OCR MEI FP2 (Further Pure Mathematics 2) 2012 January

1

1. A curve has polar equation \(r = 1 + \cos \theta\) for \(0 \leqslant \theta < 2 \pi\).
   1. Sketch the curve.
   2. Find the area of the region enclosed by the curve, giving your answer in exact form.
2. Assuming that \(x ^ { 4 }\) and higher powers may be neglected, write down the Maclaurin series approximations for \(\sin x\) and \(\cos x\) (where \(x\) is in radians). Hence or otherwise obtain an approximation for \(\tan x\) in the form \(a x + b x ^ { 3 }\).
3. Find \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 1 - \frac { 1 } { 4 } X ^ { 2 } } } \mathrm {~d} x\), giving your answer in exact form.

2

1. The infinite series \(C\) and \(S\) are defined as follows. $$\begin{aligned} & C = 1 + a \cos \theta + a ^ { 2 } \cos 2 \theta + \ldots
   & S = a \sin \theta + a ^ { 2 } \sin 2 \theta + a ^ { 3 } \sin 3 \theta + \ldots \end{aligned}$$ where \(a\) is a real number and \(| a | < 1\).
   By considering \(C + \mathrm { j } S\), show that \(C = \frac { 1 - a \cos \theta } { 1 + a ^ { 2 } - 2 a \cos \theta }\) and find a corresponding expression for \(S\).
2. Express the complex number \(z = - 1 + \mathrm { j } \sqrt { 3 }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\). Find the 4th roots of \(z\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\).
   Show \(z\) and its 4th roots in an Argand diagram.
   Find the product of the 4th roots and mark this as a point on your Argand diagram.

3

1. Show that the characteristic equation of the matrix $$\mathbf { M } = \left( \begin{array} { r r r } 3 & - 1 & 2
   - 4 & 3 & 2
   2 & 1 & - 1 \end{array} \right)$$ is \(\lambda ^ { 3 } - 5 \lambda ^ { 2 } - 7 \lambda + 35 = 0\).
2. Show that \(\lambda = 5\) is an eigenvalue of \(\mathbf { M }\), and find its other eigenvalues.
3. Find an eigenvector, \(\mathbf { v }\), of unit length corresponding to \(\lambda = 5\). State the magnitudes and directions of the vectors \(\mathbf { M } ^ { 2 } \mathbf { v }\) and \(\mathbf { M } ^ { - 1 } \mathbf { v }\).
4. Use the Cayley-Hamilton theorem to find the constants \(a , b , c\) such that $$\mathbf { M } ^ { 4 } = a \mathbf { M } ^ { 2 } + b \mathbf { M } + c \mathbf { I } .$$ Section B (18 marks)

4

1. Define tanh \(t\) in terms of exponential functions. Sketch the graph of \(\tanh t\).
2. Show that \(\operatorname { artanh } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\). State the set of values of \(x\) for which this equation is valid.
3. Differentiate the equation \(\tanh y = x\) with respect to \(x\) and hence show that the derivative of \(\operatorname { artanh } x\) is \(\frac { 1 } { 1 - x ^ { 2 } }\). Show that this result may also be obtained by differentiating the equation in part (ii).
4. By considering \(\operatorname { artanh } x\) as \(1 \times \operatorname { artanh } x\) and using integration by parts, show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } } \operatorname { artanh } x \mathrm {~d} x = \frac { 1 } { 4 } \ln \frac { 27 } { 16 }$$

5 The points \(\mathrm { A } ( - 1,0 ) , \mathrm { B } ( 1,0 )\) and \(\mathrm { P } ( x , y )\) are such that the product of the distances PA and PB is 1 . You are given that the cartesian equation of the locus of P is $$\left( ( x + 1 ) ^ { 2 } + y ^ { 2 } \right) \left( ( x - 1 ) ^ { 2 } + y ^ { 2 } \right) = 1 .$$

1. Show that this equation may be written in polar form as $$r ^ { 4 } + 2 r ^ { 2 } = 4 r ^ { 2 } \cos ^ { 2 } \theta$$ Show that the polar equation simplifies to $$r ^ { 2 } = 2 \cos 2 \theta$$
2. Give a sketch of the curve, stating the values of \(\theta\) for which the curve is defined.
3. The equation in part (i) is now to be generalised to $$r ^ { 2 } = 2 \cos 2 \theta + k$$ where \(k\) is a constant.
   (A) Give sketches of the curve in the cases \(k = 1 , k = 2\). Describe how these two curves differ at the pole.
   (B) Give a sketch of the curve in the case \(k = 4\). What happens to the shape of the curve as \(k\) tends to infinity?
4. Sketch the curve for the case \(k = - 1\). What happens to the curve as \(k \rightarrow - 2\) ? \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}