| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2019 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Find constant from definite integral |
| Difficulty | Standard +0.3 Part (i) requires straightforward integration of standard functions and algebraic rearrangement. Parts (ii) and (iii) involve routine numerical methods (interval testing and iteration) that are standard P2 content. The question is slightly above average difficulty due to the multi-part structure and need for careful algebraic manipulation, but all techniques are standard textbook exercises with no novel insight required. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Integrate to obtain form \(x^3 + k_1 \sin 2x + k_2 \cos x\) | \*M1 | |
| Obtain correct \(x^3 + 2\sin 2x + \cos x\) | A1 | |
| Apply limits correctly and equate to 2 | DM1 | |
| Confirm given result | A1 | AG; necessary detail needed |
| 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Consider sign of \(a - \sqrt[3]{3} - 2\sin 2a - \cos a\) or equivalent for \(0.5\) and \(0.75\) | M1 | |
| Obtain \(-0.26\) and \(0.10\) or equivalents and justify conclusion | A1 | AG; necessary detail needed |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use iterative process correctly at least once | M1 | Need to see a correct \(x_3\), may be implied by \(x_1 = 0.5\) so \(x_3 = 0.65256\) or \(x_1 = 0.75\) so \(x_3 = 0.64897\) OE. Must be working with radians |
| Obtain final answer \(0.651\) | A1 | |
| Show sufficient iterations to 5sf to justify answer or show a sign change in the interval \([0.6505,\ 0.6515]\) | A1 | |
| 3 |
## Question 5(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Integrate to obtain form $x^3 + k_1 \sin 2x + k_2 \cos x$ | **\*M1** | |
| Obtain correct $x^3 + 2\sin 2x + \cos x$ | **A1** | |
| Apply limits correctly and equate to 2 | **DM1** | |
| Confirm given result | **A1** | AG; necessary detail needed |
| | **4** | |
---
## Question 5(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Consider sign of $a - \sqrt[3]{3} - 2\sin 2a - \cos a$ or equivalent for $0.5$ and $0.75$ | **M1** | |
| Obtain $-0.26$ and $0.10$ or equivalents and justify conclusion | **A1** | AG; necessary detail needed |
| | **2** | |
---
## Question 5(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use iterative process correctly at least once | **M1** | Need to see a correct $x_3$, may be implied by $x_1 = 0.5$ so $x_3 = 0.65256$ or $x_1 = 0.75$ so $x_3 = 0.64897$ OE. Must be working with radians |
| Obtain final answer $0.651$ | **A1** | |
| Show sufficient iterations to 5sf to justify answer or show a sign change in the interval $[0.6505,\ 0.6515]$ | **A1** | |
| | **3** | |
---5 It is given that $\int _ { 0 } ^ { a } \left( 3 x ^ { 2 } + 4 \cos 2 x - \sin x \right) \mathrm { d } x = 2$, where $a$ is a constant.\\
(i) Show that $a = \sqrt [ 3 ] { } ( 3 - 2 \sin 2 a - \cos a )$.\\
(ii) Using the equation in part (i), show by calculation that $0.5 < a < 0.75$.\\
(iii) Use an iterative formula, based on the equation in part (i), to find the value of $a$ correct to 3 significant figures. Give the result of each iteration to 5 significant figures.\\
\hfill \mbox{\textit{CAIE P2 2019 Q5 [9]}}