OCR MEI FP2 (Further Pure Mathematics 2) 2010 June

1

1. 1. Given that \(\mathrm { f } ( t ) = \arcsin t\), write down an expression for \(\mathrm { f } ^ { \prime } ( t )\) and show that $$\mathrm { f } ^ { \prime \prime } ( t ) = \frac { t } { \left( 1 - t ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }$$
   2. Show that the Maclaurin expansion of the function \(\arcsin \left( x + \frac { 1 } { 2 } \right)\) begins $$\frac { \pi } { 6 } + \frac { 2 } { \sqrt { 3 } } x$$ and find the term in \(x ^ { 2 }\).
2. Sketch the curve with polar equation \(r = \frac { \pi a } { \pi + \theta }\), where \(a > 0\), for \(0 \leqslant \theta < 2 \pi\). Find, in terms of \(a\), the area of the region bounded by the part of the curve for which \(0 \leqslant \theta \leqslant \pi\) and the lines \(\theta = 0\) and \(\theta = \pi\).
3. Find the exact value of the integral $$\int _ { 0 } ^ { \frac { 3 } { 2 } } \frac { 1 } { 9 + 4 x ^ { 2 } } \mathrm {~d} x$$

2

1. Given that \(z = \cos \theta + \mathrm { j } \sin \theta\), express \(z ^ { n } + \frac { 1 } { z ^ { n } }\) and \(z ^ { n } - \frac { 1 } { z ^ { n } }\) in simplified trigonometric form.
   Hence find the constants \(A , B , C\) in the identity $$\sin ^ { 5 } \theta \equiv A \sin \theta + B \sin 3 \theta + C \sin 5 \theta$$
2. 1. Find the 4th roots of - 9 j in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), where \(r > 0\) and \(0 < \theta < 2 \pi\). Illustrate the roots on an Argand diagram.
   2. Let the points representing these roots, taken in order of increasing \(\theta\), be \(\mathrm { P } , \mathrm { Q } , \mathrm { R } , \mathrm { S }\). The mid-points of the sides of PQRS represent the 4th roots of a complex number \(w\). Find the modulus and argument of \(w\). Mark the point representing \(w\) on your Argand diagram.

3

1. 1. A \(3 \times 3\) matrix \(\mathbf { M }\) has characteristic equation $$2 \lambda ^ { 3 } + \lambda ^ { 2 } - 13 \lambda + 6 = 0$$ Show that \(\lambda = 2\) is an eigenvalue of \(\mathbf { M }\). Find the other eigenvalues.
   2. An eigenvector corresponding to \(\lambda = 2\) is \(\left( \begin{array} { r } 3
      - 3
      1 \end{array} \right)\). Evaluate \(\mathbf { M } \left( \begin{array} { r } 3
      - 3
      1 \end{array} \right)\) and \(\mathbf { M } ^ { 2 } \left( \begin{array} { r } 1
      - 1
      \frac { 1 } { 3 } \end{array} \right)\).
      Solve the equation \(\mathbf { M } \left( \begin{array} { l } x
      y
      z \end{array} \right) = \left( \begin{array} { r } 3
      - 3
      1 \end{array} \right)\).
   3. Find constants \(A , B , C\) such that $$\mathbf { M } ^ { 4 } = A \mathbf { M } ^ { 2 } + B \mathbf { M } + C \mathbf { I }$$
2. A \(2 \times 2\) matrix \(\mathbf { N }\) has eigenvalues -1 and 2, with eigenvectors \(\binom { 1 } { 2 }\) and \(\binom { - 1 } { 1 }\) respectively. Find \(\mathbf { N }\). Section B (18 marks)

4

1. Prove, using exponential functions, that $$\sinh 2 x = 2 \sinh x \cosh x$$ Differentiate this result to obtain a formula for \(\cosh 2 x\).
2. Sketch the curve with equation \(y = \cosh x - 1\). The region bounded by this curve, the \(x\)-axis, and the line \(x = 2\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find, correct to 3 decimal places, the volume generated. (You must show your working; numerical integration by calculator will receive no credit.)
3. Show that the curve with equation $$y = \cosh 2 x + \sinh x$$ has exactly one stationary point.
   Determine, in exact logarithmic form, the \(x\)-coordinate of the stationary point.

5 In parts (i), (ii), (iii) of this question you are required to investigate curves with the equation $$x ^ { k } + y ^ { k } = 1$$ for various positive values of \(k\).

1. Firstly consider cases in which \(k\) is a positive even integer.
   (A) State the shape of the curve when \(k = 2\).
   (B) Sketch, on the same axes, the curves for \(k = 2\) and \(k = 4\).
   (C) Describe the shape that the curve tends to as \(k\) becomes very large.
   (D) State the range of possible values of \(x\) and \(y\).
2. Now consider cases in which \(k\) is a positive odd integer.
   (A) Explain why \(x\) and \(y\) may take any value.
   (B) State the shape of the curve when \(k = 1\).
   (C) Sketch the curve for \(k = 3\). State the equation of the asymptote of this curve.
   (D) Sketch the shape that the curve tends to as \(k\) becomes very large.
3. Now let \(k = \frac { 1 } { 2 }\). Sketch the curve, indicating the range of possible values of \(x\) and \(y\).
4. Now consider the modified equation \(| x | ^ { k } + | y | ^ { k } = 1\).
   (A) Sketch the curve for \(k = \frac { 1 } { 2 }\).
   (B) Investigate the shape of the curve for \(k = \frac { 1 } { n }\) as the positive integer \(n\) becomes very large.