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

3

1. Obtain the binomial expansion of \(( 1 + x ) ^ { \frac { 1 } { 2 } }\) up to and including the term in \(x ^ { 2 }\).
2. Hence obtain the binomial expansion of \(\sqrt { 1 + \frac { 3 } { 2 } x }\) up to and including the term in \(x ^ { 2 }\).
3. Hence show that \(\sqrt { \frac { 2 + 3 x } { 8 } } \approx a + b x + c x ^ { 2 }\) for small values of \(x\), where \(a , b\) and \(c\) are constants to be found.

4

1. 1. Find the binomial expansion of \(( 1 - x ) ^ { \frac { 1 } { 2 } }\) up to and including the term in \(x ^ { 2 }\).
      (2 marks)
   2. Hence obtain the binomial expansion of \(\sqrt { 4 - x }\) up to and including the term in \(x ^ { 2 }\).
      (3 marks)
2. Use your answer to part (a)(ii) to find an approximate value for \(\sqrt { 3 }\). Give your answer to three decimal places.

3

1. 1. Find the binomial expansion of \(( 1 + x ) ^ { - \frac { 1 } { 3 } }\) up to and including the term in \(x ^ { 2 }\).
      (2 marks)
   2. Hence find the binomial expansion of \(\left( 1 + \frac { 3 } { 4 } x \right) ^ { - \frac { 1 } { 3 } }\) up to and including the term in \(x ^ { 2 }\).
2. Hence show that \(\sqrt [ 3 ] { \frac { 256 } { 4 + 3 x } } \approx a + b x + c x ^ { 2 }\) for small values of \(x\), stating the values of the constants \(a , b\) and \(c\).

4

1. Find the binomial expansion of \(( 1 + x ) ^ { - \frac { 1 } { 2 } }\) up to the term in \(x ^ { 2 }\).
2. Hence, or otherwise, obtain the binomial expansion of \(\frac { 1 } { \sqrt { 1 + 2 x } }\) up to the term in \(x ^ { 2 }\), in simplified form.
3. Use your answer to part (b) with \(x = - 0.1\) to show that \(\sqrt { 5 } \approx 2.23\).

2

1. Obtain the binomial expansion of \(( 1 - x ) ^ { - 3 }\) up to and including the term in \(x ^ { 2 }\).
2. Hence obtain the binomial expansion of \(\left( 1 - \frac { 5 } { 2 } x \right) ^ { - 3 }\) up to and including the term in \(x ^ { 2 }\).
3. Find the range of values of \(x\) for which the binomial expansion of \(\left( 1 - \frac { 5 } { 2 } x \right) ^ { - 3 }\) would be valid.
4. Given that \(x\) is small, show that \(\left( \frac { 4 } { 2 - 5 x } \right) ^ { 3 } \approx a + b x + c x ^ { 2 }\), where \(a , b\) and \(c\) are integers.

2

1. 1. Find the binomial expansion of \(( 1 + x ) ^ { - 1 }\) up to the term in \(x ^ { 3 }\).
   2. Hence, or otherwise, obtain the binomial expansion of \(\frac { 1 } { 1 + 3 x }\) up to the term in \(x ^ { 3 }\).
2. Express \(\frac { 1 + 4 x } { ( 1 + x ) ( 1 + 3 x ) }\) in partial fractions.
3. 1. Find the binomial expansion of \(\frac { 1 + 4 x } { ( 1 + x ) ( 1 + 3 x ) }\) up to the term in \(x ^ { 3 }\).
   2. Find the range of values of \(x\) for which the binomial expansion of \(\frac { 1 + 4 x } { ( 1 + x ) ( 1 + 3 x ) }\) is valid.

4

1. 1. Obtain the binomial expansion of \(( 1 - x ) ^ { \frac { 1 } { 4 } }\) up to and including the term in \(x ^ { 2 }\).
   2. Hence show that \(( 81 - 16 x ) ^ { \frac { 1 } { 4 } } \approx 3 - \frac { 4 } { 27 } x - \frac { 8 } { 729 } x ^ { 2 }\) for small values of \(x\).
2. Use the result from part (a)(ii) to find an approximation for \(\sqrt [ 4 ] { 80 }\), giving your answer to seven decimal places.

