| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2008 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from area/geometry |
| Difficulty | Standard +0.8 Part (i) requires setting up geometric relationships (perimeter of sector equals half rectangle perimeter) and algebraic manipulation to derive the given equation—a multi-step problem requiring careful reasoning. Part (ii) is routine application of fixed point iteration with a given formula. The geometry setup and equation derivation elevate this above average difficulty, but it remains accessible with standard A-level techniques. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply \(r = a \cos ec\,x\), or equivalent | B1 | |
| Using perimeters, obtain a correct equation in \(x\), e.g. \(2a \cos ec\,x + ax \cos ec\,x = 4a\), or \(2r + rx = 4a\) | B1 | |
| Deduce the given form of equation correctly | B1 | [3 marks] |
| (ii) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 0.76 | A1 | |
| Show sufficient iterations to 4 d.p. to justify its accuracy to 2 d.p., or show that there is a sign change in the value of \(\sin x - \frac{1}{4}(2+x)\) in the interval \((0.755, 0.765)\) | A1 | [3 marks] |
**(i)** State or imply $r = a \cos ec\,x$, or equivalent | B1 |
Using perimeters, obtain a correct equation in $x$, e.g. $2a \cos ec\,x + ax \cos ec\,x = 4a$, or $2r + rx = 4a$ | B1 |
Deduce the given form of equation correctly | B1 | [3 marks]
**(ii)** Use the iterative formula correctly at least once | M1 |
Obtain final answer 0.76 | A1 |
Show sufficient iterations to 4 d.p. to justify its accuracy to 2 d.p., or show that there is a sign change in the value of $\sin x - \frac{1}{4}(2+x)$ in the interval $(0.755, 0.765)$ | A1 | [3 marks]
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In the diagram, $A B C D$ is a rectangle with $A B = 3 a$ and $A D = a$. A circular arc, with centre $A$ and radius $r$, joins points $M$ and $N$ on $A B$ and $C D$ respectively. The angle $M A N$ is $x$ radians. The perimeter of the sector $A M N$ is equal to half the perimeter of the rectangle.\\
(i) Show that $x$ satisfies the equation
$$\sin x = \frac { 1 } { 4 } ( 2 + x ) \text {. }$$
(ii) This equation has only one root in the interval $0 < x < \frac { 1 } { 2 } \pi$. Use the iterative formula
$$x _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 2 + x _ { n } } { 4 } \right) ,$$
with initial value $x _ { 1 } = 0.8$, to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2008 Q3 [6]}}