Math B - Eduware
Math B
Table of Contents
I. Algebra
1. Numbers, Sets, Systems, and Operations
A. Basic Algebra
i. Comparing Mathematical Expressions………………1
ii. Undefined Fractions…………………………………… 3
B. Mathematical Systems
5
i. Properties of Operations…………………………………
6
ii. Mathematical Fields………………………………………
C. Complex Numbers
i. Powers of i …………………………………………………7
8
ii. Add and Sub of Complex Numbers……………………
11
iii. Mult & Div of Complex Numbers………………………
iv. Simplifying Complex Numbers……………………… 13
v. Mult & Add Inverses of Complex #'s…………………14
15
vi. The Complex Plane………………………………………
2. Simplifying Algebraic Expressions
A. Positive, Negative and Fractional Exponents
i. Zero & Negative Exponents…………………………… 18
ii. Fractional Exponents……………………………………19
B. Scientific Notation……………………………………………………………………21
C. Factoring Algebraic Expressions
i. Factoring using Multiple Methods…………………… 22
D. Simplifying Algebraic Fractions
i. Adding and Subtracting Algebraic Fractions……… 23
27
ii. Mult and Div Algebraic Fractions………………………
iii. Simplifying Algebraic Fractions………………………33
iv. Complex Fractions………………………………………37
v. Irrational Denominators…………………………………44
E. Operations with Irrational Numbers
48
i. Squares and Square Roots………………………………
F. Algebraic Proofs…………………………………………………………………….. 49
G. Simplifying Using Log Identities………………………………………………… 50
3. Solving Algebraic Equations With One Variable
A. Multiple Step Equations
i. Equations with Multiple Steps…………………………55
B. Solving Quadratic Equations
i. Solving Quadratic Equations by Factoring………… 65
ii. Quadratic Formula with Equations……………………68
73
iii. When One or Both Roots are Known…………………
iv. Discriminant and Nature of Roots……………………74
v. Sum and Product of Roots…………………………… 79
C. Mixed Type of Equations
i. Absolute Value Equations………………………………81
ii. Irrational Equations………………………………………84
D. Logarithms and Exponents
86
i. Solving Logarithmic Equations…………………………
ii. Exponential Equations………………………………… 93
4. Algebraic Inequalities With One Variable
A. Multiple Step Inequalities
i. Absolute Value Inequalities…………………………… 100
103
ii. Quadratic Inequalities……………………………………
106
B. Graphing Inequalities……………………………………………………………...…
5. Functions
A. Functions
i. Domain and Range of a Function………………………108
ii. Inverse of a Function……………………………………112
117
iii. Definition of a Function…………………………………
iv. Evaluating a Function………………………………… 121
v. Modeling Functions with Graphs…………………… 130
II. Geometry
1. Geometry of the Circle
A. Degrees in a Circle
i. Angles of the Circle………………………………………139
ii. Radian & Degree Conversions…………………………152
B. Lines and Circles
i. Lengths of Line Segments………………………………156
ii. Length of an Arc…………………………………………169
iii. Extended Task……………………………………………173
C. Properties of Circles…………….………….……………………………………… 189
2. Euclidian Geometry Proofs
A. Properties of Figures (Short Answer)
i. Properties of Triangles………………………………… 193
ii. Properties of Parallelograms …………………………198
iii. Properties of Trapezoids ………………………………203
B. Direct Euclidean Proofs (Ext. Task)
i. Proving Triangles Congruent ………………………… 204
ii. Other Proofs………………………………………………208
C. Indirect Proofs
i. Contradiction………………………………………………219
ii. Counterexample …………………………………………220
III. The Coordinate Plane
1. Analytic Geometry
A. Transformations
221
i. Line Reflections……………………………………………
ii. Point Reflections…………………………………………232
iii. Translations………………………………………………233
iv. Rotations…………………………………………………241
v. Dilations………………………………………..………… 246
vi. Glide Reflections……………………………………… 250
vii. Composite Transformations…………………………250
B. Symmetry
i. Line Symmetry……………………………………………256
ii. Point Symmetry………………………………………… 259
260
iii. Isometry……………………………………………………
C. Graphing Curves
i. Equations of Parabolas…………………………………261
ii. Equations of Circles…………………………………… 277
iii. Ellipses in the Form ax² + bx²=c………………………283
iv. Equations of Hyperbolas………………………………286
v. Equations of an Exponential Function………………288
vi. Linear-Quadratic Systems ……………………………294
D. Coordinate Geometry Proofs
i. Short Answer …………………………………………… 301
ii. Extended Task……………………………………………302
E. Points and Areas in the Coordinate Plane
i. Finding Points…………………………………………… 306
ii. Finding Areas…………………………………………… 307
iii. Finding Vertices of Polygons…………………………310
IV. Trigonometry
1. Trigonometry
A. Trigonometry of the Right Triangle
i. Sine, Cosine & Tangent Functions……………………311
ii. Trigonometric Funct (Ext. Task)………………………323
B. Trigonometric Functions
i. Quadrants…………………………………..………………331
ii. Express an Angle as a Positive Acute Angle………335
iii. Inverse Trigonometric Function………………………336
339
iv. Reciprocal Trigonometric Functions…………………
v. Evaluating Trigonometric Functions…………………339
vi. Special Angles & Distinct Triangles…………………344
349
vii. Converting to & from Radian Measure………………
C. Graphing Trigonometric Functions
i. Amplitude, Frequency, and Period……………………350
356
ii. The Graphs of Sin, Cos, and Tan………………………
iii. The Graphs of their Inverses…………………………377
D. Trigonometry of Acute & Obtuse Triangles
i. Law of Sines………………………………………………378
ii. Law of Cosines……………………………………………386
iii. Area of a Triangle using Trig…………………………396
E. Trigonometric Equations and Identities
401
i. Solving Trigonometric Equations………………………
ii. Pythagorean, Quotient, & Reciprocal Identities……409
iii. Functions of the Sum of Two Angles……………… 413
iv. Functions of the Difference of Two Angles…………415
417
v. Functions of the Double Angle…………………………
vi. Functions of the Half Angle……………………………420
vii. Proof………………………………………………………422
V. Ratios and Proportions
1. Mathematical Ratios
A. Using Proportions
i. Inverse Variation………………………………………… 423
ii. Other Ratios………………………………………………427
VI. Counting, Probability, and Statistics
1. Probability
A. Evaluating Simple Probabilities
431
i. The Probability of "NOT"…………………………………
B. Bernoulli Trials
i. Exactly………………………………………………………432
ii. At Most & At Least………………………………………438
C. Probability
i. Probability (Ext. Task)……………………………………442
2. Statistics
A. Basic Statistics
i. Mean, Median and Mode…………………………………448
ii. Quartiles & Percentiles…………………………………451
iii. Range………………………………………………………452
iv. Standard Deviation………………………………………454
v. Summation Notation…………………………………… 468
vi. Binomial Expansion……………………………………475
B. Scatter Plots
i. Scatter Plots ………………………………………………478
ii. Lines of Best Fit …………………………………………482
iii. Linear Correlation Coeficient …………………………487
iv. Other Regression Curves …………………………… 489
I. ALGEBRA
A. Basic Algebra
1. Numbers, Sets, Systems, and Operations
i. Comparing Mathematical Expressions
5389. The accompanying diagram represents the biological
process of cell division.
If this process continues, which expression best represents
the number of cells at any time, t?
(1) t + 2
(3) t2
(2) 2t
(4) 2t
5004. Which graph shows that soil permeability varies inversely
to runoff?
(1)
(2)
5323. If 10k = x, then 103k is equal to
(3) 3x
(1) x3
(2) 3 + x
(4) 1,000x
5053. According to Boyle’s Law, the pressure, p, of a compressed
gas is inversely proportional to the volume, v. If a pressure
of 20 pounds per square inch exists when the volume of the
gas is 500 cubic inches, what is the pressure when the gas is
compressed to 400 cubic inches?
(3) 40 lb/in2
(1) 16 lb/in2
2
(4) 50 lb/in2
(2) 25 lb/in
5025. The time it takes to travel to a location varies inversely to
the speed traveled. It takes 4 hours driving at an average
speed of 55 miles per hour to reach a location. To the
nearest tenth of an hour, how long will it take to reach the
same location driving at an average speed of 50 miles per
hour?
4.4
4915. The expression
(1) 37 + 7`2
(2)
7 7
3 – `2
(3)
(4)
is equivalent to
21 + `2
7 7
(3) 3 + `2
(4) 3 – `2
4605. Juan got a 95 on his last English test which consisted of 20
questions worth 2 points each and 20 questions worth 3
points each. How many possible ways could Juan have
scored his 95?
(1) 0
(3) 2
(4) 4
(2) 1
4604. Jenny scored 17 points in a basketball game. She attempted
8 field goals and 3 free throws. Each successful field goal
is 2 points and each successful free throw is 1 point. If she
made all of her free throws, how many field goals did she
miss?
(3) 3
(1) 1
(2) 2
(4) 4
2760. Which equation is an illustration of the distributive law?
(3) (ab)c = a(bc)
(1) a(b + c) = ab + ac
(2) (a + b) + c = a + (b + c) (4) ab + ac = ac + ab
4593. Tom scored 23 points in a basketball game. He attempted
15 field goals and 6 free throws. If each successful field
goal is 2 points and each successful free throw is 1 point, is
it possible he successfully made all 6 of his free throws?
Justify your answer.
