| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Year | 2019 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conic sections |
| Type | Hyperbola focus-directrix properties |
| Difficulty | Challenging +1.2 Part (a) is routine recall of hyperbola asymptotes. Part (b) is standard sketching. Part (c) requires setting up a volume of revolution integral about the y-axis (less common than x-axis), rearranging the hyperbola equation for x in terms of y, and evaluating a definite integral involving square roots. While multi-step, the techniques are standard for Further Maths students and the answer is given to verify against. Moderately above average difficulty due to the y-axis rotation and algebraic manipulation required. |
| Spec | 4.08d Volumes of revolution: about x and y axes |
| Answer | Marks |
|---|---|
| 5(a) | Writes the correct equations with a removed. |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | 1.1b | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 5(b) | 𝑥𝑥 = ±2𝑦𝑦 | |
| Draws the correct graph, correctly approaching the asymptotes – mark the intention. | 1.1b | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| (𝑎𝑎,0) (−𝑎𝑎,0) | 1.1b | B1 |
| Q | Marking instructions | AO |
| Answer | Marks |
|---|---|
| 5(c) | Formulates an expression for a volume generated by rotating the hyperbola about an axis |
| Answer | Marks | Guidance |
|---|---|---|
| Condone missing limits and/or and/or (or ). | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| 𝜋𝜋 | 1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Their expression must be of the form or where and are constants. | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Accept substitution of 0 not seen. | 1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| NMS scores 0/5 | 2.1 | R1 |
| Total | 8 | |
| Q | Marking instructions | AO |
Question 5:
--- 5(a) ---
5(a) | Writes the correct equations with a removed.
Accept any equivalent equations, e.g.
1 | 1.1b | B1 | 𝑥𝑥 𝑦𝑦
= ±
𝑎𝑎 2𝑎𝑎
--- 5(b) ---
5(b) | 𝑥𝑥 = ±2𝑦𝑦
Draws the correct graph, correctly approaching the asymptotes – mark the intention. | 1.1b | B1 | 𝑦𝑦 = ±2𝑥𝑥
Writes and
Accept a and –a written close to the intercepts.
(𝑎𝑎,0) (−𝑎𝑎,0) | 1.1b | B1
Q | Marking instructions | AO | Marks | Typical solution
--- 5(c) ---
5(c) | Formulates an expression for a volume generated by rotating the hyperbola about an axis
– must be a clear intent to integrate.
A volume expression must be of the form or where is a polynomial function of
degree 2.
∫f(𝑥𝑥) ∫f(𝑦𝑦) f
Condone missing limits and/or and/or (or ). | 3.1a | M1 | 2
Volume= 𝜋𝜋�𝑥𝑥 d𝑦𝑦
𝑎𝑎 2
𝑦𝑦 2
= 𝜋𝜋� � +𝑎𝑎 � d𝑦𝑦
0 4
3 𝑎𝑎
𝑦𝑦 2
= 𝜋𝜋� +𝑎𝑎 𝑦𝑦�
12 0
3
𝑎𝑎 3
= 𝜋𝜋� +𝑎𝑎 −0�
12
13𝜋𝜋 3
= 𝑎𝑎
12
3
(3 sig figs)
= 3.403𝑎𝑎
3
= 3.40𝑎𝑎
𝜋𝜋 d𝑦𝑦 𝑑𝑑𝑥𝑥
Expresses the volume as or equivalent.
2
𝑦𝑦 2
Must include – may be seen later.
𝜋𝜋∫�4 +𝑎𝑎 � d𝑦𝑦
Condone missing limits and/or .
𝜋𝜋 | 1.1b | A1
𝑑𝑑𝑦𝑦
Correctly integrates their expression.
Their expression must be of the form or where and are constants. | 1.1a | M1
2 2
𝑐𝑐𝑦𝑦 +𝑑𝑑 𝑐𝑐𝑥𝑥 +𝑑𝑑 𝑐𝑐 𝑑𝑑
Substitutes correct limits into , where p and q are positive constants.
May be unsimplified. 3
𝑝𝑝𝑦𝑦 +𝑞𝑞𝑦𝑦
Accept substitution of 0 not seen. | 1.1b | A1
Completes a rigorous mathematical argument, including either k where
or (or equivalent), that the volume can be expressed as 3 to 3 significant figures.
𝑎𝑎 𝑘𝑘 ∈ [3.4005,3.4045]
13 3 3
Must use correctly throughout.
12𝜋𝜋𝑎𝑎 3.40𝑎𝑎
Must include an appropriate reference to 3 significant figures, e.g. (3sf)
𝑑𝑑𝑦𝑦
13𝜋𝜋
Accept substitution of 0 not seen.
12 = 3.40
This mark can only be awarded if M2A2 scored.
NMS scores 0/5 | 2.1 | R1
Total | 8
Q | Marking instructions | AO | Marks | Typical solutionA hyperbola $H$ has the equation
$$\frac{x^2}{a^2} - \frac{y^2}{4a^2} = 1$$
where $a$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Write down the equations of the asymptotes of $H$. [1 mark]
\item Sketch the hyperbola $H$ on the axes below, indicating the coordinates of any points of intersection with the coordinate axes.
The asymptotes have already been drawn. [2 marks]
\item The finite region bounded by $H$, the positive $x$-axis, the positive $y$-axis and the line $y = a$ is rotated through $360°$ about the $y$-axis.
Show that the volume of the solid generated is $ma^3$, where $m = 3.40$ correct to three significant figures. [5 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 2019 Q5 [8]}}