NATIONAL UNIVERSITY OF SINGAPORE

DEPARTMENT OF MATHEMATICS

SEMESTER 2 EXAMINATION 2010-2011

GEK1505 Living with Mathematics

April 2011 – Time allowed: 2 hours

INSTRUCTIONS TO CANDIDATES

1. This is an open book examination. Candidates may use calculators.

2. This examination paper contains a total of FIVE questions and com-prises SIX printed pages including this page.

3. Attempt all questions.

4. Not all parts or sub-parts of a question carry the same number of marks.

5. Do not answer parts or sub-parts belonging to different questions(questions labeled with different numbers) on the same sheet of paper.

6. For part (a) of each question, you are only required to give your answers,which should be simplified as much as possible, and no working need tobe given. For part (a), marks will be awarded to correct answers only.

7. For part (b) of each question, you should give details of your working.Marks will be awarded for evidence of correct method or strategy evenif the answers are incorrect.

GEK1505 2

Answer all questions.

Question 1 [20 marks]

(a) [12 marks]

(i) Find the number of integers between 1 and 1000 inclusive whichare multiples of 12 or 15 but not multiples of 45.

(ii) Find the number of circular arrangements of seating 9 guests ata round table with 10 seats if two particular persons refuse to sitnext to each other and no empty seat is removed.

(iii) Find the number of bar codes that can beformed using 4 black bars of width 1, 2, 3, 4units respectively separated by white spaces ofwidth 1 or 2 units (such as in Figure 1). Figure 1

(iv) An insect moves without backtracking along thegridlines of Figure 2 from O to P . Find thenumber of paths it can take that pass throughA but not through B. r

rr r

O

A

B

P

Figure 2

(b) [8 marks]

A vertical bookcase consists of 5 horizontal shelves. Books are classifiedinto 8 different types. Find the number of ways of arranging thesebooks into the bookcase in each of the following cases. (Assume thatthe ordering of books in any shelf is not important and that each shelfhas enough space to take in books allocated to it.)

(i) The bottom shelf is left empty and not more than 2 types of booksare placed in any shelf.

(ii) No shelf is left empty and not more than 2 types of books areplaced in any shelf.

GEK1505 3

Question 2 [20 marks]

(a) [12 marks]

s ss s s ss

G1

s ss ssss

G2

s ss s s ss

G3

Figure 3

(i) Is the following statement true or false?

“If a connected graph has an Euler circuit, then its number ofedges is even.”

(ii) Are the graphs G1 and G2 in Figure 3 equivalent?

(iii) Is the graph G3 in Figure 3 planar?

(iv) Can the line drawing in Fig-ure 4 be drawn with a pen inone continuous movementby tracing each line exactlyonce and without lifting thepen?

hFigure 4

(b) [8 marks]

Find all the labeled spanningtrees of the graph G in Figure 5with vertices labeled A, B, C, D.How many of these spanning treesare not paths of G? r r

rr

A B

C

D

Figure 5

GEK1505 4

Question 3 [20 marks]

(a) [12 marks]

(i) Can the hexadecimal number (222 . . . 22)16 with more than twodigits and in which each digit is 2, have an octal representation inwhich each digit is 2?

(ii) Is the following number divisible by 9?

1 + 22 + 32 + 42 + . . . + 123456782

(Note: 1 + 22 + 32 + 42 + . . . + n2 = 16n(n + 1)(2n + 1))

(iii) Find the check digit N of the 10-digit ISBN number 025321339N .

(iv) The following Hamming (7,4) codeword is received: 1011011.Assuming at most two errors in transmission, is it possible torecover the original message sent?

(b) [8 marks]

A game of musical chairs is played among 11 children C0, C1, . . . , C10

seated in a circle as follows. A ball is passed from one child to thenext in a clockwise direction in one second as a tune is being played.Whoever holds the ball when the tune stops playing is removed anddoes not participate in the next round of the game. At the start ofthe tune for the first round, the ball is with child C0. Assume that theduration of the tune played in any round is 45 seconds and that theball is always successfully passed from one child to the next at eachstep.

(i) Which child is removed at the end of the first round?

(ii) Which children would have been removed at the end of the thirdround?

GEK1505 5

Question 4 [20 marks]

(a) [12 marks]

(i) You received the following cryptic message:

RFC KMPC UC AYPC DMP MRFCPQ RFC

EPCYRCP MSP QCLQC MD UCJJ ZCGLE

Suspecting that a shift transformation was used to encipher theoriginal message, you try frequency analysis to decipher it. Whatis the message?

(ii) An enciphering transformation y = f(x) ≡ ax + b (mod 26) on26 symbols satisfies the conditions f(5) = 11, f(10) = 6.Find the deciphering transformation in the form x = f−1(y) ≡cy + d (mod 26) where 1 ≤ c, d ≤ 25.

(iii) What is the probability that the total score of a throw of 3 fairdice is odd?

(iv) It is known that Mr Tan has 5 children of whom at least two aregirls. Find the probability that the youngest and the eldest areboth girls. (Assume that it is equally likely for a child to be a boyor a girl.)

(b) [8 marks]

0 1 2 3 4 5 6� �� Figure 6

A game is played as follows. You start with a counter g placed inSquare 0 of a row of 6 squares (Figure 6). You toss a fair die and movethe counter k squares to the right according to the outcome k of thetoss of the die. If the counter reaches Square 6, you get one dollar.If not, you toss the die a second time and move the counter from thecurrent position in the same fashion according to the outcome of thetoss. If the counter ends up at Square 6, you get 2 dollars. If not, thegame is over and you get nothing. Find

(i) the probability of winning some money in one game,

(ii) the expected payout of one game.

GEK1505 6

Question 5 [20 marks]

(a) [12 marks]

Write down the number of non-identity symmetries of each of the dia-grams (A), (B), (C), (D) in Figure 7.

(A) (B) (C) (D)

Figure 7

(b) [8 marks]

-

6

0 10 20 30 40 50 60

10

20

30

40

50

Z

YX

Figure 8

In Figure 8, X, Y, Z are semi-circles.

(i) Find an orientation-preserving rigid motion that will move X toY.

(ii) Find an orientation-reversing rigid motion that will move X to Z.

In your answers, state the parameters of the relevant rigid motionsclearly. (You may use a graph paper to present your answers.)

– END OF PAPER –