No
3892. A rectangle is said to have a golden ration when wh = hw h– h,
where w represents width and h represents height. When
w = 3, between which two consecutive integers will h lie?
1 and 2, 1 < x < 2, or 1 < 1.854 < 2
3850. Which is the correct arrangement of these terms in order of
value, from smallest to greatest?
(1) 3`2, 4 1/8, |–4.24|, _75 (3) 4 1/8, _75, |–4.24|, 3`2
(2) _75, |–4.24|, 4 1/8, 3`2 (4) 4 1/8, |–4.24|, _75, 3`2
© 1998-2009 Eduware, Inc.
1
I. ALGEBRA
C. Complex Numbers
1. Numbers, Sets, Systems, and Operations
iii. Mult & Div of Complex Numbers
4453. In an electrical circuit, the voltage, E, in volts, the current, I
, in amps, and the opposition to the flow of current, called
impedance, Z, in ohms, are related by the equation E = IZ.
A circuit has a current of (9 + 2i) amps and an impedance
of (–5 + 3i) ohms. Determine the voltage in a + bi form.
–51 + 17i
5419.
(1)
(2)
(3)
(4)
-2i
2i
`2i
2i
`5
4613. What is the area of an imaginary circle of radius 1 – 5i?
(1) (–24 – 5i)p
(3) (26 + 5i)p
(4) (26 + 10i)p
(2) (–24 – 10i)p
4612. What is the area of an imaginary triangle with a height of 2
and base of 2 + 4i?
(3) 4 + 8i
(1) 2 + 4i
(2) 4 + 6i
(4) 8
4610. What is the area of an imaginary rectangle with sides of
2 + 2i and 3i?
(1) 6 + 6i
(3) 6 – 6i
(2) –6 – 6i
(4) –6 + 6i
4608. Bill and Melanie are partners playing a game with complex
numbers. A team's score is equal to the product of its
members' individual scores. Bill has a score of 3 + 7i and
Melanie has a score of 3 – 7i. What is their team score?
(1) –40
(3) 58
(2) –40 – 7i
(4) 58 + 49i
4581. What is the product of 5 + `™36 and 1 – `™49, expressed
in simplest a + bi form?
(1) –37 + 41i
(3) 47 + 41i
(2) 5 – 71i
(4) 47 – 29i
4539. The relationship between voltage, E, current, I, and
resistance, Z, is given by the equation E = IZ. If a circuit
has a current I = 3 + 2i and a resistance Z = 2 – i, what is
the voltage of this circuit?
(3) 4 + i
(1) 8 + i
(2) 8 + 7i
(4) 4 – i
4526. In an electrical circuit, the voltage, E, in volts, the current, I
, in amps, and the opposition to the flow of current, called
impedance, Z, in ohms, are related by the equation E = IZ.
A circuit has a current of (3 + i) amps and an impedance of
(–2 + i) ohms. Determine the voltage in a + bi form.
–7 + i
4502. The relationship of distance, D, rate, r, and time, t, is given
by the equation D = rt. If the rate = 4-3i and the time = 5+2i
, what is the distance?
26-7i
4303. Given AB = C, A = 7 – i, and B = 3 + 3i, what is the value
of C?
(1) 18i + 18
(3) 41i
(2) 10 + 2i
(4) 18i + 24
4154. What is the reciprocal of 3 – `5?
(1) 3 – `5
4
(2) 3 + `5
4
(3) 3 – `5
14
(4) 3 + `5
14
3887. The expression (–1 + i)3 is equivalent to
(1) –3i
(3) –1 – i
(2) –2 – 2i
(4) 2 + 2i
3675. Where i is the imaginary unit, expand and simplify
completely (3 – i)4.
28 – 96i
2486. The value of (1 ‚ i)2 is
(1) 0
(2) 2
(3) ‚2i
(4) 2 ‚ 2i
2345. Express the product of 4 ‚ 3i and 2 + i in simplest a + bi
form.
11 ‚ 2i
2311. The product of 5 ‚ 2i and i is
(1) 7
(3) 5 ‚ 2i
(4) ‚2 + 5i
(2) 2 + 5i
2265. Expressed in a + bœ form,
5 is equivalent to
3+œ
(1) 15 – 5i
8 8
(2) 5 – 5i
3
(3) 3 – i
2 2
(4) 15 – 5i
2163. Express the
5 in simplest a + bi form.
2–i
2+i
2129. The product of (‚2 + 6i) and (3 + 4i) is
(1) ‚6 + 24i
(3) 18 + 10i
(2) ‚6 ‚ 24i
(4) ‚30 + 10i
2089. The expression (3 ‚ i)2 is equivalent to
(1) 8
(3) 10
(4) 8 + 6i
(2) 8 ‚ 6i
© 1998-2009 Eduware, Inc.
11
I. ALGEBRA
2. Simplifying Algebraic Expressions
G. Simplifying Using Log Identities
5012.
(1) 1087 + ¡ log T – log 273
(2) 1087 (¡ log T – ¡ log 273)
(3) log 1087 + ¡ log T – ¡ log 273
(4) log 1087 + 2 log (T + 273)
4836. If logb x = y, then x equals
(1) y • b
(2) y
b
(3) yb
(4) by
5331.
(1) 14, only
(2) 34, only
(3) 14 and –34
(4) –14 and 34
5321. The expression 12log m – 3 log n is equivalent to
(3)
(1)
(2)
4677. Which could be the sides of a rectangle whose perimeter is
log(A2B2)?
(3) log(AB), log(2)
(1) log(A), log(B)
(2) 2log(A), 2log(B)
(4) log(AB2), log(A)
4676. What is the volume of a cube whose sides each have a
perimeter of log(D4)?
(1) 3log(D)
(3) (log(D))3
3
(2) log(D )
(4) log(3D)
(4)
5292. If 24x + 1 = 8x + a, which expression is equivalent to x?
(1) a – 1
(2) 3a – 1
(3) a – 1
15
(4) a – 1
3
5121. A black hole is a region in space where objects seem to
disappear. A formula used in the study of black holes is the
Schwarzschild formula,
4675. Two runners are running in a race. The first place runner is
log(16x) from the starting line, and the second place runner
is log(4) from starting line. What is the distance between
the two runners?
(3) 4log(16x)
(1) log(4x)
(2) log(64x)
(4) 16x(log(4))
4674. If the side of a square room is log(5x), which of the
following could be the perimeter of the room?
(1) log(20x)
(3) 2log(25x2)
4
(4) 625log(x)
(2) (log(5x))
4331. The expression log (10(7 + x)) – log (10(x – 2)) is equivalent to
(3) 5
(1) 9
(2) –9
(4) 2x + 5
Based on the laws of logarithms, log R can be represented
by
(1) 2 log G + log M – log 2c
(2) log 2G + log M – log 2c
(3) log 2 + log G + log M – 2 log c
(4) 2 log GM – 2 log c
4204. The expression 2 log (x – 3) log (y) is equivalent to
(1)
(3)
(2)
(4)
4943. If log a = x and log b = y, what is
(1) x + 2y
(2) 2x + 2y
(3) x2 +2 y
(4) x + 22y
50
4114. If A = pr2, log A equals
(1) 2 log p + log r
(2) log p + 2 log r
(3) 2 log p + 2 log r
(4) 2p log r
3820. If log 5 = a, then log 250 can be expressed as
(1) 50a
(3) 10 + 2a
(4) 25a
(2) 2a + 1
© 1998-2009 Eduware, Inc.
I. ALGEBRA
A. Multiple Step Equations
3. Solving Algebraic Equations With One Variable
i. Equations with Multiple Steps
Base your answers to questions 4054 and 4055 on the diagram
below.
Peter worked at an ice cream shop. He was filling up a cone with
ice cream but he accidentally got a cone with a hole in the bottom.
The rate of the ice cream being poured in was 110 milliters per
second. The rate that the ice cream was flowing out was 68
milliliters per second. note: 1 mL = 1 cm3
4052. Two balls are thrown into the air. The first ball follows that
path represented by the equation h = –x2–2x+6. The path of
the other ball is represented by the equation h = x+3.
At what height do the paths of the two balls cross? Round
your answer to the nearest tenth of a foot.
3.8 feet
Base your answers to questions 4041 and 4042 on the information
below.
Michele was holding an ice cream cone for her brother. The ice
cream would melt at a rate of h(t)= –2.5t + 10, where t represents
the time the ice cream melts in minutes and h represents the height
of the ice cream in centimeters.
4041. How long will it take for the whole ice cream cone to melt?
4 minutes
4054. What is the volume of the cone? Round your answer to the
nearest whole number.
201cm3 or 201 mL
4042. How high was the ice cream originally?
10cm
4055. Estimate the time it will take for the cone to overflow.
Round your answer to the nearest second.
5 seconds
Base your answers to questions 3994 and 3995 on the information
below.
4053. Claudio was making a snowman. The diameter of the head
of the snowman, MN, is 4ft. The base of the snowman, QR,
has a diameter of 12ft.
Shelly is going to a museum. She wants to take a taxicab and
needs to decide what company to use. The Sunshine Taxicab
Company charges $3.75 for their service and $2.35 for every mile
after that. Tracy's Taxicab Company charges $5.25 for their
service and $1.62 for every mile after that.
3994. After how many miles, will the charge of both cab
companies be equal? Round your answer to the nearest
mile.
2 miles
3995. If Shelly needs to travel 12 miles to get to the museum,
which company should she choose and how much would it
cost her?
Tracy's Cab Company and it would cost her $24.69.
