OCR FP2 (Further Pure Mathematics 2) 2016 June

1

1. By first expanding \(\left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { 3 }\), or otherwise, show that \(\cosh 3 x \equiv 4 \cosh ^ { 3 } x - 3 \cosh x\).
2. Solve the equation \(\cosh 3 x = 6 \cosh x\), giving your answers in exact logarithmic form.

2 It is given that \(\mathrm { f } ( x ) = \frac { x ( x - 1 ) } { ( x + 1 ) \left( x ^ { 2 } + 1 \right) }\). Express \(\mathrm { f } ( x )\) in partial fractions and hence find the exact value of \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x\).

3 The diagram shows the curve \(y = \mathrm { f } ( x )\). Points \(A , B , C\) and \(D\) on the curve have coordinates ( \(- 1,0 ) , ( 2,0 )\), \(( 5,0 )\) and \(( 0,2 )\) respectively.
\includegraphics[max width=\textwidth, alt={}, center]{a31997f4-7890-42c1-9725-1b7058e8741f-2_593_1221_1041_406} On the copy of this diagram in the Printed Answer Book, sketch the curve \(y ^ { 2 } = \mathrm { f } ( x )\), giving the coordinates of the points where the curve crosses the axes.

4 You are given the equation \(( 2 x - 1 ) ^ { 2 } - \mathrm { e } ^ { x } = 0\).

1. Verify that 0 is a root of the equation. There are also two other roots, \(\alpha\) and \(\beta\), where \(0 < \alpha < \beta\).
2. The iterative formula \(x _ { r + 1 } = \ln \left( 2 x _ { r } - 1 \right) ^ { 2 }\) is to be used to find a root of the equation.
   (a) Sketch the line \(y = x\) and the curve \(y = \ln ( 2 x - 1 ) ^ { 2 }\) on the same axes, showing the roots \(0 , \alpha\) and \(\beta\).
   (b) By drawing a 'staircase' diagram on your sketch, starting with a value of \(x\) that is between \(\alpha\) and \(\beta\), show that this iteration does not converge to \(\alpha\).
   (c) Using this iterative formula with \(x _ { 1 } = 3.75\), find the value of \(\beta\) correct to 3 decimal places.
3. Using the Newton-Raphson method with \(x _ { 1 } = 1.6\), find the root \(\alpha\) of the equation \(( 2 x - 1 ) ^ { 2 } - \mathrm { e } ^ { x } = 0\) correct to 5 significant figures. Show the result of each iteration.

5 It is given that \(y = \tan ^ { - 1 } 2 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 x \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } = 0\).
2. Find the Maclaurin series for \(y\) up to and including the term in \(x ^ { 3 }\). Show all your working.
3. The result in part (ii), together with the value \(x = \frac { 1 } { 2 }\), is used to find an estimate for \(\pi\). Show that this estimate is only correct to 1 significant figure.

6 The equation of a curve in polar coordinates is \(r = \sin 5 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 5 } \pi\).

1. Sketch the curve and write down the equations of the tangents at the pole.
2. The line of symmetry meets the curve at the pole and at one other point \(A\). Find the equation of the line of symmetry and the cartesian coordinates of \(A\).
3. Find the area of the region enclosed by this curve.

7

1. By using a set of rectangles of unit width to approximate an area under the curve \(y = \frac { 1 } { x }\), show that \(\sum _ { x = 1 } ^ { \infty } \frac { 1 } { x }\) is infinite.
2. By using a set of rectangles of unit width to approximate an area under the curve \(y = \frac { 1 } { x ^ { 2 } }\), find an upper limit for the series \(\sum _ { x = 1 } ^ { \infty } \frac { 1 } { x ^ { 2 } }\).

8 It is given that \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sec ^ { n } x \mathrm {~d} x\) where \(n\) is a positive integer.

1. By writing \(\sec ^ { n } x = \sec ^ { n - 2 } x \sec ^ { 2 } x\), or otherwise, show that $$( n - 1 ) I _ { n } = ( \sqrt { 2 } ) ^ { n - 2 } + ( n - 2 ) I _ { n - 2 } \text { for } n > 1 .$$
2. Show that \(I _ { 8 } = \frac { 96 } { 35 }\).
3. Prove by induction that \(I _ { 2 n }\) is rational for all values of \(n > 1\). \section*{END OF QUESTION PAPER}