| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Polar curve intersection points |
| Difficulty | Challenging +1.2 This is a Further Maths polar coordinates question requiring students to find intersection points by solving 3 + 2cos(θ) = 8cos²(θ), which rearranges to a quadratic in cos(θ). While it involves multiple steps (algebraic manipulation, solving quadratic, finding all θ values in the given range, computing corresponding r values), the techniques are standard for Further Maths students. The 6 marks reflect computational work rather than conceptual difficulty. Slightly above average due to being Further Maths content and requiring careful attention to multiple solutions. |
| Spec | 4.09b Sketch polar curves: r = f(theta) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Maximum value of \(r = 5\) | B1 | States max value for \(r\) |
| Minimum value of \(r = 1\) | B1 | States min value for \(r\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Simple closed curve enclosing pole | M1 | Draws simple closed curve enclosing pole |
| Correct shape with dimple (not cusp) when \(\theta = \pi\) | A1 | Must show dimple, not cusp |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(3 + 2\cos\theta = 8\cos^2\theta\) | M1 | Equates the two expressions |
| \(8\cos^2\theta - 2\cos\theta - 3 = 0\), \((4\cos\theta - 3)(2\cos\theta + 1) = 0\) | M1 | Solves quadratic; FT only if first M1 awarded |
| \(\cos\theta = \frac{3}{4}\), \(\cos\theta = -\frac{1}{2}\); \(\theta = 0.723\) or \(\frac{2\pi}{3}\), \(\theta = 5.56\) or \(\frac{4\pi}{3}\) | A1F | Two values of \(\theta\) for each \(\cos\theta\); FT if both M1s awarded |
| Substitutes \(\cos\theta\) into polar equation to find \(r\) | M1 | FT their \(\cos\theta\) only if both M1s awarded |
| \(\cos\theta = \frac{3}{4} \Rightarrow r = \frac{9}{2}\); \(\cos\theta = -\frac{1}{2} \Rightarrow r = 2\) | A1F | Both values correct; FT their \(\cos\theta\) if both M1s awarded |
| Four intersection points: \(\left(\frac{9}{2}, 0.723\right)\), \(\left(\frac{9}{2}, 5.56\right)\), \(\left(2, \frac{2\pi}{3}\right)\), \(\left(2, \frac{4\pi}{3}\right)\) | R1 | Deduces four values of \(\theta\); points in required form |
## Question 8(a)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Maximum value of $r = 5$ | B1 | States max value for $r$ |
| Minimum value of $r = 1$ | B1 | States min value for $r$ |
## Question 8(a)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Simple closed curve enclosing pole | M1 | Draws simple closed curve enclosing pole |
| Correct shape with dimple (not cusp) when $\theta = \pi$ | A1 | Must show dimple, not cusp |
## Question 8(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $3 + 2\cos\theta = 8\cos^2\theta$ | M1 | Equates the two expressions |
| $8\cos^2\theta - 2\cos\theta - 3 = 0$, $(4\cos\theta - 3)(2\cos\theta + 1) = 0$ | M1 | Solves quadratic; FT only if first M1 awarded |
| $\cos\theta = \frac{3}{4}$, $\cos\theta = -\frac{1}{2}$; $\theta = 0.723$ or $\frac{2\pi}{3}$, $\theta = 5.56$ or $\frac{4\pi}{3}$ | A1F | Two values of $\theta$ for each $\cos\theta$; FT if both M1s awarded |
| Substitutes $\cos\theta$ into polar equation to find $r$ | M1 | FT their $\cos\theta$ only if both M1s awarded |
| $\cos\theta = \frac{3}{4} \Rightarrow r = \frac{9}{2}$; $\cos\theta = -\frac{1}{2} \Rightarrow r = 2$ | A1F | Both values correct; FT their $\cos\theta$ if both M1s awarded |
| Four intersection points: $\left(\frac{9}{2}, 0.723\right)$, $\left(\frac{9}{2}, 5.56\right)$, $\left(2, \frac{2\pi}{3}\right)$, $\left(2, \frac{4\pi}{3}\right)$ | R1 | Deduces four values of $\theta$; points in required form |
---8 A curve has polar equation $r = 3 + 2 \cos \theta$, where $0 \leq \theta < 2 \pi$\\
8
\begin{enumerate}[label=(\alph*)]
\item (i) State the maximum and minimum values of $r$.\\[0pt]
[2 marks]\\
L\\
8 (a) (ii) Sketch the curve.\\
\includegraphics[max width=\textwidth, alt={}, center]{e61d0202-49c9-4ed9-9fa3-f10734e17463-12_77_832_2037_651}
8
\item The curve $r = 3 + 2 \cos \theta$ intersects the curve with polar equation $r = 8 \cos ^ { 2 } \theta$, where $0 \leq \theta < 2 \pi$
Find all of the points of intersection of the two curves in the form $[ r , \theta ]$.\\[0pt]
[6 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 Q8 [8]}}