| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| 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 This question requires setting up and equating geometric areas (sector, triangle) involving trigonometry, algebraic manipulation to derive a transcendental equation, then applying an iterative method. The geometry setup and equation derivation demand careful reasoning about areas and angles, while the iteration is routine but requires precision. More challenging than standard calculus questions due to the geometric insight needed. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta1.09d Newton-Raphson method |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply \(CT = r \tan x\) or \(OT = r \sec x\), or equivalent | B1 | |
| Using correct area formulae, form an equation in \(r\) and \(x\) | M1 | |
| Obtain the given answer correctly | A1 | [3] |
| (ii) Use the iterative formula correctly at least once | M1 | |
| Obtain the final answer \(1.35\) | A1 | |
| Show sufficient iterations to 4 d.p. to justify its accuracy to 2 d.p., or show there is a sign change in the interval \((1.345, 1.355)\) | A1 | [3] |
**(i)** State or imply $CT = r \tan x$ or $OT = r \sec x$, or equivalent | B1 |
Using correct area formulae, form an equation in $r$ and $x$ | M1 |
Obtain the given answer correctly | A1 | [3]
**(ii)** Use the iterative formula correctly at least once | M1 |
Obtain the final answer $1.35$ | A1 |
Show sufficient iterations to 4 d.p. to justify its accuracy to 2 d.p., or show there is a sign change in the interval $(1.345, 1.355)$ | A1 | [3]4\\
\includegraphics[max width=\textwidth, alt={}, center]{76371b0f-0145-4cc4-a147-27bcd749816a-2_339_1395_1089_374}
The diagram shows a semicircle $A C B$ with centre $O$ and radius $r$. The tangent at $C$ meets $A B$ produced at $T$. The angle $B O C$ is $x$ radians. The area of the shaded region is equal to the area of the semicircle.\\
(i) Show that $x$ satisfies the equation
$$\tan x = x + \pi$$
(ii) Use the iterative formula $x _ { n + 1 } = \tan ^ { - 1 } \left( x _ { n } + \pi \right)$ to determine $x$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2011 Q4 [6]}}