# jefferson math project regents by topic

```JEFFERSON MATH PROJECT
REGENTS BY TOPIC
All 612 NY Math B Regents Exam Questions
from June 2001 to January 2007 Sorted by Topic
www.jmap.org
Dear Sir
I have to acknolege the reciept of your favor of May 14. in which you mention that you have finished the
6. first books of Euclid, plane trigonometry, surveying & algebra and ask whether I think a further
pursuit of that branch of science would be useful to you. there are some propositions in the latter books of
Euclid, & some of Archimedes, which are useful, & I have no doubt you have been made acquainted with
them. trigonometry, so far as this, is most valuable to every man, there is scarcely a day in which he will not
resort to it for some of the purposes of common life. the science of calculation also is indispensible as far as
the extraction of the square & cube roots; Algebra as far as the quadratic equation & the use of logarithms
are often of value in ordinary cases: but all beyond these is but a luxury; a delicious luxury indeed; but
not to be indulged in by one who is to have a profession to follow for his subsistence. in this light I view the
conic sections, curves of the higher orders, perhaps even spherical trigonometry, Algebraical operations
beyond the 2d dimension, and fluxions.
Letter from Thomas Jefferson to William G. Munford, Monticello, June 18, 1799.
ALGEBRA
TOPICS
NUMBERS
OPERATIONS
AND PROPERTIES
GRAPHS AND
STATISTICS
KEYWORDS/SUBTOPICS
QUESTION NUMBER
Properties of Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Imaginary Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-10
Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-28
Summations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29-41
Relating Graphs to Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Central Tendency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43-47
Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48-55
Correlation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56-59
Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60-74
Normal Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75-86
PROBABILITY
Normal Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87-89
Binomial Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90-107
EQUATIONS
Transforming Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108-109
Absolute Value Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
RATE
Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111-113
FUNCTIONS
Modeling Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Graphing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115-117
Defining Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118-129
Compositions of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 130-141
Operations with Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142-144
Inverse of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145-154
SYSTEMS
Writing Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155-156
Break Even . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157-158
Solving Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 159-169
INEQUALITIES
Absolute Value Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 170-182
Quadratic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183-188
Trigonometric Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189-191
-i-
POWERS
Solving Quadratics by Factoring . . . . . . . . . . . . . . . . . . . . . . .
Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Minimum and Maximum of Quadratics . . . . . . . . . . . . . . . . . .
Quadratic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Using the Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
192-194
195-201
202-211
212-219
220-236
Zero and Negative Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237-238
Operations with Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239-240
Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242-250
Properties of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251-262
Graphing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . 263-264
Logarithmic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265-273
Exponential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274-291
Binomial Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292-296
Rationalizing Denominators . . . . . . . . . . . . . . . . . . . . . . . . . .
Properties of Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Solving Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exponents as Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
297-306
307-309
310-321
322-332
RATIONALS
Multiplication and Division of Rationals . . . . . . . . . . . . . . . . .
Addition and Subtraction of Rationals . . . . . . . . . . . . . . . . . . .
Solving Rationals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Complex Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inverse Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
333-336
337-339
340-345
346-350
351-358
359-370
371-383
GEOMETRY
ANGLES
TRIANGLES
Unit Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Radian Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Double Angle and Angle Sum and Difference Identities . . . . .
Solving Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . .
Trigonometric Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
384-401
402-410
411-418
419-430
431-442
443-462
Pythagoras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
Perimeter and Area of Triangles . . . . . . . . . . . . . . . . . . . . . . . 464-466
Triangle Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
Basic Trigonometric Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
Using Trigonometry to Find Area . . . . . . . . . . . . . . . . . . . . . . 469-475
Law of Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476-484
Law of Sines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485-498
Using Trigonometry to Solve Triangle Inequalities . . . . . . . . . 499-507
Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508-511
-ii-
OTHER POLYGONS
CONICS
Perimeter and Area of Other Polygons . . . . . . . . . . . . . . . . . . 512-515
Circumference and Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Equations of Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Equations of Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chords Secants and Tangents . . . . . . . . . . . . . . . . . . . . . . . . .
