OCR FP2 (Further Pure Mathematics 2) 2011 June

1 Express \(\frac { 2 x + 3 } { ( x + 3 ) \left( x ^ { 2 } + 9 \right) }\) in partial fractions.

2 A curve has equation \(y = \frac { x ^ { 2 } - 6 x - 5 } { x - 2 }\).

1. Find the equations of the asymptotes.
2. Show that \(y\) can take all real values.

3 It is given that \(\mathrm { F } ( x ) = 2 + \ln x\). The iteration \(x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)\) is to be used to find a root, \(\alpha\), of the equation \(x = 2 + \ln x\).

1. Taking \(x _ { 1 } = 3.1\), find \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers correct to 5 decimal places.
2. The error \(e _ { n }\) is defined by \(e _ { n } = \alpha - x _ { n }\). Given that \(\alpha = 3.14619\), correct to 5 decimal places, use the values of \(e _ { 2 }\) and \(e _ { 3 }\) to make an estimate of \(\mathrm { F } ^ { \prime } ( \alpha )\) correct to 3 decimal places. State the true value of \(\mathrm { F } ^ { \prime } ( \alpha )\) correct to 4 decimal places.
3. Illustrate the iteration by drawing a sketch of \(y = x\) and \(y = \mathrm { F } ( x )\), showing how the values of \(x _ { n }\) approach \(\alpha\). State whether the convergence is of the 'staircase' or 'cobweb' type.

4 A curve \(C\) has the cartesian equation \(x ^ { 3 } + y ^ { 3 } = a x y\), where \(x \geqslant 0 , y \geqslant 0\) and \(a > 0\).

1. Express the polar equation of \(C\) in the form \(r = \mathrm { f } ( \theta )\) and state the limits between which \(\theta\) lies. The line \(\theta = \alpha\) is a line of symmetry of \(C\).
2. Find and simplify an expression for \(\mathrm { f } \left( \frac { 1 } { 2 } \pi - \theta \right)\) and hence explain why \(\alpha = \frac { 1 } { 4 } \pi\).
3. Find the value of \(r\) when \(\theta = \frac { 1 } { 4 } \pi\).
4. Sketch the curve \(C\).

5

1. Prove that, if \(y = \sin ^ { - 1 } x\), then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\).
2. Find the Maclaurin series for \(\sin ^ { - 1 } x\), up to and including the term in \(x ^ { 3 }\).
3. Use the result of part (ii) and the Maclaurin series for \(\ln ( 1 + x )\) to find the Maclaurin series for \(\left( \sin ^ { - 1 } x \right) \ln ( 1 + x )\), up to and including the term in \(x ^ { 4 }\).

6 It is given that \(I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } ( 1 - x ) ^ { \frac { 3 } { 2 } } \mathrm {~d} x\), for \(n \geqslant 0\).

1. Show that \(I _ { n } = \frac { 2 n } { 2 n + 5 } I _ { n - 1 }\), for \(n \geqslant 1\).
2. Hence find the exact value of \(I _ { 3 }\).

7

1. Sketch the graph of \(y = \tanh x\) and state the value of the gradient when \(x = 0\). On the same axes, sketch the graph of \(y = \tanh ^ { - 1 } x\). Label each curve and give the equations of the asymptotes.
2. Find \(\int _ { 0 } ^ { k } \tanh x \mathrm {~d} x\), where \(k > 0\).
3. Deduce, or show otherwise, that \(\int _ { 0 } ^ { \tanh k } \tanh ^ { - 1 } x \mathrm {~d} x = k \tanh k - \ln ( \cosh k )\).

8

1. Use the substitution \(x = \cosh ^ { 2 } u\) to find \(\int \sqrt { \frac { x } { x - 1 } } \mathrm {~d} x\), giving your answer in the form \(\mathrm { f } ( x ) + \ln ( \mathrm { g } ( x ) )\).
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2. Hence calculate the exact area of the region between the curve \(y = \sqrt { \frac { x } { x - 1 } }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\) (see diagram).
3. What can you say about the volume of the solid of revolution obtained when the region defined in part (ii) is rotated completely about the \(x\)-axis? Justify your answer.