Download Diagnostic Test Study Guide

Californi a State Uni versity, Sacramento
Department of Mathematics and Stati stics
1/2/12
Diagnostic Tests Study G uide
Descriptions
Study Guides
Sample Tests & Answers
Tabl e of Contents:
Introduction
1
Elementary Algebra (w. Geometry) Diagnostic Test (EGAD)
Study topics
Sample Test
Answers to Sample Test
2
4
15
Intermediate Algebra Diagnostic Test (IAD)
Study topics
Sample Test
Answers to Sample Test
18
20
26
Calculus Readiness Diagnostic Test (CR)
Study topics
Sample Test
Answers to Sample Test
28
30
38
Introduction
Some classes at CSUS require that you pass a mathematics diagnostic test in order to
remain enrolled in the class. The diagnostic test determines if you have the basic math
skills necessary to be successful in that class. This study guide has been developed to help
you review for the particular test you need to take.
There are 3 diagnostic test study guides in this booklet. The Intermediate Algebra Test
(IAD) assumes you can easily do the math in the Elementary Algebra (with Geometry)
Diagnostic Test (EGAD), and the Calculus Readiness Diagnostic Test (CR) assumes you
can easily do the math in both the EGAD and IAD Tests. So, you should review the
material in the sample diagnostic test (s) before the one you need to pass for your class.
To stay in this Math Class:
Math 9, 11
Take this Diagnostic Test:
IAD
Math 17, Math 24, Math 26A, Math
29, Math 107A, Statistics 1
IAD
Math 30
CR
1
Elementary Algebra (with Geometry) Diagnostic Test (EGAD)
The following is a list of Elementary Algebra topics covered on the EGAD. After that is an
EGAD Sample Test with Answers. If you find a section of the sample test questions difficult,
use the topics list to help you find a similar section in an Elementary Algebra textbook.
Then read and study the textbook until you understand how to do those problems.
This is also a good Sample Test to take when preparing for the IAD!
Study Guide: Elementary Algebra Diagnostic Test
Topic I – Arithmetic Operations
Arithmetic operations with fractions and decimals
Integral exponents and square roots of perfect squares
Conversion between fractions and decimals
Ordering of rational numbers
Percentages
Estimation and approximation
Topic II – Polynomials
Simplification and evaluation of polynomials
Basic operations with polynomials
Special products
Factoring—monomial factors, trinomials, and special products
Topic III – Linear Equations and Inequalities
Solving linear equations with numerical and literal coefficients
Solving equations reducible to linear equations
Solving linear inequalities
Solving two linear equations by elimination and substitution
Topic IV – Quadratic Equations
Solution of quadratic equations by factoring
Topic V – Graphing
Graphing points on the number line and in the coordinate plane
Graphs of linear inequalities
Inequalities in one unknown
Graphs of simple linear equations
Slope of a line
2
Topic VI – Rational Expressions
Simplification of rational expressions
Addition and Subtraction of rational expressions
Multiplication and Division of rational expressions
Complex rational expressions
Topic VII – Exponents and Square Roots
Exponentiation with integral exponents
Simplification, addition, subtraction, multiplication, and division of square roots
Scientific notation
Topic VIII – Geometric Measurement
Measurement formulas for perimeter, circumference, area, and volume of triangles,
squares, rectangles, parallelograms, circles, cubes, rectangular solids,
cylinders, and spheres
Sum of the interior angles of a triangle
Properties of isosceles triangles
Properties of similar triangles
Pythagorean Theorem
Topic IX – Word Problems
Problems involving arithmetic, percent, average, algebraic substitution, evaluation,
ratio, and proportion
Problems leading to one and two linear equations
Problems from geometry
3
Elementary Algebra (with Geometry) Diagnostic Sample Test Questions
Topic I – Arithmetic Operations
1.
7
as a decimal rounded to the thousandths place.
15
Express
! 5 3$
#" 8 + 4 &% ÷ 11
2.
Simplify:
3.
3 – (2 – 4.17 + .003) =
4.
(2 ) ! 2 as a power of 2.
Write
3
5
Topic II – Polynomials
(
5ab 2 2a 2 b3 ! 11a + 3
)
5.
Multiply:
6.
Factor completely:
7.
Solve for x: (x + 6)(2x – 5) = 0
8.
2 3
2
If x = 2 and y = – 3, then x y ! 3 xy + 4 x =
9a 2 ! 25b 2
(2 x ! 5 y )
2
9.
Multiply and simplify:
10.
Factor completely:
11.
Simplify:
(2m
2
3a 2 ! 7a ! 6
) (
)
! m + 1 ! m2 ! 3 m ! 1
4
(w + 4 )(v ! 1) ! 3w (v + 2 )
12.
Multiply and simplify:
13.
