| Exam Board | OCR MEI |
|---|---|
| Module | Further Numerical Methods (Further Numerical Methods) |
| Session | Specimen |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sign Change & Interval Methods |
| Type | Interval Bisection from Spreadsheet |
| Difficulty | Moderate -0.3 This question tests understanding of the bisection method through spreadsheet implementation, requiring students to interpret IF statements and calculate when error bounds are met. While it involves multiple parts, each part is straightforward: (i) recognizes interval updating logic, (ii) completes a symmetric formula pattern, and (iii) applies repeated halving (mpe = 0.03125/2^n < 5×10^-7). This is slightly easier than average as it's more about following an algorithm than mathematical problem-solving. |
| Spec | 1.09a Sign change methods: locate roots |
| A | B | C | D | E | F | G | |
| 1 | a | \(\mathrm { f } ( a )\) | b | f(b) | \(( a + b ) / 2\) | \(\mathrm { f } ( ( a + b ) / 2 )\) | mpe |
| 2 | 1 | -0.28172 | 2 | 1.389056 | 1.5 | 0.231689 | 0.5 |
| 3 | 1 | -0.28172 | 1.5 | 0.231689 | 1.25 | -0.072157 | 0.25 |
| 4 | 1.25 | -0.07216 | 1.5 | 0.231689 | 1.375 | 0.064452 | 0.125 |
| 5 | 1.25 | -0.07216 | 1.375 | 0.064452 | 1.3125 | -0.007206 | 0.0625 |
| 6 | 1.3125 | -0.00721 | 1.375 | 0.064452 | 1.34375 | 0.027728 | 0.03125 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (i) | To determine whether 1 is replaced by 1.5 |
| Answer | Marks | Guidance |
|---|---|---|
| [1] | 2.4 | Accept explanation in terms of |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (ii) | E |
| E2, C2 P | B1 | |
| [1] | 2.2a | In correct order |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (iii) | S |
| Answer | Marks |
|---|---|
| Row 22 | M1 |
| Answer | Marks |
|---|---|
| [2] | 3.1a |
| 1.1 | So n > 19.93 i.e. 20 cells further |
| down | NB ln10−6 ÷ ln0.5 |
Question 2:
2 | (i) | To determine whether 1 is replaced by 1.5 | C
B1
[1] | 2.4 | Accept explanation in terms of
method
2 | (ii) | E
E2, C2 P | B1
[1] | 2.2a | In correct order | The resulting formula must be
fully correct, so that it could be
copied to produce the results in
column C.
2 | (iii) | S
0.5 × 0.5n < 5 × 10−7 soi
Row 22 | M1
A1
[2] | 3.1a
1.1 | So n > 19.93 i.e. 20 cells further
down | NB ln10−6 ÷ ln0.5
If M0, B2 for correct answer
www2 The following spreadsheet printout shows the bisection method being applied to the equation $\mathrm { f } ( x ) = 0$, where $\mathrm { f } ( x ) = \mathrm { e } ^ { x } - x ^ { 2 } - 2$. Some values of $\mathrm { f } ( x )$ are shown in columns B and D.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
& A & B & C & D & E & F & G \\
\hline
1 & a & $\mathrm { f } ( a )$ & b & f(b) & $( a + b ) / 2$ & $\mathrm { f } ( ( a + b ) / 2 )$ & mpe \\
\hline
2 & 1 & -0.28172 & 2 & 1.389056 & 1.5 & 0.231689 & 0.5 \\
\hline
3 & 1 & -0.28172 & 1.5 & 0.231689 & 1.25 & -0.072157 & 0.25 \\
\hline
4 & 1.25 & -0.07216 & 1.5 & 0.231689 & 1.375 & 0.064452 & 0.125 \\
\hline
5 & 1.25 & -0.07216 & 1.375 & 0.064452 & 1.3125 & -0.007206 & 0.0625 \\
\hline
6 & 1.3125 & -0.00721 & 1.375 & 0.064452 & 1.34375 & 0.027728 & 0.03125 \\
\hline
\end{tabular}
\end{center}
(i) The formula in cell A 3 is $= \mathrm { IF } ( \mathrm { F } 2 > 0$, A2, E2). State the purpose of this formula.\\
(ii) The formula in cell C 3 is $= \mathrm { IF } ( \mathrm { F } 2 > 0 , \ldots , \ldots )$. What are the missing cell references?\\
(iii) In which row is the magnitude of the maximum possible error (mpe) less than $5 \times 10 ^ { - 7 }$ for the first time?
\hfill \mbox{\textit{OCR MEI Further Numerical Methods Q2 [4]}}