Binomial Theorem (positive integer n)

Find the coefficient of the \(x^4\) term in \((2 - 3x)^6\). [3]

Find the coefficient of the \(x\) term in the binomial expansion of \((3 + x)^4\) Circle your answer. [1 mark] 12 27 54 108

6

1. Find the coefficient of \(x ^ { 4 }\) in the expansion of \(( 3 x - 2 ) ^ { 10 }\).
2. In the expansion of \(( 1 + 2 x ) ^ { n }\), where \(n\) is a positive integer, the coefficients of \(x ^ { 7 }\) and \(x ^ { 8 }\) are equal. Find the value of \(n\).
3. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(\frac { 1 } { \sqrt { 4 + x } }\).

1 The coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 + k x ) ( 1 - 2 x ) ^ { 5 }\) is 20 .
Find the value of the constant \(k\).

4

1. Find the first three terms in the expansion of \(( 2 + a x ) ^ { 5 }\) in ascending powers of \(x\).
2. Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 + 2 x ) ( 2 + a x ) ^ { 5 }\) is 240 , find the possible values of \(a\).

8

1. It is given that in the expansion of \(( 4 + 2 x ) ( 2 - a x ) ^ { 5 }\), the coefficient of \(x ^ { 2 }\) is - 15 .
   Find the possible values of \(a\).
2. It is given instead that in the expansion of \(( 4 + 2 x ) ( 2 - a x ) ^ { 5 }\), the coefficient of \(x ^ { 2 }\) is \(k\). It is also given that there is only one value of \(a\) which leads to this value of \(k\). Find the values of \(k\) and \(a\).

Find expansion of

2 Find the binomial expansion of \(( 3 - 2 x ) ^ { 3 }\).

4 Using factorials, show that \(\binom { n } { r - 1 } + \binom { n } { r } = \binom { n + 1 } { r }\). Hence prove by mathematical induction that $$( a + x ) ^ { n } = \binom { n } { 0 } a ^ { n } + \binom { n } { 1 } a ^ { n - 1 } x + \ldots + \binom { n } { r } a ^ { n - r } x ^ { r } + \ldots + \binom { n } { n } x ^ { n }$$ for every positive integer \(n\).

1 In the expansion of \(( 2 + a x ) ^ { 7 }\), the coefficient of \(x\) is equal to the coefficient of \(x ^ { 2 }\). Find the value of the non-zero constant \(a\).

1 The coefficients of \(x ^ { 2 }\) and \(x ^ { 3 }\) in the expansion of \(( 3 - 2 x ) ^ { 6 }\) are \(a\) and \(b\) respectively. Find the value of \(\frac { a } { b }\).

6 In the expansion of \(\left( \frac { x } { a } + \frac { a } { x ^ { 2 } } \right) ^ { 7 }\), it is given that $$\frac { \text { the coefficient of } x ^ { 4 } } { \text { the coefficient of } x } = 3 \text {. }$$ Find the possible values of the constant \(a\).

The coefficient of \(x^3\) in the expansion of \((1 - px)^5\) is \(-2160\). Find the value of the constant \(p\). [3]

1 The term independent of \(x\) in the expansion of \(\left( 2 x + \frac { k } { x } \right) ^ { 6 }\), where \(k\) is a constant, is 540.

1. Find the value of \(k\).
2. For this value of \(k\), find the coefficient of \(x ^ { 2 }\) in the expansion.

6 The coefficient of \(\frac { 1 } { x }\) in the expansion of \(\left( 2 x + \frac { a } { x ^ { 2 } } \right) ^ { 5 }\) is 720 .

1. Find the possible values of the constant \(a\).
2. Hence find the coefficient of \(\frac { 1 } { x ^ { 7 } }\) in the expansion.

