Expand and state validity

1. (a) Use the binomial expansion, in ascending powers of \(x\), to show that

$$\sqrt { ( 4 - x ) } = 2 - \frac { 1 } { 4 } x + k x ^ { 2 } + \ldots$$ where \(k\) is a rational constant to be found. A student attempts to substitute \(x = 1\) into both sides of this equation to find an approximate value for \(\sqrt { 3 }\).
(b) State, giving a reason, if the expansion is valid for this value of \(x\).

10

1. Determine the first three terms in ascending powers of \(x\) of the binomial expansion of \(( 8 + 3 x ) ^ { \frac { 1 } { 3 } }\).
2. State the range of values of \(x\) for which this expansion is valid.

6

1. Find the first three terms in ascending powers of \(x\) of the binomial expansion of \(( 1 + 4 x ) ^ { \frac { 1 } { 2 } }\).
2. State the range of values of \(x\) for which this expansion is valid.

2 Find the first four terms of the binomial expansion of \(( 1 - 2 x ) ^ { \frac { 1 } { 2 } }\). State the set of values of \(x\) for which the expansion is valid.

1. (a) Find the binomial expansion of \(( 2 - 3 x ) ^ { - 3 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
   (b) State the set of values of \(x\) for which your expansion is valid.
   
2. A curve has the equation

$$x ^ { 2 } + 3 x y - 2 y ^ { 2 } + 17 = 0$$ (a) Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
(b) Find an equation for the normal to the curve at the point \(( 3 , - 2 )\).

1. (a) Expand \(( 1 + 4 x ) ^ { \frac { 3 } { 2 } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
   (b) State the set of values of \(x\) for which your expansion is valid.
   
2. Use the substitution \(u = 1 + \sin x\) to find the value of

$$\int _ { 0 } ^ { \frac { \pi } { 2 } } \cos x ( 1 + \sin x ) ^ { 3 } d x$$

5

1. Find the first three non-zero terms of the expansion, in ascending powers of \(x\), of \(( 4 + x ) ^ { \frac { 1 } { 2 } }\).
2. State the range of values of \(x\) for which your expansion in part (a) is valid. [1]
3. Use your expansion to determine an approximation to \(\sqrt { } 36.9\), showing all the figures on your calculator.

6. $$f ( x ) = ( 3 - 2 x ) ^ { - 4 }$$ a) Find the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), giving each coefficient as a simplified fraction.
(3)
b) For what values of \(x\) is the expansion valid?