| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2022 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Maximum/minimum distance from pole or line |
| Difficulty | Challenging +1.8 This is a challenging Further Maths polar coordinates question requiring: (a) sketching an unusual polar curve and finding maximum distance, (b) polar area integration, (c) optimization using perpendicular distance from a line in polar form leading to a transcendental equation. The combination of inverse tan, polar geometry, and implicit differentiation for optimization makes this substantially harder than typical A-level questions, though the verification in part (c) is routine. |
| Spec | 1.09a Sign change methods: locate roots4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Correct sketch with correct domain, \(r\) strictly increasing | B1 | Correct domain, \(r\) strictly increasing |
| Correct gradient when \(\theta = 0\) and \(\theta = 2\) | B1 | Correct gradient when \(\theta = 0\) and \(\theta = 2\) |
| Maximum distance of \(C\) from the pole is \(\sqrt{\frac{1}{4}\pi}\) | B1 | Must be exact: \(\frac{1}{2}\sqrt{\pi}\) |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{2}\int_0^2 \tan^{-1}\left(\frac{1}{2}\theta\right) d\theta\) | M1 | Uses \(\frac{1}{2}\int r^2\, d\theta\) with correct limits |
| \(\left[\frac{1}{2}\theta\tan^{-1}\left(\frac{1}{2}\theta\right)\right]_0^2 - \int_0^2 \frac{\theta}{\theta^2+4}\, d\theta\) | M1 A1 | Applies integration by parts |
| \(\frac{1}{2}\left[\theta\tan^{-1}\left(\frac{1}{2}\theta\right) - \ln(\theta^2+4)\right]_0^2\) | A1 | |
| \(\frac{1}{4}\pi - \frac{1}{2}\ln 8 + \frac{1}{2}\ln 4 = \frac{1}{4}\pi - \frac{1}{2}\ln 2\) | A1 | Must be exact |
| 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x = \left(\tan^{-1}\left(\frac{1}{2}\theta\right)\right)^{\frac{1}{2}}\cos\theta\) | B1 | Uses \(x = r\cos\theta\) |
| \(\frac{dx}{d\theta} = -\left(\tan^{-1}\left(\frac{1}{2}\theta\right)\right)^{\frac{1}{2}}\sin\theta + \cos\theta\left(\tan^{-1}\left(\frac{1}{2}\theta\right)\right)^{-\frac{1}{2}}\left(\theta^2+4\right)^{-1} = 0\) | M1 A1 | Sets derivative equal to zero |
| \(\left(\theta^2+4\right)\tan^{-1}\left(\frac{1}{2}\theta\right)\sin\theta - \cos\theta = 0\) | A1 | AG |
| \((0.6^2+4)\tan^{-1}(0.3)\sin 0.6 - \cos 0.6 = -0.108\) and \((0.7^2+4)\tan^{-1}(0.35)\sin 0.7 - \cos 0.7 = 0.209\) | B1 | Shows sign change |
| 5 |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct sketch with correct domain, $r$ strictly increasing | B1 | Correct domain, $r$ strictly increasing |
| Correct gradient when $\theta = 0$ and $\theta = 2$ | B1 | Correct gradient when $\theta = 0$ and $\theta = 2$ |
| Maximum distance of $C$ from the pole is $\sqrt{\frac{1}{4}\pi}$ | B1 | Must be exact: $\frac{1}{2}\sqrt{\pi}$ |
| | **3** | |
## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{2}\int_0^2 \tan^{-1}\left(\frac{1}{2}\theta\right) d\theta$ | M1 | Uses $\frac{1}{2}\int r^2\, d\theta$ with correct limits |
| $\left[\frac{1}{2}\theta\tan^{-1}\left(\frac{1}{2}\theta\right)\right]_0^2 - \int_0^2 \frac{\theta}{\theta^2+4}\, d\theta$ | M1 A1 | Applies integration by parts |
| $\frac{1}{2}\left[\theta\tan^{-1}\left(\frac{1}{2}\theta\right) - \ln(\theta^2+4)\right]_0^2$ | A1 | |
| $\frac{1}{4}\pi - \frac{1}{2}\ln 8 + \frac{1}{2}\ln 4 = \frac{1}{4}\pi - \frac{1}{2}\ln 2$ | A1 | Must be exact |
| | **5** | |
## Question 6(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x = \left(\tan^{-1}\left(\frac{1}{2}\theta\right)\right)^{\frac{1}{2}}\cos\theta$ | B1 | Uses $x = r\cos\theta$ |
| $\frac{dx}{d\theta} = -\left(\tan^{-1}\left(\frac{1}{2}\theta\right)\right)^{\frac{1}{2}}\sin\theta + \cos\theta\left(\tan^{-1}\left(\frac{1}{2}\theta\right)\right)^{-\frac{1}{2}}\left(\theta^2+4\right)^{-1} = 0$ | M1 A1 | Sets derivative equal to zero |
| $\left(\theta^2+4\right)\tan^{-1}\left(\frac{1}{2}\theta\right)\sin\theta - \cos\theta = 0$ | A1 | AG |
| $(0.6^2+4)\tan^{-1}(0.3)\sin 0.6 - \cos 0.6 = -0.108$ and $(0.7^2+4)\tan^{-1}(0.35)\sin 0.7 - \cos 0.7 = 0.209$ | B1 | Shows sign change |
| | **5** | |6 The curve $C$ has polar equation $r ^ { 2 } = \tan ^ { - 1 } \left( \frac { 1 } { 2 } \theta \right)$, where $0 \leqslant \theta \leqslant 2$.
\begin{enumerate}[label=(\alph*)]
\item Sketch $C$ and state, in exact form, the greatest distance of a point on $C$ from the pole.
\item Find the exact value of the area of the region bounded by $C$ and the half-line $\theta = 2$.\\
Now consider the part of $C$ where $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.
\item Show that, at the point furthest from the half-line $\theta = \frac { 1 } { 2 } \pi$,
$$\left( \theta ^ { 2 } + 4 \right) \tan ^ { - 1 } \left( \frac { 1 } { 2 } \theta \right) \sin \theta - \cos \theta = 0$$
and verify that this equation has a root between 0.6 and 0.7 .\\
$7 \quad$ The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & 3 \\ 4 & k & 6 \\ 7 & 8 & 9 \end{array} \right)$.\\
(a) Find the set of values of $k$ for which $\mathbf { A }$ is non-singular.\\
(b) Given that $\mathbf { A }$ is non-singular, find, in terms of $k$, the entries in the top row of $\mathbf { A } ^ { - 1 }$.\\
(c) Given that $\mathbf { B } = \left( \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \end{array} \right)$, give an example of a matrix $\mathbf { C }$ such that $\mathbf { B A C } = \left( \begin{array} { l l } 2 & 1 \\ k & 4 \end{array} \right)$.
\item Find the set of values of $k$ for which the transformation in the $x - y$ plane represented by $\left( \begin{array} { l l } 2 & 1 \\ k & 4 \end{array} \right)$ has two distinct invariant lines through the origin.\\
If you use the following page to complete the answer to any question, the question number must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2022 Q6 [13]}}