| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas Between Curves |
| Type | Trapezium Rule Approximation with Area |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard P2 techniques: trapezium rule application (routine calculation with given intervals), integration using substitution (standard exponential form), and combining results to find an area. Part (d) requires understanding of trapezium rule error, but this is bookwork knowledge. All parts are mechanical applications of learned methods with no problem-solving insight required, making it slightly easier than average. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08e Area between curve and x-axis: using definite integrals1.08h Integration by substitution1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use \(y\)-values \(3\), \(\dfrac{6}{1+\sqrt{2}}\), \(\dfrac{6}{1+\sqrt{3}}\), \(2\) or equivalents \(3,\ 6\sqrt{2}-6,\ 3\sqrt{3}-3,\ 2\) or \(3,\ 2.4853...,\ 2.19615...,\ 2\) | B1 | |
| Use correct formula, OE, with \(h=1\) | M1 | Allow 3 separate trapezia of width 1 |
| Obtain \(7.1814\) | A1 | AWRT |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Integrate to obtain form \(ke^{\frac{1}{2}x-2}\) | M1 | \(k \neq 1\); If \(k=2\), integration must be implied by use of square bracket notation or by substitution of limits |
| Obtain correct \(4e^{\frac{1}{2}x-2}\) | A1 | or exact equivalents |
| Obtain \(4 - 4e^{-\frac{3}{2}}\) | A1 | or exact equivalents |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Evaluate answer to part (a) minus answer to part (b) | M1 | |
| Obtain \(4.07\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State over-estimate with reference to top of each trapezium being above the [first] curve, or clear equivalent, e.g. concave up so over-estimate or convex down so over-estimate | B1 |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use $y$-values $3$, $\dfrac{6}{1+\sqrt{2}}$, $\dfrac{6}{1+\sqrt{3}}$, $2$ or equivalents $3,\ 6\sqrt{2}-6,\ 3\sqrt{3}-3,\ 2$ or $3,\ 2.4853...,\ 2.19615...,\ 2$ | B1 | |
| Use correct formula, OE, with $h=1$ | M1 | Allow 3 separate trapezia of width 1 |
| Obtain $7.1814$ | A1 | AWRT |
---
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate to obtain form $ke^{\frac{1}{2}x-2}$ | M1 | $k \neq 1$; If $k=2$, integration must be implied by use of square bracket notation or by substitution of limits |
| Obtain correct $4e^{\frac{1}{2}x-2}$ | A1 | or exact equivalents |
| Obtain $4 - 4e^{-\frac{3}{2}}$ | A1 | or exact equivalents |
---
## Question 6(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Evaluate answer to part **(a)** minus answer to part **(b)** | M1 | |
| Obtain $4.07$ | A1 | |
---
## Question 6(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| State over-estimate with reference to top of each trapezium being above the [first] curve, or clear equivalent, e.g. concave up so over-estimate or convex down so over-estimate | B1 | |6
\begin{enumerate}[label=(\alph*)]
\item Use the trapezium rule with three intervals to find an approximation to $\int _ { 1 } ^ { 4 } \frac { 6 } { 1 + \sqrt { x } } \mathrm {~d} x$. Give your answer correct to 5 significant figures.
\item Find the exact value of $\int _ { 1 } ^ { 4 } 2 \mathrm { e } ^ { \frac { 1 } { 2 } x - 2 } \mathrm {~d} x$.
\item \\
\includegraphics[max width=\textwidth, alt={}, center]{2d6fc4c5-70ec-4cd8-9b48-59d5ce0e39b7-11_556_805_262_705}
The diagram shows the curves $y = \frac { 6 } { 1 + \sqrt { x } }$ and $y = 2 \mathrm { e } ^ { \frac { 1 } { 2 } x - 2 }$ which meet at a point with $x$-coordinate 4. The shaded region is bounded by the two curves and the line $x = 1$.
Use your answers to parts (a) and (b) to find an approximation to the area of the shaded region. Give your answer correct to 3 significant figures.
\item State, with a reason, whether your answer to part (c) is an over-estimate or under-estimate of the exact area of the shaded region.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2021 Q6 [9]}}