| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2013 |
| 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 A-level fixed point iteration question requiring sketching graphs to show a root exists, verifying the interval, algebraic rearrangement of the equation, and applying an iterative formula. All steps are routine techniques with no novel insight required, though it involves multiple parts and careful calculator work. Slightly easier than average due to the straightforward nature of each component. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.09a Sign change methods: locate roots1.09b Sign change methods: understand failure cases1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Make a recognisable sketch of a relevant graph, e.g. \(y = \cot x\) or \(y = 4x - 2\) | B1 | |
| Sketch a second relevant graph and justify the given statement | B1 | [2] |
| (ii) Consider sign of \(4x - 2 - \cot x\) at \(x = 0.7\) and \(x = 0.9\), or equivalent | M1 | |
| Complete the argument correctly with appropriate calculations | A1 | [2] |
| (iii) Show that given equation is equivalent to \(x = \frac{1 + 2\tan x}{4\tan x}\), or vice versa | B1 | [1] |
| (iv) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 0.76 | A1 | |
| Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change in the interval (0.755, 0.765) | B1 | [3] |
(i) Make a recognisable sketch of a relevant graph, e.g. $y = \cot x$ or $y = 4x - 2$ | B1 |
Sketch a second relevant graph and justify the given statement | B1 | [2]
(ii) Consider sign of $4x - 2 - \cot x$ at $x = 0.7$ and $x = 0.9$, or equivalent | M1 |
Complete the argument correctly with appropriate calculations | A1 | [2]
(iii) Show that given equation is equivalent to $x = \frac{1 + 2\tan x}{4\tan x}$, or vice versa | B1 | [1]
(iv) Use the iterative formula correctly at least once | M1 |
Obtain final answer 0.76 | A1 |
Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change in the interval (0.755, 0.765) | B1 | [3]6 (i) By sketching a suitable pair of graphs, show that the equation
$$\cot x = 4 x - 2$$
where $x$ is in radians, has only one root for $0 \leqslant x \leqslant \frac { 1 } { 2 } \pi$.\\
(ii) Verify by calculation that this root lies between $x = 0.7$ and $x = 0.9$.\\
(iii) Show that this root also satisfies the equation
$$x = \frac { 1 + 2 \tan x } { 4 \tan x }$$
(iv) Use the iterative formula $x _ { n + 1 } = \frac { 1 + 2 \tan x _ { n } } { 4 \tan x _ { n } }$ to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P2 2013 Q6 [8]}}