Problem 1 :
What is the radius of a circle that has a circumference of π?
A) 1/4 B) 1/2 C) 1 D) 2 E) 4
Solution:
Given that the circumference of circle is π.
We know that the circumference of circle is 2πr, where r is radius of circle.
Circumference = π = 2πr
r = 1/2
So, option (B) is correct.
Problem 2 :
In the figure above, arc SBT is one quarter of a circle with center R and radius 6. If the length plus the width of rectangle ABCR is 8, then the perimeter of the shaded region is
A) 8 + 3π B) 10 + 3π C) 14 + 3π D) 1 + 6π
E) 12 + 6π
Solution:
Given, radius = 6
AS = 6 - a , CT = 6 - b , AC = RB = 6 ( radius )
Perimeter of the shaded region = AS + AC + CT + C(SBT)
Perimeter = ( 6 - a ) + 6 + ( 6 - b ) + 3π
Perimeter = 18 - ( a + b ) + 3π
length + width = 8
Perimeter = 18 - 8 + 3π
Perimeter = 10 + 3π
So, option (B) is correct.
Problem 3 :
In the xy-coordinate plane, what is the area of the square with opposite vertices at (-2, -2) and (2, 2) ?
A) 4 B) 8 C) 16 D) 32 E) 64
Solution:
By finding the distance between the above two points, we get the side length of the square.
So, option (D) is correct.
Problem 4 :
In rectangle ABCD, point E is the midpoint of BC. If the area of quadrilateral ABED is 2/3, what is the area of rectangle ABCD?
A) 1/2 B) 3/4 C) 8/9 D) 1 E) 8/3
Solution :
Area of rectangle = AB x AD
Area of quadrilateral = 1/2 x height x sum of parallel sides
1/2 x AB x (AD + BE) = 2/3
1/2 x AB x (AD + 1/2 x AD) = 2/3
1/2 x AB x (3/2 x AD) = 2/3
AB x AD = (2/3) x (4/3)
AB x AD = (8/9)
So, option C is correct.
Problem 5 :
In right circular cylinder above has diameter d and height h. Of the following expressions, which represents the volume of the smallest rectangular box that completely contains the cylinder?
A) dh B) d^{2}h C) dh^{2 }D) d^{2}h^{2 }E) (d + h)^{2}
Solution:
Diameter of the cylinder will be the width of the rectangular box, length of the rectangular box is h.
Volume of rectangular box = length x width x height
= h x d x d
= h d^{2}
So, option B is correct.
Problem 6 :
If the volume of a cube is 8, what is the shortest distance from the center of the cube to the base of the cube?
A) 1 B) 2 C) 4 D) √2 E) 2√2
Solution:
Given, volume of a cube = 8
Volume of cube = a^{3}
a^{3} = 8
a = 2
Shortest distance from the center of the cube to the base is half of its side length a
= a/2
= 2/2
= 1 unit
So, option (A) is correct.
Problem 7 :
In the figure above, point A is the center of the circle and segments BD and CE are diameters. Which of the following statements is true?
A) CA > 6 B) ED > 4 C) BA < 4 D) CA = 4 E) ED = 4
Solution:
In triangles AED and BCA.
AD = AE = AC = AB (radii)
∠EAD = ∠BAC (A)
AE = AC (S)
AD = AB (S)
Using SAS, the triangles are congruent.
So, ED = 4, option E.
Problem 8 :
In the figure above, the two circles are tangent at point B and AC = 6. If the circumference of the circle with center A is twice the circumference of the circle with center C, what is the length of BC?
A) 1 B) 2 C) 3 D) 4 E) 6
Solution:
Let BC (r) be the radius of smaller circle and AB (R) is the radius of the larger circle.
R + r = 6
Circumference of the larger circle = 2(circumference of the smaller circle)
2πR = 2(2πr)
R = 2r
R = 2(6-R)
R = 12 - 2R
3R = 12
R = 4
BC = r
AB + BC = 6
R + r = 6
BC = 6 - 4
BC = 2
Problem 9 :
The figure above shows part of a circle whose circumference is 45. If arcs of length 2 and length b continue to alternate around the entire circle so that there are 18 arcs of each length, what is the degree measure of each of the arcs of length b ?
A) 4° B) 6° C) 10° D) 16° E) 20°
Solution:
You know that 18(2 + b) = 45, so you can calculate b:
18(2 + b) = 45
36 + 18b = 45
18b = 9
b = 0.5
0.5/45 = x/360
x = 4
Problem 10 :
In the figure above, the circle with center O is inscribed in square ABCD. What is the area of the shaded portion of the circle?
Solution:
Area of sector = θ/360 πr^{2}
Side length of the square = diameter of the circle
radius = 1
= (90/360) π1(1)^{2}
= π/4
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