| Exam Board | OCR |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2011 |
| 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 | Moderate -0.8 This is a straightforward algorithmic trace question requiring systematic execution of given steps with simple arithmetic operations. Parts (i)-(ii) are pure mechanical calculation, (iii) requires recognizing the algorithm computes square roots (standard Newton-Raphson), (iv)-(v) involve observing failure cases. No complex mathematics or novel problem-solving is needed—just careful following of instructions and basic pattern recognition. |
| Spec | 7.03a Algorithm definition: input, output, deterministic, finite7.03c Working with algorithms: trace, interpret, adapt |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(N = 16\) | M1 A1 | |
| \(8, 5\) and \(4\) | M1 | |
| \(4.0012 (4.00)\) | A1 | |
| (ii) \(S\) | M1 A1 | |
| \(5\) | ||
| \(4\) | ||
| \(4.0012 (4.00)\) | ||
| Output \(1.41\) | ||
| Final value of R 1.467 (accept 1, 15, 1.42) | M1 | |
| Accuracy (accept 1, 15, 1.42) | A1 | Need not be in table form |
| Watch out for truncation on S = 1.4167... | [2] | Square root |
| (iii)(a) "Division by 0 without writing" | B1 | Do not get mark for asking about writing |
| Saves three times per round | [1] | |
| (iv) Steps when R = 0 without working | B1 | Do not continue if asking about writing |
| Previous M marks may be implied from this | ||
| Mark if they write just "0 also seen" | Allow covered listed values |
(i) $N = 16$ | M1 A1 |
| $8, 5$ and $4$ | M1 |
| $4.0012 (4.00)$ | A1 |
(ii) $S$ | M1 A1 |
| $5$ | |
| $4$ | |
| $4.0012 (4.00)$ | |
| Output $1.41$ | |
Final value of R 1.467 (accept 1, 15, 1.42) | M1 |
Accuracy (accept 1, 15, 1.42) | A1 | Need not be in table form
Watch out for truncation on S = 1.4167... | [2] | Square root
(iii)(a) "Division by 0 without writing" | B1 | Do not get mark for asking about writing
Saves three times per round | [1] |
(iv) Steps when R = 0 without working | B1 | Do not continue if asking about writing
Previous M marks may be implied from this | |
Mark if they write just "0 also seen" | | Allow covered listed values
---2 Consider the following algorithm.\\
STEP 1 Input a number $N$\\
STEP 2 Calculate $R = N \div 2$\\
STEP 3 Calculate $S = ( ( N \div R ) + R ) \div 2$\\
STEP 4 If $R$ and $S$ are the same when rounded to 2 decimal places, go to STEP 7\\
STEP 5 Replace $R$ with the value of $S$\\
STEP 6 Go to STEP 3\\
STEP 7 Output the value of $R$ correct to 2 decimal places\\
(i) Work through the algorithm starting with $N = 16$. Record the values of $R$ and $S$ each time they change and show the value of the output.\\
(ii) Work through the algorithm starting with $N = 2$. Record the values of $R$ and $S$ each time they change and show the value of the output.\\
(iii) What does the algorithm achieve for positive inputs?\\
(iv) Show that the algorithm fails when it is applied to $N = - 4$.\\
(v) Describe what happens when the algorithm is applied to $N = - 2$. Suggest how the algorithm could be improved to avoid this problem, without imposing a restriction on the allowable input values.
\hfill \mbox{\textit{OCR D1 2011 Q2 [8]}}