OCR FP2 (Further Pure Mathematics 2) Specimen

1

1. Starting from the definition of \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\), show that \(\cosh 2 x = 2 \cosh ^ { 2 } x - 1\).
2. Given that \(\cosh 2 x = k\), where \(k > 1\), express each of \(\cosh x\) and \(\sinh x\) in terms of \(k\).

2
\includegraphics[max width=\textwidth, alt={}, center]{e4e1c424-8dd5-4d18-9950-e902de0301b0-2_728_951_486_534} The diagram shows the graph of $$y = \frac { 2 x ^ { 2 } + 3 x + 3 } { x + 1 }$$

1. Find the equations of the asymptotes of the curve.
2. Prove that the values of \(y\) between which there are no points on the curve are - 5 and 3 .

3

1. Find the first three terms of the Maclaurin series for \(\ln ( 2 + x )\).
2. Write down the first three terms of the series for \(\ln ( 2 - x )\), and hence show that, if \(x\) is small, then $$\ln \left( \frac { 2 + x } { 2 - x } \right) \approx x$$

4 The equation of a curve, in polar coordinates, is $$r = 2 \cos 2 \theta \quad ( - \pi < \theta \leqslant \pi ) .$$

1. Find the values of \(\theta\) which give the directions of the tangents at the pole. One loop of the curve is shown in the diagram.
   \includegraphics[max width=\textwidth, alt={}, center]{e4e1c424-8dd5-4d18-9950-e902de0301b0-3_362_720_653_708}
2. Find the exact value of the area of the region enclosed by the loop.

5
\includegraphics[max width=\textwidth, alt={}, center]{e4e1c424-8dd5-4d18-9950-e902de0301b0-3_444_999_1258_539} The diagram shows the curve \(y = \frac { 1 } { x + 1 }\) together with four rectangles of unit width.

1. Explain how the diagram shows that $$\frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 4 } + \frac { 1 } { 5 } < \int _ { 0 } ^ { 4 } \frac { 1 } { x + 1 } \mathrm {~d} x$$ The curve \(y = \frac { 1 } { x + 2 }\) passes through the top left-hand corner of each of the four rectangles shown.
2. By considering the rectangles in relation to this curve, write down a second inequality involving \(\frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 4 } + \frac { 1 } { 5 }\) and a definite integral.
3. By considering a suitable range of integration and corresponding rectangles, show that $$\ln ( 500.5 ) < \sum _ { r = 2 } ^ { 1000 } \frac { 1 } { r } < \ln ( 1000 ) .$$

6

1. Given that \(I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \sqrt { } ( 1 - x ) \mathrm { d } x\), prove that, for \(n \geqslant 1\), $$( 2 n + 3 ) I _ { n } = 2 n I _ { n - 1 } .$$
2. Hence find the exact value of \(I _ { 2 }\).

7 The curve with equation $$y = \frac { x } { \cosh x }$$ has one stationary point for \(x > 0\).

1. Show that the \(x\)-coordinate of this stationary point satisfies the equation \(x \tanh x - 1 = 0\). The positive root of the equation \(x \tanh x - 1 = 0\) is denoted by \(\alpha\).
2. Draw a sketch showing (for positive values of \(x\) ) the graph of \(y = \tanh x\) and its asymptote, and the graph of \(y = \frac { 1 } { x }\). Explain how you can deduce from your sketch that \(\alpha > 1\).
3. Use the Newton-Raphson method, taking first approximation \(x _ { 1 } = 1\), to find further approximations \(x _ { 2 }\) and \(x _ { 3 }\) for \(\alpha\).
4. By considering the approximate errors in \(x _ { 1 }\) and \(x _ { 2 }\), estimate the error in \(x _ { 3 }\).

8

1. Use the substitution \(t = \tan \frac { 1 } { 2 } x\) to show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { \frac { 1 - \cos x } { 1 + \sin x } } \mathrm {~d} x = 2 \sqrt { } 2 \int _ { 0 } ^ { 1 } \frac { t } { ( 1 + t ) \left( 1 + t ^ { 2 } \right) } \mathrm { d } t$$
2. Express \(\frac { t } { ( 1 + t ) \left( 1 + t ^ { 2 } \right) }\) in partial fractions.
3. Hence find \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { \frac { 1 - \cos x } { 1 + \sin x } } \mathrm {~d} x\), expressing your answer in an exact form.