Edexcel A Level Further Maths Paper 2: Core Pure Mathematics 2 June 2025 question paper 9FM0/02

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Given that

show that

arg(zw) arg(z) arg(w)

An archer shoots an arrow towards a target.

In a model

• the arrow is a particle

• the flight path of the arrow is a straight line

• the target is part of a plane

Relative to a fixed origin O

• the arrow is fired from the point with position vector 3i 5j 2k

• the plane containing the target has equation 2x 4y z 3

Use the model to answer parts (a) to (d).

Determine the shortest distance that the arrow must travel to reach the plane.

The arrow hits the target at the point with position vector 6i 2j k

Determine a vector equation of the flight path of the arrow.

Determine the acute angle that the flight path of the arrow makes with the target. Give your answer to the nearest degree.

Determine the distance travelled by the arrow.

Comment on whether the actual distance travelled by the arrow is likely to match the answer to part (d), giving a reason for your answer.

Given

prove by mathematical induction that, for n

Given

describe fully the single geometrical transformation P represented by the matrix B

The transformation Q is represented by the matrix Aⁿ

The transformation P followed by the transformation Q is the transformation R, which is represented by the matrix C

Determine C in terms of n

Given that, for a particular value of n, the transformation R maps the point with coordinates (27, 1) to the point with coordinates (a, a), where a is a constant,

determine the matrix that represents the transformation Q

z₁ a bi and z₂ c di

where a, b, c and d are real constants.

Given that

• z₁ z₂ is real

show that a 2 and determine the value of each of b, c and d

On the same Argand diagram

• sketch the locus of points z which satisfy z 12 7

• sketch the locus of points w which satisfy w 5i 4

showing the coordinates of any points of intersection with the axes.

Three planes are defined by the following equations

where p and q are constants.

Given that the planes form a sheaf, determine

the value of p

the value of q

The quartic equation

where A is a real constant, has roots , , and

Determine the value of

determine the possible values of A

The curve C shown in Figure 1 has polar equation

where a is a constant.

The tangents to C at the points A and B are perpendicular to the initial line.

Use calculus to determine the polar coordinates of A and B

The curve C models the perimeter of the surface of a swimming pool.

Given that, according to the model, the distance across the pool from A to B is 10 m,

Use algebraic integration to determine the surface area of the swimming pool, according to the model.

Given that

show that

where k is a constant to be determined.

Hence determine the first three non-zero terms of the Maclaurin series for y, giving each coefficient in simplest form.

Given that

The curve C has equation

Determine giving your answer in simplest form.

Hence determine the x coordinate of the point on C at which the gradient is 4