math placement exam materials and practice exam
DESCRIPTION
Needing to take the Math Placement Exam at Summer Orientation? View this document from the Math Department to see which topics to study to help you prepare for the exam. A 15-question practice exam (starting on pg. 9) will be similar to what you will complete at Summer Orientation.TRANSCRIPT
Placement Test Practice Materials
What’s here: 1. Placement Exam Topic List—to help you know what types of things might be
on the placement test. This list is not comprehensive, but will help you get a handle on what types of topics might be included and what kinds of skills will be expected.
2. Mathematics Placement Practice Test—to help you get a feeling for the terminology and wording of problems that could be posed on the test. None of these are actual test problems, but they have the same flavor as the problems on the test.
⎯ It is important to note that the practice test has only 15 questions, whereas the actual test has 25 questions that you must complete in half an hour. Therefore, you must complete the 15 practice test questions in 18 minutes if you are trying to get a sense of how fast you must work on the actual test.
⎯ Calculators and reference books or notes, including formulas, are not allowed on the actual placement exam. If you want to simulate the experience of taking the placement exam, do not use these on the practice exam.
3. Mathematics Placement Practice Test Answers—to help you score your practice test.
4. Mathematics Placement Practice Test Solutions—to help you understand the answers to the Practice Test. Full solutions are provided for each problem. Further, each problem is identified with the corresponding topics it covers from the Placement Exam Topic list.
Recommendation on how to use the materials: If you are mainly concerned about language, but not mathematics.
• Read through the Placement Exam Topic List. Try to learn the American English words for the mathematics that you know by looking at the diagrams and formulas.
• Do not hesitate to look up mathematical terms on www.wikipedia.org/ or mathworld.wolfram.com/ . Both are fine for most mathematics.
• Take the Placement Practice Test. • Score your practice test, noting which problems you missed because of the
math and which problems you missed because of the wording. • For all problems that you missed, be sure to examine the Practice Test
solutions. Think about the problems that you missed because of wording separately. Now that you can see the solutions, do you better understand the meaning? Would you understand the meaning in another similar situation? Why or why not? Try to pose similarly worded mathematical questions.
If you are mainly concerned with brushing up on mathematics, but not with understanding English.
• Read through the Placement Exam Topic List. Mark the topics that you do not recall how to do and/or the formulas and identities that you do not recall.
• Write all formulas and identities you need to memorize on a small piece of paper or an index card. Carry this around with you, looking at it every 5 minutes or so to quiz yourself on what is written there. You should be working towards being able to recall everything on the card verbatim with your eyes closed. (You may wish to make flash cards instead.) By the way, you want this in long-‐term memory storage, not just in “until after I take the placement test” storage, as you will be expected to know these formulas and identities in subsequent courses.
• For the topics you do not remember how to do, you have a number of options.
o Go to your local library and check out the appropriate level math book for self-‐study,
o Try to buy a book at a book store to help you, o Use targeted mini-‐lectures on the web, one popular website is:
https://www.khanacademy.org, or o Use pre-‐made courseware on the web, one such website is:
http://www.saylor.org/majors/math/ .
If you don’t know the mathematics, you should take the appropriate course rather than trying to learn it on your own and place into an inappropriately high course. Knowingly placing yourself too high or too low will undercut your ability to thrive in your mathematics course and will waste your semester. Help yourself have a great mathematical experience in the Mercer Math Department with our committed faculty by placing yourself correctly. We look forward to getting to know you!
Mathematics Placement Exam Topic List The mathematics placement exam emphasizes topics from algebra, geometry, pre-‐calculus, and trigonometry. For full understanding of the Mercer Math courses resting upon the knowledge from these subjects, please do not assume that you will only need to know the limited number of concepts inventoried below. Some student learning objectives from Algebra, Geometry, Pre-‐calculus, and Trigonometry that might be tested on the Mathematics Placement Exam are listed below. This list does not promise to be comprehensive. Others may categorize the objectives into different courses. Please note that formulas are not given on the placement test. They are to be memorized or internalized. For each of the following, students will know how and be able to accurately: Algebra
1. Solve an equation to find the value of a variable. 2. Graph the equation of a line. 3. Find the slope of a line. 4. Explain how the slope of a line relates to the appearance of a line.
a. A line with zero slope is horizontal (e.g., y=7) b. A line with infinite slope is vertical (e.g., x=2) c. A line with positive slope could be described as increasing as one
moves to the right or it could be described as rising to the right (e.g., y-‐3=5(x-‐4) or y=5x-‐17 or 2y-‐10x+34=0.)
d. A line with negative slope could be described as decreasing as one moves to the right or it could be described as falling to the right (e.g., y-‐1=-‐2(x+1) or y=-‐2x+3 or 5y+10x=15.)
