| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2015 |
| Session | June |
| Marks | 8 |
| Topic | Fixed Point Iteration |
| Type | Show root in interval |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question on numerical methods. Part (i) requires simple substitution to show sign change, (ii) is routine iteration, (iii) is basic graph sketching, and (iv) requires only observing that the graphs intersect once. All parts are standard textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
**Question 5:**
**(i)** $f(1.5) = 0.497494...$, $f(2) = -0.090702...$
- **M1**: Attempt evaluation of $f(1.5)$ and $f(2)$ – evaluation must be seen so $f(1.5)>0$ is not sufficient
- **A1**: Conclude correctly – refer to sign change oe. Must have correct values for $f(1.5)$ and $f(2)$. Allow rounded or truncated values – 1sf, or better
**Total for (i): [2]**
**(ii)** e.g., starting with $x_0 = 1.5$:
$x_1 = 1.9974...$
$x_2 = 1.9103...$
$x_3 = 1.9429...$
$x = 1.93$ to 2 dp
- **B1**: Correct first iterate – must start with $1.5 \leq x \leq 2$; $f(1.75)=1.9839...$, $f(2)=1.9092...$
- **M1**: Correct iteration process (at least 3). Allow iteration in degrees (gives 1.0177...)
- **A1**: Obtain 1.93 – must be 2dp exactly. Must be clear conclusion for root so A0 for e.g. $x_6 = 1.93$
**Total for (ii): [3]**
**(iii)**
- **M1***: Sketch attempt at sine graph, with period of $2\pi$, and a positive linear graph, with negative $y$-intercept
- **A1**: Both graphs fully correct for $[0,\pi]$, with some indication of scale on both axes and with $y=x-1$ passing through $\left(\frac{\pi}{2}, \approx 0.6\right)$
**Total for (iii): [2]**
**(iv)** One point of intersection oe
- **B1 d***: Allow 'they will not cross again' or equivalent
**Total for (iv): [1]**5 (i) Show that the equation $\sin x - x + 1 = 0$ has a root between 1.5 and 2 .\\
(ii) Use the iteration $x _ { n + 1 } = 1 + \sin x _ { n }$, with a suitable starting value, to find that root correct to 2 decimal places.\\
(iii) Sketch the graphs of $y = \sin x$ and $y = x - 1$, on the same set of axes, for $0 \leqslant x \leqslant \pi$.\\
(iv) Explain why the equation $\sin x - x + 1 = 0$ has no root other than the one found in part (ii). [1]
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2015 Q5 [8]}}