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Unformatted text preview: 8004A SEMESTER 2 2009 THE UNIVERSITY OF SYDNEY
FACULTIES OF ARTS, ECONOMICS, EDUCATION,
ENGINEERING AND SCIENCE MATH 1 004
DISCRETE MATHEMATICS November 2009 TIME ALLOWED: One and a half Hours Family Name: . . . L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Names: This examination has two sections: Multiple Choice and Extended Answer. The Multiple Choice Section is worth 50% of the total examination;
there are 25 questions; the questions are of equal value;
all questions may be attempted. Answers to the Multiple Choice questions must be entered on
the Multiple Choice Answer Sheet; The Extended Answer Section is worth 50% of the total eXamination;
there are 3 questions; the questions are of equal value;
all questions may be attempted;
working must be shown. Calculators will be supplied; no other calculators are permitted. THE QUESTION PAPER MUST NOT BE REMOVED FROM THE
EXAMINATION ROOM. PAGE 1 OF 18 LECTURER: AI Molev MARKER’S USE
ONLY 8004A SEMESTER 2 2009 PAGE 10 OF 18 Extended Answer Section There are three questions in this section, each with a number of parts. Write your answers
in the space provided below each part. If you need more space there are extra pages at the
end of the examination paper. 1. A particle moves in the my—plane so that each second it moves either one unit in the positive r—direction or one unit in the positive y—direction. The initial position of the
particle is at the origin. (a) What is the number of possible trajectories of the particle during the first
10 seconds? (2 marks) (b) What is the number of possible trajectories of the particle from the origin to the
point (4,6)? » (2 marks) 8004A SEMESTER 2 2009 PAGE 11 OF 18 (c) What is the number of possible trajectories of the particle from the origin to the
point (12, 6) Without two consecutive moves in the y—direction? (2 marks) (d) What is the number of possible trajectories of the particle in the ﬁrst 7 seconds
' Without two consecutive moves in the y—direction? (2 marks) (e) What is the number of possible trajectories of the particle in the ﬁrst 7 seconds with
at most ﬁve consecutive moves in any direction? (2 marks) 8004A SEMESTER 2 2009 PAGE 12 OF 18 2. The following switching circuit represents a Boolean function f in three variables x, y, z: (a) Use the switching circuit to write down a Boolean expression representing
the function f. (2 marks) (b) Complete the table of values for the function f. (2 marks) 8004A SEMESTER 2 2009 PAGE 13 OF 18 (0) Write down the Boolean expression for f in disjunctive normal form. (2 marks) ((21) Apply the Karnaugh map method to ﬁnd
a simpler Boolean expression for f. (2 marks) (e) Draw a simpler switching Circuit representing the function f. (2 marks) 8004A SEMESTER 2 2009 PAGE 14 OF 18 3. A sequence (xn I n 2 0) satisﬁes the recurrence relation as” = 3mn_2 + 2mn_3.
(a) Write down the corresponding characteristic equation and ﬁnd its roots. (3 marks) (b) Write down the general solution of the recurrence relation. (2 marks) 8004A SEMESTER 2 2009 PAGE 15 OF 18 (c) Find the particular solution satisfying the initial conditions 230 = 1, $1 = 3 and
332 = 2. (2 marks) ((1) Using your answer to the previous part or otherwise write down a closed form of the
generating function of the sequence (5’ marks) You may use next pages for your answers ...
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