3

1. Find the binomial expansion of \(( 1 - x ) ^ { - 1 }\) up to and including the term in \(x ^ { 2 }\).
2. 1. Express \(\frac { 3 x - 1 } { ( 1 - x ) ( 2 - 3 x ) }\) in the form \(\frac { A } { 1 - x } + \frac { B } { 2 - 3 x }\), where \(A\) and \(B\) are integers.
   2. Find the binomial expansion of \(\frac { 3 x - 1 } { ( 1 - x ) ( 2 - 3 x ) }\) up to and including the term in \(x ^ { 2 }\).
3. Find the range of values of \(x\) for which the binomial expansion of \(\frac { 3 x - 1 } { ( 1 - x ) ( 2 - 3 x ) }\) is valid.

6

1. Find the first three non-zero terms of the binomial expansion of \(\frac { 1 } { \sqrt { 4 - x ^ { 2 } } }\) for \(| x | < 2\).
2. Use this result to find an approximation for \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x\), rounding your answer to
   4 significant figures.
3. Given that \(\int \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x = \arcsin \left( \frac { 1 } { 2 } x \right) + c\), evaluate \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x\), rounding your answer to 4 significant figures.

2

1. Given that $$\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) } = \frac { A } { 1 + x } + \frac { B } { ( 1 + x ) ^ { 2 } } + \frac { C } { 1 - 4 x } ,$$ where \(A , B\) and \(C\) are constants, find \(B\) and \(C\), and show that \(A = 0\).
2. Given that \(x\) is sufficiently small, find the first three terms of the binomial expansions of \(( 1 + x ) ^ { - 2 }\) and \(( 1 - 4 x ) ^ { - 1 }\). Hence find the first three terms of the expansion of \(\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) }\).

6

1. Find the first three non-zero terms of the binomial series expansion of \(\frac { 1 } { \sqrt { 1 + 4 x ^ { 2 } } }\), and state the set of values of \(x\) for which the expansion is valid.
2. Hence find the first three non-zero terms of the series expansion of \(\frac { 1 - x ^ { 2 } } { \sqrt { 1 + 4 x ^ { 2 } } }\).

1.In the binomial expansion of $$( 1 - 8 x ) ^ { p } \quad | x | < \frac { 1 } { 8 }$$ where \(p\) is a positive constant,
-the sum of the coefficient of \(x\) and the coefficient of \(x ^ { 2 }\) is equal to the coefficient of \(x ^ { 3 }\) -the coefficient of \(x ^ { 2 }\) is positive
Determine the value of \(p\) . \includegraphics[max width=\textwidth, alt={}, center]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-02_2264_56_315_1977}

1.(a)Show that \(\sqrt { \frac { 1 + x } { 1 - x } }\) can be written in the form \(\frac { 1 + x } { \sqrt { 1 - x ^ { 2 } } }\) for \(| x | < 1\) (b)Hence,or otherwise,find the expansion,in ascending powers of \(x\) up to and including the term in \(x ^ { 5 }\) ,of \(\sqrt { \frac { 1 + x } { 1 - x } }\)

5

1. Find the first three terms in the expansion of \(( 1 + p x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\).
2. The expansion of \(( 1 + q x ) ( 1 + p x ) ^ { \frac { 1 } { 3 } }\) is \(1 + x - \frac { 2 } { 9 } x ^ { 2 } + \ldots\). Find the possible values of the constants \(p\) and \(q\).

6

1. Find the first three terms, in ascending powers of \(x\), of the binomial expansion of \(\frac { 1 } { \sqrt { 4 + x } }\) 6
2. Hence, find the first three terms of the binomial expansion of \(\frac { 1 } { \sqrt { 4 - x ^ { 3 } } }\) 6 (d) (i) Edward, a student, decides to use this method to find a more accurate value for the integral by increasing the number of terms of the binomial expansion used. Explain clearly whether Edward's approximation will be an overestimate, an underestimate, or if it is impossible to tell.
   [0pt] [2 marks]
   6 (d) (ii) Edward goes on to use the expansion from part (b) to find an approximation for \(\int _ { - 2 } ^ { 0 } \frac { 1 } { \sqrt { 4 - x ^ { 3 } } } \mathrm {~d} x\) Explain why Edward's approximation is invalid.

