Product with unknown constant to determine

7

1. Write down the first four terms of the expansion, in ascending powers of \(x\), of \(( a - x ) ^ { 6 }\).
2. Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 + \frac { 2 } { a x } \right) ( a - x ) ^ { 6 }\) is - 20 , find in exact form the possible values of the constant \(a\).

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

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

6 It is given that the coefficient of \(x ^ { 3 }\) in the expansion of $$( 2 + a x ) ^ { 4 } ( 5 - a x )$$ is 432 .
Find the value of the constant \(a\).

1 The coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 + k x ) ( 1 - 2 x ) ^ { 5 }\) is 20 .
Find the value of the constant \(k\).

8

1. It is given that in the expansion of \(( 4 + 2 x ) ( 2 - a x ) ^ { 5 }\), the coefficient of \(x ^ { 2 }\) is - 15 .
   Find the possible values of \(a\).
2. It is given instead that in the expansion of \(( 4 + 2 x ) ( 2 - a x ) ^ { 5 }\), the coefficient of \(x ^ { 2 }\) is \(k\). It is also given that there is only one value of \(a\) which leads to this value of \(k\). Find the values of \(k\) and \(a\).

2

1. Find the first three terms, in ascending powers of \(x\), in the expansion of \(( 1 + a x ) ^ { 6 }\).
2. Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 - 3 x ) ( 1 + a x ) ^ { 6 }\) is - 3 , find the possible values of the constant \(a\).

4

1. Expand the following in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\).
   1. \(( 1 + 2 x ) ^ { 5 }\).
   2. \(( 1 - a x ) ^ { 6 }\), where \(a\) is a constant.
      In the expansion of \(( 1 + 2 x ) ^ { 5 } ( 1 - a x ) ^ { 6 }\), the coefficient of \(x ^ { 2 }\) is - 5 .
2. Find the possible values of \(a\).

3

1. Find the coefficients of \(x ^ { 3 }\) and \(x ^ { 4 }\) in the expansion of \(( 3 - a x ) ^ { 5 }\), where \(a\) is a constant. Give your answers in terms of \(a\).
2. Given that the coefficient of \(x ^ { 4 }\) in the expansion of \(( a x + 7 ) ( 3 - a x ) ^ { 5 }\) is 240 , find the positive value of \(a\).
   \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-05_2723_33_99_21}

6

1. Fid \(\mathbf { b }\) co fficien \(\mathrm { s } 6 x ^ { 2 }\) ad \(x ^ { 3 }\) irt \(\mathbf { b }\) e in ( \(\left. 2 x \right) ^ { 6 }\). [
2. Hen e fid b co fficien \(6 x ^ { 3 }\) in b \(\mathbf { b }\) in \(\left( 3 x + 1 ( 2 \quad x ) ^ { 6 } \right.\).

4

1. Find the first three terms in the expansion of \(( 2 + a x ) ^ { 5 }\) in ascending powers of \(x\).
2. Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 + 2 x ) ( 2 + a x ) ^ { 5 }\) is 240 , find the possible values of \(a\).

3

1. Write down the first 4 terms, in ascending powers of \(x\), of the expansion of \(( a - x ) ^ { 5 }\).
2. The coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 - a x ) ( a - x ) ^ { 5 }\) is - 200 . Find the possible values of the constant \(a\).

1

1. Find the first three terms in the expansion, in ascending powers of \(x\), of \(( 1 - 2 x ) ^ { 5 }\).
2. Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 + a x + 2 x ^ { 2 } \right) ( 1 - 2 x ) ^ { 5 }\) is 12 , find the value of the constant \(a\).

3

1. Find the first 3 terms in the expansion of \(( 2 - x ) ^ { 6 }\) in ascending powers of \(x\).
2. Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 + 2 x + a x ^ { 2 } \right) ( 2 - x ) ^ { 6 }\) is 48 , find the value of the constant \(a\).

