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Loyola University Chicago

Mathematics and Statistics

MPA: Sample Questions

Part A: Elementary Algebra and Logical Reasoning (Answers Below)

1. A pepperoni pizza costs $18.40 and is divided into 16 equal pieces. Siegfried eats six of the pieces. What is the cost of the portion of pizza that Siegfried ate? (Express your answer to the nearest cent).

(a) $1.15 (b) $6.90 (e) $8.24 (d) $7.45 (c) $9.20

2. For which value of k is the expression 2k + 5 a prime number?

(a) k = 5 (b) k = 7 (c) k = 8 (d) k = 10 (e) k = 58

3. Assume that the surface of the Earth is 70% water and that 97.5% of the surface area covered by this water is salt water. What percentage of the surface of the Earth is covered with fresh water? (Express your answer to the nearest tenth of a percent).

(a) 1.8% (b) 18% (c) 0.18% (d) 68.3% (e) 29.3%

4. Suppose that x and y are numbers such that x is 200% larger than y. If y equals 12, then x equals what value?

(a) 24 (b) 30 (c) 2,400 (d) 3,600 (e) 36

5. Simplify the expression ab(a2ba4b)5.

(a) a21b5 (b) a31b11 (c) a41b6 (d) a41b5 (e) a31b5

6. The price of an algebra book together with 8% tax is $54.95. How much is the original price of the book?

(a) $50.00 (b) $51.83 (c) $50.88 (d) $50.55 (e) $49.55

7. What is the slope of the straight line 3x – 9y = 44 ?

(a) 3 (b) 9 (c) 1/3 (d) 1/9 (e) 44

 

 

Part B: Intermediate Algebra (Answers Below)

1. The points (x,y) represented in the table below lie on a straight line. The point (25,t) lies on the same line. What is the value of t ?

x y
1 7
3 13
7 25


(a) t = 95 (b) t = 7 (c) t = 175 (d) t = 79 (e) t = 87

2. Simplify the expression (3x-1)/((x-1)(x-2))+1/(x-1)+1/(x-2).

(a) (3x-2)/((x-1)(x-2)) (b) (5x-4)/((x-1)(x-2)) (c) (3x+2)/((x-1)(x-2)) (d) (5x+4)/((x-1)(x-2)) (e)(3x+1)/((x-1)(x-2))

3. What is the maximum value achieved by the function g(x) = 8 – 4(x – 1)2 ?

(a) 1 (b) 8 (c) 4 (d) 12 (e) no maximum value exists for this function

4. Solve for y in the following system of equations.

x + 5y = 17
5x 2y = 4


(a) y = 3 (b) y = 5 (c) y = 7 (d) y = 17/2 (e) y = 17/4

5. Jack can till a garden in 8 hours. Jill can till the same garden in 6 hours. Working together, how long (in hours) would it take to till this garden? (Express your answer to the nearest tenth of an hour.)

(a) 7.0 (b) 4.0 (c) 3.0 (d) 3.9 (e) 3.4

6. Which of the following binomials is a factor of 21x2 + x – 10 ?

(a) 7x 5 (b) 3x + 2 (c) x 5 (d) 7x + 10 (e) 7x + 5

7. Two trains are 525 miles apart at 1:00 pm and moving toward one another. The faster train travels at 90 mph, the slower train at 60 mph. At what time will the trains meet?

(a) 4:00 pm (b) 4:15 pm (c) 4:30 pm (d) 5:00 pm (e) 4:45 pm

 

 

Part C: Advanced Algebra (Answers Below)

1. What is the sum of the five roots of the equation x5 – 4x3 + 5x – 11 = 0 ?

(a) 4 (b) -4 (c) 5 (d) -5 (e) 0

2. Find the remainder when the polynomial x5 – 2x4 + 3x3 – 2x + 4 is divided by x – 1.

(a) 0 (b) 1 (c) 2 (d) 3 (e) 4

3. Which of the following is a vertical asymptote of the following rational function? y=3(x-1)(x-2)4/((2x-3)(x+4)).

(a) x = -4 (b) x = -3/2 (c) x = 1 (d) x = 2 (e) x = 3

4. If g(x) = 1/x, what is the value of the expression g(x + h) – g(x) ?

(a) h/(x(x+h)) (b) -h/(x(x+h)) (c) 2h/(x(x+h)) (d) -2h/(x(x+h)) (e) 1/h

5. Simplify the following exponential expression. 3^(2log_3 5).

(a) 10 (b) 6 (c) 9 log3 5 (d) 25 (e) log3 25

6. Solve for x in the following equation. (1/3)^x=(81)(9^(x-1)).

(a) x = 2/3 (b) x = 3/2 (c) x = -2/3 (d) x = -3/2 (e) x = 8/9

7. Consider the equation x4 + ax3 + bx2 + cx + d = 0, where a, b, c, and d are integers. If three of the roots of this equation are 3 – i, 3 + i, and 5 – 2i, then the fourth root must be what value?

(a) 11 2i (b) 5 + 2i (c) -5 + 2i (d) -5 2i

(e) cannot be determined
from the given information

 

 

Part D: Precalculus (Answers Below)

1. What is the limiting behavior of the following function as x increases without bound?

y=(x+1)^3 (2x+5)^2 / (x-5)^5.

(a) y tends toward 0 (b) y tends toward 1 (c) y tends toward 4 (d) y increases without bound (e) y tends toward 2

2. The surface area of a sphere varies directly as the square of its radius. If the radius of a sphere is tripled, then the surface area of the sphere increases by what percentage?

(a) 800% (b) 100% (c) 2,700% (d) 2,600% (e) 400%

3. According to one model of population growth, the world’s population, P, in billions, t years after 1990, can be modeled by the function

P(t)=40/(1+11 exp(-0.08 t))

Using this model, one would expect the world’s population in the year 3000 to reach how many billion people?

(a) 40 (b) 20 (c) 3.3 (d) 55 (e) 47.4

4. Simplify the following complex number by writing it in the form a + bi where a and b are real numbers.

100/(3-4i)

(a) 30 + 40i (b) 30 – 40i (c) 300 + 400i (d) 12 + 16i (e) 12 – 16i

5. What is the behavior of the following function as x increases without bound? y=x^100 /exp(x).

(a) y tends toward 0 (b) y tends toward 1 (c) y tends toward 100 (d) y increases without bound (e) y oscillates between 0 and 100

6. What is the period of the function y = 4 + 5 sin(4t + 3) ?

(a) 4 (b) (c) (d) (e)

7. Find a closed form expression for the following (finite) sum.

3/5 + 3/25 + 3/125 + 3/625 + ... + 3/(5^10).

 

(a) 3/4 (b) (3/4)(1 1/510) (c) (3/4)(1 1/511) (d) (4/5)(1 1/510) (e) (4/5)(1 1/511)

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