Edexcel A Level Further Maths Paper 3A: Further Pure Mathematics 1 June 2025 question paper 9FM0/3A

Given that

complete the table below with the value of y corresponding to x = 2, giving your answer to 2 decimal places.

Use Simpson’s rule, with all the values of y in the completed table, to estimate, to one decimal place, the value of

Using your answer to part (b) and making your method clear, estimate the value of

The Taylor series expansion of f(x) about x = a is given by

Given that

show that

where a, b and c are integers to be determined.

Hence, given thaty = 1 and = 1 when x = 2

determine a Taylor series solution, in ascending powers of (x – 2), up to and including the term in (x – 2)⁴, of the differential equation (I), giving each coefficient in simplest form.

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

Use algebra to determine the values of x for which

Show that

Hence determine the Maclaurin series expansion for y, in ascending powers of x, up to and including the term in x², giving each term in simplest form.

Use the series expansion for sinx in ascending powers of x to determine, in simplest form, the first 2 non‑zero terms in ascending powers of x of the series for sin 3 x.

Use the answers to parts (b) and (c) to show that

Use the substitution t tan to show that

Hence show that

where A and B are constants to be determined and k is an arbitrary constant.

Use L’Hospital’s Rule to show that

y = x²e^3x

Use Leibnitz’s theorem to show that, fork

where A, B and C are integers to be determined.

Hence determine the values of x for which

The concentration, Pmg m^–3, of a pollutant in a reservoir, t days after the pollutant entered the reservoir, is modelled by the differential equation

Show that the transformation x = transforms equation (I) into the equation

Given that P = 0.5 when t = 0

solve differential equation (II) to show that, according to the model,

where k is a constant to be determined.

Given that the concentration of the pollutant in the reservoir, 3 days after the pollutant entered the reservoir, was 0.034 mg m^–3

comment on the reliability of the model, giving a reason for your answer.

The ellipse E has equation

The line l has equationy = mx + cwhere m and c are constants.

Show that the x coordinates of the points of intersection of E and l satisfy the equation

where k is a constant to be determined.

Hence, given that the line l

• is a tangent to E

• passes through the point (10, 20)

determine the possible equations of l

The parallelogram P has vertices E, F, G and H, with coordinates

Determine the exact area of P, giving your answer as a simplified surd.

The line l passes through E and is perpendicular to the plane containing P

Determine an equation for l giving your answer in the form (r – a)×b = 0

where a and b are constant vectors.

Given that

• P is one face of a parallelepiped

• the opposite face of the parallelepiped lies in the plane Π

• the point (25, –4, 13) lies in Π

determine the volume of the parallelepiped.