5. Find the intersection of two lines. a. Report the intersection point. b. Report only the x-‐coordinate of the intersection. c. Report only the y-‐coordinate of the intersection.
6. Solve inequalities involving quadratic functions. 7. Solve an inequality with absolute values. 8. Find the distance between two points in the Cartesian plane. 9. Simplify or expand an expression with exponents, including fractional and
negative exponents. 10. Understand the role of exponents in working with numbers./Have number
sense when it comes to exponents. 11. Factor algebraic expressions. 12. Evaluate a function at a given value. 13. Read, interpret, set up, and solve a basic word problem.
Geometry
1. Use the formulas for perimeters of standard shapes.
a. A. Square: perimeter=4s b. B. Rectangle: perimeter=2x+2y c. C. Circle: perimeter= 2πr d. D. Triangle: perimeter=a+b+c
2. Use the formulas for areas of standard shapes. a. A. Square: area= s2 b. B. Rectangle: area=xy c. C. Circle: area=πr2 d. D. Triangle: area=(1/2)bh
ac
b
h
s x
ysA. B.
C. D.r
3. Use the formulas for surface areas of standard shapes.
a. E. Box with all 6 sides: Surface Area (SA) = 2lw + 2wh + 2lh b. E. Box with open top, but closed bottom:
Surface Area (SA) = lw +2wh + 2lh c. G. Sphere: Surface Area (SA) = 4πr2 d. H. Cylinder with both bases: Surface Area (SA) = 2πr2+2πrh e. H. Cylindrical side, no bases: Surface Area (SA)= 2πrh f. Students are not expected to know the surface area of a cone given
only the height and base radius. 4. Use the formulas for volumes of standard shapes.
a. E. Box: Volume = lwh b. F. Cone: Volume = (1/3)πr2h c. G. Sphere: Volume = (4/3)πr3 d. H. Cylinder: Volume = πr2h
5. Reason using geometric logic, such as by using proportions. Pre-‐Calculus
1. Recognize the use of or employ the laws of exponents. For c,d,g real numbers and a,b > 0, x,y variables taking on real values,
a. cd+g=cdcg and ax+y=axay b. 𝑐!!! = !!
!! and 𝑎!!! = !!
!!
c. 𝑐! ! = 𝑐!"and 𝑎! ! = 𝑎!" d. cgdg=(cd)g and axbx=(ab)x
w
h
l
E.
r
h
F.
G. H.
r
r
h
2. Solve exponential growth problems, which might be presented as word problems. Contexts may include banking, population growth or decay, or radioactive decay.
3. Recognize the use of or employ the laws of logarithms. For c,d,g positive real numbers and a,b > 0, x,y variables taking on positive real values,
a. logc(dg) = logcd + logcg and loga(xy) = logax + logay b. logc(d/g) = logcd – logc g and loga(x/y)= logax -‐ logay c. logc(dg) = g(logcd) and loga(xb) = b(logax)
4. Realize that logarithms and exponential functions in the same base are inverse functions and exploit that to simplify and solve equations. (I.e., loga(ax)=x.)
5. Solve logarithmic equations, which might be presented as word problems. These problems are often part of what originally seem to be exponential growth problems.
6. Recognize and sketch the graphs of standard functions such as lines, power functions (x2, x3, x-‐1, x-‐2, xn, etc. and multiples of these), exponential functions, and logarithmic functions, all with variations.
7. Understand the role of transformations on graphs and determine the graph of a function as the transformation of a standard graph using the algebraic expression of the function.
8. Solve for the zeros of polynomials and rational functions. 9. Find the domain of rational and root functions. Trigonometry 1. Understand the meanings of period and amplitude of a function both
graphically and algebraically.
2. Use the six trigonometric (trig) functions via their definitions:
a. sine: sin 𝜃 = !!
b. cosine: cos 𝜃 = !!
c. tangent: tan 𝜃 = !!
d. cosecant: csc 𝜃 = !!
e. secant: sec 𝜃 = !!
f. cotangent: cot 𝜃 = !!