3989. A new shoe store was just opened up. The cost of opening
the store is represented by the equation, C(x)= 36x + 1,500,
where x represents the number of pairs of shoes they start
with. The revenue obtained by selling x pairs of shoes is
represented by the equation, R(x)= 42x. The total profit
earned by the shoe store is represented by they equation,
P(x)= R(x) – C(x). For the values of R(x) and C(x) given
above, what is P(x)?
(1) 6x + 1,500
(3) 6x – 1,500
(2) –6x – 1,500
(4) –6x + 1,500
What is the diameter of the middle layer OP?
8ft
3087. Solve for x: ³ = ‚ 2
‚1
60
3935. At the local music store Gavin buys two CDs and one video
for a total of $41.95. At the same time Morgan buys one
CD and two videos for a total of $39.50. How much does it
cost to buy a combination of one CD and one video?
$27.15
© 1998-2009 Eduware, Inc.
I. ALGEBRA
B. Solving Quadratic Equations
3. Solving Algebraic Equations With One Variable
ii. Quadratic Formula with Equations
4040. Base your answer on the diagram below.
There are two ways for Jimmy to get to school. If Jimmy
takes Copper Lane he travels 12x2 – 3 feet and then goes 3x
+ 4 feet down Young St. If he takes Marsha St. to school
he travels 11x + 6 feet and then goes 9x2 + 5 feet down
Greenland Ave.
3804. A homeowner wants to increase the size of a rectangular
deck that now measures 15 feet by 20 feet, but building
code laws state that a homeowner cannot have a deck larger
than 900 square feet. If the length and the width are to be
increased by the same amount, find, to the nearest tenth,
the maximum number of feet that the length of the deck
may be increased in size legally.
12.6 ft
3746. Two toy rockets are launched ten seconds apart. The height
in feet of the first rocket after 0 < t < 16 seconds is given by
h(t) = –16t2 + 256t. The height of the second one after 10 < t
< 20 seconds is given by g(t) = –16t2 + 480t – 3200. How
many seconds after the first rocket is launched are the
rockets at the same height?
14.286 seconds
3677. Solve for x and express your answer in simplest a + bi form:
¡ ) ¥i
3533. What are the values of x in the equation x2 + 4x – 1 = 0?
(1) –4 ) `5
(3) –2 ) `5
(2) –4 ) `3
(4) –2 ) `3
If both ways take him the same distance, how far does
Jimmy have to walk to the school? Round your answer to
the nearest tenth of a foot..
167.3 feet
4038. Natalie is making a rectangular rug. The rectangle is said
w
to have a golden ratio of l–w
ww =ll , where l equals the length of
the rug and w equals the width of the rug.
If w = 4 than between what two consecutive integers will l
lie?
6 and 7
3955. A rectangular patio is said to have a golden ratio when
ww
h h
2h = w + 2h, where w represents width and h represents
height. When w = 4, between which two consecutive
integers will h lie?
1 and 2
3890. Solve for x in simplest a + bi form: x2 + 8x + 25 = 0
–4 ± 3i
3865. Solve the equation x2 = 6x – 12 and express the roots in
simplest a + bi form.
3 + i`3, 3 – i`3
3512. What are the roots of the equation x2 – 3x + 1 = 0?
(1) 3 ) `5
2
(2) –3 ) `5
2
(3) 3 ) `13
2
(4) –3 ) `13
2
3504. What are the roots of the equation 3x2 + 6x – 2 = 0?
(1) 6 ) `60
6
(2) –6 ) `60
6
(3) 6 ) `12
6
(4) –6 ) `12
6
2609. Express the roots of the equation + x = 2 in simplest
a + bi form.
1 ) i`2
2604. Solve for x and express the roots in terms of i:
ax2 = 6x ‚ 5
3)i
2
3624. Express the roots of the equation 9x2 = 2(3x – 1) in simplest
a + bi form.
ii
1
3)3
© 1998-2009 Eduware, Inc.
69
I. ALGEBRA
D. Logarithms and Exponents
3. Solving Algebraic Equations With One Variable
ii. Exponential Equations
5305. The accompanying table shows the amount of water vapor, y, that will saturate 1 cubic meter of air at different temperatures, x.
Write an exponential regression equation for this set of data, rounding all values to the nearest thousandth.
Using this equation, predict the amount of water vapor that will saturate 1 cubic meter of air at a temperature of 50°C, and round
your answer to the nearest tenth of a gram.
y = 4.194(1.068)x
112.5
5281. Kathy deposits $25 into an investment account with an
annual rate of 5%, compounded annually. The amount in
her account can be determined by the formula A = P(1 +
R)t, where P is the amount deposited, R is the annual
interest rate and t is the number of years the money is
invested. If she makes no other deposits or withdrawals,
how much money will be in her account at the end of 15
years?
(1) $25.75
(3) $51.97
(2) $43.75
(4) $393.97
5272. The number of houses in Central Village, New York, grows
every year according to the function H(t) = 540(1.039)
t
, where H represents the number of houses, and t represents
the number of years since January 1995. A civil
engineering firm has suggested that a new, larger well must
be built by the village to supply its water when the number
of houses exceeds 1,000. During which year will this first
happen?
2011, and appropriate work is shown, such as
solving a logarithmic equation or trial and error
with at least three trials and appropriate checks.
4967. Solve for m:
3m + 1 – 5 = 22
2
94
5275. Water is draining from a tank maintained by the Yorkville
Fire Department. Students measured the depth of the water
in 15-second intervals and recorded the results in the
accompanying table.
Write the power regression equation for this set of data,
rounding all values to the nearest ten thousandth.
Using this equation, predict the depth of the water at 2
minutes, to the nearest tenth of a foot.
y = 42.2326x –0.4494 and 4.9, and appropriate work is
shown.
© 1998-2009 Eduware, Inc.
I. ALGEBRA
A. Functions
5. Functions
i. Domain and Range of a Function
5364. The accompanying graph illustrates the presence of a
certain strain of bacteria at various pH levels.
What is the range of this set of data?
(1) 5 % x % 9
(3) 0 % y % 70
(2) 5 % x % 70
(4) 5 % y % 70
5335. Evaluate:
5003. The effect of pH on the action of a certain enzyme is shown
on the accompanying graph.
What is the domain of this function?
(3) x & 0
(1) 4 % x % 13
(2) 4 % y % 13
(4) y & 0
4934. The accompanying graph shows the heart rate, in beats per
minute, of a jogger during a 4-minute interval.
42
5282. The accompanying graph shows the elevation of a certain
region in New York State as a hiker travels along a trail.
What is the range of the jogger’s heart rate during this
interval?
(1) 0 – 4
(3) 0 – 110
(2) 1 – 4
(4) 60 – 110
What is the domain of this function?
(1) 1,000 % x % 1,500
(3) 0 % x % 12
(2) 1,000 % y % 1,500
(4) 0 % y % 12
4260. What is the domain of f(x) = 2x?
(1) all integers
(3) x & 0
(4)
x%0
(2) all real numbers
108
4834. What is the domain of the function below?
(1)
(2)
(3)
(4)
all real numbers except 0
all real numbers except 3
all real numbers except 3 and –3
all real numbers
© 1998-2009 Eduware, Inc.
II. GEOMETRY
A. Degrees in a Circle
1. Geometry of the Circle
i. Angles of the Circle
4728. A peach pie is made with 13 peaches which are evenly
distributed throughout the pie. If Joe eats one slice of pie
that is big enough so that he eats exactly 5 peaches, what
was the angle of the slice to the nearest tenth of a degree?
138.5°
4384. Planet Z has a diameter of 6,500 miles. A moon rotates
around Planet Z at a distance of 700 miles from the planet's
surface. When the moon has traced an arc of 2,500 miles,
how many radians, to the nearest thousandth, is the angle
created from the rotation of the moon.
.633
4727. An apple pie contains 17 whole apples which are evenly
distributed throughout the pie. If Maria eats a section of the 4336. What is the measure of aABC if (ABC is isosceles?
pie that accounts for 2 radians, how many apples did she eat
to the nearest hundredth?
5.41
4726. A whole pizza has a circumference of 56 inches. If one
slice of the pizza has an arc length of 10 inches, what is the
measure of its angle in radians to the nearest hundreth?
1.12
4725. A circular cake is cut into three sections so that the ratio of
the sizes of the pieces is 4:2:1. What is the angle of the
middle sized piece in radians?
(1) .9
(3) 1.8
(2) 1.5
(4) 3.14
4724. A wedge of cheese is cut from a wheel that has a radius of 6
inches. The arc of the wedge of cheese is 3 inches long.
What is the angle of the wedge of cheese in radians?
(3) 2
(1) .5
(2) 1.57
(4) 3
4435. A car drives around a circular driveway that has a radius of
7 meters, as shown in the accompanying diagram. What
distance has the car travelled when the arc it has made is
110º? Express your answer to the nearest hundredth of a
meter.
30º
4279. An art student wants to make a string collage by connecting
six equally spaced points on the circumference of a circle to
its center with string. What would be the radian measure of
the angle between two adjacent pieces of string, in simplest
form?
4225. In the accompanying diagram, the length of ´B¼ is 322π
radians.
What is maABC?
(1) 36
(2) 45
13.44
4378. A slice of a pizza pie has a radius of 6.46 inches and the
measure of the edge around the crust of the slice is 2.15
inches. What is the angle measure of the pointed end of the
slice of pizza, to the nearest hundredth of a radian?
