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 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 \(a\).

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 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 \(k\).

The coefficient of \(x^3\) in the expansion of \((3 + ax)^6\) is 160.

1. Find the value of the constant \(a\). [2]
2. Hence find the coefficient of \(x^5\) in the expansion of \((3 + ax)^6(1 - 2x)\). [3]

In the expansion of \(\left(kx+\frac{2}{x}\right)^4\), 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 \(x^5\) in the expansion. [4]

The coefficient of \(x^3\) in the expansion of \((a + x)^5 + (2 - x)^6\) is \(90\). Find the value of the positive constant \(a\). [5]

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

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

The coefficient of \(x^2\) in the binomial expansion of \((1 + ax)^6\) is \(\frac{20}{3}\) Find the two possible values of \(a\) [3 marks]

In the expansion of \((3 + ax)^n\), where \(a\) and \(n\) are integers, the coefficient of \(x^2\) is 4860

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