| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Year | 2024 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Show polar curve has Cartesian form |
| Difficulty | Standard +0.8 This is a Further Maths polar coordinates question requiring algebraic manipulation to convert to Cartesian form. Students must square both sides, substitute r² = x² + y², and use x = r cos θ, y = r sin θ to obtain an ellipse equation. While the steps are systematic, the manipulation requires careful handling of the denominator and recognizing the final form. The 4 marks and Further Maths context place it moderately above average difficulty. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Correctly removes the square root sign, or correctly isolates \(r^2\) (or \(r\)) if moving from Cartesian to polar form | M1 (3.1a) | |
| Correctly uses \(x = r\cos\theta\) or \(y = r\sin\theta\) | M1 (1.1a) | |
| Correctly uses \(x = r\cos\theta\) and \(y = r\sin\theta\) to obtain \((r\cos\theta)^2 + 4(r\sin\theta)^2 = 4\) | M1 (1.1a) | |
| Completes reasoned argument to obtain \(\frac{x^2}{2^2} + \frac{y^2}{1^2} = 1\); accept \(\frac{x^2}{4} + y^2 = 1\) if \(a=2\) and \(b=1\) seen | A1 (3.2a) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Draws an ellipse centred on the origin; accept a circle | M1 (1.1a) | |
| Identifies correct intercepts at \(2\) and \(-2\) and \(1\) and \(-1\); FT their \(a\) and \(b\); accept \(\pm a\) and \(\pm b\) if no values found in part (a) | A1F (1.1b) |
## Question 16(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Correctly removes the square root sign, or correctly isolates $r^2$ (or $r$) if moving from Cartesian to polar form | M1 (3.1a) | |
| Correctly uses $x = r\cos\theta$ **or** $y = r\sin\theta$ | M1 (1.1a) | |
| Correctly uses $x = r\cos\theta$ **and** $y = r\sin\theta$ to obtain $(r\cos\theta)^2 + 4(r\sin\theta)^2 = 4$ | M1 (1.1a) | |
| Completes reasoned argument to obtain $\frac{x^2}{2^2} + \frac{y^2}{1^2} = 1$; accept $\frac{x^2}{4} + y^2 = 1$ if $a=2$ and $b=1$ seen | A1 (3.2a) | |
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## Question 16(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Draws an ellipse centred on the origin; accept a circle | M1 (1.1a) | |
| Identifies correct intercepts at $2$ and $-2$ and $1$ and $-1$; FT their $a$ and $b$; accept $\pm a$ and $\pm b$ if no values found in part (a) | A1F (1.1b) | |
---16 The curve $C$ has the polar equation
$$r = \frac { 2 } { \sqrt { \cos ^ { 2 } \theta + 4 \sin ^ { 2 } \theta } } \quad - \pi < \theta \leq \pi$$
16
\begin{enumerate}[label=(\alph*)]
\item Show that the Cartesian equation of $C$ can be written as
$$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$
where $a$ and $b$ are positive integers to be determined.\\[0pt]
[4 marks]\\
16
\item Hence sketch the graph of $C$ on the axes below.
Indicate the value of any intercepts of the curve with the axes.\\
\includegraphics[max width=\textwidth, alt={}, center]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-23_1122_1121_452_447}
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 2024 Q16 [6]}}