Two equations from coefficients

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 \(k\).

3 The coefficient of \(x ^ { 4 }\) in the expansion of \(\left( 2 x ^ { 2 } + \frac { k ^ { 2 } } { x } \right) ^ { 5 }\) is \(a\). The coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 k x - 1 ) ^ { 4 }\) is \(b\).

1. Find \(a\) and \(b\) in terms of the constant \(k\).
2. Given that \(a + b = 216\), find the possible values of \(k\).

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

$$\left( 2 - \frac { x } { 8 } \right) ^ { 10 }$$ giving each term in its simplest form. $$\mathrm { f } ( x ) = \left( 2 - \frac { x } { 8 } \right) ^ { 10 } ( a + b x ) , \text { where } a \text { and } b \text { are constants }$$ Given that the first two terms, in ascending powers of \(x\) in the series expansion of \(\mathrm { f } ( x )\), are 256 and \(352 x\),
(b) find the value of \(a\),
(c) find the value of \(b\).

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.

5. (a) Write down the first 4 terms of the binomial expansion, in ascending powers of \(x\), of $$( 1 + a x ) ^ { n } , n > 2 .$$ Given that, in this expansion, the coefficient of \(x\) is 8 and the coefficient of \(x ^ { 2 }\) is 30 ,
(b) calculate the value of \(n\) and the value of \(a\),
(c) find the coefficient of \(x ^ { 3 }\).
[0pt] [P2 November 2003 Question 3]

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]

1. Write down the first four terms of the binomial expansion, in ascending powers of x, of \((1 + ax)^n\), where \(n > 2\). [2]

Given that, in this expansion, the coefficient of x is 8 and the coefficient of x² is 30,

1. find the value of n and the value of a, [4]
2. find the coefficient of x³. [2]