AQA A Level Further Maths Paper 1 June 2024 question paper 7367/1

The roots of the equation 20x³ – 16x² – 4x + 7 = 0 are α, β and γ

Find the value of αβ + βγ + γα

Circle your answer.

Which one of the following is a real number?

Circle your answer.

z⁴ z⁵ z⁶ z⁷

Which one of the following statements is correct?

Tick (✓) one box.

Find a vector that is normal to the plane ∏ 5 (a)

Find a Cartesian equation of the plane ∏ 5 (b)

The ellipse E has equation

The line with equation y = mx + 4 is a tangent to E

Without using differentiation show that m = ±

It is given that

Starting from the exponential definition of the sinh function, show that sinh p = r

Solve the equation cosh² x = 2sinh x + 16

Give your answers in logarithmic form.

The complex numbers z and w are defined by

By evaluating the product zw, show that

Show that the angle between L₁ and L₂ is 0.701 radians, correct to three decimal places.

Find the shortest distance between L₁ and L₂

Give your answer in an exact form.

Use de Moivre’s theorem to show that

cos3θ = 4cos³θ – 3cosθ

Use de Moivre’s theorem to express sin3θ in terms of sinθ

Hence show that

Solve the differential equation

given that y = 3 when x = ln2

Give your answer in an exact form.

A curve is defined parametrically by the equations

Show that the arc length of the curve from t = 0 to t = 2 is equal to 26 units.

The curve C has polar equation r = 2 + tanθ The curve C meets the line θ = at the point A The point B has polar coordinates (4, 0)

The diagram shows part of the curve C, and the points A and B

Find the area of the shaded region.

Give your answer in an exact form.

By making a suitable substitution, show that

Find the extension of each string when the system is in equilibrium.

The ball is pulled a distance of 0.6 metres downwards from its equilibrium position towards C, and released from rest.

Show that during the subsequent motion the particle satisfies the differential equation

where x metres is the displacement of the particle below the equilibrium position at time t seconds after the particle is released.

Find x in terms of t

State one limitation of the model used in part (b)