State validity only

1.

1. Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of $$\sqrt { 1 + 4 x }$$ giving each coefficient in its simplest form. The expansion can be used to find an approximation for \(\sqrt { 26 }\)
2. Explain why \(x = \frac { 25 } { 4 }\) should not be used in the expansion to find an approximation for \(\sqrt { 26 }\)

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.

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

1 State the range of values of \(x\) for which the binomial expansion of $$\sqrt { 1 - \frac { x } { 4 } }$$ is valid. Circle your answer.
[0pt] [1 mark] $$| x | < \frac { 1 } { 4 } \quad | x | < 1 \quad | x | < 2 \quad | x | < 4$$