| Exam Board | Edexcel |
|---|---|
| Module | F2 (Further Pure Mathematics 2) |
| Year | 2024 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area of region with line boundary |
| Difficulty | Challenging +1.2 This is a standard polar area calculation requiring integration of r²/2 with a moderately complex integrand (10cos θ + tan θ)². While it involves Further Maths content (polar coordinates) and requires careful algebraic manipulation of trigonometric identities and integration, it follows a completely standard template with clear boundaries and a given answer form to verify against. The algebraic work is somewhat tedious but routine for F2 students. |
| Spec | 1.08h Integration by substitution4.09c Area enclosed: by polar curve |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(r^2 = 100\cos^2\theta + 20\cos\theta\tan\theta + \tan^2\theta\) | B1 | Any correct expression for \(r^2\) |
| \(\left\{\frac{1}{2}\right\}\int_0^{\frac{\pi}{3}} r^2\, d\theta = \left\{\frac{1}{2}\right\}\int_0^{\frac{\pi}{3}}\left(100\cos^2\theta + 20\sin\theta + \tan^2\theta\right)\{d\theta\}\) | M1 | Attempts area formula with their \(r^2\); condone missing \(\frac{1}{2}\) and limits not required |
| \(= \frac{1}{2}\int_0^{\frac{\pi}{3}}\left(50(1+\cos 2\theta) + 20\sin\theta + \sec^2\theta - 1\right)\{d\theta\}\) | M1 | Uses \(\cos^2\theta = \pm\frac{1}{2} \pm \frac{1}{2}\cos 2\theta\) in \(r^2\) |
| M1 | Uses both \(\cos^2\theta = \pm\frac{1}{2} \pm \frac{1}{2}\cos 2\theta\) and \(\tan^2\theta = \pm\sec^2\theta \pm 1\) in \(r^2\) | |
| A1 | Correct integral following \(\cos^2\theta = \frac{1}{2}+\frac{1}{2}\cos 2\theta\) and \(\tan^2\theta = \sec^2\theta - 1\); \(\cos\theta\tan\theta\) must be written as \(\sin\theta\) | |
| \(= \frac{1}{2}\left[49\theta + 25\sin 2\theta - 20\cos\theta + \tan\theta\right]_0^{\frac{\pi}{3}}\) | M1 | Achieves three of: \(k\to k\theta\), \(\cos 2\theta\to\ldots\sin 2\theta\), \(\sin\theta\to\ldots\cos\theta\), \(\sec^2\theta\to\ldots\tan\theta\) |
| A1 | Correct integration including \(\frac{1}{2}\); limits not required | |
| \(= \frac{1}{2}\left(\frac{49\pi}{3} + 25\sin\frac{2\pi}{3} - 20\cos\frac{\pi}{3} + \tan\frac{\pi}{3} - (0+0-20+0)\right)\) | M1 | Applies correct limits to expression of form \(p\theta + q\sin 2\theta + r\cos\theta + s\tan\theta\), \((p,q,r,s\neq 0)\) |
| \(= \frac{1}{12}\left(98\pi + 81\sqrt{3} + 60\right)\) | A1 | Correct answer; values for \(a\), \(b\) and \(c\) |
# Question 5:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $r^2 = 100\cos^2\theta + 20\cos\theta\tan\theta + \tan^2\theta$ | B1 | Any correct expression for $r^2$ |
| $\left\{\frac{1}{2}\right\}\int_0^{\frac{\pi}{3}} r^2\, d\theta = \left\{\frac{1}{2}\right\}\int_0^{\frac{\pi}{3}}\left(100\cos^2\theta + 20\sin\theta + \tan^2\theta\right)\{d\theta\}$ | M1 | Attempts area formula with their $r^2$; condone missing $\frac{1}{2}$ and limits not required |
| $= \frac{1}{2}\int_0^{\frac{\pi}{3}}\left(50(1+\cos 2\theta) + 20\sin\theta + \sec^2\theta - 1\right)\{d\theta\}$ | M1 | Uses $\cos^2\theta = \pm\frac{1}{2} \pm \frac{1}{2}\cos 2\theta$ in $r^2$ |
| | M1 | Uses both $\cos^2\theta = \pm\frac{1}{2} \pm \frac{1}{2}\cos 2\theta$ **and** $\tan^2\theta = \pm\sec^2\theta \pm 1$ in $r^2$ |
| | A1 | Correct integral following $\cos^2\theta = \frac{1}{2}+\frac{1}{2}\cos 2\theta$ and $\tan^2\theta = \sec^2\theta - 1$; $\cos\theta\tan\theta$ must be written as $\sin\theta$ |
| $= \frac{1}{2}\left[49\theta + 25\sin 2\theta - 20\cos\theta + \tan\theta\right]_0^{\frac{\pi}{3}}$ | M1 | Achieves **three** of: $k\to k\theta$, $\cos 2\theta\to\ldots\sin 2\theta$, $\sin\theta\to\ldots\cos\theta$, $\sec^2\theta\to\ldots\tan\theta$ |
| | A1 | Correct integration including $\frac{1}{2}$; limits not required |
| $= \frac{1}{2}\left(\frac{49\pi}{3} + 25\sin\frac{2\pi}{3} - 20\cos\frac{\pi}{3} + \tan\frac{\pi}{3} - (0+0-20+0)\right)$ | M1 | Applies correct limits to expression of form $p\theta + q\sin 2\theta + r\cos\theta + s\tan\theta$, $(p,q,r,s\neq 0)$ |
| $= \frac{1}{12}\left(98\pi + 81\sqrt{3} + 60\right)$ | A1 | Correct answer; values for $a$, $b$ and $c$ |
---5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{d5eb0ca8-92ba-466f-84f5-8fc36c168695-16_669_817_296_625}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of the curve with polar equation
$$r = 10 \cos \theta + \tan \theta \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$
The point $P$ lies on the curve where $\theta = \frac { \pi } { 3 }$\\
The region $R$, shown shaded in Figure 1, is bounded by the initial line, the curve and the line $O P$, where $O$ is the pole.
Use algebraic integration to show that the exact area of $R$ is
$$\frac { 1 } { 12 } ( a \pi + b \sqrt { 3 } + c )$$
where $a$, $b$ and $c$ are integers to be determined.
\hfill \mbox{\textit{Edexcel F2 2024 Q5 [9]}}