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

6

1. Find the first three terms in ascending powers of \(x\) of the binomial expansion of \(( 1 + 4 x ) ^ { \frac { 1 } { 2 } }\).
2. State the range of values of \(x\) for which this expansion is valid.

7 A student is trying to find the binomial expansion of \(\sqrt { 1 - x ^ { 3 } }\).
She gets the first three terms as \(1 - \frac { x ^ { 3 } } { 2 } + \frac { x ^ { 6 } } { 8 }\).
She draws the graphs of the curves \(y = \sqrt { 1 - x ^ { 3 } } , y = 1 - \frac { x ^ { 3 } } { 2 }\) and \(y = 1 - \frac { x ^ { 3 } } { 2 } + \frac { x ^ { 6 } } { 8 }\) using software. \includegraphics[max width=\textwidth, alt={}, center]{c30a926b-d832-46f5-aa65-0066ef482c3d-6_901_1265_516_248}

1. Explain why \(1 - \frac { x ^ { 3 } } { 2 } + \frac { x ^ { 6 } } { 8 } \geqslant 1 - \frac { x ^ { 3 } } { 2 }\) for all values of \(x\).
2. Explain why the graphs suggest that the student has made a mistake in the binomial expansion.
3. Find the first four terms in the binomial expansion of \(\sqrt { 1 - x ^ { 3 } }\).
4. State the set of values of \(x\) for which the binomial expansion in part (c) is valid.
5. Sketch the curve \(y = 2.5 \sqrt { 1 - x ^ { 3 } }\) on the grid in the Printed Answer Booklet. \section*{(f) In this question you must show detailed reasoning.} The end of a bus shelter is modelled by the area between the curve \(\mathrm { y } = 2.5 \sqrt { 1 - x ^ { 3 } }\), the lines \(x = - 0.75 , x = 0.75\) and the \(x\)-axis. Lengths are in metres. Calculate, using your answer to part (c), an approximation for the area of the end of the bus shelter as given by this model.

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

7. 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.

3

1. Express \(\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }\) in the form \(\frac { A } { 1 + x } + \frac { B } { 3 + 5 x }\), where \(A\) and \(B\) are integers.
2. Hence, or otherwise, find the binomial expansion of \(\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }\) up to and including the term in \(x ^ { 2 }\).
3. Find the range of values of \(x\) for which the binomial expansion of \(\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }\) is valid.
   (2 marks)

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)

2 It is given that \(\mathrm { f } ( x ) = \frac { 7 x - 1 } { ( 1 + 3 x ) ( 3 - x ) }\).

1. Express \(\mathrm { f } ( x )\) in the form \(\frac { A } { 3 - x } + \frac { B } { 1 + 3 x }\), where \(A\) and \(B\) are integers.
   (3 marks)
2. 1. Find the first three terms of the binomial expansion of \(\mathrm { f } ( x )\) in the form \(a + b x + c x ^ { 2 }\), where \(a\), \(b\) and \(c\) are rational numbers.
      (7 marks)
   2. State why the binomial expansion cannot be expected to give a good approximation to \(\mathrm { f } ( x )\) at \(x = 0.4\).
      (1 mark)

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 \(( 1 + 4 x ) ^ { \frac { 1 } { 2 } }\) up to and including the term in \(x ^ { 2 }\).
   (2 marks)
2. 1. Find the binomial expansion of \(( 4 - x ) ^ { - \frac { 1 } { 2 } }\) up to and including the term in \(x ^ { 2 }\).
   2. State the range of values of \(x\) for which the expansion in part (b)(i) is valid.
3. Find the binomial expansion of \(\sqrt { \frac { 1 + 4 x } { 4 - x } }\) up to and including the term in \(x ^ { 2 }\).
   (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)

3

1. Find the binomial expansion of \(( 1 - 4 x ) ^ { \frac { 1 } { 4 } }\) up to and including the term in \(x ^ { 2 }\).
   [0pt] [2 marks]
2. Find the binomial expansion of \(( 2 + 3 x ) ^ { - 3 }\) up to and including the term in \(x ^ { 2 }\).
3. Hence find the binomial expansion of \(\frac { ( 1 - 4 x ) ^ { \frac { 1 } { 4 } } } { ( 2 + 3 x ) ^ { 3 } }\) up to and including the term in \(x ^ { 2 }\).
   [0pt] [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]

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).

4.

