| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule with stated number of strips |
| Difficulty | Moderate -0.8 Part (a) is a straightforward application of the trapezium rule formula with clearly specified ordinates—purely procedural calculation requiring no problem-solving. Part (b) tests basic recognition of vertical translation and combined horizontal stretch/vertical stretch transformations, which are standard C2 topics. The question requires only routine recall and application of standard techniques with no novel insight or multi-step reasoning. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(h = 3\) | B1 |
| (a) | \(f(x) = \sqrt{x^2 + 9}\) | M1 |
| (a) | \(\frac{h}{2}\) with \([\sqrt{13} + \sqrt{130} + 2(\sqrt{34} + \sqrt{73})]\) | A1 |
| (a) | \(= \frac{h}{2}[3.60(5...) + 11.4(0...) + 2[5.83(09...) + 8.54(4...)]]\) | A1 |
| (a) | \(= \frac{h}{2}\{15.0(07...) + 28.7(499...)]\}\) | |
| (a) | \((I \approx 1.5 \times 43.7(57...)) = (65.63(58...))\) | |
| (a) | \(I = 65.6\) (to 1 dp) | |
| Total | 8 | |
| (b)(i) | Translation \(\begin{bmatrix} 0 \\ 5 \end{bmatrix}\) | E2,1,0 |
| (b)(ii) | Stretch (I) in x-direction (II) scale factor \(\frac{1}{3}\) (III) | E2,1,0 |
(a) | $h = 3$ | B1 | $h = 3$ OE stated or used. (PI by y-values 2, 5, 8, 11 provided no contradiction)
(a) | $f(x) = \sqrt{x^2 + 9}$ | M1 | $h/2 \{f(2) + f(11) + 2[f(5) + f(8)]\}$ seen or used OE Summing of areas of the "trapezla".. ($M0$ if using an incorrect f(x))
(a) | $\frac{h}{2}$ with $[\sqrt{13} + \sqrt{130} + 2(\sqrt{34} + \sqrt{73})]$ | A1 | OE Accept 3sf or better evidence for surds. Can be implied by later correct work provided >1 term or a single term for I which rounds to 65.6
(a) | $= \frac{h}{2}[3.60(5...) + 11.4(0...) + 2[5.83(09...) + 8.54(4...)]]$ | A1 | 4 marks | CAO Must be 65.6
(a) | $= \frac{h}{2}\{15.0(07...) + 28.7(499...)]\}$ | | SC 4 strips used: Max B0M1A0; 65.5 A1
(a) | $(I \approx 1.5 \times 43.7(57...)) = (65.63(58...))$ |
(a) | $I = 65.6$ (to 1 dp) |
| **Total** | **8** |
(b)(i) | Translation $\begin{bmatrix} 0 \\ 5 \end{bmatrix}$ | E2,1,0 | 2 marks | E2: 'translat...' and $\begin{bmatrix} 0 \\ 5 \end{bmatrix}$... If not E2 award E1 for 'translat... in y-dir' OE. E0 if more than one transformation
(b)(ii) | Stretch (I) in x-direction (II) scale factor $\frac{1}{3}$ (III) | E2,1,0 | 2 marks | Need (I) and (II) and (III) for E2. If not E2 then award E1 for seeing $3\sqrt{x^2 + 1} = \sqrt{(3x)^2 + 9}$ OE together with (I) and either (II) or (III). E0 if more than one transformation
---5
\begin{enumerate}[label=(\alph*)]
\item Use the trapezium rule with four ordinates (three strips) to find an approximate value for $\int _ { 2 } ^ { 11 } \sqrt { x ^ { 2 } + 9 } \mathrm {~d} x$. Give your answer to one decimal place.
\item Describe the geometrical transformation that maps the graph of $y = \sqrt { x ^ { 2 } + 9 }$ onto the graph of :
\begin{enumerate}[label=(\roman*)]
\item $y = 5 + \sqrt { x ^ { 2 } + 9 }$;
\item $y = 3 \sqrt { x ^ { 2 } + 1 }$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C2 2016 Q5 [8]}}