OCR GCE AS Level Further Maths B (MEI) Core Pure November 2021 question paper Y410

The equation 3x² - 4x + 2 = 0 has roots α and β.

Find an equation with integer coefficients whose roots are 3 - 2 α and 3 - 2 β.

Three planes have the following equations.

2 x - 3 y + z =- 3, x - 4 y + 2 z = 1, - 3 x - 2 y + 3 z = 14.

Write the system of equations in matrix form.

Hence find the point of intersection of the planes.

In this question you must show detailed reasoning.

Find the acute angle between the planes 2 x - 3 y + z =- 3 and x - 4 y + 2 z = 1.

Anika thinks that, for two square matrices A and B, the inverse of AB is A^-1B^-1. Her attempted proof of this is as follows.

(AB)(A^-1B^-1) = A(BA^-1)B^-1

= I × I = I Hence (AB)^-1 = A^-1B^-1

Explain the error in Anika’s working.

State the correct inverse of the matrix AB and amend Anika’s working to prove this.

Show that T reverses orientation.

State, in terms of λ, the area scale factor of T.

Show that M² - λ²I = 0.

Hence specify the transformation equivalent to two applications of T.

In the case where λ = 1, T is equivalent to a transformation S followed by a reflection in the x-axis.

Determine the matrix associated with S.

Hence describe the transformation S.

Find the modulus and argument of z₁, where z₁ = 1 + i.

Given that |z | = 2 and arg(z) = , express z in a + bi form, where a and b are exact

In this question you must show detailed reasoning.

• the roots of the equation,

• the value of k.

On a single Argand diagram, sketch the loci defined by

In this question you must show detailed reasoning.

The point of intersection of the two loci in part (a) represents the complex number w.

Find w, giving your answer in exact form.