Edexcel A Level Further Maths Core Pure 1 June 2024 question paper 9FM0/01

f(z) = z⁴ 6z³ az² bz 145

where a and b are real constants.

Given that 2 + 5i is a root of the equation f(z) = 0

determine the other roots of the equation f(z) = 0

Show all the roots of f(z) = 0 on a single Argand diagram.

The roots of the equation

are α, β and γ

Without solving the equation,

Use the answers to part (a) to determine the value of

(α – 1)(β – 1)(γ – 1)

α² + β² + γ²

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1 shows the design for a bathing pool.

The pool, P, shown unshaded in Figure 1, is surrounded by a tiled area, T, shown shaded in Figure 1.

The tiled area is bounded by the edge of the pool and by a circle, C, with radius 6 m.

The centre of the pool and the centre of the circle are the same point.

The edge of the pool is modelled by the curve with polar equation

where a is a positive constant.

Given that the shortest distance between the edge of the pool and the circle C is 0.5 m,

determine the value of a.

Hence, using algebraic integration, determine, according to the model, the exact area of T.

The complex number z = e^iθ, where θ is real.

Show that

where n is a positive integer.

Show that

Hence, making your reasoning clear, determine all the solutions of

cos 5 θ + 5 cos 3 θ + 12 cos θ = 0

in the interval 0 θ < 2π

A raindrop falls from rest from a cloud. The velocity, vm s^–1 vertically downwards, of the raindrop, t seconds after the raindrop starts to fall, is modelled by the differential equation

where k is a positive constant.

Solve the differential equation to show that

Given that v = 4 whent = 2

determine, according to the model, the velocity of the raindrop 5 seconds after it starts to fall.

Comment on the validity of the model for very large values of t

Prove by induction that, for all positive integers n,

The line l₁ has equation

r = i – 2 j + 3 k + λ (2 i + j – 4 k)

and the line l₂ has equation

r = 5 i + p j – 7 k + μ (6 i + j + 8 k)

where λ and μ are scalar parameters and p is a constant.

The plane Π contains l₁ and l₂

Show that the vector 3i – 10j – kis perpendicular to Π

Hence determine a Cartesian equation of Π

Hence determine the value of p

Given that

• the lines l₁ and l₂ intersect at the point A

• the point B has coordinates (12, –11, 6)

determine, to the nearest degree, the acute angle between AB and Π

A scientist is studying the effect of introducing a population of type A bacteria into a population of type B bacteria.

At time t days, the number of type A bacteria, x, and the number of type B bacteria, y, are modelled by the differential equations

Show that

Determine a general solution for the number of type A bacteria at time t days.

Determine a general solution for the number of type B bacteria at time t days.

The model predicts that, at time T hours, the number of bacteria in the two populations will be equal.

Given thatx = 100 and y = 275 when t = 0

determine the value of T, giving your answer to 2 decimal places.

Suggest a limitation of the model.