1.04c Extend binomial expansion: rational n, |x|<1

2. \(\quad \mathrm { f } ( x ) = ( 2 + k x ) ^ { - 3 } , \quad | k x | < 2\), where \(k\) is a positive constant The binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) is $$A + B x + \frac { 243 } { 16 } x ^ { 2 }$$ where \(A\) and \(B\) are constants.

1. Write down the value of \(A\).
2. Find the value of \(k\).
3. Find the value of \(B\).

1. (a) Find the binomial series expansion of

$$\sqrt { 4 - 9 x } , | x | < \frac { 4 } { 9 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) Give each coefficient in its simplest form.
(b) Use the expansion from part (a), with a suitable value of \(x\), to find an approximate value for \(\sqrt { 310 }\) Show all your working and give your answer to 3 decimal places.

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.

1. Given that \(k\) is a constant and the binomial expansion of

$$\sqrt { 1 + k x } \quad | k x | < 1$$ in ascending powers of \(x\) up to the term in \(x ^ { 3 }\) is $$1 + \frac { 1 } { 8 } x + A x ^ { 2 } + B x ^ { 3 }$$

1. 1. find the value of \(k\),
   2. find the value of the constant \(A\) and the constant \(B\).
2. Use the expansion to find an approximate value to \(\sqrt { 1.15 }\) Show your working and give your answer to 6 decimal places.

1. The binomial expansion of

$$( 3 + k x ) ^ { - 2 } \quad | k x | < 3$$ where \(k\) is a non-zero constant, may be written in the form $$A + B x + C x ^ { 2 } + D x ^ { 3 } + \ldots$$ where \(A\), \(B\), \(C\) and \(D\) are constants.

1. Find the value of \(A\) Given that \(C = 3 B\)
2. show that $$k ^ { 2 } + 6 k = 0$$
3. Hence (i) find the value of \(k\) (ii) find the value of \(D\)

1. (a) Find the first 4 terms of the binomial expansion, in ascending powers of \(x\), of

$$\left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { - \frac { 3 } { 2 } } \quad | x | < \frac { 1 } { 2 }$$ giving each term in simplest form. Given that $$\left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { n } \left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { - \frac { 3 } { 2 } } = \left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { \frac { 1 } { 2 } }$$ (b) write down the value of \(n\).
(c) Hence, or otherwise, find the first 3 terms of the binomial expansion, in ascending powers of \(x\), of $$\left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { \frac { 1 } { 2 } } \quad | x | < \frac { 1 } { 2 }$$ giving each term in simplest form.

1. (a) Use the binomial expansion to expand

$$( 4 - 5 x ) ^ { - \frac { 1 } { 2 } } \quad | x | < \frac { 4 } { 5 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) giving each coefficient as a fully simplified fraction. $$f ( x ) = \frac { 2 + k x } { \sqrt { 4 - 5 x } } \quad \text { where } k \text { is a constant and } | x | < \frac { 4 } { 5 }$$ Given that the series expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), is $$1 + \frac { 3 } { 10 } x + m x ^ { 2 } + \ldots \quad \text { where } m \text { is a constant }$$ (b) find the value of \(k\),
(c) find the value of \(m\).

4. $$\mathrm { f } ( x ) = \sqrt { 1 - 4 x ^ { 2 } } \quad | x | < \frac { 1 } { 2 }$$

1. Find, in ascending powers of \(x\), the first four non-zero terms of the binomial expansion of \(\mathrm { f } ( x )\). Give each coefficient in simplest form.
2. By substituting \(x = \frac { 1 } { 4 }\) into the binomial expansion of \(\mathrm { f } ( x )\), obtain an approximation for \(\sqrt { 3 }\) Give your answer to 4 decimal places. \includegraphics[max width=\textwidth, alt={}, center]{08756c4b-6619-42da-ac8a-2bf065c01de8-13_42_63_2606_1852}

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.

1. Use the binomial series to find the expansion of

$$\frac { 1 } { ( 2 + 5 x ) ^ { 3 } } \quad | \boldsymbol { x } | < \frac { 2 } { 5 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\) Give each coefficient as a fraction in its simplest form.

