Edexcel A Level Further Maths Paper 2: Core Pure Mathematics 2 November 2020 question paper 9FM0/02

The curve C has equation

Determine, in terms of natural logarithms, the exact x coordinates of the stationary points of C.

In an Argand diagram, the points A and B are represented by the complex numbers −3 + 2i and 5 − 4i respectively. The points A and B are the end points of a diameter of a circle C.

Find the equation of C, giving your answer in the form

The circle D, with equation |z− 2 − 3i| = 2, intersects C at the points representing the complex numbers z₁ and z₂

Find the complex numbers z₁ and z₂

A scientist is investigating the concentration of antibodies in the bloodstream of a patient following a vaccination. The concentration of antibodies, x, measured in micrograms (μg) per millilitre (ml) of blood, is modelled by the differential equation

where t is the number of weeks since the vaccination was given.

Find a general solution of the differential equation.

Initially, • there are no antibodies in the bloodstream of the patient • the concentration of antibodies is estimated to be increasing at 10 μg/ml per week

Find, according to the model, the maximum concentration of antibodies in the bloodstream of the patient after the vaccination.

A second dose of the vaccine has to be given to try to ensure that it is fully effective. It is only safe to give the second dose if the concentration of antibodies in the bloodstream of the patient is less than 5 μg/ml.

Determine whether, according to the model, it is safe to give the second dose of the vaccine to the patient exactly 10 weeks after the first dose.

Use de Moivre’s theorem to prove that

Hence find the distinct roots of the equation

giving your answer to 3 decimal places where appropriate.

y= tan⁻¹x

Assuming the derivative of tanx, prove that

Show that

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

Given that k ≠4, find, in terms of k, the inverse of the matrix M.

Find, in terms of p, the coordinates of the point where the following planes intersect.

2 x + 5 y + 7 z = 1 x + y + z = p 2 x + y − z = 2

Find the value of q for which the following planes intersect in a straight line.

4 x + 5 y + 7 z = 1 x + y + z = q 2 x + y − z = 2

For this value of q, determine a vector equation for the line of intersection.

A student wants to make plastic chess pieces using a 3D printer. Figure 1 shows the central vertical cross-section of the student’s design for one chess piece. The plastic chess piece is formed by rotating the region bounded by the y-axis, the x-axis, the line with equation x = 1, the curve C₁ and the curve C₂ through 360° about the y-axis. The point A has coordinates (1, 0.5) and the point B has coordinates (0.5, 2.5) where the units are centimetres.

The curve C₁ is modelled by the equation

Determine the value of a and the value of b according to the model.

The curve C₂ is modelled to be an arc of the circle with centre (0, 3).

Use calculus to determine the volume of plastic required to make the chess piece according to the model.