| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| Session | June |
| 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 This is a straightforward multi-part question requiring standard integration techniques (logarithm and polynomial), algebraic rearrangement, and simple iteration. Part (a) involves routine integration and algebra, part (b) is basic substitution to verify bounds, and part (c) is mechanical iteration with no conceptual challenge. Slightly above average due to the multi-step nature and iteration component, but all techniques are standard A-level fare. |
| Spec | 1.08d Evaluate definite integrals: between limits1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Integrate to obtain the form \(k_1\ln(2x+1) + k_2x^2\) | *M1 | |
| Obtain correct \(2\ln(2x+1) + 4x^2\) | A1 | |
| Use limits correctly and attempt rearrangement | DM1 | |
| Confirm \(a = \sqrt{2.5 - 0.5\ln(2a+1)}\) | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Consider sign of \(a - \sqrt{2.5 - 0.5\ln(2a+1)}\) or equivalent for 1 and 2 | M1 | |
| Obtain \(-0.3...\) and \(0.6...\) or equivalents and justify conclusion | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use iteration process correctly at least once | M1 | |
| Obtain final answer \(1.358\) | A1 | |
| Show sufficient iterations to 6 sf to justify answer or show a sign change in the interval \([1.3575, 1.3585]\) | A1 |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate to obtain the form $k_1\ln(2x+1) + k_2x^2$ | *M1 | |
| Obtain correct $2\ln(2x+1) + 4x^2$ | A1 | |
| Use limits correctly and attempt rearrangement | DM1 | |
| Confirm $a = \sqrt{2.5 - 0.5\ln(2a+1)}$ | A1 | AG |
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Consider sign of $a - \sqrt{2.5 - 0.5\ln(2a+1)}$ or equivalent for 1 and 2 | M1 | |
| Obtain $-0.3...$ and $0.6...$ or equivalents and justify conclusion | A1 | |
## Question 7(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use iteration process correctly at least once | M1 | |
| Obtain final answer $1.358$ | A1 | |
| Show sufficient iterations to 6 sf to justify answer or show a sign change in the interval $[1.3575, 1.3585]$ | A1 | |
---7 It is given that $\int _ { 0 } ^ { a } \left( \frac { 4 } { 2 x + 1 } + 8 x \right) \mathrm { d } x = 10$, where $a$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Show that $a = \sqrt { 2.5 - 0.5 \ln ( 2 a + 1 ) }$.
\item Using the equation in part (a), show by calculation that $1 < a < 2$.
\item Use an iterative formula, based on the equation in part (a), to find the value of $a$ correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2020 Q7 [9]}}