516-523
524-530
531-538
539-558
SOLIDS AND
SIMILARITY
Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
TRANSFORMATIONS
Identifying Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562-563
Dilations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564-567
Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568-571
Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572-577
Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578
Compositions of Transformations . . . . . . . . . . . . . . . . . . . . . . 579-588
LOGIC
Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589-612
-iii-
Math B Regents Exam Questions by Topic
Page 1
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NUMBERS OPERATIONS
AND PROPERTIES
8. The expression i 0 ⋅ i 1 ⋅ i 2 ⋅ i 3 ⋅ i 4 is equal to
PROPERTIES OF INTEGERS
9. The expression
1. 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
IMAGINARY NUMBERS
[B] i
[B] 1
[A] 1
[C] -i
[D] -1
i 16
is equivalent to
i3
[B] -i
[C] i
[D] -1
10. What is the multiplicative inverse of 3i?
[A] −
i
3
[B]
1
3
[C] − 3i
[D] − 3
COMPLEX NUMBERS
2. The expression i 25 is equivalent to
[A] − i
[A] i
[D] − 1
[C] 1
11. Fractal geometry uses the complex number
plane to draw diagrams, such as the one
shown in the accompanying graph.
3. Mrs. Donahue made up a game to help her
class learn about imaginary numbers. The
winner will be the student whose expression
is equivalent to -i. Which expression will win
the game?
[A] i 48
[B] i 49
[C] i 47
[D] i 46
4. Expressed in simplest form, i 16 + i 6 − 2i 5 + i 13
[A] − i
[B] 1
[C] − 1
[D] i
5. When simplified, i 27 + i 34 is equal to
[A] i-1
[B] i
[D] i 61
[C] -i-1
Which number is not included in the shaded
area?
6. What is the value of i − i ?
99
[A] i 96
[B] 1
7. What is the sum of
3
[C] − i
− 2 and
[D] 0
− 18 ?
[A] 5i 2
[B] 6i
[C] 4i 2
[D] 2i 5
[A] -0.9
[B] -0.5i
[C] -0.5 - 0.5i
[D] -0.9 - 0.9i
Math B Regents Exam Questions by Topic
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12. Two complex numbers are graphed below.
14. On the accompanying set of axes, graphically
represent the sum of 3 + 4i and − 1 + 2i.
What is the sum of w and u, expressed in
standard complex number form?
[A] 7 + 3i
[B] 5 + 7i
[C] 3 + 7i
[D] -5 + 3i
13. Find the sum of -2 + 3i and -1 - 2i.
Graph the resultant on the accompanying set
of axes.
15. Melissa and Joe are playing a game with
complex numbers. If Melissa has a score of
5 − 4i and Joe has a score of 3 + 2i, what is
their total score?
[A] 8 - 6i
[B] 8 + 6i
[C] 8 - 2i
[D] 8 + 2i
16. Express − 48 + 35
. + 25 + − 27 in
simplest a + bi form.
17. What is the sum of 2 − − 4 and − 3 + − 16
expressed in simplest a + bi form?
[A] − 1 + i 20
[B] − 1 + 12i
[C] − 1 + 2i
[D] − 14 + i
18. When expressed as a monomial in terms of i,
2 − 32 − 5 − 8 is equivalent to
[A] 18i 2
[B] 2i 2
[C] 2 2i
[D] − 2i 2
Math B Regents Exam Questions by Topic
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19. What is the product of 5 + − 36 and
1 − − 49 , expressed in simplest a + bi form?
[A] -37 + 41i
[B] 47 - 29i
[C] 5 - 71i
[D] 47 + 41i
20. Show that the product of a + bi and its
conjugate is a real number.
21. 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.
22. 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?
[A] 4 + i
[B] 8 + i
[C] 8 + 7i
27. The expression
[A]
7+i
10
[B]
7 − 5i
10
[C]
6+i
8
[D]
6 + 5i
8
28. Impedance measures the opposition of an
electrical circuit to the flow of electricity.
The total impedance in a particular circuit is
ZZ
given by the formula ZT = 1 2 . What is
Z1 + Z2
the total impedance of a circuit, ZT , if
Z1 = 1 + 2i and Z2 = 1 − 2i ?
[A] −
3
2
[B] 0
[C]
5
2
[D] 1
SUMMATIONS
5
29. Evaluate:
[D] 4 - i
23. The complex number c + di is equal to
(2 + i ) 2 . What is the value of c?
