Coefficient zero after multiplying binomial

Questions where a binomial expansion is multiplied by a linear or simple factor, and a specific coefficient in the product is set to zero to find unknown constants.

9 questions

CAIE P1 2023 June Q3
3
  1. Give the complete expansion of \(\left( x + \frac { 2 } { x } \right) ^ { 5 }\).
  2. In the expansion of \(\left( a + b x ^ { 2 } \right) \left( x + \frac { 2 } { x } \right) ^ { 5 }\), the coefficient of \(x\) is zero and the coefficient of \(\frac { 1 } { x }\) is 80 . Find the values of the constants \(a\) and \(b\).
CAIE P1 2005 June Q4
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 2009 June Q3
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 Q6
6
  1. Find the first 3 terms in the expansion of \(( 1 + a x ) ^ { 5 }\) in ascending powers of \(x\).
  2. Given that there is no term in \(x\) in the expansion of \(( 1 - 2 x ) ( 1 + a x ) ^ { 5 }\), find the value of the constant \(a\).
  3. For this value of \(a\), find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 - 2 x ) ( 1 + a x ) ^ { 5 }\).
CAIE P1 2011 June Q2
2
  1. Find the terms in \(x ^ { 2 }\) and \(x ^ { 3 }\) in the expansion of \(\left( 1 - \frac { 3 } { 2 } x \right) ^ { 6 }\).
  2. Given that there is no term in \(x ^ { 3 }\) in the expansion of \(( k + 2 x ) \left( 1 - \frac { 3 } { 2 } x \right) ^ { 6 }\), find the value of the constant \(k\).
CAIE P1 2013 June Q2
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 2016 March Q1
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 2013 November Q1
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 \(a\).
Edexcel C12 2017 October Q11
11. \(\mathrm { f } ( x ) = ( a - x ) ( 3 + a x ) ^ { 5 }\), where \(a\) is a positive constant
  1. Find the first 3 terms, in ascending powers of \(x\), in the binomial expansion of $$( 3 + a x ) ^ { 5 }$$ Give each term in its simplest form. Given that in the expansion of \(\mathrm { f } ( x )\) the coefficient of \(x\) is zero,
  2. find the exact value of \(a\).