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

2 In the expansion of \(( x + 2 k ) ^ { 7 }\), where \(k\) is a non-zero constant, the coefficients of \(x ^ { 4 }\) and \(x ^ { 5 }\) are equal. Find the value of \(k\).

2 Find the coefficient of \(x\) in the expansion of \(\left( \frac { x } { 3 } + \frac { 9 } { x ^ { 2 } } \right) ^ { 7 }\).

2 Find the term independent of \(x\) in the expansion of \(\left( 2 x + \frac { 1 } { 2 x ^ { 3 } } \right) ^ { 8 }\).

4 In the expansion of \(( 3 - 2 x ) \left( 1 + \frac { x } { 2 } \right) ^ { n }\), the coefficient of \(x\) is 7 . Find the value of the constant \(n\) and hence find the coefficient of \(x ^ { 2 }\).

1 Find the term independent of \(x\) in the expansion of \(\left( 2 x - \frac { 1 } { 4 x ^ { 2 } } \right) ^ { 9 }\).

3

1. Find the term independent of \(x\) in the expansion of \(\left( \frac { 2 } { x } - 3 x \right) ^ { 6 }\).
2. Find the value of \(a\) for which there is no term independent of \(x\) in the expansion of $$\left( 1 + a x ^ { 2 } \right) \left( \frac { 2 } { x } - 3 x \right) ^ { 6 }$$

1 Find the coefficient of \(\frac { 1 } { x ^ { 3 } }\) in the expansion of \(\left( x - \frac { 2 } { x } \right) ^ { 7 }\).

1 Find the term independent of \(x\) in the expansion of \(\left( 2 x + \frac { 1 } { 4 x ^ { 2 } } \right) ^ { 6 }\).

1 The coefficient of \(x ^ { 2 }\) in the expansion of \(( 4 + a x ) \left( 1 + \frac { x } { 2 } \right) ^ { 6 }\) is 3 . Find the value of the constant \(a\). [4]

1

1. Expand \(( 1 + y ) ^ { 6 }\) in ascending powers of \(y\) as far as the term in \(y ^ { 2 }\).
2. In the expansion of \(\left( 1 + \left( p x - 2 x ^ { 2 } \right) \right) ^ { 6 }\) the coefficient of \(x ^ { 2 }\) is 48 . Find the value of the positive constant \(p\).

1 In the expansion of \(\left( 1 - \frac { 2 x } { a } \right) ( a + x ) ^ { 5 }\), where \(a\) is a non-zero constant, show that the coefficient of \(x ^ { 2 }\) is zero.

3 Find the coefficient of \(x ^ { 3 }\) in the binomial expansion of \(( 3 + x ) \sqrt { 1 + 4 x }\).

4 By considering the binomial expansions of \(\left( z + \frac { 1 } { z } \right) ^ { 5 }\) and \(\left( z - \frac { 1 } { z } \right) ^ { 5 }\), where \(z = \cos \theta + \mathrm { i } \sin \theta\), use de Moivre's theorem to show that $$\tan ^ { 5 } \theta = \frac { \sin 5 \theta - \mathrm { a } \sin 3 \theta + \mathrm { b } \sin \theta } { \cos 5 \theta + \mathrm { a } \cos 3 \theta + \mathrm { b } \cos \theta }$$ where \(a\) and \(b\) are integers to be determined.

8

1. Find \(\int \sin \theta \cos ^ { n } \theta d \theta\), where \(n \neq - 1\).
   Let \(I _ { m , n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { m } \theta \cos ^ { n } \theta d \theta\).
2. Show that, for \(m \geqslant 2\) and \(n \geqslant 0\), $$I _ { m , n } = \frac { m - 1 } { m + n } I _ { m - 2 , n }$$
3. By considering the binomial expansion of \(\left( z + \frac { 1 } { z } \right) ^ { 5 }\), where \(z = \cos \theta + i \sin \theta\), use de Moivre's theorem to show that $$\cos ^ { 5 } \theta = a \cos 5 \theta + b \cos 3 \theta + c \cos \theta$$ where \(a\), \(b\) and \(c\) are constants to be determined.
4. Using the results given in parts (b) and (c), find the exact value of \(I _ { 2,5 }\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

