| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Iterative formula from integral equation |
| Difficulty | Standard +0.8 This question requires integration by parts to establish the equation, then numerical iteration to solve a transcendental equation. While the integration is standard P3 material, the combination with iterative methods and the need to manipulate the result into a suitable form for iteration elevates it above typical textbook exercises. The multi-part structure and requirement for numerical work adds moderate complexity. |
| Spec | 1.08i Integration by parts1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Commence integration and reach \(a\sqrt{x}\ln x + b\int\sqrt{x} \cdot \frac{1}{x}\,dx\), or equivalent | \*M1 | |
| Obtain \(2\sqrt{x}\ln x - \int 2\sqrt{x} \cdot \frac{1}{x}\,dx\), or equivalent | A1 | |
| Obtain integral \(2\sqrt{x}\ln x - 4\sqrt{x}\), or equivalent | A1 | |
| Substitute limits and equate result to 6 | DM1 | |
| Rearrange and obtain \(a = \exp\!\left(\frac{1}{\sqrt{a}}+2\right)\) | A1 | Obtain given answer from full and correct working. |
| Total: 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Calculate the values of a relevant expression or pair of expressions at \(a=9\) and \(a=11\) | M1 | e.g. \(\begin{cases}9 < 10.31\\11 > 9.99\end{cases}\) or \(1.31>0,\,-1.01<0\) |
| Complete the argument correctly with correct values | A1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Use the iterative process \(a_{n+1} = \exp\!\left(\frac{1}{\sqrt{a_n}}+2\right)\) correctly at least once | M1 | |
| Obtain answer 10.12 | A1 | |
| Show sufficient iterations to 4dp to justify 10.12 to 2dp, or show there is a sign change in the interval (10.115, 10.125) | A1 | e.g. 10, 10.1374, 10.1156, 10.1190, ... / 9, 10.3123, 10.0886, 10.1233, 10.1178, ... / 11, 9.9893, 10.1391, 10.1153, 10.1191, ... |
| Total: 3 |
## Question 8(a):
| Commence integration and reach $a\sqrt{x}\ln x + b\int\sqrt{x} \cdot \frac{1}{x}\,dx$, or equivalent | \*M1 | |
| Obtain $2\sqrt{x}\ln x - \int 2\sqrt{x} \cdot \frac{1}{x}\,dx$, or equivalent | A1 | |
| Obtain integral $2\sqrt{x}\ln x - 4\sqrt{x}$, or equivalent | A1 | |
| Substitute limits and equate result to 6 | DM1 | |
| Rearrange and obtain $a = \exp\!\left(\frac{1}{\sqrt{a}}+2\right)$ | A1 | Obtain **given answer** from full and correct working. |
| **Total: 5** | | |
## Question 8(b):
| Calculate the values of a relevant expression or pair of expressions at $a=9$ and $a=11$ | M1 | e.g. $\begin{cases}9 < 10.31\\11 > 9.99\end{cases}$ or $1.31>0,\,-1.01<0$ |
| Complete the argument correctly with correct values | A1 | |
| **Total: 2** | | |
## Question 8(c):
| Use the iterative process $a_{n+1} = \exp\!\left(\frac{1}{\sqrt{a_n}}+2\right)$ correctly at least once | M1 | |
| Obtain answer 10.12 | A1 | |
| Show sufficient iterations to 4dp to justify 10.12 to 2dp, or show there is a sign change in the interval (10.115, 10.125) | A1 | e.g. 10, 10.1374, 10.1156, 10.1190, ... / 9, 10.3123, 10.0886, 10.1233, 10.1178, ... / 11, 9.9893, 10.1391, 10.1153, 10.1191, ... |
| **Total: 3** | | |
---8 The constant $a$ is such that $\int _ { 1 } ^ { a } \frac { \ln x } { \sqrt { x } } \mathrm {~d} x = 6$.
\begin{enumerate}[label=(\alph*)]
\item Show that $a = \exp \left( \frac { 1 } { \sqrt { a } } + 2 \right)$.\\
$\left[ \exp ( x ) \right.$ is an alternative notation for $\left. \mathrm { e } ^ { x } .\right]$
\item Verify by calculation that $a$ lies between 9 and 11 .
\item Use an iterative formula based on the equation in part (a) to determine $a$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q8 [10]}}