| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| Session | Specimen |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive stationary point equation |
| Difficulty | Standard +0.3 This is a standard A-level calculus question requiring quotient rule differentiation, setting derivative to zero, and applying a given iterative formula. While it involves multiple steps, each component (differentiation, solving tan equation via iteration) is routine for P2 level with no novel insight required. Slightly easier than average due to the iterative formula being provided. |
| Spec | 1.05o Trigonometric equations: solve in given intervals1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempt use of quotient rule or equivalent | M1 | |
| Obtain \(\frac{2(x+2)\cos 2x - \sin 2x}{(x+2)^2}\) or equivalent | A1 | |
| Equate numerator to zero and attempt rearrangement | M1 | |
| Confirm given result \(\tan 2x = 2x + 4\) | A1 | AG – Showing necessary detail |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Consider sign of \(\tan 2x - 2x - 4\) for \(0.6\) and \(0.7\) or equivalent | M1 | |
| Obtain \(-2.63\) and \(0.40\) or equivalent and justify conclusion | A1 | |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use iteration process correctly at least once | M1 | |
| Obtain final answer \(0.694\) | A1 | |
| Show sufficient iterations to 5 decimal places to justify answer OR show a sign change in the interval \((0.6935, 0.6945)\) | A1 | |
| Total | 3 |
# Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt use of quotient rule or equivalent | M1 | |
| Obtain $\frac{2(x+2)\cos 2x - \sin 2x}{(x+2)^2}$ or equivalent | A1 | |
| Equate numerator to zero and attempt rearrangement | M1 | |
| Confirm given result $\tan 2x = 2x + 4$ | A1 | AG – Showing necessary detail |
| **Total** | **4** | |
# Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Consider sign of $\tan 2x - 2x - 4$ for $0.6$ and $0.7$ or equivalent | M1 | |
| Obtain $-2.63$ and $0.40$ or equivalent and justify conclusion | A1 | |
| **Total** | **2** | |
# Question 5(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use iteration process correctly at least once | M1 | |
| Obtain final answer $0.694$ | A1 | |
| Show sufficient iterations to 5 decimal places to justify answer **OR** show a sign change in the interval $(0.6935, 0.6945)$ | A1 | |
| **Total** | **3** | |
---5\\
\includegraphics[max width=\textwidth, alt={}, center]{0af2714b-d3eb-4112-a869-eda5cf266cd8-08_410_977_274_543}
The diagram shows the curve $y = \frac { \sin 2 x } { x + 2 }$ for $0 \leqslant x \leqslant \frac { 1 } { 2 } \pi$. The $x$-coordinate of the maximum point $M$ is denoted by $\alpha$.\\
(a) Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and show that $\alpha$ satisfies the equation $\tan 2 x = 2 x + 4$.\\
(b) Show by calculation that $\alpha$ lies between 0.6 and 0.7 .\\
(c) Use the iterative formula $x _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( 2 x _ { n } + 4 \right)$ to find the value of $\alpha$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.\\
\hfill \mbox{\textit{CAIE P2 2020 Q5 [9]}}