| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2010 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Simpson's rule application |
| Difficulty | Standard +0.3 This is a multi-part question testing standard C3 techniques: mid-ordinate rule (routine numerical integration), algebraic rearrangement (trivial), volume of revolution about y-axis (standard formula application), and function transformations (bookwork). Each part is straightforward application of learned methods with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.09f Trapezium rule: numerical integration4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x\) values: 1.5, 4.5, 7.5, 10.5 correct | B1 | \(x\) values correct |
| \(y\) values: 1.98100, 3.22883, 4.11496, 4.74710 | M1 | 3+ \(y\) values correct to 2sf or better |
| A1 | 1.981, 3.228/9, 4.114/5, 4.747 for \(y\) (or better) | |
| \(\int = 3 \times \sum y = 42.2\) | A1 | Note: 42.2 with evidence of mid-ordinate rule with four strips scores 4/4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(y = \ln(x^2 + 5)\), so \(e^y = x^2 + 5\), so \(x^2 = e^y - 5\) | B1 | AG Must see middle line, and no errors |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((\pi)\int(e^y - 5)\,(dy)\) | M1 | Condone omission of brackets around \(f(y)\) throughout |
| \(= (\pi)\left[e^y - 5y\right]_{(5)}^{(10)}\) | A1 | |
| \(= (\pi)\left[(e^{10} - 50) - (e^5 - 25)\right]\) | m1 | \(F(10) - F(5)\) |
| \(V = \pi\left[e^{10} - e^5 - 25\right]\) | A1 | CSO including correct notation — must see \(dy\); ISW if evaluated |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((y =)\ln\left[\left(\frac{x}{4}\right)^2 + 5\right] + 3\) | M1 | \(\frac{x}{4}\) seen, condone \(\ln\frac{x^2}{4} + \ldots\) |
| B1 | \(\ldots + 3\) | |
| A1 | CSO mark final answer (no ISW) |
## Question 5(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x$ values: 1.5, 4.5, 7.5, 10.5 correct | B1 | $x$ values correct |
| $y$ values: 1.98100, 3.22883, 4.11496, 4.74710 | M1 | 3+ $y$ values correct to 2sf or better |
| | A1 | 1.981, 3.228/9, 4.114/5, 4.747 for $y$ (or better) |
| $\int = 3 \times \sum y = 42.2$ | A1 | Note: 42.2 with evidence of mid-ordinate rule with four strips scores 4/4 |
## Question 5(b)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y = \ln(x^2 + 5)$, so $e^y = x^2 + 5$, so $x^2 = e^y - 5$ | B1 | AG Must see middle line, and no errors |
## Question 5(b)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(\pi)\int(e^y - 5)\,(dy)$ | M1 | Condone omission of brackets around $f(y)$ throughout |
| $= (\pi)\left[e^y - 5y\right]_{(5)}^{(10)}$ | A1 | |
| $= (\pi)\left[(e^{10} - 50) - (e^5 - 25)\right]$ | m1 | $F(10) - F(5)$ |
| $V = \pi\left[e^{10} - e^5 - 25\right]$ | A1 | CSO including correct notation — must see $dy$; ISW if evaluated |
## Question 5(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(y =)\ln\left[\left(\frac{x}{4}\right)^2 + 5\right] + 3$ | M1 | $\frac{x}{4}$ seen, condone $\ln\frac{x^2}{4} + \ldots$ |
| | B1 | $\ldots + 3$ |
| | A1 | CSO mark final answer (no ISW) |
---5
\begin{enumerate}[label=(\alph*)]
\item Use the mid-ordinate rule with four strips to find an estimate for $\int _ { 0 } ^ { 12 } \ln \left( x ^ { 2 } + 5 \right) \mathrm { d } x$, giving your answer to three significant figures.
\item A curve has equation $y = \ln \left( x ^ { 2 } + 5 \right)$.
\begin{enumerate}[label=(\roman*)]
\item Show that this equation can be rewritten as $x ^ { 2 } = \mathrm { e } ^ { y } - 5$.
\item The region bounded by the curve, the lines $y = 5$ and $y = 10$ and the $y$-axis is rotated through $360 ^ { \circ }$ about the $y$-axis. Find the exact value of the volume of the solid generated.
\end{enumerate}\item The graph with equation $y = \ln \left( x ^ { 2 } + 5 \right)$ is stretched with scale factor 4 parallel to the $x$-axis, and then translated through $\left[ \begin{array} { l } 0 \\ 3 \end{array} \right]$ to give the graph with equation $y = \mathrm { f } ( x )$. Write down an expression for $\mathrm { f } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2010 Q5 [12]}}