| Exam Board | OCR |
|---|---|
| Module | H240/01 (Pure Mathematics) |
| Year | 2019 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from area/geometry |
| Difficulty | Standard +0.3 This is a structured multi-part question combining circle geometry with fixed point iteration. Part (a) requires standard sector/segment area formulas (routine A-level). Parts (b)-(d) are guided applications of iteration theory with explicit formulas given—students follow prescribed steps rather than devise their own approach. The convergence check in (b) is straightforward differentiation and evaluation. This is slightly easier than average due to heavy scaffolding and standard techniques. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{1}{2} \times 6^2 \times \theta\) | B1 | Correct area of sector soi. Could be part of attempt at area of segment. Allow unsimplified, inc \(\pi \times 6^2 \times \frac{\theta}{2\pi}\) |
| \(\frac{1}{2} \times 6^2 \times (\theta - \sin\theta) = 7.2\) | M1 | Attempt area of segment and equate to \(7.2\). Any equivalent method eg sector area = triangle area + 7.2. Correct formula for area of triangle. Area of sector must be \(\left(\frac{1}{2}\right) \times 6^2 \times \theta\) |
| \(\theta - \sin\theta = \frac{7.2}{18} = 0.4\), \(\theta = 0.4 + \sin\theta\) AG | A1 | Rearrange to obtain given answer. At least one line of working needed after equating to \(7.2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(F'(1.2) = \cos(1.2) = 0.362\) | B1* | Correct \(F'(1.2)\). Allow \(0.36\) |
| \( | F'(1.2) | < 1\) so iteration will converge |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(1.3320\) | B1 | Correct first iterate. Allow \(1.332\) |
| \(1.3716, 1.3802, 1.3819, 1.3822, 1.3823\ldots\) | M1 | Attempt correct process. At least 3 iterations in total. Must be given to at least 4 sf |
| hence \(\theta = 1.382\) | A1 | Obtain \(\theta = 1.382\) (must be 4sf). Following at least 2 iterations that are \(1.382\) to 4sf. A0 if given as \(\theta_k = \ldots\) Allow recovery from an incorrect interim value (but B0M1A1 if first iterate is wrong) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(1.3815 - 0.4 - \sin 1.3815 = -0.000637\) and \(1.3825 - 0.4 - \sin 1.3825 = 0.000175\) | M1 | Attempt relevant values either side of their root, using relevant \(\theta\)s. Allow other valid methods as long as they have one value either side of the root (\(1.382284\ldots\)) and both round to \(1.382\), to 4sf. Could use \(\sin\theta + 0.4 = \theta\) and compare inequality signs |
| A1 | Obtain both correct values. For their \(\theta\) values. At least 1 sf. Allow truncating not rounding | |
| \(f(1.3815) < 0\) and \(f(1.3825) > 0\), by sign change \(1.3815 < \theta < 1.3825\) so must \(1.382\) correct to 4sf | B1 | Must refer to sign change. Must be fully correct to award B1 |
# Question 10:
## Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{1}{2} \times 6^2 \times \theta$ | B1 | Correct area of sector soi. Could be part of attempt at area of segment. Allow unsimplified, inc $\pi \times 6^2 \times \frac{\theta}{2\pi}$ |
| $\frac{1}{2} \times 6^2 \times (\theta - \sin\theta) = 7.2$ | M1 | Attempt area of segment and equate to $7.2$. Any equivalent method eg sector area = triangle area + 7.2. Correct formula for area of triangle. Area of sector must be $\left(\frac{1}{2}\right) \times 6^2 \times \theta$ |
| $\theta - \sin\theta = \frac{7.2}{18} = 0.4$, $\theta = 0.4 + \sin\theta$ **AG** | A1 | Rearrange to obtain given answer. At least one line of working needed after equating to $7.2$ |
## Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $F'(1.2) = \cos(1.2) = 0.362$ | B1* | Correct $F'(1.2)$. Allow $0.36$ |
| $|F'(1.2)| < 1$ so iteration will converge | B1d* | Identify correct condition. Must be $|F'(1.2)| < 1$ or $-1 < F'(1.2) < 1$ oe. $F'(1.2) < 1$ is B0 as condition is incomplete, but condone $0 < F'(1.2) < 1$ oe in words |
## Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $1.3320$ | B1 | Correct first iterate. Allow $1.332$ |
| $1.3716, 1.3802, 1.3819, 1.3822, 1.3823\ldots$ | M1 | Attempt correct process. At least 3 iterations in total. Must be given to at least 4 sf |
| hence $\theta = 1.382$ | A1 | Obtain $\theta = 1.382$ (must be 4sf). Following at least 2 iterations that are $1.382$ to 4sf. A0 if given as $\theta_k = \ldots$ Allow recovery from an incorrect interim value (but B0M1A1 if first iterate is wrong) |
## Part (d):
| Answer | Mark | Guidance |
|--------|------|----------|
| $1.3815 - 0.4 - \sin 1.3815 = -0.000637$ and $1.3825 - 0.4 - \sin 1.3825 = 0.000175$ | M1 | Attempt relevant values either side of their root, using relevant $\theta$s. Allow other valid methods as long as they have one value either side of the root ($1.382284\ldots$) and both round to $1.382$, to 4sf. Could use $\sin\theta + 0.4 = \theta$ and compare inequality signs |
| | A1 | Obtain both correct values. For their $\theta$ values. At least 1 sf. Allow truncating not rounding |
| $f(1.3815) < 0$ and $f(1.3825) > 0$, by sign change $1.3815 < \theta < 1.3825$ so must $1.382$ correct to 4sf | B1 | Must refer to sign change. Must be fully correct to award B1 |
---10\\
\includegraphics[max width=\textwidth, alt={}, center]{05bec6d6-b526-4b6f-86f3-39aa38cbf5c6-6_405_661_251_703}
The diagram shows a sector $A O B$ of a circle with centre $O$ and radius 6 cm .\\
The angle $A O B$ is $\theta$ radians.\\
The area of the segment bounded by the chord $A B$ and the $\operatorname { arc } A B$ is $7.2 \mathrm {~cm} ^ { 2 }$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\theta = 0.4 + \sin \theta$.
\item Let $\mathrm { F } ( \theta ) = 0.4 + \sin \theta$.
By considering the value of $\mathrm { F } ^ { \prime } ( \theta )$ where $\theta = 1.2$, explain why using an iterative method based on the equation in part (a) will converge to the root, assuming that 1.2 is sufficiently close to the root.
\item Use the iterative formula $\theta _ { n + 1 } = 0.4 + \sin \theta _ { n }$ with a starting value of 1.2 to find the value of $\theta$ correct to 4 significant figures.\\
You should show the result of each iteration.
\item Use a change of sign method to show that the value of $\theta$ found in part (c) is correct to 4 significant figures.
\end{enumerate}
\hfill \mbox{\textit{OCR H240/01 2019 Q10 [11]}}