Factor completely: 15a3 b2 z 5 ! 9ab 3 z 2 + 24 a2 b 3 z 3
Topic III – Linear Equations and Inequalities
2x + 7x
= 5 + 2x
7
14.
Solve for x:
15.
Solve for x: 2x – 3a = c
16.
Solve for x:
5
8
!
=1
2x ! 3 3 ! 2x
Solve for x and y:
"2x = y
#
$ 3 x ! 4y = 5
18.
Solve for x and y:
"2x + y = 4
#
$ x ! 3y = 9
19.
Solve for x: 4 – 7x < 8
20.
Solve for x: 4x – 7 = 13
17.
Topic IV – Quadratic Equations
5
"
%
21.
Solve for x:
(x + 5 )$# x ! 32 '& = 0
22.
Solve for x:
4 x 2 + 7 x = 12 x
23.
Solve for x:
24.
Solve for x:
25.
Solve for x:
x2 + 6x = 0
26.
Solve for x:
x 2 ! x ! 12 = 0
27.
Solve for x:
x! 5 = !
(
)
x 2x ! 7 = ! 6
x2 ! 4x + 2 = 0
6
x
Topic V – Graphing
28.
Graph the point ( – 2, 3)
29.
Express the graph shown in interval notation.
–6
–2
0
2
30.
Graph 3 x ! 2y " ! 2 and y # 0 on the same set of axes.
31.
Find the midpoint of this line segment.
( – 2, 5)
(4, – 1)
6
32.
What if anything can be said about the following lines:
A.
(I) is parallel to (II)
B.
(II) is parallel to (III)
C.
(I) is perpendicular to (III)
D.
E.
F.
G.
33.
(I) is perpendicular to (II)
Both A and B are true
Both B and D are true
None of the above are true.
I.
y = 3x ! 7
II. x + 3 y = 6
III. y = !
1
x!3
3
If (a, b) denotes the coordinates of the point P in the graph below, then where is the
point with coordinates (a, – b)?
P
34.
What is the equation of this line?
y
5
0
-5
-5
x
0
35.
Graph
y > 3x – 2
36.
Graph
x + 2y ! 2
5
7
37.
5
Where does 13 graph on this number line?
0
1
4
1
2
3
4
1
!5 " 3 x < 2
38.
Graph the solution set for this inequality:
39.
Find the slope of the line passing through the points (– 2, 3) and (– 5, – 1).
40.
Which line is the graph of y = –2x?
L2
L3
L1
Topic VI – Rational Expressions
41.
42.
43.
44.
45.
xy 2 ! 2 x
=
2
3
x
!
xy
If x = 2 and y = – 3, then
Simplify:
(2 x + 1)(x ! 2) =
6x + 3
Simplify:
x2 + x ! 6
=
x!2
Simplify:
a 2 b + ab2 + a 3 b 4
=
a2 b
Find the quotient of the following:
x 3 ! 3 x 2 ! 4 x + 12
x!2
8
2
1
+
x +1 x ! 1
46.
Add and simplify:
47.
Subtract and simplify:
48.
2x ! 1
2
!
x+2
x!3
x ! 2y
xy ! 2 y 2
÷ 2
2x + 2y
x ! y2
Divide and simplify:
Topic VII – Exponents and Square Roots
2
Evaluate:
4
46
50.
Evaluate:
(3 )(2 )(a )
(a )(3 )(2 )
51.
Simplify:
52.
Evaluate:
2 !3
53.
Expand:
(! x y )
54.
Multiply:
(a b )(!b a )
55.
Expand and simplify:
56.
If x > 0, then
57.
Simplify:
58.
Simplify:
49.
59.
3
5
6
5
2
2
48
2
3
2
3
5
(
3! x
)
2
14 x ! 7 x 3 =
36a 2 + 16b 2
2 8 + 32 ! 50
Simplify and express with positive exponents:
Simplify:
x !3 y 2 x
x 2 y !2 y !1
(7a b )(3a b )
2
60.
2
3
2
a 3 b3
9
61.
Simplify and express with positive exponents:
x !3 y 2 z !4
x 2 y !2 z !5
62.
Simplify and express with positive exponents:
(x
63.
Express the following in scientific notation:
a)
b)
2,365,000
– .000456
64.
Express the following in standard notation:
a)
b)
– 2.51 x 104
6.156 x 10 – 2
!7
)(
y 2 x 4 y !3
)
!1
Topic VIII – Geometric Measurement
65.
In the rectangle ABCD shown, find the length of AC.
C
B
3
D
A
66.
5
Find the measure of ! in the triangle ABC.
C
!
9
9
66º
A
67.
B
Find the surface area of the rectangular prism pictured here.
Find the volume also.
8
5
10
10
25!
. Find the radius of the circle.
4
68.
The area of a circle is
69.