.33
(3) 53
(4) 72
2694. In circle O, PA and PB are tangent to the circle from point
P. If the ratio of the measure of major arc AB to the
measure of minor arc AB is 5:1, then maP is
(1) 60
(3) 120
(2) 90
(4) 180
© 1998-2009 Eduware, Inc.
141
II. GEOMETRY
B. Lines and Circles
1. Geometry of the Circle
ii. Length of an Arc
4732. Between the hours of 5 PM and 10 PM, the hour hand of a
clock moves through an arc of length 17 in. What is
the length of the hour hand to the nearest hundredth of an
inch?
6.49 in
4468. The accompanying diagram represents a wheel of cheese.
A wedge of 26º is cut out. What is the length of the arc of
the wedge that is cut out? Round your answer to the
nearest tenth.
4731. A cake with a radius of 8 inches is cut into 6 equal pieces.
What is the distance around the outer edge of each piece to
the nearest hundredth of an inch?
8.38 in
4576. A dog has a 20-foot leash attached to the corner where a
garage and a fence meet, as shown in the accompanying
diagram. When the dog pulls the leash tight and walks from
the fence to the garage, the arc the leash makes is 55.8 feet.
2.3 cm
What is the measure of angle ‡ between the garage and the
fence, in radians?
(1) 0.36
(3) 3.14
(4) 160
(2) 2.79
4510. Ilana buys a large circular pizza that is divided into eight
equal slices. She measures along the outer edge of the crust
from one piece and finds it to be 512 inches. What is the
diameter of the pizza to the nearest inch?
(3) 7
(1) 14
(2) 8
(4) 4
4170. In a circle, an arc of length 5 is subtended by a central
angle of 53 radians. What is the radius of the circle?
(1) 25
3
(2) 3
25
(3) 3
(4) 5
3990. A merry-go-round rotates in a circle as shown in the
diagram below. The radius of the circle made is 25in.
4485. Anthony buys a pizza pie for his friends, and cuts it into
eight equal slices. The measure of the crust of each slice is
6 inches. Find the diameter of the pizza to the nearest tenth
of an inch.
15.3 in
4380. A circular clock has two hands, each of which is the length
of the radius of the clock. It is 5:00. The arc of the clock
from the minute hand to the hour hand is 13 inches. What
is the length of one hand of the clock, to the nearest
hundredth of an inch?
4.97 inches
4101. In a circle with a radius of 4 centimeters, what is the
number of radians in a central angle that intercepts an arc of
24 centimeters?
6
Lily is riding on the black horse. If the subtended arc is
145º, how far did her horse travel around the circle?
Express your answer to the nearest hundredth of an inch.
63.27 in
231. In a circle with radius 4.5 centimeters, find, in centimeters,
the length of the arc intercepted by a central angle of 3
radians.
13.5
193. In a circle of radius 6, find the length of the arc intercepted
by a central angle of 2 radians.
12
170
© 1998-2009 Eduware, Inc.
II. GEOMETRY
B. Direct Euclidean Proofs (Extended Task)
5414. In the accompanying diagram of circle O, AD is a diameter
with AD parallel to chord BC , chords AB and CD are
drawn, and chords BD and AC intersect at E.
Prove: BE ? CE.
2. Euclidean Geometry Proofs
ii. Other Proofs
5097. In the accompanying diagram of circle O, diameter AOB is
drawn, tangent CB is drawn to the circle at B, E is a point
on the circle, and BE || ADC.
Prove: (ABE ' (CAB
5380. Given: PROE is a rhombus, SEO, PEV, aSPR ? aVOR
A complete and correct proof that includes a
concluding statement is written.
4094. Given: aH ? aJ, K is the midpoint of HJ, and IF ? IG.
Prove SE ? EV
A complete and correct proof that includes a
conclusion is written.
4531. A picnic table in the shape of a regular octagon is shown in
the accompanying diagram. If the length of AE is 6 feet,
find the length of one side of the table to the nearest tenth
of a foot, and find the area of the table’s surface to the
nearest tenth of a square foot.
Prove: FK ? GK
Proof.
3681. For an isosceles triangle, TRI, aT is the vertex angle, and U
is the midpoint of RI. Prove that median TU bisects aT.
proof
3680. For an isosceles triangle ABC, prove that the altitude to the
base, AD, is also the median.
proof
The side equals 2.3 and the area equals 25.5
210
© 1998-2009 Eduware, Inc.
III. THE COORDINATE PLANE
A. Transformations
1. Analytic Geometry
i. Line Reflections
661. In the accompanying diagram, p and q are symmetry lines
for rectangle ABCD.
Find rp † rq † rp(A).
D
573. In the accompanying figure, p and q are symmetry lines for
the figure ABCDEF.
Find rq † rp † rq(A).
C
629. When point A(‚2,5) is reflected in the line x = 1, the image
is
(1) (5,2)
(3) (4,5)
(2) (‚2,‚3)
(4) (0,5)
530. In the accompanying figure, lines … and m are lines of
symmetry. What is rm † r… (BC)?
613. In the accompanying figure, … and m are symmetry lines for
regular pentagon ABCDE.
(1) HA
(2) GF
Find r…
C
†
(3) DE
(4) BC
254. The equation y = tanx is graphed in the interval
0 % x % p/2 and is reflected over the x–axis. On this
reflection, point (p/4,y) has which value for y?
(1) 1
(2) ‚1
(3) 0
(4) `3
3
rm(A)
583. A line reflection preserves
(1) distance and orientation
(2) angle measurement and orientation
(3) distance, but not angle measurement
(4) distance and angle measurement
210. If … and m are parallel lines, then r… † rm(AB) is equivalent to
a
(1) rotation
(3) translation
(2) dilation
(4) glide-reflection
189. Point P(‚1,‚5) is reflected over the line y = ‚x. What are
the coordinates of P', the image of P?
(5,1)
146. If B(‚2,5) is reflected over the line y = x, what are the
coordinates of the image of B?
(5,‚2)
© 1998-2009 Eduware, Inc.
229
III. THE COORDINATE PLANE
A. Transformations
1. Analytic Geometry
iv. Rotations
1418. Which is the image of A under the transformation
rx-axis † R90°
(3)
(1)
(2)
474. In the accompanying diagram, regular hexagon ABCDEF is
inscribed in circle O. With O as the center of rotation find R
–120º † R240º (A).
(4)
1377. What is the image of (1,0) after a counterclockwise rotation
of 60º?
(3)
(1)
(2)
E
(4)
398. Figure B is the image of figure A under which single
transformation?
1158. Write the coordinates of P', the image of P(5,‚1) after a
clockwise rotation of 180º about the origin.
(‚5,1)
935. The point (‚2,1) is rotated 180† about the origin in a
clockwise direction. What are the coordinates of its image?
(2,‚1)
779. Which transformation is equivalent to the composite line
reflections ry-axis † ry = x (AB)?
(3) a translation
(1) a rotation
(2) a dilation
(4) a glide reflection
729. What are the coordinates of M', the image of M(2,4), after a
counterclockwise rotation of 90º about the origin?
(1) (‚2,4)
(3) (‚4,2)
(2) (‚2,‚4)
(4) (‚4,‚2)
668. If the letter P is rotated 180 degrees, which is the resulting
figure?
(3)
(1)
(2)
(4)
528. Which rotation about the origin is equivalent to R–200°?
(1) R200º
(2) R–160º
(3) R160º
(4) R560º
437. If the point (3,0) is rotated 270º counter-clockwise about
the origin (R270º), its image is on the line
(3) y = x
(1) x = 0
(2) y = 0
(4) y = ‚x
249. Which geometric figure has 120† rotational symmetry?
(1) square
(3) regular pentagon
(2) rhombus
(4) equilateral triangle
244
(1) line reflection
(2) translation
(3) rotation
(4) glide reflection
305. a On graph paper, graph and label triangle ABC whose
vertices have coordinates A(4,0), B(8,1), and C(8,4).
b Graph and state the coordinates of (A'B'C', the image of
(ABC after the composite transformation rx=0 † ry = x
((ABC).
c Which single type of transformation maps (ABC onto
(A'B'C'?
(1) rotation
(3) glide reflection
(2) dilation
(4) translation
d Graph and state the coordinates of (A''B''C'', the image of
(ABC after the composite transformation ry =-4 † ry = 0
((ABC).
e Which single type of transformation maps (ABC onto
A''B''C''?
(1) rotation
(3) glide reflection
(2) dilation
(4) translation
b A'(0,4) B'(‚1,8) C'(‚4,8)
c 1
d A''(4,‚8) B''(8,‚7) C''(8,‚4)
e 4
101. What is the image of the point (2,‚3) under a clockwise
rotation of 90° about the origin?
(‚3,‚2)
© 1998-2009 Eduware, Inc.
III. THE COORDINATE PLANE
C. Graphing Curves
1. Analytic Geometry
i. Equations of Parabolas
3188. Which is an equation of the axis of symmetry of the graph
of the equation y = 2x2 ‚ 5x + 3?
(1) x = ‚5
2
(2) x = 5
2
(3) x = ‚5
4
(4) x = 5
4
3093. Which equation defines the graph in the diagram below?
3166. Which is an equation of the axis of symmetry for the
parabola whose equation is
y = 2x2 ‚ 3x + 4?
(3) x = ¦
(1) x = ¥
(2) x = ‚¥
(4) x = ‚¦
3141. What is an equation of the axis of symmetry of the graph of
the parabola y = 2x2 + 3x + 5?
