OCR FP2 (Further Pure Mathematics 2) 2010 January

1 It is given that \(\mathrm { f } ( x ) = x ^ { 2 } - \sin x\).

1. The iteration \(x _ { n + 1 } = \sqrt { \sin x _ { n } }\), with \(x _ { 1 } = 0.875\), is to be used to find a real root, \(\alpha\), of the equation \(\mathrm { f } ( x ) = 0\). Find \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving the answers correct to 6 decimal places.
2. The error \(e _ { n }\) is defined by \(e _ { n } = \alpha - x _ { n }\). Given that \(\alpha = 0.876726\), correct to 6 decimal places, find \(e _ { 3 }\) and \(e _ { 4 }\). Given that \(\mathrm { g } ( x ) = \sqrt { \sin x }\), use \(e _ { 3 }\) and \(e _ { 4 }\) to estimate \(\mathrm { g } ^ { \prime } ( \alpha )\).

2 It is given that \(\mathrm { f } ( x ) = \tan ^ { - 1 } ( 1 + x )\).

1. Find \(\mathrm { f } ( 0 )\) and \(\mathrm { f } ^ { \prime } ( 0 )\), and show that \(\mathrm { f } ^ { \prime \prime } ( 0 ) = - \frac { 1 } { 2 }\).
2. Hence find the Maclaurin series for \(\mathrm { f } ( x )\) up to and including the term in \(x ^ { 2 }\).

3
\includegraphics[max width=\textwidth, alt={}, center]{63afce50-e15f-4634-b2f1-ad5d78ab8bf5-2_597_1006_973_571} A curve with no stationary points has equation \(y = \mathrm { f } ( x )\). The equation \(\mathrm { f } ( x ) = 0\) has one real root \(\alpha\), and the Newton-Raphson method is to be used to find \(\alpha\). The tangent to the curve at the point \(\left( x _ { 1 } , \mathrm { f } \left( x _ { 1 } \right) \right)\) meets the \(x\)-axis where \(x = x _ { 2 }\) (see diagram).

1. Show that \(x _ { 2 } = x _ { 1 } - \frac { \mathrm { f } \left( x _ { 1 } \right) } { \mathrm { f } ^ { \prime } \left( x _ { 1 } \right) }\).
2. Describe briefly, with the help of a sketch, how the Newton-Raphson method, using an initial approximation \(x = x _ { 1 }\), gives a sequence of approximations approaching \(\alpha\).
3. Use the Newton-Raphson method, with a first approximation of 1 , to find a second approximation to the root of \(x ^ { 2 } - 2 \sinh x + 2 = 0\).

4 The equation of a curve, in polar coordinates, is $$r = \mathrm { e } ^ { - 2 \theta } , \quad \text { for } 0 \leqslant \theta \leqslant \pi .$$

1. Sketch the curve, stating the polar coordinates of the point at which \(r\) takes its greatest value.
2. The pole is \(O\) and points \(P\) and \(Q\), with polar coordinates ( \(r _ { 1 } , \theta _ { 1 }\) ) and ( \(r _ { 2 } , \theta _ { 2 }\) ) respectively, lie on the curve. Given that \(\theta _ { 2 } > \theta _ { 1 }\), show that the area of the region enclosed by the curve and the lines \(O P\) and \(O Q\) can be expressed as \(k \left( r _ { 1 } ^ { 2 } - r _ { 2 } ^ { 2 } \right)\), where \(k\) is a constant to be found.

1. Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x \equiv 1$$ Deduce that \(1 - \tanh ^ { 2 } x \equiv \operatorname { sech } ^ { 2 } x\).
2. Solve the equation \(2 \tanh ^ { 2 } x - \operatorname { sech } x = 1\), giving your answer(s) in logarithmic form.
3. Express \(\frac { 4 } { ( 1 - x ) ( 1 + x ) \left( 1 + x ^ { 2 } \right) }\) in partial fractions.
4. Show that \(\int _ { 0 } ^ { \frac { 1 } { \sqrt { 3 } } } \frac { 4 } { 1 - x ^ { 4 } } \mathrm {~d} x = \ln \left( \frac { \sqrt { 3 } + 1 } { \sqrt { 3 } - 1 } \right) + \frac { 1 } { 3 } \pi\).

7
\includegraphics[max width=\textwidth, alt={}, center]{63afce50-e15f-4634-b2f1-ad5d78ab8bf5-3_591_1131_986_507} The diagram shows the curve with equation \(y = \sqrt [ 3 ] { x }\), together with a set of \(n\) rectangles of unit width.

1. By considering the areas of these rectangles, explain why $$\sqrt [ 3 ] { 1 } + \sqrt [ 3 ] { 2 } + \sqrt [ 3 ] { 3 } + \ldots + \sqrt [ 3 ] { n } > \int _ { 0 } ^ { n } \sqrt [ 3 ] { x } \mathrm {~d} x$$
2. By drawing another set of rectangles and considering their areas, show that $$\sqrt [ 3 ] { 1 } + \sqrt [ 3 ] { 2 } + \sqrt [ 3 ] { 3 } + \ldots + \sqrt [ 3 ] { n } < \int _ { 1 } ^ { n + 1 } \sqrt [ 3 ] { x } \mathrm {~d} x$$
3. Hence find an approximation to \(\sum _ { n = 1 } ^ { 100 } \sqrt [ 3 ] { n }\), giving your answer correct to 2 significant figures.

8 The equation of a curve is $$y = \frac { k x } { ( x - 1 ) ^ { 2 } } ,$$ where \(k\) is a positive constant.

1. Write down the equations of the asymptotes of the curve.
2. Show that \(y \geqslant - \frac { 1 } { 4 } k\).
3. Show that the \(x\)-coordinate of the stationary point of the curve is independent of \(k\), and sketch the curve.

9

1. Given that \(y = \tanh ^ { - 1 } x\), for \(- 1 < x < 1\), prove that \(y = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\).
2. It is given that \(\mathrm { f } ( x ) = a \cosh x - b \sinh x\), where \(a\) and \(b\) are positive constants.
   (a) Given that \(b \geqslant a\), show that the curve with equation \(y = \mathrm { f } ( x )\) has no stationary points.
   (b) In the case where \(a > 1\) and \(b = 1\), show that \(\mathrm { f } ( x )\) has a minimum value of \(\sqrt { a ^ { 2 } - 1 }\).