| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2004 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Rearrange to iterative form |
| Difficulty | Moderate -0.3 This is a standard fixed-point iteration question requiring routine sketching, interval verification, algebraic rearrangement (cot x = 1/tan x leads directly to the given form), and applying a given iterative formula. All steps are straightforward applications of A-level techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02m Graphs of functions: difference between plotting and sketching1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Make recognisable sketch of appropriate trig curve, e.g. \(y = \cot x\), for \(0 < x < \frac{1}{2}\pi\) | B1 | |
| Sketch appropriate second curve e.g. \(y = x\) correctly and justify given statement | B1 | Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) | Consider sign of \(\cot x - x\) at \(x = 0.8\) and \(x = 0.9\), or equivalent | M1 |
| Complete the argument correctly with appropriate calculations | A1 | [2] |
| (iii) | Show, using \(\cot x \equiv \frac{1}{\tan x}\), that \(\cot x = x\) is equivalent to \(x = \arctan\left(\frac{1}{x}\right)\) (or vice versa) | B1 |
| (iv) | Use the iterative formula correctly at least once | M1 |
| Obtain final answer \(0.86\) | A1 | |
| Show sufficient iterations to justify its accuracy to 2 decimal places, or show that there is a sign change in \((0.855, 0.865)\) | B1 | [3] |
## Question 6(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Make recognisable sketch of appropriate trig curve, e.g. $y = \cot x$, for $0 < x < \frac{1}{2}\pi$ | B1 | |
| Sketch appropriate second curve e.g. $y = x$ correctly and justify given statement | B1 | **Total: 2** |
# Question 6 (continued):
**(ii)** | Consider sign of $\cot x - x$ at $x = 0.8$ and $x = 0.9$, or equivalent | M1 | |
Complete the argument correctly with appropriate calculations | A1 | **[2]**
**(iii)** | Show, using $\cot x \equiv \frac{1}{\tan x}$, that $\cot x = x$ is equivalent to $x = \arctan\left(\frac{1}{x}\right)$ (or vice versa) | B1 | **[1]**
**(iv)** | Use the iterative formula correctly at least once | M1 | |
Obtain final answer $0.86$ | A1 | |
Show sufficient iterations to justify its accuracy to 2 decimal places, or show that there is a sign change in $(0.855, 0.865)$ | B1 | **[3]**
---6 (i) By sketching a suitable pair of graphs, show that there is only one value of $x$ in the interval $0 < x < \frac { 1 } { 2 } \pi$ that is a root of the equation
$$\cot x = x$$
(ii) Verify by calculation that this root lies between 0.8 and 0.9 radians.\\
(iii) Show that this value of $x$ is also a root of the equation
$$x = \tan ^ { - 1 } \left( \frac { 1 } { x } \right)$$
(iv) Use the iterative formula
$$x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { x _ { n } } \right)$$
to determine this root correct to 2 decimal places, showing the result of each iteration.
\hfill \mbox{\textit{CAIE P2 2004 Q6 [8]}}