The coefficients of \(x\) and \(x^2\) in the expansion of \((2 + ax)^7\) are equal. Find the value of the non-zero constant \(a\). [3]

2. (a) 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 \(x\),
(b) find the value of \(k\).

$$f(x) = \left(1 + \frac{x}{k}\right)^n, \quad k, n \in \mathbb{N}, \quad n > 2.$$ Given that the coefficient of \(x^3\) is twice the coefficient of \(x^2\) in the binomial expansion of f(x),

1. prove that \(n = 6k + 2\). Given also that the coefficients of \(x^4\) and \(x^5\) are equal and non-zero, [3]
2. form another equation in \(n\) and \(k\) and hence show that \(k = 2\) and \(n = 14\). Using these values of \(k\) and \(n\), [4]
3. expand f(x) in ascending powers of \(x\), up to and including the term in \(x^5\). Give each coefficient as an exact fraction in its lowest terms [4]

Find the coefficient of \(\frac{1}{x}\) in the expansion of \(\left(x - \frac{2}{x}\right)^5\). [3]

Find the coefficient of \(\frac{1}{x}\) in the expansion of \(\left(x - \frac{2}{x}\right)^5\). [3]

2 Find the coefficient of \(x ^ { 6 }\) in the expansion of \(\left( 2 x ^ { 3 } - \frac { 1 } { x ^ { 2 } } \right) ^ { 7 }\).

Find the coefficient of \(x^2\) in the expansion of $$(1 + x)(1 - x)^6.$$ [4]

Find the coefficient of \(x\) in the expansion of $$(4x^3 - 5x^2 + 3x - 2)(x^5 + 4x + 1)$$ Circle your answer. $$-5 \quad -2 \quad 7 \quad 11$$ [1 mark]

7

1. The expression \(( 1 - 2 x ) ^ { 5 }\) can be written in the form $$1 + p x + q x ^ { 2 } + r x ^ { 3 } + 80 x ^ { 4 } - 32 x ^ { 5 }$$ By using the binomial expansion, or otherwise, find the values of the coefficients \(p , q\) and \(r\).
2. Find the value of the coefficient of \(x ^ { 10 }\) in the expansion of \(( 1 - 2 x ) ^ { 5 } ( 2 + x ) ^ { 7 }\).
   [0pt] [5 marks]

4

1. Expand \(( 1 + x ) ^ { 4 }\).
2. Use your expansion to determine the exact value of \(1002 ^ { 4 }\).

4

1. Expand \(( 1 + x ) ^ { 4 }\).
2. Use your expansion to determine the exact value of \(1002 ^ { 4 }\).

1. The binomial expansion of

$$( 1 + a x ) ^ { 12 }$$ up to and including the term in \(x ^ { 2 }\) is $$1 - \frac { 15 } { 2 } x + k x ^ { 2 }$$ where \(a\) and \(k\) are constants.

1. Show that \(a = - \frac { 5 } { 8 }\)
2. Hence find the value of \(k\) Using the expansion and making your method clear,
3. find an estimate for the value of \(\left( \frac { 17 } { 16 } \right) ^ { 12 }\), giving your answer to 4 decimal places.

6 Find the constant term in the expansion of \(\left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 15 }\).

3 Find the term independent of \(x\) in the expansion of \(\left( 4 x ^ { 3 } + \frac { 1 } { 2 x } \right) ^ { 8 }\).

4 The binomial expansion of \(\left( 2 x + \frac { 5 } { x } \right) ^ { 6 }\) has a term which is a constant. Find this term.

The first three terms in the expansion, in ascending powers of \(x\), of \((1 + px)^n\), are \(1 - 18x + 36p^2x^2\). Given that \(n\) is a positive integer, find the value of \(n\) and the value of \(p\). [7]

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 \(q\).

$$f(x) = \left(1 + \frac{x}{k}\right)^n, \quad k, n \in \mathbb{N}, \quad n > 2.$$ Given that the coefficient of \(x^3\) is twice the coefficient of \(x^2\) in the binomial expansion of \(f(x)\),

1. prove that \(n = 6k + 2\). [3]

Given also that the coefficients of \(x^4\) and \(x^5\) are equal and non-zero,

1. form another equation in \(n\) and \(k\) and hence show that \(k = 2\) and \(n = 14\). [4]

Using these values of \(k\) and \(n\),

1. expand \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^5\). Give each coefficient as an exact fraction in its lowest terms [4]

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

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 curve \(C\) has equation \(y = \mathrm { f } ( x )\) where

$$f ( x ) = ( 2 - k x ) ^ { 5 }$$ and \(k\) is a constant.
Given that when \(\mathrm { f } ( x )\) is divided by \(( 4 x - 5 )\) the remainder is \(\frac { 243 } { 32 }\)

1. show that \(k = \frac { 2 } { 5 }\)
2. Find the first three terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 2 - \frac { 2 } { 5 } x \right) ^ { 5 }$$ giving each term in simplest form. Using the solution to part (b) and making your method clear,
3. find the gradient of \(C\) at the point where \(x = 0\)

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

1

1. Expand \(\left( 1 - \frac { 1 } { 2 x } \right) ^ { 2 }\).
2. Find the first four terms in the expansion, in ascending powers of \(x\), of \(( 1 + 2 x ) ^ { 6 }\).
3. Hence find the coefficient of \(x\) in the expansion of \(\left( 1 - \frac { 1 } { 2 x } \right) ^ { 2 } ( 1 + 2 x ) ^ { 6 }\).

