Binomial with negative or fractional powers of x

Questions asking for a coefficient in the expansion of expressions like (x + k/x^m)^n or (kx^p + m/x^q)^n, requiring careful tracking of powers when terms contain x in denominators or non-unit powers.

16 questions · Moderate -0.3

1.04a Binomial expansion: (a+b)^n for positive integer n
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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 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 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 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 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 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
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 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 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 2015 November Q2
4 marks Standard +0.3
2 Find the coefficient of \(x\) in the expansion of \(\left( \frac { x } { 3 } + \frac { 9 } { x ^ { 2 } } \right) ^ { 7 }\).
CAIE P1 2018 November Q1
3 marks Moderate -0.3
1 Find the coefficient of \(\frac { 1 } { x ^ { 3 } }\) in the expansion of \(\left( x - \frac { 2 } { x } \right) ^ { 7 }\).
Edexcel C2 Q4
8 marks Standard +0.3
4. The first three terms in the expansion in descending powers of \(x\) of $$\left( x + \frac { k } { x ^ { 2 } } \right) ^ { 15 } ,$$ where \(k\) is a constant, are $$x ^ { 15 } + 30 x ^ { 12 } + A x ^ { 9 } .$$
  1. Find the values of \(k\) and \(A\).
  2. Find the value of the term independent of \(x\) in the expansion.
CAIE P1 2018 November Q1
4 marks Standard +0.3
Find the coefficient of \(\frac{1}{x^2}\) in the expansion of \(\left(3x + \frac{2}{3x^2}\right)^7\). [4]
CAIE P3 2018 June Q2
3 marks Easy -1.2
Find the coefficient of \(\frac{1}{x}\) in the expansion of \(\left(x - \frac{2}{x}\right)^5\). [3]
AQA Paper 2 2020 June Q3
3 marks Moderate -0.3
Find the coefficient of \(x^2\) in the binomial expansion of \(\left(2x - \frac{3}{x}\right)^8\) [3 marks]