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Final Examination: December 2007 Number of pages: 3
Course: Math 1P66 Number of students: 85
Examination date: 12 December 2007 Number of hours: 3
Time of Examination: 14:00—17:00 Instructor: S. D’Agostino No aids are permitted except for a nonprogrammable, non—graphing calculator. Solve all problems in the exam booklets provided. All questions are worth 10 points, even
though some questions are easier than others. Make Sure to show your work if you wish to receive partial credit for incorrect ﬁnal results. Total number of marks: 100. 1. (a) Express 114 552 as a product of prime factors.
(b) Determine the highest common factor (HCF) of 14 760 and 21 240. 2. (a) Complete the following table by changing numbers from one base to another.
Quote results correct to two decimal places, if necessary. Base 2 Base 6 Base 10 Base 16 I 111 100.11
554.3 79.8 AE.4 (b) The equation 2” = 553 has a solution between I = 4 and as = 5. Use the bisection
method to estimate this solution correct to two decimal places. 3. Let X be the set of all prime numbers less than 50. Determine the number of: (a) elements in X. (Remember, the number 1 is not prime.)
(b 4—sequences on X.
(
( c 6—permutations on X. )
) d) subsets of X.
) (e subsets of X that contain exactly 5 elements. Math 1P66 December 2007 Page 2 of 3 4. (a) Determine the number of automobile license plates (having 4 letters followed by
3 digits) that either start with an A or end with a 9. (b) Consider the set of all 8sequences on the set {0,1}. How many such sequences
do not contain at least 6 consecutive 0s? ((3) Determine the sum of the following inﬁnite series, if it exists. If the sum does not
exist, brieﬂy explain why. 4+8+1—6+32+
3 9 27 (d) Determine the sum of the series 3+9+15++477 5. (a) Use truth tables to test the equivalence of the two statements.
(PﬂQ) —>R vs. (NR) —>[(~ P)U(~ 62)]
(b) Use truth tables to determine whether the following argument form is valid. P —> (QUR)
R <——> Q
NC) —> (POR) 6. (a) Prove that the square of an odd number is also odd. (b) Prove that there are an inﬁnite number of prime numbers.
(c) Prove that W is irrational.
7. (a) Draw two different (that is, not isomorphic) graphs that each have degree sequence
4, 4, 2, 2.
(b) For ONE of the graphs in part (a), write the adjacency matrix, A.
(c) Calculate A2.
(d) Brieﬂy explain what the entries in A and A2 mean. (e) Does your graph contain an Euler cycle? Show that it does, or briefly explain
why not. (f) Does your graph contain a Hamiltonian cycle? Show that it does, or brieﬂy explain
why not. 8. Consider the list of numbers 4, 17, 3, 11, 3, 8, 6, 9. Use the Bubblesort algorithm to place
the numbers in numerical order. List the results of the algorithm after each iteration. Math 1P66 December 2007 Page 3 of 3 9. Prove each statement. (Hint: Using mathematical induction may be helpful.) (a) For every natural number n, n(n + 1)(n + 2) <1>(2)+<2)(3>+<3>(4>+..+n(n+i)I 3 (b) For every natural number n, 11"+2 + 122n+1 is divisible by 133. 10. The entry in row i and column j of the following matrix indicates the weight of the
edge that joins vertex 2‘ to vertex j in a weighted graph. If there is no entry in row 2’
and column j, then vertices i and j are not directly connected by an edge. (a) Draw the corresponding weighted graph. (b) Use Dijkstra’s algorithm to select the path from vertex 1 to vertex 4 that has
minimum weight. Show your work. ...
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