Section A
15 Multiple Choice Questions
(+3)
1
geometry
medium
In the figure below, point O is the center of the circle, AC is a circle diameter and has a length of 24 inches, point B is on the circle. AB has the same length as the circle radius. What is the length of the arc AB in inches?
A
Not enough information
B
π inches
C
2π inches
D
2.5π inches
E
4π inches
Show answer
Answer: E
The radius is AC/2 = 12 inches, so OA = OB = 12. Since AB is also given to equal the radius, AB = 12 too, making triangle OAB equilateral, so the central angle ∠AOB = 60°. The arc AB has length (60/360)×2π(12) = (1/6)(24π) = 4π inches. The answer is E.
2
algebra
medium
Let T_n be the sum of all the positive integers in {1,2,3,⋯,n−1,n}, where n is a positive integer. What is √(8T_n+1)?
A
n−1
B
n+1
C
2n+1
D
2n−1
E
3n+1
Show answer
Answer: C
T_n = n(n+1)/2, so 8T_n+1 = 4n(n+1)+1 = 4n²+4n+1 = (2n+1)². Taking the square root gives 2n+1. The answer is C.
3
probability
medium
A round table has a diameter of 12 feet. If two people choose to sit randomly around the table. What is the probability that they will be at least 6 feet apart?
A
1/2
B
1/3
C
2/3
D
1/4
E
3/4
Show answer
Answer: C
The table has radius 6 feet. Fix one person's position; let φ be the angular separation of the second person, uniform on [0°,360°). The straight-line (chord) distance between them is 2(6)sin(φ/2) = 12 sin(φ/2). We need this ≥ 6, i.e. sin(φ/2) ≥ 1/2, i.e. φ/2 ∈ [30°,150°], i.e. φ ∈ [60°,300°], an interval of 240° out of 360° possible. Probability = 240/360 = 2/3. The answer is C.
4
algebra
medium
Define two functions f(x) = 2^x − 15 and g(x) = 5x+6 over real numbers. What is the value of the following expression? f⁻¹(g(1/5+2))
A
16
B
5
C
17
D
1/5
E
19
Show answer
Answer: B
First, 1/5+2 = 11/5, so g(11/5) = 5(11/5)+6 = 11+6 = 17. Now find f⁻¹(17): solve 2^x−15 = 17, so 2^x = 32 = 2⁵, giving x = 5. So f⁻¹(17) = 5. The answer is B.
5
geometry
medium
In the figure below, two poles are mounted upright on the ground. Ropes are then used to connect the top of one pole to the foot of the other. The ropes are tight, intersecting at point K. We know that AB is six feet tall and CD is four feet tall. What is the height of point K with respect to the ground?
A
Not enough information
B
2 feet
C
2.4 feet
D
2.5 feet
E
3.2 feet
Show answer
Answer: C
This is the classic crossed-ladders configuration: for two poles of heights h1 and h2 with ropes crossing between the top of each pole and the foot of the other, the height of the intersection point above the ground is h1·h2/(h1+h2), independent of the distance between the poles. Here h1=6, h2=4, so the height is (6×4)/(6+4) = 24/10 = 2.4 feet. The answer is C.
6
algebra
medium
If the following equation is true, what is the sum of values of A and B? (16x−3)/(10x²−13x−30) = A/(2x−5) + B/(5x+6)
A
4
B
5
C
6
D
7
E
8
Show answer
Answer: B
Factor the denominator: 10x²−13x−30 = (2x−5)(5x+6). So 16x−3 = A(5x+6)+B(2x−5) = (5A+2B)x + (6A−5B). Matching coefficients: 5A+2B=16 and 6A−5B=−3. From the first, B = 8−2.5A; substituting into the second: 6A−5(8−2.5A) = −3 → 6A−40+12.5A = −3 → 18.5A = 37 → A=2, so B = 8−5=3. A+B = 5. The answer is B.
7
probability
medium
David is throwing darts at a circular target shown in the figure. There are two concentric circles on the target. Circle C₂ has a diameter of 9 inches; circle C₁ has a diameter of 3 inches. Assuming all the darts will hit the big circle and will land at a random point. What is the probability that a dart will land within the shaded circle C₁?
A
1/3
B
1/4
C
1/6
D
1/9
E
1/π²
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Answer: D
The probability equals the ratio of areas, which equals the ratio of the squares of the radii (or diameters): (3/9)² = (1/3)² = 1/9. The answer is D.
8
algebra
medium
In the figure below, the graphs of two functions, f(x) and g(x), are shown. Both functions are defined over real numbers. What is the value of g(f(2))?
