OCR Further Pure Core 2 (Further Pure Core 2) 2021 June

2 A 2-D transformation \(T\) is a shear which leaves the \(y\)-axis invariant and which transforms the object point \(( 2,1 )\) to the image point \(( 2,9 )\). \(A\) is the matrix which represents the transformation \(T\).

1. Find A .
2. By considering the determinant of A , explain why the area of a shape is invariant under T .

3 A particle of mass 2 kg moves along the \(x\)-axis. At time \(t\) seconds the velocity of the particle is \(v \mathrm {~ms} ^ { - 1 }\). The particle is subject to two forces.

• One acts in the positive \(x\)-direction with magnitude \(\frac { 1 } { 2 } t \mathrm {~N}\).
• One acts in the negative \(x\)-direction with magnitude \(v \mathrm {~N}\).
  1. Show that the motion of the particle can be modelled by the differential equation

$$\frac { \mathrm { d } v } { \mathrm {~d} t } + \frac { 1 } { 2 } v = \frac { 1 } { 4 } t$$ The particle is at rest when \(t = 0\).

Find \(v\) in terms of \(t\).

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.

Refine the differential equation given in part (a) to model the motion for \(t \geqslant 2\).

Use the refined model from part (d) to find an exact expression for \(v\) in terms of \(t\) for \(t \geqslant 2\).

4 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. 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 \(\pi \approx 3 \sqrt { 3 \ln \left( \frac { 3 } { 2 } \right) }\).