OCR MEI FP2 (Further Pure Mathematics 2) 2011 June

1

1. A curve has polar equation \(r = a ( 1 - \sin \theta )\), where \(a > 0\) and \(0 \leqslant \theta < 2 \pi\).
   1. Sketch the curve.
   2. Find, in an exact form, the area of the region enclosed by the curve.
2. 1. Find, in an exact form, the value of the integral \(\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 1 } { 1 + 4 x ^ { 2 } } \mathrm {~d} x\).
   2. Find, in an exact form, the value of the integral \(\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 1 } { \left( 1 + 4 x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x\).

2

1. Use de Moivre's theorem to find expressions for \(\sin 5 \theta\) and \(\cos 5 \theta\) in terms of \(\sin \theta\) and \(\cos \theta\).
   Hence show that, if \(t = \tan \theta\), then $$\tan 5 \theta = \frac { t \left( t ^ { 4 } - 10 t ^ { 2 } + 5 \right) } { 5 t ^ { 4 } - 10 t ^ { 2 } + 1 }$$
2. 1. Find the 5th roots of \(- 4 \sqrt { 2 }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\). These 5th roots are represented in the Argand diagram, in order of increasing \(\theta\), by the points A , \(\mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\).
   2. Draw the Argand diagram, making clear which point is which. The mid-point of AB is the point P which represents the complex number \(w\).
   3. Find, in exact form, the modulus and argument of \(w\).
   4. \(w\) is an \(n\)th root of a real number \(a\), where \(n\) is a positive integer. State the least possible value of \(n\) and find the corresponding value of \(a\).

3

1. Find the value of \(k\) for which the matrix $$\mathbf { M } = \left( \begin{array} { r r r } 1 & - 1 & k
   5 & 4 & 6
   3 & 2 & 4 \end{array} \right)$$ does not have an inverse.
   Assuming that \(k\) does not take this value, find the inverse of \(\mathbf { M }\) in terms of \(k\).
2. In the case \(k = 3\), evaluate $$\mathbf { M } \left( \begin{array} { r } - 3
   3
   1 \end{array} \right)$$
3. State the significance of what you have found in part (ii).
4. Find the value of \(t\) for which the system of equations $$\begin{array} { r } x - y + 3 z = t
   5 x + 4 y + 6 z = 1
   3 x + 2 y + 4 z = 0 \end{array}$$ has solutions. Find the general solution in this case and describe the solution geometrically.

4

1. Given that \(\cosh y = x\), show that \(y = \pm \ln \left( x + \sqrt { x ^ { 2 } - 1 } \right)\) and that \(\operatorname { arcosh } x = \ln \left( x + \sqrt { x ^ { 2 } - 1 } \right)\).
2. Find \(\int _ { \frac { 4 } { 5 } } ^ { 1 } \frac { 1 } { \sqrt { 25 x ^ { 2 } - 16 } } \mathrm {~d} x\), expressing your answer in an exact logarithmic form.
3. Solve the equation $$5 \cosh x - \cosh 2 x = 3$$ giving your answers in an exact logarithmic form.

5 In this question, you are required to investigate the curve with equation $$y = x ^ { m } ( 1 - x ) ^ { n } , \quad 0 \leqslant x \leqslant 1 ,$$ for various positive values of \(m\) and \(n\).

1. On separate diagrams, sketch the curve in each of the following cases.
   (A) \(m = 1 , n = 1\),
   (B) \(m = 2 , n = 2\),
   (C) \(m = 2 , n = 4\),
   (D) \(m = 4 , n = 2\).
2. What feature does the curve have when \(m = n\) ? What is the effect on the curve of interchanging \(m\) and \(n\) when \(m \neq n\) ?
3. Describe how the \(x\)-coordinate of the maximum on the curve varies as \(m\) and \(n\) vary. Use calculus to determine the \(x\)-coordinate of the maximum.
4. Find the condition on \(m\) for the gradient to be zero when \(x = 0\). State a corresponding result for the gradient to be zero when \(x = 1\).
5. Use your calculator to investigate the shape of the curve for large values of \(m\) and \(n\). Hence conjecture what happens to the value of the integral \(\int _ { 0 } ^ { 1 } x ^ { m } ( 1 - x ) ^ { n } \mathrm {~d} x\) as \(m\) and \(n\) tend to infinity.
6. Use your calculator to investigate the shape of the curve for small values of \(m\) and \(n\). Hence conjecture what happens to the shape of the curve as \(m\) and \(n\) tend to zero. OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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