| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Find equation satisfied by limit |
| Difficulty | Moderate -0.3 This is a straightforward iterative formula question requiring mechanical calculation followed by algebraic manipulation. Part (i) is pure computation with a calculator, and part (ii) involves the standard technique of setting x_{n+1} = x_n = α to find the fixed point equation, then solving a simple quadratic. Both parts are routine applications of well-practiced methods with no novel insight required, making it slightly easier than average. |
| Spec | 1.06g Equations with exponentials: solve a^x = b1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 3.6840 | A1 | |
| Show sufficient iterations to at least 6 d.p. to justify 3.6840, or show there is a sign change in the interval (3.68395, 3.68405) | A1 | [3] |
| (ii) State a suitable equation, e.g. \(x = \frac{x(x^3 + 100)}{2(x^3 + 25)}\) | B1 | |
| State that the value of \(\alpha\) is \(3\sqrt{50}\), or exact equivalent | B1 | [2] |
**(i)** Use the iterative formula correctly at least once | M1 |
Obtain final answer 3.6840 | A1 |
Show sufficient iterations to at least 6 d.p. to justify 3.6840, or show there is a sign change in the interval (3.68395, 3.68405) | A1 | [3]
**(ii)** State a suitable equation, e.g. $x = \frac{x(x^3 + 100)}{2(x^3 + 25)}$ | B1 |
State that the value of $\alpha$ is $3\sqrt{50}$, or exact equivalent | B1 | [2]The sequence of values given by the iterative formula
$$x_{n+1} = \frac{x_n(x_n^2 + 100)}{2(x_n^2 + 25)},$$
with initial value $x_1 = 3.5$, converges to $\alpha$.
\begin{enumerate}[label=(\roman*)]
\item Use this formula to calculate $\alpha$ correct to 4 decimal places, showing the result of each iteration to 6 decimal places. [3]
\item State an equation satisfied by $\alpha$ and hence find the exact value of $\alpha$. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2013 Q2 [5]}}