Edexcel A Level Further Maths Paper 4A: Further Pure Mathematics 2 November 2021 question paper 9FM0/4A

In this question you must show detailed reasoning.

Without performing any division, explain why n = 20 210 520 is divisible by 66

A binary operation on the set of non‑negative integers, ⁺, is defined by

Explain why ⁺₀ is closed under the operation

Show that 0 is an identity for (⁺₀, )

Show that all elements of ⁺₀ have an inverse under

Determine if ⁺₀ forms a group under , giving clear justification for your answer.

Use the Euclidean Algorithm to find integers a and b such that

125 a + 87 b = 1

Hence write down a multiplicative inverse of 87 modulo 125

Solve the linear congruence

87 x ≡ 16 (mod 125)

Let G be a group of order 46⁴⁶ + 47⁴⁷

Using Fermat’s Little Theorem and explaining your reasoning, determine which of the following are possible orders for a subgroup of G

11

21

The point P in the complex plane represents a complex number z such that

Given that, as z varies, the locus of P is a circle,

determine the centre and radius of this circle.

Shade on an Argand diagram the region defined by the set

A recurrence system is defined by

u_n+2 = 9(n + 1)² u_n − 3u_n+1 n 1

u₁ = −3, u₂ = 18

Prove by induction that, for n ,

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

You must not use the integration facility on your calculator.

Show that, for n 1

The curve shown in Figure 1 is defined by the parametric equations

This curve is rotated through 2π radians about the x‑axis to form a hollow open shell.

Show that the external surface area of the shell is given by

Using the results in parts (a) and (b) and making each step of your working clear,

determine the value of the external surface area of the shell, giving your answer to 3 significant figures.

determine the eigenvalue corresponding to this eigenvector

hence show that p = 2

determine the remaining eigenvalues and corresponding eigenvectors of A

Write down a matrix P and a diagonal matrix D such that A = PDP⁻¹

Solve the differential equation u˙ = ku, where k is a constant.

With respect to a fixed origin O, the velocity of a particle moving through space is modelled by

determine a general solution for the displacement of the particle.