OCR MEI FP2 (Further Pure Mathematics 2) 2011 January

1

1. A curve has polar equation \(r = 2 ( \cos \theta + \sin \theta )\) for \(- \frac { 1 } { 4 } \pi \leqslant \theta \leqslant \frac { 3 } { 4 } \pi\).
   1. Show that a cartesian equation of the curve is \(x ^ { 2 } + y ^ { 2 } = 2 x + 2 y\). Hence or otherwise sketch the curve.
   2. Find, by integration, the area of the region bounded by the curve and the lines \(\theta = 0\) and \(\theta = \frac { 1 } { 2 } \pi\). Give your answer in terms of \(\pi\).
2. 1. Given that \(\mathrm { f } ( x ) = \arctan \left( \frac { 1 } { 2 } x \right)\), find \(\mathrm { f } ^ { \prime } ( x )\).
   2. Expand \(\mathrm { f } ^ { \prime } ( x )\) in ascending powers of \(x\) as far as the term in \(x ^ { 4 }\). Hence obtain an expression for \(\mathrm { f } ( x )\) in ascending powers of \(x\) as far as the term in \(x ^ { 5 }\).

2

1. 1. Given that \(z = \cos \theta + \mathrm { j } \sin \theta\), express \(z ^ { n } + z ^ { - n }\) and \(z ^ { n } - z ^ { - n }\) in simplified trigonometrical form.
   2. By considering \(\left( z + z ^ { - 1 } \right) ^ { 6 }\), show that $$\cos ^ { 6 } \theta = \frac { 1 } { 32 } ( \cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta + 10 )$$
   3. Obtain an expression for \(\cos ^ { 6 } \theta - \sin ^ { 6 } \theta\) in terms of \(\cos 2 \theta\) and \(\cos 6 \theta\).
2. The complex number \(w\) is \(8 \mathrm { e } ^ { \mathrm { j } \pi / 3 }\). You are given that \(z _ { 1 }\) is a square root of \(w\) and that \(z _ { 2 }\) is a cube root of \(w\). The points representing \(z _ { 1 }\) and \(z _ { 2 }\) in the Argand diagram both lie in the third quadrant.
   1. Find \(z _ { 1 }\) and \(z _ { 2 }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\). Draw an Argand diagram showing \(w , z _ { 1 }\) and \(z _ { 2 }\).
   2. Find the product \(z _ { 1 } z _ { 2 }\), and determine the quadrant of the Argand diagram in which it lies.
   3. Show that the characteristic equation of the matrix $$\mathbf { M } = \left( \begin{array} { r r r } 1 & - 4 & 5
      2 & 3 & - 2
      - 1 & 4 & 1 \end{array} \right)$$ is \(\lambda ^ { 3 } - 5 \lambda ^ { 2 } + 28 \lambda - 66 = 0\).
   4. Show that \(\lambda = 3\) is an eigenvalue of \(\mathbf { M }\), and determine whether or not \(\mathbf { M }\) has any other real eigenvalues.
   5. Find an eigenvector, \(\mathbf { v }\), of unit length corresponding to \(\lambda = 3\). State the magnitude of the vector \(\mathbf { M } ^ { n } \mathbf { v }\), where \(n\) is an integer.
   6. Using the Cayley-Hamilton theorem, obtain an equation for \(\mathbf { M } ^ { - 1 }\) in terms of \(\mathbf { M } ^ { 2 } , \mathbf { M }\) and \(\mathbf { I }\).

4

1. Solve the equation $$\sinh t + 7 \cosh t = 8$$ expressing your answer in exact logarithmic form. A curve has equation \(y = \cosh 2 x + 7 \sinh 2 x\).
2. Using part (i), or otherwise, find, in an exact form, the coordinates of the points on the curve at which the gradient is 16 . Show that there is no point on the curve at which the gradient is zero.
   Sketch the curve.
3. Find, in an exact form, the positive value of \(a\) for which the area of the region between the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = a\) is \(\frac { 1 } { 2 }\).

5 A curve has parametric equations $$x = t + a \sin t , \quad y = 1 - a \cos t$$ where \(a\) is a positive constant.

1. Draw, on separate diagrams, sketches of the curve for \(- 2 \pi < t < 2 \pi\) in the cases \(a = 1 , a = 2\) and \(a = 0.5\). By investigating other cases, state the value(s) of \(a\) for which the curve has
   (A) loops,
   (B) cusps.
2. Suppose that the point \(\mathrm { P } ( x , y )\) lies on the curve. Show that the point \(\mathrm { P } ^ { \prime } ( - x , y )\) also lies on the curve. What does this indicate about the symmetry of the curve?
3. Find an expression in terms of \(a\) and \(t\) for the gradient of the curve. Hence find, in terms of \(a\), the coordinates of the turning points on the curve for \(- 2 \pi < t < 2 \pi\) and \(a \neq 1\).
4. In the case \(a = \frac { 1 } { 2 } \pi\), show that \(t = \frac { 1 } { 2 } \pi\) and \(t = \frac { 3 } { 2 } \pi\) give the same point. Find the angle at which the curve crosses itself at this point.