Factoring out constants first

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

1 Expand \(\frac { 1 } { \sqrt { } ( 4 + 3 x ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

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

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

1 Expand \(\frac { 16 } { ( 2 + x ) ^ { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

1 Expand \(( 9 - 3 x ) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

1. $$f ( x ) = ( 3 - 2 x ) ^ { - 4 } , \quad | x | < \frac { 3 } { 2 }$$ Find the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), giving each coefficient as a simplified fraction.

1. Use the binomial theorem to expand

$$\sqrt { } ( 4 - 9 x ) , \quad | x | < \frac { 4 } { 9 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying each term.

1. $$f ( x ) = \frac { 1 } { \sqrt { ( 4 + x ) } } , \quad | x | < 4$$ Find the binomial expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient as a simplified fraction.
(6)

Use the binomial theorem to expand \(( 4 - 3 x ) ^ { - \frac { 1 } { 2 } }\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient as a simplified fraction.

7

1. Show that \(\frac { 1 } { \sqrt { 25 - x } } = \frac { 1 } { 5 } \left( 1 - \frac { x } { 25 } \right) ^ { - \frac { 1 } { 2 } }\).
2. Hence expand \(\frac { 1 } { \sqrt { 25 - x } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\).
3. Write down the range of values of \(x\) for which the expansion is valid.

2 Find the first three terms in the binomial expansion of \(( 4 + x ) ^ { \frac { 3 } { 2 } }\). State the set of values of \(x\) for which the expansion is valid.

6 Find the first three terms in the binomial expansion of \(\overline { 4 + x }\) in ascending powers of \(x\).
State the set of values of \(x\) for which the expansion is valid.

5

1. Expand \(( 1 + x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
2. (a) Hence, or otherwise, expand \(( 8 + 16 x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
   (b) State the set of values of \(x\) for which the expansion in part (ii) (a) is valid.

2 Find the first three terms in the expansion of \(( 9 - 16 x ) ^ { \frac { 3 } { 2 } }\) in ascending powers of \(x\), and state the set of values for which this expansion is valid.

4

1. Find the first three terms in the binomial expansion of \(( 8 - 9 x ) ^ { \frac { 2 } { 3 } }\) in ascending powers of \(x\).
2. State the set of values of \(x\) for which this expansion is valid.

2 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.

7. a. Use the binomial theorem to expand $$( 8 - 3 x ) ^ { \frac { 2 } { 3 } }$$ in ascending powers of \(x\), up to and including the term \(x ^ { 3 }\), as a fully simplifying each term. Edward, a student decides to use the expansion with \(x = \frac { 1 } { 3 }\) to find an approximation for \(( 7 ) ^ { \frac { 2 } { 3 } }\). Using the answer to part (a) and without doing any calculations, b. explain clearly whether Edward's approximation will be an overestimate, or, an underestimate.

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

$$\sqrt { 4 - 9 x }$$ writing each term in simplest form. A student uses this expansion with \(x = \frac { 1 } { 9 }\) to find an approximation for \(\sqrt { 3 }\)
Using the answer to part (a) and without doing any calculations,
(b) state whether this approximation will be an overestimate or an underestimate of \(\sqrt { 3 }\) giving a brief reason for your answer.

13 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = \sqrt [ 3 ] { 27 - 8 x ^ { 3 } }\). Jenny uses her scientific calculator to create a table of values for \(\mathrm { f } ( x )\) and \(\mathrm { f } ^ { \prime } ( x )\).

1. Use calculus to find an expression for \(\mathrm { f } ^ { \prime } ( x )\) and hence explain why the calculator gives an error for \(\mathrm { f } ^ { \prime } ( 1.5 )\).
2. Find the first three terms of the binomial expansion of \(\mathrm { f } ( x )\).
3. Jenny integrates the first three terms of the binomial expansion of \(\mathrm { f } ( x )\) to estimate the value of \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). Explain why Jenny's method is valid in this case. (You do not need to evaluate Jenny's approximation.)
4. Use the trapezium rule with 4 strips to obtain an estimate for \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). The calculator gives 2.92117438 for \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). The graph of \(y = \mathrm { f } ( x )\) is shown in Fig. 13. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-08_490_906_1505_568} \captionsetup{labelformat=empty} \caption{Fig. 13}
   \end{figure}
5. Explain why the trapezium rule gives an underestimate.

3

1. Find the binomial expansion of \(( 1 + 6 x ) ^ { \frac { 2 } { 3 } }\) up to and including the term in \(x ^ { 2 }\).
   (2 marks)
2. Find the binomial expansion of \(( 8 + 6 x ) ^ { \frac { 2 } { 3 } }\) up to and including the term in \(x ^ { 2 }\).
   (3 marks)
3. Use your answer from part (b) to find an estimate for \(\sqrt [ 3 ] { 100 }\) in the form \(\frac { a } { b }\), where \(a\) and \(b\) are integers.
   (2 marks)

4

1. 1. Find the binomial expansion of \(( 1 + x ) ^ { \frac { 3 } { 2 } }\) up to and including the term in \(x ^ { 2 }\).
   2. Find the binomial expansion of \(( 16 + 9 x ) ^ { \frac { 3 } { 2 } }\) up to and including the term in \(x ^ { 2 }\).
2. Use your answer to part (a)(ii) to show that \(13 ^ { \frac { 3 } { 2 } } \approx 46 + \frac { a } { b }\), stating the values of the integers \(a\) and \(b\).

3

1. 1. Find the binomial expansion of \(( 1 - x ) ^ { \frac { 1 } { 3 } }\) up to and including the term in \(x ^ { 2 }\).
   2. Hence, or otherwise, show that $$( 125 - 27 x ) ^ { \frac { 1 } { 3 } } \approx 5 + \frac { m } { 25 } x + \frac { n } { 3125 } x ^ { 2 }$$ for small values of \(x\), stating the values of the integers \(m\) and \(n\).
2. Use your result from part (a)(ii) to find an approximate value of \(\sqrt [ 3 ] { 119 }\), giving your answer to five decimal places.
   (2 marks)

3

1. Find the binomial expansion of \(( 1 + 6 x ) ^ { - \frac { 1 } { 3 } }\) up to and including the term in \(x ^ { 2 }\).
2. 1. Find the binomial expansion of \(( 27 + 6 x ) ^ { - \frac { 1 } { 3 } }\) up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
   2. Given that \(\sqrt [ 3 ] { \frac { 2 } { 7 } } = \frac { 2 } { \sqrt [ 3 ] { 28 } }\), use your binomial expansion from part (b)(i) to obtain an approximation to \(\sqrt [ 3 ] { \frac { 2 } { 7 } }\), giving your answer to six decimal places.
      (2 marks)

4

1. Find the binomial expansion of \(( 1 + 5 x ) ^ { \frac { 1 } { 5 } }\) up to and including the term in \(x ^ { 2 }\).
2. 1. Find the binomial expansion of \(( 8 + 3 x ) ^ { - \frac { 2 } { 3 } }\) up to and including the term in \(x ^ { 2 }\).
   2. Use your expansion from part (b)(i) to find an estimate for \(\sqrt [ 3 ] { \frac { 1 } { 81 } }\), giving your answer to four decimal places.
      [0pt] [2 marks]