3

1. Find the term independent of \(x\) in the expansion of \(\left( \frac { 2 } { x } - 3 x \right) ^ { 6 }\).
2. Find the value of \(a\) for which there is no term independent of \(x\) in the expansion of $$\left( 1 + a x ^ { 2 } \right) \left( \frac { 2 } { x } - 3 x \right) ^ { 6 }$$

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

2. One of the terms in the binomial expansion of \(( 3 + a x ) ^ { 6 }\), where \(a\) is a constant, is \(540 x ^ { 4 }\)

1. Find the possible values of \(a\).
2. Hence find the term independent of \(x\) in the expansion of $$\left( \frac { 1 } { 81 } + \frac { 1 } { x ^ { 6 } } \right) ( 3 + a x ) ^ { 6 }$$

4. (a) Find the first three terms, in ascending powers of \(x\), of the binomial expansion of $$( 2 + p x ) ^ { 6 }$$ where \(p\) is a constant. Give each term in simplest form. Given that in the expansion of $$\left( 3 - \frac { 1 } { 2 } x \right) ( 2 + p x ) ^ { 6 }$$ the coefficient of \(x ^ { 2 }\) is \(- \frac { 3 } { 4 }\)
(b) find the possible values of \(p\).
\includegraphics[max width=\textwidth, alt={}, center]{52c90d0e-a5e4-45fa-95a4-9523287e7588-11_2255_50_314_34}

3. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 2 - \frac { k x } { 4 } \right) ^ { 8 }$$ where \(k\) is a non-zero constant. Give each term in simplest form. $$f ( x ) = ( 5 - 3 x ) \left( 2 - \frac { k x } { 4 } \right) ^ { 8 }$$ In the expansion of \(\mathrm { f } ( x )\), the constant term is 3 times the coefficient of \(x\).
(b) Find the value of \(k\).
T

6

1. Find and simplify the first four terms in the expansion of \(( 1 + 4 x ) ^ { 7 }\) in ascending powers of \(x\).
2. In the expansion of $$( 3 + a x ) ( 1 + 4 x ) ^ { 7 }$$ the coefficient of \(x ^ { 2 }\) is 1001 . Find the value of \(a\).
3. (a) Sketch the graph of \(y = 2 \cos x\) for values of \(x\) such that \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), indicating the coordinates of any points where the curve meets the axes.
   (b) Solve the equation \(2 \cos x = 0.8\), giving all values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
4. Solve the equation \(2 \cos x = \sin x\), giving all values of \(x\) between \(- 180 ^ { \circ }\) and \(180 ^ { \circ }\).

3

1. Find and simplify the first three terms in the expansion of \(( 2 + 5 x ) ^ { 6 }\) in ascending powers of \(x\).
2. In the expansion of \(( 3 + c x ) ^ { 2 } ( 2 + 5 x ) ^ { 6 }\), the coefficient of \(x\) is 4416. Find the value of \(c\).

4

1. Find and simplify the first three terms in the binomial expansion of \(( 2 + a x ) ^ { 6 }\) in ascending powers of \(x\).
2. In the expansion of \(( 3 - 5 x ) ( 2 + a x ) ^ { 6 }\), the coefficient of \(x\) is 64 . Find the value of \(a\).

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

$$\left( 2 - \frac { x } { 16 } \right) ^ { 9 }$$ giving each term in its simplest form. $$f ( x ) = ( a + b x ) \left( 2 - \frac { x } { 16 } \right) ^ { 9 } , \text { where } a \text { and } b \text { are constants }$$ Given that the first two terms, in ascending powers of \(x\), in the series expansion of \(\mathrm { f } ( x )\) are 128 and \(36 x\),
(b) find the value of \(a\),
(c) find the value of \(b\).

4

1. Find and simplify the first three terms in the expansion of \(( 2 - 5 x ) ^ { 5 }\) in ascending powers of \(x\).
2. In the expansion of \(( 1 + a x ) ^ { 2 } ( 2 - 5 x ) ^ { 5 }\), the coefficient of \(x\) is 48 . Find the value of \(a\).

4

1. Find and simplify the first three terms in the expansion, in ascending powers of \(x\), of \(\left( 2 + \frac { 1 } { 3 } k x \right) ^ { 6 }\), where \(k\) is a constant.
2. In the expansion of \(( 3 - 4 x ) \left( 2 + \frac { 1 } { 3 } k x \right) ^ { 6 }\), the constant term is equal to the coefficient of \(x ^ { 2 }\). Determine the exact value of \(k\), given that \(k\) is positive.

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 $$( 2 - x ) ( 1 + k x ) ^ { 7 }$$