OCR FP2 (Further Pure Mathematics 2) 2009 June

1
\includegraphics[max width=\textwidth, alt={}, center]{cf77e51a-1d3f-423a-be59-96ec60fbeb67-2_568_959_269_593} The diagram shows the curve with equation \(y = \ln ( \cos x )\), for \(0 \leqslant x \leqslant 1.5\). The region bounded by the curve, the \(x\)-axis and the line \(x = 1.5\) has area \(A\). The region is divided into five strips, each of width 0.3 .

1. By considering the set of rectangles indicated in the diagram, find an upper bound for \(A\). Give the answer correct to 3 decimal places.
2. By considering another set of five suitable rectangles, find a lower bound for \(A\). Give the answer correct to 3 decimal places.
3. How could you reduce the difference between the upper and lower bounds for \(A\) ?

2 Given that \(y = \frac { x ^ { 2 } + x + 1 } { ( x - 1 ) ^ { 2 } }\), prove that \(y \geqslant \frac { 1 } { 4 }\) for all \(x \neq 1\).

3

1. Given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { \sin x }\), find \(\mathrm { f } ^ { \prime } ( 0 )\) and \(\mathrm { f } ^ { \prime \prime } ( 0 )\).
2. Hence find the first three terms of the Maclaurin series for \(\mathrm { f } ( x )\).

4 Express \(\frac { x ^ { 3 } } { ( x - 2 ) \left( x ^ { 2 } + 4 \right) }\) in partial fractions.

5 It is given that \(I = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { \cos \theta } { 1 + \cos \theta } \mathrm { d } \theta\).

1. By using the substitution \(t = \tan \frac { 1 } { 2 } \theta\), show that \(I = \int _ { 0 } ^ { 1 } \left( \frac { 2 } { 1 + t ^ { 2 } } - 1 \right) \mathrm { d } t\).
2. Hence find \(I\) in terms of \(\pi\).

6 Given that $$\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 16 + 9 x ^ { 2 } } } \mathrm {~d} x + \int _ { 0 } ^ { 2 } \frac { 1 } { \sqrt { 9 + 4 x ^ { 2 } } } \mathrm {~d} x = \ln a$$ find the exact value of \(a\).

7

1. Sketch the graph of \(y = \operatorname { coth } x\), and give the equations of any asymptotes.
2. It is given that \(\mathrm { f } ( x ) = x \tanh x - 2\). Use the Newton-Raphson method, with a first approximation \(x _ { 1 } = 2\), to find the next three approximations \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\) to a root of \(\mathrm { f } ( x ) = 0\). Give the answers correct to 4 decimal places.
3. If \(\mathrm { f } ( x ) = 0\), show that \(\operatorname { coth } x = \frac { 1 } { 2 } x\). Hence write down the roots of \(\mathrm { f } ( x ) = 0\), correct to 4 decimal places.

8

1. Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that
   (a) \(\cosh ( \ln a ) \equiv \frac { a ^ { 2 } + 1 } { 2 a }\), where \(a > 0\),
   (b) \(\cosh x \cosh y - \sinh x \sinh y \equiv \cosh ( x - y )\).
2. Use part (i)(b) to show that \(\cosh ^ { 2 } x - \sinh ^ { 2 } x \equiv 1\).
3. Given that \(R > 0\) and \(a > 1\), find \(R\) and \(a\) such that $$13 \cosh x - 5 \sinh x \equiv R \cosh ( x - \ln a )$$
4. Hence write down the coordinates of the minimum point on the curve with equation \(y = 13 \cosh x - 5 \sinh x\).

9

1. It is given that, for non-negative integers \(n\), $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { n } \theta \mathrm {~d} \theta$$ Show that, for \(n \geqslant 2\), $$n I _ { n } = ( n - 1 ) I _ { n - 2 } .$$
2. The equation of a curve, in polar coordinates, is $$r = \sin ^ { 3 } \theta , \quad \text { for } 0 \leqslant \theta \leqslant \pi$$ (a) Find the equations of the tangents at the pole and sketch the curve.
   (b) Find the exact area of the region enclosed by the curve. RECOGNISING ACHIEVEMENT