Lines L and L are parallel and AB = EF. What is true about the areas of ABDC and
EFHG?
70.
A
B
C
D
E
G
F
H
L1
L2
Find the measure of angle X. Lines m1 and m2 are parallel.
m1
55º
X
77º
m2
71.
Find the area of the shaded region.
9
2
6
3
72.
Find the measure of angle A.
85º
A
35º
11
73.
Find the measure of X in this figure.
2X
96º
75º
X
74.
Find the measure of x in this right triangle.
14
x
60º
75.
Find the length of x in this right triangle.
45º
6
x
76.
77.
The radius of the circle is 5.
What is the perimeter of the square?
What is the area of the square?
•
Find the length of x in the figure shown here.
x
15
7
12
78.
These triangles are similar. Find the measure of x.
6
15
79.
14
x
Find the measure of
!BEC if !CED is a right angle.
C
B
50°
A
80.
E
D
Triangle ABC is similar to triangle DEF. Find the length of DF.
E
B
9
7
A
C
5
D
81.
F
OMIT
13
Topic IX – Word Problems
82.
The length of a rectangle is 3 times its width plus 3 inches. If the perimeter is 54
inches, how wide is the rectangle?
83.
An item is sold for $138. If this price is 15% more than the cost, what is the cost of
the item?
84.
A number is divided by 5 and then multiplied by 7. The result is 14. Find the
number.
85.
Two integers have a sum of 22 and a product of 105. What are the numbers?
86.
Ruth has $1.15 in change consisting of only nickels and dimes. If she has 15 coins
altogether, how many nickels and how many dimes does she have?
87.
A triangle has an area of 16. If its base is half of its height, what is the base of the
triangle?
88.
$9000 is invested in two accounts. Account A pays 4% annual interest and account
B pays 6% annual interest. In one year a total of $383 is earned in interest. How
much was invested in each account?
89.
The sum of three consecutive integers is 24. Find the numbers.
14
Answers to the EGAD Sample Diagnostic Test
1.
3.
4.
5.
6.
.46 and .467
1
8
5.167
216
10a3 b5 ! 55a 2 b 2 + 15ab 2
3 a + 5b 3a ! 5 b
7.
x = ! 6, x =
8.
9.
– 74
4 x 2 ! 20 xy + 25 y 2
2.
(
)(
11.
12.
13.
m 2 + 2m + 2
!2wv ! 7w + 4v ! 4
3ab 2 z 2 5a 2 z3 ! 3b + 8abz
14.
x=–7
3a + c
x=
2
x=8
x = – 1, y = – 2
x = 3, y = – 2
4
x>!
7
x=5
3
x = ! 5, x =
2
5
x = 0, x =
4
3
x= , x= 2
2
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
31.
32.
33.
(1, 2)
F
5
2
(3 a + 2 )(a ! 3 )
16.
17.
18.
[ – 6, – 2)
)
10.
15.
29.
30.
(
)
(a, – b) P = (a, b) --
34.
y=!
3
x + 3 or 3 x + 2 y = 6
2
35.
x= 2± 2
x = 0, x = – 6
x = 4, x = – 3
x = 2, x = 3
•
15
36.
49.
50.
53.
54.
4 3
1
8
! x6 y 9
! a 4 b6
55.
3 ! 2 3x + x
56.
7x2 2
57.
2 9a 2 + 4b 2
58.
3 2
y5
x4
63a5
y4z
x5
y5
x11
a)
b)
a)
b)
51.
52.
37.
59.
0
38.
1
4
5
13
1
2
1
60.
61.
5
!
3
2
3
62.
63.
39.
40.
41.
4.
43.
44.
45.
46.
47.
48.
4
3
L2
7
9
x!2
3
x+3
b
1+ + ab3
a
2
x ! x! 6
3x ! 1
1
44
24a
64.
2.365 x 106
– 4.56 x 10 – 4
– 25,000
.06156
69.
70.
71.
72.
73.
34
48º
Surface area: 340 square units
Volume: 400 cubic units
5
2
They are equal.
48º
12 square units
60º
63º
74.
7 3
75.
3 2
(x + 2 )(x ! 3)
76.
Perimeter: 20 2 units
Area: 50 square units
x!y
2y
77.
78.
4 11
35
(x + 1)(x ! 1)
2x2 ! 9x ! 1
65.
66.
67.
68.
16
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
40º
35
9
(7, 8)
6 inches
$120
10
7 and 15
7 nickels and 8 dimes
4
$7850 in account A
$1150 in account B
7, 8, 9
17
Intermedi ate Al gebra Diagnosti c Test (IAD)
The Intermediate Algebra Diagnostic Test (IAD) is required in order to stay in these Math
classes:
Math 17, 24, 26A, 29, 107A, and Statistics 1.
The IAD has 45 multiple choice questions. The test time is 60 minutes.
