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

375 questions

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OCR MEI C1 Q17
4 marks Easy -1.2
17 Calculate \({ } ^ { 6 } \mathrm { C } _ { 3 }\).
Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 - 2 x ) ^ { 6 }\).
OCR MEI C1 Q18
4 marks Easy -1.2
18 Find the binomial expansion of \(( 2 + x ) ^ { 4 }\), writing each term as simply as possible.
OCR C2 2005 January Q1
5 marks Moderate -0.5
1 Simplify \(( 3 + 2 x ) ^ { 3 } - ( 3 - 2 x ) ^ { 3 }\).
OCR C2 2006 January Q3
6 marks Moderate -0.8
3
  1. Find the first three terms of the expansion, in ascending powers of \(x\), of \(( 1 - 2 x ) ^ { 12 }\).
  2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of $$( 1 + 3 x ) ( 1 - 2 x ) ^ { 12 } .$$
OCR C2 2008 January Q10
12 marks Moderate -0.3
10
  1. Find the binomial expansion of \(( 2 x + 5 ) ^ { 4 }\), simplifying the terms.
  2. Hence show that \(( 2 x + 5 ) ^ { 4 } - ( 2 x - 5 ) ^ { 4 }\) can be written as $$320 x ^ { 3 } + k x$$ where the value of the constant \(k\) is to be stated.
  3. Verify that \(x = 2\) is a root of the equation $$( 2 x + 5 ) ^ { 4 } - ( 2 x - 5 ) ^ { 4 } = 3680 x - 800$$ and find the other possible values of \(x\).
OCR C2 2005 June Q6
8 marks Moderate -0.8
6
  1. Find the binomial expansion of \(\left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 3 }\), simplifying the terms.
  2. Hence find \(\int \left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 3 } \mathrm {~d} x\).
OCR C2 2006 June Q1
4 marks Moderate -0.8
1 Find the binomial expansion of \(( 3 x - 2 ) ^ { 4 }\).
OCR C2 2007 June Q2
5 marks Easy -1.2
2 Expand \(\left( x + \frac { 2 } { x } \right) ^ { 4 }\) completely, simplifying the terms.
OCR C2 Q4
8 marks Moderate -0.8
4. The coefficient of \(x ^ { 2 }\) in the binomial expansion of \(( 1 + k x ) ^ { 7 }\), where \(k\) is a positive constant, is 525.
  1. Find the value of \(k\). Using this value of \(k\),
  2. show that the coefficient of \(x ^ { 3 }\) in the expansion is 4375 ,
  3. find the first three terms in the expansion in ascending powers of \(x\) of $$( 2 - x ) ( 1 + k x ) ^ { 7 }$$
OCR C2 Q3
7 marks Moderate -0.3
3.
  1. Expand \(( 2 + y ) ^ { 6 }\) in ascending powers of \(y\) as far as the term in \(y ^ { 3 }\), simplifying each coefficient.
  2. Hence expand \(\left( 2 + x - x ^ { 2 } \right) ^ { 6 }\) in ascending powers of \(x\) as far as the term in \(x ^ { 3 }\), simplifying each coefficient.
OCR C2 Q6
9 marks Moderate -0.5
  1. (a) Expand \(( 1 + x ) ^ { 4 }\) in ascending powers of \(x\).
    (b) Using your expansion, express each of the following in the form \(a + b \sqrt { 2 }\), where \(a\) and \(b\) are integers.
    1. \(( 1 + \sqrt { 2 } ) ^ { 4 }\)
    2. \(( 1 - \sqrt { 2 } ) ^ { 8 }\)
    3. The second and fifth terms of an arithmetic sequence are 26 and 41 repectively.
OCR C2 Q2
5 marks Moderate -0.8
  1. Find the coefficient of \(x ^ { 2 }\) in the expansion of
$$( 1 + x ) ( 1 - x ) ^ { 6 }$$
OCR MEI C2 Q1
12 marks Moderate -0.8
1
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{1a6d059d-8ab8-41e0-8bf3-54e248f820e4-1_650_759_252_762} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure} Fig. 12 shows part of the curve \(y = x ^ { 4 }\) and the line \(y = 8 x\), which intersect at the origin and the point P .
    (A) Find the coordinates of P , and show that the area of triangle OPQ is 16 square units.
    (B) Find the area of the region bounded by the line and the curve.
  2. You are given that \(\mathrm { f } ( x ) = x ^ { 4 }\).
    (A) Complete this identity for \(\mathrm { f } ( x + h )\). $$\mathrm { f } ( x + h ) = ( x + h ) ^ { 4 } = x ^ { 4 } + 4 x ^ { 3 } h + \ldots$$ (B) Simplify \(\frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
    (C) Find \(\lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
    (D) State what this limit represents.
OCR MEI FP2 2007 January Q2
18 marks Challenging +1.8
2
  1. You are given the complex numbers \(w = 3 \mathrm { e } ^ { - \frac { 1 } { 12 } \pi \mathrm { j } }\) and \(z = 1 - \sqrt { 3 } \mathrm { j }\).
