OCR A Level Further Maths A Pure Core 2 June 2019 question paper

In this question you must show detailed reasoning.

In this question you must show detailed reasoning.

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.

Find A.

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

A particle of mass 2 kg moves along the x-axis. At time t seconds the velocity of the particle is v ms^-1.

The particle is subject to two forces.

• One acts in the positive x-direction with magnitude N.

• One acts in the negative x-direction with magnitude v N.

Show that the motion of the particle can be modelled by the differential equation

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

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

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

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 m.

Find the displacement of P from M at 0.75 seconds after release.

In an Argand diagram the points representing the numbers 2 + 3i and 1 - i are two adjacent vertices of a square, S.

Find the area of S.

Find all the possible pairs of numbers represented by the other two vertices of S.

In this question you must show detailed reasoning.

By writing sinθ in terms of e^iθ and e^-iθ show that

In this question you must show detailed reasoning.

Find the exact area enclosed by the curve.

Use differentiation to find the first two non-zero terms of the Maclaurin expansion of