OCR Further Pure Core 2 (Further Pure Core 2) 2019 June

1 In this question you must show detailed reasoning.

1. By using partial fractions show that \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } + 3 r + 2 } = \frac { 1 } { 2 } - \frac { 1 } { n + 2 }\).
2. Hence determine the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 2 } + 3 r + 2 }\).

2

1. A plane \(\Pi\) has the equation \(\mathbf { r } \cdot \left( \begin{array} { r } 3
   6
   - 2 \end{array} \right) = 15 . C\) is the point \(( 4 , - 5,1 )\).
   Find the shortest distance between \(\Pi\) and \(C\).
2. Lines \(l _ { 1 }\) and \(l _ { 2 }\) have the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 4

3
1 \end{array} \right) + \lambda \left( \begin{array} { r } - 2
4
- 2 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l }

4
3
1 \end{array} \right) + \lambda \left( \begin{array} { r } - 2
4
- 2 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l }

5
2
4 \end{array} \right) + \mu \left( \begin{array} { r } 1
- 2
1 \end{array} \right) \end{aligned}$$ Find, in exact form, the distance between \(l _ { 1 }\) and \(l _ { 2 }\).

1. Show that the motion of the particle can be modelled by the differential equation \end{itemize} $$\frac { \mathrm { d } v } { \mathrm {~d} t } + \frac { 1 } { 2 } v = \frac { 1 } { 4 } t$$ The particle is at rest when \(t = 0\).
2. Find \(v\) in terms of \(t\).
3. Find the velocity of the particle when \(t = 2\). When \(t = 2\) the force acting in the positive \(x\)-direction is replaced by a constant force of magnitude \(\frac { 1 } { 2 } \mathrm {~N}\) in the same direction.
4. Refine the differential equation given in part (a) to model the motion for \(t \geqslant 2\).
5. Use the refined model from part (d) to find an exact expression for \(v\) in terms of \(t\) for \(t \geqslant 2\).
   \(6 \quad A\) is a fixed point on a smooth horizontal surface. A particle \(P\) is initially held at \(A\) and released from rest. It subsequently performs simple harmonic motion in a straight line on the surface. After its release it is next at rest after 0.2 seconds at point \(B\) whose displacement is 0.2 m from \(A\). The point \(M\) is halfway between \(A\) and \(B\). The displacement of \(P\) from \(M\) at time \(t\) seconds after release is denoted by \(x \mathrm {~m}\).
6. On the axes provided in the Printed Answer Booklet, sketch a graph of \(x\) against \(t\) for \(0 \leqslant t \leqslant 0.4\).
7. Find the displacement of \(P\) from \(M\) at 0.75 seconds after release.

7 In an Argand diagram the points representing the numbers \(2 + 3 \mathrm { i }\) and \(1 - \mathrm { i }\) are two adjacent vertices of a square, \(S\).

1. Find the area of \(S\).
2. Find all the possible pairs of numbers represented by the other two vertices of \(S\).

8 In this question you must show detailed reasoning.

1. By writing \(\sin \theta\) in terms of \(\mathrm { e } ^ { \mathrm { i } \theta }\) and \(\mathrm { e } ^ { - \mathrm { i } \theta }\) show that $$\sin ^ { 6 } \theta = \frac { 1 } { 32 } ( 10 - 15 \cos 2 \theta + 6 \cos 4 \theta - \cos 6 \theta ) .$$
2. Hence show that \(\sin \frac { 1 } { 8 } \pi = \frac { 1 } { 2 } \sqrt [ 6 ] { 20 - 14 \sqrt { 2 } }\).

1. Find the exact area enclosed by the curve.
2. Show that the greatest value of \(r\) on the curve is \(\sqrt { \frac { \sqrt { 3 } } { 2 } } \mathrm { e } ^ { \frac { 1 } { 6 } }\).

10

1. Use differentiation to find the first two non-zero terms of the Maclaurin expansion of \(\ln \left( \frac { 1 } { 2 } + \cos x \right)\).
2. By considering the root of the equation \(\ln \left( \frac { 1 } { 2 } + \cos x \right) = 0\) deduce that \(\pi \approx 3 \sqrt { 3 \ln \left( \frac { 3 } { 2 } \right) }\). \section*{END OF QUESTION PAPER}