Edexcel A Level Further Maths Paper 3A: Further Pure Mathematics 1 October 2020 question paper 9FM0/3A

Use l’Hospital’s Rule to show that

Figure 1 shows a sketch of the vertical cross-section of the entrance to a tunnel. The width at the base of the tunnel entrance is 2 metres and its maximum height is 3 metres.

The shape of the cross-section can be modelled by the curve with equation y= f (x) where

A wooden door of uniform thickness 85 mm is to be made to seal the tunnel entrance.

Use Simpson’s rule with 6 intervals to estimate the volume of wood required for this door, giving your answer in m³ to 4 significant figures.

The points A, B and C, with position vectors a= 3i− 2j+k, b=i+ 4j+ 5k and c = −2 i + 3 j + 3 k respectively, lie on the plane Π

Find ABAC

Find an equation for Π in the form r.n=p

The point D has position vector 8i+ 7j+ 5k

Determine the volume of the tetrahedron ABCD

Use Leibnitz’s theorem to show that the coefficient of (x−π)⁸ in the Taylor series expansion of f (x) about π is

where a and b are integers to be determined.

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

The ellipse E has equation

The points S and S′ are the foci of E.

Find the coordinates of S and S′

Show that for any point P on E, the triangle PSS′ has constant perimeter and determine its value.

A physics student is studying the movement of particles in an electric field. In one experiment, the distances in micrometres of two moving particles, A and B, from a fixed point O are modelled by

respectively, where t is the time in seconds after motion begins.

Use algebra to find the range of time for which particle A is further away from O than particle B is from O.

It was recorded that the distance of particle B from O was less than the distance of particle A from O for approximately 4 seconds.

Use this information to assess the validity of the model.

The points P(9p², 18p) and Q(9q², 18q), p q, lie on the parabola C with equation

The line l passes through the points P and Q

Show that an equation for the line l is

(p + q) y = 2(x + 9 pq)

The normal to C at P and the normal to C at Q meet at the point A.

Show that the coordinates of A are

Given that the points P and Q vary such that l always passes through the point (12, 0)

find, in the form y²= f (x), an equation for the locus of A, giving f (x) in simplest form.

show that f (x) can be written in the form

Hence solve, for 0 x 2π, the equation

giving your answers to 2 decimal places where appropriate.

Use the result of part (a) to show that

where K is a constant to be determined.