OCR FP2 (Further Pure Mathematics 2) 2009 January

1. Write down and simplify the first three terms of the Maclaurin series for \(e^{2x}\). [2]
2. Hence show that the Maclaurin series for $$\ln(e^{2x} + e^{-2x})$$ begins \(\ln a + bx^2\), where \(a\) and \(b\) are constants to be found. [4]

It is given that \(\alpha\) is the only real root of the equation \(x^3 + 2x - 28 = 0\) and that \(1.8 < \alpha < 2\).

1. The iteration \(x_{n+1} = \sqrt[3]{28 - 2x_n}\), with \(x_1 = 1.9\), is to be used to find \(\alpha\). Find the values of \(x_2\), \(x_3\) and \(x_4\), giving the answers correct to 7 decimal places. [3]
2. The error \(e_n\) is defined by \(e_n = \alpha - x_n\). Given that \(\alpha = 1.891 574 9\), correct to 7 decimal places, evaluate \(\frac{e_3}{e_2}\) and \(\frac{e_4}{e_3}\). Comment on these values in relation to the gradient of the curve with equation \(y = \sqrt[3]{28 - 2x}\) at \(x = \alpha\). [3]

1. Prove that the derivative of \(\sin^{-1} x\) is \(\frac{1}{\sqrt{1-x^2}}\). [3]
2. Given that $$\sin^{-1} 2x + \sin^{-1} y = \frac{1}{2}\pi,$$ find the exact value of \(\frac{dy}{dx}\) when \(x = \frac{1}{4}\). [4]

1. By means of a suitable substitution, show that $$\int \frac{x^2}{\sqrt{x^2-1}} dx$$ can be transformed to \(\int \cosh^2 \theta \, d\theta\). [2]
2. Hence show that \(\int \frac{x^2}{\sqrt{x^2-1}} dx = \frac{1}{2}\sqrt{x^2-1} + \frac{1}{2}\cosh^{-1} x + c\). [4]

\includegraphics{figure_5} The diagram shows the curve with equation \(y = f(x)\), where $$f(x) = 2x^3 - 9x^2 + 12x - 4.36.$$ The curve has turning points at \(x = 1\) and \(x = 2\) and crosses the \(x\)-axis at \(x = \alpha\), \(x = \beta\) and \(x = \gamma\), where \(0 < \alpha < \beta < \gamma\).

1. The Newton-Raphson method is to be used to find the roots of the equation \(f(x) = 0\), with \(x_1 = k\).
   1. To which root, if any, would successive approximations converge in each of the cases \(k < 0\) and \(k = 1\)? [2]
   2. What happens if \(1 < k < 2\)? [2]
2. Sketch the curve with equation \(y^2 = f(x)\). State the coordinates of the points where the curve crosses the \(x\)-axis and the coordinates of any turning points. [4]

1. Using the definitions of \(\cosh x\) and \(\sinh x\) in terms of \(e^x\) and \(e^{-x}\), show that $$1 + 2\sinh^2 x = \cosh 2x.$$ [3]
2. Solve the equation $$\cosh 2x - 5\sinh x = 4,$$ giving your answers in logarithmic form. [5]

\includegraphics{figure_7} The diagram shows the curve with equation, in polar coordinates, $$r = 3 + 2\cos \theta, \quad \text{for } 0 \leq \theta < 2\pi.$$ The points \(P\), \(Q\), \(R\) and \(S\) on the curve are such that the straight lines \(POR\) and \(QOS\) are perpendicular, where \(O\) is the pole. The point \(P\) has polar coordinates \((r, \alpha)\).

1. Show that \(OP + OQ + OR + OS = k\), where \(k\) is a constant to be found. [3]
2. Given that \(\alpha = \frac{1}{4}\pi\), find the exact area bounded by the curve and the lines \(OP\) and \(OQ\) (shaded in the diagram). [5]

\includegraphics{figure_8} The diagram shows the curve with equation \(y = \frac{1}{x+1}\). A set of \(n\) rectangles of unit width is drawn, starting at \(x = 0\) and ending at \(x = n\), where \(n\) is an integer.

1. By considering the areas of these rectangles, explain why $$\frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n+1} < \ln(n+1).$$ [5]
2. By considering the areas of another set of rectangles, show that $$1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} > \ln(n+1).$$ [2]
3. Hence show that $$\ln(n+1) + \frac{1}{n+1} < \sum_{r=1}^{n+1} \frac{1}{r} < \ln(n+1) + 1.$$ [2]
4. State, with a reason, whether \(\sum_{r=1}^{\infty} \frac{1}{r}\) is convergent. [2]

A curve has equation $$y = \frac{4x - 3a}{2(x^2 + a^2)},$$ where \(a\) is a positive constant.

1. Explain why the curve has no asymptotes parallel to the \(y\)-axis. [2]
2. Find, in terms of \(a\), the set of values of \(y\) for which there are no points on the curve. [5]
3. Find the exact value of \(\int_a^{2a} \frac{4x - 3a}{2(x^2 + a^2)} dx\), showing that it is independent of \(a\). [5]