OCR MEI FP2 (Further Pure Mathematics 2) 2009 June

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. [4]
   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. [4]
2. A curve has polar equation \(r = \frac{a}{1 + \sin \theta}\) for \(0 \leq \theta \leq \pi\), 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. [3]
   2. Show that, for this curve, \(r + y = a\) and hence find the cartesian equation of the curve. [5]

1. Obtain the characteristic equation for the matrix \(\mathbf{M}\) where $$\mathbf{M} = \begin{pmatrix} 3 & 1 & -2 \\ 6 & -1 & 0 \\ 2 & 0 & 1 \end{pmatrix}.$$ Hence or otherwise obtain the value of \(\det(\mathbf{M})\). [3]
2. 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. [9] $$3x + y - 2z = -0.1$$ $$-y = 0.6$$ $$2x + z = 0.1$$
3. 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 \(\mathbf{I}\). [4]
4. 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 \(-1 \leq x \leq 1\). [1] Find \(\frac{dy}{dx}\), justifying the sign of your answer by reference to your sketch. [4]
   2. Find the exact value of the integral \(\int_0^1 \frac{1}{\sqrt{2 - x^2}} dx\). [3]
2. The infinite series \(C\) and \(S\) are defined as follows. $$C = \cos \theta + \frac{1}{3}\cos 3\theta + \frac{1}{5}\cos 5\theta + \ldots$$ $$S = \sin \theta + \frac{1}{3}\sin 3\theta + \frac{1}{5}\sin 5\theta + \ldots$$ By considering \(C + jS\), show that $$C = \frac{3\cos \theta}{5 - 3\cos 2\theta},$$ and find a similar expression for \(S\). [11]

1. Prove, from definitions involving exponentials, that $$\cosh 2u = 2\cosh^2 u - 1.$$ [3]
2. Prove that \(\arsinh y = \ln\left(y + \sqrt{y^2 + 1}\right)\). [4]
3. Use the substitution \(x = 2\sinh u\) to show that $$\int \sqrt{x^2 + 4} dx = 2\arsinh \frac{x}{2} + \frac{x}{2}\sqrt{x^2 + 4} + c,$$ where \(c\) is an arbitrary constant. [6]
4. By first expressing \(t^2 + 2t + 5\) in completed square form, show that $$\int_{-1}^1 \sqrt{t^2 + 2t + 5} dt = 2\left(\ln(1 + \sqrt{2}) + \sqrt{2}\right).$$ [5]

Fig. 5 shows a circle with centre C \((a, 0)\) and radius \(a\). B is the point \((0, 1)\). The line BC intersects the circle at P and Q. P is above the \(x\)-axis and Q is below. \includegraphics{figure_5}

1. Show that, in the case \(a = 1\), P has coordinates \(\left(1 - \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\). Write down the coordinates of Q. [3]
2. Show that, for all positive values of \(a\), the coordinates of P are $$x = a\left(1 - \frac{a}{\sqrt{a^2 + 1}}\right), \quad y = \frac{a}{\sqrt{a^2 + 1}} \quad (*)$$ Write down the coordinates of Q in a similar form. [4] 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 \(a \to \infty\) and as \(a \to -\infty\). 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. [8] (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? [3]