Jan 2, 2012 ... The diagnostic test determines if you have the basic math skills necessary to be
successful ... Study Guide: Elementary Algebra Diagnostic Test.
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
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
