OCR FP2 (Further Pure Mathematics 2)

1

1. Write down and simplify the first three non-zero terms of the Maclaurin series for \(\ln ( 1 + 3 x )\).
2. Hence find the first three non-zero terms of the Maclaurin series for $$\mathrm { e } ^ { x } \ln ( 1 + 3 x )$$ simplifying the coefficients.

2 Use the Newton-Raphson method to find the root of the equation \(\mathrm { e } ^ { - x } = x\) which is close to \(x = 0.5\). Give the root correct to 3 decimal places.

3 Express \(\frac { x + 6 } { x \left( x ^ { 2 } + 2 \right) }\) in partial fractions.

4 Answer the whole of this question on the insert provided. The sketch shows the curve with equation \(y = \mathrm { F } ( x )\) and the line \(y = x\). The equation \(x = \mathrm { F } ( x )\) has roots \(x = \alpha\) and \(x = \beta\) as shown.

1. Use the copy of the sketch on the insert to show how an iteration of the form \(x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)\), with starting value \(x _ { 1 }\) such that \(0 < x _ { 1 } < \alpha\) as shown, converges to the root \(x = \alpha\).
2. State what happens in the iteration in the following two cases.
   1. \(x _ { 1 }\) is chosen such that \(\alpha < x _ { 1 } < \beta\).
   2. \(x _ { 1 }\) is chosen such that \(x _ { 1 } > \beta\). \section*{Jan 2006} 4
   3. \includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-03_873_1259_274_484}
   4. (a) \(\_\_\_\_\) (b) \(\_\_\_\_\) \section*{Jan 2006}

5

1. Find the equations of the asymptotes of the curve with equation $$y = \frac { x ^ { 2 } + 3 x + 3 } { x + 2 }$$
2. Show that \(y\) cannot take values between - 3 and 1 .

6

1. It is given that, for non-negative integers \(n\), $$I _ { n } = \int _ { 0 } ^ { 1 } \mathrm { e } ^ { - x } x ^ { n } \mathrm {~d} x$$ Prove that, for \(n \geqslant 1\), $$I _ { n } = n I _ { n - 1 } - \mathrm { e } ^ { - 1 } .$$
2. Evaluate \(I _ { 3 }\), giving the answer in terms of e.

7 \includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-04_673_1285_1176_429} The diagram shows the curve with equation \(y = \sqrt { x }\). A set of \(N\) rectangles of unit width is drawn, starting at \(x = 1\) and ending at \(x = N + 1\), where \(N\) is an integer (see diagram).

1. By considering the areas of these rectangles, explain why $$\sqrt { 1 } + \sqrt { 2 } + \sqrt { 3 } + \ldots + \sqrt { N } < \int _ { 1 } ^ { N + 1 } \sqrt { x } \mathrm {~d} x$$
2. By considering the areas of another set of rectangles, explain why $$\sqrt { 1 } + \sqrt { 2 } + \sqrt { 3 } + \ldots + \sqrt { N } > \int _ { 0 } ^ { N } \sqrt { x } \mathrm {~d} x$$
3. Hence find, in terms of \(N\), limits between which \(\sum _ { r = 1 } ^ { N } \sqrt { r }\) lies. \section*{Jan 2006}

8 The equation of a curve, in polar coordinates, is $$r = 1 + \cos 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$

1. State the greatest value of \(r\) and the corresponding values of \(\theta\).
2. Find the equations of the tangents at the pole.
3. Find the exact area enclosed by the curve and the lines \(\theta = 0\) and \(\theta = \frac { 1 } { 2 } \pi\).
4. Find, in simplified form, the cartesian equation of the curve.

9

1. Using the definitions of \(\cosh x\) and \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), prove that $$\sinh 2 x = 2 \sinh x \cosh x$$
2. Show that the curve with equation $$y = \cosh 2 x - 6 \sinh x$$ has just one stationary point, and find its \(x\)-coordinate in logarithmic form. Determine the nature of the stationary point.