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

375 questions

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CAIE P1 2018 June Q1
5 marks Moderate -0.3
1 The coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 2 + \frac { x } { 2 } \right) ^ { 6 } + ( a + x ) ^ { 5 }\) is 330 . Find the value of the constant \(a\).
CAIE P1 2018 June Q2
3 marks Moderate -0.8
2 Find the coefficient of \(\frac { 1 } { x }\) in the expansion of \(\left( x - \frac { 2 } { x } \right) ^ { 5 }\).
CAIE P1 2019 June Q1
5 marks Moderate -0.8
1 The term independent of \(x\) in the expansion of \(\left( 2 x + \frac { k } { x } \right) ^ { 6 }\), where \(k\) is a constant, is 540.
  1. Find the value of \(k\).
  2. For this value of \(k\), find the coefficient of \(x ^ { 2 }\) in the expansion.
CAIE P1 2019 June Q1
3 marks Moderate -0.5
1 Find the coefficient of \(x\) in the expansion of \(\left( \frac { 2 } { x } - 3 x \right) ^ { 5 }\).
CAIE P1 2016 March Q1
4 marks Moderate -0.8
1
  1. Find the coefficients of \(x ^ { 4 }\) and \(x ^ { 5 }\) in the expansion of \(( 1 - 2 x ) ^ { 5 }\).
  2. It is given that, when \(( 1 + p x ) ( 1 - 2 x ) ^ { 5 }\) is expanded, there is no term in \(x ^ { 5 }\). Find the value of the constant \(p\).
CAIE P1 2017 March Q2
4 marks Standard +0.3
2 In the expansion of \(\left( \frac { 1 } { a x } + 2 a x ^ { 2 } \right) ^ { 5 }\), the coefficient of \(x\) is 5 . Find the value of the constant \(a\).
CAIE P1 2002 November Q1
3 marks Moderate -0.8
1 Find the value of the term which is independent of \(x\) in the expansion of \(\left( x + \frac { 3 } { x } \right) ^ { 4 }\).
CAIE P1 2004 November Q1
4 marks Moderate -0.5
1 Find the coefficient of \(x\) in the expansion of \(\left( 3 x - \frac { 2 } { x } \right) ^ { 5 }\).
CAIE P1 2006 November Q1
3 marks Moderate -0.8
1 Find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( x + \frac { 2 } { x } \right) ^ { 6 }\).
CAIE P1 2007 November Q3
5 marks Moderate -0.8
3
  1. Find the first three terms in the expansion of \(( 2 + u ) ^ { 5 }\) in ascending powers of \(u\).
  2. Use the substitution \(u = x + x ^ { 2 }\) in your answer to part (i) to find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 2 + x + x ^ { 2 } \right) ^ { 5 }\).
CAIE P1 2008 November Q1
3 marks Moderate -0.3
1 Find the value of the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( \frac { x } { 2 } + \frac { 2 } { x } \right) ^ { 6 }\).
CAIE P1 2009 November Q3
6 marks Moderate -0.3
3
  1. Find the first 3 terms in the expansion of \(( 2 - x ) ^ { 6 }\) in ascending powers of \(x\).
  2. Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 + 2 x + a x ^ { 2 } \right) ( 2 - x ) ^ { 6 }\) is 48 , find the value of the constant \(a\).
CAIE P1 2009 November Q1
4 marks Moderate -0.8
1 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { \sqrt { x } } - x\). Given that the curve passes through the point (4,6), find the equation of the curve.
CAIE P1 2010 November Q1
3 marks Easy -1.2
1 Find \(\int \left( x + \frac { 1 } { x } \right) ^ { 2 } \mathrm {~d} x\).
CAIE P1 2010 November Q1
4 marks Moderate -0.8
1
  1. Find the first 3 terms in the expansion, in ascending powers of \(x\), of \(\left( 1 - 2 x ^ { 2 } \right) ^ { 8 }\).
  2. Find the coefficient of \(x ^ { 4 }\) in the expansion of \(\left( 2 - x ^ { 2 } \right) \left( 1 - 2 x ^ { 2 } \right) ^ { 8 }\).
CAIE P1 2010 November Q1
3 marks Moderate -0.3
1 Find the term independent of \(x\) in the expansion of \(\left( x - \frac { 1 } { x ^ { 2 } } \right) ^ { 9 }\).
CAIE P1 2011 November Q1
3 marks Moderate -0.3
1 Find the term independent of \(x\) in the expansion of \(\left( 2 x + \frac { 1 } { x ^ { 2 } } \right) ^ { 6 }\).
CAIE P1 2011 November Q1
5 marks Moderate -0.3
1
  1. Find the first 3 terms in the expansion of \(( 2 - y ) ^ { 5 }\) in ascending powers of \(y\).
  2. Use the result in part (i) to find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 2 - \left( 2 x - x ^ { 2 } \right) \right) ^ { 5 }\).
CAIE P1 2012 November Q4
5 marks Moderate -0.3
4
  1. Find the first 3 terms in the expansion of \(\left( 2 x - x ^ { 2 } \right) ^ { 6 }\) in ascending powers of \(x\).
  2. Hence find the coefficient of \(x ^ { 8 }\) in the expansion of \(( 2 + x ) \left( 2 x - x ^ { 2 } \right) ^ { 6 }\).
CAIE P1 2012 November Q1
3 marks Moderate -0.8
1 In the expansion of \(\left( x ^ { 2 } - \frac { a } { x } \right) ^ { 7 }\), the coefficient of \(x ^ { 5 }\) is - 280 . Find the value of the constant \(a\).
CAIE P1 2012 November Q1
3 marks Moderate -0.8
1 Find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 2 - \frac { 1 } { 2 } x \right) ^ { 7 }\).
CAIE P1 2013 November Q1
5 marks Moderate -0.8
1
  1. Find the first three terms when \(( 2 + 3 x ) ^ { 6 }\) is expanded in ascending powers of \(x\).
  2. In the expansion of \(( 1 + a x ) ( 2 + 3 x ) ^ { 6 }\), the coefficient of \(x ^ { 2 }\) is zero. Find the value of \(a\).
CAIE P1 2013 November Q8
8 marks Moderate -0.3
8
  1. Find the coefficient of \(x ^ { 8 }\) in the expansion of \(\left( x + 3 x ^ { 2 } \right) ^ { 4 }\).
  2. Find the coefficient of \(x ^ { 8 }\) in the expansion of \(\left( x + 3 x ^ { 2 } \right) ^ { 5 }\).
  3. Hence find the coefficient of \(x ^ { 8 }\) in the expansion of \(\left[ 1 + \left( x + 3 x ^ { 2 } \right) \right] ^ { 5 }\).
CAIE P1 2014 November Q1
3 marks Moderate -0.8
1 In the expansion of \(( 2 + a x ) ^ { 7 }\), the coefficient of \(x\) is equal to the coefficient of \(x ^ { 2 }\). Find the value of the non-zero constant \(a\).
CAIE P1 2015 November Q1
3 marks Standard +0.3
1 In the expansion of \(\left( 1 - \frac { 2 x } { a } \right) ( a + x ) ^ { 5 }\), where \(a\) is a non-zero constant, show that the coefficient of \(x ^ { 2 }\) is zero.