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

1 Simplify \(( 3 + 2 x ) ^ { 3 } - ( 3 - 2 x ) ^ { 3 }\).

6

1. Using the binomial expansion, or otherwise:
   1. express \(( 1 + x ) ^ { 3 }\) in ascending powers of \(x\);
   2. express \(( 1 + x ) ^ { 4 }\) in ascending powers of \(x\).
2. Hence, or otherwise:
   1. express \(( 1 + 4 x ) ^ { 3 }\) in ascending powers of \(x\);
   2. express \(( 1 + 3 x ) ^ { 4 }\) in ascending powers of \(x\).
3. Show that the expansion of $$( 1 + 3 x ) ^ { 4 } - ( 1 + 4 x ) ^ { 3 }$$ can be written in the form $$p x ^ { 2 } + q x ^ { 3 } + r x ^ { 4 }$$ where \(p , q\) and \(r\) are integers.

5. (a) Given that \(( 2 + x ) ^ { 5 } + ( 2 - x ) ^ { 5 } = A + B x ^ { 2 } + C x ^ { 4 }\), find the values of the constants \(A , B\) and \(C\).
(b) Using the substitution \(y = x ^ { 2 }\) and your answers to part (a), solve, $$( 2 + x ) ^ { 5 } + ( 2 - x ) ^ { 5 } = 349$$

4

1. Find the binomial expansion of \(( 2 + x ) ^ { 5 }\), simplifying the terms.
2. Hence find the coefficient of \(y ^ { 3 }\) in the expansion of \(\left( 2 + 3 y + y ^ { 2 } \right) ^ { 5 }\).

4

1. Expand \(( 1 + a ) ^ { 5 }\) in ascending powers of \(a\) up to and including the term in \(a ^ { 3 }\).
2. Hence expand \(\left[ 1 + \left( x + x ^ { 2 } \right) \right] ^ { 5 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying your answer.

4. Alison and Gemma play table tennis. Alison starts by serving for the first five points. The probability that she wins a point when serving is \(p\).

1. Show that the probability that Alison is ahead at the end of her five serves is given by $$p ^ { 3 } \left( 6 p ^ { 2 } - 15 p + 10 \right) .$$
2. Evaluate this probability when \(p = 0.6\).

1 Find the coefficient of \(x ^ { 2 }\) in the expansion of $$( 2 - 5 x ) ( 1 + 3 x ) ^ { 10 }$$

1 Find the coefficient of \(x ^ { 2 }\) in the expansion of $$( 2 - 5 x ) ( 1 + 3 x ) ^ { 10 }$$

4

1. Find the first 3 terms in the expansion of \(\left( 2 x - x ^ { 2 } \right) ^ { 6 }\) in ascending powers of \(x\).
2. Hence find the coefficient of \(x ^ { 8 }\) in the expansion of \(( 2 + x ) \left( 2 x - x ^ { 2 } \right) ^ { 6 }\).

4

1. Find the first 3 terms in the expansion of \(( 2 - x ) ^ { 6 }\) in ascending powers of \(x\).
2. Find the value of \(k\) for which there is no term in \(x ^ { 2 }\) in the expansion of \(( 1 + k x ) ( 2 - x ) ^ { 6 }\).

1

1. Find the coefficients of \(x ^ { 4 }\) and \(x ^ { 5 }\) in the expansion of \(( 1 - 2 x ) ^ { 5 }\).
2. It is given that, when \(( 1 + p x ) ( 1 - 2 x ) ^ { 5 }\) is expanded, there is no term in \(x ^ { 5 }\). Find the value of the constant \(p\).

