OCR MEI C4 (Core Mathematics 4)

1 Solve the equation \(\frac { 2 x } { x + 1 } - \frac { 1 } { x - 1 } = 1\).

2 Express \(\frac { x + 1 } { ( 2 x - 1 ) }\) in partial fractions.

3 Express \(\frac { 3 x + 2 } { x \left( x ^ { 2 } + 1 \right) }\) in partial fractions.

4 Express \(\frac { 4 } { x \left( x ^ { 2 } + 4 \right) }\) in partial fractions.

5 Solve the equation \(\frac { 2 x } { x - 2 } - \frac { 4 x } { x + 1 } = 3\).

6

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. State the set of values of \(x\) for which the expansion is valid.

7 A skydiver drops from a helicopter. Before she opens her parachute, her speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after time \(t\) seconds is modelled by the differential equation $$\frac { \mathrm { d } v } { \mathrm {~d} t } = 10 \mathrm { e } ^ { - \frac { 1 } { 2 } t }$$ When \(t = 0 , v = 0\).

1. Find \(v\) in terms of \(t\).
2. According to this model, what is the speed of the skydiver in the long term? She opens her parachute when her speed is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Her speed \(t\) seconds after this is \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and is modelled by the differential equation $$\frac { \mathrm { d } w } { \mathrm {~d} t } = - \frac { 1 } { 2 } ( w - 4 ) ( w + 5 )$$
3. Express \(\frac { 1 } { ( w - 4 ) ( w + 5 ) }\) in partial fractions.
4. Using this result, show that \(\frac { w - 4 } { w + 5 } = 0.4 \mathrm { e } ^ { - 4.5 t }\).
5. According to this model, what is the speed of the skydiver in the long term?