    1. Find the modulus and argument of each of the complex numbers \(w , z\) and \(\frac { w } { z }\).
    2. Hence write \(\frac { w } { z }\) in the form \(a + b \mathrm { j }\), giving the exact values of \(a\) and \(b\).
  2. In this part of the question, \(n\) is a positive integer and \(\theta\) is a real number with \(0 < \theta < \frac { \pi } { n }\).
    1. Express \(\mathrm { e } ^ { - \frac { 1 } { 2 } \mathrm { j } \theta } + \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { j } \theta }\) in simplified trigonometric form, and hence, or otherwise, show that $$1 + \mathrm { e } ^ { \mathrm { j } \theta } = 2 \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { j } \theta } \cos \frac { 1 } { 2 } \theta$$ Series \(C\) and \(S\) are defined by $$\begin{aligned} & C = 1 + \binom { n } { 1 } \cos \theta + \binom { n } { 2 } \cos 2 \theta + \binom { n } { 3 } \cos 3 \theta + \ldots + \binom { n } { n } \cos n \theta \\ & S = \binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta + \binom { n } { 3 } \sin 3 \theta + \ldots + \binom { n } { n } \sin n \theta \end{aligned}$$
    2. Find \(C\) and \(S\), and show that \(\frac { S } { C } = \tan \frac { 1 } { 2 } n \theta\).
Edexcel AEA 2002 Specimen Q6
18 marks Hard +2.3
6.Given that the coefficients of \(x , x ^ { 2 }\) and \(x ^ { 4 }\) in the expansion of \(( 1 + k x ) ^ { n }\) ,where \(n \geq 4\) and \(k\) is a positive constant,are the consecutive terms of a geometric series,
  1. show that \(k = \frac { 6 ( n - 1 ) } { ( n - 2 ) ( n - 3 ) }\) .
  2. Given further that both \(n\) and \(k\) are positive integers,find all possible pairs of values for \(n\) and \(k\) .You should show clearly how you know that you have found all possible pairs of values.
  3. For the case where \(k = 1.4\) ,find the value of the positive integer \(n\) .
  4. Given that \(k = 1.4 , n\) is a positive integer and that the first term of the geometric series is the coefficient of \(x\) ,estimate how many terms are required for the sum of the geometric series to exceed \(1.12 \times 10 ^ { 12 }\) .[You may assume that \(\log _ { 10 } 4 \approx 0.6\) and \(\log _ { 10 } 5 \approx 0.7\) .]
Edexcel AEA 2013 June Q1
6 marks Challenging +1.2
1.In the binomial expansion of $$\left( 1 + \frac { 12 n } { 5 } x \right) ^ { n }$$ the coefficients of \(x ^ { 2 }\) and \(x ^ { 3 }\) are equal and non-zero.
  1. Find the possible values of \(n\) .
    (4)
  2. State,giving a reason,which value of \(n\) gives a valid expansion when \(x = \frac { 1 } { 2 }\) (2)
OCR MEI C1 2007 January Q5
3 marks Easy -1.2
5 Calculate the coefficient of \(x ^ { 4 }\) in the expansion of \(( x + 5 ) ^ { 6 }\).
OCR MEI C1 2010 January Q8
4 marks Easy -1.2
8 Find the binomial expansion of \(\left( x + \frac { 5 } { x } \right) ^ { 3 }\), simplifying the terms.
OCR MEI C1 2011 January Q6
4 marks Easy -1.2
6 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 2013 January Q6
4 marks Moderate -0.3
6 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 2014 June Q7
4 marks Moderate -0.8
7 Find the coefficient of \(x ^ { 4 }\) in the binomial expansion of \(( 5 + 2 x ) ^ { 7 }\).
OCR MEI C1 2016 June Q6
4 marks Easy -1.2
6 Find the binomial expansion of \(( 1 - 5 x ) ^ { 4 }\), expressing the terms as simply as possible.
OCR C2 2009 January Q7
9 marks Moderate -0.3
7 In the binomial expansion of \(( k + a x ) ^ { 4 }\) the coefficient of \(x ^ { 2 }\) is 24 .
  1. Given that \(a\) and \(k\) are both positive, show that \(a k = 2\).
  2. Given also that the coefficient of \(x\) in the expansion is 128 , find the values of \(a\) and \(k\).
  3. Hence find the coefficient of \(x ^ { 3 }\) in the expansion.
OCR C2 2010 January Q3
6 marks Moderate -0.8
3
  1. Find and simplify the first four terms in the expansion of \(( 2 - x ) ^ { 7 }\) in ascending powers of \(x\).
  2. Hence find the coefficient of \(w ^ { 6 }\) in the expansion of \(\left( 2 - \frac { 1 } { 4 } w ^ { 2 } \right) ^ { 7 }\).
OCR C2 2011 January Q1
6 marks Moderate -0.8
1
  1. Find and simplify the first three terms, in ascending powers of \(x\), in the binomial expansion of \(( 1 + 2 x ) ^ { 7 }\).
  2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 - 5 x ) ( 1 + 2 x ) ^ { 7 }\).