NO CALCULATORS are allowed.
What score do you need?
For this Math course:
If your score is…
Math 17 or 107A
Math 24, 26A, 29
Stats 1
Then…
8 or less
9 to 19
20 to 23
Take LS 7AB or LS10
Take Math 9
Take Math 9 or Math 11
24 to 28
29 or more
Advisory qualification for course
Qualifies for course
8 or less
9 to 19
20 to 26
Take LS 7AB or LS10
Take Math 9
Take Math 11
27 to 31
32 or more
Advisory qualification for course
Qualifies for course
The following is a list of topics covered on the IAD. That is followed by an IAD Sample Test
with answers, which has problems in the same order as the topics list. If you find a section
difficult, use the IAD Study Topics list to help you find a similar section in an Intermediate
Algebra textbook. Read and work problems in the text until you understand how to do these
problems. You may also find it helpful to do the Elementary Algebra Diagnostic (with
Geometry) Test (EGAD) Sample Test as well.
Study Gui de: Intermediate Algebra Diagnosti c Test
Topic I – Elementary Operations
Simplification of algebraic expressions
Laws of Exponents with positive integral exponents
Scientific notation
Absolute Value
Topic II – Rational Expressions
Simplification, addition, subtraction, multiplication, division or rational expressions
Complex rational expressions
18
Topic III – Exponents and Radicals
Laws of exponents with rational and literal exponents
Numerical and algebraic calculations with rational exponents and radicals
Factorization and simplification of algebraic expressions with rational and literal
exponents and radicals
Rationalization techniques
Topic IV – Linear Equations and Inequalities
Solution on one and two linear equations
Solution of equations reducible to a linear equation
Solution of linear equalities
Topic V – Quadratic Polynomials, Equations, and Inequalities
Polynomial algebra
Factorization
Completing the square
Solution of quadratic equations by factoring and the quadratic formula
Solution of quadratic inequalities
Complex numbers
Topic VI – Graphing and Geometry
Points in the coordinate plane
Distance in the coordinate plane
Slope and intercepts and quadratic equations
Graphs of linear and quadratic equations
Topic VII – Logarithms and Functions
Numerical and literal function evaluation
Definition and rules of logarithms
Logarithmic and exponential equations
Topic VIII – Word Problems
Problems involving percents, average, ratio, and proportions
Problems leading to linear and quadratic equations
Problems from geometry
19
Intermediate Algebra Diagnostic Sample Test Questions
Topic I—Elementary Operations
1.
Express in scientific notation:
(5.7 x 10 )(2.3 x 10 )
Express in scientific notation:
3.81 x 10
1.05 x 10 !7
!3
6
!5
2.
x 2y 3 9
!
!
x 24 y 4
4
3.
Simplify:
4.
Add and simplify:
5.
(x ) " a
If x = (– 1) and a = 2, then
6.
If a = – 3, evaluate: 1! a ! 2 a + 2 + 1
2[(y – x) – 5] – 3[4 – (x – y)]
3 !1
4
a2 x 2
=
Topic II –Rational Expressions
7.
8.
Add and simplify:
xy 2 ! x 2 y
+ 2x
xy + y 2
Add and simplify:
xy 3
y 5 + xy 4
+
y!x
xy + y
y!x
"
4x 2
9.
Simplify:
10.
Simplify:
1
x+3
3x ! 1
! 2
+ 2
x + 2 x + 3x + 2 x !1
Simplify:
! 3 1$
#" y + 4 &%
!
1 $
#" y ' 4 y &%
11.
(2 + 2 x ) (x
2
2
! y2
)
20
12.
Simplify:
" !1
x%
$# x + 1 + 2 '&
" 1
3%
$# x + 1 ! x '&
Topic III – Exponents and Radicals
Assume all variables are positive real numbers
!2
13.
Evaluate:
14.
Simplify:
10 ! 20
15.
Simplify:
16 x 2 + 24y 2 + 36 z 2
8
3
( ab )( a b)
1
2
3
16.
Simplify:
17.
Simplify:
18.
Simplify:
19.
Simplify:
20.
Simplify:
ab
(x
!3
y2
3
2
) (x
2
!2
2
4
!6
y2
)
!2
8a 3 + a 32a + 3 16 a3
" 43 x 3 2 y 4 3 %
$
'
$# 16 x !1 2 y 2 '&
a(
3 x +2
a(
4! x
!1
2
)
)
z 2 x !4 y 3
y !2 z x !2
21.
Simplify:
22.
Express without radical signs:
23.
Express without radical signs:
3
x2 ! 5 x
3
z7
21
5 ! 3y
4
Solve for y:
2x =
25.
Simplify:
10
2x
26.
Rationalize the denominator:
24.
x
2
3! 7
Topic IV – Linear Equations and Inequalities
27.
