| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from integral condition |
| Difficulty | Challenging +1.2 This question requires integration by parts to evaluate the integral, then algebraic manipulation to derive the given equation. While it involves multiple steps (integration, applying limits, rearranging), these are all standard A-level techniques with no novel insight required. The iterative part (iii) is routine calculator work. Slightly above average difficulty due to the integration by parts and algebraic manipulation needed, but well within typical P3 scope. |
| Spec | 1.08i Integration by parts1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Commence integration by parts, reaching \(ax\sin\frac{1}{3}x - b\int\sin\frac{1}{3}x\,dx\) | *M1 | |
| Obtain \(3x\sin\frac{1}{3}x - 3\int\sin\frac{1}{3}x\,dx\) | A1 | |
| Complete integration and obtain \(3x\sin\frac{1}{3}x + 9\cos\frac{1}{3}x\) | A1 | |
| Substitute limits correctly and equate result to \(3\) in an integral of the form \(px\sin\frac{1}{3}x + q\cos\frac{1}{3}x\) | DM1 | \(3 = 3a\sin\frac{a}{3}+9\cos\frac{a}{3}(-0)-9\) |
| Obtain \(a = \dfrac{4-3\cos\frac{a}{3}}{\sin\frac{a}{3}}\) correctly | A1 | With sufficient evidence to show how they reach the given equation |
| Total | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Calculate values at \(a=2.5\) and \(a=3\) of a relevant expression or pair of expressions | M1 | \(2.5 < 2.679\) and \(3 > 2.827\). If using \(2.679\) and \(2.827\) must be linked explicitly to \(2.5\) and \(3\). Solving \(f(a)=0\): \(f(2.5)=0.179\) and \(f(3)=-0.173\) or if \(f(a)=a\sin\frac{1}{3}a+3\cos\frac{1}{3}a-4 \Rightarrow f(2.5)=-0.13\ldots, f(3)=0.145\ldots\) |
| Complete the argument correctly with correct calculated values | A1 | Accept values to 1 s.f. or better |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use the iterative process \(a_{n+1} = a_{n+1}\dfrac{4 - 3\cos\frac{1}{3}a_n}{\sin\frac{1}{3}a_n}\) correctly at least once | M1 | |
| Show sufficient iterations to at least 5 d.p. to justify 2.736 to 3d.p., or show a sign change in the interval \((2.7355, 2.7365)\) | A1 | |
| Obtain final answer \(2.736\) | A1 |
## Question 9(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Commence integration by parts, reaching $ax\sin\frac{1}{3}x - b\int\sin\frac{1}{3}x\,dx$ | *M1 | |
| Obtain $3x\sin\frac{1}{3}x - 3\int\sin\frac{1}{3}x\,dx$ | A1 | |
| Complete integration and obtain $3x\sin\frac{1}{3}x + 9\cos\frac{1}{3}x$ | A1 | |
| Substitute limits correctly and equate result to $3$ in an integral of the form $px\sin\frac{1}{3}x + q\cos\frac{1}{3}x$ | DM1 | $3 = 3a\sin\frac{a}{3}+9\cos\frac{a}{3}(-0)-9$ |
| Obtain $a = \dfrac{4-3\cos\frac{a}{3}}{\sin\frac{a}{3}}$ correctly | A1 | With sufficient evidence to show how they reach the given equation |
| **Total** | **5** | |
---
## Question 9(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Calculate values at $a=2.5$ and $a=3$ of a relevant expression or pair of expressions | M1 | $2.5 < 2.679$ and $3 > 2.827$. If using $2.679$ and $2.827$ must be linked explicitly to $2.5$ and $3$. Solving $f(a)=0$: $f(2.5)=0.179$ and $f(3)=-0.173$ or if $f(a)=a\sin\frac{1}{3}a+3\cos\frac{1}{3}a-4 \Rightarrow f(2.5)=-0.13\ldots, f(3)=0.145\ldots$ |
| Complete the argument correctly with correct calculated values | A1 | Accept values to 1 s.f. or better |
| **Total** | **2** | |
## Question 9(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use the iterative process $a_{n+1} = a_{n+1}\dfrac{4 - 3\cos\frac{1}{3}a_n}{\sin\frac{1}{3}a_n}$ correctly at least once | M1 | |
| Show sufficient iterations to at least 5 d.p. to justify 2.736 to 3d.p., or show a sign change in the interval $(2.7355, 2.7365)$ | A1 | |
| Obtain final answer $2.736$ | A1 | |
---9 It is given that $\int _ { 0 } ^ { a } x \cos \frac { 1 } { 3 } x \mathrm {~d} x = 3$, where the constant $a$ is such that $0 < a < \frac { 3 } { 2 } \pi$.\\
(i) Show that $a$ satisfies the equation
$$a = \frac { 4 - 3 \cos \frac { 1 } { 3 } a } { \sin \frac { 1 } { 3 } a }$$
(ii) Verify by calculation that $a$ lies between 2.5 and 3 .\\
(iii) Use an iterative formula based on the equation in part (i) to calculate $a$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.\\
\hfill \mbox{\textit{CAIE P3 2019 Q9 [10]}}