| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from area/geometry |
| Difficulty | Challenging +1.2 This is a multi-part question combining geometry with numerical methods. Part (i) requires setting up an equation from geometric relationships (arc length and tangent lengths), which is moderately challenging but follows standard A-level techniques. Parts (ii) and (iii) are routine applications of interval verification and fixed-point iteration—standard Pure 3 content. The geometric setup requires some insight but the execution is straightforward, placing this slightly above average difficulty. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta1.09a Sign change methods: locate roots1.09b Sign change methods: understand failure cases1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply \(AT = r\tan x\) or \(BT = r\tan x\) | B1 | |
| Use correct arc formula and form an equation in \(r\) and \(x\) | M1 | |
| Rearrange in the given form | A1 | [3] |
| (ii) Calculate values of a relevant expression or expressions at \(x = 1\) and \(x = 1.3\) | M1 | |
| Complete the argument correctly with correct calculated values | A1 | [2] |
| (iii) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 1.11 | A1 | |
| Show sufficient iterations to 4 d.p. to justify 1.11 to 2 d.p., or show there is a sign change in the interval (1.105, 1.115) | A1 | [3] |
**(i)** State or imply $AT = r\tan x$ or $BT = r\tan x$ | B1 |
Use correct arc formula and form an equation in $r$ and $x$ | M1 |
Rearrange in the given form | A1 | [3]
**(ii)** Calculate values of a relevant expression or expressions at $x = 1$ and $x = 1.3$ | M1 |
Complete the argument correctly with correct calculated values | A1 | [2]
**(iii)** Use the iterative formula correctly at least once | M1 |
Obtain final answer 1.11 | A1 |
Show sufficient iterations to 4 d.p. to justify 1.11 to 2 d.p., or show there is a sign change in the interval (1.105, 1.115) | A1 | [3]5\\
\includegraphics[max width=\textwidth, alt={}, center]{d1377d66-73c8-4d97-9cae-d784b41fb0a8-2_519_800_1359_669}
The diagram shows a circle with centre $O$ and radius $r$. The tangents to the circle at the points $A$ and $B$ meet at $T$, and the angle $A O B$ is $2 x$ radians. The shaded region is bounded by the tangents $A T$ and $B T$, and by the minor $\operatorname { arc } A B$. The perimeter of the shaded region is equal to the circumference of the circle.\\
(i) Show that $x$ satisfies the equation
$$\tan x = \pi - x .$$
(ii) This equation has one root in the interval $0 < x < \frac { 1 } { 2 } \pi$. Verify by calculation that this root lies between 1 and 1.3.\\
(iii) Use the iterative formula
$$x _ { n + 1 } = \tan ^ { - 1 } \left( \pi - x _ { n } \right)$$
to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2015 Q5 [8]}}