| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Deduce related integral from numerical approximation |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard P2 skills: sketching an exponential curve with asymptote, applying the trapezium rule with given values (pure calculation), then using linearity of integration to deduce related integral values. Part (c)(ii) requires recognizing symmetry properties but is still routine manipulation. Slightly easier than average due to scaffolding and computational rather than conceptual demands. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.06b Gradient of e^(kx): derivative and exponential model1.09f Trapezium rule: numerical integration |
| \(x\) | - 4 | - 1.5 | 1 | 3.5 | 6 | 8.5 |
| \(y\) | 13 | 6.280 | 4.577 | 4.146 | 4.037 | 4.009 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Correct shape (exponentially decreasing curve in quadrants 1 and 2, not touching x-axis) | B1 | Does not appear in quadrants 3 or 4; do not penalise parts which appear linear |
| Correct intercept at \((0, 5)\) | B1 | Condone invisible brackets; may have 5 marked on y-axis; if labelled \((5,0)\) condone if clearly a y-intercept above x-axis |
| Correct equation of asymptote \(y = 4\) | B1 | Must be an equation; asymptote does not need to be drawn but must be clear it refers to the graph not the y-intercept; if more than one horizontal asymptote given, B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Strip width \(= 2.5\) | B1 | May be implied by \(1.25\) or equivalent in front of bracket |
| \(\int_{-4}^{8.5}\left(3^{-\frac{1}{2}x}+4\right)dx \approx \frac{5}{4}\{13+4.009+2\times(6.280+4.577+4.146+4.037)\}\) | M1 | Correct application of trapezium rule with all y-values and their \(h\); condone missing trailing bracket; also award for individual trapezia summed |
| \(= 68.86125\) (= awrt 69) | A1 | Exact answer is \(68.86125\); note calculator answer for the integral is \(66.367...\) which scores 0 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int_{-4}^{8.5}\left(3^{-\frac{1}{2}x}\right)dx = 69 - 4\times(8.5--4)\) = awrt 19 | M1 | Attempts \(\text{"69"}\pm 4\times(8.5--4)\) or \(\text{"69"}\pm\int_{-4}^{8.5}4\,dx\) or \((b)\pm 50\) |
| awrt 19 | A1ft | Follow through on \((b)-50\) |
| Note: Attempts at redoing the trapezium rule with new values is M0A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int_{-4}^{8.5}\left(3^{-\frac{1}{2}x}+4\right)dx + \int_{-8.5}^{4}\left(3^{\frac{1}{2}x}+4\right)dx\) = awrt 138 | B1ft | Follow through on \(2\times(b)\) |
## Question 4:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct shape (exponentially decreasing curve in quadrants 1 and 2, not touching x-axis) | B1 | Does not appear in quadrants 3 or 4; do not penalise parts which appear linear |
| Correct intercept at $(0, 5)$ | B1 | Condone invisible brackets; may have 5 marked on y-axis; if labelled $(5,0)$ condone if clearly a y-intercept above x-axis |
| Correct equation of asymptote $y = 4$ | B1 | Must be an equation; asymptote does not need to be drawn but must be clear it refers to the graph not the y-intercept; if more than one horizontal asymptote given, B0 |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Strip width $= 2.5$ | B1 | May be implied by $1.25$ or equivalent in front of bracket |
| $\int_{-4}^{8.5}\left(3^{-\frac{1}{2}x}+4\right)dx \approx \frac{5}{4}\{13+4.009+2\times(6.280+4.577+4.146+4.037)\}$ | M1 | Correct application of trapezium rule with all y-values and their $h$; condone missing trailing bracket; also award for individual trapezia summed |
| $= 68.86125$ (= awrt 69) | A1 | Exact answer is $68.86125$; note calculator answer for the integral is $66.367...$ which scores 0 marks |
### Part (c)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int_{-4}^{8.5}\left(3^{-\frac{1}{2}x}\right)dx = 69 - 4\times(8.5--4)$ = awrt 19 | M1 | Attempts $\text{"69"}\pm 4\times(8.5--4)$ or $\text{"69"}\pm\int_{-4}^{8.5}4\,dx$ or $(b)\pm 50$ |
| awrt 19 | A1ft | Follow through on $(b)-50$ |
| **Note:** Attempts at redoing the trapezium rule with new values is M0A0 | | |
### Part (c)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int_{-4}^{8.5}\left(3^{-\frac{1}{2}x}+4\right)dx + \int_{-8.5}^{4}\left(3^{\frac{1}{2}x}+4\right)dx$ = awrt 138 | B1ft | Follow through on $2\times(b)$ |
---\begin{enumerate}
\item (a) Sketch the curve with equation
\end{enumerate}
$$y = a ^ { - x } + 4$$
where $a$ is a constant and $a > 1$\\
On your sketch show
\begin{itemize}
\item the coordinates of the point of intersection of the curve with the $y$-axis
\item the equation of the asymptote to the curve.
\end{itemize}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
$x$ & - 4 & - 1.5 & 1 & 3.5 & 6 & 8.5 \\
\hline
$y$ & 13 & 6.280 & 4.577 & 4.146 & 4.037 & 4.009 \\
\hline
\end{tabular}
\end{center}
The table above shows corresponding values of $x$ and $y$ for $y = 3 ^ { - \frac { 1 } { 2 } x } + 4$\\
The values of $y$ are given to four significant figures, as appropriate.\\
Using the trapezium rule with all the values of $y$ in the table,\\
(b) find an approximate value for
$$\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) d x$$
giving your answer to two significant figures.\\
(c) Using the answer to part (b), find an approximate value for\\
(i) $\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } \right) \mathrm { d } x$\\
(ii) $\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) \mathrm { d } x + \int _ { - 8.5 } ^ { 4 } \left( 3 ^ { \frac { 1 } { 2 } x } + 4 \right) \mathrm { d } x$
\hfill \mbox{\textit{Edexcel P2 2024 Q4 [9]}}