| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2005 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Rearrange to iterative form |
| Difficulty | Standard +0.3 This is a standard fixed-point iteration question requiring routine sketching, verification by substitution, algebraic rearrangement (cosec x = (x+2)/2 → x = arcsin(2/(x+2))), and iterative calculation. All steps are procedural with no novel insight required, making it slightly easier than average. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Make recognisable sketch of a relevant graph over the given range, e.g. \(y = \cosec x\) | B1 | |
| Sketch the other relevant graph, e.g. \(y = \frac{1}{2}x + 1\), and justify the given statement | B1 | 2 |
| (ii) Consider sign of \(\cosec x - \frac{1}{2}x - 1\) at \(x = 0.5\) and \(x = 1\), or equivalent | M1 | |
| Complete the argument correctly with appropriate calculations | A1 | 2 |
| (iii) Rearrange \(\cosec x = \frac{1}{2}x + 1\) in the given form, or vice versa | B1 | 1 |
| (iv) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer \(x = 0.80\) | A1 | |
| Show sufficient iterations to at least 3 d.p. to justify its accuracy to 2 d.p., or show there is a sign change in the interval \((0.795, 0.805)\) | A1 | 3 |
**(i)** Make recognisable sketch of a relevant graph over the given range, e.g. $y = \cosec x$ | B1 |
Sketch the other relevant graph, e.g. $y = \frac{1}{2}x + 1$, and justify the given statement | B1 | 2
**(ii)** Consider sign of $\cosec x - \frac{1}{2}x - 1$ at $x = 0.5$ and $x = 1$, or equivalent | M1 |
Complete the argument correctly with appropriate calculations | A1 | 2
**(iii)** Rearrange $\cosec x = \frac{1}{2}x + 1$ in the given form, or vice versa | B1 | 1
**(iv)** Use the iterative formula correctly at least once | M1 |
Obtain final answer $x = 0.80$ | A1 |
Show sufficient iterations to at least 3 d.p. to justify its accuracy to 2 d.p., or show there is a sign change in the interval $(0.795, 0.805)$ | A1 | 37 (i) By sketching a suitable pair of graphs, show that the equation
$$\operatorname { cosec } x = \frac { 1 } { 2 } x + 1$$
where $x$ is in radians, has a root in the interval $0 < x < \frac { 1 } { 2 } \pi$.\\
(ii) Verify, by calculation, that this root lies between 0.5 and 1 .\\
(iii) Show that this root also satisfies the equation
$$x = \sin ^ { - 1 } \left( \frac { 2 } { x + 2 } \right)$$
(iv) Use the iterative formula
$$x _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 2 } { x _ { n } + 2 } \right)$$
with initial value $x _ { 1 } = 0.75$, to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2005 Q7 [8]}}