(1) y = ‚¦
(3) x = ‚¦
(2) y = ‚¥
(4) x = ‚¥
3138. The graph of the equation y = x2 is a
(1) circle
(3) point
(4) straight line
(2) parabola
3128. Which is a point of intersection of the equations y = x and y
= x2 + x ‚ 1?
(1) (0,0)
(3) (‚1,0)
(2) (1,0)
(4) (‚1,‚1)
3127. What is an equation of the axis of symmetry of the graph of
the equation y = 2x2 ‚ 3x ‚ 1?
(1) x = ¦
(3) x = ¥
(2) y = ‚¦
(4) y = ¥
3114. a On graph paper, draw the graph of the
equation y = x2 + 4, including all values of x in
the interval ‚3 % x % 3.
b Write the coordinates of the turning point of
the graph drawn in part a.
c Indicate whether the point in part b is a
minimum or a maximum point.
d On the same set of axes, draw the graph of the
image of the graph drawn in part a after a
reflection in the x-axis.
b 0, 4 c minimum
3109. Which is an equation of the axis of symmetry for the
parabola whose equation is y = 2x2 + 8x ‚ 1?
(3) x = ‚4
(1) x = ‚2
(2) x = 2
(4) x = 4
3091. Which is the axis of symmetry of the graph of the equation
y = ‚x2 ‚ 2x ‚ 1?
(3) x = 1
(1) x = ‚1
(2) y = ‚1
(4) y = 1
(1) y = x2 + 6x + 1
(2) y = ‚x2 + 6x + 1
(3) y = x2 + 3x
(4) y = ‚x2 + 3x ‚ 1
3081. a On graph paper, draw the graph of. the
parabola y = x2 + 6x + 5, including all values
of x in the interval ‚6 % x % 0.
b On the same set of axes, draw the image of the
parabola drawn in part a after a translation of
(x + 3, y ‚ 3).
c Using the graph, write the coordinates of the
point of intersection of the parabolas drawn in
parts a and b.
c (–2, –3)
3080. The turning point of the graph of the function of
y = 2x2 + 4x + 3 is
(3) (1,‚1)
(1) (‚1,1)
(2) (‚1,‚1)
(4) (1,1)
3058. What is the y-intercept of the parabola whose equation is
y = x2 + 7x + 5?
(1) ‚7
2
(2) 5
(3) 3
(4) 7
2
3047. Which is the turning point of the parabola whose equation
is y = x2 ‚ 4x + 4?
(1) (2,‚4)
(3) (‚2,16)
(4) (‚2,0)
(2) (2,0)
© 1998-2009 Eduware, Inc.
271
III. THE COORDINATE PLANE
C. Graphing Curves
1. Analytic Geometry
ii. Equations of Circles
2850. In the accompanying figure, point S(‚3,4) lies on circle O
with center (0,0). Line @ASB[ and radius OS are drawn.
2219.
(1)
(2)
a Find the length of OS.
b Write an equation of circle O.
c If @AB[ ; OS, find the slope of @AB[.
d Write an equation of line @ASB[.
e Find the coordinates of any point on @AB[
other than S.
a 5 b x2 + y2 = 25 c ¥
d y ‚ 4 = ¥ (x + 3)
or
4y = 3x + 25
or
y = 34x + 25
44
(3)
2785. Which point lies on the circle x2 + y2 = 49?
(1) (5,24)
(3) (‚7,0)
(2) (‚4,3)
(4) (0,0)
(4)
2776. Which is an equation of a circle whose center has
coordinates (4,‚3) and whose radius has length 6?
(1) (x + 4)2 + (y ‚ 3)2 = 36 (3) (x + 4)2 + (y ‚ 3)2 = 6
(2) (x ‚ 4)2 + (y + 3)2 = 36 (4) (x ‚ 4)2 + (y + 3)2 = 6
2768. What are the coordinates of the center of the circle whose
equation is (x ‚ 3)2 + (y + 2)2 = 12?
(3,‚2) or x = 3, y = ‚2
2763. An equation of a circle with center at (2,‚3) and radius 5 is
(1) (x ‚ 2)2 + (y + 3)2 = 25 (3) (x + 2)2 + (y ‚ 3)2 = 25
(2) (x ‚ 2)2 + (y + 3)2 = 5
(4) (x + 2)2 + (y ‚ 3)2 = 5
2186.
(1) a circle
(2) an ellipse
(3) a hyperbola
(4) a parabola
1881. The graph of the equation y2 = 4 ‚ x2 is
(1) an ellipse
(3) a circle
(2) a hyperbola
(4) a parabola
282
© 1998-2009 Eduware, Inc.
III. THE COORDINATE PLANE
E. Points and Areas in the Coordinate Plane
1. Analytic Geometry
ii. Finding Areas
5303. The accompanying diagram shows the peak of a roof that is in the shape of an isosceles triangle. A base angle of the triangle is 50º
and each side of the roof is 20.4 feet. Determine, to the nearest tenth of a square foot, the area of this triangular region.
204.9
5448. Firefighters dug three trenches in the shape of a triangle to
prevent a fire from completely destroying a forest. The
lengths of the trenches were 250 feet, 312 feet, and 490
feet.
5411. In the accompanying diagram, CD is an altitude of (ABC.
If CD = 8, maA = 45, and maB = 30, find the perimeter of
(ABC in simplest radical form.
Find, to the nearest degree, the smallest angle formed by
the trenches.
Find the area of the plot of land within the trenches, to the
nearest square foot.
26 and 33,443, and appropriate work is shown.
[Allow full credit if student uses 26 and finds A =
33,509.]
5413. The accompanying diagram shows a triangular plot of land
located in Moira's garden.
24 + 8`2 + 8`3
4712. A homeowner's property consists of a coordinate grid. A
triangular portion of the grid having vertices at (1,2), (3,10)
and (7, 2) has been reserved for a garden. What is the area
of the garden?
24
4060. What is the area of the triangle whose vertices are (3,1),
(7,1), and (6,4)?
6
3492. Find the area of pentagon CANDY with vertices C(–6,8),
A(3,8), N(6,–2), D(–4,–1), and Y(–7,4).
97.5
3465. The vertices of a pentagon are A(–2,–1), B(1,3), C(3,4),
D(5,0), and E(3,–2). Find the area of pentagon ABCDE.
23.5
3420. Trapezoid ABCD, which has coordinates A(0,9),
B(12,9), C(8,4), and D(0,4).
Find the area of the plot of land, and round your answer to
the nearest hundred square feet.
8,200
4713. When designing a house, an architect forms a coordinate
grid to use on his blueprints. A triangular room has two
vertices at (1, 5) and (5, 12). If the room is to have an area
of 14 and be in the shape of a right triangle, what are the
coordinates of the third vertex?
(5,5)
Find the perimeter of ABCD to the nearest integer.
31
3419. Trapezoid ABCD, which has coordinates A(0,9),
B(12,9), C(8,4), and D(0,4).
Find the area of trapezoid ABCD.
50
© 1998-2009 Eduware, Inc.
307
IV. TRIGONOMETRY
A. Trigonometry of the Right Triangle
1. Trigonometry
i. Sine, Cosine, & Tangent Functions
3274. In the accompanying diagram of right triangle ABC, the
hypotenuse is AB , AC = 3, BC = 4, and AB = 5.
Sin B is equal to
(1) sin A
(2) cos A
3240. A 20-foot ladder is leaning against a wall. The foot of the
ladder makes an angle of 58° with the ground. Find, to the
nearest foot, the vertical distance from the top of the ladder
to the ground.
(3) tan A
(4) cos B
17 ft
3272. In the accompanying diagram of right triangle ABC, b = 40
centimeters, maA = 60º, and maC = 90º. Find the number
of centimeters in the length of side c.
3215. In the diagram below, maC = 90º, maA and
CA = 10.
Which equation can be used to find AB?
(3)
(1)
80 cm
3244. In the accompanying diagram of right triangle ABC, what is
tan C?
(4)
(2)
3179. In the accompanying diagram of right triangle ABC,
maC = 90º, maA = 45º, and AC = 1. Find, in radical form,
the length of AB.
(1) 2
3
(2) `13
3
(3) 3
2
(4) 2
`13
316
`2
3177. If sin A = 0.3642, find the measure of aA to the nearest
degree.
21º
© 1998-2009 Eduware, Inc.
IV. TRIGONOMETRY
B. Trigonometric Functions
1. Trigonometry
i. Quadrants
5139. If sin x < 0 and tan x > 0, then x must be in
Quadrant
(1) I
(3) III
(2) II
(4) IV
4162. If sin ‡ is less than 0 and sec ‡ is greater than 0, in which
quadrant does the terminal side of ‡ lie?
(1) I
(3) III
(2) II
(4) IV
4977.
5
4111. If sin A = 513
and cos A > 0, angle A terminates in Quadrant
(1) I
(3) III
(2) II
(4) IV
3831. If the sine of an angle is © and the angle is not in
Quadrant I, what is the value of the cosine of the angle?
–0.8
3644. If sin A < 0 and tan A > 0, in which quadrant does the
terminal side of aA lie?
(1) I
(3) III
(2) II
(4) IV
3548. In which quadrant are both tangent and cosecant negative?
IV
(1) 210
(2) 225
2746. If sec x < 0 and tan x < 0, then the terminal side
of angle x is located in Quadrant
(1) I
(3) III
(4) IV
(2) II
(3) 233
(4) 240
4935. If sin ‡ is negative and cos ‡ is negative, in which quadrant
does the terminal side of ‡ lie?
(1) I
(3) III
(2) II
(4) IV
2697. In the accompanying diagram, point P(–0.6,–0.8) is on unit
circle O.
