| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2020 |
| Session | Specimen |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Show polar curve has Cartesian form |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question on polar coordinates requiring: (a) sketching a cardioid, (b) computing area using polar integration, and (c) algebraic manipulation to derive the Cartesian form. Part (c) requires systematic substitution of r²=x²+y², x=r cos θ, and algebraic rearrangement—moderately challenging but follows standard techniques for Further Maths polar coordinate conversions. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((4,0)\) and \((0,\pi)\) lie on \(C\) | B1 | Shows \((4,0)\) and \((0,\pi)\) lie on \(C\) |
| Section \(\frac{\pi}{2} < \theta < \pi\) correct | B1 | |
| Sketch showing curve perpendicular to initial line at \((4,0)\), with \(\theta=0\) at 4 and \(\theta=\pi\) at origin | B1 | Correct shape with curve perpendicular to initial line at \((4,0)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{2}\int_0^{\pi}(4 + 8\cos\theta + 4\cos^2\theta)\,d\theta\) | M1 | Uses correct formula |
| \(= \int_0^{\pi}(3 + 4\cos\theta + \cos 2\theta)\,d\theta\) | M1 | Uses double angle formula |
| \(\left[3\theta + 4\sin\theta + \frac{1}{2}\sin 2\theta\right]_0^{\pi} = 3\pi\) | A1A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\sqrt{x^2+y^2} = 2 + 2\dfrac{x}{\sqrt{x^2+y^2}}\) | M1A1 | |
| \(2\sqrt{x^2+y^2} = x^2+y^2-2x \Rightarrow 4(x^2+y^2) = (x^2+y^2-2x)^2\) | A1 | AG Intermediate step required |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(4,0)$ and $(0,\pi)$ lie on $C$ | B1 | Shows $(4,0)$ and $(0,\pi)$ lie on $C$ |
| Section $\frac{\pi}{2} < \theta < \pi$ correct | B1 | |
| Sketch showing curve perpendicular to initial line at $(4,0)$, with $\theta=0$ at 4 and $\theta=\pi$ at origin | B1 | Correct shape with curve perpendicular to initial line at $(4,0)$ |
---
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{2}\int_0^{\pi}(4 + 8\cos\theta + 4\cos^2\theta)\,d\theta$ | M1 | Uses correct formula |
| $= \int_0^{\pi}(3 + 4\cos\theta + \cos 2\theta)\,d\theta$ | M1 | Uses double angle formula |
| $\left[3\theta + 4\sin\theta + \frac{1}{2}\sin 2\theta\right]_0^{\pi} = 3\pi$ | A1A1 | |
---
## Question 3(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\sqrt{x^2+y^2} = 2 + 2\dfrac{x}{\sqrt{x^2+y^2}}$ | M1A1 | |
| $2\sqrt{x^2+y^2} = x^2+y^2-2x \Rightarrow 4(x^2+y^2) = (x^2+y^2-2x)^2$ | A1 | AG Intermediate step required |
# Mark Scheme Extraction3 The curve $C$ has polar equation $r = 2 + 2 \cos \theta$, for $0 \leqslant \theta \leqslant \pi$.\\
(a) Sketch $C$.\\
(b) Find the area of the region enclosed by $C$ and the initial line.\\
(c) Show that the Cartesian equation of $C$ can be expressed as $4 \left( x ^ { 2 } + y ^ { 2 } \right) = \left( x ^ { 2 } + y ^ { 2 } - 2 x \right) ^ { 2 }$.\\
\hfill \mbox{\textit{CAIE Further Paper 1 2020 Q3 [10]}}