| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Sketch graphs to show root existence |
| Difficulty | Moderate -0.3 This is a slightly below-average C3 question. Part (a) requires sketching two simple curves and making a basic observation about their intersection—straightforward graphical reasoning. Part (b) is routine iteration with no complications: the formula is given, convergence is guaranteed, and students simply apply it mechanically until achieving the required accuracy. No problem-solving insight or technical difficulty beyond standard procedure. |
| Spec | 1.02m Graphs of functions: difference between plotting and sketching1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
\begin{enumerate}[label=(\alph*)]
\item It is given that $a$ and $b$ are positive constants. By sketching graphs of
$$y = x^5 \quad \text{and} \quad y = a - bx$$
on the same diagram, show that the equation
$$x^5 + bx - a = 0$$
has exactly one real root. [3]
\item Use the iterative formula $x_{n+1} = \sqrt[5]{53 - 2x_n}$, with a suitable starting value, to find the real root of the equation $x^5 + 2x - 53 = 0$. Show the result of each iteration, and give the root correct to 3 decimal places. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 Q3 [12]}}