Direct single expansion substitution

4. (a) Find the binomial expansion of $$\sqrt [ 3 ] { ( 8 - 9 x ) , \quad } \quad | x | < \frac { 8 } { 9 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient as a simplified fraction.
(b) Use your expansion to estimate an approximate value for \(\sqrt [ 3 ] { 7100 }\), giving your answer to 4 decimal places. State the value of \(x\), which you use in your expansion, and show all your working.

1. (a) Find the binomial expansion of

$$( 4 + 5 x ) ^ { \frac { 1 } { 2 } } , \quad | x | < \frac { 4 } { 5 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
Give each coefficient in its simplest form.
(b) Find the exact value of \(( 4 + 5 x ) ^ { \frac { 1 } { 2 } }\) when \(x = \frac { 1 } { 10 }\) Give your answer in the form \(k \sqrt { 2 }\), where \(k\) is a constant to be determined.
(c) Substitute \(x = \frac { 1 } { 10 }\) into your binomial expansion from part (a) and hence find an approximate value for \(\sqrt { 2 }\) Give your answer in the form \(\frac { p } { q }\) where \(p\) and \(q\) are integers.

6. $$f ( x ) = \frac { 3 } { \sqrt { 1 - x } } , | x | < 1$$

1. Show that \(\mathrm { f } \left( \frac { 1 } { 10 } \right) = \sqrt { 10 }\).
2. Expand \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
3. Use your expansion to find an approximate value for \(\sqrt { 10 }\), giving your answer to 8 significant figures.
4. Find, to 1 significant figure, the percentage error in your answer to part (c).

5

1. Find the first three terms, in ascending powers of \(x\), in the binomial expansion of \(( 1 + 6 x ) ^ { \frac { 1 } { 3 } }\)
   5
2. Use the result from part (a) to obtain an approximation to \(\sqrt [ 3 ] { 1.18 }\) giving your answer to 4 decimal places.
   5
3. Explain why substituting \(x = \frac { 1 } { 2 }\) into your answer to part (a) does not lead to a valid approximation for \(\sqrt [ 3 ] { 4 }\).

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

$$( 1 + 8 x ) ^ { \frac { 1 } { 2 } }$$ giving each term in simplest form.
(b) Explain how you could use \(x = \frac { 1 } { 32 }\) in the expansion to find an approximation for \(\sqrt { 5 }\) There is no need to carry out the calculation.

1. (a) Show that the binomial expansion of

$$( 4 + 5 x ) ^ { \frac { 1 } { 2 } }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) is $$2 + \frac { 5 } { 4 } x + k x ^ { 2 }$$ giving the value of the constant \(k\) as a simplified fraction.
(b) (i) Use the expansion from part (a), with \(x = \frac { 1 } { 10 }\), to find an approximate value for \(\sqrt { 2 }\) Give your answer in the form \(\frac { p } { q }\) where \(p\) and \(q\) are integers.
(ii) Explain why substituting \(x = \frac { 1 } { 10 }\) into this binomial expansion leads to a valid approximation.

5. (a) Prove that, when \(x = \frac { 1 } { 15 }\), the value of \(( 1 + 5 x ) ^ { - \frac { 1 } { 2 } }\) is exactly equal to \(\sin 60 ^ { \circ }\).
(3)
(b) Expand \(( 1 + 5 x ) ^ { - \frac { 1 } { 2 } } , | x | < 0.2\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each term.
(c) Use your answer to part (b) to find an approximation for \(\sin 60 ^ { \circ }\).
(d) Find the difference between the exact value of \(\sin 60 ^ { \circ }\) and the approximation in part (c).

2. $$f ( x ) = \frac { 3 } { \sqrt { 1 - x } } , | x | < 1$$

1. Show that \(\mathrm { f } \left( \frac { 1 } { 10 } \right) = \sqrt { 10 }\).
2. Expand \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
3. Use your expansion to find an approximate value for \(\sqrt { 10 }\), giving your answer to 8 significant figures.
4. Find, to 1 significant figure, the percentage error in your answer to part (c).

1. (a) Show that \(\left( 1 \frac { 1 } { 24 } \right) ^ { - \frac { 1 } { 2 } } = k \sqrt { 6 }\), where \(k\) is rational.
   (b) Expand \(\left( 1 + \frac { 1 } { 2 } x \right) ^ { - \frac { 1 } { 2 } } , | x | < 2\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
   (c) Use your answer to part (b) with \(x = \frac { 1 } { 12 }\) to find an approximate value for \(\sqrt { 6 }\), giving your answer to 5 decimal places.
2. continued
3. Relative to a fixed origin, two lines have the equations

$$\mathbf { r } = ( 7 \mathbf { j } - 4 \mathbf { k } ) + s ( 4 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } )$$ and $$\mathbf { r } = ( - 7 \mathbf { i } + \mathbf { j } + 8 \mathbf { k } ) + t ( - 3 \mathbf { i } + 2 \mathbf { k } )$$ where \(s\) and \(t\) are scalar parameters.
(a) Show that the two lines intersect and find the position vector of the point where they meet.
(b) Find, in degrees to 1 decimal place, the acute angle between the lines.