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 \(k\).

2 The coefficient of \(x ^ { 4 }\) in the expansion of \(( x + a ) ^ { 6 }\) is \(p\) and the coefficient of \(x ^ { 2 }\) in the expansion of \(( a x + 3 ) ^ { 4 }\) is \(q\). It is given that \(p + q = 276\). Find the possible values of the constant \(a\).

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.

10. 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.
a) Show that \(n k ( n - 1 ) = 252\)
b) Find the value of \(k\) and the value of \(n\).
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19. \(\quad \mathrm { f } ( x ) = \left( 1 + \frac { x } { k } \right) ^ { n } , \quad k , n \in \mathbb { N } , \quad n > 2\). Given that the coefficient of \(x ^ { 3 }\) is twice the coefficient of \(x ^ { 2 }\) in the binomial expansion of \(\mathrm { f } ( x )\),

1. prove that \(n = 6 k + 2\). Given also that the coefficients of \(x ^ { 4 }\) and \(x ^ { 5 }\) are equal and non-zero,
2. form another equation in \(n\) and \(k\) and hence show that \(k = 2\) and \(n = 14\). Using these values of \(k\) and \(n\),
3. expand \(\mathrm { f } ( x )\) 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
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6 In the expansion of \(( 3 + a x ) ^ { n }\), where \(a\) and \(n\) are integers, the coefficient of \(x ^ { 2 }\) is 4860 6

1. Show that $$3 ^ { n } a ^ { 2 } n ( n - 1 ) = 87480$$ [3 marks]
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2. The constant term in the expansion is 729 The coefficient of \(x\) in the expansion is negative. 6
3. 1. Verify that \(n = 6\)
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4. (ii) Find the value of \(a\)