OCR A Level Further Maths A Pure Core 1 June 2025 question paper Y540

Write down the 2 × 2 matrix that represents a rotation of 90° anticlockwise about the origin.

Write down the 3 × 3 matrix that represents a reflection in the x – z plane.

The equation 5x³ - 4x² + 10 = 0 has roots α, β and γ.

Write down the values of α + β + γ, αβ + βγ + γα and αβγ.

By expanding (αβ + βγ + γα)², determine the value of α²β² + β²γ² + γ²α².

By expanding a suitable expression, find the value of α² + β² + γ².

Hence find a cubic equation with integer coefficients that has roots α², β², and γ².

Verify that a cartesian equation of Π₁ is x - y + 2z = 3.

For some real constant a, cartesian equations of planes Π₂ and Π₃ are

Π₂: x - 3z = 1 Π₃: ax - y - z = 4

By considering a suitable matrix, show that Π₁, Π₂ and Π₃ intersect at a single point for all

values of a except a = 2.

Use the matrix from part (b) to find the coordinates of the point of intersection of Π₁, Π₂

and Π₃ in the case where a = 3.

In the case where a = 2, determine the geometrical arrangement of Π₁, Π₂ and Π₃.

Show that the cables do not intersect.

To access the cables for maintenance, a ladder can be used. The base of the ladder is placed at a fixed point on the ground.

The ladder is modelled as a straight-line segment. The base of the ladder is modelled as being located at the point (4, 5, 0).

Determine the minimum length of the ladder required so that it reaches cable T₁. Give your

answer in centimetres to the nearest centimetre.

Identify a modelling assumption used in part (a) that is unrealistic, and which could affect

your answer to this part.

In this question you must show detailed reasoning.

In this question, arguments of complex numbers are in the interval [ 0, 2 π).

Given that z =- 5 + 5i, express z in exact exponential form.

A pendulum comprises an object P of mass m kg and a string of length 7 m. One end of the string is attached to P and the other end is attached to a fixed point A.

At time t seconds, t ⩾ 0, the string forms an angle of θ radians, measured anti-clockwise, from the downward vertical through A, as shown in the diagram.

When t = 0, θ = θ₀ > 0 and P is released from rest. You may assume that in the subsequent motion P moves along the arc of a circle, centre A and radius 7 m, and that |θ| ⩽ θ₀ for all t ⩾ 0.

The motion of P is modelled by the following differential equation.

In some situations, it is appropriate to approximate (∗) with the following differential equation.

An additional force now acts on P. It can be shown that it is now appropriate to model the motion of P with the differential equation + + = 0 where k > 0.

Find the range of values of m, in terms of k, for which the motion of the pendulum is

overdamped.

In this question you must show detailed reasoning.

Use de Moivre’s Theorem to show that, if cos5θ ≠ 0,

By considering the equation tan5 θ = 1, use the result in part (a) to find the exact roots of the equation

Give the roots in the form t = tan φ where 0 < φ < π.

By first expressing t⁴ - 4t³ - 14t² - 4t + 1 = 0 in the form (t - 1)⁴ = kt², where k is a constant to be determined, show that