OCR A Level Further Maths A Pure Core 1 June 2019 question paper

In this question you must show detailed reasoning.

The quadratic equation x² - 2x + 5 = 0 has roots α and β.

Write down the values of α + β and αβ.

Hence find a quadratic equation with roots α + and β +

Indicate by shading on an Argand diagram the region

{ z: | z | ⩽ | z - 4 | } ∩ { z: | z - 3 - 2 i | ⩽ 2 }.

In this question you must show detailed reasoning.

You are given that x = 2 + 5i is a root of the equation x³ - 2x² + 21x + 58 = 0.

Solve the equation.

The diagram shows part of the curve y = 5 cosh x + 3 sinh x.

Solve the equation 5 cosh x + 3 sinh x = 4 giving your solution in exact form.

The function sech x is defined by sech x =

The equation of a plane is 4 x + 2 y + z = 7. The point A has coordinates (9, 6, 1) and the point B is the reflection of A in the plane.

Find the coordinates of the point B.

In this question you must show detailed reasoning.

You are given the complex number ω = cos + i sin and the equation z⁵ = 1.

Show that ω is a root of the equation.

Write down the other four roots of the equation.

Show that ω + ω² + ω³ + ω⁴ = - 1.

Find, in terms of a, the determinant of A, simplifying your answer.

Hence find the values of a for which A is singular.

A particle is suspended in a resistive medium from one end of a light spring. The other end of the spring is attached to a point which is made to oscillate in a vertical line.

The displacement of the particle may be modelled by the differential equation

where x is the displacement of the particle below the equilibrium position at time t.

When t = 0 the particle is stationary and its displacement is 2.

Find the particular solution of the differential equation.

Write down an approximate equation for the displacement when t is large.