OCR Further Pure Core 2 (Further Pure Core 2) 2017 Specimen

2 In this question you must show detailed reasoning. The finite region \(R\) is enclosed by the curve with equation \(y = \frac { 8 } { \sqrt { 16 + x ^ { 2 } } }\), the \(x\)-axis and the lines \(x = 0\) and \(x = 4\). Region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the exact value of the volume generated. [4]

3

1. Find \(\sum _ { r = 1 } ^ { n } \left( \frac { 1 } { r } - \frac { 1 } { r + 2 } \right)\).
2. What does the sum in part (i) tend to as \(n \rightarrow \infty\) ? Justify your answer.

4 Express \(\frac { 5 x ^ { 2 } + x + 12 } { x ^ { 3 } + 4 x }\) in partial fractions.

5 In this question you must show detailed reasoning. Evaluate \(\int _ { 0 } ^ { \infty } 2 x \mathrm { e } ^ { - x } \mathrm {~d} x\).
[0pt] [You may use the result \(\lim _ { x \rightarrow \infty } x \mathrm { e } ^ { - x } = 0\).]

6 The equation of a plane \(\Pi\) is \(x - 2 y - z = 30\).

1. Find the acute angle between the line \(\mathbf { r } = \left( \begin{array} { c } 3
   2
   - 5 \end{array} \right) + \lambda \left( \begin{array} { r } - 5
   3
   2 \end{array} \right)\) and \(\Pi\).
2. Determine the geometrical relationship between the line \(\mathbf { r } = \left( \begin{array} { l } 1
   4
   2 \end{array} \right) + \mu \left( \begin{array} { r } 3
   - 1
   5 \end{array} \right)\) and \(\Pi\).

7

1. Use the Maclaurin series for \(\sin x\) to work out the series expansion of \(\sin x \sin 2 x \sin 4 x\) up to and including the term in \(x ^ { 3 }\).
2. Hence find, in exact surd form, an approximation to the least positive root of the equation \(2 \sin x \sin 2 x \sin 4 x = x\).