Ratio of coefficients condition

1 The coefficient of \(x ^ { 4 }\) in the expansion of \(( 3 + x ) ^ { 5 }\) is equal to the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 2 x + \frac { a } { x } \right) ^ { 6 }\). Find the value of the positive constant \(a\).

1 The coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 - 4 x ) ^ { 6 }\) is 12 times the coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 + a x ) ^ { 5 }\). Find the value of the positive constant \(a\).

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 \(a\).

5 In the expansion of \(\left( 2 x ^ { 2 } + \frac { a } { x } \right) ^ { 6 }\), the coefficients of \(x ^ { 6 }\) and \(x ^ { 3 }\) are equal.

1. Find the value of the non-zero constant \(a\).
2. Find the coefficient of \(x ^ { 6 }\) in the expansion of \(\left( 1 - x ^ { 3 } \right) \left( 2 x ^ { 2 } + \frac { a } { x } \right) ^ { 6 }\).

1 The coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 + 2 a x ) ^ { 5 }\) is six times the coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 + a x ) ^ { 6 }\). Find the value of the 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 \(\frac { a } { b }\).

1 The coefficients of \(x\) and \(x ^ { 2 }\) in the expansion of \(( 2 + a x ) ^ { 7 }\) are equal. Find the value of the non-zero constant \(a\).

2

1. Find, in terms of the non-zero constant \(k\), the first 4 terms in the expansion of \(( k + x ) ^ { 8 }\) in ascending powers of \(x\).
2. Given that the coefficients of \(x ^ { 2 }\) and \(x ^ { 3 }\) in this expansion are equal, find the value of \(k\).

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 In the expansion of \(( 2 + a x ) ^ { 6 }\), the coefficient of \(x ^ { 2 }\) is equal to the coefficient of \(x ^ { 3 }\). Find the value of the non-zero constant \(a\).

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 \(k\).

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 \(a\).

5. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 3 - \frac { a x } { 2 } \right) ^ { 5 }$$ where \(a\) is a positive constant. Give each term in its simplest form. Given that, in the expansion, the coefficient of \(x\) is equal to the coefficient of \(x ^ { 3 }\),
(b) find the exact value of \(a\) in its simplest form.

3

1. Find the binomial expansion of \(( 3 + k x ) ^ { 3 }\), simplifying the terms.
2. It is given that, in the expansion of \(( 3 + k x ) ^ { 3 }\), the coefficient of \(x ^ { 2 }\) is equal to the constant term. Find the possible values of \(k\), giving your answers in an exact form.

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

$$( 1 + k x ) ^ { 10 }$$ where \(k\) is a non-zero constant. Write each coefficient as simply as possible. Given that in the expansion of \(( 1 + k x ) ^ { 10 }\) the coefficient \(x ^ { 3 }\) is 3 times the coefficient of \(x\), (b) find the possible values of \(k\).

1. In the binomial expansion of \(( 2 - k x ) ^ { 10 }\) where \(k\) is a non-zero positive constant.

The coefficient of \(x ^ { 4 }\) is 256 times the coefficient of \(x ^ { 6 }\).
Find the value of \(k\).

1. (a) Write down the first four terms of the binomial expansion, in ascending powers of \(x\), of \(( 1 + 3 x ) ^ { n }\), where \(n > 2\).

Given that the coefficient of \(x ^ { 3 }\) in this expansion is ten times the coefficient of \(x ^ { 2 }\),
(b) find the value of \(n\),
(c) find the coefficient of \(x ^ { 4 }\) in the expansion.

3. For the binomial expansion in ascending powers of \(x\) of \(\left( 1 + \frac { 1 } { 4 } x \right) ^ { n }\), where \(n\) is an integer and \(n \geq 2\),

1. find and simplify the first three terms,
2. find the value of \(n\) for which the coefficient of \(x\) is equal to the coefficient of \(x ^ { 2 }\).

6. (a) Find the first 4 terms, in ascending powers of \(x\), in the binomial expansion of $$( 1 + k x ) ^ { 10 }$$ where \(k\) is a non-zero constant. Write each coefficient as simply as possible. Given that in the expansion of \(( 1 + k x ) ^ { 10 }\) the coefficient \(x ^ { 3 }\) is 3 times the coefficient of \(x\), (b) find the possible values of \(k\).

10. (a) Find the first 4 terms, in ascending powers of \(x\), in the binomial expansion of $$( 1 + k x ) ^ { 10 }$$ where \(k\) is a non-zero constant. Write each coefficient as simply as possible. Given that in the expansion of \(( 1 + k x ) ^ { 10 }\) the coefficient \(x ^ { 3 }\) is 3 times the coefficient of \(x\), (b) find the possible values of \(k\).
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1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable. $$f ( x ) = \left( 2 + \frac { k x } { 8 } \right) ^ { 7 } \quad \text { where } k \text { is a non-zero constant }$$

1. Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(\mathrm { f } ( x )\). Give each term in simplest form. Given that, in the binomial expansion of \(\mathrm { f } ( x )\), the coefficients of \(x , x ^ { 2 }\) and \(x ^ { 3 }\) are the first 3 terms of an arithmetic progression,
2. find, using algebra, the possible values of \(k\).
   (Solutions relying entirely on calculator technology are not acceptable.)

10. (a) Find the first 4 terms, in ascending powers of \(x\), in the binomial expansion of $$( 1 + k x ) ^ { 10 }$$ where \(k\) is a non-zero constant. Write each coefficient as simply as possible. Given that in the expansion of \(( 1 + k x ) ^ { 10 }\) the coefficient \(x ^ { 3 }\) is 3 times the coefficient of \(x\), (b) find the possible values of \(k\).
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3. For the binomial expansion in ascending powers of \(x\) of \(\left( 1 + \frac { 1 } { 4 } x \right) ^ { n }\), where \(n\) is an integer and \(n \geq 2\),

1. find and simplify the first three terms,
2. find the value of \(n\) for which the coefficient of \(x\) is equal to the coefficient of \(x ^ { 2 }\).

10 In the binomial expansion of \(( 1 + x ) ^ { n }\), where \(n \geq 4\), the coefficient of \(x ^ { 4 }\) is \(1 \frac { 1 } { 2 }\) times the sum of the coefficients of \(x ^ { 2 }\) and \(x ^ { 3 }\) Find the value of \(n\).