OCR FP2 (Further Pure Mathematics 2) 2009 January

1

1. Write down and simplify the first three terms of the Maclaurin series for \(\mathrm { e } ^ { 2 x }\).
2. Hence show that the Maclaurin series for $$\ln \left( \mathrm { e } ^ { 2 x } + \mathrm { e } ^ { - 2 x } \right)$$ begins \(\ln a + b x ^ { 2 }\), where \(a\) and \(b\) are constants to be found.

2 It is given that \(\alpha\) is the only real root of the equation \(x ^ { 5 } + 2 x - 28 = 0\) and that \(1.8 < \alpha < 2\).

1. The iteration \(x _ { n + 1 } = \sqrt [ 5 ] { 28 - 2 x _ { n } }\), with \(x _ { 1 } = 1.9\), is to be used to find \(\alpha\). Find the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving the answers correct to 7 decimal places.
2. The error \(e _ { n }\) is defined by \(e _ { n } = \alpha - x _ { n }\). Given that \(\alpha = 1.8915749\), correct to 7 decimal places, evaluate \(\frac { e _ { 3 } } { e _ { 2 } }\) and \(\frac { e _ { 4 } } { e _ { 3 } }\). Comment on these values in relation to the gradient of the curve with equation \(y = \sqrt [ 5 ] { 28 - 2 x }\) at \(x = \alpha\).
3. Prove that the derivative of \(\sin ^ { - 1 } x\) is \(\frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\).
4. Given that $$\sin ^ { - 1 } 2 x + \sin ^ { - 1 } y = \frac { 1 } { 2 } \pi$$ find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = \frac { 1 } { 4 }\).
5. By means of a suitable substitution, show that $$\int \frac { x ^ { 2 } } { \sqrt { x ^ { 2 } - 1 } } \mathrm {~d} x$$ can be transformed to \(\int \cosh ^ { 2 } \theta \mathrm {~d} \theta\).
6. Hence show that \(\int \frac { x ^ { 2 } } { \sqrt { x ^ { 2 } - 1 } } \mathrm {~d} x = \frac { 1 } { 2 } x \sqrt { x ^ { 2 } - 1 } + \frac { 1 } { 2 } \cosh ^ { - 1 } x + c\).

5
\includegraphics[max width=\textwidth, alt={}, center]{b9f29713-bc86-4869-9e54-195208e5e81d-3_661_734_267_703} The diagram shows the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = 2 x ^ { 3 } - 9 x ^ { 2 } + 12 x - 4.36$$ The curve has turning points at \(x = 1\) and \(x = 2\) and crosses the \(x\)-axis at \(x = \alpha , x = \beta\) and \(x = \gamma\), where \(0 < \alpha < \beta < \gamma\).

1. The Newton-Raphson method is to be used to find the roots of the equation \(\mathrm { f } ( x ) = 0\), with \(x _ { 1 } = k\).
   (a) To which root, if any, would successive approximations converge in each of the cases \(k < 0\) and \(k = 1\) ?
   (b) What happens if \(1 < k < 2\) ?
2. Sketch the curve with equation \(y ^ { 2 } = \mathrm { f } ( x )\). State the coordinates of the points where the curve crosses the \(x\)-axis and the coordinates of any turning points.
3. Using the definitions of \(\cosh x\) and \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that $$1 + 2 \sinh ^ { 2 } x \equiv \cosh 2 x .$$
4. Solve the equation $$\cosh 2 x - 5 \sinh x = 4$$ giving your answers in logarithmic form.

7
\includegraphics[max width=\textwidth, alt={}, center]{b9f29713-bc86-4869-9e54-195208e5e81d-4_511_609_264_769} The diagram shows the curve with equation, in polar coordinates, $$r = 3 + 2 \cos \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi .$$ The points \(P , Q , R\) and \(S\) on the curve are such that the straight lines \(P O R\) and \(Q O S\) are perpendicular, where \(O\) is the pole. The point \(P\) has polar coordinates ( \(r , \alpha\) ).

1. Show that \(O P + O Q + O R + O S = k\), where \(k\) is a constant to be found.
2. Given that \(\alpha = \frac { 1 } { 4 } \pi\), find the exact area bounded by the curve and the lines \(O P\) and \(O Q\) (shaded in the diagram).

8
\includegraphics[max width=\textwidth, alt={}, center]{b9f29713-bc86-4869-9e54-195208e5e81d-5_579_1363_267_390} The diagram shows the curve with equation \(y = \frac { 1 } { x + 1 }\). A set of \(n\) rectangles of unit width is drawn, starting at \(x = 0\) and ending at \(x = n\), where \(n\) is an integer.

1. By considering the areas of these rectangles, explain why $$\frac { 1 } { 2 } + \frac { 1 } { 3 } + \ldots + \frac { 1 } { n + 1 } < \ln ( n + 1 ) .$$
2. By considering the areas of another set of rectangles, show that $$1 + \frac { 1 } { 2 } + \frac { 1 } { 3 } + \ldots + \frac { 1 } { n } > \ln ( n + 1 ) .$$
3. Hence show that $$\ln ( n + 1 ) + \frac { 1 } { n + 1 } < \sum _ { r = 1 } ^ { n + 1 } \frac { 1 } { r } < \ln ( n + 1 ) + 1$$
4. State, with a reason, whether \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r }\) is convergent.

9 A curve has equation $$y = \frac { 4 x - 3 a } { 2 \left( x ^ { 2 } + a ^ { 2 } \right) }$$ where \(a\) is a positive constant.

1. Explain why the curve has no asymptotes parallel to the \(y\)-axis.
2. Find, in terms of \(a\), the set of values of \(y\) for which there are no points on the curve.
3. Find the exact value of \(\int _ { a } ^ { 2 a } \frac { 4 x - 3 a } { 2 \left( x ^ { 2 } + a ^ { 2 } \right) } \mathrm { d } x\), showing that it is independent of \(a\).