4871. If the tangent of an angle is negative and its secant is
positive, in which quadrant does the angle terminate?
(1) I
(3) III
(2) II
(4) IV
4636. A landscaper uses a coordinate grid to design gardens. He
puts one row of plants along the x-axis and wants to put
another row of plants so that the angle they form with the
first row has a secant that is positive and a sine that is
negative. In what quadrant of the garden must the second
row of plants be placed?
IV
4537. If sin ‡ > 0 and sec ‡ < 0, in which quadrant does the
terminal side of angle ‡ lie?
(1) I
(3) III
(4) IV
(2) II
What is the measure of angle ‡ to the nearest degree?
(1) 143
(3) 225
(2) 217
(4) 233
2675. If sin A < 0 and cot A > 0, in which quadrant does the
terminal side of aA lie?
III
4332. If (csc x – 7)(9csc x – 5) = 0, then x terminates in
(1) Quadrant I, only
(2) Quadrants I and II, only
(3) Quadrants I and IV, only
(4) Quadrants I, II, III, and IV
2584. If ‡ is in Quadrant II and cos ‡ = ‚¥, find an exact value
for sin 2‡.
‚3`7
8
4198. Which trigonometric function is positive in Quadrant IV?
(1) sin x
(3) csc x
(4) cot x
(2) sec x
2561. If tan x = ‚`3, in which quadrant could angle x terminate?
(1) I and III
(3) II and IV
(2) II and III
(4) III and IV
© 1998-2009 Eduware, Inc.
331
IV. TRIGONOMETRY
B. Trigonometric Functions
1. Trigonometry
v. Evaluating Trigonometric Functions
5273. Find all values of x in the interval 0º % x < 360º that satisfy
the equation 3 cos x + sin 2x = 0.
90° and 270°, and appropriate work is shown, such
as solving the equation
3 cos x + 2 sin x cos x = 0 or sketching a graph and
finding the x-intercepts.
5045. The expression
is equivalent to
(1) 1
(2) –1
(3) sin x
(4) cos x
4880. The path traveled by a roller coaster is modeled by the
equation y = 27 sin 13x + 30. What is the maximum altitude
of the roller coaster?
(1) 13
(3) 30
(2) 27
(4) 57
4811. If x is an acute angle and sin x = 12
13, then cos 2x equals
(1) 25
169
(2) 119
169
(3) – 25
169
(4) – 119
169
4642. What is the length, to the nearest tenth of a foot, of the
slope of a hill with an angle of inclination of 30° if the
change in elevation from the top to the bottom is 47 feet
94.0 ft
4639. The electric current in a RC circuit oscillates according to
the equation I(t) = 5 cos(3t), where I is the current and t is
the time in seconds. What is the period of oscillation of the
current?
2p/3 s
4638. A mass on a spring is oscillating according to the equation
x(t) = 3 sin 3t, where x is the distance from the equilibrium
position in centimeters and t is the time in seconds. What is
the distance from equilibrium when t = p/6?
3 cm
4637. A pendulum oscillates according to the equation
x = 3 cos 4t, where t is the time in seconds. What is the
frequency of oscillation in s–1?
(1) p/2
(3) 2/p
(2) 8p
(4) 1/8p
3836. If sin x = .8, where 0° < x < 90°, find the value of
cos (x + 180°).
–0.6 or an equivalent answer
342
4463. A gardener moves around the lawn following the equation
y = sin x + 1. A fly flies around the lawn according to the
equation y = 4 cos 5x + 4. For how many values of x on the
intervals 0 % x % 2p do the gardener and the fly collide
(intersect)?
(1) 5
(3) 9
(2) 8
(4) 10
3872. An object that weighs 2 pounds is suspended in a liquid.
When the object is depressed 3 feet from its equilibrium
point, it will oscillate according to the formula
x = 3 cos 8t, where t is the number of seconds after the
object is released. How many seconds are in the period of
oscillation?
(1) p
4
(2) p
(3) 3
(4) 2p
3626. Find all values of ‡ in the interval 0 % ‡ % 360º that satisfy
the equation sin ‡ = 2 + 3 cos 2‡. Express your answer to
the nearest ten minutes or nearest tenth of a degree.
56.4º, 123.6º, 270º
or
56º30', 123º30', 270º
3615. As angle ‡ increases from p radians to 2p radians, the
cosine of ‡
(1) increases throughout the interval
(2) decreases throughout the interval
(3) increases, then decreases
(4) decreases, then increases
2692. If f(x) = sin (Arc tan x), the value of f(1) is
(1) `2
(2) `2
2
(3) `3
2
(4) `3
3
2689. As angle x increases from p/2 to p, the value of
sin x will
(1) increase from –1 to 0
(3) decrease from 0 to –1
(2) increase from 0 to 1
(4) decrease from 1 to 0
2648. If sin ‡ = –© and cos ‡ > 0, what is the value of tan ‡?
(1) ¥
(3) 43
(4) –43
(2) –¥
2542. If f(x) = sin x + cos x, evaluate f(2p).
1
2505. If f(x) = sin ¡x + 2 cos x, evaluate f(p).
‚1
© 1998-2009 Eduware, Inc.
IV. TRIGONOMETRY
C. Graphing Trigonometric Functions
1. Trigonometry
ii. The Graphs of Sine, Cosine, and Tangent
1882. a On the same set of axes, sketch and label the
graphs of the equations y = sin ¡x and
y = 2 cos x in the interval 0 % x % 2p.
b Use the graphs sketched in part a to determine
the number of points in the interval 0 % x % 2p
that satisfy the equation sin ¡x = 2 cos x.
b 2
1835. a Graph the equation y = 3 sin x in the domain
‚p/2 % x % p/2.
b On the same set of axes, reflect the graph
drawn in part a in the line y = x, and label the
graph b.
c (1) Is the relation graphed in part b a function?
(2) State a mathematical justification for your
answer.
d Write an equation that represents the graph
drawn in part b.
c (1) Yes d x = 3 sin y
1793. a On the same set of axes, sketch and label the
graphs of the equations y = 2 sin ¡x and
y = cos 2x in the interval 0 < x < 2p.
b Use the graphs from part a to determine how
many values of x in the interval 0 < x < 2p
satisfy the equation 2 sin ¡x = cos 2x.
b 2
1790. For which value of u is the expression below undefined?
(1) 0
(2) 45
(3) 50
(4) 110
1749. a On the same set of axes, sketch and label the
graphs of the equations y = sin ¡x and
y = ¡ cos x for the values of x in the interval
‚p % x % p.
b In which interval is sin ¡x always greater than
¡ cos x?
(1) ‚p % x % p/2
(3) 0 % x % p/2
(2) ‚p/2 % x % 0
(4) p/2 % x % p
b 4
1661. a On the same set of axes, sketch and label the
graphs of the equations y = 2 cos x and
y = sin 2x as x varies from ‚p to p radians.
b Use the graphs drawn in part a to determine all
values of x in the interval ‚p % x % p that
satisfy the equation 2 cos x = sin 2x.
a Graph b ‚Œ, Œ
287. As ‡ increases from p/2 to p, which statement is true?
(1) sin ‡ decreases from 0 to ‚1
(2) cos ‡ decreases from 0 to ‚1
(3) cos ‡ increases from ‚1 to 0
(4) sin ‡ increases from ‚1 to 0
1694. Which equation is represented by the graph below?
(1) y = 2 sin ¡x
(2) y = ¡ sin ¡x
(3) y = ¡ sin 2x
(4) y = ‚¡ cos 2x
1622. a Sketch the graph of the equation y = 2 sin x in
the interval ‚p % x % p.
b On the same set of axes, reflect the graph draw
in part a in the y-axis and label the graph b.
c Write an equation of the graph drawn in part b.
d Using the equation from part c, find the value
of y when x = p/6.
c y = ‚2 sin x d ‚1
1573. a On the same set of axes, sketch the graphs of
the equations y = 2 cos ¡x and y = ‚sin x in
the interval 0 % x % 2p.
b From the graphs drawn in part a, find all
values of x that satisfy the equation
2 cos ¡x = ‚sin x.
b p
1478. a Graph the equation y = cos ¡x for values of x on the
interval –p % x % p.
b On the same set of axes, sketch the transformation of the
and label it b.
graph drawn in part a under T(
p,0)
c If 64 cos x = –14 and x is in the second quadrant, find cos
¡x and express in simplest form.
c 5
8
1438. a On the same set of axes, sketch and label the
graphs of y = 2 sin x and y = cos 2x for the
values of x in the interval ‚p % x % p.
b Based on the graphs drawn in part a, which
value of x in the interval ‚p % x % p satisfies
the equation 2 sin x ‚ cos 2x = 3?
b Œ
1378. As ‡ increases from p to 3p, the value of cos ‡
2
2
(1) decreases, only
(2) increases, only
(3) decreases and then increases
(4) increases and then decreases
© 1998-2009 Eduware, Inc.
373
IV. TRIGONOMETRY
D. Trigonometry of Acute & Obtuse Triangles
2066. The sides of a triangle have lengths 58, 92, and 124.
a Find, to the nearest ten minutes, the largest
angle of the triangle.
b Find, to the nearest integer, the area of the
triangle.
a 109º 30' b 2515
2054. If a = 5`2, b = 8, and maA = 45, how many distinct
triangles can be constructed?