2+i
is equivalent to
3+i
2 ∑ (2n − 1)
n =1
5
30. What is the value of ∑ ( −2n + 100) ?
n =1
[A] 530
[B] 130
[C] 470
[D] 70
24. The expression ( −1 + i ) 3 is equivalent to
5
[A] -3i
[B] 2 + 2i
[C] -1 - i
[D] -2 - 2i
31. What is the value of ∑ (m2 − 1) ?
m= 2
[A] 58
25. If f ( x ) = x 3 − 2 x 2 , then f (i ) is equivalent to
[A] 2 + i
[B] -2 + i
[C] 2 - i
[D] -2 - i
26. What is the value of x in the equation
5 − 2 x = 3i ?
[A] 1
[B] 7
[C] 4
[D] -2
[B] 50
[C] 53
[D] 54
5
32. Evaluate:
2
∑ ( n + n)
n =1
3
33. What is the value of ∑ (2m + 1) m−1 ?
m =1
[A] 245
[B] 57
[C] 55
[D] 15
Math B Regents Exam Questions by Topic
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34. The projected total annual profits, in dollars,
for the Nutyme Clothing Company from 2002
to 2004 can be approximated by the model
2
∑ (13,567n + 294), where n is the year and
n=0
n = 0 represents 2002. Use this model to
find the company's projected total annual
profits, in dollars, for the period 2002 to
2004.
35. A ball is dropped from a height of 8 feet and
allowed to bounce. Each time the ball
bounces, it bounces back to half its previous
height. The vertical distance the ball travels,
k
n 1
d, is given by the formula d = 8 + 16 ∑ ( ) ,
k =1 2
where n is the number of bounces. Based on
this formula, what is the total vertical distance
that the ball has traveled after four bounces?
[A] 15.0 ft
[B] 23.0 ft
[C] 22.0 ft
[D] 8.9 ft
( −1) k −1
k =1 ( 2 k − 1)!
2
36. Evaluate: ∑
37. If n Cr represents the number of combinations
of n items taken r at a time, what is the
41. Jonathan's teacher required him to express the
2 3 4 5 6
sum + + + + using sigma notation.
3 4 5 6 7
Which of these four answers is not correct?
k +1
k =1 k + 2
[B] ∑
k
k =2 k + 1
[D] ∑
5
[A] ∑
6
[C] ∑
k −1
k =3 k
7
k
k =1 k + 1
5
GRAPHS & STATISTICS
RELATING GRAPHS TO EVENTS
42. A bug travels up a tree, from the ground, over
a 30-second interval. It travels fast at first
and then slows down. It stops for 10 seconds,
then proceeds slowly, speeding up as it goes.
Which sketch best illustrates the bug's
distance (d) from the ground over the 30second interval (t)?
[A]
[B]
[C]
[D]
3
value of ∑ 4 Cr ?
r=1
[A] 4
[B] 24
[C] 6
[D] 14
CENTRAL TENDENCY
43. What is the mean of the data in the
accompanying table?
4
38. The value of ∑ 5 Cr is
r =2
[A] 45
[B] 25
[C] 10
[D] 5
3
39. Evaluate: ∑ (3 cos kπ + 1)
k =0
3
40. What is the value of ∑ (2 − (b)i ) ?
b=0
[A] 2-6i
[B] 8-6i
[C] 2-5i
[D] 8-5i
[A] 16
[B] 15
[C] 14.5
[D] 11
Math B Regents Exam Questions by Topic
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44. Two social studies classes took the same
current events examination that was scored on
the basis of 100 points. Mr. Wong's class had
a median score of 78 and a range of 4 points,
while Ms. Rizzo's class had a median score of
78 and a range of 22 points. Explain how
these classes could have the same median
score while having very different ranges.
46. Data collected during an experiment are
shown in the accompanying graph.
45. The accompanying graph shows the heart
rate, in beats per minute, of a jogger during a
4-minute interval.
What is the range of this set of data?
[A] 1 ≤ x ≤ 10
[B] 0 ≤ y ≤ 100
[C] 2.5 ≤ y ≤ 9.5
[D] 2.5 ≤ x ≤ 9.5
47. The effect of pH on the action of a certain
enzyme is shown on the accompanying graph.
What is the range of the jogger's heart rate
during this interval?
[A] 60-110
[B] 0-4
[C] 1-4
[D] 0-110
What is the domain of this function?
