| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Year | 2023 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Convert Cartesian to polar equation |
| Difficulty | Moderate -0.5 Parts (a) and (b) are trivial recall of standard polar coordinate definitions. Part (c)(i) is a routine algebraic manipulation using r²=x²+y² and x=r cos θ, requiring careful but straightforward algebra. Part (c)(ii) requires recognizing the symmetry between the two equations (swapping x and y), which is a standard transformation. This is easier than average for Further Maths as it's mostly procedural with minimal problem-solving insight required. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02j Cubic/quartic equations: conjugate pairs and factor theorem4.02q De Moivre's theorem: multiple angle formulae |
| Answer | Marks | Guidance |
|---|---|---|
| \(r = \sqrt{x^2 + y^2}\) | B1 | Writes a correct expression for \(r\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = r\cos\theta\) | B1 | Writes a correct expression for \(x\) |
| Answer | Marks | Guidance |
|---|---|---|
| Substitutes (a) and (b) to form equation in \(x\) and \(y\) only: \(r(2 + \cos\theta) = 1 \Rightarrow 2r + r\cos\theta = 1\) | M1 | |
| \(\Rightarrow 2\sqrt{x^2+y^2} + x = 1 \Rightarrow 2\sqrt{x^2+y^2} = 1-x\) | M1 | Must be equation in terms of \(x\) and \(y\) only; correctly removes square root |
| \(\Rightarrow 4(x^2+y^2) = (1-x)^2 \Rightarrow 4x^2+4y^2 = 1-2x+x^2 \Rightarrow 4y^2 = 1-2x-3x^2\) | A1 | Correct equation without roots |
| \(\Rightarrow 4y^2 = (1-3x)(1+x)\) | A1 | Correct equation in required form |
| Answer | Marks | Guidance |
|---|---|---|
| Identifies a reflection; must also specify a line of reflection (which could be wrong for this mark), or identifies a rotation about \((0,0)\) | M1 | |
| \(y\) is replaced with \(x\), and vice versa; Reflection in \(y = x\) | A1 | Accept \(90°\) anticlockwise rotation about \((0,0)\); if rotation direction included it must be anticlockwise |
## Question 11:
**Part 11(a):**
| $r = \sqrt{x^2 + y^2}$ | B1 | Writes a correct expression for $r$ |
**Part 11(b):**
| $x = r\cos\theta$ | B1 | Writes a correct expression for $x$ |
**Part 11(c)(i):**
| Substitutes (a) and (b) to form equation in $x$ and $y$ only: $r(2 + \cos\theta) = 1 \Rightarrow 2r + r\cos\theta = 1$ | M1 | |
| $\Rightarrow 2\sqrt{x^2+y^2} + x = 1 \Rightarrow 2\sqrt{x^2+y^2} = 1-x$ | M1 | Must be equation in terms of $x$ and $y$ only; correctly removes square root |
| $\Rightarrow 4(x^2+y^2) = (1-x)^2 \Rightarrow 4x^2+4y^2 = 1-2x+x^2 \Rightarrow 4y^2 = 1-2x-3x^2$ | A1 | Correct equation without roots |
| $\Rightarrow 4y^2 = (1-3x)(1+x)$ | A1 | Correct equation in required form |
**Part 11(c)(ii):**
| Identifies a reflection; must also specify a line of reflection (which could be wrong for this mark), or identifies a rotation about $(0,0)$ | M1 | |
| $y$ is replaced with $x$, and vice versa; Reflection in $y = x$ | A1 | Accept $90°$ anticlockwise rotation about $(0,0)$; if rotation direction included it must be anticlockwise |
---11 A point has Cartesian coordinates $( x , y )$ and polar coordinates $( r , \theta )$ where $r \geq 0$ and $- \pi < \theta \leq \pi$
11
\begin{enumerate}[label=(\alph*)]
\item Express $r$ in terms of $x$ and $y$
11
\item Express $x$ in terms of $r$ and $\theta$
11
\item The curve $C _ { 1 }$ has the polar equation
$$r ( 2 + \cos \theta ) = 1 \quad - \pi < \theta \leq \pi$$
11 (c) (i) Show that the Cartesian equation of $C _ { 1 }$ can be written as
$$a y ^ { 2 } = ( 1 + b x ) ( 1 + x )$$
where $a$ and $b$ are integers to be determined.\\
11 (c) (ii) The curve $C _ { 2 }$ has the Cartesian equation
$$a x ^ { 2 } = ( 1 + b y ) ( 1 + y )$$
where $a$ and $b$ take the same values as in part (c)(i).
Describe fully a single transformation that maps the curve $C _ { 1 }$ onto the curve $C _ { 2 }$
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 2023 Q11 [8]}}