5 Fig. 5 shows part of the curve $y = \operatorname { cosec } x$ together with the $x$ - and $y$-axes.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{a13f7a05-e2d3-4354-a0c7-ef7283eff514-06_732_625_317_244}
\captionsetup{labelformat=empty}
\caption{Fig. 5}
\end{center}
\end{figure}
\begin{enumerate}[label=(\alph*)]
\item For the section of the curve which is shown in Fig. 5, write down
\begin{enumerate}[label=(\roman*)]
\item the equations of the two vertical asymptotes,
\item the coordinates of the minimum point.
\end{enumerate}\item Show that the equation $x = \operatorname { cosec } x$ has a root which lies between $x = 1$ and $x = 2$.
\item Use the iteration $\mathrm { x } _ { \mathrm { n } + 1 } = \operatorname { cosec } \left( \mathrm { x } _ { \mathrm { n } } \right)$, with $x _ { 0 } = 1$, to find
\begin{enumerate}[label=(\roman*)]
\item the values of $x _ { 1 }$ and $x _ { 2 }$, correct to 5 decimal places,
\item this root of the equation, correct to 3 decimal places.
\end{enumerate}\item There is another root of $x = \operatorname { cosec } x$ which lies between $x = 2$ and $x = 3$.

Determine whether the iteration $\mathrm { x } _ { \mathrm { n } + 1 } = \operatorname { cosec } \left( \mathrm { x } _ { \mathrm { n } } \right)$ with $x _ { 0 } = 2.5$ converges to this root.
\item Sketch the staircase or cobweb diagram for the iteration, starting with $x _ { 0 } = 2.5$, on the diagram in the Printed Answer Booklet.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Paper 3 2020 Q5 [11]}}