- The complex number \(z\) satisfies the equation \(z + 2 \mathrm { i } z ^ { * } = 12 + 9 \mathrm { i }\). Find \(z\), giving your answer in the form \(x + \mathrm { i } y\).
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[BLANK PAGE] - (a) Use binomial expansions to show that \(\sqrt { \frac { 1 + 4 x } { 1 - x } } \approx 1 + \frac { 5 } { 2 } x - \frac { 5 } { 8 } x ^ { 2 }\)
A student substitutes \(x = \frac { 1 } { 2 }\) into both sides of the approximation shown in part (a) in an attempt to find an approximation to \(\sqrt { 6 }\)
(b) Give a reason why the student should not use \(x = \frac { 1 } { 2 }\)
(c) Substitute \(x = \frac { 1 } { 11 }\) into
$$\sqrt { \frac { 1 + 4 x } { 1 - x } } = 1 + \frac { 5 } { 2 } x - \frac { 5 } { 8 } x ^ { 2 }$$
to obtain an approximation to \(\sqrt { 6 }\). Give your answer as a fraction in its simplest form.
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