| Exam Board | OCR MEI |
|---|---|
| Module | Further Numerical Methods (Further Numerical Methods) |
| Year | 2022 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule applied to real-world data |
| Difficulty | Standard +0.8 This Further Maths numerical methods question requires constructing difference tables, deriving cubic and quartic interpolating polynomials, and evaluating model appropriateness. While the techniques are systematic, the multi-part nature (6 parts), extended reasoning about model validity, and refinement from cubic to quartic elevate it above standard A-level fare but remain within established Further Maths procedures. |
| Spec | 1.04e Sequences: nth term and recurrence relations4.05a Roots and coefficients: symmetric functions |
| \(t\) | 0 | 10 | 20 | 30 |
| \(M\) | 88.3 | 80.05 | 78.7 | 78.85 |
| \(t\) | \(M\) | \(\Delta M\) | \(\Delta ^ { 2 } M\) | \(\Delta ^ { 3 } M\) |
| 0 | 88.3 | |||
| 10 | 80.05 | |||
| 20 | 78.7 | |||
| 30 | 78.85 |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (a) | t M ΔM Δ²M Δ³M |
| Answer | Marks | Guidance |
|---|---|---|
| 30 78.85 | B1 | |
| [1] | 1.1 | |
| 7 | (b) | 8.25𝑡 6.9𝑡(𝑡−10) |
| Answer | Marks |
|---|---|
| 88.3 | M1 |
| Answer | Marks |
|---|---|
| [4] | 3.3 |
| Answer | Marks |
|---|---|
| 1.1 | soi; allow bracket and/or sign |
| Answer | Marks |
|---|---|
| 2000 | FT their ‒5.4 and their |
| Answer | Marks | Guidance |
|---|---|---|
| t | M | ΔM |
| 7 | (c) | evaluation of M at t = 40 and t = 50 |
| Answer | Marks |
|---|---|
| so a poor fit for these data oe | M1 |
| Answer | Marks |
|---|---|
| [2] | 3.4 |
| 3.5a | FT their cubic, dependent on |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (d) | t W ΔW Δ²W Δ³W Δ4W |
| Answer | Marks |
|---|---|
| 50 80.05 | B1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 2.4 | difference table correct |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (e) | −0.0009𝑡3 +0.0615𝑡2 −1.35𝑡+88.3 |
| Answer | Marks |
|---|---|
| 1.44𝑡+88.3 | M1 |
| Answer | Marks |
|---|---|
| [3] | 3.5c |
| Answer | Marks |
|---|---|
| 1.1 | for addition of their cubic to extra |
| Answer | Marks |
|---|---|
| condone use of other variable | NB |
| Answer | Marks | Guidance |
|---|---|---|
| t | W | ΔW |
| 7 | (f) | the model predicts that Sam’s weight will |
| Answer | Marks |
|---|---|
| so not appropriate (in long run) | M1 |
| Answer | Marks |
|---|---|
| [2] | 3.5a |
| 3.5b | must be in context; dependent on |
| award of 2nd M1 in (e) | do not allow eg |
Question 7:
7 | (a) | t M ΔM Δ²M Δ³M
0 88.3
‒8.25
10 80.05 6.9
‒1.35 ‒5.4
20 78.7 1.5
0.15
30 78.85 | B1
[1] | 1.1
7 | (b) | 8.25𝑡 6.9𝑡(𝑡−10)
𝑀 = 88.3− + −
10 2!×102
5.4𝑡(𝑡−10)(𝑡−20)
3!×10³
𝑀 = −0.0009𝑡3 +0.0615𝑡2 −1.35𝑡+
88.3 | M1
A1
A1
A1
[4] | 3.3
1.1
1.1
1.1 | soi; allow bracket and/or sign
errors; allow use of different
variable
two correct coefficients in cubic
three correct coefficients
all correct
123
NB 𝑏 =
2000 | FT their ‒5.4 and their
6.9
FT first A1 only
A0 if different variable;
must see “M =” at some
stage
t | M | ΔM | Δ²M | Δ³M
