Edexcel A Level Further Maths Paper 1: Core Pure Mathematics 1 June 2025 question paper 9FM0/01

Determine the values of a for which the matrix M is singular.

Given that matrix M is non-singular and that

• det(M) = −108

• a 0

determine the matrix M⁻¹

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Determine the exact values of x for which

The complex number z a bi where a and b are real constants.

Given that is purely imaginary,

show that a² = b²

Given also that zz* 50

determine the possible complex numbers z

Determine the general solution of the differential equation

giving your answer in the form y = f (x)

Given that y 5 and 12 when x = 0

determine the particular solution of the differential equation.

Use the method of differences to prove that for n 2

where p and q are constants to be determined.

f (z) = z³ + az² + bz + c where a, b and c are real constants

The roots of the equation f (z) = 0 are z₁, z₂ and z₃

When plotted on an Argand diagram, the points representing these roots form the vertices of a triangle.

Given that

• z₁ 2 4i

• the area of the triangle is 12

Determine the two possible functions f (z)

Express

in partial fractions.

Hence, show that

A rambler starts a walk at the bottom of a hill at 10 am.

The rambler walks all the way to the top of the hill and then turns around and walks back down to the bottom of the hill.

The differential equation

where A is a constant, is used to model the vertical displacement, x metres, of the rambler from the bottom of the hill, t hours after the start of the walk.

Given that after 1 hour

• the rambler has a vertical displacement of 213 m

determine the value of A to 3 significant figures.

Hence, determine the particular solution of the differential equation, giving your answer in the form x = f (t)

Use the model to determine the time at which the rambler will return to the bottom of the hill.

Given that the vertical displacement of the top of the hill is 300.68 m

use the model to find the value of t when the rambler reaches the top of the hill.

Using your answers to parts (c) and (d), give a limitation of the model.

The curves with equations

intersect at just one point P

Use algebra to show that the x coordinate of P satisfies the equation

Show that e^x 3 is a solution of this equation.

Hence state the exact coordinates of P.

Show that

Given that, for n

show that

Figure 1 Figure 2

Figure 1 shows the central vertical cross-section of a solid wooden ornament.

Figure 2 shows the curve with equation

The region R, shown shaded in Figure 2, is bounded by the curve, the line with equation y = and the y-axis.

The ornament is modelled by the solid of revolution formed when R is rotated 360° about the y-axis. The units are centimetres.

Using algebraic integration and the result in part (a), determine, in cm³, the volume of wood needed to make the ornament, according to the model. Give your answer to 2 significant figures.

[ Solutions based entirely on calculator technology are not acceptable.]

Given that

• the density of the wood is 0.85 g/cm³

• the mass of the ornament is 6 grams

comment on the suitability of the model.