State validity only

1 The first three terms, in ascending powers of \(x\), of the binomial expansion of \(( 9 + 2 x ) ^ { \frac { 1 } { 2 } }\) are given by $$( 9 + 2 x ) ^ { \frac { 1 } { 2 } } \approx a + \frac { x } { 3 } - \frac { x ^ { 2 } } { 54 }$$ where \(a\) is a constant. 1

1. State the range of values of \(x\) for which this expansion is valid.
   Circle your answer. \(| x | < \frac { 2 } { 9 }\) \(| x | < \frac { 2 } { 3 }\) \(| x | < 1\) \(| x | < \frac { 9 } { 2 }\) 1
2. Find the value of \(a\).
   Circle your answer.
   [0pt] [1 mark]
   1239

9

1. Find the first three terms, in ascending powers of \(x\), of the binomial expansion of $$( 1 + x ) ^ { - \frac { 1 } { 2 } }$$ 9
2. A student substitutes \(x = 2\) into the expansion of \(( 1 + x ) ^ { - \frac { 1 } { 2 } }\) to find an approximation for \(\frac { 1 } { \sqrt { 3 } }\) Explain the mistake in the student's approach.
   [0pt] [1 mark] 9
3. By substituting \(x = - \frac { 1 } { 4 }\) in your expansion for \(( 1 + x ) ^ { - \frac { 1 } { 2 } }\) find an approximation for \(\frac { 1 } { \sqrt { 3 } }\) Give your answer to three significant figures.

State the values of \(|x|\) for which the binomial expansion of \((3 + 2x)^{-4}\) is valid. Circle your answer. [1 mark] \(|x| < \frac{2}{3}\) \(\quad\) \(|x| < 1\) \(\quad\) \(|x| < \frac{3}{2}\) \(\quad\) \(|x| < 3\)

State the range of values of \(x\) for which the binomial expansion of $$\sqrt{1 - \frac{x}{4}}$$ is valid. Circle your answer. [1 mark] \(|x| < \frac{1}{4}\) \quad\quad \(|x| < 1\) \quad\quad \(|x| < 2\) \quad\quad \(|x| < 4\)