OCR MEI FP2 (Further Pure Mathematics 2) 2010 January

1

1. Given that \(y = \arctan \sqrt { x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer in terms of \(x\). Hence show that $$\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { x } ( x + 1 ) } \mathrm { d } x = \frac { \pi } { 2 }$$
2. A curve has cartesian equation $$x ^ { 2 } + y ^ { 2 } = x y + 1$$
   1. Show that the polar equation of the curve is $$r ^ { 2 } = \frac { 2 } { 2 - \sin 2 \theta }$$
   2. Determine the greatest and least positive values of \(r\) and the values of \(\theta\) between 0 and \(2 \pi\) for which they occur.
   3. Sketch the curve.

2

1. Use de Moivre's theorem to find the constants \(a , b , c\) in the identity $$\cos 5 \theta \equiv a \cos ^ { 5 } \theta + b \cos ^ { 3 } \theta + c \cos \theta$$
2. Let $$\begin{aligned} C & = \cos \theta + \cos \left( \theta + \frac { 2 \pi } { n } \right) + \cos \left( \theta + \frac { 4 \pi } { n } \right) + \ldots + \cos \left( \theta + \frac { ( 2 n - 2 ) \pi } { n } \right)
   \text { and } S & = \sin \theta + \sin \left( \theta + \frac { 2 \pi } { n } \right) + \sin \left( \theta + \frac { 4 \pi } { n } \right) + \ldots + \sin \left( \theta + \frac { ( 2 n - 2 ) \pi } { n } \right) \end{aligned}$$ where \(n\) is an integer greater than 1 .
   By considering \(C + \mathrm { j } S\), show that \(C = 0\) and \(S = 0\).
3. Write down the Maclaurin series for \(\mathrm { e } ^ { t }\) as far as the term in \(t ^ { 2 }\). Hence show that, for \(t\) close to zero, $$\frac { t } { \mathrm { e } ^ { t } - 1 } \approx 1 - \frac { 1 } { 2 } t$$

3

1. Find the inverse of the matrix $$\left( \begin{array} { r r r } 1 & 1 & a
   2 & - 1 & 2
   3 & - 2 & 2 \end{array} \right)$$ where \(a \neq 4\).
   Show that when \(a = - 1\) the inverse is $$\frac { 1 } { 5 } \left( \begin{array} { r r r } 2 & 0 & 1
   2 & 5 & - 4
   - 1 & 5 & - 3 \end{array} \right)$$
2. Solve, in terms of \(b\), the following system of equations. $$\begin{aligned} x + y - z & = - 2
   2 x - y + 2 z & = b
   3 x - 2 y + 2 z & = 1 \end{aligned}$$
3. Find the value of \(b\) for which the equations $$\begin{aligned} x + y + 4 z & = - 2
   2 x - y + 2 z & = b
   3 x - 2 y + 2 z & = 1 \end{aligned}$$ have solutions. Give a geometrical interpretation of the solutions in this case. Section B (18 marks)

4

1. Prove, using exponential functions, that $$\cosh 2 x = 1 + 2 \sinh ^ { 2 } x$$ Differentiate this result to obtain a formula for \(\sinh 2 x\).
2. Solve the equation $$2 \cosh 2 x + 3 \sinh x = 3$$ expressing your answers in exact logarithmic form.
3. Given that \(\cosh t = \frac { 5 } { 4 }\), show by using exponential functions that \(t = \pm \ln 2\). Find the exact value of the integral $$\int _ { 4 } ^ { 5 } \frac { 1 } { \sqrt { x ^ { 2 } - 16 } } \mathrm {~d} x$$

5 A line PQ is of length \(k\) (where \(k > 1\) ) and it passes through the point ( 1,0 ). PQ is inclined at angle \(\theta\) to the positive \(x\)-axis. The end Q moves along the \(y\)-axis. See Fig. 5. The end P traces out a locus. \begin{figure}[h] \end{figure}

1. Show that the locus of P may be expressed parametrically as follows. $$x = k \cos \theta \quad y = k \sin \theta - \tan \theta$$ You are now required to investigate curves with these parametric equations, where \(k\) may take any non-zero value and \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\).
2. Use your calculator to sketch the curve in each of the cases \(k = 2 , k = 1 , k = \frac { 1 } { 2 }\) and \(k = - 1\).
3. For what value(s) of \(k\) does the curve have
   (A) an asymptote (you should state what the asymptote is),
   (B) a cusp,
   (C) a loop?
4. For the case \(k = 2\), find the angle at which the curve crosses itself.
5. For the case \(k = 8\), find in an exact form the coordinates of the highest point on the loop.
6. Verify that the cartesian equation of the curve is $$y ^ { 2 } = \frac { ( x - 1 ) ^ { 2 } } { x ^ { 2 } } \left( k ^ { 2 } - x ^ { 2 } \right) .$$