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

375 questions

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OCR MEI C1 2008 January Q7
4 marks Easy -1.3
7
  1. Find the value of \({ } ^ { 8 } \mathrm { C } _ { 3 }\).
  2. Find the coefficient of \(x ^ { 3 }\) in the binomial expansion of \(\left( 1 - \frac { 1 } { 2 } x \right) ^ { 8 }\).
OCR MEI C1 2009 January Q5
5 marks Easy -1.2
5
  1. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( x ^ { 2 } - 3 \right) \left( x ^ { 3 } + 7 x + 1 \right)\).
  2. Find the coefficient of \(x ^ { 2 }\) in the binomial expansion of \(( 1 + 2 x ) ^ { 7 }\).
OCR MEI C1 2007 June Q6
4 marks Moderate -0.8
6 Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 - 2 x ) ^ { 5 }\).
OCR MEI C1 2008 June Q8
4 marks Moderate -0.8
8 Find the coefficient of \(x ^ { 3 }\) in the binomial expansion of \(( 5 - 2 x ) ^ { 5 }\).
OCR MEI C1 2015 June Q7
4 marks Easy -1.2
7 Find and simplify the binomial expansion of \(( 3 x - 2 ) ^ { 4 }\).
OCR MEI C1 Q12
12 marks Moderate -0.8
12
  1. Expand \(( 1 + 2 x ) ^ { 6 }\), simplifying all the terms.
  2. Hence find an expression for \(\mathrm { f } ( x ) = ( 1 + 2 x ) ^ { 6 } + ( 1 - 2 x ) ^ { 6 }\) in its simplest form.
  3. Substituting \(x = 0.01\) into the first two terms of \(\mathrm { f } ( x )\) gives an approximate value, z for \(1.02 ^ { 6 } + 0.98 ^ { 6 }\). Find \(z\). By considering the value of the third term, comment on the accuracy of \(z\) as an approximation for \(1.02 ^ { 6 } + 0.98 ^ { 6 }\).
OCR MEI C1 Q3
3 marks Moderate -0.3
3 Find the term independent of \(x\) in the expansion of \(\left( x - \frac { 2 } { x } \right) ^ { 4 }\).
OCR MEI C1 Q6
4 marks Moderate -0.8
6 Find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 3 - 2 x ) ^ { 5 }\).
OCR MEI C1 Q3
3 marks Moderate -0.8
3 Find the term which has the highest coefficient in the expansion of \(( 1 + x ) ^ { 8 }\).
OCR MEI C1 Q1
4 marks Moderate -0.8
1 Find and simplify the binomial expansion of \(( 3 x - 2 ) ^ { 4 }\).
OCR MEI C1 Q2
4 marks Moderate -0.8
2 Find the coefficient of \(x ^ { 4 }\) in the binomial expansion of \(( 5 + 2 x ) ^ { 7 }\).
OCR MEI C1 Q3
4 marks Moderate -0.8
3 Find the coefficient of \(x ^ { 3 }\) in the binomial expansion of \(( 2 - 4 x ) ^ { 5 }\).
OCR MEI C1 Q4
4 marks Standard +0.3
4 The binomial expansion of \(\left( 2 x + \frac { 5 } { x } \right) ^ { 6 }\) has a term which is a constant. Find this term.
OCR MEI C1 Q5
5 marks Easy -1.2
5
  1. Evaluate \({ } ^ { 5 } \mathrm { C } _ { 3 }\).
  2. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 - 2 x ) ^ { 5 }\).
OCR MEI C1 Q6
4 marks Moderate -0.8
6 Find the coefficient of \(x ^ { 4 }\) in the binomial expansion of \(( 5 + 2 x ) ^ { 6 }\).
OCR MEI C1 Q7
4 marks Moderate -0.8
7 Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \(( 2 - 3 x ) ^ { 5 }\), simplifying each term.
OCR MEI C1 Q8
5 marks Moderate -0.3
8 You are given that
  • the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 5 + 2 x ^ { 2 } \right) \left( x ^ { 3 } + k x + m \right)\) is 29 ,
  • when \(x ^ { 3 } + k x + m\) is divided by ( \(x - 3\) ), the remainder is 59 .
Find the values of \(k\) and \(m\).
OCR MEI C1 Q9
4 marks Easy -1.3
9 Expand \(\left( 1 + \frac { 1 } { 2 } x \right) ^ { 4 }\), simplifying the coefficients.
OCR MEI C1 Q10
4 marks Easy -1.2
10 Find the binomial expansion of \(\left( x + \frac { 5 } { x } \right) ^ { 3 }\), simplifying the terms.
OCR MEI C1 Q11
4 marks Easy -1.2
11
  1. Calculate \({ } ^ { 5 } \mathrm { C } _ { 3 }\).
  2. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 + 2 x ) ^ { 5 }\).
OCR MEI C1 Q12
5 marks Easy -1.2
12
  1. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( x ^ { 2 } - 3 \right) \left( x ^ { 3 } + 7 x + 1 \right)\).
  2. Find the coefficient of \(x ^ { 2 }\) in the binomial expansion of \(( 1 + 2 x ) ^ { 7 }\).
OCR MEI C1 Q13
4 marks Moderate -0.8
13 Find the coefficient of \(x ^ { 3 }\) in the binomial expansion of \(( 5 - 2 x ) ^ { 5 }\).
OCR MEI C1 Q14
4 marks Easy -1.2
14
  1. Find the value of \({ } ^ { 8 } \mathrm { C } _ { 3 }\).
  2. Find the coefficient of \(x ^ { 3 }\) in the binomial expansion of \(\left( 1 - \frac { 1 } { 2 } x \right) ^ { 8 }\).
OCR MEI C1 Q15
4 marks Moderate -0.8
15 Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 - 2 x ) ^ { 5 }\).
OCR MEI C1 Q16
3 marks Easy -1.2
16 Calculate the coefficient of \(x ^ { 4 }\) in the expansion of \(( x + 5 ) ^ { 6 }\).