Solve this system of equations:
"1
$ x + 3 y = !5
#2
$ 5 x ! 2y = 14
%
28.
Solve this system of equations:
!4x = 5y
"
# 2 x + 3 y = 22
29.
Solve for x:
2 ! 1! x = ! 3
30.
Solve for m:
17 – 19m > – 4(m + 5)
31.
Solve for x:
3 5 ! 4 x ! 3 ! 7x = ! 2 + 9 x
(
)(
) (
)
Topic V – Quadratic Polynomials, Equations, and Inequalities
32.
Multiply:
(x
33.
Factor:
4a 2 ! 9a 2 b 4
34.
Solve for x:
x2 +
35.
Solve for x:
x4 + x2 ! 2 = 0
36.
If you solve this equation by completing the square, what do you add to each side of
x2 + 3x = 4
the equation?
37.
Solve for x:
2
)(
! 5 x 4x2 ! 3x + 2
)
5
3
x=
2
2
2x2 + 4 x + 1 = 0
OMIT 38.
OMIT 39.
22
40.
Simplify:
i
2 ! 5i
41.
Solve for z:
3z 2 ! 4z + 2 = 0
42.
Solve for a:
43.
For what values of m is
3!
1
1
=
1+ 2a a ! 1
(2m ! 1)(m ! 5 )> 0 ?
Topic VI – Graphing and Geometry
44.
Find an equation of the line which passes through the points (1, 3) and (7, – 2).
45.
Find an equation of the line with a slope of 3 which passes through the point (0, –2).
46.
Find the distance between points A and B in the graph shown below.
y
5
0
x
-5
-5
0
5
47.
What is the midpoint of the line segment between the points ( – 5, 3) and (7, – 2)?
48.
Find the quadratic function whose graph is shown in the figure below.
y
5
0
-5
-5
x
0
5
23
49.
Find the slope of the line, L, shown in the figure below.
y
5
0
x
-5
-5
0
5
50.
Graph the equation: y = 4
51.
Graph the equation: 3x – 4y – 5 = 0
52.
Graph the equation:
53.
OMIT
54.
Find the exact value of x:
y = 2x2 ! 1
7
10
x
55.
OMIT
56.
OMIT
57.
OMIT
58.
OMIT
59.
OMIT
60.
OMIT
24
Topic VII – Logarithms and Functions
61.
62.
()
3
If f (h )= h + + 2 ,
h
( )
find f ( ! 3 ).
If f x = 3 x 2 ! 4 x + 5 , then f u ! 2 equals…
63.
Write logb a = x in exponential form.
64.
Solve for x:
4 3 x = 16
65.
Solve for x:
log x 125 = 3
66.
Evaluate:
67.
Solve for x:
log5 2 x + 5 = 3
68.
Solve for x:
log2 x + log2 x ! 7 = 3
69.
Simplify:
log64 16
(
)
(
log5 25 ! log4
)
1
! log7 7 =
16
Topic VIII – Word Problems
70.
You can exchange $2.00 for £1.7. How many pounds (£) can you get for $15?
71.
The side of a square is doubled. How much does the area change?
72.
If 73% of a number is 365, what is the number?
73.
10 is to 28 as 35 is to what number?
74.
The product of two numbers is – 48. Their sum is 13. What are the numbers?
75.
A rectangle has an area of 40 and a perimeter of 26. What are the dimensions of
the rectangle?
76.
Five times a positive number is equal to the difference of the square of that number
and 36. What is the number?
25
Answers to the Intermediate Algebra Sample Test
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
1.311 x 104
3.63 x 102
3x
2y
x ! y ! 22
–4
3
x 2 + 3 xy
x+y
xy 3 + y 5
y=
25.
! 10 $
x
#" 2 &% = 5
x
32.
33.
! 3! 7
2
x = 2, y = ! 2
x = 5, y = 4
x = – 24
37
m<
15
!7
x=
2
4
4 x ! 23 x 3 + 17 x 2 ! 10 x
a2 2 + 3b 2 2 ! 3b 2
34.
x=
26.
27.
28.
29.
(y ! x )(x + 1)
30.
!2 x 2
(1+ x )(x + y )
2
31.
3x
2
x + x!2
12 + y
4y ! 1
2
(
)(
)
!x x + 2 x !1
(
2 2x + 3
)
13.
1
4
14.
10 2
15.
2 4 x 2 + 6 y 2 + 9z 2
16.
3
4 3
16a b
17.
x6
18.
6a 2a + 2a 3 2
35.
36.
37.
38.
39.
40.
1
19.
20.
21.
y 3
2x
a4 x !2
y 5z
x2
2
1
22.
x 3 !x
23.
