| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2022 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Deduce related integral from numerical approximation |
| Difficulty | Moderate -0.8 This is a straightforward application of the trapezium rule followed by simple manipulations using standard integral properties. Part (a) requires only substituting given values into the trapezium rule formula. Parts (b)(i) and (b)(ii) use basic rules: multiplying by a constant (since 2^{6-√x} = 2·2^{5-√x}) and adding a constant respectively. No problem-solving insight is needed, just routine application of memorized techniques with arithmetic. |
| Spec | 1.09f Trapezium rule: numerical integration |
| \(x\) | 5 | 5.5 | 6 | 6.5 | 7 |
| \(y\) | 6.792 | 6.298 | 5.858 | 5.466 | 5.113 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(h = 0.5\) | B1 | Correct strip width; correct value for \(h\) stated or implied by \(\frac{1}{4}\{...\}\) |
| \(A \approx \frac{1}{2} \times \frac{1}{2}\{6.792 + 5.113 + 2(6.298 + 5.858 + 5.466)\}\) | M1 | Correct application of trapezium rule with their \(h\); must include the \(\frac{1}{2} \times "h"\) but may be implied by any multiple of the bracket |
| \(= 11.79\) | A1 | cao; must be to 2 d.p. (actual value is 11.78 to 2 d.p. — scores no marks if just this is seen) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(A \approx 2 \times \text{"11.79"}\) | M1 | Multiplies their answer to (a) by 2; sight of \(2 \times \text{"their }(a)\text{"}\) is sufficient |
| \(= 23.58\) | A1ft | For 23.58 or awrt 23.57 following a correct part (a); accurate answer is 23.56 to 2 d.p., which is M0A0 if no method shown |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(A \approx \text{"11.79"} + 6\) | M1 | Adds 6 to their answer to (a); the 6 need not be simplified if evaluated from an integral with correct substitution |
| \(= 17.79\) | A1ft | 17.79 or follow through their answer to (a) \(+ 6\); accurate answer is 17.78 to 2 d.p., which is M0A0 if no method shown |
## Question 1:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $h = 0.5$ | B1 | Correct strip width; correct value for $h$ stated or implied by $\frac{1}{4}\{...\}$ |
| $A \approx \frac{1}{2} \times \frac{1}{2}\{6.792 + 5.113 + 2(6.298 + 5.858 + 5.466)\}$ | M1 | Correct application of trapezium rule with their $h$; must include the $\frac{1}{2} \times "h"$ but may be implied by any multiple of the bracket |
| $= 11.79$ | A1 | cao; must be to 2 d.p. (actual value is 11.78 to 2 d.p. — scores no marks if just this is seen) |
### Part (b)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $A \approx 2 \times \text{"11.79"}$ | M1 | Multiplies their answer to (a) by 2; sight of $2 \times \text{"their }(a)\text{"}$ is sufficient |
| $= 23.58$ | A1ft | For 23.58 or awrt 23.57 following a correct part (a); accurate answer is 23.56 to 2 d.p., which is M0A0 if no method shown |
### Part (b)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $A \approx \text{"11.79"} + 6$ | M1 | Adds 6 to their answer to (a); the 6 need not be simplified if evaluated from an integral with correct substitution |
| $= 17.79$ | A1ft | 17.79 or follow through their answer to (a) $+ 6$; accurate answer is 17.78 to 2 d.p., which is M0A0 if no method shown |
---\begin{enumerate}
\item The table below shows corresponding values of $x$ and $y$ for
\end{enumerate}
$$y = 2 ^ { 5 - \sqrt { x } }$$
The values of $y$ are given to 3 decimal places.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 5 & 5.5 & 6 & 6.5 & 7 \\
\hline
$y$ & 6.792 & 6.298 & 5.858 & 5.466 & 5.113 \\
\hline
\end{tabular}
\end{center}
Using the trapezium rule with all the values of $y$ in the given table,\\
(a) obtain an estimate for
$$\int _ { 5 } ^ { 7 } 2 ^ { 5 - \sqrt { x } } \mathrm {~d} x$$
giving your answer to 2 decimal places.\\
(b) Using your answer to part (a) and making your method clear, estimate\\
(i) $\quad \int _ { 5 } ^ { 7 } 2 ^ { 6 - \sqrt { x } } \mathrm {~d} x$\\
(ii) $\int _ { 5 } ^ { 7 } \left( 3 + 2 ^ { 5 - \sqrt { x } } \right) \mathrm { d } x$
\hfill \mbox{\textit{Edexcel P2 2022 Q1 [7]}}