| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2019 |
| Session | October |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Deduce related integral from numerical approximation |
| Difficulty | Standard +0.3 Part (a) is a routine sketch of an exponential decay curve. Part (b) is a standard trapezium rule application with values provided. Part (c) requires recognizing that x(x-3) = x² - 3x, then using linearity of integration to relate it to part (b), requiring one additional integral calculation. This is slightly above average due to the algebraic manipulation and connection between parts, but remains a standard P2 exercise. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.09f Trapezium rule: numerical integration |
| \(x\) | 2 | 2.5 | 3 | 3.5 | 4 |
| \(y\) | 4.25 | 6.427 | 9.125 | 12.34 | 16.06 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Sketch: shape in quadrants 1 and 2 only, or \(y\)-intercept at 1 | M1 | Condone slips of pen; gradient tending to 0 as \(x\) increases; condone \((1,0)\) on correct axis |
| Fully correct graph with \((0,1)\) marked, asymptote not appearing to be \(y=1\) | A1 | Shape in Q1 and Q2 only, \(y\)-intercept at 1; asymptote at least half way below \(y\)-intercept |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(h=0.5\) | B1 | Seen or implied by sight of \(\frac{0.5}{2}\) in front of bracket |
| Area \(\approx \frac{0.5}{2}\{4.25+16.06+2\times(6.427+9.125+12.34)\}\) | M1 | Correct bracket structure, condoning slips copying from table or omission of final bracket |
| \(= \text{awrt } 19.0\) | A1 | 19.0235 but awrt 19.0; calculator answer 18.937… is A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int_2^4\left(x(x-3)+\left(\frac{1}{2}\right)^x\right)dx = \int_2^4\left(x^2+\left(\frac{1}{2}\right)^x-3x\right)dx = (b)-\left[\frac{3}{2}x^2\right]_2^4\) | M1 | Correct method |
| \(= \text{awrt } 1.0\) | A1ft | Follow through |
## Question 5:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Sketch: shape in quadrants 1 and 2 only, or $y$-intercept at 1 | M1 | Condone slips of pen; gradient tending to 0 as $x$ increases; condone $(1,0)$ on correct axis |
| Fully correct graph with $(0,1)$ marked, asymptote not appearing to be $y=1$ | A1 | Shape in Q1 and Q2 only, $y$-intercept at 1; asymptote at least half way below $y$-intercept |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $h=0.5$ | B1 | Seen or implied by sight of $\frac{0.5}{2}$ in front of bracket |
| Area $\approx \frac{0.5}{2}\{4.25+16.06+2\times(6.427+9.125+12.34)\}$ | M1 | Correct bracket structure, condoning slips copying from table or omission of final bracket |
| $= \text{awrt } 19.0$ | A1 | 19.0235 but awrt 19.0; **calculator answer 18.937… is A0** |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int_2^4\left(x(x-3)+\left(\frac{1}{2}\right)^x\right)dx = \int_2^4\left(x^2+\left(\frac{1}{2}\right)^x-3x\right)dx = (b)-\left[\frac{3}{2}x^2\right]_2^4$ | M1 | Correct method |
| $= \text{awrt } 1.0$ | A1ft | Follow through |5. (a) Given $0 < a < 1$, sketch the curve with equation
$$y = a ^ { x }$$
showing the coordinates of the point at which the curve crosses the $y$-axis.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 2 & 2.5 & 3 & 3.5 & 4 \\
\hline
$y$ & 4.25 & 6.427 & 9.125 & 12.34 & 16.06 \\
\hline
\end{tabular}
\end{center}
The table above shows corresponding values of $x$ and $y$ for $y = x ^ { 2 } + \left( \frac { 1 } { 2 } \right) ^ { x }$
The values of $y$ are given to 4 significant figures as appropriate.\\
Using the trapezium rule with all the values of $y$ in the given table,\\
(b) obtain an estimate for $\int _ { 2 } ^ { 4 } \left( x ^ { 2 } + \left( \frac { 1 } { 2 } \right) ^ { x } \right) \mathrm { d } x$
Using your answer to part (b) and making your method clear, estimate\\
(c) $\quad \int _ { 2 } ^ { 4 } \left( x ( x - 3 ) + \left( \frac { 1 } { 2 } \right) ^ { x } \right) \mathrm { d } x$
\hfill \mbox{\textit{Edexcel P2 2019 Q5 [7]}}