7 | (c) | evaluation of M at t = 40 and t = 50
75.1 and 62.05 obtained respectively,
so a poor fit for these data oe | M1
A1FT
[2] | 3.4
3.5a | FT their cubic, dependent on
award of M1 in (b)
may be stated separately
7 | (d) | t W ΔW Δ²W Δ³W Δ4W
0 88.3
‒8.25
10 80.05 6.9
‒1.35 ‒5.4
20 78.7 1.5 3.6
0.15 ‒1.8
30 78.85 ‒0.3 3.6
‒0.15 1.8
40 78.7 1.5
1.35
50 80.05 | B1
B1
[2] | 1.1
2.4 | difference table correct
4th differences same (so quartic
may be good model)
7 | (e) | −0.0009𝑡3 +0.0615𝑡2 −1.35𝑡+88.3
3.6𝑡(𝑡−10)(𝑡−20)(𝑡−30)
+
4!×104
𝑊 = 0.000015𝑡4 −0.0018𝑡3 +0.078𝑡2 −
1.44𝑡+88.3 | M1
M1
A1
[3] | 3.5c
3.3
1.1 | for addition of their cubic to extra
term
quartic term; FT their 3.6
condone use of other variable | NB
may see
3 9 39 36
,− , ,−
200000 5000 500 25
t | W | ΔW | Δ²W | Δ³W | Δ4W
7 | (f) | the model predicts that Sam’s weight will
continue to increase oe
allow eg 𝑡 → ∞,𝑀 → ∞
so not appropriate (in long run) | M1
A1
[2] | 3.5a
3.5b | must be in context; dependent on
award of 2nd M1 in (e) | do not allow eg
increases exponentially
PMT
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Please get in touch if you want to discuss the accessibility of resources we offer to support you in delivering our qualifications.7 Sam decided to go on a high-protein diet. Sam's mass in $\mathrm { kg } , M$, after $t$ days of following the diet is recorded in Fig. 7.1.
\begin{table}[h]
\begin{center}
\begin{tabular}{ | l | c | c | c | c | }
\hline
$t$ & 0 & 10 & 20 & 30 \\
\hline
$M$ & 88.3 & 80.05 & 78.7 & 78.85 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 7.1}
\end{center}
\end{table}
A difference table for the data is shown in Fig. 7.2.
\begin{table}[h]
\begin{center}
\begin{tabular}{ | c | r | r | r | r | }
\hline
$t$ & $M$ & $\Delta M$ & $\Delta ^ { 2 } M$ & $\Delta ^ { 3 } M$ \\
\hline
0 & 88.3 & & & \\
\hline
& & & & \\
\hline
10 & 80.05 & & & \\
\hline
& & & & \\
\hline
20 & 78.7 & & & \\
\hline
& & & & \\
\hline
30 & 78.85 & & & \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 7.2}
\end{center}
\end{table}
\begin{enumerate}[label=(\alph*)]
\item Complete the copy of the difference table in the Printed Answer Booklet.
Sam's doctor uses these data to construct a cubic interpolating polynomial to model Sam's mass at time $t$ days after starting the diet.
\item Find the model in the form $\mathrm { M } = \mathrm { at } ^ { 3 } + \mathrm { bt } ^ { 2 } + \mathrm { ct } + \mathrm { d }$, where $a , b , c$ and $d$ are constants to be determined.
Subsequently it is found that when $\mathrm { t } = 40 , \mathrm { M } = 78.7$ and when $\mathrm { t } = 50 , \mathrm { M } = 80.05$.
\item Determine whether the model is a good fit for these data.
\item By completing the extended copy of Fig. 7.2 in the Printed Answer Booklet, explain why a quartic model may be more appropriate for the data.
\item Refine the doctor's model to include a quartic term.
\item Explain whether the new model for Sam's mass is likely to be appropriate over a longer period of time.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Numerical Methods 2022 Q7 [14]}}