Edexcel A Level Further Maths Paper 3A: Further Pure Mathematics 1 June 2019 question paper 9FM0/3A

Use Simpson’s rule with 4 intervals to estimate

Given that k is a real non-zero constant and that

use Leibnitz’s theorem to show that

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

Show that

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

Hence find a series solution, in ascending powers of x as far as the term in x⁵, of the differential equation (I), given that y = 1 at x = 0

The parabola C has equation

The distinct points P(p², 4p) and Q(q², 4q) lie on C, where pq

The tangent to C at P and the tangent to C at Q meet at the point R (–28, 6).

Show that the area of triangle PQR is 1331

where k is an arbitrary constant.

The concentration of a drug in the bloodstream of a patient, t hours after the drug has been administered, where t 6, is modelled by the differential equation

where C is measured in micrograms per litre.

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

Hence find the general solution for the concentration C at time t hours.

Given that when t = 6, C = 0 and = 36

find the maximum concentration of the drug in the bloodstream of the patient.

With respect to a fixed origin O, the points A, B and C have coordinates (3, 4, 5), (10, –1, 5) and (4, 7, –9) respectively.

The plane has equation 4x – 8y + z = 2

The line segment AB meets the plane at the point P and the line segment BC meets the plane at the point Q.

Show that, to 3 significant figures, the area of quadrilateral APQC is 38.5

The point D has coordinates (k, 4, –1), where k is a constant. Given that the vectors AB, AC and AD form three edges of a parallelepiped of volume 226

find the possible values of the constant k.

The hyperbola H has equation

The line l₁ is the tangent to H at the point P(4cosh , 3sinh ).

The line l₁ meets the x-axis at the point A.

The line l₂ is the tangent to H at the point (4, 0).

The lines l₁ and l₂ meet at the point B and the midpoint of AB is the point M.

Show that, as θ varies, a Cartesian equation for the locus of M is

where p and q are values to be determined.

Let S be the focus of H that lies on the positive x-axis.

Show that the distance from M to S is greater than 1