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

313 questions

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CAIE P3 2002 June Q2
4 marks Moderate -0.8
2 Expand \(( 1 - 3 x ) ^ { - \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
CAIE P3 2005 June Q1
4 marks Moderate -0.8
1 Expand \(( 1 + 4 x ) ^ { - \frac { 1 } { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
CAIE P3 2007 June Q1
4 marks Moderate -0.8
1 Expand \(( 2 + 3 x ) ^ { - 2 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2009 June Q5
7 marks Standard +0.3
5 When \(( 1 + 2 x ) ( 1 + a x ) ^ { \frac { 2 } { 3 } }\), where \(a\) is a constant, is expanded in ascending powers of \(x\), the coefficient of the term in \(x\) is zero.
  1. Find the value of \(a\).
  2. When \(a\) has this value, find the term in \(x ^ { 3 }\) in the expansion of \(( 1 + 2 x ) ( 1 + a x ) ^ { \frac { 2 } { 3 } }\), simplifying the coefficient.
CAIE P3 2011 June Q1
4 marks Moderate -0.8
1 Expand \(\sqrt [ 3 ] { } ( 1 - 6 x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
CAIE P3 2011 June Q8
10 marks Standard +0.3
8
  1. Express \(\frac { 5 x - x ^ { 2 } } { ( 1 + x ) \left( 2 + x ^ { 2 } \right) }\) in partial fractions.
  2. Hence obtain the expansion of \(\frac { 5 x - x ^ { 2 } } { ( 1 + x ) \left( 2 + x ^ { 2 } \right) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
CAIE P3 2012 June Q2
5 marks Standard +0.3
2
  1. Expand \(\frac { 1 } { \sqrt { } ( 1 - 4 x ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
  2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { 1 + 2 x } { \sqrt { } ( 4 - 16 x ) }\).
CAIE P3 2012 June Q3
5 marks Moderate -0.3
3 Expand \(\sqrt { } \left( \frac { 1 - x } { 1 + x } \right)\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2012 June Q1
4 marks Moderate -0.3
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.
CAIE P3 2013 June Q2
4 marks Moderate -0.3
2 Expand \(\frac { 1 + 3 x } { \sqrt { } ( 1 + 2 x ) }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2014 June Q9
10 marks Standard +0.3
9
  1. Express \(\frac { 4 + 12 x + x ^ { 2 } } { ( 3 - x ) ( 1 + 2 x ) ^ { 2 } }\) in partial fractions.
  2. Hence obtain the expansion of \(\frac { 4 + 12 x + x ^ { 2 } } { ( 3 - x ) ( 1 + 2 x ) ^ { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2014 June Q2
4 marks Moderate -0.8
2 Expand \(( 1 + 3 x ) ^ { - \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
CAIE P3 2015 June Q3
6 marks Standard +0.3
3 Show that, for small values of \(x ^ { 2 }\), $$\left( 1 - 2 x ^ { 2 } \right) ^ { - 2 } - \left( 1 + 6 x ^ { 2 } \right) ^ { \frac { 2 } { 3 } } \approx k x ^ { 4 }$$ where the value of the constant \(k\) is to be determined.
CAIE P3 2016 June Q8
10 marks Standard +0.3
8 Let \(\mathrm { f } ( x ) = \frac { 4 x ^ { 2 } + 12 } { ( x + 1 ) ( x - 3 ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2016 June Q2
4 marks Moderate -0.3
2 Expand \(\frac { 1 } { \sqrt { ( 1 - 2 x ) } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
CAIE P3 2016 June Q10
10 marks Standard +0.3
10 Let \(\mathrm { f } ( x ) = \frac { 10 x - 2 x ^ { 2 } } { ( x + 3 ) ( x - 1 ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2017 June Q2
4 marks Moderate -0.3
2 Expand \(\frac { 1 } { \sqrt [ 3 ] { } ( 1 + 6 x ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
CAIE P3 2017 June Q2
4 marks Moderate -0.8
2 Expand \(( 3 + 2 x ) ^ { - 3 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2019 June Q8
10 marks Standard +0.3
8 Let \(f ( x ) = \frac { 16 - 17 x } { ( 2 + x ) ( 3 - x ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2019 June Q1
4 marks Standard +0.3
1 Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 - x ) ( 1 + 3 x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\).
CAIE P3 2019 June Q9
10 marks Standard +0.8
9 Let \(\mathrm { f } ( x ) = \frac { 2 x ( 5 - x ) } { ( 3 + x ) ( 1 - x ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\).
CAIE P3 2017 March Q9
10 marks Standard +0.8
9 Let \(\mathrm { f } ( x ) = \frac { x ( 6 - x ) } { ( 2 + x ) \left( 4 + x ^ { 2 } \right) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2019 March Q8
10 marks Standard +0.3
8 Let \(\mathrm { f } ( x ) = \frac { 12 + 12 x - 4 x ^ { 2 } } { ( 2 + x ) ( 3 - 2 x ) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2002 November Q6
9 marks Standard +0.3
6 Let \(f ( x ) = \frac { 6 + 7 x } { ( 2 - x ) \left( 1 + x ^ { 2 } \right) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Show that, when \(x\) is sufficiently small for \(x ^ { 4 }\) and higher powers to be neglected, $$f ( x ) = 3 + 5 x - \frac { 1 } { 2 } x ^ { 2 } - \frac { 15 } { 4 } x ^ { 3 }$$
CAIE P3 2003 November Q2
4 marks Moderate -0.8
2 Expand \(\left( 2 + x ^ { 2 } \right) ^ { - 2 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 4 }\), simplifying the coefficients.