(1) 1
(3) 3
(4) 0
(2) 2
2022. a Find, to the nearest degree, the measure of the
largest angle of a triangle whose sides measure
22, 34, and 50.
b Find, to the nearest integer, the area of the
triangle described in part a.
a 125 b 306
2017. If a = 5, b = 7, and maA = 30, how many distinct triangles
can be constructed?
(1) 1
(3) 3
(4) 4
(2) 2
2009. The sides of a triangle measure 6, 7, and 9. What is the
cosine of the largest angle?
(1) _ 4
84
(2) 81
(3) 4
84
(4) _ 1
81
1971. In (ABC, maA = 42†20', AC = 2.0 feet, and
AB = 18 inches.
a Find BC to the nearest tenth.[Indicate the unit of
measure.]
b Find the area of (ABC to the nearest tenth.
[Indicate the unit of measure]
a 16.2 in or 1.3 ft
b 145.5 in2 or 1.0 ft2
1918. In (ABC if a = 8, b = 5, and c = 9, then cos A is
(1) 7
15
(2) _ 7
15
(3) 1
4
(4) _ 1
4
1826. In (ABC, maA = 30, a = 4, and b = 6. Which type of
angle is aB?
(1) either acute or obtuse (3) acute, only
(2) obtuse, only
(4) right
392
1. Trigonometry
ii. Law of Cosines
1885. In parallelogram ABCD, AD = 11, diagonal AC = 15, and
maBAD = 63†50'.
a Find, to the nearest ten minutes, the measure
of aACD.
b Find, to the nearest integer, the area of
parallelogram ABCD.
a 41†10' b 64
1810. In (ABC, a = 5, b = 6, and c = 8. Find cos A.
75
96
1747. In (ABC, a = 1, b = 1, and c = `2. What is the value of
cos C?
(1) 1
(3) ¡`2
(2) `2
(4) 0
1709. A side of rhombus ABCD measures 100 feet. The measure
of aABC = 110†20'.
a Find, to the nearest foot, the measure of
diagonal AC.
b Find, to the nearest square foot, the area of
rhombus ABCD.
a 164 b 9377
1664. Two forces act on a body at an angle of 120º. The forces
are 28 pounds and 35 pounds.
a Find the magnitude of the resultant force to the
nearest tenth of a pound.
b Find the angle formed by the greater of the
two forces and resultant force to the nearest
degree.
a 32.1 b 49
1646. In (ABC, a = 6, b = 5, and c = 8. Cos A equals
(1) 75
80
(2) 53
80
(3) ‚3
80
(4) 53
60
1623. One angle of a rhombus measures 100†, and the longer
diagonal measures 5.8 meters.
To the nearest tenth of a meter, find the length of
a each side of the rhombus
b the shorter diagonal
a 3.8 b 4.9
1576. In parallelogram ABCD, AD = 10, AB = 12, and diagonal
BD = 18. Find the measure of angle A to the nearest ten
minutes.
109†30'
© 1998-2009 Eduware, Inc.
IV. TRIGONOMETRY
D. Trigonometry of Acute & Obtuse Triangles
1. Trigonometry
iii. Area of a Triangle using Trig
1157. In (ABC, a = 6, b = 8, and sin C = £. Find the area of
(ABC.
6
520. In (ABC, maA = 150, b = 8, and c = 10. Find the number
of square units in the area of (ABC.
20
1149. In the accompanying diagram of (ABC, AC = 30
centimeters, maB = 100, and maA = 50. Find the area of
(ABC to the nearest square centimeter.
438. In triangle ABC, a = 20, and maC = 30. For which value of
b is the area of triangle ABC equal to 100 square units?
(1) 10
(2) 20
(3) 20`3
3
(4) 25
376. In the accompanying diagram of (RST, ST = 3 and RT = 4.
If maT = 30, find the area of (RST.
175
1138. The area of (ABC is 100 square centimeters. If
c = 20 centimeters and maA = 30, then b is equal to
(3) 20`3 cm
(1) 20 cm
(2) 500 cm
(4) 10`2 cm
1083. Find the area of (ABC if m<A = 30, b = 10, and
c = 5.
12¡
991. In (PQR, PQ = 5 cm, QR = 6 cm, and maQ = 30. Find the
area of (PQR in square centimeters.
7.5
865. Two sides of a triangle measure 6 and 8, and the measure of
the included angle is 150†. The area of the triangle is
(1) 24`3
(3) 12`3
(2) 24
(4) 12
690. a In (ABC, AB = 15 cm, BC = 10 cm, and
AC = 6 cm. Find the measure of angle B to the
nearest degree.
b Using the answer to part a, find the area of
(ABC to the nearest square centimeter.
a 16º
b 21
104. In the accompanying figure of (ABC, a = 10,
b = 10, and maA = 75. Find the area of (ABC.
3
282. Find, to the nearest integer, the area of parallelogram
ABCD if AB = 10, BC = 8, and maA = 50.
61
236. In triangle ABC, maC = 90, side c = 13 centimeters, and
side a = 5 centimeters. Find the area of the triangle in
square centimeters.
30
219. a Two sides of a triangular plot measure 30
meters and 18 meters, respectively. If the
angle opposite the 30-meter side measures
58†, find, to the nearest degree, the measure
of the angle opposite the 18-meter side.
b Using the answer to part a, find the area of the
triangle to the nearest square meter.
a 31 b 270
149. In triangle ABC, maA = 30, side b = 10, and side c = 20.
Find the area of triangle ABC.
50
64. In the accompanying diagram, ABCD is a parallelogram
with AB = 20, AD = 10, and maA = 30. Find the area of
parallelogram ABCD.
100
655. In triangle ABC, a = 6, b = 8, and sin C = £. Find the area
of triangle ABC.
6
25
400
616. In (ABC, a = 6, c = 4, and maB = 150. Find the number of
square units in the area of the triangle.
6
© 1998-2009 Eduware, Inc.
IV. TRIGONOMETRY
E. Trigonometric Equations and Identities
4794. The expression cos 40° cos 10° + sin 40° sin 10° is
equivalent to
(3) sin 30°
(1) cos 30°
(2) cos 50°
(4) sin 50°
3
4583. If sin x = 12
13 , cos y = 5 , and x and y are acute angles, the
value of cos (x – y) is
(1) 21
65
(2) 63
65
(3) – 14
65
(4) – 33
65
4110. The value of sin 170° cos 20° – cos 170° sin 20° is
(1) ¡
(2) –¡
(3) `3
2
(4) –`3
2
3643. The expression cos 80° cos 70° + sin 80° sin 70° is
equivalent to
(3) sin 10°
(1) cos 10°
(2) cos 150°
(4) sin 150°
1. Trigonometry
iv. Functions of the Difference of Two Angles
4
1920. If cos x = 12
13 and sin y = 5, then sin(x ‚ y) equals
(1) 72
65
(2) 56
65
(3) ‚16
65
(4) ‚33
65
1606. The value of sin 60† cos 45† ‚ sin 45† cos 60† is
(1) 1
(2) 0
(3) `6 ‚ `2
4
(4) ¡
1520. If sin A = ©, sin B = ¤, and aA and aB are
acute angles, what is the value of cos (A ‚ B)?
(1)
(3)
(2)
(4)
8
3629. When sin x = –817
and x lies in Quadrant III and
cos y = –45 and y lies in Quadrant II, what is cos (x – y)?
36
85
2702. The expression cos (270º – A) is equivalent to
(1) cos A
(3) sin A
(2) – cos A
(4) – sin A
2471. The expression cos 70º cos 10º + sin 70º sin 10º is
equivalent to
(3) sin 60º
(1) cos 60º
(2) cos 80º
(4) sin 80º
2433. The expression cos2 40 ‚ sin2 40 has the same value as
(1) sin 20
(3) cos 80
(2) sin 80
(4) cos 20
2293. Express sin 75º cos 15º ‚ cos 75º sin 15º as a single
trigonometric function of a positive acute angle.
sin 60º
2216. The expression cos 80† cos 20† ‚ sin 80† sin 20† is
equivalent to
(1) cos 60†
(3) sin 100†
(4) sin 60†
(2) cos 100†
1861. If A = ‚2 + 4i and B = 3 ‚ 2i, in which quadrant does the
graph of (A ‚ B) lie?
II
1559. If A and B are acute angles, sin A = 1/2, and
sin B = `3/2, what is the value of sin (A ‚ B)?
(1) 1
(3) ¡
(2) ‚1
(4) ‚¡
1390. Find the exact value for cos 15º using the formula for cos(x
– y). (Hint: max = 45º, may = 30º)
`6 + `2
4
5
and sin B = ª,
1345. If A and B are both acute angles, sin A = 513
then sin (A ‚ B) is
(1) ‚33
65
(2) 63
65
(3) 33
65
(4) 43
65
1238. Express in radical form:
sin 90º cos 30º ‚ cos 90º sin 30º
`3
2
963. The expression cos y (csc y ‚ sec y) is equivalent to
(3) 1 ‚ tan y
(1) cot y ‚ 1
(2) tan y ‚ 1
(4) ‚cos y
821. The expression sin (90º ‚ ‡) is equivalent to
(3) ‚cos ‡
(1) cos ‡
(2) sin ‡
(4) ‚sin ‡
© 1998-2009 Eduware, Inc.
415
V. RATIOS AND PROPORTIONS
A. Using Proportions
1. Mathematical Ratios
ii. Other Ratios
Base your answers to questions 4046 through 4048 on the picture
below.
Katalina has two cans. One can is a soda pop can and has a
height of 8 inches. The radius of its top is 3 inches. The other
can is a soup can. Its raduis is 34 the radius of the soda can, and it
has half the height.
