OCR MEI FP2 (Further Pure Mathematics 2) 2007 January

1

1. A curve has polar equation \(r = a \mathrm { e } ^ { - k \theta }\) for \(0 \leqslant \theta \leqslant \pi\), where \(a\) and \(k\) are positive constants. The points A and B on the curve correspond to \(\theta = 0\) and \(\theta = \pi\) respectively.
   1. Sketch the curve.
   2. Find the area of the region enclosed by the curve and the line AB .
2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 } { 3 + 4 x ^ { 2 } } \mathrm {~d} x\).
3. 1. Find the Maclaurin series for \(\tan x\), up to the term in \(x ^ { 3 }\).
   2. Use this Maclaurin series to show that, when \(h\) is small, \(\int _ { h } ^ { 4 h } \frac { \tan x } { x } \mathrm {~d} x \approx 3 h + 7 h ^ { 3 }\).

2

1. You are given the complex numbers \(w = 3 \mathrm { e } ^ { - \frac { 1 } { 12 } \pi \mathrm { j } }\) and \(z = 1 - \sqrt { 3 } \mathrm { j }\).
   1. Find the modulus and argument of each of the complex numbers \(w , z\) and \(\frac { w } { z }\).
   2. Hence write \(\frac { w } { z }\) in the form \(a + b \mathrm { j }\), giving the exact values of \(a\) and \(b\).
2. In this part of the question, \(n\) is a positive integer and \(\theta\) is a real number with \(0 < \theta < \frac { \pi } { n }\).
   1. Express \(\mathrm { e } ^ { - \frac { 1 } { 2 } \mathrm { j } \theta } + \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { j } \theta }\) in simplified trigonometric form, and hence, or otherwise, show that $$1 + \mathrm { e } ^ { \mathrm { j } \theta } = 2 \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { j } \theta } \cos \frac { 1 } { 2 } \theta$$ Series \(C\) and \(S\) are defined by $$\begin{aligned} & C = 1 + \binom { n } { 1 } \cos \theta + \binom { n } { 2 } \cos 2 \theta + \binom { n } { 3 } \cos 3 \theta + \ldots + \binom { n } { n } \cos n \theta
      & S = \binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta + \binom { n } { 3 } \sin 3 \theta + \ldots + \binom { n } { n } \sin n \theta \end{aligned}$$
   2. Find \(C\) and \(S\), and show that \(\frac { S } { C } = \tan \frac { 1 } { 2 } n \theta\).

3 Let \(\mathbf { P } = \left( \begin{array} { r r r } 4 & 2 & k
1 & 1 & 3
1 & 0 & - 1 \end{array} \right) (\) where \(k \neq 4 )\) and \(\mathbf { M } = \left( \begin{array} { r r r } 2 & - 2 & - 6
- 1 & 3 & 1
1 & - 2 & - 2 \end{array} \right)\).

1. Find \(\mathbf { P } ^ { - 1 }\) in terms of \(k\), and show that, when \(k = 2 , \mathbf { P } ^ { - 1 } = \frac { 1 } { 2 } \left( \begin{array} { r r r } - 1 & 2 & 4
   4 & - 6 & - 10
   - 1 & 2 & 2 \end{array} \right)\).
2. Verify that \(\left( \begin{array} { l } 4
   1
   1 \end{array} \right) , \left( \begin{array} { l } 2
   1
   0 \end{array} \right)\) and \(\left( \begin{array} { r } 2
   3
   - 1 \end{array} \right)\) are eigenvectors of \(\mathbf { M }\), and find the corresponding eigenvalues.
3. Show that \(\mathbf { M } ^ { n } = \left( \begin{array} { r r r } 4 & - 6 & - 10
   2 & - 3 & - 5
   0 & 0 & 0 \end{array} \right) + 2 ^ { n - 1 } \left( \begin{array} { r r r } - 2 & 4 & 4
   - 3 & 6 & 6
   1 & - 2 & - 2 \end{array} \right)\). Section B (18 marks)

4

1. Show that \(\operatorname { arcosh } x = \ln \left( x + \sqrt { x ^ { 2 } - 1 } \right)\).
2. Find \(\int _ { 2.5 } ^ { 3.9 } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 9 } } \mathrm {~d} x\), giving your answer in the form \(a \ln b\), where \(a\) and \(b\) are rational numbers.
3. There are two points on the curve \(y = \frac { \cosh x } { 2 + \sinh x }\) at which the gradient is \(\frac { 1 } { 9 }\). Show that one of these points is \(\left( \ln ( 1 + \sqrt { 2 } ) , \frac { 1 } { 3 } \sqrt { 2 } \right)\), and find the coordinates of the other point, in a similar form.

5 Cartesian coordinates \(( x , y )\) and polar coordinates \(( r , \theta )\) are set up in the usual way, with the pole at the origin and the initial line along the positive \(x\)-axis, so that \(x = r \cos \theta\) and \(y = r \sin \theta\). A curve has polar equation \(r = k + \cos \theta\), where \(k\) is a constant with \(k \geqslant 1\).

1. Use your graphical calculator to obtain sketches of the curve in the three cases $$k = 1 , k = 1.5 \text { and } k = 4$$
2. Name the special feature which the curve has when \(k = 1\).
3. For each of the three cases, state the number of points on the curve at which the tangent is parallel to the \(y\)-axis.
4. Express \(x\) in terms of \(k\) and \(\theta\), and find \(\frac { \mathrm { d } x } { \mathrm {~d} \theta }\). Hence find the range of values of \(k\) for which there are just two points on the curve where the tangent is parallel to the \(y\)-axis. The distance between the point ( \(r , \theta\) ) on the curve and the point ( 1,0 ) on the \(x\)-axis is \(d\).
5. Use the cosine rule to express \(d ^ { 2 }\) in terms of \(k\) and \(\theta\), and deduce that \(k ^ { 2 } \leqslant d ^ { 2 } \leqslant k ^ { 2 } + 1\).
6. Hence show that, when \(k\) is large, the shape of the curve is very nearly circular.