OCR H240/01 2018 December — Question 10 13 marks

Exam BoardOCR
ModuleH240/01 (Pure Mathematics)
Year2018
SessionDecember
Marks13
TopicFunction Transformations
TypeSequence of transformations order
DifficultyStandard +0.3 This is a multi-part question covering standard A-level topics (inverse trig functions, transformations, iteration) but each part is routine. Part (a) requires substituting x=0 and y=0; part (b) is textbook transformation sequencing; parts (c-e) are standard numerical methods with clear instructions. The most challenging aspect is correctly ordering transformations, but this is a well-practiced skill at this level, making it slightly easier than average overall.
Spec1.02w Graph transformations: simple transformations of f(x)1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams
10 \includegraphics[max width=\textwidth, alt={}, center]{a16ab26f-21fb-4a73-8b94-c16bef611bcb-7_524_714_274_246} The diagram shows the graph of \(y = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x - \frac { 1 } { 3 } \pi \right)\), which crosses the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\).
  1. Determine the coordinates of the points \(A\) and \(B\).
  2. Give full details of a sequence of three geometrical transformations which transform the graph of \(y = \tan ^ { - 1 } x\) to the graph of \(y = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x - \frac { 1 } { 3 } \pi \right)\). The equation \(x = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x - \frac { 1 } { 3 } \pi \right)\) has only one root.
  3. Show by calculation that this root lies between \(x = 0\) and \(x = 1\).
  4. Use the iterative formula \(x _ { n + 1 } = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x _ { n } - \frac { 1 } { 3 } \pi \right)\), with a suitable starting value, to find the root correct to 3 significant figures. Show the result of each iteration.
  5. Using the diagram in the Printed Answer Booklet, show how the iterative process converges to the root.
10\\
\includegraphics[max width=\textwidth, alt={}, center]{a16ab26f-21fb-4a73-8b94-c16bef611bcb-7_524_714_274_246}

The diagram shows the graph of $y = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x - \frac { 1 } { 3 } \pi \right)$, which crosses the $x$-axis at the point $A$ and the $y$-axis at the point $B$.
\begin{enumerate}[label=(\alph*)]
\item Determine the coordinates of the points $A$ and $B$.
\item Give full details of a sequence of three geometrical transformations which transform the graph of $y = \tan ^ { - 1 } x$ to the graph of $y = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x - \frac { 1 } { 3 } \pi \right)$.

The equation $x = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x - \frac { 1 } { 3 } \pi \right)$ has only one root.
\item Show by calculation that this root lies between $x = 0$ and $x = 1$.
\item Use the iterative formula $x _ { n + 1 } = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x _ { n } - \frac { 1 } { 3 } \pi \right)$, with a suitable starting value, to find the root correct to 3 significant figures. Show the result of each iteration.
\item Using the diagram in the Printed Answer Booklet, show how the iterative process converges to the root.
\end{enumerate}

\hfill \mbox{\textit{OCR H240/01 2018 Q10 [13]}}
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