6) Let U = 2 * 2004^2005, V = 2004 ^ 2005, W = 2003 * 2004^2004, X = 2 * 2004^2004, Y = 2004^2004, and Z = 2004^2003. Which of the following is the largest?
U - V
V - W
W - X
X - Y
Y - Z
Easy.
U-V = 2004^2005
V-W = 2003 * 2004^2004
W-X = 2001 * 2004^2004
X-Y = 2004^2004
Y-Z = 2004^2004 - 2004^2003
They're already in size order. U-V is the largest.
Triangles EAB and CBA share base AB. If <EAB and <ABC are right angles, AB = 4,, BC = 6, AE = 8, and line AC and line BE intersect at D. What is the difference between the areas of triangle ADE and triangle BDC?
2
4
5
8
9
ADE Area = 1/2 (6*4) + 1/2 (8*4) + 1/2 (28*20) - 1/2(22*20)
BDC Area = 1/2 (28*20) - 1/2(22*20)
Difference in area = 1/2 (6*4) + 1/2 (8*4) = 12 + 16 = 28.
Hmmm. I've crapped up somewhere there. :unsure:
14) A sequence of three real numbers forms an arithmetic progression with a first term of 9. If 2 is added to the second term and 20 is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term of the geometric progression?
1
4
36
49
81
Oooh, this one uses algebra. Fun.
9 9
B B+2
C C+20
(C+20)/(B+2) = (B+2)/9
C-B = B-9
B+B = C+9
B = (C+9)/2
(C+20)/(((C+9)/2)+2) = (((C+9)/2)+2)/9
9*((C+20)/(((C+9)/2)+2)) = (((C+9)/2)+2)
(9C+180) = ((C/2)+6.5)*((C/2)+6.5)
9C+180 = (1/4)C² + 42.25 + 6.5C
(1/4)C² -2.5C - 137.75 = 0
Use Quadratic Formula
C=-19
B = (9-19)/2
B = -5
Arithmetic Progression: 9, -5, -19. (Change: -14)
Geometric Progression: 9, -3, 1. (Multiplier: -1/3)
Answer: 1
21) If the sum of (cos(x))^(2n), with lowerlimit n = 0 and upper limit infinity, is equal to 5, what is the value of cos(2x)?
1/5
2/5
rt(5)/5
3/5
4/5
Hmmm, can you re-check the wording in that question? I'm getting odd answers. :/
22) Three mutually tangent spheres of radius 1 rest on a horizontal plane. A sphere of radius 2 rests on them. What is the distance from the plane to the top of the larger sphere?
3 + rt(30) / 2
3 + rt(69) / 3
3 + rt(123) / 4
52 / 9
3 + 2 * rt(2)
Blah, that's just trig - the three spheres form a 2x2x2 equilateral triangle, with the large one forming it into a pyramid. I can't be arsed to calculate the exact value, it's about five.
24) A plane contains points A and B with AB = 1. Let S be the union of all disks of radius 1 in the plane that cover line AB. What is the area of S?
2pi + rt(3)
8pi / 3
3pi - rt(3)/2
10pi / 3 - rt(3)
4pi - 2 * rt(3)
Is that the center of the disk is on the line, or just some part of the disk is on the line? It's too vague.
