| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2024 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Find maximum or minimum value |
| Difficulty | Standard +0.3 This is a standard P3/C3 harmonic form question following a well-established template: use double angle identities, convert to R sin(θ+α) form, then find maximum and corresponding x-value. All steps are routine applications of learned techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| States or uses \(\sin 2x = 2\sin x\cos x\) or \(\cos 2x = \pm2\cos^2 x \pm 1\) | M1 | Either identity stated; can be implied by \(a=4\), or \(b=\pm2\) and \(c=-5\) or \(-1\) |
| Uses both \(\sin 2x = 2\sin x\cos x\) and \(\cos 2x = \pm2\cos^2 x \pm 1\) in \(f(x)\) | dM1 | Produces form \(a\sin 2x + b\cos 2x + c\); condone slips when substituting |
| \(f(x) = 4\sin 2x + 2\cos 2x - 1\) | A1 | Correct answer; must be in terms of \(x\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(R^2 = a^2 + b^2 \Rightarrow R = \sqrt{20}\) or \(2\sqrt{5}\) | B1ft | Follow through on their \(a\) or \(b\); can be implied by correct exact value |
| \(\tan\alpha = \frac{b}{a} \Rightarrow \alpha = \ldots\) (= awrt 0.464) | M1 | Uses correct method with their \(a\) and \(b\); allow degrees: \(\alpha=\) awrt \(26.6°\) |
| \(f(x) = 2\sqrt{5}\sin(2x + 0.464) - 1\) | A1 | Full marks only if correct \(a\), \(b\), \(c\) found in (a) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Maximum value \(= 2\sqrt{5} - 1\) | B1ft | e.g. \(\sqrt{20}-1\); follow through on \(R+c\) as long as \(R\) correctly found; \(c\) cannot be 0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Solves \(2x + \alpha = \frac{5\pi}{2} \Rightarrow x = \ldots\) | M1 | Using their \(\alpha\) from (b); \(\frac{5\pi}{2} = 7.85\ldots\) may be used |
| \(x =\) awrt \(3.69\) (or \(x =\) awrt \(3.70\)) | A1 | If several angles found, must indicate final answer |
# Question 4:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| States or uses $\sin 2x = 2\sin x\cos x$ or $\cos 2x = \pm2\cos^2 x \pm 1$ | M1 | Either identity stated; can be implied by $a=4$, or $b=\pm2$ and $c=-5$ or $-1$ |
| Uses both $\sin 2x = 2\sin x\cos x$ and $\cos 2x = \pm2\cos^2 x \pm 1$ in $f(x)$ | dM1 | Produces form $a\sin 2x + b\cos 2x + c$; condone slips when substituting |
| $f(x) = 4\sin 2x + 2\cos 2x - 1$ | A1 | Correct answer; must be in terms of $x$ |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $R^2 = a^2 + b^2 \Rightarrow R = \sqrt{20}$ or $2\sqrt{5}$ | B1ft | Follow through on their $a$ or $b$; can be implied by correct exact value |
| $\tan\alpha = \frac{b}{a} \Rightarrow \alpha = \ldots$ (= awrt 0.464) | M1 | Uses correct method with their $a$ and $b$; allow degrees: $\alpha=$ awrt $26.6°$ |
| $f(x) = 2\sqrt{5}\sin(2x + 0.464) - 1$ | A1 | Full marks only if correct $a$, $b$, $c$ found in (a) |
## Part (c)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Maximum value $= 2\sqrt{5} - 1$ | B1ft | e.g. $\sqrt{20}-1$; follow through on $R+c$ as long as $R$ correctly found; $c$ cannot be 0 |
## Part (c)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Solves $2x + \alpha = \frac{5\pi}{2} \Rightarrow x = \ldots$ | M1 | Using their $\alpha$ from (b); $\frac{5\pi}{2} = 7.85\ldots$ may be used |
| $x =$ awrt $3.69$ (or $x =$ awrt $3.70$) | A1 | If several angles found, must indicate final answer |
---4.
$$f ( x ) = 8 \sin x \cos x + 4 \cos ^ { 2 } x - 3$$
\begin{enumerate}[label=(\alph*)]
\item Write $\mathrm { f } ( x )$ in the form
$$a \sin 2 x + b \cos 2 x + c$$
where $a$, $b$ and $c$ are integers to be found.
\item Use the answer to part (a) to write $\mathrm { f } ( x )$ in the form
$$R \sin ( 2 x + \alpha ) + c$$
where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$\\
Give the exact value of $R$ and give the value of $\alpha$ in radians to 3 significant figures.
\item Hence, or otherwise,
\begin{enumerate}[label=(\roman*)]
\item state the maximum value of $\mathrm { f } ( x )$
\item find the second smallest positive value of $x$ at which a maximum value of $\mathrm { f } ( x )$ occurs. Give your answer to 3 significant figures.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel P3 2024 Q4 [9]}}