Download 2012ApolloR2Test - Rocket City Math League

Name________________________________________
Rocket City Math League
2012-2013
Round 2
Apollo Test
Answers must be written inside the adjacent answer boxes. All answers must be written in exact, reduced, simplified, and rationalized form. All
decimals and mixed numbers must be written as proper or improper fractions. No calculators, books, or other aides may be used. Time Limit: 45
minutes.
1. Jorge Getson and his wife, Gane, decided to take their kids Jelroy and Eudy on a space vacation to Moon X.
They were flying against the wind on the way to Moon X and with the wind on the way back from Moon X. With
no wind, the space-vehicle they take travels at a rate of 45 mph. If the trip to Moon X took eighteen hours, and the
trip back from Moon X took fourteen hours, what was the wind’s speed in mph?
(1 point)
2. Find cos
2
  sin 2 
for the following triangle (not drawn to scale).
68
25 2
7
0
3. Find the determinant of the matrix 
1

0

5 8 4
0 3 0
.
8 4 3

2 9 1
(1 point)
(1 point)
4. Simplify: 3log 4 + log 18 – 2log 6.
(1 point)
5. Angela wishes to take an intergalactic vacation this summer to Mercury. However, before doing so, she needs
to change 65 moops left over from last year’s visit to Saturn to Mercurian coins, which are zoops, froops, goops,
and bloops. Eleven moops equal five zoops, and five moops equal eighteen froops. Thirteen zoops also equal eight
froops. Eight froops equal nine bloops. Twelve bloops equal six goops. If Angela wants the same number of
zoops, froops, goops, and bloops, how many should she have of each?
(2 points)
6. Secret agent Bill has allowed himself to be trapped in the dungeons of a Martian castle. The only way to escape
is to enter the correct code into the keypad on the dungeon’s door. Luckily, there are instructions for finding the
six-digit code carved into the walls of the dungeon, which look like this (where an is the nth term of a sequence and
r is the common ratio):
First, enter the sum of the infinite geometric sequence 81 + 162/3 + 36 + 72/3 …
Second, enter the common difference in an arithmetic progression if a18 = 122 and a37 = 255.
Third, enter the first term of a geometric series where a4 = 375 and r = 5/3.
Find the six-digit sequence to free Bill.
(2 points)
7. Determine the remainder when 9x7 + 4x6 + 3x5 + 18x4 + 13x2 + x is divided by x+1.
(2 points)
8. Find the focal width of the conic section described by the equation 4x2 + 32x + 6y2 – 36y = 14.
(3 points)
9. Gojidsfo has forgotten the entry code for his laboratory. The code is equal to P+AS, where P is the period, A
x 
  . What is Gojidsfo’s
2 6
the amplitude, and S the (least positive) phase shift of the function y  3 sin 
entry code?
(3 points)
10. Once in his lab, Gojidsfo begins a computer simulation in which an avatar is placed in an environment with
infinitely many rooms. While in a room, the avatar cannot see anything outside the room. Every room has 7 tubes
connecting it to other rooms, and all tubes are identical in appearance. One tube in every room takes him to a
treasure room containing $150; all other tubes take the avatar to a room other than a treasure room. In the process
of traveling to a room that is not a treasure room, the avatar is knocked unconscious and robbed of $50; he awakes
in the next room unaware of which tube brought him to the room. The simulation ends when the avatar claims the
money in a treasure room. If the avatar starts with $1000 and can have negative money, what is the expected
amount of money the avatar will have at the end of the simulation?
(4 points)
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