| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive stationary point equation |
| Difficulty | Standard +0.3 This is a straightforward calculus and iteration question requiring standard differentiation (quotient rule), solving dy/dx = 0 algebraically, verifying bounds by substitution, then applying a given iterative formula. All techniques are routine for P3 level with no novel insight required, making it slightly easier than average. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use correct quotient or product rule | M1 | — |
| Obtain correct derivative in any form, e.g. \(\frac{1}{x(x+1)} - \frac{\ln x}{(x+1)^2}\) | A1 | — |
| Equate derivative to zero and obtain the given equation correctly | A1 | — |
| Consider the sign of \(x - \frac{4}{\ln x}\) at \(x = 3\) and \(x = 4\), or equivalent | M1 | — |
| Complete the argument with correct calculated values | A1 | [5] |
| (ii) Use the iterative formula correctly at least once, using or reaching a value in the interval \((3, 4)\) | M1 | — |
| Obtain final answer 3.59 | A1 | — |
| Show sufficient iterations to at least 4 d.p. to justify its accuracy to 2 d.p., or show there is a sign change in the interval \((3.585, 3.595)\) | A1 | [3] |
**(i)** Use correct quotient or product rule | M1 | —
Obtain correct derivative in any form, e.g. $\frac{1}{x(x+1)} - \frac{\ln x}{(x+1)^2}$ | A1 | —
Equate derivative to zero and obtain the given equation correctly | A1 | —
Consider the sign of $x - \frac{4}{\ln x}$ at $x = 3$ and $x = 4$, or equivalent | M1 | —
Complete the argument with correct calculated values | A1 | [5]
**(ii)** Use the iterative formula correctly at least once, using or reaching a value in the interval $(3, 4)$ | M1 | —
Obtain final answer 3.59 | A1 | —
Show sufficient iterations to at least 4 d.p. to justify its accuracy to 2 d.p., or show there is a sign change in the interval $(3.585, 3.595)$ | A1 | [3]
---The curve $y = \frac{\ln x}{x + 1}$ has one stationary point.
\begin{enumerate}[label=(\roman*)]
\item Show that the $x$-coordinate of this point satisfies the equation
$$x = \frac{x + 1}{\ln x},$$
and that this $x$-coordinate lies between 3 and 4. [5]
\item Use the iterative formula
$$x_{n+1} = \frac{x_n + 1}{\ln x_n}$$
to determine the $x$-coordinate correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2010 Q6 [8]}}