OCR GCE AS Level Further Maths B (MEI) Core Pure June 2023 question paper Y410

The transformation R of the plane is reflection in the line x = 0.

Write down the matrix M associated with R.

Find M².

Interpret the result of part (b) in terms of the transformation R.

In this question you must show detailed reasoning.

The equation x² - kx + 2k = 0, where k is a non-zero constant, has roots α and β. Find + in terms of k, simplifying your answer.

The Argand diagram below shows the points representing 1 and z, where | z | = 2.

Mark the points representing the following complex numbers on the copy of the diagram in the Printed Answer Booklet, labelling them clearly.

• z *

• 1 + z

• i z

Calculate M², M³ and M⁴.

Hence make a conjecture about the matrix Mⁿ.

Prove your conjecture.

In this question you must show detailed reasoning.

Express z in modulus-argument form.

Hence find z⁵ in modulus-argument form.

Use this result to verify your answers to part (a).

The equations of three planes are

2 x + y + 3 z = 3, 3 x - y - 2 z = 2, - 4 x + 3 y + 7 z = k,

where k is a constant.

By considering a suitable determinant, show that the planes do not meet at a single point.

Given that the planes form a sheaf, determine the value of k.

Show that, for all values of k, T has no invariant lines through the origin.

The plane P has normal vector 2 i + a j - k, where a is a positive constant, and the point (3, - 1, 1) lies in P. The plane x - z = 3 makes an angle of 45° with P.

Find the cartesian equation of P.