[A] x ≥ 0
[B] y ≥ 0
[C] 4 ≤ x ≤ 13
[D] 4 ≤ y ≤ 13
Math B Regents Exam Questions by Topic
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STANDARD DEVIATION
48. Jean's scores on five mathematics tests were
98, 97, 99, 98, and 96. Her scores on five
English tests were 78, 84, 95, 72, and 79.
Which statement is true about the standard
deviations for the scores?
51. The number of children of each of the first 41
United States presidents is given in the
accompanying table. For this population,
determine the mean and the standard
deviation to the nearest tenth.
How many of these presidents fall within one
standard deviation of the mean?
the relationship between the standard
deviations.
[B] The standard deviation for the math
scores is greater than the standard
deviation for the English scores.
[C] The standard deviations for both sets of
scores are equal.
[D] The standard deviation for the English
scores is greater than the standard
deviation for the math scores.
49. On a nationwide examination, the Adams
School had a mean score of 875 and a
standard deviation of 12. The Boswell School
had a mean score of 855 and a standard
deviation of 20. In which school was there
greater consistency in the scores? Explain
50. The term “snowstorms of note” applies to all
snowfalls over 6 inches. The snowfall
amounts for snowstorms of note in Utica,
New York, over a four-year period are as
follows: 7.1, 9.2, 8.0, 6.1, 14.4, 8.5, 6.1, 6.8,
7.7, 21.5, 6.7, 9.0, 8.4, 7.0, 11.5, 14.1, 9.5, 8.6
What are the mean and population standard
deviation for these data, to the nearest
hundredth?
[A] mean = 9.46; standard deviation = 3.85
[B] mean = 9.45; standard deviation = 3.74
[C] mean = 9.46; standard deviation = 3.74
[D] mean = 9.45; standard deviation = 3.85
52. Beth's scores on the six Earth science tests
she took this semester are 100, 95, 55, 85, 75,
and 100. For this population, how many
scores are within one standard deviation of
the mean?
53. From 1984 to 1995, the winning scores for a
golf tournament were 276, 279, 279, 277,
278, 278, 280, 282, 285, 272, 279, and 278.
Using the standard deviation for the sample,
S x , find the percent of these winning scores
that fall within one standard deviation of the
mean.
Math B Regents Exam Questions by Topic
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54. An electronics company produces a
into the distance between the top of people's
heads and the top of their ears produced the
following data, in inches:
4.5, 4.8, 6.2, 5.5, 5.6, 5.4, 5.8, 6.0, 5.8, 6.2,
4.6, 5.0, 5.4, 5.8
The company decides to design their
deviations from the mean. Find, to the nearest
tenth, the mean, the standard deviation, and
the range of distances that must be
accommodated.
55. On a standardized test, a score of 86 falls
exactly 1.5 standard deviations below the
mean. If the standard deviation for the test is
2, what is the mean score for this test?
[A] 87.5
[B] 89
[C] 84.5
58. Which scatter diagram shows the strongest
positive correlation?
[A]
[B]
[C]
[D]
59. Which graph represents data used in a linear
regression that produces a correlation
coefficient closest to − 1?
[A]
[B]
[C]
[D]
[D] 84
CORRELATION COEFFICIENT
56. A linear regression equation of best fit
between a student's attendance and the degree
of success in school is h = 0.5x + 68.5. The
correlation coefficient, r, for these data would
be
[A] 0 < r < 1
[B] -1 < r < 0
[C] r = 0
[D] r = -1
REGRESSION
60. The 1999 win-loss statistics for the American
League East baseball teams on a particular
date is shown in the accompanying chart.
57. The relationship of a woman's shoe size and
length of a woman's foot, in inches, is given
in the accompanying table.
The linear correlation coefficient for this
relationship is
[A] 0
[B] -1
[C] 1
[D] 0.5
Find the mean for the number of wins, W,
and the mean for the number of losses, L,
and determine if the point (W, L ) is a point
Math B Regents Exam Questions by Topic
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61. A real estate agent plans to compare the price
of a cottage, y, in a town on the seashore to
the number of blocks, x, the cottage is from
the beach. The accompanying table shows a
random sample of sales and location data.
Write a linear regression equation that relates
the price of a cottage to its distance from the
beach.
