THE UNIVERSITY OF BRITISH COLUMBIA. Mathematics 414 Section 201. FINAL EXAM December 9, 2011

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1 Page 1 of 11 THE UNIVERSITY OF BRITISH COLUMBIA Mathematics 414 Section 201 No calculators allowed Final exam begins at 12 noon and ends at 2:30 pm FINAL EXAM December 9, 2011 NAME STUDENT NUMBER

2 Page 2 of 11 [10] 1. Two walls and the ceiling of a room meet at right angles at a point P. A fly is in midair near this corner, 1 metre from one of the walls, 8 metres from the second wall, and 9 metres from P. How many metres below the ceiling is the fly?

3 Page 3 of [5] (a) April 24, 2012, is a Tuesday. What day of the week was April 24, 2010? Answer the same question for April 24, Explain how you arrive at your answer. [5] (b) You started a long mathematics exam at noon on December 9, You were told that you could work as long as you liked. You worked 475 hours straight. At what time of day did you finish? At the time of finishing how long did you have to get ready for your New Years Eve party which started at 9pm on December 31 st?

4 Page 4 of 11 [10] 3. The distances from a point inside a square to 3 consecutive vertices of the square are 9, 10, and 7. Find (exactly) the area of the square. (Hint: The law of cosines says that if T is a triangle with sides of length a, b, and c, then c squared equals a squared plus b squared minus 2ab cosc where C is the angle opposite side c).

5 Page 5 of 11 [10] 4. Roberta Maples received a birthday cheque for y dollars and z cents from her boyfriend. When she tried to cash it at the ABC Department store she found that due to a special store promotion she received instead z dollars and y cents. This was 99 cents less than twice the amount of the original birthday cheque. How much was the original cheque for?

6 Page 6 of 11 [10] 5. A voter with a rowboat finds herself in an unusual situation. On one side of a river is Bob Dole, Bill Clinton, and a big bag of Burger King Whoppers. The voter must get Dole, Clinton, and the bag of burgers across the river to the other side. Her boat, however, is large enough for her and just one other item or person. If she leaves Dole alone with Clinton, then Dole (an old war veteran) will beat up Clinton. If she leaves Clinton alone with the bag of Whoppers, then Clinton (a big burger lover) will devour the entire contents of the bag. Of course, Dole, Clinton, and the bag are all incapable of rowing the boat across the river. Is it possible for this voter to get all her cargo across the river? If so, carefully explain her method; if not, carefully explain why not.

7 Page 7 of 11 [10] 6. Suppose (log 3x)(log 5x) = Given that this equation has 2 distinct roots, find the product of the two roots.

8 Page 8 of Suppose the course offerings at the University of Texas at Austin are simplified. Only 16 classes are offered, each meets at a different hour each week in the stadium, and there is no enrollment limit on any course. Each student is required to take exactly four courses. There are no prerequisites, and every student can take any class. UT at Austin has about 44,000 students. [5] (a) If the students choose independently, how many different course schedules are possible? [5] (b) Will some students have to have the exact same program or not? Justify your answer.

9 Page 9 of 11 [10] 8. Bernadette is playing a game at Hogwarts. There are 10 devil s snares in front of her in a line. Starting from the first snare, in the first round of the game she casts lumos on every second snare (i.e second snare dies). In the second round she casts on every third snare, and in the third round she casts on every fourth snare. She then repeats the process for the following rounds. However, after 1 round has passed, a dead snare will come back to life, and after 2 rounds have passed, a live snare will die(assume that the change happens before she casts any spells for that round). If every time she casts lumos on a live snare, it dies, and every time she casts on a dead snare it comes back to life, how many snares are alive after the 4 th round, assuming all the snares started out alive?

10 Page 10 of 11 [10] 9. A sequence consists of 2010 terms. Each term after the first is 1 larger than the previous term. The sum of the 2010 terms is If every second term is added up, starting with the first term and ending with the second last term, then what is the sum of these 1005 terms?

11 Page 11 of 11 [10] 10. Let n be the largest integer for which 14n has exactly 100 digits. Counting from right to left, what is the 68th digit of n? Show work.

Furman University Wylie Mathematics Tournament Ciphering Competition. March 11, 2006

Furman University Wylie Mathematics Tournament Ciphering Competition March 11, 2006 House Rules 1. All answers are integers(!) 2. All answers must be written in standard form. For example, 8 not 2 3, and