SPS SPS SM Pure 2025 February — Question 10 7 marks

Exam BoardSPS
ModuleSPS SM Pure (SPS SM Pure)
Year2025
SessionFebruary
Marks7
TopicNewton-Raphson method
TypeIntersection of curves via iteration
DifficultyStandard +0.3 This is a straightforward application of Newton-Raphson with clear setup. Part (a) is trivial recall (a=1), part (b) requires understanding inverse functions but the reasoning is guided, and part (c) is a standard three-iteration calculation with f(x)=x-cos(x), f'(x)=1+sin(x). The conceptual demand is low and the execution is routine textbook practice.
Spec1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.09d Newton-Raphson method
10. The function f is defined by \(\mathrm { f } ( x ) = \arccos x\) for \(0 \leq x \leq a\) The curve with equation \(y = \mathrm { f } ( x )\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{bc7fb499-9462-40ae-88f4-87fc60f6a005-22_769_771_317_648}
  1. State the value of \(a\)
    1. On the diagram above, sketch the curve with equation $$y = \cos x \text { for } 0 \leq x \leq \frac { \pi } { 2 }$$ and
      sketch the line with equation $$y = x \text { for } 0 \leq x \leq \frac { \pi } { 2 }$$
    2. Explain why the solution to the equation $$x - \cos x = 0$$ must also be a solution to the equation $$\cos x = \arccos x$$
  3. Use the Newton-Raphson method with \(x _ { 0 } = 0\) to find an approximate solution, \(x _ { 3 }\), to the equation $$x - \cos x = 0$$ Give your answer to four decimal places. CONTINUE YOUR ANSWER HERE CONTINUE YOUR ANSWER HERE
10.

The function f is defined by $\mathrm { f } ( x ) = \arccos x$ for $0 \leq x \leq a$\\
The curve with equation $y = \mathrm { f } ( x )$ is shown below.\\
\includegraphics[max width=\textwidth, alt={}, center]{bc7fb499-9462-40ae-88f4-87fc60f6a005-22_769_771_317_648}
\begin{enumerate}[label=(\alph*)]
\item State the value of $a$
\item \begin{enumerate}[label=(\roman*)]
\item On the diagram above, sketch the curve with equation

$$y = \cos x \text { for } 0 \leq x \leq \frac { \pi } { 2 }$$

and\\
sketch the line with equation

$$y = x \text { for } 0 \leq x \leq \frac { \pi } { 2 }$$
\item Explain why the solution to the equation

$$x - \cos x = 0$$

must also be a solution to the equation

$$\cos x = \arccos x$$
\end{enumerate}\item Use the Newton-Raphson method with $x _ { 0 } = 0$ to find an approximate solution, $x _ { 3 }$, to the equation

$$x - \cos x = 0$$

Give your answer to four decimal places.

CONTINUE YOUR ANSWER HERE

CONTINUE YOUR ANSWER HERE
\end{enumerate}

\hfill \mbox{\textit{SPS SPS SM Pure 2025 Q10 [7]}}
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