| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2021 |
| Session | May |
| Marks | 13 |
| Topic | Fixed Point Iteration |
| Type | Rearrange to iterative form |
| Difficulty | Challenging +1.2 This is a structured multi-part question involving trigonometric identities, algebraic manipulation, and numerical methods. Part (i) requires the double angle formula for tan and algebraic rearrangement (routine A-level technique). Parts (ii) and (iii) are standard numerical methods (sign change and iteration) commonly examined at A-level. Part (iv) is a straightforward application of the earlier result using substitution. While it requires competence across multiple techniques, each step is well-scaffolded and uses standard A-level methods without requiring novel insight, placing it slightly above average difficulty. |
| Spec | 1.05o Trigonometric equations: solve in given intervals1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
It is given that there is exactly one value of $x$, where $0 < x < \pi$, that satisfies the equation
$$3\tan 2x - 8\tan x = 4.$$
\begin{enumerate}[label=(\roman*)]
\item Show that $t = \sqrt[3]{\frac{1}{2} + \frac{1}{3}t - \frac{1}{3}t^2}$, where $t = \tan x$. [3]
\item Show by calculation that the value of $t$ satisfying the equation in part (i) lies between 0.7 and 0.8. [2]
\item Use an iterative process based on the equation in part (i) to find the value of $t$ correct to 4 significant figures. Use a starting value of 0.75 and show the result of each iteration. [3]
\item Solve the equation $3\tan 4y - 8\tan 2y = 4$ for $0 < y < \frac{1}{4}\pi$. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2021 Q7 [13]}}