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

Edexcel C2 2011 January Q5

4 marks Moderate -0.8

Given that \(\binom { 40 } { 4 } = \frac { 40 ! } { 4 ! b ! }\),

1. write down the value of \(b\). In the binomial expansion of \(( 1 + x ) ^ { 40 }\), the coefficients of \(x ^ { 4 }\) and \(x ^ { 5 }\) are \(p\) and \(q\) respectively.
2. Find the value of \(\frac { q } { p }\).
   

Edexcel C2 2012 January Q3

7 marks Moderate -0.8

3.

1. Find the first 4 terms of the binomial expansion, in ascending powers of \(x\), of $$\left( 1 + \frac { x } { 4 } \right) ^ { 8 }$$ giving each term in its simplest form.
2. Use your expansion to estimate the value of \(( 1.025 ) ^ { 8 }\), giving your answer to 4 decimal places.
   

Edexcel C2 2013 January Q1

4 marks Easy -1.2

1. Find the first 3 terms, in ascending powers of \(x\), in the binomial expansion of

$$( 2 - 5 x ) ^ { 6 }$$ Give each term in its simplest form.

Edexcel C2 2014 January Q1

5 marks Moderate -0.8

1. The first three terms in ascending powers of \(x\) in the binomial expansion of \(( 1 + p x ) ^ { 12 }\) are given by

$$1 + 18 x + q x ^ { 2 }$$ where \(p\) and \(q\) are constants.
Find the value of \(p\) and the value of \(q\).

Edexcel C2 2006 June Q1

4 marks Easy -1.2

Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \(( 2 + x ) ^ { 6 }\), giving each term in its simplest form.

Edexcel C2 2007 June Q3

6 marks Moderate -0.3

3.

1. Find the first four terms, in ascending powers of \(x\), in the binomial expansion of \(( 1 + k x ) ^ { 6 }\), where \(k\) is a non-zero constant. Given that, in this expansion, the coefficients of \(x\) and \(x ^ { 2 }\) are equal, find
2. the value of \(k\),
3. the coefficient of \(x ^ { 3 }\).
   

Edexcel C2 2008 June Q3

6 marks Moderate -0.3

3.

1. Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + a x ) ^ { 10 }\), where \(a\) is a non-zero constant. Give each term in its simplest form. Given that, in this expansion, the coefficient of \(x ^ { 3 }\) is double the coefficient of \(x ^ { 2 }\),
2. find the value of \(a\).
   

Edexcel C2 2009 June Q2

6 marks Moderate -0.8

2.

1. Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 2 + k x ) ^ { 7 }$$ where \(k\) is a constant. Give each term in its simplest form. Given that the coefficient of \(x ^ { 2 }\) is 6 times the coefficient of \(x\),
2. find the value of \(k\).
   

Edexcel C2 2010 June Q4

6 marks Moderate -0.8

4.

1. Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + a x ) ^ { 7 }\), where \(a\) is a constant. Give each term in its simplest form. Given that the coefficient of \(x ^ { 2 }\) in this expansion is 525 ,
2. find the possible values of \(a\).
   

Edexcel C2 2011 June Q2

6 marks Moderate -0.8

2.

1. Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 3 + b x ) ^ { 5 }$$ where \(b\) is a non-zero constant. Give each term in its simplest form. Given that, in this expansion, the coefficient of \(x ^ { 2 }\) is twice the coefficient of \(x\),
2. find the value of \(b\).
   

Edexcel C2 2012 June Q1

4 marks Easy -1.2

Find expansion of

Edexcel C2 2013 June Q3

4 marks Easy -1.2

3. Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 2 - \frac { 1 } { 2 } x \right) ^ { 8 }$$ giving each term in its simplest form.

Edexcel C2 2013 June Q2

5 marks Easy -1.2

2.

1. Use the binomial theorem to find all the terms of the expansion of $$( 2 + 3 x ) ^ { 4 }$$ Give each term in its simplest form.
2. Write down the expansion of $$( 2 - 3 x ) ^ { 4 }$$ in ascending powers of \(x\), giving each term in its simplest form.
   

Edexcel C2 2014 June Q1

4 marks Moderate -0.8

1. Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of

$$\left( 1 + \frac { 3 x } { 2 } \right) ^ { 8 }$$ giving each term in its simplest form.

Edexcel C2 2014 June Q3

7 marks Moderate -0.8

3.

1. Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 2 - 3 x ) ^ { 6 }$$ giving each term in its simplest form.
2. Hence, or otherwise, find the first 3 terms, in ascending powers of \(x\), of the expansion of $$\left( 1 + \frac { x } { 2 } \right) ( 2 - 3 x ) ^ { 6 }$$

Edexcel C2 2015 June Q1

4 marks Moderate -0.8

1. Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of

$$\left( 2 - \frac { x } { 4 } \right) ^ { 10 }$$ giving each term in its simplest form.

Edexcel C2 2016 June Q5

9 marks Moderate -0.3

1. Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 2 - 9 x ) ^ { 4 }$$ giving each term in its simplest form. $$f ( x ) = ( 1 + k x ) ( 2 - 9 x ) ^ { 4 } , \text { where } k \text { is a constant }$$ The expansion, in ascending powers of \(x\), of \(\mathrm { f } ( x )\) up to and including the term in \(x ^ { 2 }\) is $$A - 232 x + B x ^ { 2 }$$ where \(A\) and \(B\) are constants.
2. Write down the value of \(A\).
3. Find the value of \(k\).
4. Hence find the value of \(B\).

Edexcel C2 2017 June Q1

4 marks Easy -1.2

1. Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of

$$\left( 3 - \frac { 1 } { 3 } x \right) ^ { 5 }$$ giving each term in its simplest form. \includegraphics[max width=\textwidth, alt={}, center]{752efc6c-8d0e-46a6-b75d-5125956969d8-03_104_107_2631_1774}

Edexcel C2 2018 June Q2

7 marks Moderate -0.8

1. Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of $$( 2 + k x ) ^ { 7 }$$ where \(k\) is a non-zero constant. Give each term in its simplest form. Given that the coefficient of \(x ^ { 3 }\) in this expansion is 1890
2. find the value of \(k\).
   

Edexcel C2 Specimen Q1

4 marks Moderate -0.8

Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \(( 2 + 3 x ) ^ { 6 }\).
(4)

Edexcel F2 2021 October Q9

9 marks Standard +0.8

1. Show that $$n ^ { 5 } - ( n - 1 ) ^ { 5 } \equiv 5 n ^ { 4 } - 10 n ^ { 3 } + 10 n ^ { 2 } - 5 n + 1$$
2. Hence, using the method of differences, show that for all integer values of \(n\), $$\sum _ { r = 1 } ^ { n } r ^ { 4 } = \frac { 1 } { 30 } n ( n + 1 ) ( 2 n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be determined.

Edexcel C12 Specimen Q2

4 marks Easy -1.2

Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 3 - x ) ^ { 6 }$$ and simplify each term.

Edexcel C2 2005 June Q4

6 marks Standard +0.3

1. Write down the first three terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + p x ) ^ { 12 }\), where \(p\) is a non-zero constant. Given that, in the expansion of \(( 1 + p x ) ^ { 12 }\), the coefficient of \(x\) is \(( - q )\) and the coefficient of \(x ^ { 2 }\) is \(11 q\),
2. find the value of \(p\) and the value of \(q\).

CAIE Further Paper 2 2024 November Q8

14 marks Hard +2.3

8

1. By considering the binomial expansion of \(\left( z + \frac { 1 } { z } \right) ^ { 7 }\) ,where \(z = \cos \theta + \mathrm { i } \sin \theta\) ,use de Moivre's theorem to show that $$\cos ^ { 7 } \theta = a \cos 7 \theta + b \cos 5 \theta + c \cos 3 \theta + d \cos \theta$$ where \(a , b , c\) and \(d\) are constants to be determined.
   Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { n } \theta \mathrm {~d} \theta\).
2. Show that $$n I _ { n } = 2 ^ { - \frac { 1 } { 2 } n } + ( n - 1 ) I _ { n - 2 }$$ \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-18_2718_42_107_2007}
3. Using the results given in parts (a) and (b), find the exact value of \(I _ { 9 }\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

OCR C1 2006 January Q2

5 marks Easy -1.2

2

1. Simplify \(( 3 x + 1 ) ^ { 2 } - 2 ( 2 x - 3 ) ^ { 2 }\).
2. Find the coefficient of \(x ^ { 3 }\) in the expansion of $$\left( 2 x ^ { 3 } - 3 x ^ { 2 } + 4 x - 3 \right) \left( x ^ { 2 } - 2 x + 1 \right)$$