| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive stationary point equation |
| Difficulty | Standard +0.3 Part (i) requires differentiating a quotient and applying the quotient rule to show x = tan x at stationary points—a standard calculus exercise. Part (ii) is straightforward iteration with a calculator. This is routine A-level pure maths requiring no novel insight, slightly easier than average due to its mechanical nature. |
| Spec | 1.07n 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) Use correct quotient or product rule | M1 | |
| Obtain correct derivative in any form | A1 | |
| Equate derivative to zero and solve for \(x\) | M1 | |
| Obtain the given answer correctly | A1 | [4] |
| (ii) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer \(4.49\) | A1 | |
| Show sufficient iterations to at least 4 d.p. to justify its accuracy to 2 d.p., or show that there is a sign change in the interval \((4.485, 4.495)\) | A1 | [3] |
**(i)** Use correct quotient or product rule | M1 |
Obtain correct derivative in any form | A1 |
Equate derivative to zero and solve for $x$ | M1 |
Obtain the given answer correctly | A1 | [4]
**(ii)** Use the iterative formula correctly at least once | M1 |
Obtain final answer $4.49$ | A1 |
Show sufficient iterations to at least 4 d.p. to justify its accuracy to 2 d.p., or show that there is a sign change in the interval $(4.485, 4.495)$ | A1 | [3]4\\
\includegraphics[max width=\textwidth, alt={}, center]{20de5ba6-9426-4431-99af-6e8e62607f3e-2_513_895_1055_625}
The diagram shows the curve $y = \frac { \sin x } { x }$ for $0 < x \leqslant 2 \pi$, and its minimum point $M$.\\
(i) Show that the $x$-coordinate of $M$ satisfies the equation
$$x = \tan x$$
(ii) The iterative formula
$$x _ { n + 1 } = \tan ^ { - 1 } \left( x _ { n } \right) + \pi$$
can be used to determine the $x$-coordinate of $M$. Use this formula to determine the $x$-coordinate of $M$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2010 Q4 [7]}}