3. Use the definitions of the trig functions to reduce trigonometric expressions
x
y
h
O
und=undefined; deg=degrees, rad=radians Θ deg
Θ rad
x y sin(θ)
cos(θ)
tan(θ)
csc(θ)
sec(θ)
cot(θ)
0 0 1 0 0 1 0 und 1 und 30 π/6 𝟑
𝟐 𝟏𝟐
𝟏𝟐
𝟑𝟐
13 2 2
3 3
45 π/4 𝟐𝟐
𝟐𝟐
𝟐𝟐
𝟐𝟐
1 2 2 1
60 π/3 𝟏𝟐
𝟑𝟐
𝟑𝟐
𝟏𝟐
3 23 2 1
3
90 π/2 0 1 1 0 und 1 und 0 120
2π/3 −12
32
32
−12
− 3 23 -‐2 −1
3
135
3π/4 − 22
22
22
− 22
-‐1 2 − 2 -‐1
150
5π/6 − 32
12
12
− 32
−13 2 −2
3 − 3
180
π -‐1 0 0 -‐1 0 und -‐1 und
210
7π/6 − 32
−12
−12
− 32
13 -‐2 −2
3 3
O
1-1
(x,y)
225
5π/4 − 22
− 22
− 22
− 22
1 − 2 − 2 1
240
4π/3 −12
− 32
− 32
−12
3 −23 -‐2 1
3
270
3π/2 0 -‐1 -‐1 0 Und -‐1 und 0
300
5π/3 12
− 32
− 32
12
− 3 −23 2 −1
3
315
7π/4 22
− 22
− 22
22
-‐1 − 2 2 -‐1
330
11π/6 32
−12
−12
32
−13 -‐2 2
3 − 3
360
2π 1 0 0 1 0 und 1 Und
-‐30 -‐π/6 =11π/6
32
−12
−12
32
-‐2 23 − 3
etc.
4. Use values of the trig functions at the standard angles (in radians) and their multiples to evaluate trigonometric expressions without a calculator. (Students should be able to find the value of any of the six trig functions at any standard angle based on having memorized the value of the functions sin(θ) and cos(θ) at x=0, x=π/6, x=π/4, x=π/3, and x=π/2. These are the bolded entries in the table.)
5. Graph the six trigonometric functions and their algebraic transformations. 6. Use the double and half angle fomulas for sine and cosine.
a. sin(2θ) = 2sin(θ)cos(θ) b. cos(2θ) = cos2(θ)-‐sin2(θ) = 2 cos2(θ) – 1 = 1 -‐ 2sin2(θ) c. sin! 𝜃 = !!!"# (!!)
!
d. cos! 𝜃 = !!!"# (!!)!
7. Use the main trig identities and their variants that arise from moving terms
from one side of the equality to the other through subtraction. a. sin2θ + cos2θ = 1 b. tan2θ + 1 = sec2θ c. 1 + cot2θ = csc2θ d. sin(-‐θ) = -‐sin(θ) e. cos(-‐θ) = cos(θ) f. tan(-‐θ) = -‐tan(θ) g. sin !
!− 𝜃 = cos 𝜃
h. cos !!− 𝜃 = sin 𝜃
Mathematics Placement Practice Test (The actual test has radio buttons for multiple choice selections, 25 questions, and allows exactly 30 minutes for completion. No calculators are allowed.)
1. What is the y-‐coordinate of the intersection point of the two lines given by the equations:
y=3x-‐2 and y=5x+6 a. 4 b. -‐4 c. 14 d. -‐14 e. 10
2. In a standard coordinate system, the graph of 7y+2x=15 is a. A horizontal line b. A vertical line c. A line rising to the right d. A line falling to the right e. A single point
3. A box has a square base. The height of the box is three times its width. If the volume of the box is 3000 cm3, what is the width of the box?
a. 10 or -‐10 b. 10 only c. 300 d. 500 e. 3000!
4. The area of a triangle is 81 inches squared. Its base is half its height. What is the height of the triangle?
a. !!
b. 9 c. 9 2 d. 18 e. !"
!
5. Suppose that the number of cells in a petri dish is 600 one hour after the cells are plated into the dish. Three hours after being plated the cell population has grown to 3000. If the population of cells in the petri dish is growing exponentially, what is the number of cells in the dish 7 hours after they were plated?
a. 725 b. 7800 c. 600 5! !
d. 75,000 e. 600(2400)3
6. A function is said to have period B if f(x+B)=f(x) for all values of x, and B is the smallest such positive number. What is the period of the graph shown below?
a. 0.2π b. 0.4π c. 0.5π d. 0.8π e. 2π
7. The inequality 0 < 3𝑥 + 2 < 𝑑 is equivalent to a. 𝑥 < !!!
!
b. –!!!!
< 𝑥 < !!!!
c. –!!!!
< 𝑥 < !!!!
d. –!!!!
