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MATH PLACEMENT Sample Questions
Northern Virginia Community College – Manassas Campus
Testing Center, MH Building, Room 112 (703-257-6645)
ƒ BEFORE YOU CAN TEST, you must be admitted to the college and have a student ID
number (EMPL ID). Go to http://www.nvcc.edu/novaconnect/.
ƒ You must have a current photo ID.
ƒ An appointment is not required to test.
ƒ You must start the test at least 2 hours before closing.
ƒ The Math Placement test is untimed.
ƒ Math test scores are valid for one year.
ƒ You may not retest if you are currently enrolled in a developmental math course.
YOU ARE RESPONSIBLE for being prepared to test and testing in a serious manner.
YOU will NOT be allowed to retest because you rushed through this test.
► Scratch paper/pen/online calculators are provided.
►Personal calculators may NOT be used.
►Cellphones must be turned OFF while testing.
►COLLEGE POLICY: The Math Placement Test may be taken once per semester (once every 3
months) but not more than 3 times per year.
Students with disabilities:
You are subject to the same testing policy stated above.
If you require accommodations related to audio/visual needs, please go to Counseling before
testing.
Northern Virginia Community College requires a minimum skill level for entry into the majority of its
mathematics courses as well as some of the science and computer science courses. The primary tool
for determining your skill level is an interactive computer generated test called COMPASS, developed
by American College Testing in Iowa City, IA.
You are strongly encouraged to review your skills prior to taking the test. The following pages will
give you a basic idea of what you can expect to see on the test.
The test is divided into five levels of material: prealgebra, algebra, college algebra, geometry, and
trigonometry. The test will begin with prealgebra questions.
As long as your answers are correct, the computer will keep giving you harder questions. Your
correct/incorrect answers will determine the number and type of problems that you will see. When the
computer program determines your placement level, the test will stop.
MATHEMATICS PLACEMENT TEST: WHAT IS THE TEST LIKE?
The following is a basic description of the type of material that should be mastered before
entering each course level. Simple examples have been given to illustrate the terminology.
These examples are not indicative of the complexity of the problems you will see on the
placement test. They merely indicate the ideas, which you should review before taking the test.
MATH 001, 002 – NO PLACEMENT TEST REQUIRED
MATH 003, 103, 120, 126 – PROFICIENCY IN PRE-ALGEBRA REQUIRED
MATERIAL IN PRE-ALGEBRA
1. Arithmetic with integers:
75 × 123 (answer: 9225);
1023 + 123 + 85 (answer: 1231);
39596 ÷ 76 (answer: 521)
5280 – 1760 (answer: 3520);
2. Arithmetic of fractions:
2 5
19
+
(answer:
);
3 2
6
5 3
1
−
(answer:
);
6 4
12
2 5
5
× (answer: )
3 2
3
3. Represent decimal numbers as fractions in simplest form and fractions as decimal numbers:
3
(answer: 0.75);
4
0.25 (answer :
4. Represent percentages as decimal values:
1
)
4
15
1
% (answer: 0.155 )
2
5. Addition, subtraction, multiplication and division of decimal numbers:
0.3 × 1.4 (answer: 0.42)
6. Solve basic word problems:
Jack bought 3 items that cost $25 each, and 5 items that cost $40 each. What was the total cost,
including 5% sales tax? (answer: $288.75)
If you score 72% on a 25-question test, how many questions did you answer correctly? (answer:
18)
If a book, which costs $100, is discounted 20%, and then the discounted price is raised by 20%,
what is the final price of the book? (answer:$96)
If you travel for 1 hour at 10 miles per hour and for 2 hours at 25 miles per hour, what is your
average speed for the entire trip? (answer: 20 miles per hour)
7. Addition, subtraction, multiplication and division using signed numbers:
(−3) × (−12) (answer: 36);
−4 +15 –5 (answer: 6)
8. Using ratios and proportion to solve problems: If 6 gallons of unleaded gasoline cost $9.72,
how much would 9 gallons cost?
(answer: $14.58)
MATH 004, 060, 151, 152, 157, 241 - PROFICIENCY IN ALGEBRA AND ALL
PREVIOUS LEVELS REQUIRED
MATERIAL IN ALGEBRA
9. Substituting values of variables in expressions:If u = 3 and v =1 then u 2 + 5 uv = ? (answer: 24)
10. Solve linear equations with a single variable: How much is x if 9x –14 = 4? (answer:2)
If w + a +2c = 9, how much is w ? (answer: 9 – a – 2c)
11. Simplify polynomial expressions :
(5 x2 – 7x + 4) + (x2 + 2x – 1) (answer: 6x2 –5x +3)
12. Binomial products and factoring trinomial expressions:
(x + 3) ( x – 1 ) (answer: x2 + 2x – 3 );
x2 + 4x +4 (answer: (x + 2) (x + 2))
13. Division of polynomials:
x3 + 3x2 – 4x + 3 (answer: x2 + 2x – 6, remainder 9)
x+1
14. Distance between two points in the plane:
How far is it from (1,3) to (2,4)? (answer:
2)
15. Simplification, multiplication and division of square roots:
27 (answer:
3
);
2
2
8 (answer: 4)
12
16. Understanding graphs of linear equations and intersections of lines:
Identify the graph of the line having equation 2x – y = 3.
