Expand and state validity

2

1. Expand \(( 2 - 3 x ) ^ { - 2 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
2. State the set of values of \(x\) for which the expansion is valid.

2

1. Expand \(\left( 2 - x ^ { 2 } \right) ^ { - 2 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 4 }\), simplifying the coefficients.
2. State the set of values of \(x\) for which the expansion is valid.

2

1. Find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 x - 5 ) \sqrt { 4 - x }\).
2. State the set of values of \(x\) for which the expansion in part (a) is valid.

2

1. Expand \(\sqrt [ 3 ] { 1 + 6 x }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
2. State the set of values of \(x\) for which the expansion is valid.

1. (a) Find the first four terms, in ascending powers of \(x\), of the binomial expansion of

$$\frac { 8 } { ( 2 - 5 x ) ^ { 2 } }$$ writing each term in simplest form.
(b) Find the range of values of \(x\) for which this expansion is valid.

4

1. Expand \(( 2 + x ) ^ { - 2 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), and state the set of values of \(x\) for which the expansion is valid.
2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(\frac { 1 + x ^ { 2 } } { ( 2 + x ) ^ { 2 } }\).

2

1. Expand \(( 1 - 2 x ) ^ { - \frac { 1 } { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
2. State the set of values for which the expansion in part (i) is valid.

5 Find the first four terms in the binomial expansion of \(( 1 + 3 x ) ^ { \frac { 1 } { 3 } }\).
State the range of values of \(x\) for which the expansion is valid.

3 Find the first three terms of the binomial expansion of \(\frac { 1 } { 2 - 3 x }\).
Give the range of values of \(x\) for which the expansion is valid.

5. (i) Expand \(( 4 - x ) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\), simplifying each coefficient.
(ii) State the set of values of \(x\) for which your expansion is valid.
(iii) Use your expansion with \(x = 0.01\) to find the value of \(\sqrt { 399 }\), giving your answer to 9 significant figures.

2. (i) Expand \(( 1 + 4 x ) ^ { \frac { 3 } { 2 } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
(ii) State the set of values of \(x\) for which your expansion is valid.

2. (i) Find the binomial expansion of \(( 2 - 3 x ) ^ { - 3 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
(ii) State the set of values of \(x\) for which your expansion is valid.

4 Find the first four terms in the binomial expansion of \(\sqrt { 1 + 2 x }\). State the set of values of \(x\) for which the expansion is valid.

5 Find the first three terms in the binomial expansion of \(\sqrt [ 3 ] { 1 + 3 x }\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid.

7 Find the first 4 terms in the binomial expansion of \(\sqrt { 4 + 2 x }\). State the range of values of \(x\) for which the expansion is valid.

3 Find the first three terms in the binomial expansion of \(\frac { 1 } { ( 3 - 2 x ) ^ { 3 } }\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid.
[0pt] [7]

5 Show that \(( 1 + 2 x ) ^ { \frac { 1 } { 3 } } = 1 + \frac { 2 } { 3 } x - \frac { 4 } { 9 } x ^ { 2 } + \ldots\), and find the next term in the expansion.
State the set of values of \(x\) for which the expansion is valid.

2

1. Ed \(( 1 + 3 x ) ^ { - \frac { 1 } { 3 } }\) in asced g N ers \(6 x\), p to ad in lid g th term in \(x ^ { 2 }\), simp ify g th co fficien s.
2. State the set \(\mathbf { 6 }\) \& le s \(\mathbf { 6 }\) x fo wh cht b e nsin s valid

3

1. Expand \(\frac { 1 + x ^ { 2 } } { \sqrt { 1 + 4 x } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
2. State the set of values of \(x\) for which this expansion is valid.

2 Show that \(( 1 + 2 x ) ^ { \frac { 1 } { 3 } } = 1 + \frac { 2 } { 3 } x - \frac { 4 } { 9 } x ^ { 2 } + \ldots\), and find the next term in the expansion.
State the set of values of \(x\) for which the expansion is valid.

2 Find the first four terms of the binomial expansion of \(\sqrt [ 3 ] { 1 - 2 x }\). State the set of values of \(x\) for which the expansion is valid.

2 Find the first three terms in the binomial expansion of \(\sqrt [ 3 ] { 1 + 3 x }\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid.

2 Find the first four terms in the binomial expansion of \(\sqrt { 1 + 2 x }\). State the set of values of \(x\) for which the expansion is valid.

2

1. Find the coefficient of \(x ^ { 5 }\) in the expansion of \(( 3 - 2 x ) ^ { 8 }\).
2. 1. Expand \(( 1 + 3 x ) ^ { 0.5 }\) as far as the term in \(x ^ { 3 }\).
   2. State the range of values of \(x\) for which your expansion is valid. A student suggests the following check to determine whether the expansion obtained in part (b)(i) may be correct.
      "Use the expansion to find an estimate for \(\sqrt { 103 }\), correct to five decimal places, and compare this with the value of \(\sqrt { 103 }\) given by your calculator."
   3. Showing your working, carry out this check on your expansion from part (b)(i).

1. (a) Find, in ascending powers of \(x\), the first four terms of the binomial expansion of

$$( 1 - 9 x ) ^ { \frac { 1 } { 2 } }$$ giving each term in simplest form.
(b) Give a reason why \(x = - \frac { 2 } { 9 }\) should not be used in the expansion to find an approximation to \(\sqrt { 3 }\)