< 𝑥 < !!!! and 𝑥 ≠ !!
!
e. 𝑥 < –!!!! or !!!
!< 𝑥
8. Find interval(s) on which 5x2-‐9x+2 >0. a. −2 < 𝑥 < !
!
b. 𝑥 < −2 and !!< 𝑥
c. !! !"!"
< 𝑥 < !! !"!"
d. 𝑥 < !! !"!"
and !! !"!"
< 𝑥 e. The function 5x2-‐9x+2 is always positive.
-1.5π -π -0.5π 0 0.5π π 1.5π
-3
-2
-1
1
2
3
9. Find the distance between points P and Q in the following figure.
a. 2 b. 5 c. 3.5 d. 13 e. 5
10. At which value of θ, 0 ≤ θ ≤ !! , does csc(θ) = 2?
a. 0 b. !
!
c. !!
d. !!
e. !!
11. If log3(2x+5)=4, then x= a. 4.5 b. 29.5 c. 35.5 d. 38 e. 43
12. Let 𝑔 𝑡 = 2𝑡 − 3. Find ! !!! !!(!)!
. Choose the most reduced answer. a. -‐1 b. !
! !
c. 𝑡 − 1 d. !!!!! !!!!
!
e. !!!!! !!!!!
-1 0 1 2 3 4 5 6
1
2
3
4
P
Q
13. A function, f, is increasing on an interval (a,b) if for x,y in (a,b) with x<y, f(x)<f(y). Determine the x-‐interval(s) on which the pictured graph is increasing.
a. (-‐6,-‐2) b. (-‐2,2) c. (2,6) d. (-‐6,-‐2) and (2,6) e. Only the points x=-‐2 and x=2
14. Which of the following best represents the graph of 𝑦 = 2− 𝑒!!!?
a.
-6 -4 -2 0 2 4 6
-2
2
-5 -4 -3 -2 -1 0 1 2 3 4 5
-3
-2
-1
1
2
3
b.
c.
-5 -4 -3 -2 -1 0 1 2 3 4 5
-3
-2
-1
1
2
3
-5 -4 -3 -2 -1 0 1 2 3 4 5
-3
-2
-1
1
2
3
d.
e.
-5 -4 -3 -2 -1 0 1 2 3 4 5
1
2
3
4
5
6
-5 -4 -3 -2 -1 0 1 2 3 4 5
-3
-2
-1
1
2
3
15. A cylinder in which the radius equals the height contains one largest possible sphere. How much volume is inside of the cylinder, but outside of the sphere? Because r=h, give your answer in terms of r.
a.!!𝜋𝑟!
b.!!𝜋𝑟!
c.!!𝜋𝑟!
d.!!𝜋𝑟!
e. There is not enough information to answer this question.
Mathematics Placement Practice Test Answers:
1. d 2. d 3. b 4. d 5. d 6. b 7. d 8. d 9. d 10. b 11. d 12. e 13. b 14. a 15. d
Mathematics Placement Practice Test Solutions The letter answer from the practice test is in parentheses after the solution. The corresponding topics from the mathematics placement exam topic list are listed after that.
1. What is the y-‐coordinate of the intersection point of the two lines given by the equations:
y=3x-‐2 and y=5x+6 In order for the two lines to intersect, the x-‐ and y-‐coordinates must be simultaneously equal. Because both equations are currently written in terms of y, we may set the other sides equal to each other:
3x-‐2=5x+6 and solve for x: x=-‐4. Plugging back into either equation to solve for y, yields
y=3(-‐4)-‐2=5(-‐4)+6=-‐14. Therefore, we report that the y-‐coordinate of the intersection point is -‐14. (d) Algebra 1,5
2. In a standard coordinate system, the graph of 7y+2x=15 is the same as the graph of the line written in y=mx+b form, which is y=(-‐2/7)x+(15/7). In this form, we see that the slope is -‐2/7. Because the slope is negative, the line falls to the right. (d) Algebra 2,3,4
3. A box has a square base. The height of the box is three times its width. If the volume of the box is 3000 cm3, what is the width of the box? The volume of a box is length times width times height. In this case, length and width are equal, because of the square base. Call this length, s. The height is three times as big, so it is 3s. The total volume is then 3s3=3000. So s3=1000 or s=10. The width of the box is then 10. (b) Algebra 10, 13, Geometry 4
4. The area of a triangle is 81 inches squared. Its base is half its height. What is the height of the triangle? Call the base of the triangle b and the height h. Then the area of the triangle is (1/2)bh. But b=(1/2)h. So the area is (1/2)(1/2)hh by substitution for b. Also, the area is given to be 81. Hence 81=(1/4)h2. That is, h2=4*81. So h=18. (d) Algebra 10, 13, Geometry 2
5. Suppose that the number of cells in a petri dish is 600 one hour after the cells are plated into the dish. Three hours after being plated the cell population has grown to 3000. If the population of cells in the petri dish is growing exponentially, what is the number of cells in the dish 7 hours after they were plated? The general formula for an exponential function is P(t)=Cat, where P is the size of the function at time t, C is the size of the function at time 0, and a is some value greater than 0. We know that 600=Ca and 3000=Ca3. Dividing the last equation by the previous, we find that a2=3000/600 = 5. Therefore, P(7)=Ca7=Ca3(a2)2=3000(5)2=3000(25)=75,000. (d) Algebra 9, Pre-‐calculus 1,2
6. A function is said to have period B if f(x+B)=f(x) for all values of x, and B is the smallest such positive number. What is the period of the graph shown below?
