Direct quotient expansion

3 Expand \(\sqrt { } \left( \frac { 1 - x } { 1 + x } \right)\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

2 Expand \(\sqrt { \frac { 1 + 2 x } { 1 - 2 x } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

6. (a) Use binomial expansions to show that, for \(| x | < \frac { 1 } { 2 }\)
(b) Find the exact value of \(\sqrt { \frac { 1 + 2 x } { 1 - x } }\) when \(x = \frac { 1 } { 10 }\) Give your answer in the form \(k \sqrt { 3 }\), where \(k\) is a constant to be determined.
(c) Substitute \(x = \frac { 1 } { 10 }\) into the expansion given in part (a) and hence find an approximate value for \(\sqrt { 3 }\) Give your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers. $$\sqrt { \frac { 1 + 2 x } { 1 - x } } \approx 1 + \frac { 3 } { 2 } x + \frac { 3 } { 8 } x ^ { 2 }$$

5

1. Show that \(\sqrt { \frac { 1 - x } { 1 + x } } \approx 1 - x + \frac { 1 } { 2 } x ^ { 2 }\), for \(| x | < 1\).
2. By taking \(x = \frac { 2 } { 7 }\), show that \(\sqrt { 5 } \approx \frac { 111 } { 49 }\).

5.

1. Show that \(\sqrt { \frac { 1 - x } { 1 + x } } \approx 1 - x + \frac { 1 } { 2 } x ^ { 2 }\), for \(| x | < 1\).
2. By taking \(x = \frac { 2 } { 7 }\), show that \(\sqrt { 5 } \approx \frac { 111 } { 49 }\).
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9. (i) Use the binomial expansion to show that \(( 1 - 2 x ) ^ { - \frac { 1 } { 2 } } \approx 1 + x + \frac { 3 } { 2 } x ^ { 2 }\) for sufficiently small values of \(x\).
(ii) For what values of \(x\) is the expansion valid?
(iii) Find the expansion of \(\sqrt { \frac { 1 + 2 x } { 1 - 2 x } }\) in ascending powers of \(x\) as far as the term in \(x ^ { 2 }\).
(iv) Use \(x = \frac { 1 } { 20 }\) in your answer to part (iii) to find an approximate value for \(\sqrt { 11 }\).
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1. 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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2. (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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