OCR FP2 (Further Pure Mathematics 2) 2013 January

1 Express \(\frac { 5 x } { ( x - 1 ) \left( x ^ { 2 } + 4 \right) }\) in partial fractions.

2 The equation of a curve is \(y = \frac { x ^ { 2 } - 3 } { x - 1 }\).

1. Find the equations of the asymptotes of the curve.
2. Write down the coordinates of the points where the curve cuts the axes.
3. Show that the curve has no stationary points.
4. Sketch the curve and the asymptotes.

3 By first expressing \(\cosh x\) and \(\sinh x\) in terms of exponentials, solve the equation $$3 \cosh x - 4 \sinh x = 7$$ giving your answer in an exact logarithmic form.

4 You are given that \(I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { 2 x } \mathrm {~d} x\) for \(n \geqslant 0\).

1. Show that \(I _ { n } = \frac { 1 } { 2 } \mathrm { e } ^ { 2 } - \frac { 1 } { 2 } n I _ { n - 1 }\) for \(n \geqslant 1\).
2. Find \(I _ { 3 }\) in terms of e.

5 You are given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { - x } \sin x\).

1. Find \(f ( 0 )\) and \(f ^ { \prime } ( 0 )\).
2. Show that \(\mathrm { f } ^ { \prime \prime } ( x ) = - 2 \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x )\) and hence, or otherwise, find \(\mathrm { f } ^ { \prime \prime } ( 0 )\).
3. Find a similar expression for \(\mathrm { f } ^ { \prime \prime \prime } ( x )\) and hence, or otherwise, find \(\mathrm { f } ^ { \prime \prime \prime } ( 0 )\).
4. Find the Maclaurin series for \(\mathrm { f } ( x )\) up to and including the term in \(x ^ { 3 }\).

6 By first completing the square, find \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { x ^ { 2 } + 4 x + 8 } } \mathrm {~d} x\), giving your answer in an exact logarithmic form.

7 A curve has polar equation \(r = 5 \sin 2 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. Sketch the curve, indicating the line of symmetry and stating the polar coordinates of the point \(P\) on the curve which is furthest away from the pole.
2. Calculate the area enclosed by the curve.
3. Find the cartesian equation of the tangent to the curve at \(P\).
4. Show that a cartesian equation of the curve is \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 } = ( 10 x y ) ^ { 2 }\).

8 It is required to solve the equation \(\ln ( x - 1 ) - x + 3 = 0\).
You are given that there are two roots, \(\alpha\) and \(\beta\), where \(1.1 < \alpha < 1.2\) and \(4.1 < \beta < 4.2\).

1. The root \(\beta\) can be found using the iterative formula $$x _ { n + 1 } = \ln \left( x _ { n } - 1 \right) + 3$$
   1. Using this iterative formula with \(x _ { 1 } = 4.15\), find \(\beta\) correct to 3 decimal places. Show all your working.
   2. Explain with the aid of a sketch why this iterative formula will not converge to \(\alpha\) whatever initial value is taken.
   3. (a) Show that the Newton-Raphson iterative formula for this equation can be written in the form $$x _ { n + 1 } = \frac { 3 - 2 x _ { n } - \left( x _ { n } - 1 \right) \ln \left( x _ { n } - 1 \right) } { 2 - x _ { n } }$$ (b) Use this formula with \(x _ { 1 } = 1.2\) to find \(\alpha\) correct to 3 decimal places.