| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | October |
| Marks | 9 |
| 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 transformations. Part (a) is textbook harmonic form conversion, parts (b-d) apply standard function transformations to find max/min values and locations. All steps follow well-practiced procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(R=17\) | B1 | For 17 only |
| \(\tan\alpha = \frac{15}{8}\) | M1 | Attempts equation in \(\alpha\); accept \(\tan\alpha=\pm\frac{15}{8}\) or \(\pm\frac{8}{15}\); implied by correct \(\alpha\) |
| \(\alpha = 1.081\) | A1 | Must be in radians |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Min \(f(x) = \frac{15}{41+2\times 17} = \frac{1}{5}\) | M1 | Attempts to apply result from (a); allow \(\frac{15}{41\pm 2\times\text{"their }R\text{"}}\) |
| \(=\frac{1}{5}\) | A1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\sin(x-1.081)=1 \Rightarrow x-1.081=\frac{\pi}{2} \Rightarrow x=\ldots\) | M1 | Attempts to solve \(x\pm\text{"}\alpha\text{"}=\frac{\pi}{2}\) |
| \(x \approx 2.65\) | A1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(-\frac{23}{5}\) (or \(-4.6\)) | B1ft | Follow through: \(2\times\left(\text{their }\frac{1}{5}\right)-5\); no other solutions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Awrt \(1.33\) | B1ft | Follow through \(0.5\times\) their \(2.65\); no other solutions |
# Question 8:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $R=17$ | B1 | For 17 only |
| $\tan\alpha = \frac{15}{8}$ | M1 | Attempts equation in $\alpha$; accept $\tan\alpha=\pm\frac{15}{8}$ or $\pm\frac{8}{15}$; implied by correct $\alpha$ |
| $\alpha = 1.081$ | A1 | Must be in radians |
## Part (b)(i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Min $f(x) = \frac{15}{41+2\times 17} = \frac{1}{5}$ | M1 | Attempts to apply result from (a); allow $\frac{15}{41\pm 2\times\text{"their }R\text{"}}$ |
| $=\frac{1}{5}$ | A1 | cao |
## Part (b)(ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sin(x-1.081)=1 \Rightarrow x-1.081=\frac{\pi}{2} \Rightarrow x=\ldots$ | M1 | Attempts to solve $x\pm\text{"}\alpha\text{"}=\frac{\pi}{2}$ |
| $x \approx 2.65$ | A1 | cao |
## Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $-\frac{23}{5}$ (or $-4.6$) | B1ft | Follow through: $2\times\left(\text{their }\frac{1}{5}\right)-5$; no other solutions |
## Part (d)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Awrt $1.33$ | B1ft | Follow through $0.5\times$ their $2.65$; no other solutions |\begin{enumerate}[label=(\alph*)]
\item Express $8 \sin x - 15 \cos x$ in the form $R \sin ( x - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$
Give the exact value of $R$, and give the value of $\alpha$, in radians, to 4 significant figures.
$$\mathrm { f } ( x ) = \frac { 15 } { 41 + 16 \sin x - 30 \cos x } \quad x > 0$$
\item Find
\begin{enumerate}[label=(\roman*)]
\item the minimum value of $\mathrm { f } ( x )$
\item the smallest value of $x$ at which this minimum value occurs.
\end{enumerate}\item State the $y$ coordinate of the minimum points on the curve with equation
$$y = 2 \mathrm { f } ( x ) - 5 \quad x > 0$$
\item State the smallest value of $x$ at which a maximum point occurs for the curve with equation
$$y = - \mathrm { f } ( 2 x ) \quad x > 0$$
\section*{8. In this question you must show all stages of your working. \\
In this question you must show all stages of your working.}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P3 2022 Q8 [9]}}