3

1. By considering the binomial expansion of \(\left( z + z ^ { - 1 } \right) ^ { 4 }\), where \(z = \cos \theta + \mathrm { i } \sin \theta\), use de Moivre's theorem to show that \(\cos ^ { 4 } \theta = \frac { 1 } { 8 } ( \cos 4 \theta + 4 \cos 2 \theta + 3 )\).
2. Use the substitution \(x = \sin \theta\) to find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } \mathrm {~d} x\).

3 By considering the binomial expansions of \(\left( z + \frac { 1 } { z } \right) ^ { 4 }\) and \(\left( z - \frac { 1 } { z } \right) ^ { 4 }\), where \(z = \cos \theta + i \sin \theta\), use de Moivre's theorem to show that $$\cot ^ { 4 } \theta = \frac { \cos 4 \theta + a \cos 2 \theta + b } { \cos 4 \theta - a \cos 2 \theta + b }$$ where \(a\) and \(b\) are integers to be determined.

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

$$\left( 2 - \frac { x } { 2 } \right) ^ { 6 }$$ giving each term in its simplest form.

1. (a) Find the first 4 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.
(b) Use your expansion to find an estimated value for \(2.025 ^ { 10 }\), stating the value of \(x\) which you have used and showing your working.

7. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + k x ) ^ { 8 }\), 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 1512
(b) find the value of \(k\).

10. The first 3 terms, in ascending powers of \(x\), in the binomial expansion of \(( 1 + a x ) ^ { 20 }\) are given by $$1 + 4 x + p x ^ { 2 }$$ where \(a\) and \(p\) are constants.

1. Find the value of \(a\).
2. Find the value of \(p\). One of the terms in the binomial expansion of \(( 1 + a x ) ^ { 20 }\) is \(q x ^ { 4 }\), where \(q\) is a constant.
3. Find the value of \(q\).

15. The binomial expansion, in ascending powers of \(x\), of \(( 1 + k x ) ^ { n }\) is $$1 + 36 x + 126 k x ^ { 2 } + \ldots$$ where \(k\) is a non-zero constant and \(n\) is a positive integer.

1. Show that \(n k ( n - 1 ) = 252\)
2. Find the value of \(k\) and the value of \(n\).
3. Using the values of \(k\) and \(n\), find the coefficient of \(x ^ { 3 }\) in the binomial expansion of \(( 1 + k x ) ^ { n }\)

1. (a) Use the binomial theorem to find the first 4 terms, in ascending powers of \(x\), of the expansion of

$$\left( 1 - \frac { x } { 2 } \right) ^ { 8 }$$ Give each term in its simplest form.
(b) Use the answer to part (a) to find an approximate value to \(0.9 ^ { 8 }\) Write your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers.

8. Given that $$1 + 12 x + 70 x ^ { 2 } + \ldots$$ is the binomial expansion, in ascending powers of \(x\) of \(( 1 + b x ) ^ { n }\), where \(n \in \mathbb { N }\) and \(b\) is a constant,

1. show that \(n b = 12\)
2. find the values of the constants \(b\) and \(n\).

6. (a) Find the first 3 terms in ascending powers of \(x\) of the binomial expansion of $$( 2 + a x ) ^ { 6 }$$ where \(a\) is a non-zero constant. Give each term in its simplest form. Given that, in the expansion, the coefficient of \(x\) is equal to the coefficient of \(x ^ { 2 }\) (b) find the value of \(a\).

1. The first three terms in ascending powers of \(x\) in the binomial expansion of \(( 1 + p x ) ^ { 8 }\) are given by

$$1 + 12 x + q x ^ { 2 }$$ where \(p\) and \(q\) are constants.
Find the value of \(p\) and the value of \(q\).