A
−1
B
0
C
1
D
2
E
3
Show answer
Answer: C
Reading the graph: f(x) is a parabola with vertex (0,−1) passing through (1,0) and (2,3), consistent with f(x)=x²−1. g(x) is a V-shape with vertex (2.5,0) and slope magnitude 2, consistent with g(x)=2|x−2.5|. First, f(2) = 4−1 = 3. Then g(3) = 2|3−2.5| = 2(0.5) = 1. The answer is C.
9
algebra
medium
In the figure below, an object is located at point A. There are two forces represented by vectors u=AF₁ and v=AF₂ acting on the object. What is the combined effect of u and v in terms of a vector?
A
〈3,7〉
B
〈8,−1〉
C
〈√5,−2〉
D
〈6,−4〉
E
〈1,8〉
Show answer
Answer: B
Reading the grid, u = AF₁ = 〈2,3〉 (2 right, 3 up from A) and v = AF₂ = 〈6,−4〉 (6 right, 4 down from A). The combined effect is u+v = 〈2+6, 3−4〉 = 〈8,−1〉. The answer is B.
10
probability
hard
Two players, Adam and Bob, are playing five rounds of a chess game. The first player to win three rounds wins the whole game and takes the award. Based on the history of the players' performance, Adam has a probability of 0.6 winning a round. Bob has a probability of 0.4 winning a round. What is the probability, rounded to the second decimal place, that Bob wins the chess game of five rounds?
A
0.21
B
0.24
C
0.32
D
0.41
E
0.64
Show answer
Answer: C
Bob (p=0.4) can win the match in exactly 3, 4, or 5 games, always winning the final game. P(win in exactly n games) = C(n−1,2)×0.4³×0.6^(n−3). n=3: C(2,2)(0.4)³=0.064. n=4: C(3,2)(0.4)³(0.6)=3(0.064)(0.6)=0.1152. n=5: C(4,2)(0.4)³(0.6)²=6(0.064)(0.36)=0.13824. Total = 0.064+0.1152+0.13824 = 0.31744 ≈ 0.32. The answer is C.
11
algebra
medium
What is the remainder of the following polynomial division? (x⁵−5x⁴+11x³−4x²−24x+31)/(x−2)
A
x−1
B
0
C
-2
D
5
E
7
Show answer
Answer: E
By the Remainder Theorem, the remainder of dividing by (x−2) equals f(2). f(2) = 32−80+88−16−48+31 = 7. The answer is E.
12
algebra
medium
What is the product of all the non-real roots of the following equation? x⁴−16=0
A
4
B
−4
C
4i
D
−4i
E
8
Show answer
Answer: A
x⁴=16 gives x = 2, −2, 2i, −2i. The non-real roots are 2i and −2i, and their product is (2i)(−2i) = −4i² = 4. The answer is A.
13
algebra
medium
What is the solution to the following system of equations represented in a matrix form? [[2,−3],[3,2]][x,y] = [22,7]
A
[x,y]=[2,3]
B
[x,y]=[5,−4]
C
[x,y]=[−5,4]
D
[x,y]=[1,0]
E
[x,y]=[−22,−7]
Show answer
Answer: B
The system is 2x−3y=22 and 3x+2y=7. Multiply the first by 2 and the second by 3: 4x−6y=44 and 9x+6y=21. Adding gives 13x=65, so x=5. Substituting into 3x+2y=7 gives 15+2y=7, so y=−4. The answer is B.
14
probability
hard
Let S=A+B+C. If A, B, C are each randomly assigned a number from {1,3,5,7}, what is the probability that S=11?
A
5/32
B
27/64
C
5/128
D
1/8
E
3/16
Show answer
Answer: E
There are 4³=64 equally likely outcomes. The unordered triples from {1,3,5,7} summing to 11 are {1,3,7} (6 orderings, all distinct), {1,5,5} (3 orderings), and {3,3,5} (3 orderings), giving 6+3+3=12 favorable outcomes. Probability = 12/64 = 3/16. The answer is E.
15
number-theory
medium
In the base-ten number system, ten digits {0,1,2,3,4,5,7,8,9} are used for number representation. The base-ten number 923 therefore means 9×10²+2×10+3×1. Similarly, in the base-five number system, five digits {0,1,2,3,4} are used for number representation. A base-five number 321₅ means 3×5²+2×5+1×1. What is the base-five representation of the base-ten number 237?
A
3212
B
1422
C
2214
D
4122
E
4321
Show answer
Answer: B
Divide repeatedly by powers of 5: 237 = 1×125 + 112; 112 = 4×25 + 12; 12 = 2×5 + 2; 2 = 2×1. So 237 in base five is 1422. The answer is B.