OCR D1 2012 June — Question 6 14 marks

Exam BoardOCR
ModuleD1 (Decision Mathematics 1)
Year2012
SessionJune
Marks14
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TopicNewton-Raphson method
TypeAlgorithm or flowchart implementation
DifficultyModerate -0.5 This is a straightforward flowchart tracing exercise testing Newton-Raphson implementation. Students must follow given steps mechanically with simple arithmetic (evaluating cubic and derivative), requiring no problem-solving insight beyond recognizing division-by-zero issues. Easier than average A-level as it's purely procedural execution rather than mathematical reasoning.
Spec1.09d Newton-Raphson method
6 The following flow chart has been written to find a root of the cubic equation \(x ^ { 3 } + A x ^ { 2 } + B x + C = 0\), given a starting value \(X\) that is thought to be near the root. \includegraphics[max width=\textwidth, alt={}, center]{ccb12789-cd5f-40dc-9f10-f8bb45399580-8_1410_1648_324_212}
  1. Work through the algorithm, recording the values of \(X , Y , Z\) and \(W\) each time they change, for the equation \(x ^ { 3 } - 4 x ^ { 2 } + 5 x + 1 = 0\), with a starting value of \(X = 0\).
  2. Show what happens when the algorithm is used for the equation \(x ^ { 3 } - 4 x ^ { 2 } + 5 x + 1 = 0\), with a starting value of \(X = 1\).
  3. Show what happens when the algorithm is used for the equation \(x ^ { 3 } - 4 x ^ { 2 } + 5 x + 1 = 0\), with a starting value of \(X = - 1\).
  4. Identify a possible problem with using this algorithm.
Question 6:
M1 ✓
A1 ✓
Question 6:
M1 ✓
A1 ✓
6 The following flow chart has been written to find a root of the cubic equation $x ^ { 3 } + A x ^ { 2 } + B x + C = 0$, given a starting value $X$ that is thought to be near the root.\\
\includegraphics[max width=\textwidth, alt={}, center]{ccb12789-cd5f-40dc-9f10-f8bb45399580-8_1410_1648_324_212}\\
(i) Work through the algorithm, recording the values of $X , Y , Z$ and $W$ each time they change, for the equation $x ^ { 3 } - 4 x ^ { 2 } + 5 x + 1 = 0$, with a starting value of $X = 0$.\\
(ii) Show what happens when the algorithm is used for the equation $x ^ { 3 } - 4 x ^ { 2 } + 5 x + 1 = 0$, with a starting value of $X = 1$.\\
(iii) Show what happens when the algorithm is used for the equation $x ^ { 3 } - 4 x ^ { 2 } + 5 x + 1 = 0$, with a starting value of $X = - 1$.\\
(iv) Identify a possible problem with using this algorithm.

\hfill \mbox{\textit{OCR D1 2012 Q6 [14]}}
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Q1 9 Q2 9 Q3 13 Q4 14 Q5 13 Q6 14