By sketching a suitable pair of graphs, show that the equation
$$\cot x = 4 x - 2$$
where \(x\) is in radians, has only one root for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
Verify by calculation that this root lies between \(x = 0.7\) and \(x = 0.9\).
Show that this root also satisfies the equation
$$x = \frac { 1 + 2 \tan x } { 4 \tan x }$$
Use the iterative formula \(x _ { n + 1 } = \frac { 1 + 2 \tan x _ { n } } { 4 \tan x _ { n } }\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
6 (i) By sketching a suitable pair of graphs, show that the equation
$$\cot x = 4 x - 2$$
where $x$ is in radians, has only one root for $0 \leqslant x \leqslant \frac { 1 } { 2 } \pi$.\\
(ii) Verify by calculation that this root lies between $x = 0.7$ and $x = 0.9$.\\
(iii) Show that this root also satisfies the equation
$$x = \frac { 1 + 2 \tan x } { 4 \tan x }$$
(iv) Use the iterative formula $x _ { n + 1 } = \frac { 1 + 2 \tan x _ { n } } { 4 \tan x _ { n } }$ to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P2 2013 Q6}}