1.04a Binomial expansion: (a+b)^n for positive integer n

375 questions

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CAIE P1 2020 Specimen Q6
5 marks Moderate -0.5
6
  1. Find the coefficients of \(x ^ { 2 }\) and \(x ^ { 3 }\) in the expansion of \(( 2 - x ) ^ { 6 }\).
  2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 x + 1 ) ( 2 - x ) ^ { 6 }\).
CAIE P1 2003 June Q1
3 marks Moderate -0.5
1 Find the value of the coefficient of \(\frac { 1 } { x }\) in the expansion of \(\left( 2 x - \frac { 1 } { x } \right) ^ { 5 }\).
CAIE P1 2004 June Q4
6 marks Moderate -0.8
4 Find the coefficient of \(x ^ { 3 }\) in the expansion of
  1. \(( 1 + 2 x ) ^ { 6 }\),
  2. \(( 1 - 3 x ) ( 1 + 2 x ) ^ { 6 }\).
CAIE P1 2005 June Q4
5 marks Moderate -0.3
4
  1. Find the first 3 terms in the expansion of \(( 2 - x ) ^ { 6 }\) in ascending powers of \(x\).
  2. Find the value of \(k\) for which there is no term in \(x ^ { 2 }\) in the expansion of \(( 1 + k x ) ( 2 - x ) ^ { 6 }\).
CAIE P1 2006 June Q4
5 marks Moderate -0.8
4 The first three terms in the expansion of \(( 2 + a x ) ^ { n }\), in ascending powers of \(x\), are \(32 - 40 x + b x ^ { 2 }\). Find the values of the constants \(n , a\) and \(b\).
CAIE P1 2008 June Q3
6 marks Moderate -0.8
3
  1. Find the first 3 terms in the expansion, in ascending powers of \(x\), of \(\left( 2 + x ^ { 2 } \right) ^ { 5 }\).
  2. Hence find the coefficient of \(x ^ { 4 }\) in the expansion of \(\left( 1 + x ^ { 2 } \right) ^ { 2 } \left( 2 + x ^ { 2 } \right) ^ { 5 }\).
CAIE P1 2009 June Q3
5 marks Moderate -0.3
3
  1. Find the first 3 terms in the expansion of \(( 2 + 3 x ) ^ { 5 }\) in ascending powers of \(x\).
  2. Hence find the value of the constant \(a\) for which there is no term in \(x ^ { 2 }\) in the expansion of \(( 1 + a x ) ( 2 + 3 x ) ^ { 5 }\).
CAIE P1 2010 June Q2
5 marks Moderate -0.3
2
  1. Find the first 3 terms in the expansion of \(\left( 2 x - \frac { 3 } { x } \right) ^ { 5 }\) in descending powers of \(x\).
  2. Hence find the coefficient of \(x\) in the expansion of \(\left( 1 + \frac { 2 } { x ^ { 2 } } \right) \left( 2 x - \frac { 3 } { x } \right) ^ { 5 }\).
CAIE P1 2010 June Q2
5 marks Moderate -0.8
2
  1. Find the first three terms, in descending powers of \(x\), in the expansion of \(\left( x - \frac { 2 } { x } \right) ^ { 6 }\).
  2. Find the coefficient of \(x ^ { 4 }\) in the expansion of \(\left( 1 + x ^ { 2 } \right) \left( x - \frac { 2 } { x } \right) ^ { 6 }\).
CAIE P1 2011 June Q1
3 marks Moderate -0.3
1 Find the coefficient of \(x\) in the expansion of \(\left( x + \frac { 2 } { x ^ { 2 } } \right) ^ { 7 }\).
CAIE P1 2011 June Q1
5 marks Moderate -0.5
1 The coefficient of \(x ^ { 3 }\) in the expansion of \(( a + x ) ^ { 5 } + ( 1 - 2 x ) ^ { 6 }\), where \(a\) is positive, is 90 . Find the value of \(a\).
CAIE P1 2012 June Q2
4 marks Standard +0.3
2 Find the coefficient of \(x ^ { 6 }\) in the expansion of \(\left( 2 x ^ { 3 } - \frac { 1 } { x ^ { 2 } } \right) ^ { 7 }\).
