| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Sketch graphs to show root existence |
| Difficulty | Standard +0.3 This is a standard multi-part question on iterative methods requiring graph sketching, algebraic manipulation, and numerical iteration. While it involves cosec and requires several steps, each part follows routine A-level procedures: sketching standard functions, rearranging equations algebraically, and applying a given iterative formula. The symmetry insight in part (iii)(b) is elegant but accessible. Slightly easier than average due to the structured guidance through each step. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Sketch \(y = \cos ecx\) for at least \(0, x, \pi\) | B1 | |
| Sketch \(y = x(\pi - x)\) for at least \(0, x, \pi\) | B1 | |
| Justify statement concerning two roots, with evidence of \(1\) and \(\frac{1}{4}\pi^2\) for \(y\)-values on graph via scales | B1 | [3] |
| (ii) Use \(\cos ecx = \frac{1}{\sin x}\) and commence rearrangement | M1 | |
| Obtain given equation correctly, showing sufficient detail | A1 | [2] |
| (iii) (a) Use iterative formula correctly at least once | M1 | |
| Obtain final answer \(0.66\) | A1 | |
| Show sufficient iterations to 4 decimal places to justify answer or show a sign change in the interval \((0.655, 0.665)\) | A1 | [3] |
| (b) Obtain \(2.48\) | B1 | [1] |
(i) Sketch $y = \cos ecx$ for at least $0, x, \pi$ | B1 |
Sketch $y = x(\pi - x)$ for at least $0, x, \pi$ | B1 |
Justify statement concerning two roots, with evidence of $1$ and $\frac{1}{4}\pi^2$ for $y$-values on graph via scales | B1 | [3]
(ii) Use $\cos ecx = \frac{1}{\sin x}$ and commence rearrangement | M1 |
Obtain given equation correctly, showing sufficient detail | A1 | [2]
(iii) (a) Use iterative formula correctly at least once | M1 |
Obtain final answer $0.66$ | A1 |
Show sufficient iterations to 4 decimal places to justify answer or show a sign change in the interval $(0.655, 0.665)$ | A1 | [3]
(b) Obtain $2.48$ | B1 | [1]8 (i) By sketching each of the graphs $y = \operatorname { cosec } x$ and $y = x ( \pi - x )$ for $0 < x < \pi$, show that the equation
$$\operatorname { cosec } x = x ( \pi - x )$$
has exactly two real roots in the interval $0 < x < \pi$.\\
(ii) Show that the equation $\operatorname { cosec } x = x ( \pi - x )$ can be written in the form $x = \frac { 1 + x ^ { 2 } \sin x } { \pi \sin x }$.\\
(iii) The two real roots of the equation $\operatorname { cosec } x = x ( \pi - x )$ in the interval $0 < x < \pi$ are denoted by $\alpha$ and $\beta$, where $\alpha < \beta$.
\begin{enumerate}[label=(\alph*)]
\item Use the iterative formula
$$x _ { n + 1 } = \frac { 1 + x _ { n } ^ { 2 } \sin x _ { n } } { \pi \sin x _ { n } }$$
to find $\alpha$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\item Deduce the value of $\beta$ correct to 2 decimal places.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2014 Q8 [9]}}