| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2020 |
| Session | October |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Deduce related integral from numerical approximation |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question requiring standard trapezium rule application, then recognizing a constant factor can be pulled out of an integral (√9 = 3), and finally comparing estimates. The 'deduction' in part (b) is a simple algebraic manipulation rather than genuine problem-solving, making this slightly easier than average. |
| Spec | 1.09f Trapezium rule: numerical integration |
| \(x\) | 0.5 | 1 | 1.5 | 2 | 2.5 |
| \(y\) | 0.5774 | 0.7071 | 0.7746 | 0.8165 | 0.8452 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(h = 0.5\) | B1 | AO 1.1a |
| \(A \approx \frac{0.5}{2}\{0.5774 + 0.8452 + 2(0.7071 + 0.7746 + 0.8165)\}\) | M1 | AO 1.1b |
| \(= \text{awrt } 1.50\) | A1 | AO 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(3 \times \text{their (a)}\) | B1ft | AO 2.2a |
| If (a) correct, allow awrt 4.50 or awrt 4.51 even with no working. Only allow 4.5 if (a) is correct and working shown e.g. \(3 \times 1.5\) | If (a) incorrect allow \(3 \times \text{their (a)}\) to at least 3sf |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Sensible comment about accuracy, e.g. answer accurate to 2sf or 1dp; \(4.535 \approx 4.50\); very accurate as 4.535 to 2sf is 4.5; it is an underestimate but quite close; less than 1% out | B1 | AO 3.2b |
| OR calculates percentage error correctly using awrt 4.50, 4.51, or 4.5: \(\frac{\ | 4.535-4.50\ | }{4.535}\times 100 = 0.77\%\) or \(\frac{\ |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(h = 0.5\) | B1 | May be implied by \(\frac{1}{4} \times \{\)... below |
| \(\frac{1}{2}h\times\{0.5774 + 0.8452 + 2(0.7071 + 0.7746 + 0.8165)\}\) | M1 | Correct attempt at trapezium rule; must use all \(y\) values with no repeats; clear attempt at \(\frac{1}{2}h \times(\text{first } y + \text{last } y + 2\times\text{"sum of rest"})\) |
| \(\approx 1.50\) | A1 | Correct answers with no working – send to review |
# Question 1:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $h = 0.5$ | B1 | AO 1.1a |
| $A \approx \frac{0.5}{2}\{0.5774 + 0.8452 + 2(0.7071 + 0.7746 + 0.8165)\}$ | M1 | AO 1.1b |
| $= \text{awrt } 1.50$ | A1 | AO 1.1b |
**Total: 3 marks**
---
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3 \times \text{their (a)}$ | B1ft | AO 2.2a |
| If (a) correct, allow awrt 4.50 or awrt 4.51 even with no working. Only allow 4.5 if (a) is correct and working shown e.g. $3 \times 1.5$ | | If (a) incorrect allow $3 \times \text{their (a)}$ to at least 3sf |
**Total: 1 mark**
---
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Sensible comment about accuracy, e.g. answer accurate to 2sf or 1dp; $4.535 \approx 4.50$; very accurate as 4.535 to 2sf is 4.5; it is an underestimate but quite close; less than 1% out | B1 | AO 3.2b |
| **OR** calculates percentage error correctly using awrt 4.50, 4.51, or 4.5: $\frac{\|4.535-4.50\|}{4.535}\times 100 = 0.77\%$ or $\frac{\|4.535-4.51\|}{4.535}\times 100 = 0.55\%$ or $\frac{\|4.535-4.51425\|}{4.535}\times 100 = 0.46\%$ or $\left\|\frac{4.50}{4.535}\right\|\times 100 = 99\%$ | | This mark depends on B1 in (b) having been awarded with awrt 4.5. **Withheld if contradictory statements present.** |
**Total: 1 mark**
**Question 1 Total: 5 marks**
# Question 1 (Trapezium Rule):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $h = 0.5$ | B1 | May be implied by $\frac{1}{4} \times \{$... below |
| $\frac{1}{2}h\times\{0.5774 + 0.8452 + 2(0.7071 + 0.7746 + 0.8165)\}$ | M1 | Correct attempt at trapezium rule; must use all $y$ values with no repeats; clear attempt at $\frac{1}{2}h \times(\text{first } y + \text{last } y + 2\times\text{"sum of rest"})$ |
| $\approx 1.50$ | A1 | Correct answers with no working – send to review |
---1 The table below shows corresponding values of $x$ and $y$ for $y = \sqrt { \frac { x } { 1 + x } }$\\
The values of $y$ are given to 4 significant figures.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 0.5 & 1 & 1.5 & 2 & 2.5 \\
\hline
$y$ & 0.5774 & 0.7071 & 0.7746 & 0.8165 & 0.8452 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Use the trapezium rule, with all the values of $y$ in the table, to find an estimate for
$$\int _ { 0.5 } ^ { 2.5 } \sqrt { \frac { x } { 1 + x } } \mathrm {~d} x$$
giving your answer to 3 significant figures.
\item Using your answer to part (a), deduce an estimate for $\int _ { 0.5 } ^ { 2.5 } \sqrt { \frac { 9 x } { 1 + x } } \mathrm {~d} x$
Given that
$$\int _ { 0.5 } ^ { 2.5 } \sqrt { \frac { 9 x } { 1 + x } } \mathrm {~d} x = 4.535 \text { to } 4 \text { significant figures }$$
\item comment on the accuracy of your answer to part (b).
\end{enumerate}
\hfill \mbox{\textit{Edexcel Paper 2 2020 Q1 [5]}}