3981. Shelly bought a shirt for $24.95 with a 20% discount. How
much did Shelly pay for the shirt?
(3) $19.95
(1) $19.96
(2) $4.99
(4) $4.95
3957. Rick fills a square box with six inch sides with his rock
collection. He wants to move his collection into a cylinder
container that has a radius of 3 inches and a height of 5
inches. Will all of his rock collection fit into the cylinder
container? Explain your answer.
No, because the volume of the box is greater then
the volume of the cylinder container.
3943. Juan and Leah are building triangle picture frames out of
popsicle sticks. They have one 4-inch popsicle stick, one 5inch popsicle stick, one 6-inch popsicle stick, and one 7inch popsicle stick. What is the maximum number of
different picture frames that can be made using these
popsicle sticks as sides?
(1) 1
(3) 3
(2) 2
(4) 4
4046. What is the diameter and height of the soup can?
diameter = 4.50 height = 4in
3940. On the way to a rock concert, John drove 50 miles per hour
for 3 hours and then drove 30 miles per hour for 1 hour.
What is John's average speed, in miles per hour, for the
whole trip?
(1) 35
(3) 45
(2) 40
(4) 50
4047. What are the volumes of the two cans? Round your answer
to the nearest whole number.
volume of soda can = 226 square inches
volume of soup can = 64 square inches
3868. If an arc of 60° on circle A has the same length as an arc of
45° on circle B, what is the ratio of the area of circle B to
the area of circle A?
16:9
4048. How many times bigger is the soda can then the soup can?
Round your answer to the nearest tenth.
3.5 times bigger
3829. If Jamar can run © of a mile in 2 minutes 30 seconds, what
is his rate in miles per minute?
(3)
(1)
4045. the information below.
(2)
Every year less and less people use audio tape players. The
equation P(t)=7(5.6).23t represents the decline of audio tape
players in one state since 1990, where t is time in years and
P represents the number of tape players in use in ten
thousands.
In what year, did the number of audio tape players in use
decrease to below 5,000?
1997
3982. Terrianne had a rectangular deck. Its length was three
times its width. She decided to rebuild her deck so that it
was 4 feet longer and 4 feet wider than her first deck. If x
represents the original width of the deck, which expression
represents the difference between the area of her new deck
and the area of the original deck?
(3) 13
(1) 3x2
2
(2) x + 13x + 16
(4) 16x + 16
428
(4)
3821. On a trip, a student drove 40 miles per hour for 2 hours and
then drove 30 miles per hour for 3 hours. What is the
student's average rate of speed, in miles per hour, for the
whole trip?
(3) 36
(1) 34
(2) 35
(4) 37
3782. The circumference of a circular plot of land is increased by
10%. What is the best estimate of the total percentage that
the area of the plot increased?
(1) 10%
(3) 25%
(4) 31%
(2) 21%
© 1998-2009 Eduware, Inc.
VI. COUNTING, PROBABILITY, AND STATISTICS
B. Bernoulli Trials
Base your answers to questions 4056 through 4058 on the
information below.
Kylie is outside watching a meteor shower for 1 hour. The
probability of her seeing a shooting star is 25.
4056. If there are 6 shooting stars within the hour Kylie is
outside, what is the probability she will see at most 3
shooting stars?
12,825
15,625
1. Probability
ii. At Most & At Least
3897. Team A and team B are playing in a league. They will play
each other five times. If the probability that team A wins a
game is ¢, what is the probability that team A will win at
least three of the five games?
51
243
3838. As shown in the accompanying diagram, a circular target
with a radius of 9 inches has a bull's-eye that has a radius of
3 inches. If five arrows randomly hit the target, what is the
probability that at least four hit the bull's-eye?
4058. What is the probability she will see at least 4 shooting stars?
2800
2800
15,625
4008. Base your answer to the following question on the diagram
below.
Antonio was at a carnival playing a game at the dunk
machine. He had to throw the ball and hit the bull's-eye of
the target to dunk the person who was sitting in the
machine. There were two circles, an outer one with a
radius of 12 inches and an inner one with a radius of 3
inches. He gets three chances to try and hit the inner target.
41
59,049
3632. The circle in the accompanying diagram is divided into six
regions of equal area and has a spinner. The regions are
labeled 1, 3, 6, 9, 12, and 15. If the spinner is spun five
times, what is the probability that it will land in an evennumbered region at most two times?
If he hits the area within the outer circle each time what is
the probablility that at most two of his balls would hit the
bulls eye.
4646
4096
4007. Base your answer to the following question on the
information below.
Five kids from Bayview school were trying out for a travel
soccer team. The probability of each of them making the
team is 1 out of 3.
What is the probability that at least four of the kids from
Bayview school make the team?
11
11
243
3951. Two soccer teams, the Fury and the Ladyhawks, have a
strong rivalry. The probability that the Fury will win each
game is 12. If they play each other six times, what is the
probability that the Fury will win at least 4 games?
22
64
192
243
3585. In a baseball game, the probability that Peter gets on base
safely is 37. If he comes to bat four times, what is the
probability that he will get on base safely at least three
times?
513
2401
2750. Five marbles are in a jar. Two are red and three are white.
Four marbles are selected at random with replacement.
(1) Find the probability that at most two red marbles are
selected.
(2) Find the probability that at least three red
marbles are selected.
(1) 513 (2) 112
625
625
© 1998-2009 Eduware, Inc.
439
VI. COUNTING, PROBABILITY, AND STATISTICS
A. Basic Statistics
4775. Car engine life is normally distributed for large samples.
The mean life time is 50,000 miles and the standard
deviation is 5,000 miles. Approximately what percentage
of engines last between 45,000 and 55,000 miles?
(1) 19%
(3) 68%
(2) 34%
(4) 81%
4774. Light bulb life is normally distributed for large samples.
The mean lifetime in 900 hours and the standard deviation
is 50 hours. Approximately what percentage of light bulbs
last less than 800 hours?
(3) 80%
(1) 2%
(2) 6%
(4) 98%
4771. A student has an average of 95 on her math tests and 90 on
her English tests. She took 4 tests in math and 7 tests in
English. Which of the following is true of the standard
deviations for her test scores?
(1) The standard deviation for the English scores is greater
than the standard deviation for the math scores.
(2) The standard deviation for the math scores is greater
than the standard deviation for the English scores.
(3) The standard deviations for both sets of scores are
equal.
(4) More information is needed to determine the
relationship between the standard deviations.
4528. A shoe manufacturer collected data regarding men’s shoe
sizes and found that the distribution of sizes exactly fits the
normal curve. If the mean shoe size is 11 and the standard
deviation is 1.5, find:
a the probability that a man’s shoe size is greater than or
equal to 11
b the probability that a man’s shoe size is greater than or
equal to 12.5
c P(size & 12.5)
P(size & 8)
a 12
b 159
1000
c 159
977
4415. The amount of time it takes for a high school student to get
ready for school in the morning is normally distributed. If
the average time taken is 35 minutes, with a standard
deviation of 4 minutes, what is the probability of a high
school student getting ready for school in between 29 and
41 minutes?
.866 or 86.6%
4366. The scores an exam are normally distributed. If the interval
from 84 to 88 contain approximately 68% of the scores,
which value can be the standard deviation?
(1) 8
(3) 2
(2) 4
(4) 1
456
2. Statistics
iv. Standard Deviation
4365. On a quiz, there is a mean score of 83 and a standard
deviation of 2.7. Which of the following score can be
expected to occur less than 5% of the time?
(3) 83
(1) 77
(2) 80
(4) 87
4364. The amount of money that college students make per hour
at summer jobs is normally distributed. If the mean amount
is $7.75 and an amount of $8.25 represents the 84th
percentile, which value best approximates the standard
deviation for this distribution?
(1) $0.05
(3) $0.25
(2) $0.15
(4) $0.50
4348. A chair company produces a chair that can be adjusted to
accomodate different preferences for the height of the chair.
Research into the preferences has produced the following
data, in inches: 30, 24, 21, 18, 20, 36, 32, 26, 28, 35. What
is the mean and standard deviation of this data?
mean, 27; standard deviation, 3
4341. On the first test of the quarter, the mean score was 80 and
the standard deviation was 3. On the second test of the
quarter, the mean score was 85 and the standard deviation
was 2. On which test was there greater consistency of
scores? Explain how you arrived at your answer.
There was greater consistency of scores on the
second test of the quarter because the standard
deviation is lower than on the first test.
4320. At Lauren's new job, the ages of all of the current
employees are normally distributed. Within this curve,
68.2% of the ages, centered about the mean, are between
28.7 and 45.3 years. Find the standard deviation of the data.
(1) 4.15
(3) 12.45
(4) 16.6
(2) 8.3
4319. At Lauren's new job, the ages of all of the current
employees are normally distributed. Within this curve,
68.2% of the ages, centered about the mean, are between
28.7 and 45.3 years. Find the mean age, in years.
(1) 35
(3) 37
(2) 36
(4) 38
4296. The class mean for a math test was 80. The standard
deviation was 8. What percentage of the class scored
between a 76 and 84?
(1) 68.2%
(3) 26.4%
(2) 52.8%
(4) 38.2%
4278. The amount of time that a teenager plays video games in
any given week is normally distributed. If a teenager plays
video games an average of 15 hours per week, with a
standard deviation of 3 hours, what is the probability of a
teenager playing video games between 15 and 18 hours a
week?
0.341 or 34.1%
© 1998-2009 Eduware, Inc.
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement