OCR GCE AS Level Further Maths B (MEI) Core Pure June 2019 question paper Y410

In this question you must show detailed reasoning.

The roots of the equation 3x² - x + 2 = 0 are α and β.

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

Hence find, in terms of the constant k, the point of intersection of the planes

x + 2 y + 3 z = 19, - x + y + 2 z = 4,

- 2 x + y + 2 z = k.

In this question you must show detailed reasoning.

Find the acute angle between the planes x + 2 y + 3 z = 19 and - x + y + 2 z = 4.

You are given that det M > 0 for all values of λ.

Find the range of possible values of k.

What is the significance of the condition det M > 0 for the transformation T?

For the remainder of this question, take k = - 2.

Determine whether there are any lines through the origin that are invariant lines for the

transformation T.

The transformation T is applied to a triangle with area 3 units². The area of the resulting image triangle is 15 units².

Find the possible values of λ.

Sketch on a single Argand diagram

the set of points for which | z - 1 - 3 i | = 3,

the set of points for which arg(z + 4) = .

Find, in exact form, the two values of z for which |z - 1 - 3i| = 3 and arg(z + 4) = .

In this question you must show detailed reasoning.

You are given that i is a root of the equation z⁴ - 2z³ + 3z² + az + b = 0, where a and b are real constants.

Show that a = - 2 and b = 2.

Find the other roots of this equation.