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Loyola University Chicago

Mathematics and Statistics

Sample math placement questions

Part A: Elementary Algebra and Logical Reasoning (Click Here for Answers)

 

 Answers: 1(B),2(B),3(A),4(E),5(B),6(C),7(C)

 

 

Part B: Intermediate Algebra (Click Here for Answers)

 

ANSWERS: 1(D),2(B),3(B),4(A),5(E),6(E),7(C)

 

Part C: Advanced Algebra (Click Here for Answers)

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

 

ANSWERS: 1(E), 2(E),3(A),4(B),5(D),6(C),7(B)

 

Part D: Precalculus (Click Here for Answers)

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)
 
 
ANSWERS: 1(C),2(A),3(A),4(D),5(A),6(E),7(B)


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