Use the equation to predict the price of a
cottage, to the nearest dollar, located three
blocks from the beach.
62. The availability of leaded gasoline in New
York State is decreasing, as shown in the
accompanying table.
Determine a linear relationship for x (years)
versus y (gallons available), based on the data
given. The data should be entered using the
year and gallons available (in thousands),
such as (1984,150).
If this relationship continues, determine the
number of gallons of leaded gasoline
available in New York State in the year 2005.
If this relationship continues, during what
year will leaded gasoline first become
unavailable in New York State?
63. The accompanying table illustrates the
number of movie theaters showing a popular
film and the film's weekly gross earnings, in
millions of dollars.
Write the linear regression equation for this
set of data, rounding values to five decimal
places.
Using this linear regression equation, find the
approximate gross earnings, in millions of
dollars, generated by 610 theaters. Round
Find the minimum number of theaters that
would generate at least 7.65 million dollars in
gross earnings in one week.
Math B Regents Exam Questions by Topic
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64. In a mathematics class of ten students, the
teacher wanted to determine how a homework
grade influenced a student's performance on
the subsequent test. The homework grade and
subsequent test grade for each student are
given in the accompanying table.
65. The table below shows the results of an
experiment that relates the height at which a
ball is dropped, x, to the height of its first
bounce, y.
Find x , the mean of the drop heights.
Find y, the mean of the bounce heights.
Find the linear regression equation that best
fits the data.
Show that ( x , y ) is a point on the line of
regression. [The use of the grid is optional.]
a Give the equation of the linear regression
line for this set of data.
b A new student comes to the class and earns
a homework grade of 78. Based on the
equation in part a, what grade would the
teacher predict the student would receive on
the subsequent test, to the nearest integer?
Math B Regents Exam Questions by Topic
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66. Two different tests were designed to measure
understanding of a topic. The two tests were
given to ten students with the following
results:
67. Since 1990, fireworks usage nationwide has
grown, as shown in the accompanying table,
where t represents the number of years since
1990, and p represents the fireworks usage
per year, in millions of pounds.
Construct a scatter plot for these scores, and
then write an equation for the line of best fit
(round slope and intercept to the nearest
hundredth).
Find the equation of the linear regression
model for this set of data, where t is the
independent variable. Round values to four
decimal places.
Using this equation, determine in what year
fireworks usage would have reached 99
million pounds.
Based on this linear model, how many
millions of pounds of fireworks would be
the nearest tenth.
Find the correlation coefficient.
Predict the score, to the nearest integer, on
test y for a student who scored 87 on test x.
Math B Regents Exam Questions by Topic
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68. A factory is producing and stockpiling metal
sheets to be shipped to an automobile
manufacturing plant. The factory ships only
when there is a minimum of 2,050 sheets in
stock. The accompanying table shows the
day, x, and the number of sheets in stock, f(x).
Write the linear regression equation for this
set of data, rounding the coefficients to four
decimal places.
Use this equation to determine the day the
sheets will be shipped.
69. A box containing 1,000 coins is shaken, and
the coins are emptied onto a table. Only the
coins that land heads up are returned to the
box, and then the process is repeated. The
accompanying table shows the number of
trials and the number of coins returned to the
box after each trial.
Write an exponential regression equation,
rounding the calculated values to the nearest
ten-thousandth.
Use the equation to predict how many coins
would be returned to the box after the eighth
trial.
70. The table below, created in 1996, shows a
history of transit fares from 1955 to 1995. On
the accompanying grid, construct a scatter
plot where the independent variable is years.
State the exponential regression equation with
the coefficient and base rounded to the
nearest thousandth. Using this equation,
determine the prediction that should have
been made for the year 1998, to the nearest
cent.
Math B Regents Exam Questions by Topic
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71. The breaking strength, y, in tons, of steel
cable with diameter d, in inches, is given in
the table below.
On the accompanying grid, make a scatter
plot of these data. Write the exponential
regression equation, expressing the regression
coefficients to the nearest tenth.
72. The accompanying table shows the average
salary of baseball players since 1984. Using
the data in the table, create a scatter plot on
the grid and state the exponential regression
equation with the coefficient and base
rounded to the nearest hundredth.
estimate the salary of a baseball player in the
year 2005, to the nearest thousand dollars.
