OCR FP2 (Further Pure Mathematics 2) 2012 January

Given that \(f(x) = \ln(\cos 3x)\), find \(f'(0)\) and \(f''(0)\). Hence show that the first term in the Maclaurin series for \(f(x)\) is \(ax^2\), where the value of \(a\) is to be found. [4]

By first completing the square in the denominator, find the exact value of $$\int_{\frac{1}{2}}^{\frac{1}{2}} \frac{1}{4x^2 - 4x + 5} dx.$$ [5]

Express \(\frac{2x^3 + x + 12}{(2x - 1)(x^2 + 4)}\) in partial fractions. [7]

\includegraphics{figure_4} The diagram shows the curve \(y = e^{-\frac{1}{x}}\) for \(0 < x \leq 1\). A set of \((n - 1)\) rectangles is drawn under the curve as shown.

1. Explain why a lower bound for \(\int_0^1 e^{-\frac{1}{x}} dx\) can be expressed as $$\frac{1}{n}\left(e^{-n} + e^{-\frac{n}{2}} + e^{-\frac{n}{3}} + \ldots + e^{-\frac{n}{n-1}}\right).$$ [2]
2. Using a set of \(n\) rectangles, write down a similar expression for an upper bound for \(\int_0^1 e^{-\frac{1}{x}} dx\). [2]
3. Evaluate these bounds in the case \(n = 4\), giving your answers correct to 3 significant figures. [2]
4. When \(n > N\), the difference between the upper and lower bounds is less than 0.001. By expressing this difference in terms of \(n\), find the least possible value of \(N\). [3]

It is given that \(f(x) = x^3 - k\), where \(k > 0\), and that \(\alpha\) is the real root of the equation \(f(x) = 0\). Successive approximations to \(\alpha\), using the Newton-Raphson method, are denoted by \(x_1, x_2, \ldots, x_n, \ldots\).

1. Show that \(x_{n+1} = \frac{2x_n^3 + k}{3x_n^2}\). [2]
2. Sketch the graph of \(y = f(x)\), giving the coordinates of the intercepts with the axes. Show on your sketch how it is possible for \(|x_2 - x_1|\) to be greater than \(|x_1|\). [3]

It is now given that \(k = 100\) and \(x_1 = 5\).

1. Write down the exact value of \(\alpha\) and find \(x_2\) and \(x_3\) correct to 5 decimal places. [3]
2. The error \(e_n\) is defined by \(e_n = \alpha - x_n\). By finding \(e_1\), \(e_2\) and \(e_3\), verify that \(e_3 \approx \frac{e_2^2}{e_1}\). [3]

1. Prove that the derivative of \(\cos^{-1} x\) is \(-\frac{1}{\sqrt{1 - x^2}}\). [3]

A curve has equation \(y = \cos^{-1}(1 - x^2)\), for \(0 < x < \sqrt{2}\).

1. Find and simplify \(\frac{dy}{dx}\), and hence show that $$(2 - x^2)\frac{d^2y}{dx^2} = x\frac{dy}{dx}.$$ [5]

1. Given that \(y = \sinh^{-1} x\), prove that \(y = \ln\left(x + \sqrt{x^2 + 1}\right)\). [3]
2. It is given that \(x\) satisfies the equation \(\sinh^{-1} x - \cosh^{-1} x = \ln 2\). Use the logarithmic forms for \(\sinh^{-1} x\) and \(\cosh^{-1} x\) to show that $$\sqrt{x^2 + 1} - 2\sqrt{x^2 - 1} = x.$$ Hence, by squaring this equation, find the exact value of \(x\). [5]

\includegraphics{figure_8} The diagram shows two curves, \(C_1\) and \(C_2\), which intersect at the pole \(O\) and at the point \(P\). The polar equation of \(C_1\) is \(r = \sqrt{2}\cos\theta\) and the polar equation of \(C_2\) is \(r = \sqrt{2}\sin 2\theta\). For both curves, \(0 \leq \theta \leq \frac{1}{2}\pi\). The value of \(\theta\) at \(P\) is \(\alpha\).

1. Show that \(\tan\alpha = \frac{1}{2}\). [2]
2. Show that the area of the region common to \(C_1\) and \(C_2\), shaded in the diagram, is \(\frac{1}{4}\pi - \frac{1}{2}\alpha\). [7]

1. Show that \(\tanh(\ln n) = \frac{n^2 - 1}{n^2 + 1}\). [2]

It is given that, for non-negative integers \(n\), \(I_n = \int_0^{\ln 2} \tanh^n u du\).

1. Show that \(I_n - I_{n-2} = -\frac{1}{n-1}\left(\frac{3}{5}\right)^{n-1}\), for \(n \geq 2\). [3]
2. Find the value of \(I_3\), giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are constants. [4]
3. Use the method of differences on the result of part (ii) to find the sum of the infinite series $$\frac{1}{2}\left(\frac{3}{5}\right)^2 + \frac{1}{4}\left(\frac{3}{5}\right)^4 + \frac{1}{6}\left(\frac{3}{5}\right)^6 + \ldots.$$ [2]