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
2. [Maximum mark: 10]
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]
4. [Maximum mark: 8]
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]
5. [Maximum mark: 9]
(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]
7. [Maximum mark: 7]
1 1 n Diagonalize the matrix A = ; hence find an expression for A . [7] 2 4
8. [Maximum mark: 9]
(a) Solve 3x 6mod9 . [3]
(b) Hence, solve the system 5
3x 6mod9 . 51x 2mod38 . [6] 6
9. [Maximum mark: 8]
Let S be a relation on NN, 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 ZZ) [2]
10. [Maximum mark: 10]
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]
11. [Maximum mark: 8]
(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
12. [Maximum mark: 8]
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]
13. [Maximum mark: 8]
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) an1 6an 8, a1 3 [4]
(b) bn2 6bn1 8bn 0 , b1 8 , b2 40 [5]
(c) cn2 6cn1 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 xR.
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]