Examining the graph shows that it repeats a full cycle every 0.4π. (b) Trigonometry 1
7. The inequality 0 < 3𝑥 + 2 < 𝑑 is equivalent to 0 ≠ 3𝑥 + 2 and 3𝑥 + 2 < 𝑑, which is equivalent to 𝑥 ≠ !!
! and −𝑑 < 3𝑥 + 2 < 𝑑, which is equivalent to
𝑥 ≠ !!! and –𝑑 − 2 < 3𝑥 < 𝑑 − 2, which is equivalent to
𝑥 ≠ !!! and –!!!
!< 𝑥 < !!!
!. (d) Algebra 7
8. Find interval(s) on which 5x2-‐9x+2 >0.
The function 5x2-‐9x+2 is an upward facing parabola, so it will only be less than or equal to 0 between its zeros. For other x-‐values the function will be greater than 0. To find these zeros, we can use the quadratic formula: 𝑥 = !± !!!!×!×!
!×!= !± !"!!"
!"= !± !"
!". (d) Algebra 6, 11
9. Find the distance between points P and Q in the following figure.
-1.5π -π -0.5π 0 0.5π π 1.5π
-3
-2
-1
1
2
3
-1 0 1 2 3 4 5 6
1
2
3
4
P
Q
We can think of this as the distance formula or as the Pythagorean Theorem. The distance between P and Q= (5− 2)! + (4− 2)! = 9+ 4 = 13. (d) Algebra 8
10. At which value of of θ, 0 ≤ θ ≤ !! , does csc(θ) = 2?
Because csc(θ)=1/ sin(θ), this is the same question as: For what value of θ, 0 ≤ θ ≤ !
! , does sin(θ) = ½? The answer is !
!. (b) Trigonometry 2, 4
11. If log3(2x+5)=4, then x= 2x+5=34=81, so 2x=81-‐5=76. Hence, x=76/2 = 38. (d) Algebra 1, Pre-‐calculus 4,5
12. Let 𝑔 𝑡 = 2𝑡 − 3. Find ! !!! !!(!)!
. Choose the most reduced answer. ! !!! !!(!)
!= ! !!! !!! !!!!
!= !!!!!!! !!!!
!= !!!!! !!!!
! . (d) Algebra 11,
12 13. A function, f, is increasing on an interval (a,b) if for x,y in (a,b) with x<y,
f(x)<f(y). Determine the x-‐interval(s) on which the pictured graph is increasing.
(-‐2,2) (b)
-6 -4 -2 0 2 4 6
-2
2
14. Which of the following best represents the graph of 𝑦 = 2− 𝑒!!!?
(a) Algebra 12, Pre-‐calulus 6, 7
15. A cylinder in which the radius equals the height contains one largest possible sphere. How much volume is inside of the cylinder, but outside of the sphere? Because r=h, give your answer in terms of r.
The volume of a cylinder is 𝜋𝑟!ℎ, but because r=h, the volume of this cylinder is 𝜋𝑟!. The volume of the sphere is (4/3)πR3, where R is the radius of the sphere. Because the sphere is the largest one that fits into the given cylinder, we know its diameter to equal r, so R=r/2. Therefore, the volume of the sphere is (4/3)π(r/2)3=(4/3)πr3/8=πr3/6. The volume of the cylinder outside of the sphere is then 𝜋𝑟!-‐ πr3/6 = 5πr3/6. (d) Geometry 4,5
-5 -4 -3 -2 -1 0 1 2 3 4 5
-3
-2
-1
1
2
3