| Exam Board | OCR MEI |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2007 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Tangent parallel/perpendicular to initial line |
| Difficulty | Challenging +1.2 This is a structured multi-part question on polar coordinates requiring calculator sketching, differentiation of parametric forms, and geometric reasoning. While it covers several techniques (implicit differentiation, cosine rule, inequalities), each part is guided and uses standard Further Maths methods without requiring novel insights. The difficulty is above average due to the Further Maths content and multi-step nature, but the scaffolding and routine application of techniques keep it from being highly challenging. |
| Spec | 1.07s Parametric and implicit differentiation4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta) |
| Answer | Marks | Guidance |
|---|---|---|
| Part (i): Sketches for \(k=1,\ 1.5,\ 4\) | B1×5 | Correct shapes for each |
| Part (ii): Special feature when \(k=1\) | Cardioid | B1 |
| Answer | Marks |
|---|---|
| \(k=1\): 2, \(k=1.5\): 3, \(k=4\): 3 (or as graphically determined) | B1, B1 |
| Answer | Marks |
|---|---|
| \(\frac{dx}{d\theta} = -(k+\cos\theta)\sin\theta - \cos\theta\sin\theta = -\sin\theta(k+2\cos\theta)\) | M1, A1 |
| \(\frac{dx}{d\theta}=0 \Rightarrow \sin\theta=0\) or \(\cos\theta=-\frac{k}{2}\) | M1 |
| Just two points: \(\frac{k}{2}>1 \Rightarrow k>2\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(d^2 = (k+\cos\theta)^2 + 1 - 2(k+\cos\theta)\cos\theta\) | M1 | Cosine rule |
| \(= k^2 + 2k\cos\theta+\cos^2\theta+1-2k\cos\theta-2\cos^2\theta\) | M1 | |
| \(= k^2 + 1 - \cos^2\theta\) | A1 | |
| Since \(0\leq\cos^2\theta\leq 1\): \(k^2 \leq d^2 \leq k^2+1\) | A1 | |
| Part (vi): When \(k\) large, \(k^2 \leq d^2 \leq k^2+1\), so \(d \approx k\) (constant), curve nearly circular | B1, B1 |
## Question 5 (Option 2):
**Part (i):** Sketches for $k=1,\ 1.5,\ 4$ | B1×5 | Correct shapes for each |
**Part (ii):** Special feature when $k=1$ | Cardioid | B1 |
**Part (iii):** Number of points where tangent parallel to $y$-axis
| $k=1$: 2, $k=1.5$: 3, $k=4$: 3 (or as graphically determined) | B1, B1 | |
**Part (iv):** $x = r\cos\theta = (k+\cos\theta)\cos\theta$
| $\frac{dx}{d\theta} = -(k+\cos\theta)\sin\theta - \cos\theta\sin\theta = -\sin\theta(k+2\cos\theta)$ | M1, A1 | |
| $\frac{dx}{d\theta}=0 \Rightarrow \sin\theta=0$ or $\cos\theta=-\frac{k}{2}$ | M1 | |
| Just two points: $\frac{k}{2}>1 \Rightarrow k>2$ | A1 | |
**Part (v):** $d^2 = r^2 + 1 - 2r\cos\theta$
| $d^2 = (k+\cos\theta)^2 + 1 - 2(k+\cos\theta)\cos\theta$ | M1 | Cosine rule |
| $= k^2 + 2k\cos\theta+\cos^2\theta+1-2k\cos\theta-2\cos^2\theta$ | M1 | |
| $= k^2 + 1 - \cos^2\theta$ | A1 | |
| Since $0\leq\cos^2\theta\leq 1$: $k^2 \leq d^2 \leq k^2+1$ | A1 | |
**Part (vi):** When $k$ large, $k^2 \leq d^2 \leq k^2+1$, so $d \approx k$ (constant), curve nearly circular | B1, B1 | |5 Cartesian coordinates $( x , y )$ and polar coordinates $( r , \theta )$ are set up in the usual way, with the pole at the origin and the initial line along the positive $x$-axis, so that $x = r \cos \theta$ and $y = r \sin \theta$.
A curve has polar equation $r = k + \cos \theta$, where $k$ is a constant with $k \geqslant 1$.\\
(i) Use your graphical calculator to obtain sketches of the curve in the three cases
$$k = 1 , k = 1.5 \text { and } k = 4$$
(ii) Name the special feature which the curve has when $k = 1$.\\
(iii) For each of the three cases, state the number of points on the curve at which the tangent is parallel to the $y$-axis.\\
(iv) Express $x$ in terms of $k$ and $\theta$, and find $\frac { \mathrm { d } x } { \mathrm {~d} \theta }$. Hence find the range of values of $k$ for which there are just two points on the curve where the tangent is parallel to the $y$-axis.
The distance between the point ( $r , \theta$ ) on the curve and the point ( 1,0 ) on the $x$-axis is $d$.\\
(v) Use the cosine rule to express $d ^ { 2 }$ in terms of $k$ and $\theta$, and deduce that $k ^ { 2 } \leqslant d ^ { 2 } \leqslant k ^ { 2 } + 1$.\\
(vi) Hence show that, when $k$ is large, the shape of the curve is very nearly circular.
\hfill \mbox{\textit{OCR MEI FP2 2007 Q5 [18]}}