OCR MEI C4 (Core Mathematics 4)

1

1. Find the first three terms of the binomial expansion of \(\frac { 1 } { \sqrt [ 3 ] { 1 - 2 x } }\). State the set of values of \(x\) for which
   the expansion is valid.
2. Hence find \(a\) and \(b\) such that \(\frac { 1 - 3 x } { \sqrt [ 3 ] { 1 - 2 x } } = 1 + a x + b x ^ { 2 } + \ldots\).

2 Find the first three terms in the binomial expansion of \(( 4 + x ) ^ { \frac { 3 } { 2 } }\). State the set of values of \(x\) for which the expansion is valid.

3

1. Express \(\frac { x } { ( 1 + x ) ( 1 - 2 x ) }\) in partial fractions.
2. Hence use binomial expansions to show that \(\frac { x } { ( 1 + x ) ( 1 - 2 x ) } = a x + b x ^ { 2 } + \ldots\), where \(a\) and \(b\) are
   constants to be determined. constants to be determined. State the set of values of \(x\) for which the expansion is valid.

4 Find the first four terms in the binomial expansion of \(\sqrt { 1 + 2 x }\). State the set of values of \(x\) for which the expansion is valid.

5 Find the first three terms in the binomial expansion of \(\sqrt [ 3 ] { 1 + 3 x }\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid.

6

1. Given that $$\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) } = \frac { A } { 1 + x } + \frac { B } { ( 1 + x ) ^ { 2 } } + \frac { C } { 1 - 4 x }$$ where \(A , B\) and \(C\) are constants, find \(B\) and \(C\), and show that \(A = 0\).
2. Given that \(x\) is sufficiently small, find the first three terms of the binomial expansions of \(( 1 + x ) ^ { - 2 }\) and \(( 1 - 4 x ) ^ { - 1 }\). Hence find the first three terms of the expansion of \(\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) }\).

7 Find the first 4 terms in the binomial expansion of \(\sqrt { 4 + 2 x }\). State the range of values of \(x\) for which the expansion is valid.