8. Given that for small values of \(x\) $$( 1 + a x ) ^ { n } \approx 1 - 24 x + 270 x ^ { 2 }$$ where \(n\) is an integer and \(n > 1\),

1. show that \(n = 16\) and find the value of \(a\),
2. use your value of \(a\) and a suitable value of \(x\) to estimate the value of (0.9985) \({ } ^ { 16 }\), giving your answer to 5 decimal places.

7 The expression \(\frac { 11 + 8 x } { ( 2 - x ) ( 1 + x ) ^ { 2 } }\) is denoted by \(\mathrm { f } ( x )\).

1. Express \(\mathrm { f } ( x )\) in the form \(\frac { A } { 2 - x } + \frac { B } { 1 + x } + \frac { C } { ( 1 + x ) ^ { 2 } }\), where \(A , B\) and \(C\) are constants.
2. Given that \(| x | < 1\), find the first 3 terms in the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\).

6

1. Expand \(( 1 + a x ) ^ { - 4 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
2. The coefficients of \(x\) and \(x ^ { 2 }\) in the expansion of \(( 1 + b x ) ( 1 + a x ) ^ { - 4 }\) are 1 and - 2 respectively. Given that \(a > 0\), find the values of \(a\) and \(b\).

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 \(\frac { 1 + x ^ { 2 } } { ( 2 + x ) ^ { 2 } }\).

5

1. Show that \(\sqrt { \frac { 1 - x } { 1 + x } } \approx 1 - x + \frac { 1 } { 2 } x ^ { 2 }\), for \(| x | < 1\).
2. By taking \(x = \frac { 2 } { 7 }\), show that \(\sqrt { 5 } \approx \frac { 111 } { 49 }\).

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.

2

1. Find the first three terms in the binomial expansion of \(\frac { 1 } { \sqrt { 1 - 2 x } }\). State the set of values of \(x\) for which the expansion is valid.
2. Hence find the first three terms in the series expansion of \(\frac { 1 + 2 x } { \sqrt { 1 - 2 x } }\).

$$+ \mu \left( \begin{array} { r } 1 \\ 0 \\ - 2 \end{array} \right)$$ Find the acute angle between the lines. 6 Two students are trying to evaluate the integral \(\int _ { 1 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { - x } } \mathrm {~d} x\).
Sarah uses the trapezium rule with 2 strips, and starts by constructing the following table.

1. Complete the calculation, giving your answer to 3 significant figures. Anish uses a binomial approximation for \(\sqrt { 1 + \mathrm { e } ^ { - x } }\) and then integrates this.
2. Show that, provided \(\mathrm { e } ^ { - x }\) is suitably small, \(\left( 1 + \mathrm { e } ^ { - x } \right) ^ { \frac { 1 } { 2 } } \approx 1 + \frac { 1 } { 2 } \mathrm { e } ^ { - x } - \frac { 1 } { 8 } \mathrm { e } ^ { - 2 x }\).
3. Use this result to evaluate \(\int _ { 1 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { - x } } \mathrm {~d} x\) approximately, giving your answer to 3 significant figures.

4 Find the first three terms in the binomial expansion of \(\sqrt { 4 + x }\) in ascending powers of \(x\).
State the set of values of \(x\) for which the expansion is valid.
show that \(\frac { y - 2 } { y + 1 } = A \mathrm { e } ^ { x ^ { 3 } }\), where \(A\) is a constant.
(ii) Hence, given that \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } ( y - 2 ) ( y + 1 ) ,$$ 6 Solve the equation \(\tan \left( \theta + 45 ^ { \circ } \right) = 1 - 2 \tan \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 90 ^ { \circ }\). Section B (36 marks)
7 A straight pipeline AB passes through a mountain. With respect to axes \(\mathrm { O } x y z\), with \(\mathrm { O } x\) due East, \(\mathrm { O } y\) due North and \(\mathrm { O } z\) vertically upwards, A has coordinates \(( - 200,100,0 )\) and B has coordinates \(( 100,200,100 )\), where units are metres.

2 Given that \(\left( 1 + \frac { x } { p } \right) ^ { q } = 1 - x + \frac { 3 } { 4 } x ^ { 2 } + \ldots\), find \(p\) and \(q\), and state the set of values of \(x\) for which the expansion is valid.

1 Find the coefficient of the term in \(x ^ { 3 }\) in the expansion of \(\frac { 1 } { ( 2 + 3 x ) ^ { 2 } }\).

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.

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.