Math B Regents Exam Questions by Topic
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73. Jean invested \$380 in stocks. Over the next
5 years, the value of her investment grew, as
shown in the accompanying table.
Write the exponential regression equation for
this set of data, rounding all values to two
decimal places.
Using this equation, find the value of her
stock, to the nearest dollar, 10 years after her
initial purchase.
74. The accompanying table shows the number of
new cases reported by the Nassau and Suffolk
County Police Crime Stoppers program for
the years 2000 through 2002.
NORMAL DISTRIBUTIONS
75. Twenty high school students took an
scores:
70, 60, 75, 68, 85, 86, 78, 72, 82, 88, 88, 73,
74, 79, 86, 82, 90, 92, 93, 73
Determine what percent of the students scored
within one standard deviation of the mean.
Do the results of the examination approximate
76. Mrs. Ramírez is a real estate broker. Last
month, the sale prices of homes in her area
approximated a normal distribution with a
mean of \$150,000 and a standard deviation of
\$25,000.
A house had a sale price of \$175,000. What
is the percentile rank of its sale price, to the
nearest whole number? Explain what that
percentile means.
Mrs. Ramírez told a customer that most of the
houses sold last month had selling prices
between \$125,000 and \$175,000. Explain
why she is correct.
77. On a standardized test, the distribution of
scores is normal, the mean of the scores is 75,
and the standard deviation is 5.8. If a student
scored 83, the student's score ranks
[A] below the 75th percentile
[B] above the 97th percentile
If x = 1 represents the year 2000, and y
represents the number of new cases, find the
equation of best fit using a power regression,
rounding all values to the nearest thousandth.
Using this equation, find the estimated
number of new cases, to the nearest whole
number, for the year 2007.
[C] between the 75th percentile and the 84th
percentile
[D] between the 84th percentile and the 97th
percentile
Math B Regents Exam Questions by Topic
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78. In a New York City high school, a survey
revealed the mean amount of cola consumed
each week was 12 bottles and the standard
deviation was 2.8 bottles. Assuming the
survey represents a normal distribution, how
many bottles of cola per week will
approximately 68.2% of the students drink?
[A] 6.4 to 12
[B] 12 to 20.4
[C] 9.2 to 14.8
[D] 6.4 to 17.6
79. The amount of juice dispensed from a
machine is normally distributed with a mean
of 10.50 ounces and a standard deviation of
0.75 ounce. Which interval represents the
amount of juice dispensed about 68.2% of the
time?
[A] 9.00-12.00
[B] 9.75-11.25
[C] 10.50-11.25
[D] 9.75-10.50
80. The mean of a normally distributed set of data
is 56, and the standard deviation is 5. In
which interval do approximately 95.4% of all
cases lie?
[A] 51-61
[B] 46-66
[C] 56-71
[D] 46-56
[B] 38.2%
[C] 68.2%
[D] 52.8%
[B] 84%
[C] 16%
[B] 100
[C] 80
[D] 5
84. Professor Bartrich has 184 students in her
mathematics class. The scores on the final
examination are normally distributed and
have a mean of 72.3 and a standard deviation
of 8.9. How many students in the class can be
expected to receive a score between 82 and
90?
85. In a certain school district, the ages of all new
teachers hired during the last 5 years are
normally distributed. Within this curve,
95.4% of the ages, centered about the mean,
are between 24.6 and 37.4 years. Find the
mean age and the standard deviation of the
data.
[A] 32
[B] 25
[C] 60
[D] 67
PROBABILITY
NORMAL PROBABILITY
82. Battery lifetime is normally distributed for
large samples. The mean lifetime is 500 days
and the standard deviation is 61 days.
Approximately what percent of batteries have
[A] 34%
[A] 16
86. The mean score on a normally distributed
exam is 42 with a standard deviation of 12.1.
Which score would be expected to occur less
than 5% of the time?
81. The national mean for verbal scores on an
exam was 428 and the standard deviation was
113. Approximately what percent of those
taking this test had verbal scores between 315
and 541?
[A] 26.4%
83. The amount of ketchup dispensed from a
machine at Hamburger Palace is normally
distributed with a mean of 0.9 ounce and a
standard deviation of 0.1 ounce. If the
machine is used 500 times, approximately
how many times will it be expected to
dispense 1 or more ounces of ketchup?
