| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2024 |
| Session | November |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Show convergence to specific root |
| Difficulty | Standard +0.8 This question requires understanding of fixed point iteration convergence theory and graphical analysis of transcendental equations. Part (a) is straightforward sketching, but part (b) requires recognizing that convergence of the iteration implies x_{n+1} = x_n at the limit, then algebraically showing this satisfies the original equation—a conceptual step beyond routine iteration problems. The manipulation of inverse trig functions adds moderate technical difficulty. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.05i 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 |
|---|---|---|
| 2(a) | Sketch a relevant graph, e.g. y = cot 2x | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| statement | B1 | Need to mark intersection with a dot, a cross, or |
| Answer | Marks |
|---|---|
| 2(b) | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Note: If using the alternative approach in (a), can stop at tan2x=cosx | B1 | Should see tan2x=cosxbefore the given |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 2:
--- 2(a) ---
2(a) | Sketch a relevant graph, e.g. y = cot 2x | B1 | Alt: use tan2xand cosx. And only one root in
π
range. (also cross at )
2
Sketch a second relevant graph on the same axes, e.g. y = sec x and justify the given
statement | B1 | Need to mark intersection with a dot, a cross, or
say roots at points of intersection, OE.
2
--- 2(b) ---
2(b) | 1
State x= tan−1(cosx)
2
and rearrange to the given equation cot 2x = sec x
Note: If using the alternative approach in (a), can stop at tan2x=cosx | B1 | Should see tan2x=cosxbefore the given
conclusion.
1
Or rearrange cot 2x = sec x to x= tan−1(cosx)
2
and state iterative formula
1
x = tan−1(cosx ).
n+1 2 n
1
Question | Answer | Marks | Guidance\begin{enumerate}[label=(\alph*)]
\item By sketching a suitable pair of graphs, show that the equation $\cot 2x = \sec x$ has exactly one root in the interval $0 < x < \frac{1}{2}\pi$. [2]
\item Show that if a sequence of real values given by the iterative formula
$$x_{n+1} = \frac{1}{2}\tan^{-1}(\cos x_n)$$
converges, then it converges to the root in part (a). [1]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2024 Q2 [3]}}