| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2013 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Single transformation between given equations |
| Difficulty | Moderate -0.3 This is a straightforward C2 question testing routine application of trapezium rule, recognition of a horizontal stretch transformation, and application of a translation. Part (a) is mechanical calculation, part (b) requires identifying that √(8x³+1) = √(x³+1) when x is replaced by x/2 (stretch factor 2 in x-direction), and part (c) is direct substitution after translation. All parts are standard textbook exercises with no problem-solving required, making it slightly easier than average. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| \(h = 0.5\) | B1 | \(h = 0.5\) stated or used. |
| \(f(x) = \sqrt[3]{8x^3 + 1}\) | ||
| \(I \approx \frac{h}{2}\{f(0) + f(2) + 2[f(0.5) + f(1) + f(1.5)]\}\) | M1 | \(I \approx \frac{h}{2}\{f(0) + f(2) + 2[f(0.5) + f(1) + f(1.5)]\}\) OE |
| \(\frac{h}{2}\) with {...} = \(\sqrt[3]{1} + \sqrt[3]{65} + 2(\sqrt[3]{2} + \sqrt[3]{9} + \sqrt[3]{28})\) | A1 | OE Accept 1dp evidence. Can be implied by later correct work provided more than one term or a single term which rounds to 7.12 |
| \(= 1 + 8.06... + 2(1.41... + 3 + 5.29...)\) | ||
| \(= 9.0622... + 2×9.7057...\) | ||
| \((I \approx) 0.25[28.47...] = \{7.118...\} = 7.12\) (to 3sf) | A1 | CAO Must be 7.12 |
| Answer | Marks | Guidance |
|---|---|---|
| Stretch(I) in x-direction(II) scale factor 2 (III) | M1 | Need (I) and either (II) or (III) |
| A1 | Need (I) and (II) and (III) More than 1 transformation scores 0/2 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(g(x) = \sqrt{(x-2)^3 + 1 - 0.7}\) | M1 | \(\sqrt{(x-2)^3 + 1 - 0.7}\) or \(\sqrt{(x-2)^3 + 1 + 0.7}\) or \(\sqrt{(x+2)^3 + 1 - 0.7}\) or \(\sqrt{(x-2)^3 + 1 - 0.7}\) or their equivalents |
| A1 | \(\sqrt{(x-2)^3 + 1 - 0.7}\) OE | |
| \(g(4) = 2.3\) | A1 | 2.3 OE |
| Answer | Marks | Guidance |
|---|---|---|
| \((4, ...)\) on \(y = g(x)\) comes from translating \((2, 3)\) on \(y = \sqrt[3]{x^3 + 1}\) | (M1) | from \((2, ...)\) on \(y = \sqrt[3]{x^3 + 1}\) |
| (A1) | from \((2, 3)\) on \(y = \sqrt[3]{x^3 + 1}\) | |
| \((2, 3)\) after translation becomes \((4, 2.3)\) so \(g(4) = 2.3\) | (A1) | 2.3 OE |
| Total | 9 |
**5(a)**
$h = 0.5$ | B1 | $h = 0.5$ stated or used.
$f(x) = \sqrt[3]{8x^3 + 1}$ | |
$I \approx \frac{h}{2}\{f(0) + f(2) + 2[f(0.5) + f(1) + f(1.5)]\}$ | M1 | $I \approx \frac{h}{2}\{f(0) + f(2) + 2[f(0.5) + f(1) + f(1.5)]\}$ OE
$\frac{h}{2}$ with {...} = $\sqrt[3]{1} + \sqrt[3]{65} + 2(\sqrt[3]{2} + \sqrt[3]{9} + \sqrt[3]{28})$ | A1 | OE Accept 1dp evidence. Can be implied by later correct work provided more than one term or a single term which rounds to 7.12
$= 1 + 8.06... + 2(1.41... + 3 + 5.29...)$ | |
$= 9.0622... + 2×9.7057...$ | |
$(I \approx) 0.25[28.47...] = \{7.118...\} = 7.12$ (to 3sf) | A1 | CAO Must be 7.12 | 4
**5(b)**
Stretch(I) in x-direction(II) scale factor 2 (III) | M1 | Need (I) and either (II) or (III)
| A1 | Need (I) and (II) and (III) More than 1 transformation scores 0/2 | 2
**5(c)**
$g(x) = \sqrt{(x-2)^3 + 1 - 0.7}$ | M1 | $\sqrt{(x-2)^3 + 1 - 0.7}$ or $\sqrt{(x-2)^3 + 1 + 0.7}$ or $\sqrt{(x+2)^3 + 1 - 0.7}$ or $\sqrt{(x-2)^3 + 1 - 0.7}$ or their equivalents
| A1 | $\sqrt{(x-2)^3 + 1 - 0.7}$ OE
$g(4) = 2.3$ | A1 | 2.3 OE | 3
**Altn**
$(4, ...)$ on $y = g(x)$ comes from translating $(2, 3)$ on $y = \sqrt[3]{x^3 + 1}$ | (M1) | from $(2, ...)$ on $y = \sqrt[3]{x^3 + 1}$
| (A1) | from $(2, 3)$ on $y = \sqrt[3]{x^3 + 1}$
$(2, 3)$ after translation becomes $(4, 2.3)$ so $g(4) = 2.3$ | (A1) | 2.3 OE | (3)
Total | | 9
---5
\begin{enumerate}[label=(\alph*)]
\item Use the trapezium rule with five ordinates (four strips) to find an approximate value for $\int _ { 0 } ^ { 2 } \sqrt { 8 x ^ { 3 } + 1 } \mathrm {~d} x$, giving your answer to three significant figures.
\item Describe the single transformation that maps the graph of $y = \sqrt { 8 x ^ { 3 } + 1 }$ onto the graph of $y = \sqrt { x ^ { 3 } + 1 }$.
\item The curve with equation $y = \sqrt { x ^ { 3 } + 1 }$ is translated by $\left[ \begin{array} { c } 2 \\ - 0.7 \end{array} \right]$ to give the curve with equation $y = \mathrm { g } ( x )$. Find the value of $\mathrm { g } ( 4 )$.\\
(3 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2013 Q5 [9]}}