Find the point of intersection of the two lines x + y = 2 and 2x – y = 1.
(answer: (1,1).)
17. Adding, subtracting, multiplying and dividing rational expressions:
2
5 x + 10
x
.
+
2
x+2 3
x
(answer:
2 x + 15
)
3x
18. Using the quadratic formula:
Find all numbers x such that 2x2 + 4x +1 =0
(answer: x is either
−2+ 2
−2− 2
or
)
2
2
19. Pythagorean Theorem and its uses: If a right triangle has a hypotenuse of 8 cm and a side of
4 cm, what is the length of the third side? (answer: 4 3 cm)
20. Find all numbers x such that |3x –4| < 4.
(answer: 0< x <
8
)
3
21. Understand and use exponents (including negative and fractional exponents)
(22 –32)3 (answer: -125);
8 -2/3 (answer: 1/4
22. Solving equations with rational expressions:
);
8
2 (answer: 4)
Find all numbers x such that x =
5
3
–
2
2x
(answer: 3/2 or 1)
23. Find the slope of the line having equation 3x +4y = 5. (answer: -3/4 )
MATH 007, 115, 150, 163, 166, 181 – PROFICIENCY IN GEOMETRY AND ALL
PREVIOUS LEVELS REQUIRED
MATERIAL IN GEOMETRY
Pythagorean Theorem: ABC is a right triangle if and only if a2 + b2 = c2, where ∠ C = 90°; c is the
hypotenuse (the side opposite the 90° angle); a is the side opposite ∠ A; b is the side opposite ∠B.
In a right triangle with two 45° angles, the 2 legs are equal. The Pythagorean Theorem can then be
used to find the lengths of all sides, if you are given the length of any one side.
In a right triangle with 30° and 60° angles, the leg opposite the 30° angle is always half as long as the
hypotenuse. The Pythagorean Theorem can again be used to find the lengths of all sides, if you are
given the length of any one side.
An isosceles triangle contains two equal angles, and the sides opposite these equal angles have equal
length. An equilateral triangle has three sides of the same length, and all three angles measure 60°.
In any triangle, the sum of all the angles is always 180°.
Two triangles are called similar if their angles have the same measurements. In the two triangles,
sides opposite a pair of equal angles are called corresponding sides. The ratios of corresponding sides
a b c
= =
d e f
are all equal:
C
F
b
a
e
d
D
A
E
f
B
c
Area Formulas:
Rectangle: A = L·W, where L = length; W = width
Circle: A = π r2 , where r = radius
Triangle: A =
1
bh
2
where b = base; h = height
h
b
Perimeter formulas:
h
h
b
b
Triangle: P = a + b + c where a , b and c are the three sides
Rectangle: P = 2L + 2W, where L = length; W = width
Circle: C = 2 π r or C = π d where C = circumference (a synonym for perimeter for circles); r =
radius; d = diameter)
24. Given that the legs of a right triangle are 4 and 3, find the length of the hypotenuse.
(answer: 5)
25. Given that one leg of a right triangle is 7 and the hypotenuse is 8, find the length of the other leg.
(answer:
15 )
26. In the isosceles triangle shown, a = b and ∠ B = 40°. Find the measure of ∠ C.
100°)
C
b
A
(answer:
a
c
B
27. In a triangle, ∠ A = 15° and ∠ C = 70°. Find the measure of ∠ B.
28. Find the area of a circle if the diameter is 10 cm.
(answer:
95°)
(answer: 25 π cm2 )
29. Find the area of a rectangle if the length is 5 yards and the width is 2 feet.
(answer: 30 ft2 or
10 2
yd. )
3
30. In a right triangle, ∠ A = 60° and side a is opposite ∠ A. If the hypotenuse c is 1 cm, find the
lengths of the legs a and b.
(answer: a =
3 / 2 cm , b =
1
cm)
2
31. Find the area of a right triangle if the lengths of the legs are 4 cm. and 7 cm., respectively.
(answer: 14 cm2 )
32. Triangle ABC is similar to triangle DEF where ∠A = ∠D, ∠B = ∠E and ∠C = ∠F.
If AB = 3, BC = 5,
AC = 6 and DE = 8, find the lengths of sides EF and DF.