(z )
7
1
3
2
5
13
=x
=z
7
(
)(
(
)(
)
!2 ± 2
2
1, 25
a = 27 (only 27 checks)
!5 + 2i
29
x=
2±i 2
3
42.
a=
1± 3
2
44.
)(
x = ± i 2, ± 1
9
Add
4
z=
15
)
1
or x = ! 3
2
x2 + 2 x + 1 x ! 1 = 0
41.
43.
6
5 ! 8x
3
24.
1
,m>5
2
5
23
y = ! x+
or 5 x + 6 y = 23
6
6
m<
26
52.
45.
y = 3x – 2
46.
26
! 1$
#" 1, 2 &%
47.
48.
y= !
49.
3
m=
2
y
5
1 2
x +2
2
0
x
50.
-5
-5
y
0
5
5
0
53.
OMIT
54.
51
x
55. through 60.
-5
-5
0
61.
62.
5
63.
64.
"
"5 %
5%
0,
!
and
$#
$# 3 , 0'&
4 '&
51.
65.
66.
y
5
0
-5
-5
x
0
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
OMITTED
3u 2 ! 16u + 25
–2
a = bx
2
x=
3
x=5
2
3
x = 60
x = 8 (x cannot be – 1)
3
£12.75
4 times
500
98
– 3 and 16
5x8
9
5
27
Calculus Readiness Diagnostic Test (CR)
The Calculus Readiness Diagnostic Test (CR) is required in order to remain enrolled in
Math 30. There are 60 multiple choice questions on this exam and the test has a 90 minute
time limit. No calculators are allowed.
What score do you need:
If your score is…
19 or less
20 to 26
27 to 31
32 to 35
36 to 40
41 or more
Then…
Take Math 9
Take Math 11
Advisory qualification for Math 29
Qualifies for Math 29
Advisory qualification for Math 30
Qualifies for Math 30
The following Precalculus Study Topics list covers the topics on the Calculus Readiness
Test. This section is then followed by sample test questions arranged in the same topical
order. If you find a section difficult, use the Precalculus Study Topics List to help you find a
similar section in a Precalculus textbook. Then read and work problems in the text until you
understand how to do these problems. You may also find it helpful to work through the
EGAD and IAD Sample Tests.
Calculus Readiness Test Study Topics
Topic I – Rational Expressions
Addition, Subtraction, Multiplication, Division of rational expressions
Complex rational expressions
Topic II – Exponents and Radicals
Laws of Exponents
Calculations with exponents and radicals
Conversion between radicals and exponents
Rationalization techniques
Topic III – Linear Equations, Inequalities, Absolute Value
Solution of one and two linear equations
Solution of equations reducible to linear equations
Graphs of linear inequalities
Solution of linear inequalities
Simplification of expressions involving absolute value
Solution of equations and inequalities involving absolute value
Topic IV – Polynomials and Polynomial Equations
Polynomial algebra
Factorization
Completing the square
28
Solution of polynomial equations by factoring
Solution of quadratic equations by quadratic formula
Graphs of quadratic equations
Solution of quadratic inequalities
Topic V – Functions
Function concept and notation
Function evaluation and composition
Graphs of translations, reflections, and absolute value functions
Topic VI – Trigonometry
Right angle trigonometry
Trigonometric functions and circular functions
Radian and degree measure
Special angles
Trigonometric identities
Graphs of trigonometric functions
Topic VII – Logarithmic and Exponential Functions
Definition and laws of logarithms
Evaluation of logarithmic expressions
Graphs of logarithmic and exponential functions
Inverse relation between logarithm and exponential functions
Logarithmic and exponential equations
Topic VIII – Word Problems
Problems involving percent, average, ratio, and proportion
Problems leading to linear and quadratic equations
Problems from geometry
29
Calculus Readiness Sample Test Questions
Topic I – Rational Expressions
Simplify the following:
1.
2.
" 9 x 2 ! 12 x % "
2x2 + 2 %
$# x 2 + 1 '& $# 9 x 2 + 6 x ! 24 '&
"
3%
$# 1 ! x '&
9 ! x2
3.
! 6v 3 $ ! 'uv $ ! 1 $
#" u 2 &% #" 2v &% #" 7v 2 &%
4.
!4
2x + 1
! 2
2
9! x
x ! 3x
3
+
2
+
1
5.
(x + 2b )(x ! b ) (3b ! x )(x + 2b ) (x ! 3b )(b ! x )
6.
" 2a b %
$# b ! a '&
7.
8.
!2
! xy 2 $
#" x + y &%
! x' y $
#" x 2 ' y 2 &%
" m
%
2m
!
$# m ! 3
m 2 ! 2m ! 3 '&
" 2
1%
!
$# m + 1 m '&
30
Topic II – Exponents and Radicals
9.
!3
81
Evaluate:
4
Simplify the following:
10.
6 15
11.
(a + b )(a
12.
