OCR A Level Further Maths A Pure Core 1 November 2020 question paper Y540

Find the mean value of f(x) = x² + 6x over the interval [0, 3].

Find A⁴.

Describe the transformation that A represents.

The matrix B represents a reflection in the plane x = 0.

Write down the matrix B.

The point P has coordinates (2, 3, 4). The point P ′ is the image of P under the transformation represented by B.

Find the coordinates of P ′.

In this question you must show detailed reasoning.

Determine the square roots of 25i in the form re^iθ, where 0 ⩽ θ < 2π.

Illustrate the number 25i and its square roots on an Argand diagram.

Prove by induction that the sum of the cubes of three consecutive positive integers is divisible by 9.

Using exponentials, show that cosh 2u ≡ 2 sinh²u + 1.

By differentiating both sides of the identity in part (a) with respect to u, show that

sinh 2u ≡ 2 sinh u cosh u.

You are given that the cubic equation 2x³ + px² + qx - 3 = 0, where p and q are real numbers, has a

Write down a second complex root, β.

Determine the third root, γ.

Find the value of p and the value of q.

Show that the motion of the particle can be modelled by the following differential equation.

Solve the differential equation in part (a), giving the particular solution for v in terms of t,

λ and μ.

Without solving the differential equation in part (d), write down what will happen to the

velocity in the long term according to this refined model.

Find f (θ).

Hence find the line of symmetry of C.

Find the value of r when θ = .