| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule applied to real-world data |
| Difficulty | Moderate -0.3 This is a straightforward application of the trapezium rule with a minor twist of squaring values first. Students must calculate V², apply the standard trapezium rule formula with h=0.25, then take a square root. It's slightly easier than average because it's purely procedural with clear instructions and no conceptual challenges beyond basic formula application. |
| Spec | 1.09f Trapezium rule: numerical integration |
| \(t\) | 0 | 0.25 | 0.5 | 0.75 | 1 |
| \(V\) | - 48 | 207 | 37 | - 161 | - 29 |
| \(V ^ { 2 }\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(V^2\) values: \((-48)^2=2304\), \(207^2=42849\), \(37^2=1369\), \((-161)^2=25921\), \((-29)^2=841\) | B1 | At least 4 correct \(V^2\) values |
| Trapezium rule: \(M^2 \approx \frac{1}{2}(0.25)[(2304+841)+2(42849+1369+25921)]\) | M1 | Correct application with \(h=0.25\) |
| \(= \frac{0.25}{2}[3145 + 2(70139)]\) | ||
| \(= \frac{0.25}{2}[143423]\) | ||
| \(M^2 \approx 17927.875\) | A1 | Correct value |
| \(M \approx 133.9\) | A1 | \(M\) found by square root |
# Question 4:
| $V^2$ values: $(-48)^2=2304$, $207^2=42849$, $37^2=1369$, $(-161)^2=25921$, $(-29)^2=841$ | B1 | At least 4 correct $V^2$ values |
| Trapezium rule: $M^2 \approx \frac{1}{2}(0.25)[(2304+841)+2(42849+1369+25921)]$ | M1 | Correct application with $h=0.25$ |
| $= \frac{0.25}{2}[3145 + 2(70139)]$ | | |
| $= \frac{0.25}{2}[143423]$ | | |
| $M^2 \approx 17927.875$ | A1 | Correct value |
| $M \approx 133.9$ | A1 | $M$ found by square root |
---\begin{enumerate}
\item A measure of the effective voltage, $M$ volts, in an electrical circuit is given by
\end{enumerate}
$$M ^ { 2 } = \int _ { 0 } ^ { 1 } V ^ { 2 } \mathrm {~d} t$$
where $V$ volts is the voltage at time $t$ seconds. Pairs of values of $V$ and $t$ are given in the following table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$t$ & 0 & 0.25 & 0.5 & 0.75 & 1 \\
\hline
$V$ & - 48 & 207 & 37 & - 161 & - 29 \\
\hline
$V ^ { 2 }$ & & & & & \\
\hline
\end{tabular}
\end{center}
Use the trapezium rule with five values of $V ^ { 2 }$ to estimate the value of $M$.\\
(6)\\
\hfill \mbox{\textit{Edexcel C4 Q4 [6]}}