| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from integral condition |
| Difficulty | Challenging +1.2 This question requires integration by parts (a standard C3/P3 technique), algebraic manipulation to reach the given form, then straightforward application of fixed-point iteration. While it involves multiple steps and some algebraic dexterity to derive the iterative formula, the techniques are all standard for this level. The iteration itself is mechanical once the formula is established. Slightly above average due to the integration by parts setup and algebraic manipulation required, but not requiring 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 |
|---|---|---|
| (i) Use integration by parts to obtain \(axe^{-\frac{1}{2}x} + \int be^{-\frac{1}{2}x} dx\) | M1* | |
| Obtain \(-8xe^{-\frac{1}{2}x} + \int 8e^{-\frac{1}{2}x} dx\) or unsimplified equivalent | A1 | |
| Obtain \(-8xe^{-\frac{1}{2}x} - 16e^{-\frac{1}{2}x}\) | A1 | |
| Use limits correctly and equate to \(9\) | M1(d*M) | |
| Obtain given answer \(p = 2\ln\left(\frac{8p+16}{7}\right)\) correctly | A1 | [5] |
| (ii) Use correct iteration formula correctly at least once | M1 | |
| Obtain final answer \(3.77\) | A1 | |
| Show sufficient iterations to \(5sf\) or better to justify accuracy \(3.77\) or show sign change in interval \((3.765, 3.775)\) | A1 | [3] |
**(i)** Use integration by parts to obtain $axe^{-\frac{1}{2}x} + \int be^{-\frac{1}{2}x} dx$ | M1* |
Obtain $-8xe^{-\frac{1}{2}x} + \int 8e^{-\frac{1}{2}x} dx$ or unsimplified equivalent | A1 |
Obtain $-8xe^{-\frac{1}{2}x} - 16e^{-\frac{1}{2}x}$ | A1 |
Use limits correctly and equate to $9$ | M1(d*M) |
Obtain given answer $p = 2\ln\left(\frac{8p+16}{7}\right)$ correctly | A1 | [5]
**(ii)** Use correct iteration formula correctly at least once | M1 |
Obtain final answer $3.77$ | A1 |
Show sufficient iterations to $5sf$ or better to justify accuracy $3.77$ or show sign change in interval $(3.765, 3.775)$ | A1 | [3]5 It is given that $\int _ { 0 } ^ { p } 4 x \mathrm { e } ^ { - \frac { 1 } { 2 } x } \mathrm {~d} x = 9$, where $p$ is a positive constant.\\
(i) Show that $p = 2 \ln \left( \frac { 8 p + 16 } { 7 } \right)$.\\
(ii) Use an iterative process based on the equation in part (i) to find the value of $p$ correct to 3 significant figures. Use a starting value of 3.5 and give the result of each iteration to 5 significant figures.
\hfill \mbox{\textit{CAIE P3 2013 Q5 [8]}}