Single coefficient given directly

1 The coefficient of \(x ^ { 3 }\) in the expansion of \(\left( p + \frac { 1 } { p } x \right) ^ { 4 }\) is 144 .
Find the possible values of the constant \(p\).

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.

1 In the expansion of \(\left( k x + \frac { 2 } { x } \right) ^ { 4 }\) ,where \(k\) is a positive constant,the term independent of \(x\) is equal to 150 .
Find the value of \(k\) and hence determine the coefficient of \(x ^ { 2 }\) in the expansion.
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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.

2 In the expansion of \(\left( \frac { 1 } { a x } + 2 a x ^ { 2 } \right) ^ { 5 }\), the coefficient of \(x\) is 5 . Find the value of the constant \(a\).

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

2 In the expansion of \(( 1 + a x ) ^ { 6 }\), where \(a\) is a constant, the coefficient of \(x\) is - 30 . Find the coefficient of \(x ^ { 3 }\).

1 The coefficient of \(x ^ { 2 }\) in the expansion of \(\left( k + \frac { 1 } { 3 } x \right) ^ { 5 }\) is 30 . Find the value of the constant \(k\).

1 In the expansion of \(\left( x ^ { 2 } - \frac { a } { x } \right) ^ { 7 }\), the coefficient of \(x ^ { 5 }\) is - 280 . Find the value of the constant \(a\).

4. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + a x ) ^ { 7 }\), where \(a\) is a constant. Give each term in its simplest form. Given that the coefficient of \(x ^ { 2 }\) in this expansion is 525 ,
(b) find the possible values of \(a\).

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

$$( 2 + k x ) ^ { 7 }$$ 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 1890
(b) find the value of \(k\).

4. The coefficient of \(x ^ { 2 }\) in the binomial expansion of \(( 1 + k x ) ^ { 7 }\), where \(k\) is a positive constant, is 525.

1. Find the value of \(k\). Using this value of \(k\),
2. show that the coefficient of \(x ^ { 3 }\) in the expansion is 4375 ,
3. find the first three terms in the expansion in ascending powers of \(x\) of $$( 2 - x ) ( 1 + k x ) ^ { 7 }$$

7 The coefficient of \(x ^ { 8 }\) in the expansion of \(( 2 x + k ) ^ { 12 }\), where \(k\) is a positive integer, is 79200000.
Determine the value of \(k\).

2. The coefficient of \(x ^ { 8 }\) in the expansion of \(( 2 x + k ) ^ { 12 }\), where \(k\) is a positive integer, is 79200000.
Determine the value of \(k\).

3 The coefficient of \(x ^ { 2 }\) in the binomial expansion of \(( 1 + a x ) ^ { 6 }\) is \(\frac { 20 } { 3 }\)