16 x 2 + 24y 2 + 36 z 2
! b2
(3xy ) (4x y )
1
13.
2
2
2
xy
3
!2
2
14.
a !2 b 3 c
a !1b !2 c 0
15.
a 3 x +2
a 4! x
16.
(m )
17.
" 43 x 3 2 y 4 3 %
$
'
$# 16 x ! 1 2 y 2 '&
18.
(!125 x
19.
( m ) (2m m )
20.
2 12 + 27 !
b+1
4
)
b !1
18
y 24
!1
2
)
!2
3
6
48
Rationalize the following:
x
21.
5
4
31
3
22.
15
23.
x
x2
35 x +2 = 27 x !4
Solve for x:
Topic III – Linear Equations, Inequalities, and Absolute Value
3x =
24.
Solve for x:
25.
Solve for a:
26.
Solve for x:
a
x!1
b
(
)
a! 2 +3 = 7
3
2
5x + 3
!
=
x
x+1
x +1
27.
Solve this system of equations:
"2x + 3y = ! 5
#
$ 5 x ! 2y = 8
28.
Solve this system of equations:
!3x = 4y
"
# 2 x + 3 y = 17
29.
If m = – 3, evaluate
30.
Find an equation of this line:
3! m + !4 ! !m
y
5
0
x
-5
-5
31.
Solve for x:
0
5
5 ! 2x " 3
32
32.
Given the points (1, 1) and (3, 4), find the area of the right triangle formed by the line
through those points, the x-axis, and x = 3.
Topic IV – Polynomials and Polynomial Equations
1
33.
Solve for x:
x ! 6x 2 + 5 = 0
34.
Solve for x:
!4 x 2 + 12 x + 3 = 0
35.
Solve for x:
36.
If you wanted to complete the square for x ! 3 x = 2 , what would you add to both
sides?
37.
Solve for x:
2 = 2x ! 5 !
38.
Simplify:
x 4 ! 14 x 2 + 3 x + 34
x!2
39.
Find the values of a so that the following has 2 distinct real roots:
40.
41.
Solve for x: x ! x ! 42 > 0
2
Graph: y = x ! x ! 6
42.
Graph:
x+ 2 = y
43.
Graph:
y = 2x + 1
44.
Graph:
y=
45.
x 2 ! xy 2 z
If x = 3, y = – 1, and z = – 2, evaluate
z 2 y 2 ! xz
46.
Find the vertical and horizontal asymptotes of f x =
(x ! 2 )(x ! 3 )= 2
2
x!2
2ax 2 ! 12 x ! 7
2
2
x
x+1
( ) (2 x ! 1)(x + 3 )
Topic V – Functions
()
3x3 ! 4 x + 5
when f x =
ax 2 + bx ! 2
()
47.
Find f 0
48.
If f x = !
49.
2
If f x = x ! 1 and g x = 2x + 1 , find a) f g x
()
()
" 1 %
2
, find f $
.
x!1
# x + 3 '&
()
( ( ))
()
and b) g ! f x
33
50.
3x + 2
If f x = 2 x ! 1 , for what x does f(x) = – 3?
()
4
()
()
if f t =
t 2 !1
2t ! 2
51.
Find
52.
Graph:
f x =
53.
Graph:
f x = x2 ! 2
54.
Find the domain and range of g x =
55.
56.
f 3
()
1
2x + 1
()
()
x2 ! 2x ! 5
() (
2
Express the domain and range of g x = 25 + 4 x ! x
exact answers, no decimals).
)
1
2
in interval notation (with
Find the area of the interior region of this figure:
12
10
8
6
4
2
0
0
57.
1
()
2
3
4
() (
5
)
Let f x = 3 x ! 5. If 3f a = f 2a + 1 , then a =
Topic VI – Trigonometry
58.
Find Cos ! in the given triangle.
w
!
2
2
59.
w –9
Use this triangle to find cotΘsinΘ.
a
!
b
34
(
cos ! + "
60.
Evaluate:
61.
Verify this identity:
62.
63.
64.
)
sec ! " cos ! = tan ! sin !
3
Graph the function: y = 2 cos 2 x
! x$
y = sin # &
Graph:
" 2%
( )
In the figure shown, find cscΘ.
!
( – 2, – 5)
65.
Express 114º in radians.
66.
Which trig functions are even?
67.
" 12 %
!1
sin
$
' .
Find
# 4 &
68.
" 5! %
Find cos $
.
# 3 '&
Topic VII – Logarithmic and Exponential Functions
69.
Solve for b: loga b = x
70.
Solve for x:
2x = 5
71.
Evaluate:
log36 3 6
Evaluate:
! 96 $
log9 # &
" 27 %
72.
35
( )
( )
Simplify:
74.
Solve for x:
53 x !
75.
Solve for y:
log4 3 y ! 5 = 2
76.
