| Exam Board | AQA |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2023 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Simpson's rule application |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard A-level techniques. Part (a) requires differentiating e^(sin x) using chain rule and solving cos x = 0, which is routine. Part (b) is a direct application of Simpson's rule formula with given ordinates. Part (c) asks for standard knowledge about improving Simpson's rule accuracy (use more ordinates). All parts are textbook exercises requiring no problem-solving insight, though it's slightly above average difficulty due to being Further Maths content. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07n Stationary points: find maxima, minima using derivatives1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| 8(a) | Deduces x-coordinates of | |
| stationary points | 2.2a | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Subtotal | 1 | |
| Q | Marking instructions | AO |
| Answer | Marks |
|---|---|
| 8(b) | Obtains exactly three |
| Answer | Marks | Guidance |
|---|---|---|
| unsimplified. | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Condone 5 ordinates. | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| AWRT 6.74 | 1.1b | A1 |
| Subtotal | 3 | |
| x | 0 | π/2 |
| y | 1 | e |
| Q | Marking instructions | AO |
| Answer | Marks |
|---|---|
| 8(c) | Infers that a larger number |
| Answer | Marks | Guidance |
|---|---|---|
| give a more accurate result | 2.2b | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| Subtotal | 1 | |
| Question total | 5 | |
| Q | Marking instructions | AO |
Question 8:
--- 8(a) ---
8(a) | Deduces x-coordinates of
stationary points | 2.2a | B1 | π 3π
and
2 2
Subtotal | 1
Q | Marking instructions | AO | Marks | Typical solution
--- 8(b) ---
8(b) | Obtains exactly three
values of y for correct
values of x , can be
unsimplified. | 1.1a | M1 | x 0 π/2 π
y 1 e 1
1 π
Estimate = × (1+1+4e)
3 2
= 6.74
Uses Simpson’s rule
correctly
Condone 5 ordinates. | 1.1a | M1
Obtains correct value
AWRT 6.74 | 1.1b | A1
Subtotal | 3
x | 0 | π/2 | π
y | 1 | e | 1
Q | Marking instructions | AO | Marks | Typical solution
--- 8(c) ---
8(c) | Infers that a larger number
of ordinates/strips could
give a more accurate result | 2.2b | E1 | By using Simpson’s rule with 5
ordinates
Subtotal | 1
Question total | 5
Q | Marking instructions | AO | Marks | Typical solutionThe function g is defined by
$$g(x) = \mathrm{e}^{\sin x} \quad (0 \leq x \leq 2\pi)$$
The diagram below shows the graph of $y = g(x)$
\includegraphics{figure_8}
\begin{enumerate}[label=(\alph*)]
\item Find the $x$-coordinate of each of the stationary points of the graph of $y = g(x)$, giving your answers in exact form.
[1 mark]
\item Use Simpson's rule with 3 ordinates to estimate
$$\int_0^\pi g(x) \, \mathrm{d}x$$
giving your answer to two decimal places.
[3 marks]
\item Explain how Simpson's rule could be used to find a more accurate estimate of the integral in part (b).
[1 mark]
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 1 2023 Q8 [5]}}