1. Expand \(( 1 + a x ) ^ { - 3 } , | a x | < 1\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\). Give each coefficient as simply as possible in terms of the constant \(a\). Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { 6 - x } { ( 1 + a x ) ^ { 3 } } , | a x | < 1\), is 3 ,
2. find the two possible values of \(a\). Given also that \(a < 0\),
3. show that the coefficient of \(x ^ { 3 }\) in the expansion of \(\frac { 6 - x } { ( 1 + a x ) ^ { 3 } }\) is \(\frac { 14 } { 9 }\).
   4. continued
   

7. $$f ( x ) = \frac { 8 - x } { ( 1 + x ) ( 2 - x ) } , \quad | x | < 1$$

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x = \ln k$$ where \(k\) is an integer to be found.
3. Find the series expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
   7. continued
   7. continued

3.

1. Express \(\frac { 2 + 20 x } { 1 + 2 x - 8 x ^ { 2 } }\) as a sum of partial fractions.
2. Hence find the series expansion of \(\frac { 2 + 20 x } { 1 + 2 x - 8 x ^ { 2 } } , | x | < \frac { 1 } { 4 }\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
   3. continued
   

1. The first four terms in the series expansion of \(( 1 + a x ) ^ { n }\) in ascending powers of \(x\) are

$$1 - 4 x + 24 x ^ { 2 } + k x ^ { 3 }$$ where \(a , n\) and \(k\) are constants and \(| a x | < 1\).

1. Find the values of \(a\) and \(n\).
2. Show that \(k = - 160\).
   3. continued
   

2.

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

2.

1. Expand \(( 1 - 3 x ) ^ { - 2 } , | x | < \frac { 1 } { 3 }\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
2. Hence, or otherwise, show that for small \(x\), $$\left( \frac { 2 - x } { 1 - 3 x } \right) ^ { 2 } \approx 4 + 20 x + 85 x ^ { 2 } + 330 x ^ { 3 }$$

9

1. Express \(\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in three partial fractions.
2. Hence find the first three terms in the expansion of \(\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in ascending powers of \(x\).
3. State the set of values for which the expansion in part (ii) is valid.

6

1. Find the coefficient of \(x ^ { 4 }\) in the expansion of \(( 3 x - 2 ) ^ { 10 }\).
2. In the expansion of \(( 1 + 2 x ) ^ { n }\), where \(n\) is a positive integer, the coefficients of \(x ^ { 7 }\) and \(x ^ { 8 }\) are equal. Find the value of \(n\).
3. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(\frac { 1 } { \sqrt { 4 + x } }\).

8

1. Expand \(\sqrt { 1 + 2 x }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
2. Hence expand \(\frac { \sqrt { 1 + 2 x } } { 1 + 9 x ^ { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
3. Determine the range of values of \(x\) for which the expansion in part (b) is valid.

5

1. 1. Obtain the binomial expansion of \(( 1 - x ) ^ { - 1 }\) up to and including the term in \(x ^ { 2 }\).
   2. Hence, or otherwise, show that $$\frac { 1 } { 3 - 2 x } \approx \frac { 1 } { 3 } + \frac { 2 } { 9 } x + \frac { 4 } { 27 } x ^ { 2 }$$ for small values of \(x\).
2. Obtain the binomial expansion of \(\frac { 1 } { ( 1 - x ) ^ { 2 } }\) up to and including the term in \(x ^ { 2 }\).
3. Given that \(\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }\) can be written in the form \(\frac { A } { ( 3 - 2 x ) } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 1 - x ) ^ { 2 } }\), find the values of \(A , B\) and \(C\).
4. Hence find the binomial expansion of \(\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }\) up to and including the term in \(x ^ { 2 }\).

5

1. 1. Obtain the binomial expansion of \(( 1 - x ) ^ { - 1 }\) up to and including the term in \(x ^ { 2 }\).
      (2 marks)
   2. Hence, or otherwise, show that $$\frac { 1 } { 3 - 2 x } \approx \frac { 1 } { 3 } + \frac { 2 } { 9 } x + \frac { 4 } { 27 } x ^ { 2 }$$ for small values of \(x\).
2. Obtain the binomial expansion of \(\frac { 1 } { ( 1 - x ) ^ { 2 } }\) up to and including the term in \(x ^ { 2 }\).
3. Given that \(\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }\) can be written in the form \(\frac { A } { ( 3 - 2 x ) } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 1 - x ) ^ { 2 } }\), find the values of \(A , B\) and \(C\).
4. Hence find the binomial expansion of \(\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }\) up to and including the term in \(x ^ { 2 }\).

5

1. Find the binomial expansion of \(( 1 + x ) ^ { \frac { 1 } { 3 } }\) up to the term in \(x ^ { 2 }\).
2. 1. Show that \(( 8 + 3 x ) ^ { \frac { 1 } { 3 } } \approx 2 + \frac { 1 } { 4 } x - \frac { 1 } { 32 } x ^ { 2 }\) for small values of \(x\).
   2. Hence show that \(\sqrt [ 3 ] { 9 } \approx \frac { 599 } { 288 }\).