| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2019 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule applied to real-world data |
| Difficulty | Moderate -0.8 This is a straightforward application of the trapezium rule to estimate distance from speed-time data, with standard 5-second intervals and clear tabulated values. Part (b) requires understanding that smooth acceleration means the curve is convex, making trapeziums an underestimate—a concept typically taught alongside the trapezium rule. The calculation is routine and the reasoning is standard textbook material. |
| Spec | 1.08g Integration as limit of sum: Riemann sums3.02c Interpret kinematic graphs: gradient and area |
| Time \(( \mathrm { s } )\) | 0 | 5 | 10 | 15 | 20 | 25 |
| Speed \(\left( \mathrm { m } \mathrm { s } ^ { - 1 } \right)\) | 2 | 5 | 10 | 18 | 28 | 42 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Uses an allowable method to estimate area under curve, e.g. trapezium rule attempt | M1 | 3.1a — Low-level problem-solving mark. Condone slip on one speed |
| \(\frac{1}{2}\times(5)\times[2 + 2(5+10+18+28) + 42]\) or \(\frac{1}{2}\times[\text{"315"} + \text{"515"}]\) | M1 | Correct trapezium rule with \(h=5\); '2' and '42' in correct place |
| \(= 415\) {m} | A1 | Units not required; do not accept 415 km or 415 ms\(^{-1}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Overestimate and relevant explanation, e.g. tops of trapezia lie above curve; area of trapezia \(>\) area under curve; curve is convex; \(\frac{d^2y}{dx^2}>0\); acceleration is continually increasing; gradient of curve is continually increasing | B1ft | Depends on both an answer to (a) being obtained and first M in (a). Do not allow "curve is concave" or "gradient is positive" or references to friction/air resistance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Underestimate and relevant explanation, e.g. all the rectangles are below the curve | B1ft | Depends on both an answer to (a) and first M in (a) |
# Question 2:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Uses an allowable method to estimate area under curve, e.g. trapezium rule attempt | M1 | 3.1a — Low-level problem-solving mark. Condone slip on one speed |
| $\frac{1}{2}\times(5)\times[2 + 2(5+10+18+28) + 42]$ or $\frac{1}{2}\times[\text{"315"} + \text{"515"}]$ | M1 | Correct trapezium rule with $h=5$; '2' and '42' in correct place |
| $= 415$ {m} | A1 | Units not required; do not accept 415 km or 415 ms$^{-1}$ |
## Part (b) Alt 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Overestimate **and** relevant explanation, e.g. tops of trapezia lie above curve; area of trapezia $>$ area under curve; curve is convex; $\frac{d^2y}{dx^2}>0$; acceleration is continually increasing; gradient of curve is continually increasing | B1ft | Depends on both an answer to (a) being obtained and first M in (a). Do not allow "curve is concave" or "gradient is positive" or references to friction/air resistance |
## Part (b) Alt 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Underestimate **and** relevant explanation, e.g. all the rectangles are below the curve | B1ft | Depends on both an answer to (a) and first M in (a) |
---\begin{enumerate}
\item The speed of a small jet aircraft was measured every 5 seconds, starting from the time it turned onto a runway, until the time when it left the ground.
\end{enumerate}
The results are given in the table below with the time in seconds and the speed in $\mathrm { ms } ^ { - 1 }$.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
Time $( \mathrm { s } )$ & 0 & 5 & 10 & 15 & 20 & 25 \\
\hline
Speed $\left( \mathrm { m } \mathrm { s } ^ { - 1 } \right)$ & 2 & 5 & 10 & 18 & 28 & 42 \\
\hline
\end{tabular}
\end{center}
Using all of this information,\\
(a) estimate the length of runway used by the jet to take off.
Given that the jet accelerated smoothly in these 25 seconds,\\
(b) explain whether your answer to part (a) is an underestimate or an overestimate of the length of runway used by the jet to take off.
\hfill \mbox{\textit{Edexcel Paper 2 2019 Q2 [4]}}