| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2003 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Find equation satisfied by limit |
| Difficulty | Standard +0.3 This is a straightforward multi-part question combining standard calculus (finding stationary points) with basic fixed point iteration. Part (i) requires routine differentiation and second derivative test. Part (ii) involves recognizing that at convergence x_{n+1} = x_n = α, leading to a simple algebraic manipulation. Part (iii) is mechanical calculator work. The connection between the iteration and the curve is explicit and requires no novel insight—slightly easier than average A-level work. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07n Stationary points: find maxima, minima using derivatives1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State derivative \(\frac{1}{x} - \frac{2}{x^2}\), or equivalent | B1 | Equate 2-term derivative to zero and attempt to solve for x |
| (ii) State or imply the equation \(\alpha = \frac{2}{3 - \ln \alpha}\) | B1 | Rearrange this as \(3 = \ln \alpha + \frac{2}{\alpha}\) (or vice versa) |
| (iii) Use the iterative formula correctly at least once | M1 | Obtain final answer 0.56 |
**(i)** State derivative $\frac{1}{x} - \frac{2}{x^2}$, or equivalent | B1 | Equate 2-term derivative to zero and attempt to solve for x | M1 | Obtain coordinates of stationary point $(2, \ln 2 + 1)$, or equivalent | A1+A1 | Determine by any method that it is a minimum point, with no incorrect work seen | A1 | **[5]**
**(ii)** State or imply the equation $\alpha = \frac{2}{3 - \ln \alpha}$ | B1 | Rearrange this as $3 = \ln \alpha + \frac{2}{\alpha}$ (or vice versa) | B1 | **[2]**
**(iii)** Use the iterative formula correctly at least once | M1 | Obtain final answer 0.56 | A1 | Show sufficient iterations to justify its accuracy to 2 d.p., or show there is a sign change in the interval (0.555, 0.565) | A1 | **[3]**8 The equation of a curve is $y = \ln x + \frac { 2 } { x }$, where $x > 0$.\\
(i) Find the coordinates of the stationary point of the curve and determine whether it is a maximum or a minimum point.\\
(ii) The sequence of values given by the iterative formula
$$x _ { n + 1 } = \frac { 2 } { 3 - \ln x _ { n } }$$
with initial value $x _ { 1 } = 1$, converges to $\alpha$. State an equation satisfied by $\alpha$, and hence show that $\alpha$ is the $x$-coordinate of a point on the curve where $y = 3$.\\
(iii) Use this iterative formula to find $\alpha$ correct to 2 decimal places, showing the result of each iteration.
\hfill \mbox{\textit{CAIE P3 2003 Q8 [10]}}