Edexcel A Level Further Maths Paper 1: Core Pure Mathematics 1 October 2020 question paper 9FM0/01

f (z) = 3z³+pz²+ 57z+q

where p and q are real constants.

show all the roots of f(z) = 0 on a single Argand diagram,

find the value of p and the value of q.

Prove that

where a and b are rational numbers to be determined.

Figure 1 shows a sketch of two curves C₁ and C₂ with polar equations

The region R lies inside C₁ and outside C₂ and is shown shaded in Figure 1. Show that the area of R is

where p and q are integers to be determined.

The plane Π₁ has equation

r = 2 i + 4 j – k + λ (i + 2 j – 3 k) + μ (– i + 2 j + k)

where λ and μ are scalar parameters.

Find a Cartesian equation for Π₁

The line l has equation

Find the coordinates of the point of intersection of l with Π₁

The plane Π₂ has equation

r. (2 i – j + 3 k) = 5

Find, to the nearest degree, the acute angle between Π₁ and Π₂

Two compounds, X and Y, are involved in a chemical reaction. The amounts in grams of these compounds, t minutes after the reaction starts, are x and y respectively and are modelled by the differential equations

Show that

Find, according to the model, a general solution for the amount in grams of compound X present at time t minutes.

Find, according to the model, a general solution for the amount in grams of compound Y present at time t minutes.

Given that x = 2 and y = 5 when t = 0

find

the particular solution for x,

the particular solution for y.

A scientist thinks that the chemical reaction will have stopped after 8 minutes.

Explain whether this is supported by the model.

Prove by induction that for n ⁺

Prove by induction that for all positive odd integers n

is divisible by 15

A sample of bacteria in a sealed container is being studied.

The number of bacteria, P, in thousands, is modelled by the differential equation

where t is the time in hours after the start of the study.

Initially, there are exactly 5000 bacteria in the container.

Determine, according to the model, the number of bacteria in the container 8 hours after the start of the study.

Find, according to the model, the rate of change of the number of bacteria in the container 4 hours after the start of the study.

State a limitation of the model.