Graph:
77.
Graph:
()
log a2 b ! log 2a + log b
73.
! 1$
y =# &
" 3%
1
=0
25
(
)
'x
(
x = log 2 y + 3
)
Topic VIII – Word Problems
78.
A bakery has a special on peanut butter cookies and chocolate chip cookies. There
are 12 dozen cookies on special. If there are 40% more chocolate chip cookies
than peanut butter cookies, how many chocolate chip cookies are there?
79.
A 24w by 36L inch poster is enlarged so that its length is 5 ft. What is its width?
80.
The circumference of a circle is quadrupled. How much is the area increased?
81.
The sum of two numbers is 179½. Six times the first number minus seven times the
second number is 778. Find the numbers.
82.
The cube root of a number is squared and the result is 16. What is the number?
83.
The price of an airplane ticket is decreased 16% to $193.20. What was the original
price?
84.
If the radius of a circle is decreased by 15%, what is the percent decrease in the
area of the circle?
85.
What is the surface area of the given triangular prism?
15
4
5
3
36
86.
Find the area of the given figure to the nearest tenth of a unit.
12
3
5
3
7
12
87.
How long does it take something to travel 500 meters at 42 meters per second?
88.
A tree twenty-five feet tall casts a shadow that is 35 feet long. If the shadow of a
nearby building is 119 feet long, how tall is the building?
37
Answers to the Calculus Readiness Sample Test Questions
1.
2.
3.
4.
5.
6.
2x
x+2
!1
(
x x+ 3
24.
(
)(
x x+ 3 3 ! x
25.
26.
)
9b
(x + 2b )(3 b ! x )(x ! b )
a 2b 2
(2a
2
! b2
9.
10.
3 10
8.
)
!3v
7u
2x2 + 3x + 3
xy2
m2
m!3
1
27
7.
1
22.
23.
)
2
11.
(a + b )
12.
2 4 x 2 + 6 y 2 + 9z 2
13.
14.
15.
16.
3
16 x 4 y 3
b 5c
a
a4 x !2
m
" 2 %
$ b !1'
#
&
1
3
17.
y
2x
18.
19.
1
25 x12 y 16
2m 3
20.
3 3
21.
x5 8
2
27.
28.
29.
30.
31.
32.
a!b
33.
34.
35.
36.
37.
38.
39.
40.
x 5
x=–7
!a
a
or
3b ! a
a ! 3b
a = 18
3
5
14
!41
x=
,y =
19
19
x = 4, y = 3
7
3x + 4y = 12
1! x ! 4
The line formed by the 2 points is
3x – 2y = 1. This line intersects
the
!1 $
x-axis at # ,0 & , so the base of the
"3 %
x=
8
right triangle is 3 units in length.
16
The area is then 3 sq units.
x = 1, 25
3±2 3
2
x = 1 or 4
9
Add 4 to both sides.
x = 27 (3 doesn’t work)
x 3 + 2 x 2 ! 10 x ! 17
!18
7
x < – 6 or x > 7
a>
38
41.
y
0
x
Vertical:
47.
Horizontal:
5
!
2
2 x+3
(
48.
0
42.
y
1
, x = !3
2
y=0
x=
46.
)
49.
x+2
a) 4x 2 + 4x
50.
x=
51.
52.
2
b) 2x 2 ! 1
1
9
y
0
x
0
x
0
43.
0
53.
y
y
0
x
0
x
0
0
44.
y
54.
(
domain:
x ! 1" 6
0
) or x # (1+ 6 )
()
range: 0 ! g x ! "
x
55.
domain:
0
range:
45.
3
2
56.
25
" 2 ! 29, 2 + 29 $
#
%
!0, 29 #
"
$
39
a=
57.
13
3
3 2w 2 " 9
cos ! =
w2
b
= cos !
a2 + b2
58.
59.
! cos "
60.
61.
69.
19!
30
cosine and secant
!
or 30°
3
1
2
b = ax
70.
x=
65.
1
" cos !
cos !
1" cos 2 !
=
cos !
sin2 !
=
cos !
sin !
=
# sin !
cos !
= tan ! sin !
sec ! " cos ! =
62.
66.
67.
68.
log 5
log 2
or
ln 5
ln 2
1
6
9
2
71.
72.
73.
! ab 2 $
log #
" 2 &%
74.
x=!
75.
y=7
2
3
y
76.
y
0
x
0
x
0
0
63.
77.
y
y
0
x
0
-12
0
x
12
0
64.
! 29
5
78.
p + 1.40p = 12; 7 dozen chocolate
chip cookies
40
79.
80.
82.
83.
84.
40 inches
16 times
1
156 and 23
2
±64
$230
27.75%
85.
86.
87.
88.
227 + 15 41
173.6
11.9 sec
85 ft.
81.
41