| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Solve trigonometric equation via iteration |
| Difficulty | Challenging +1.2 Part (i) requires setting up a geometric area equation involving sectors and segments, using standard circle formulas (area = πr²/2, sector area, chord length from cosine rule). This is multi-step but follows a clear template. Part (ii) is straightforward iteration with a given formula—purely computational with no problem-solving required. The geometry setup is moderately challenging but well within standard P3 scope, making this slightly above average difficulty overall. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta1.08e Area between curve and x-axis: using definite integrals1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply \(AB=2r\cos\theta\) or \(AB^{2}=2r^{2}-2r^{2}\cos(\pi-2\theta)\) | B1 | |
| Use correct formula to express the area of sector \(ABC\) in terms of \(r\) and \(\theta\) | M1 | |
| Use correct area formulae to express the area of a segment in terms of \(r\) and \(\theta\) | M1 | |
| State a correct equation in \(r\) and \(\theta\) in any form | A1 | |
| Obtain the given answer | A1 | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 0.95 | A1 | |
| Show sufficient iterations to 4 d.p. to justify 0.95 to 2 d.p., or show there is a sign change in the interval (0.945, 0.955) | A1 | [3] |
**(i)** State or imply $AB=2r\cos\theta$ or $AB^{2}=2r^{2}-2r^{2}\cos(\pi-2\theta)$ | B1 |
Use correct formula to express the area of sector $ABC$ in terms of $r$ and $\theta$ | M1 |
Use correct area formulae to express the area of a segment in terms of $r$ and $\theta$ | M1 |
State a correct equation in $r$ and $\theta$ in any form | A1 |
Obtain the given answer | A1 | [5]
[SR: If the complete equation is approached by adding two sectors to the shaded area above $BO$ and $OC$ give the first M1 as on the scheme, and the second M1 for using correct area formulae for a triangle $AOB$ or $AOC$, and a sector $AOB$ or $AOC$.]
**(ii)** Use the iterative formula correctly at least once | M1 |
Obtain final answer 0.95 | A1 |
Show sufficient iterations to 4 d.p. to justify 0.95 to 2 d.p., or show there is a sign change in the interval (0.945, 0.955) | A1 | [3]6\\
\includegraphics[max width=\textwidth, alt={}, center]{dd7b2aee-4318-48e8-97c0-541e47f2e83a-2_551_567_1416_788}
In the diagram, $A$ is a point on the circumference of a circle with centre $O$ and radius $r$. A circular arc with centre $A$ meets the circumference at $B$ and $C$. The angle $O A B$ is $\theta$ radians. The shaded region is bounded by the circumference of the circle and the arc with centre $A$ joining $B$ and $C$. The area of the shaded region is equal to half the area of the circle.\\
(i) Show that $\cos 2 \theta = \frac { 2 \sin 2 \theta - \pi } { 4 \theta }$.\\
(ii) Use the iterative formula
$$\theta _ { n + 1 } = \frac { 1 } { 2 } \cos ^ { - 1 } \left( \frac { 2 \sin 2 \theta _ { n } - \pi } { 4 \theta _ { n } } \right)$$
with initial value $\theta _ { 1 } = 1$, to determine $\theta$ correct to 2 decimal places, showing the result of each iteration to 4 decimal places.\\
$7 \quad$ Let $\mathrm { f } ( x ) = \frac { 2 x ^ { 2 } - 7 x - 1 } { ( x - 2 ) \left( x ^ { 2 } + 3 \right) }$.\\
(i) Express $\mathrm { f } ( x )$ in partial fractions.\\
(ii) Hence obtain the expansion of $\mathrm { f } ( x )$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$.
\hfill \mbox{\textit{CAIE P3 2013 Q6 [8]}}