b) (i) the maximum value of $\mathrm { H } ( \theta )$,\\
(ii) the smallest value of $\theta$, for $0 \leqslant \theta < \pi$, at which this maximum value occurs.
Find\\
(c) (i) the minimum value of $\mathrm { H } ( \theta )$,\\
(ii) the largest value of $\theta$, for $0 \leqslant \theta < \pi$, at which this minimum value occurs.
\hfill \mbox{\textit{Edexcel C3 2014 Q9 [9]}}