Expansion up to x^2 term

1. Find the first 3 terms in ascending powers of \(x\) of

$$\left( 2 - \frac { x } { 2 } \right) ^ { 6 }$$ giving each term in its simplest form.

1. A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where

$$f ( x ) = ( 2 - k x ) ^ { 5 }$$ and \(k\) is a constant.
Given that when \(\mathrm { f } ( x )\) is divided by \(( 4 x - 5 )\) the remainder is \(\frac { 243 } { 32 }\)

1. show that \(k = \frac { 2 } { 5 }\)
2. Find the first three terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 2 - \frac { 2 } { 5 } x \right) ^ { 5 }$$ giving each term in simplest form. Using the solution to part (b) and making your method clear,
3. find the gradient of \(C\) at the point where \(x = 0\)

Find the first three terms, in ascending powers of \(x\), of the binomial expansion of \(( 3 + 2 x ) ^ { 5 }\), giving each term in its simplest form.
(4)

2. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 - 2 x ) ^ { 5 }\). Give each term in its simplest form.
(b) If \(x\) is small, so that \(x ^ { 2 }\) and higher powers can be ignored, show that $$( 1 + x ) ( 1 - 2 x ) ^ { 5 } \approx 1 - 9 x$$ DU

Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \(( 3 - 2 x ) ^ { 5 }\), giving each term in its simplest form.
(4)

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

$$( 3 - x ) ^ { 6 }$$ and simplify each term.

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

$$( 2 - 5 x ) ^ { 6 }$$ Give each term in its simplest form.

Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \(( 2 + x ) ^ { 6 }\), giving each term in its simplest form.

1. Find the first 3 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.

Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \(( 2 + 3 x ) ^ { 6 }\).
(4)

Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 3 - x ) ^ { 6 }$$ and simplify each term.

7 Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \(( 2 - 3 x ) ^ { 5 }\), simplifying each term.

6 Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \(( 2 - 3 x ) ^ { 5 }\), simplifying each term.