3

1. Give the complete expansion of \(\left( x + \frac { 2 } { x } \right) ^ { 5 }\).
2. In the expansion of \(\left( a + b x ^ { 2 } \right) \left( x + \frac { 2 } { x } \right) ^ { 5 }\), the coefficient of \(x\) is zero and the coefficient of \(\frac { 1 } { x }\) is 80 . Find the values of the constants \(a\) and \(b\).

The first three terms in the expansion of \((1 - 2x)^2(1 + ax)^6\), in ascending powers of \(x\), are \(1 - x + bx^2\). Find the values of the constants \(a\) and \(b\). [6]

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.

4 The coefficient of \(x\) in the expansion of \(\left( 4 x + \frac { 10 } { x } \right) ^ { 3 }\) is \(p\). The coefficient of \(\frac { 1 } { x }\) in the expansion of \(\left( 2 x + \frac { k } { x ^ { 2 } } \right) ^ { 5 }\) is \(q\). Given that \(p = 6 q\), find the possible values of \(k\).

4

1. Find the binomial expansion of \(\left( x ^ { 2 } - 5 \right) ^ { 3 }\), simplifying the terms.
2. Hence find \(\int \left( x ^ { 2 } - 5 \right) ^ { 3 } \mathrm {~d} x\).

In the binomial expansion of \((\sqrt{3} + \sqrt{2})^4\) there are two irrational terms. Find the difference between these two terms. [3 marks]

The coefficient of \(x^3\) in the expansion of \((1 - 3x)^6 + (1 + ax)^5\) is 100. Find the value of the constant \(a\). [4]

4 The first three terms in the expansion of \(( 2 + a x ) ^ { n }\), in ascending powers of \(x\), are \(32 - 40 x + b x ^ { 2 }\). Find the values of the constants \(n , a\) and \(b\).

5 In the expansion of \(( a + b x ) ^ { 7 }\), where \(a\) and \(b\) are non-zero constants, the coefficients of \(x , x ^ { 2 }\) and \(x ^ { 4 }\) are the first, second and third terms respectively of a geometric progression. Find the value of \(\frac { a } { b }\).

2 Expand \(\left( x + \frac { 2 } { x } \right) ^ { 4 }\) completely, simplifying the terms.

4 Find the term that is independent of \(x\) in the expansion of

1. \(\left( x - \frac { 2 } { x } \right) ^ { 6 }\),
2. \(\left( 2 + \frac { 3 } { x ^ { 2 } } \right) \left( x - \frac { 2 } { x } \right) ^ { 6 }\).

Find the first four terms, in ascending powers of \(x\), of the binomial expansion of $$\left(2 + \frac{3}{8}x\right)^{10}$$ Give each coefficient as an integer. [4]

1

1. Expand \(( 1 + 3 x ) ^ { 6 }\) in ascending powers of \(x\) up to, and including, the term in \(x ^ { 2 }\).
2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 - 7 x + x ^ { 2 } \right) ( 1 + 3 x ) ^ { 6 }\).

5

1. Using the binomial expansion, or otherwise, express \(( 1 - x ) ^ { 3 }\) in ascending powers of \(x\).
2. Show that the expansion of $$( 1 + y ) ^ { 4 } - ( 1 - y ) ^ { 3 }$$ is $$7 y + p y ^ { 2 } + q y ^ { 3 } + y ^ { 4 }$$ where \(p\) and \(q\) are constants to be found.
3. Hence find \(\int \left[ ( 1 + \sqrt { x } ) ^ { 4 } - ( 1 - \sqrt { x } ) ^ { 3 } \right] \mathrm { d } x\), expressing each coefficient in its simplest form.

Find the term which is independent of \(x\) in the expansion of \(\frac{(2-3x)^5}{x^3}\). [4]

3 One of the terms in the binomial expansion of \(( 4 + a x ) ^ { 6 }\) is \(160 x ^ { 3 }\).

1. Find the value of \(a\).
2. Using this value of \(a\), find the first two terms in the expansion of \(( 4 + a x ) ^ { 6 }\) in ascending powers of \(x\).

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.

1. In the binomial expansion of \(\left(2x - \frac{1}{2x}\right)^5\), the first three terms are \(32x^5 - 40x^3 + 20x\). Find the remaining three terms of the expansion. [3]
2. Hence find the coefficient of \(x\) in the expansion of \((1 + 4x^2)\left(2x - \frac{1}{2x}\right)^5\). [2]