OCR MEI FP2 (Further Pure Mathematics 2) 2009 June

1

1. 1. Use the Maclaurin series for \(\ln ( 1 + x )\) and \(\ln ( 1 - x )\) to obtain the first three non-zero terms in the Maclaurin series for \(\ln \left( \frac { 1 + x } { 1 - x } \right)\). State the range of validity of this series.
   2. Find the value of \(x\) for which \(\frac { 1 + x } { 1 - x } = 3\). Hence find an approximation to \(\ln 3\), giving your answer to three decimal places.
2. A curve has polar equation \(r = \frac { a } { 1 + \sin \theta }\) for \(0 \leqslant \theta \leqslant \pi\), where \(a\) is a positive constant. The points on the curve have cartesian coordinates \(x\) and \(y\).
   1. By plotting suitable points, or otherwise, sketch the curve.
   2. Show that, for this curve, \(r + y = a\) and hence find the cartesian equation of the curve.
   3. Obtain the characteristic equation for the matrix \(\mathbf { M }\) where $$\mathbf { M } = \left( \begin{array} { r r r } 3 & 1 & - 2
      0 & - 1 & 0
      2 & 0 & 1 \end{array} \right)$$ Hence or otherwise obtain the value of \(\operatorname { det } ( \mathbf { M } )\).
   4. Show that - 1 is an eigenvalue of \(\mathbf { M }\), and show that the other two eigenvalues are not real. Find an eigenvector corresponding to the eigenvalue - 1 .
      Hence or otherwise write down the solution to the following system of equations. $$\begin{aligned} 3 x + y - 2 z & = - 0.1
      - y & = 0.6
      2 x + z & = 0.1 \end{aligned}$$
   5. State the Cayley-Hamilton theorem and use it to show that $$\mathbf { M } ^ { 3 } = 3 \mathbf { M } ^ { 2 } - 3 \mathbf { M } - 7 \mathbf { I }$$ Obtain an expression for \(\mathbf { M } ^ { - 1 }\) in terms of \(\mathbf { M } ^ { 2 } , \mathbf { M }\) and \(\mathbf { I }\).
   6. Find the numerical values of the elements of \(\mathbf { M } ^ { - 1 }\), showing your working.

3

1. 1. Sketch the graph of \(y = \arcsin x\) for \(- 1 \leqslant x \leqslant 1\). Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), justifying the sign of your answer by reference to your sketch.
   2. Find the exact value of the integral \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 2 - x ^ { 2 } } } \mathrm {~d} x\).
2. The infinite series \(C\) and \(S\) are defined as follows. $$\begin{gathered} C = \cos \theta + \frac { 1 } { 3 } \cos 3 \theta + \frac { 1 } { 9 } \cos 5 \theta + \ldots
   S = \sin \theta + \frac { 1 } { 3 } \sin 3 \theta + \frac { 1 } { 9 } \sin 5 \theta + \ldots \end{gathered}$$ By considering \(C + \mathrm { j } S\), show that $$C = \frac { 3 \cos \theta } { 5 - 3 \cos 2 \theta }$$ and find a similar expression for \(S\). Section B (18 marks)

4

1. Prove, from definitions involving exponentials, that $$\cosh 2 u = 2 \cosh ^ { 2 } u - 1$$
2. Prove that \(\operatorname { arsinh } y = \ln \left( y + \sqrt { y ^ { 2 } + 1 } \right)\).
3. Use the substitution \(x = 2 \sinh u\) to show that $$\int \sqrt { x ^ { 2 } + 4 } \mathrm {~d} x = 2 \operatorname { arsinh } \frac { 1 } { 2 } x + \frac { 1 } { 2 } x \sqrt { x ^ { 2 } + 4 } + c$$ where \(c\) is an arbitrary constant.
4. By first expressing \(t ^ { 2 } + 2 t + 5\) in completed square form, show that $$\int _ { - 1 } ^ { 1 } \sqrt { t ^ { 2 } + 2 t + 5 } \mathrm {~d} t = 2 ( \ln ( 1 + \sqrt { 2 } ) + \sqrt { 2 } )$$ \section*{[Question 5 is printed overleaf.]}

5 Fig. 5 shows a circle with centre \(\mathrm { C } ( a , 0 )\) and radius \(a\). B is the point \(( 0,1 )\). The line BC intersects the circle at P and \(\mathrm { Q } ; \mathrm { P }\) is above the \(x\)-axis and Q is below. \begin{figure}[h] \end{figure}

1. Show that, in the case \(a = 1 , \mathrm { P }\) has coordinates \(\left( 1 - \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } \right)\). Write down the coordinates of Q .
2. Show that, for all positive values of \(a\), the coordinates of P are $$x = a \left( 1 - \frac { a } { \sqrt { a ^ { 2 } + 1 } } \right) , \quad y = \frac { a } { \sqrt { a ^ { 2 } + 1 } } .$$ Write down the coordinates of Q in a similar form. Now let the variable point P be defined by the parametric equations \(( * )\) for all values of the parameter \(a\), positive, zero and negative. Let Q be defined for all \(a\) by your answer in part (ii).
3. Using your calculator, sketch the locus of P as \(a\) varies. State what happens to P as \(a \rightarrow \infty\) and as \(a \rightarrow - \infty\). Show algebraically that this locus has an asymptote at \(y = - 1\).
   On the same axes, sketch, as a dotted line, the locus of Q as \(a\) varies.
   (The single curve made up of these two loci and including the point B is called a right strophoid.)
4. State, with a reason, the size of the angle POQ in Fig. 5. What does this indicate about the angle at which a right strophoid crosses itself? \section*{OCR
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