| Exam Board | OCR MEI |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2009 |
| Session | January |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area of region with line boundary |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question combining polar area integration (requiring the formula ½∫r²dθ and integration of tan²θ) with eigenvalue/eigenvector computation and diagonalization. While each component uses standard techniques, the polar area calculation with tan θ requires careful setup and the full question demands proficiency across multiple FM topics within time constraints. |
| Spec | 4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Curve with \(r\) increasing with \(\theta\) | G1 | \(r\) increasing with \(\theta\) |
| Correct for \(0 \leq \theta \leq \pi/3\) (ignore extra) | G1 | |
| Gradient less than 1 at O | G1 | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Area \(= \int_0^{\frac{\pi}{4}} \dfrac{1}{2}r^2\,d\theta = \dfrac{1}{2}a^2\int_0^{\frac{\pi}{4}}\tan^2\theta\,d\theta\) | M1 | Integral expression involving \(\tan^2\theta\) |
| \(= \dfrac{1}{2}a^2\int_0^{\frac{\pi}{4}}\sec^2\theta - 1\,d\theta\) | M1 | Attempt to express \(\tan^2\theta\) in terms of \(\sec^2\theta\) |
| \(= \dfrac{1}{2}a^2\left[\tan\theta - \theta\right]_0^{\frac{\pi}{4}}\) | A1 | \(\tan\theta - \theta\) and limits \(0,\ \dfrac{\pi}{4}\) |
| \(= \dfrac{1}{2}a^2\left(1 - \dfrac{\pi}{4}\right)\) | A1 | A0 if e.g. triangle − this answer |
| Region marked on graph | G1 | |
| Total | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Characteristic equation: \((0.2-\lambda)(0.7-\lambda) - 0.24 = 0\) | M1 | |
| \(\lambda^2 - 0.9\lambda - 0.1 = 0 \Rightarrow \lambda = 1,\ -0.1\) | A1 | |
| When \(\lambda=1\): \(\begin{pmatrix}-0.8 & 0.8 \\ 0.3 & -0.3\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}\) | \((\mathbf{M}-\lambda\mathbf{I})\mathbf{x}=\mathbf{x}\) M0 below | |
| \(-0.8x + 0.8y = 0,\ 0.3x - 0.3y = 0\) | M1 | At least one equation relating \(x\) and \(y\) |
| \(x - y = 0\), eigenvector \(\begin{pmatrix}1\\1\end{pmatrix}\) o.e. | A1 | |
| When \(\lambda=-0.1\): \(\begin{pmatrix}0.3 & 0.8 \\ 0.3 & 0.8\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}\) | ||
| \(0.3x + 0.8y = 0\) | M1 | At least one equation relating \(x\) and \(y\) |
| Eigenvector \(\begin{pmatrix}8\\-3\end{pmatrix}\) o.e. | A1 | |
| Total | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\mathbf{Q} = \begin{pmatrix}1 & 8 \\ 1 & -3\end{pmatrix}\) | B1ft | B0 if Q is singular. Must label correctly |
| \(\mathbf{D} = \begin{pmatrix}1 & 0 \\ 0 & -0.1\end{pmatrix}\) | B1ft, B1 | If order consistent. Dep on B1B1 earned |
| Total | 3 |
# Question 3:
## Part (a)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| Curve with $r$ increasing with $\theta$ | G1 | $r$ increasing with $\theta$ |
| Correct for $0 \leq \theta \leq \pi/3$ (ignore extra) | G1 | |
| Gradient less than 1 at O | G1 | |
| **Total** | **3** | |
## Part (a)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Area $= \int_0^{\frac{\pi}{4}} \dfrac{1}{2}r^2\,d\theta = \dfrac{1}{2}a^2\int_0^{\frac{\pi}{4}}\tan^2\theta\,d\theta$ | M1 | Integral expression involving $\tan^2\theta$ |
| $= \dfrac{1}{2}a^2\int_0^{\frac{\pi}{4}}\sec^2\theta - 1\,d\theta$ | M1 | Attempt to express $\tan^2\theta$ in terms of $\sec^2\theta$ |
| $= \dfrac{1}{2}a^2\left[\tan\theta - \theta\right]_0^{\frac{\pi}{4}}$ | A1 | $\tan\theta - \theta$ and limits $0,\ \dfrac{\pi}{4}$ |
| $= \dfrac{1}{2}a^2\left(1 - \dfrac{\pi}{4}\right)$ | A1 | A0 if e.g. triangle − this answer |
| Region marked on graph | G1 | |
| **Total** | **5** | |
## Part (b)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| Characteristic equation: $(0.2-\lambda)(0.7-\lambda) - 0.24 = 0$ | M1 | |
| $\lambda^2 - 0.9\lambda - 0.1 = 0 \Rightarrow \lambda = 1,\ -0.1$ | A1 | |
| When $\lambda=1$: $\begin{pmatrix}-0.8 & 0.8 \\ 0.3 & -0.3\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}$ | | $(\mathbf{M}-\lambda\mathbf{I})\mathbf{x}=\mathbf{x}$ M0 below |
| $-0.8x + 0.8y = 0,\ 0.3x - 0.3y = 0$ | M1 | At least one equation relating $x$ and $y$ |
| $x - y = 0$, eigenvector $\begin{pmatrix}1\\1\end{pmatrix}$ o.e. | A1 | |
| When $\lambda=-0.1$: $\begin{pmatrix}0.3 & 0.8 \\ 0.3 & 0.8\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}$ | | |
| $0.3x + 0.8y = 0$ | M1 | At least one equation relating $x$ and $y$ |
| Eigenvector $\begin{pmatrix}8\\-3\end{pmatrix}$ o.e. | A1 | |
| **Total** | **6** | |
## Part (b)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\mathbf{Q} = \begin{pmatrix}1 & 8 \\ 1 & -3\end{pmatrix}$ | B1ft | B0 if **Q** is singular. Must label correctly |
| $\mathbf{D} = \begin{pmatrix}1 & 0 \\ 0 & -0.1\end{pmatrix}$ | B1ft, B1 | If order consistent. Dep on B1B1 earned |
| **Total** | **3** | |
---3
\begin{enumerate}[label=(\alph*)]
\item A curve has polar equation $r = a \tan \theta$ for $0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi$, where $a$ is a positive constant.
\begin{enumerate}[label=(\roman*)]
\item Sketch the curve.
\item Find the area of the region between the curve and the line $\theta = \frac { 1 } { 4 } \pi$. Indicate this region on your sketch.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Find the eigenvalues and corresponding eigenvectors for the matrix $\mathbf { M }$ where
$$\mathbf { M } = \left( \begin{array} { l l }
0.2 & 0.8 \\
0.3 & 0.7
\end{array} \right)$$
\item Give a matrix $\mathbf { Q }$ and a diagonal matrix $\mathbf { D }$ such that $\mathbf { M } = \mathbf { Q D } \mathbf { Q } ^ { - 1 }$.
Section B (18 marks)
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP2 2009 Q3 [17]}}