[D] 68%
87. A set of normally distributed student test
scores has a mean of 80 and a standard
deviation of 4. Determine the probability that
a randomly selected score will be between 74
and 82.
Math B Regents Exam Questions by Topic
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88. 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?
89. 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
P( size ≥ 12.5)
c
P( size ≥ 8)
BINOMIAL PROBABILITY
90. The probability that Kyla will score above a
4
90 on a mathematics test is . What is the
5
probability that she will score above a 90 on
three of the four tests this quarter?
[A]
[B]
[C]
[D]
91. The Hiking Club plans to go camping in a
State park where the probability of rain on
any given day is 0.7. Which expression can
be used to find the probability that it will rain
on exactly three of the seven days they are
there?
[A] 4 C3 (0.7) 3 (0.7) 4
[B] 7 C3 (0.3) 3 (0.7) 4
[C] 4 C3 (0.4) 4 (0.3) 3
[D] 7 C3 (0.7) 3 (0.3) 4
92. Which fraction represents the probability of
obtaining exactly eight heads in ten tosses of a
fair coin?
[A]
45
1,024
[B]
90
1,024
[C]
180
1,024
[D]
64
1,024
93. At a certain intersection, the light for
eastbound traffic is red for 15 seconds, yellow
for 5 seconds, and green for 30 seconds.
Find, to the nearest tenth, the probability that
out of the next eight eastbound cars that arrive
randomly at the light, exactly three will be
stopped by a red light.
94. After studying a couple's family history, a
doctor determines that the probability of any
child born to this couple having a gene for
disease X is 1 out of 4. If the couple has three
children, what is the probability that exactly
two of the children have the gene for disease
X?
95. If the probability that it will rain on any given
day this week is 60%, find the probability it
will rain exactly 3 out of 7 days this week.
96. The Coolidge family's favorite television
channels are 3, 6, 7, 10, 11, and 13. If the
Coolidge family selects a favorite channel at
random to view each night, what is the
probability that they choose exactly three
even-numbered channels in five nights?
decimal rounded to four decimal places.
Math B Regents Exam Questions by Topic
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97. During a recent survey, students at Franconia
College were asked if they drink coffee in the
morning. The results showed that two-thirds
of the students drink coffee in the morning
and the remainder do not. What is the
probability that of six students selected at
random, exactly two of them drink coffee in
fraction or as a decimal rounded to four
decimal places.
98. Ginger and Mary Anne are planning a
vacation trip to the island of Capri, where the
probability of rain on any day is 0.3. What is
the probability that during their five days on
the island, they have no rain on exactly three
of the five days?
99. 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?
100. Team A and team B are playing in a league.
They will play each other five times. If the
1
probability that team A wins a game is ,
3
what is the probability that team A will win at
least three of the five games?
101. On any given day, the probability that the
entire Watson family eats dinner together is
2
. Find the probability that, during any 75
day period, the Watsons eat dinner together at
least six times.
102. Tim Parker, a star baseball player, hits one
home run for every ten times he is at bat. If
Parker goes to bat five times during tonight's
game, what is the probability that he will hit
at least four home runs?
103. The probability that a planted watermelon
3
seed will sprout is . If Peyton plants seven
4
seeds from a slice of watermelon, find, to the
nearest ten thousandth, the probability that at
least five will sprout.
104. On mornings when school is in session in
January, Sara notices that her school bus is
late one-third of the time. What is the
probability that during a 5-day school week in
January her bus will be late at least three
times?
105. A board game has a spinner on a circle that
has five equal sectors, numbered 1, 2, 3, 4,
and 5, respectively. If a player has four spins,
find the probability that the player spins an
even number no more than two times on those
four spins.
106. Dr. Glendon, the school physician in charge
of giving sports physicals, has compiled his
information and has determined that the
probability a student will be on a team is 0.39.
Yesterday, Dr. Glendon examined five
students chosen at random.
Find, to the nearest hundredth, the probability
that at least four of the five students will be
on a team.
Find, to the nearest hundredth, the probability
that exactly one of the five students will not
be on a team.
Math B Regents Exam Questions by Topic
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107. When Joe bowls, he can get a strike (knock
down all the pins) 60% of the time. How
many times more likely is it for Joe to bowl at
least three strikes out of four tries as it is for
him to bowl zero strikes out of four tries?
number.
