OCR MEI FP2 (Further Pure Mathematics 2) 2009 January

1

1. 1. By considering the derivatives of \(\cos x\), show that the Maclaurin expansion of \(\cos x\) begins $$1 - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 24 } x ^ { 4 }$$
   2. The Maclaurin expansion of \(\sec x\) begins $$1 + a x ^ { 2 } + b x ^ { 4 }$$ where \(a\) and \(b\) are constants. Explain why, for sufficiently small \(x\), $$\left( 1 - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 24 } x ^ { 4 } \right) \left( 1 + a x ^ { 2 } + b x ^ { 4 } \right) \approx 1$$ Hence find the values of \(a\) and \(b\).
2. 1. Given that \(y = \arctan \left( \frac { x } { a } \right)\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { a } { a ^ { 2 } + x ^ { 2 } }\).
   2. Find the exact values of the following integrals. $$\begin{aligned} & \text { (A) } \int _ { - 2 } ^ { 2 } \frac { 1 } { 4 + x ^ { 2 } } \mathrm {~d} x
      & \text { (B) } \int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 4 } { 1 + 4 x ^ { 2 } } \mathrm {~d} x \end{aligned}$$

1. Write down the modulus and argument of the complex number \(\mathrm { e } ^ { \mathrm { j } \pi / 3 }\).
2. The triangle OAB in an Argand diagram is equilateral. O is the origin; A corresponds to the complex number \(a = \sqrt { 2 } ( 1 + \mathrm { j } ) ; \mathrm { B }\) corresponds to the complex number \(b\). Show A and the two possible positions for B in a sketch. Express \(a\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\). Find the two possibilities for \(b\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\).
3. Given that \(z _ { 1 } = \sqrt { 2 } \mathrm { e } ^ { \mathrm { j } \pi / 3 }\), show that \(z _ { 1 } ^ { 6 } = 8\). Write down, in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), the other five complex numbers \(z\) such that \(z ^ { 6 } = 8\). Sketch all six complex numbers in a new Argand diagram. Let \(w = z _ { 1 } \mathrm { e } ^ { - \mathrm { j } \pi / 12 }\).
4. Find \(w\) in the form \(x + \mathrm { j } y\), and mark this complex number on your Argand diagram.
5. Find \(w ^ { 6 }\), expressing your answer in as simple a form as possible.

3

1. A curve has polar equation \(r = a \tan \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\), where \(a\) is a positive constant.
   1. Sketch the curve.
   2. Find the area of the region between the curve and the line \(\theta = \frac { 1 } { 4 } \pi\). Indicate this region on your sketch.
2. 1. Find the eigenvalues and corresponding eigenvectors for the matrix \(\mathbf { M }\) where $$\mathbf { M } = \left( \begin{array} { l l } 0.2 & 0.8
      0.3 & 0.7 \end{array} \right)$$
   2. Give a matrix \(\mathbf { Q }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { M } = \mathbf { Q D } \mathbf { Q } ^ { - 1 }\). Section B (18 marks)

4

1. 1. Prove, from definitions involving exponentials, that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$$
   2. Given that \(\sinh x = \tan y\), where \(- \frac { 1 } { 2 } \pi < y < \frac { 1 } { 2 } \pi\), show that
      (A) \(\tanh x = \sin y\),
      (B) \(x = \ln ( \tan y + \sec y )\).
2. 1. Given that \(y = \operatorname { artanh } x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\). Hence show that \(\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 1 } { 1 - x ^ { 2 } } \mathrm {~d} x = 2 \operatorname { artanh } \frac { 1 } { 2 }\).
   2. Express \(\frac { 1 } { 1 - x ^ { 2 } }\) in partial fractions and hence find an expression for \(\int \frac { 1 } { 1 - x ^ { 2 } } \mathrm {~d} x\) in terms of logarithms.
   3. Use the results in parts (i) and (ii) to show that \(\operatorname { artanh } \frac { 1 } { 2 } = \frac { 1 } { 2 } \ln 3\).

5 The limaçon of Pascal has polar equation \(r = 1 + 2 a \cos \theta\), where \(a\) is a constant.

1. Use your calculator to sketch the curve when \(a = 1\). (You need not distinguish between parts of the curve where \(r\) is positive and negative.)
2. By using your calculator to investigate the shape of the curve for different values of \(a\), positive and negative,
   (A) state the set of values of \(a\) for which the curve has a loop within a loop,
   (B) state, with a reason, the shape of the curve when \(a = 0\),
   (C) state what happens to the shape of the curve as \(a \rightarrow \pm \infty\),
   (D) name the feature of the curve that is evident when \(a = 0.5\), and find another value of \(a\) for which the curve has this feature.
3. Given that \(a > 0\) and that \(a\) is such that the curve has a loop within a loop, write down an equation for the values of \(\theta\) at which \(r = 0\). Hence show that the angle at which the curve crosses itself is \(2 \arccos \left( \frac { 1 } { 2 a } \right)\). Obtain the cartesian equations of the tangents at the point where the curve crosses itself. Explain briefly how these equations relate to the answer to part (ii)(A).