Product with reciprocal term binomial

1

1. Expand \(\left( 1 - \frac { 1 } { 2 x } \right) ^ { 2 }\).
2. Find the first four terms in the expansion, in ascending powers of \(x\), of \(( 1 + 2 x ) ^ { 6 }\).
3. Hence find the coefficient of \(x\) in the expansion of \(\left( 1 - \frac { 1 } { 2 x } \right) ^ { 2 } ( 1 + 2 x ) ^ { 6 }\).

2

1. Find the first 3 terms in the expansion of \(\left( 2 x - \frac { 3 } { x } \right) ^ { 5 }\) in descending powers of \(x\).
2. Hence find the coefficient of \(x\) in the expansion of \(\left( 1 + \frac { 2 } { x ^ { 2 } } \right) \left( 2 x - \frac { 3 } { x } \right) ^ { 5 }\).

2

1. Find the first three terms, in descending powers of \(x\), in the expansion of \(\left( x - \frac { 2 } { x } \right) ^ { 6 }\).
2. Find the coefficient of \(x ^ { 4 }\) in the expansion of \(\left( 1 + x ^ { 2 } \right) \left( x - \frac { 2 } { x } \right) ^ { 6 }\).

2 Find the coefficient of \(x ^ { 2 }\) in the expansion of

1. \(\left( 2 x - \frac { 1 } { 2 x } \right) ^ { 6 }\),
2. \(\left( 1 + x ^ { 2 } \right) \left( 2 x - \frac { 1 } { 2 x } \right) ^ { 6 }\).

2 Find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 + x ^ { 2 } \right) \left( \frac { x } { 2 } - \frac { 4 } { x } \right) ^ { 6 }\).

1

1. Find the coefficient of \(x\) in the expansion of \(\left( 2 x - \frac { 1 } { x } \right) ^ { 5 }\).
2. Hence find the coefficient of \(x\) in the expansion of \(\left( 1 + 3 x ^ { 2 } \right) \left( 2 x - \frac { 1 } { x } \right) ^ { 5 }\).

2

1. In the binomial expansion of \(\left( 2 x - \frac { 1 } { 2 x } \right) ^ { 5 }\), the first three terms are \(32 x ^ { 5 } - 40 x ^ { 3 } + 20 x\). Find the remaining three terms of the expansion.
2. Hence find the coefficient of \(x\) in the expansion of \(\left( 1 + 4 x ^ { 2 } \right) \left( 2 x - \frac { 1 } { 2 x } \right) ^ { 5 }\).
   \includegraphics[max width=\textwidth, alt={}, center]{f462c036-45d3-4679-ad53-4edbf99df76d-04_385_655_262_744} The diagram shows triangle \(A B C\) which is right-angled at \(A\). Angle \(A B C = \frac { 1 } { 5 } \pi\) radians and \(A C = 8 \mathrm {~cm}\). The points \(D\) and \(E\) lie on \(B C\) and \(B A\) respectively. The sector \(A D E\) is part of circle with centre \(A\) and is such that \(B D C\) is the tangent to the \(\operatorname { arc } D E\) at \(D\).
3. Find the length of \(A D\).
4. Find the area of the shaded region.

6. (a) Find, in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), the binomial expansion of $$\left( 1 + \frac { 1 } { 4 } x \right) ^ { 12 }$$ giving each term in its simplest form.
(b) Hence find the coefficient of \(x\) in the expansion of $$\left( 3 + \frac { 2 } { x } \right) ^ { 2 } \left( 1 + \frac { 1 } { 4 } x \right) ^ { 12 }$$

3. (a) Find the first 4 terms, in ascending powers of \(x\), in the binomial expansion of $$\left( 1 + \frac { x } { 4 } \right) ^ { 12 }$$ giving each coefficient in its simplest form.
(b) Find the term independent of \(x\) in the expansion of $$\left( \frac { x ^ { 2 } + 8 } { x ^ { 5 } } \right) \left( 1 + \frac { x } { 4 } \right) ^ { 12 }$$

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

$$\left( 2 - \frac { x } { 4 } \right) ^ { 10 }$$ giving each term in its simplest form.
(b) Hence find the constant term in the series expansion of $$\left( 3 - \frac { 1 } { x } \right) ^ { 2 } \left( 2 - \frac { x } { 4 } \right) ^ { 10 }$$

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

$$\left( 3 - \frac { 2 x } { 9 } \right) ^ { 8 }$$ giving each term in simplest form. $$f ( x ) = \left( \frac { x - 1 } { 2 x } \right) \left( 3 - \frac { 2 x } { 9 } \right) ^ { 8 }$$ (b) Find the coefficient of \(x ^ { 2 }\) in the series expansion of \(\mathrm { f } ( x )\), giving your answer as a simplified fraction.

1. Find, in simplest form, the coefficient of \(x ^ { 5 }\) in the expansion of

$$\left( 5 + 8 x ^ { 2 } \right) \left( 3 - \frac { 1 } { 2 } x \right) ^ { 6 }$$

8

1. Expand \(\left( 1 + \frac { 4 } { x } \right) ^ { 2 }\).
2. The first four terms of the binomial expansion of \(\left( 1 + \frac { x } { 4 } \right) ^ { 8 }\) in ascending powers of \(x\) are \(1 + a x + b x ^ { 2 } + c x ^ { 3 }\). Find the values of the constants \(a , b\) and \(c\).
3. Hence find the coefficient of \(x\) in the expansion of \(\left( 1 + \frac { 4 } { x } \right) ^ { 2 } \left( 1 + \frac { x } { 4 } \right) ^ { 8 }\).

1. a) Find the first four terms, in ascending powers of \(x\), of the binomial expansion of \(\left( 1 + \frac { x } { 2 } \right) ^ { 7 }\), giving each coefficient in exact simplified form.
b) Hence determine the coefficient of \(x\) in the expansion of $$\left( 1 + \frac { 2 } { x } \right) ^ { 2 } \left( 1 + \frac { x } { 2 } \right) ^ { 7 }$$ [BLANK PAGE]