| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from integral condition |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring integration by parts (standard technique), algebraic rearrangement, and mechanical application of an iterative formula. All steps are routine for P3 level with no novel insight required, making it slightly easier than average. |
| Spec | 1.08i Integration by parts1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Integrate and reach \(\pm x\sin x \mp \int \sin x \, dx\) | M1* | |
| Obtain integral \(x\sin x + \cos x\) | A1 | |
| Substitute limits correctly, must be seen since AG, and equate result to 0.5 | M1(dep*) | |
| Obtain the given form of the equation | A1 | 4 |
| (ii) EITHER: Consider the sign of a relevant expression at \(a = 1\) and at another relevant value, e.g. \(a = 1.5 \leq \frac{\pi}{2}\) | M1 | |
| OR: Using limits correctly, consider the sign of \([x\sin x + \cos x]_0^a - 0.5\), or compare the value of \([x\sin x + \cos x]_0^a\) with 0.5, for \(a = 1\) AND for another relevant value, e.g. \(a = 1.5 \leq \frac{\pi}{2}\) | M1 | |
| Complete the argument, so change of sign, or above and below stated, both with correct calculated values | A1 | 2 |
| (iii) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 1.2461 | A1 | |
| Show sufficient iterations to 6 d.p. to justify 1.2461 to 4 d.p., or show there is a sign change in the interval (1.24605, 1.24615) | A1 | 3 |
**(i)** Integrate and reach $\pm x\sin x \mp \int \sin x \, dx$ | M1* |
Obtain integral $x\sin x + \cos x$ | A1 |
Substitute limits correctly, must be seen since AG, and equate result to 0.5 | M1(dep*) |
Obtain the given form of the equation | A1 | 4 |
**(ii)** **EITHER:** Consider the sign of a relevant expression at $a = 1$ and at another relevant value, e.g. $a = 1.5 \leq \frac{\pi}{2}$ | M1 |
**OR:** Using limits correctly, consider the sign of $[x\sin x + \cos x]_0^a - 0.5$, or compare the value of $[x\sin x + \cos x]_0^a$ with 0.5, for $a = 1$ AND for another relevant value, e.g. $a = 1.5 \leq \frac{\pi}{2}$ | M1 |
Complete the argument, so change of sign, or above and below stated, both with correct calculated values | A1 | 2 |
**(iii)** Use the iterative formula correctly at least once | M1 |
Obtain final answer 1.2461 | A1 |
Show sufficient iterations to 6 d.p. to justify 1.2461 to 4 d.p., or show there is a sign change in the interval (1.24605, 1.24615) | A1 | 3 |6 It is given that $\int _ { 0 } ^ { a } x \cos x \mathrm {~d} x = 0.5$, where $0 < a < \frac { 1 } { 2 } \pi$.\\
(i) Show that $a$ satisfies the equation $\sin a = \frac { 1.5 - \cos a } { a }$.\\
(ii) Verify by calculation that $a$ is greater than 1 .\\
(iii) Use the iterative formula
$$a _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 1.5 - \cos a _ { n } } { a _ { n } } \right)$$
to determine the value of $a$ correct to 4 decimal places, giving the result of each iteration to 6 decimal places.
\hfill \mbox{\textit{CAIE P3 2015 Q6 [9]}}