Further Mathematics: Discrete Mathematics Topical Past Paper Questions

The following diagram shows the weighted graph G. (a) Write down a Hamiltonian circuit in $G$ starting at P. 2023 (b) (i) State what feature of $G$ enables you to deduce that it is not bipartite. (ii) Explain why G does not have an Eulerian trail. (iii) An edge is removed from $G…

[Maximum mark: 15] Consider the second-degree recurrence relation $ u_{n}=4u_{n-1}-5u_{n-2} $. (a) (i) Write down and solve the auxiliary equation. (ii) Hence write down the general solution to this recurrence relation. (iii) Given the initial conditions $ u_{0}=u_{1}=2 $, show t…

[Maximum mark: 19] The set of all integers from 0 to 99 inclusive is denoted by S. The binary operations $ * $ and $ \circ $ are defined on S by $$ a*b=[a+b+20](\bmod100) $$ $$ a\circ b=[a+b-20](\bmod100). $$ (a) Find the identity element of S with respect to $ * $. [3] (b) Show…