| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2018 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Complete table then apply trapezium rule |
| Difficulty | Easy -1.2 This is a straightforward two-part question requiring only routine application of basic techniques: (a) substituting x=15 into a simple formula, and (b) applying the trapezium rule formula with given values. Both are standard textbook exercises with no problem-solving or conceptual challenge, making it easier than average for A-level. |
| Spec | 1.09f Trapezium rule: numerical integration |
| \(x\) | 0 | 3 | 6 | 9 | 12 | 15 |
| \(y\) | 1 | 0.5 | 0.378 | 0.316 | 0.277 |
| Answer | Marks | Guidance |
|---|---|---|
| [answer/working] | [mark, e.g. M1/A1/B1] | [guidance notes] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y = 0.25\) when \(x = 15\) | B1 [1] | 0.25 or exact equivalent; allow 0.250; may not be in the table |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State \(h = 3\), or use of \(\frac{1}{2} \times 3\) | B1 | For using \(\frac{1}{2} \times 3\) or \(h=3\) or sight of \(1.5\{......(...)\}\); award for \(\frac{15-0}{5}\) or similar |
| Sight of expression \(1 + 0.25 + 2(0.5 + 0.378 + 0.316 + 0.277)\) | M1 | Look for correct sequence of terms within expression; condone bracketing error \(1.5 \times (1+0.25) + 2(0.5+0.378+0.316+0.277)\); M0 if all values in brackets are \(x\) values instead of \(y\) values |
| Area \(= \frac{1}{2} \times 3 \times \{1 + 0.25 + 2(0.5 + 0.378 + 0.316 + 0.277)\}\) = awrt 6.29 | A1ft, A1 [4] | A1ft: correct unsimplified expression with correct bracketing using their \(h\) and 0.25; A1: answer which rounds to 6.29 |
Question 1:
[answer/working] | [mark, e.g. M1/A1/B1] | [guidance notes]
```
# Question 1:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y = 0.25$ when $x = 15$ | B1 [1] | 0.25 or exact equivalent; allow 0.250; may not be in the table |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| State $h = 3$, or use of $\frac{1}{2} \times 3$ | B1 | For using $\frac{1}{2} \times 3$ or $h=3$ or sight of $1.5\{......(...)\}$; award for $\frac{15-0}{5}$ or similar |
| Sight of expression $1 + 0.25 + 2(0.5 + 0.378 + 0.316 + 0.277)$ | M1 | Look for correct sequence of terms within expression; condone bracketing error $1.5 \times (1+0.25) + 2(0.5+0.378+0.316+0.277)$; M0 if all values in brackets are $x$ values instead of $y$ values |
| Area $= \frac{1}{2} \times 3 \times \{1 + 0.25 + 2(0.5 + 0.378 + 0.316 + 0.277)\}$ = awrt 6.29 | A1ft, A1 [4] | A1ft: correct unsimplified expression with correct bracketing using their $h$ and 0.25; A1: answer which rounds to 6.29 |
---\begin{enumerate}
\item The table below shows corresponding values of $x$ and $y$ for $y = \frac { 1 } { \sqrt { ( x + 1 ) } }$, with the values\\
for $y$ rounded to 3 decimal places where necessary.
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
$x$ & 0 & 3 & 6 & 9 & 12 & 15 \\
\hline
$y$ & 1 & 0.5 & 0.378 & 0.316 & 0.277 & \\
\hline
\end{tabular}
\end{center}
(a) Complete the table by giving the value of $y$ corresponding to $x = 15$\\
(b) Use the trapezium rule with all the values of $y$ from the completed table to find an approximate value for
$$\int _ { 0 } ^ { 15 } \frac { 1 } { \sqrt { ( x + 1 ) } } \mathrm { d } x$$
giving your answer to 2 decimal places.\\
\hfill \mbox{\textit{Edexcel C12 2018 Q1 [5]}}