Full Marks Are Not Necessarily Awarded for a Correct Answer with No Working

JUNE17/FURMA/HP1/ENG

FURTHER MATHEMATICS Student’s Name HIGHER LEVEL PAPER 1

Wednesday 14 June 2017 (morning)

2 hours 30 minutes

INSTRUCTIONS TO CANDIDATES

 Do not open this examination paper until instructed to do so.  Answer all questions.  Unless otherwise stated in the question, all numerical answers must be given exactly or correct to three significant figures.  A graphic display calculator is required for this paper.  A clean copy of the Mathematics HL and Further Mathematics HL formula booklet is required for this paper.  The maximum mark for this examination paper is [150 marks].

7 pages

IB PSYCHICO COLLEGE 2017 2

Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. In particular, solutions found from a graphic display calculator should be supported by suitable working. For example, if graphs are used to find a solution, you should sketch these as part of your answer. Where an answer is incorrect, some marks may be given for a correct method, provided this shown by written working. You are therefore advised to show all working.

1. [Maximum mark: 10]

Consider the following graph

(a) Show that this graph has

(i) an Eulerian circuit;

(ii) a Hamiltonian cycle. [3]

(b) The edge joining V2 and V6 is removed. Does the graph still have an Eulerian circuit and a Hamiltonian cycle? Give reasons for your answers. [3]

(c) Replace the edge joining V2 and V6 , and remove the edge joining V1 and V2.

(i) Find an Eulerian trail.

(ii) Find a Hamiltonian path. [4] 3

The square matrix X is such that X 3= O. (i.e. the zero matrix). Show that

(a) detX = 0 [1]

(b) the inverse of the matrix (I – X) is (I + X + X 2). [2]

n(n 1) (c) (I – X)n = I – nX + X 2 for n Z  , by using mathematical induction. [7] 2

3. [Maximum mark: 6] The permutation P is given by

1 2 3 4 5 6 P    3 4 5 6 2 1 (a) Determine the order of P, justifying your answer [2]

(b) The permutation group G is generated by P. Determine the element of G that is of order 2, giving your answer in cycle notation. [4]

Determine, with reasons, whether the following functions are injective or surjective.

(a) f : R  R where f (x)  x2  3x  2 [2]

(b) g : R  R  R  R where g(x, y)  (x  y, xy) [3]

(c) h : Z   Z   Z   Z  where h(a,b,c)  2a 3b 5c [3]

(a) Prove that there are infinitely many prime numbers. [5]

(b) Explain why 24!+17 is not a prime number. [1]

(c) Find 80 consecutive composite (non-prime) numbers. [3] 4

Turn over 6. [Maximum mark: 16]

1 n Let M be the set of all matrices of the form   where n  Z . 0 1

(a) Show that (M,+) is not a group. [1]

(b) Show that M form an abelian group under matrix multiplication. (you may assume that matrix multiplication is associative) [5]

(c) Show that f : (M, ∙ ) → (Z,+) given by

1 n f :    n 0 1 is an isomorphism. [4]

(d) Write down the kernel of f. [1]

1 3k  Let K be the set of all matrices of the form   where k  Z . 0 1 

(e) Show that (K, ∙ ) is a subgroup of (M, ∙ ) [3]

(f) Write down the (left) cosets of K. [2]

1 1   n Diagonalize the matrix A =  ; hence find an expression for A . [7]  2 4

(a) Solve 3x  6mod9 . [3]

(b) Hence, solve the system 5

3x  6mod9 . 51x  2mod38 . [6] 6

Let S be a relation on NN, where N ={0,1,2,…} is the set of natural numbers, such that

(a,b) S (c,d) if and only if max{a,b}  max{c,d}

(a) Find all pairs related to (1,2) [2]

(b) Show that S is an equivalence relation. [4]

(c) Describe the equivalence classes (i.e. the partition of ZZ) [2]

In the vector space R 3

1 1 0       (a) Prove that the vectors u1  1 , u2  0 and u3  1 are linearly independent [4]       0 1 1

(b) Prove that three vectors w1 , w2 and w3 are linearly independent if and only if any

vector in R 3 can be expressed uniquely as a linear combination of them. [6]

(a) Represent the cube as a planar graph and verify Euler’s relation v – e + f = 2 [2]

(b) For a simple connected bipartite graph explain why 4f ≤ 2e and deduce the inequality e ≤ 2v – 4. [4]

(c) Explain why the equality e = 2v – 4 holds for the cube in question (a); draw one more example where the equality holds. [2]

Turn over 7

Consider the matrix  3 2 1     1  3 4  M= 1 1  2      8 5 3 

(a) Explain how it can be seen immediately that the column vectors of M are linearly dependent. [2]

(b) Determine the null space of M. [3]

(c) Explain briefly how your results verify the rank-nullity theorem. [3]

Consider the weighted graph

Use Dijkstra’s Algorithm to find the length of the shortest path between the vertices P and T. Show all the steps used by the algorithm and write down the shortest path. 8

14. [Maximum mark: 13] Solve the following difference equations

(a) an1  6an  8, a1  3 [4]

(b) bn2  6bn1  8bn  0 , b1  8 , b2  40 [5]

(c) cn2  6cn1  9cn , c1  0 , c2  9 [4]

15. [Maximum mark: 20] Consider the set G of the differentiable functions f : R  R such that

f (x)  6 f (x)  8 f (x)  0 , for all xR.

2x (a) Show that the function f1 (x)  e lies in G. [3]

x (b) Given that f 2 (x)  e is another function in G, show that   4 . [4] (c) Show the sum of those two functions, i.e. f (x)  e2x  e4x , also lies in G. [2]

It is given that G consists exactly of all linear combinations of f1 and f2 , that is of the functions of the form f (x)  ae2x  be4x , where a,b  R ,

(d) Show that G is an abelian group under the addition of functions [4]

(e) Let f (x)  ae2x  be4x be a function in G. f (0) (i) If a and b are coprime integers show that f (0) and are also coprime. 2 (ii) Find a and b given that f has a stationary point at A(0,1). [5]

(f) Explain why the group (G,+) is isomorphic to the additive group (R2,+) (where R2 is the standard 2-dimensional vector space). [2]