(answer: DF = 16 and EF =
40
)
3
MATH 270, 271 – PROFICIENCY IN COLLEGE ALGEBRA AND ALL PREVIOUS
LEVELS REQUIRED
MATERIAL IN COLLEGE ALGEBRA
33. Quadratic equations and inequalities:
(answer: x <3/2 or x>3)
Find all numbers x such that 2x2 –9x > –9.
34. Completing the square and graphs of parabolas
Find the vertex of the parabola: y = x2 + 6x +5 (answer: (-3, -4) )
35. Exponential equations:
36. Factorial notation:
How much is x if 3x = 81?
(answer: x = 4)
How much is 6!/4!? (answer: 30)
37. Understanding what is meant by a logarithm: How much is y if log2 y = –4?
(Answer: y =
1
)
16
38. Understand and use the equation of a circle: Find the center and radius of the circle having
equation (x – 1)2 + (y + 3)2 = 4
(answer: Center (1, -3), radius 2)
39. Use of functional notation, including composite functions:
If f and g are functions such that f(x) = x2 + x + 1 and g(x) = x + 1, find f (g(3)) (answer: 21)
If g is a function such that g(x) = ax2 –3 and g(3) = 3, find g(6). (answer: 21)
40. Be able to find the domain and range of a function: What is the domain of f(x) = 2 + x ?
(answer: All real numbers greater than or equal to –2.)
What is the range of g(x) = x2 – 1? (answer: All real numbers greater than or equal to –1.)
41. Be able to find the inverse of a function: Find the inverse function for f(x)= 2x + 3.
(answer:
f- -1 (x) =
x−3
)
2
42. Recognize and be able to graph basic functions: constant functions, linear functions,
quadratic functions, polynomial functions, rational functions, exponential functions, and
logarithmic functions.
43. Be able to solve problems in which a particular concept is defined for you;
If i is a number for which i2 = –1, find the product (3 − 2i ) ⋅ (5 + 2i ) .
Answer: 19 – 4i
a
(where a is the first term
1− r
2 1 1
and r is the common ratio), find the sum of the geometric series + + + L
3 3 6
If the sum of the geometric series a + a ⋅ r + a r 2 + … is
(answer: 4/3)
MATH 173, 213 – PROFICIENCY IN TRIGONOMETRY AND ALL PREVIOUS
LEVELS
MATERIAL IN TRIGONOMETRY
44. Be able to recognize the graphs of each of the trigonometric functions sin, cos, tan, cot,
sec,and csc: Which of the following functions has the greatest period:
(a ) y = (1 / 2)sin x, (b ) y = sin( x / 2), (c ) y = 2 sin x, (d ) y = sin 2 x, (e) y = 5 + sin x
(answer: (b ) )
45. Understand the right triangle definitions of all six trigonometric functions:
If θ is an angle between 0º and 90º such that sin θ = 3/5, how much is sec θ ?
(answer: 5/4)
46. Understand degree and radian measure of angles:
radians does it measure? (answer: 2π / 3 )
If an angle measures 120º, how many
47. Know the values of all six trigonometric functions for basic angles (such as 0 degrees,
30 º ( π / 6 ), 45 º ( π / 4 ), 60 º ( π / 3 ), 90 º ( π / 2 ), 180 º (π ) , 270 º (3π / 2) , 360 º (2π )) as well
as angles having the above as reference angles:
Sin 30° ( Answer : 1 / 2); cos180°( Answer : −1); cos 4π / 3( Answer : −1 / 2); sin 120°
(Answer :
3 /2
)
48. Understand and use the Law of Sines: If a triangle has angles 30 degrees, 70 degrees and 80
degrees, what is the length of the side opposite the 30 degree angle if the length of the side
opposite the 70 degree angle is 1? (answer: sin 30° / sin 70°)
49. Understand and use the Law of Cosines:
If a triangle has sides of length 3, 4, and 6, what is the cosine of the angle between the sides
of length 4 and 6?
(answer: 43/48)
50. Understand and use various identities such as basic identities, the Sum and Difference identity, Half-.
Angle identities, and Double-Angle identities.
1 + sin θ
cos θ
?
+
cos θ
1 + sin θ
(a )2 secθ , (b)2 cosθ , (c ) tanθ , (d )1 + sin θ , (e) cosθ
(answer: (a ) )
Which of the following five is the same as cos4 x – sin4 x?
(a ) cos2x, (b ) sin2x, (c ) cos2x, (d ) tan4x, (e) tan22x.
(answer: (a ) )
Which of the following five is the same as
Which of the following is the same as cos2 x- sin2 x?
tan2x
(answer: (d ) )
(a) 1, (b) 2, (c ) cos2x-1, (d) 1-2 sin2 x, (e)
If x is an angle between 0 and π /2 such that sin x = 1/3, how much is sin2x ?
(Answer:
2 8
)
9