| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Iterative formula from integral equation |
| Difficulty | Challenging +1.2 This question combines integration by parts with numerical methods (iterative formula). Part (i) requires standard integration by parts technique with u=ln(x), which is routine for P3. Parts (ii) and (iii) involve straightforward substitution and iteration with no conceptual challenges. The multi-step nature and combination of techniques elevates it slightly above average, but each component is standard textbook material requiring no novel insight. |
| Spec | 1.08i Integration by parts1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Integrate by parts and reach \(ax^{\frac{3}{2}}\ln x + b\int x^{\frac{3}{2}}\cdot\dfrac{1}{x}\,dx\) | *M1 | |
| Obtain \(\dfrac{2}{3}x^{\frac{3}{2}}\ln x - \dfrac{2}{3}\int x^{\frac{1}{2}}\,dx\) | A1 | |
| Obtain integral \(\dfrac{2}{3}x^{\frac{3}{2}}\ln x - \dfrac{4}{9}x^{\frac{3}{2}}\), or equivalent | A1 | |
| Substitute limits correctly and equate to 2 | DM1 | |
| Obtain the given answer correctly | A1 | AG |
| Total | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Evaluate a relevant expression or pair of expressions at \(x=2\) and \(x=4\) | M1 | |
| Complete the argument correctly with correct calculated values | A1 | |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 3.031 | A1 | |
| Show sufficient iterations to 5 d.p. to justify 3.031 to 3 d.p., or show there is a sign change in the interval \((3.0305, 3.0315)\) | A1 | |
| Total | 3 |
# Question 9(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate by parts and reach $ax^{\frac{3}{2}}\ln x + b\int x^{\frac{3}{2}}\cdot\dfrac{1}{x}\,dx$ | *M1 | |
| Obtain $\dfrac{2}{3}x^{\frac{3}{2}}\ln x - \dfrac{2}{3}\int x^{\frac{1}{2}}\,dx$ | A1 | |
| Obtain integral $\dfrac{2}{3}x^{\frac{3}{2}}\ln x - \dfrac{4}{9}x^{\frac{3}{2}}$, or equivalent | A1 | |
| Substitute limits correctly and equate to 2 | DM1 | |
| Obtain the given answer correctly | A1 | AG |
| **Total** | **5** | |
---
# Question 9(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Evaluate a relevant expression or pair of expressions at $x=2$ and $x=4$ | M1 | |
| Complete the argument correctly with correct calculated values | A1 | |
| **Total** | **2** | |
---
# Question 9(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 3.031 | A1 | |
| Show sufficient iterations to 5 d.p. to justify 3.031 to 3 d.p., or show there is a sign change in the interval $(3.0305, 3.0315)$ | A1 | |
| **Total** | **3** | |
---9 It is given that $\int _ { 1 } ^ { a } x ^ { \frac { 1 } { 2 } } \ln x \mathrm {~d} x = 2$, where $a > 1$.\\
(i) Show that $a ^ { \frac { 3 } { 2 } } = \frac { 7 + 2 a ^ { \frac { 3 } { 2 } } } { 3 \ln a }$.\\
(ii) Show by calculation that $a$ lies between 2 and 4 .\\
(iii) Use the iterative formula
$$a _ { n + 1 } = \left( \frac { 7 + 2 a _ { n } ^ { \frac { 3 } { 2 } } } { 3 \ln a _ { n } } \right) ^ { \frac { 2 } { 3 } }$$
to determine $a$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.\\
\hfill \mbox{\textit{CAIE P3 2017 Q9 [10]}}