OCR MEI FP2 (Further Pure Mathematics 2) 2013 January

1

1. 1. Differentiate with respect to \(x\) the equation \(a \tan y = x\) (where \(a\) is a constant), and hence show that the derivative of \(\arctan \frac { x } { a }\) is \(\frac { a } { a ^ { 2 } + x ^ { 2 } }\).
   2. By first expressing \(x ^ { 2 } - 4 x + 8\) in completed square form, evaluate the integral \(\int _ { 0 } ^ { 4 } \frac { 1 } { x ^ { 2 } - 4 x + 8 } \mathrm {~d} x\), giving your answer exactly.
   3. Use integration by parts to find \(\int \arctan x \mathrm {~d} x\).
2. 1. A curve has polar equation \(r = 2 \cos \theta\), for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Show, by considering its cartesian equation, that the curve is a circle. State the centre and radius of the circle.
   2. Another circle has radius 2 and its centre, in cartesian coordinates, is ( 0,2 ). Find the polar equation of this circle.

1. 1. Show that $$1 + \mathrm { e } ^ { \mathrm { j } 2 \theta } = 2 \cos \theta ( \cos \theta + \mathrm { j } \sin \theta )$$
   2. The series \(C\) and \(S\) are defined as follows. $$\begin{aligned} & C = 1 + \binom { n } { 1 } \cos 2 \theta + \binom { n } { 2 } \cos 4 \theta + \ldots + \cos 2 n \theta
      & S = \binom { n } { 1 } \sin 2 \theta + \binom { n } { 2 } \sin 4 \theta + \ldots + \sin 2 n \theta \end{aligned}$$ By considering \(C + \mathrm { j } S\), show that $$C = 2 ^ { n } \cos ^ { n } \theta \cos n \theta$$ and find a corresponding expression for \(S\).
2. 1. Express \(\mathrm { e } ^ { \mathrm { j } 2 \pi / 3 }\) in the form \(x + \mathrm { j } y\), where the real numbers \(x\) and \(y\) should be given exactly.
   2. An equilateral triangle in the Argand diagram has its centre at the origin. One vertex of the triangle is at the point representing \(2 + 4 \mathrm { j }\). Obtain the complex numbers representing the other two vertices, giving your answers in the form \(x + \mathrm { j } y\), where the real numbers \(x\) and \(y\) should be given exactly.
   3. Show that the length of a side of the triangle is \(2 \sqrt { 15 }\).

3 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r r } 1 & 3 & 0
3 & - 2 & - 1
0 & - 1 & 1 \end{array} \right)\).

1. Show that the characteristic equation of \(\mathbf { M }\) is $$\lambda ^ { 3 } - 13 \lambda + 12 = 0 .$$
2. Find the eigenvalues and corresponding eigenvectors of \(\mathbf { M }\).
3. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { M } ^ { n } = \mathbf { P D P } ^ { - 1 } .$$ (You are not required to calculate \(\mathbf { P } ^ { - 1 }\).)

4

1. Show that the curve with equation $$y = 3 \sinh x - 2 \cosh x$$ has no turning points.
   Show that the curve crosses the \(x\)-axis at \(x = \frac { 1 } { 2 } \ln 5\). Show that this is also the point at which the gradient of the curve has a stationary value.
2. Sketch the curve.
3. Express \(( 3 \sinh x - 2 \cosh x ) ^ { 2 }\) in terms of \(\sinh 2 x\) and \(\cosh 2 x\). Hence or otherwise, show that the volume of the solid of revolution formed by rotating the region bounded by the curve and the axes through \(360 ^ { \circ }\) about the \(x\)-axis is $$\pi \left( 3 - \frac { 5 } { 4 } \ln 5 \right) .$$ Option 2: Investigation of curves \section*{This question requires the use of a graphical calculator.}

5 This question concerns the curves with polar equation $$r = \sec \theta + a \cos \theta ,$$ where \(a\) is a constant which may take any real value, and \(0 \leqslant \theta \leqslant 2 \pi\).

1. On a single diagram, sketch the curves for \(a = 0 , a = 1 , a = 2\).
2. On a single diagram, sketch the curves for \(a = 0 , a = - 1 , a = - 2\).
3. Identify a feature that the curves for \(a = 1 , a = 2 , a = - 1 , a = - 2\) share.
4. Name a distinctive feature of the curve for \(a = - 1\), and a different distinctive feature of the curve for \(a = - 2\).
5. Show that, in cartesian coordinates, equation (*) may be written $$y ^ { 2 } = \frac { a x ^ { 2 } } { x - 1 } - x ^ { 2 }$$ Hence comment further on the feature you identified in part (iii).
6. Show algebraically that, when \(a > 0\), the curve exists for \(1 < x < 1 + a\). Find the set of values of \(x\) for which the curve exists when \(a < 0\).