| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | June |
| Marks | 8 |
| 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 P3 harmonic form question with routine application of the R sin(x - α) method. Part (a) uses the standard formula R = √(a² + b²) and tan α = b/a. Parts (b) and (c) require recognizing that the range of R sin(x - α) is [-R, R] and applying this to find minima and ranges. While multi-part, each step follows textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(R = 13\) | B1 | \(R=\pm13\) is B0 |
| \(\tan\alpha = \frac{5}{12} \Rightarrow \alpha = \text{awrt } 0.395\) | M1, A1 | M1: \(\tan\alpha = \pm\frac{5}{12}\), \(\tan\alpha=\pm\frac{12}{5}\); or if \(R\) used: \(\sin\alpha=\pm\frac{5}{R}\) or \(\cos\alpha=\pm\frac{12}{R}\). A1: \(\alpha=\text{awrt } 0.395\); degree equivalent \(22.6°\) is A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Minimum value is \(-3\) | B1ft | States value of \(10-R\) following through their \(R\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(2\theta - \frac{\pi}{6} - 0.395 = \frac{3\pi}{2} \Rightarrow \theta = \text{awrt } 2.82\) | M1, A1 | M1: Attempts to solve \(2\theta - \frac{\pi}{6} \pm\text{"0.395"} = \frac{3\pi}{2}\). A1: \(\theta=\text{awrt }2.82\); no other values should be given |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(h(\beta) = 10 - 169\sin^2(\beta - 0.395)\) | ||
| \(-159 \leqslant h \leqslant 10\) | M1, A1 | M1: Achieves one end value, either \(-159\) (or \(10-(\text{their }R)^2\)) or \(10\). A1: Fully correct range \(-159\leqslant h\leqslant 10\); equivalent correct ranges accepted e.g. \([-159,10]\) |
## Question 9:
**Part (a):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $R = 13$ | B1 | $R=\pm13$ is B0 |
| $\tan\alpha = \frac{5}{12} \Rightarrow \alpha = \text{awrt } 0.395$ | M1, A1 | M1: $\tan\alpha = \pm\frac{5}{12}$, $\tan\alpha=\pm\frac{12}{5}$; or if $R$ used: $\sin\alpha=\pm\frac{5}{R}$ or $\cos\alpha=\pm\frac{12}{R}$. A1: $\alpha=\text{awrt } 0.395$; degree equivalent $22.6°$ is A0 |
**Part (b)(i):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Minimum value is $-3$ | B1ft | States value of $10-R$ following through their $R$ |
**Part (b)(ii):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $2\theta - \frac{\pi}{6} - 0.395 = \frac{3\pi}{2} \Rightarrow \theta = \text{awrt } 2.82$ | M1, A1 | M1: Attempts to solve $2\theta - \frac{\pi}{6} \pm\text{"0.395"} = \frac{3\pi}{2}$. A1: $\theta=\text{awrt }2.82$; no other values should be given |
**Part (c):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $h(\beta) = 10 - 169\sin^2(\beta - 0.395)$ | | |
| $-159 \leqslant h \leqslant 10$ | M1, A1 | M1: Achieves one end value, either $-159$ (or $10-(\text{their }R)^2$) or $10$. A1: Fully correct range $-159\leqslant h\leqslant 10$; equivalent correct ranges accepted e.g. $[-159,10]$ |\begin{enumerate}
\item (a) Express $12 \sin x - 5 \cos x$ in the form $R \sin ( x - \alpha )$, where $R$ and $\alpha$ are constants, $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$. Give the exact value of $R$ and give the value of $\alpha$ in radians, to 3 decimal places.
\end{enumerate}
The function g is defined by
$$g ( \theta ) = 10 + 12 \sin \left( 2 \theta - \frac { \pi } { 6 } \right) - 5 \cos \left( 2 \theta - \frac { \pi } { 6 } \right) \quad \theta > 0$$
Find\\
(b) (i) the minimum value of $\mathrm { g } ( \theta )$\\
(ii) the smallest value of $\theta$ at which the minimum value occurs.
The function h is defined by
$$\mathrm { h } ( \beta ) = 10 - ( 12 \sin \beta - 5 \cos \beta ) ^ { 2 }$$
(c) Find the range of h .\\
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\hfill \mbox{\textit{Edexcel P3 2021 Q9 [8]}}