ARML Practice – December 7, 2014
AMC 10/12 Practice Quiz Problems
1. Rhombus ABCD is similar to rhombus BFDE. The area
of ABCD is 24, and m BAD  60 . What is the area
of rhombus BFDE?
(A) 6
(B) 4 3
(C) 8
(E) 6 3
(D) 9
2. In rectangle ABCD, we have A = (6, -22), B = (2006, 178), and D = (8, y), for some
integer y. What is the area of rectangle ABCD?
(A) 4000
(B) 4040
(C) 4400
(D) 40,000
(E) 40,400
3. Square ABCD has side length s, a circle centered at
E has radius r, and r and s are both rational. The
circle passes through D, and D lies on BE . Point F
lies on the circle, on the same side of BE as A.
Segment AF is tangent to the circle, and
AF  9  5 2 . What is r ?
s
(A)
1
2
(B)
5
9
(C)
3
5
(D)
5
3
(E)
9
5
4. Regular hexagon ABCDEF has vertices A and C at (0, 0) and (7, 1), respectively. What is
its area?
(A) 20 3
(B) 22 3
(C) 25 3
(D) 27 3
(E) 50
S1 
 x, y  | log 1  x
and S2 
 x, y  | log  2  x
5. Let:
2
10
10
2

 y 2   1  log10  x  y 

 y 2   2  log10  x  y 
What is the ratio of the area of S 2 to the area of S1 ?
(A) 98
(B) 99
(C) 100
(D) 101
(E) 102
6. A circle of radius r is concentric with and outside a regular hexagon of side length 2. The
probability that three entire sides of the hexagon are visible from a randomly chosen
point on the circle is 1 . What is r?
2
(A) 2 2  2 3
(B) 3 3  2
(C) 2 6  3
(D) 3 2  6
(E) 6 2  3
7. Rectangle ABCD has area 2006. An ellipse with area 2006 passes through A and C and
has foci at B and D. What is the perimeter of the rectangle? (The area of an ellipse is
 ab , where 2a and 2b are the lengths of its axes.)
(A)
16 2006

(B)
1003
4
(D) 6 2006
(C) 8 1003
(E)
32 1003

8. A unit square is rotated 45 about its center. What is the area of the region swept out by
the interior of the square?
(A) 1 
2 

2
4
(B)
1 

2 4
(C) 2  2 

4
(D)
2 

2
4
(E) 1 
2 

4
8
9. Two tangents to a circle are drawn from a point A. The points of contact B and C divide
the circle into arcs with lengths in the ratio 2:3. What is the degree measure of BAC ?
(A) 24
(B) 30
(C) 36
(D) 48
(E) 60
10. Three circles with radius 2 are mutually tangent. What is the total area of the circles and
the region bounded by them, as shown in the figure?
(C) 12  3
(A) 10  4 3 (B) 13  3
(D) 10  9 (E) 13
11. Jesse cuts a circular paper disk of radius 12 along two radii to form two sectors, the
smaller having a central angle of 120 degrees. He makes two circular cones, using each
sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller
cone to that of the larger?
(A)
1
8
(B)
1
4
(C)
10
10
(D)
5
6
(E)
10
5
12. Mary divides a circle into 12 sectors. The central angle of these sectors, measured in
degrees, are all integers and they form an arithmetic sequence. What is the degree of the
smallest possible sector angle?
(A) 5
(B) 6
(C) 8
(D) 10
(E) 12
AMC 10/12 Practice Quiz Problems – Solutions
Answers: 1-C 2-E 3-B 4-C 5-E 6-D 7-C 8-C 9-C 10-A 11-C 12-C
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