CAIE P1 2013 June Q2
5 marks Moderate -0.3
2
  1. In the expression \(( 1 - p x ) ^ { 6 } , p\) is a non-zero constant. Find the first three terms when \(( 1 - p x ) ^ { 6 }\) is expanded in ascending powers of \(x\).
  2. It is given that the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 - x ) ( 1 - p x ) ^ { 6 }\) is zero. Find the value of \(p\).
CAIE P1 2013 June Q2
5 marks Moderate -0.3
2 Find the coefficient of \(x ^ { 2 }\) in the expansion of
  1. \(\left( 2 x - \frac { 1 } { 2 x } \right) ^ { 6 }\),
  2. \(\left( 1 + x ^ { 2 } \right) \left( 2 x - \frac { 1 } { 2 x } \right) ^ { 6 }\).
CAIE P1 2013 June Q4
6 marks Moderate -0.8
4
  1. Find the first three terms in the expansion of \(( 2 + a x ) ^ { 5 }\) in ascending powers of \(x\).
  2. Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 + 2 x ) ( 2 + a x ) ^ { 5 }\) is 240 , find the possible values of \(a\).
CAIE P1 2014 June Q3
4 marks Moderate -0.8
3 Find the term independent of \(x\) in the expansion of \(\left( 4 x ^ { 3 } + \frac { 1 } { 2 x } \right) ^ { 8 }\).
CAIE P1 2014 June Q2
5 marks Standard +0.3
2 Find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 + x ^ { 2 } \right) \left( \frac { x } { 2 } - \frac { 4 } { x } \right) ^ { 6 }\).
CAIE P1 2014 June Q1
3 marks Moderate -0.5
1 Find the coefficient of \(x\) in the expansion of \(\left( x ^ { 2 } - \frac { 2 } { x } \right) ^ { 5 }\).
CAIE P1 2015 June Q3
6 marks Standard +0.3
3
  1. Write down the first 4 terms, in ascending powers of \(x\), of the expansion of \(( a - x ) ^ { 5 }\).
  2. The coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 - a x ) ( a - x ) ^ { 5 }\) is - 200 . Find the possible values of the constant \(a\).
CAIE P1 2016 June Q1
3 marks Moderate -0.5
1 Find the term independent of \(x\) in the expansion of \(\left( x - \frac { 3 } { 2 x } \right) ^ { 6 }\).
CAIE P1 2016 June Q4
6 marks Moderate -0.3
4 Find the term that is independent of \(x\) in the expansion of
  1. \(\left( x - \frac { 2 } { x } \right) ^ { 6 }\),
  2. \(\left( 2 + \frac { 3 } { x ^ { 2 } } \right) \left( x - \frac { 2 } { x } \right) ^ { 6 }\).
CAIE P1 2016 June Q1
3 marks Moderate -0.5
1 Find the coefficient of \(x\) in the expansion of \(\left( \frac { 1 } { x } + 3 x ^ { 2 } \right) ^ { 5 }\).
CAIE P1 2017 June Q1
4 marks Moderate -0.8
1 The coefficients of \(x ^ { 2 }\) and \(x ^ { 3 }\) in the expansion of \(( 3 - 2 x ) ^ { 6 }\) are \(a\) and \(b\) respectively. Find the value of \(\frac { a } { b }\).
CAIE P1 2017 June Q1
6 marks Moderate -0.8
1
  1. Find the coefficient of \(x\) in the expansion of \(\left( 2 x - \frac { 1 } { x } \right) ^ { 5 }\).
  2. Hence find the coefficient of \(x\) in the expansion of \(\left( 1 + 3 x ^ { 2 } \right) \left( 2 x - \frac { 1 } { x } \right) ^ { 5 }\).
CAIE P1 2018 June Q1
5 marks Moderate -0.8
1
  1. Find the first three terms in the expansion, in ascending powers of \(x\), of \(( 1 - 2 x ) ^ { 5 }\).
  2. Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 + a x + 2 x ^ { 2 } \right) ( 1 - 2 x ) ^ { 5 }\) is 12 , find the value of the constant \(a\).