| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | November |
| Marks | 8 |
| 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 derive the given equation, then algebraic manipulation to reach the required form, followed by iterative numerical methods. While integration by parts of x ln x is standard, the algebraic rearrangement and iterative solution adds moderate complexity beyond routine exercises, placing it somewhat above average difficulty. |
| Spec | 1.08i Integration by parts1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| (i) Either: Use integration by parts and reach an expression \(kx^2\ln x \pm n\int x^2.\frac{1}{x}\,dx\) | M1 | |
| Obtain \(\frac{1}{2}x^2\ln x - \int\frac{1}{2}x\,dx\) or equivalent | A1 | |
| Obtain \(\frac{1}{2}x^2\ln x - \frac{1}{4}x^2\) | A1 | |
| Or: Use integration by parts and reach an expression \(kx(x\ln x - x) \pm m\int x\ln x - x\,dx\) | M1 | |
| Obtain \(I = (x^2\ln x - x^2) - I + \int x\,dx\) | A1 | |
| Obtain \(\frac{1}{2}x^2\ln x - \frac{1}{4}x^2\) | A1 | |
| Substitute limits correctly and equate to 22, having integrated twice | DM1* | |
| Rearrange and confirm given equation \(a = \sqrt{\dfrac{87}{2\ln a - 1}}\) | A1 | [5] |
| (ii) Use iterative process correctly at least once | M1 | |
| Obtain final answer 5.86 | A1 | |
| Show sufficient iterations to 4 d.p. to justify 5.86 or show a sign change in the interval (5.855, 5.865) | A1 | |
| \((6 \to 5.8030 \to 5.8795 \to 5.8491 \to 5.8611 \to 5.8564)\) | [3] |
## Question 5:
| Answer/Working | Mark | Guidance |
|---|---|---|
| **(i) Either:** Use integration by parts and reach an expression $kx^2\ln x \pm n\int x^2.\frac{1}{x}\,dx$ | M1 | |
| Obtain $\frac{1}{2}x^2\ln x - \int\frac{1}{2}x\,dx$ or equivalent | A1 | |
| Obtain $\frac{1}{2}x^2\ln x - \frac{1}{4}x^2$ | A1 | |
| **Or:** Use integration by parts and reach an expression $kx(x\ln x - x) \pm m\int x\ln x - x\,dx$ | M1 | |
| Obtain $I = (x^2\ln x - x^2) - I + \int x\,dx$ | A1 | |
| Obtain $\frac{1}{2}x^2\ln x - \frac{1}{4}x^2$ | A1 | |
| Substitute limits correctly and equate to 22, having integrated twice | DM1* | |
| Rearrange and confirm given equation $a = \sqrt{\dfrac{87}{2\ln a - 1}}$ | A1 | [5] |
| **(ii)** Use iterative process correctly at least once | M1 | |
| Obtain final answer 5.86 | A1 | |
| Show sufficient iterations to 4 d.p. to justify 5.86 or show a sign change in the interval (5.855, 5.865) | A1 | |
| $(6 \to 5.8030 \to 5.8795 \to 5.8491 \to 5.8611 \to 5.8564)$ | | [3] |
---5 It is given that $\int _ { 1 } ^ { a } x \ln x \mathrm {~d} x = 22$, where $a$ is a constant greater than 1 .\\
(i) Show that $a = \sqrt { } \left( \frac { 87 } { 2 \ln a - 1 } \right)$.\\
(ii) Use an iterative formula based on the equation in part (i) to find the value of $a$ correct to 2 decimal places. Use an initial value of 6 and give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2011 Q5 [8]}}