| Exam Board | OCR MEI |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Apply iteration to find root (pure fixed point) |
| Difficulty | Easy -1.8 This is a straightforward algorithm trace through a given flowchart with explicit step-by-step instructions. It requires only careful bookkeeping and basic arithmetic (finding line intersections), with no problem-solving, insight, or understanding of iteration methods despite the topic label. The algorithm essentially finds pairwise intersections of three lines, which is routine computational work well below average A-level difficulty. |
| Spec | 7.03a Algorithm definition: input, output, deterministic, finite7.03b Algorithm awareness: uses and practical limitations7.03c Working with algorithms: trace, interpret, adapt |
| Step 1 | Set \(i = 1\) |
| Step 2 | Input \(m _ { i }\) and \(c _ { i }\) |
| Step 3 | If \(i = 3\) then go to Step 6 |
| Step 4 | Set \(i = i + 1\) |
| Step 5 | Go to Step 2 |
| Step 6 | Set \(j = 1\) |
| Step 7 | Set \(a = j + 1\) |
| Step 8 | If \(a > 3\) then set \(a = a - 3\) |
| Step 9 | Set \(b = j + 2\) |
| Step 10 | If \(b > 3\) then set \(b = b - 3\) |
| Step 11 | Set \(d _ { j } = m _ { b } - m _ { a }\) |
| Step 12 | If \(d _ { j } = 0\) then go to Step 20 |
| Step 13 | Set \(x _ { j } = \frac { c _ { a } - c _ { b } } { d _ { j } }\) |
| Step 14 | Set \(y _ { j } = m _ { a } \times x _ { j } + c _ { a }\) |
| Step 15 | Record \(\left( x _ { j } , y _ { j } \right)\) in the print area |
| Step 16 | If \(j = 3\) then go to Step 19 |
| Step 17 | Set \(j = j + 1\) |
| Step 18 | Go to Step 7 |
| Step 19 | Stop |
| Step 20 | Record "parallel" in the print area |
| Step 21 | Go to Step 16 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(i=1,\ 2,\ 3\); \(m_1=2\), \(c_1=8\); \(m_2=2\), \(c_2=5\); \(m_3=4\), \(c_3=3\); \(j=1,2,3\); \(a=2,3,4,1\); \(b=3,4,1,5,2\); \(d_1=2\), \(x_1=1\) | M1 | \(j\ 1\), \(a\ 2\), \(b\ 3\) |
| \(a\)s and \(b\)s correct | A1 | \(a\)s and \(b\)s (4's and 5's not essential) |
| \(y_1=7\) | B1 | for 1 and 7 |
| \(d_2=-2\), \(x_2=2.5\), \(y_2=13\) | B1 | for 2.5 and 13 |
| \(d_3=0\) | B1 | for 0 |
| Print area: \((1,7)\), \((2.5, 13)\), parallel | M1 | use of print area |
| 3 copied, including "parallel" | \(A1\sqrt{}\) | 3 copied, inc "parallel" |
| [7] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Finds the line intersections | B1 | |
| [1] |
# Question 2:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $i=1,\ 2,\ 3$; $m_1=2$, $c_1=8$; $m_2=2$, $c_2=5$; $m_3=4$, $c_3=3$; $j=1,2,3$; $a=2,3,4,1$; $b=3,4,1,5,2$; $d_1=2$, $x_1=1$ | M1 | $j\ 1$, $a\ 2$, $b\ 3$ |
| $a$s and $b$s correct | A1 | $a$s and $b$s (4's and 5's not essential) |
| $y_1=7$ | B1 | for 1 and 7 |
| $d_2=-2$, $x_2=2.5$, $y_2=13$ | B1 | for 2.5 and 13 |
| $d_3=0$ | B1 | for 0 |
| Print area: $(1,7)$, $(2.5, 13)$, parallel | M1 | use of print area |
| 3 copied, including "parallel" | $A1\sqrt{}$ | 3 copied, inc "parallel" |
| **[7]** | | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Finds the line intersections | B1 | |
| **[1]** | | |
---2 The following algorithm operates on the equations of 3 straight lines, each in the form $y = m _ { i } x + c _ { i }$.
\begin{center}
\begin{tabular}{|l|l|}
\hline
Step 1 & Set $i = 1$ \\
\hline
Step 2 & Input $m _ { i }$ and $c _ { i }$ \\
\hline
Step 3 & If $i = 3$ then go to Step 6 \\
\hline
Step 4 & Set $i = i + 1$ \\
\hline
Step 5 & Go to Step 2 \\
\hline
Step 6 & Set $j = 1$ \\
\hline
Step 7 & Set $a = j + 1$ \\
\hline
Step 8 & If $a > 3$ then set $a = a - 3$ \\
\hline
Step 9 & Set $b = j + 2$ \\
\hline
Step 10 & If $b > 3$ then set $b = b - 3$ \\
\hline
Step 11 & Set $d _ { j } = m _ { b } - m _ { a }$ \\
\hline
Step 12 & If $d _ { j } = 0$ then go to Step 20 \\
\hline
Step 13 & Set $x _ { j } = \frac { c _ { a } - c _ { b } } { d _ { j } }$ \\
\hline
Step 14 & Set $y _ { j } = m _ { a } \times x _ { j } + c _ { a }$ \\
\hline
Step 15 & Record $\left( x _ { j } , y _ { j } \right)$ in the print area \\
\hline
Step 16 & If $j = 3$ then go to Step 19 \\
\hline
Step 17 & Set $j = j + 1$ \\
\hline
Step 18 & Go to Step 7 \\
\hline
Step 19 & Stop \\
\hline
Step 20 & Record "parallel" in the print area \\
\hline
Step 21 & Go to Step 16 \\
\hline
\end{tabular}
\end{center}
(i) Run the algorithm for the straight lines $y = 2 x + 8 , y = 2 x + 5$ and $y = 4 x + 3$ using the table given in your answer book. The first five steps have been completed, so you should continue from Step 6. [7]\\
(ii) Describe what the algorithm achieves.
\hfill \mbox{\textit{OCR MEI D1 2015 Q2 [8]}}