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

6

1. Find the first two terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 1 - \frac { x } { 2 } \right) ^ { \frac { 1 } { 2 } }$$ 6
2. Hence, for small values of \(x\), show that $$\sin 4 x + \sqrt { \cos x } \approx A + B x + C x ^ { 2 }$$ where \(A , B\) and \(C\) are constants to be found.

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.

4 One end of a light elastic string of natural length \(l \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\) is attached to a particle \(A\) of mass \(m \mathrm {~kg}\). The other end of the string is attached to a fixed point \(O\) which is on a horizontal surface. The surface is modelled as being smooth and \(A\) moves in a circular path around \(O\) with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The extension of the string is denoted by \(x \mathrm {~m}\).

1. Show that \(x\) satisfies \(\lambda x ^ { 2 } + \lambda l x - l m v ^ { 2 } = 0\).
2. By solving the equation in part (a) and using a binomial series, show that if \(\lambda\) is very large then \(\lambda x \approx m v ^ { 2 }\).
3. By considering the tension in the string, explain how the result obtained when \(\lambda\) is very large relates to the situation when the string is inextensible. The nature of the horizontal surface is such that the modelling assumption that it is smooth is justifiable provided that the speed of the particle does not exceed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the case where \(m = 0.16\) and \(\lambda = 260\), the extension of the string is measured as being 3.0 cm .
4. Estimate the value of \(v\).
5. Explain whether the value of \(v\) means that the modelling assumption is necessarily justifiable in this situation.

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

$$( 8 - 3 x ) ^ { - \frac { 1 } { 3 } } \quad | x | < \frac { 8 } { 3 }$$ giving each coefficient as a simplified fraction.
(b) Use the answer from part (a) with \(x = \frac { 2 } { 3 }\) to find a rational approximation to \(\sqrt [ 3 ] { 6 }\)

7

1. Show that the first three terms in the expansion of \(( 1 - 2 x ) ^ { \frac { 1 } { 2 } }\) are \(1 - x - \frac { 1 } { 2 } x ^ { 2 }\) and find the next term.
2. State the range of values of \(x\) for which this expansion is valid.
3. Hence show that the first four terms in the expansion of \(( 2 + x ) ( 1 - 2 x ) ^ { \frac { 1 } { 2 } }\) are \(2 - x + a x ^ { 2 } + b x ^ { 3 }\) and state the values of \(a\) and \(b\).

11

1. Expand \(( 1 + x ) ^ { - 1 }\) up to and including the term in \(x ^ { 2 }\).
2. (a) Expand \(\sqrt { 2 + 3 x ^ { 2 } }\) up to and including the term in \(x ^ { 4 }\).
   (b) For what range of values of \(x\) is this expansion valid?
3. Find the first three terms of the expansion of \(\frac { \sqrt { 2 + 3 x ^ { 2 } } } { 1 + x }\) in ascending powers of \(x\) and hence show that \(\int _ { 0 } ^ { 0.1 } \frac { \sqrt { 2 + 3 x ^ { 2 } } } { 1 + x } \mathrm {~d} x \approx 0.135\).

6

1. Expand \(( 1 + x ) ^ { \frac { 1 } { 2 } }\), for \(| x | < 1\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
2. In the expansion of \(( 2 + k x ) ( 1 + x ) ^ { \frac { 1 } { 2 } }\) the coefficient of \(x ^ { 3 }\) is 1 . Find the value of \(k\).

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

Find the first three terms in the binomial expansion of \(\frac { 2 - x } { \sqrt { 1 + 3 x } }\) in ascending powers of \(x\). State the range of values of \(x\) for which the expansion is valid. By writing \(x = \frac { 1 } { 22 }\) in your expansion, find an approximate value for \(\sqrt { 22 }\) in the form \(\frac { a } { b }\), where \(a , b\) are integers whose values are to be found.