112. 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?
[A] 36
[B] 34
[C] 37
3
of a mile in 2 minutes 30
5
seconds, what is his rate in miles per minute?
113. If Jamar can run
EQUATIONS
TRANSFORMING FORMULAS
108. If x − a = b, x > a , which expression is
equivalent to x?
[A] 4
1
6
[B]
4
5
[C]
[A] b 2 − a
[B] b − a
FUNCTIONS
[C] b + a
[D] b 2 + a
MODELING RELATIONSHIPS
109. The volume of any spherical balloon can be
4
found by using the formula V = π r 3 .
3
Write an equation for r in terms of V and π .
ABSOLUTE VALUE EQUATIONS
110. What is the solution set of the equation
x 2 − 2 x = 3x − 6 ?
[A] {±3}
[B] {2,±3}
[C] {2,3}
[D] {2}
RATE
SPEED
111. On her first trip, Sari biked 24 miles in T
hours. The following week Sari biked 32
miles in T hours. Determine the ratio of her
average speed on her second trip to her
average speed on her first trip.
[A]
3
4
[B]
[D] 35
3
2
[C]
2
3
[D]
4
3
6
25
[D] 3
1
10
114. A store advertises that during its Labor Day
sale \$15 will be deducted from every
purchase over \$100. In addition, after the
deduction is taken, the store offers an earlybird discount of 20% to any person who
makes a purchase before 10 a.m. If Hakeem
makes a purchase of x dollars, x>100, at 8
a.m., what, in terms of x, is the cost of
Hakeem's purchase?
[A] 0.80x - 12
[B] 0.20x - 3
[C] 0.20x - 15
[D] 0.85x - 20
Math B Regents Exam Questions by Topic
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GRAPHING FUNCTIONS
DEFINING FUNCTIONS
115. Which equation is represented by the
accompanying graph?
[A] y = ( x − 3) 2 + 1
[B] y = x + 3 − 1
[C] y = x − 3 + 1
[D] y = x − 3
116. The graph below represents f ( x ) .
118. Which graph is not a function?
[A]
[B]
[C]
[D]
119. Which graph does not represent a function of
x?
[A]
[B]
[C]
[D]
Which graph best represents f ( x ) ?
[A]
[C]
[B]
120. Each graph below represents a possible
relationship between temperature and
pressure. Which graph does not represent a
function?
[A]
[B]
[C]
[D]
[D]
117. Given the function y = f ( x ), such that the
entire graph of the function lies above the xaxis. Explain why the equation f ( x ) = 0 has
no real solutions.
Math B Regents Exam Questions by Topic
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121. Which set of ordered pairs is not a function?
127. Which relation is a function?
[A] {(4,1), (5,1), (6,1), (7,1)}
[A] x = 7
[B] x 2 + y 2 = 7
[B] {(1,2), (3,4), (4,5), (5,6)}
[C] xy = 7
[D] x 2 − y 2 = 7
[C] {(3,1), (2,1), (1,2), (3,2)}
[D] {(0,0), (1,1), (2,2), (3,3)}
122. Which relation is not a function?
[A] y = 2 x + 4
[B] y = x 2 − 4 x + 3
[C] x = 3 y − 2
[D] x = y 2 + 2 x − 3
128. Which diagram represents a relation in which
each member of the domain corresponds to
only one member of its range?
[A]
123. Which equation does not represent a
function?
[A] y = 4
[B] y = x 2 + 5x
[C] x = π
[D] y = x
124. On the accompanying diagram, draw a
mapping of a relation from set A to set B that
is not a function. Explain why the
relationship you drew is not a function.
125. Which relation is a function?
[A] y = sin x
[B] x = y 2 + 1
[C] x = 4
[D] x 2 + y 2 = 16
126. Which equation represents a function?
[A] y = x 2 − 3x − 4
[B] x = y 2 − 6 x + 8
[C] x 2 + y 2 = 4
[D] 4 y 2 = 36 − 9 x 2
[C]
[D]
[B]
Math B Regents Exam Questions by Topic
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129. Which diagram represents a one-to-one
function?
[A]
133. If f ( x ) = 5x 2 and g( x ) = 2 x , what is the
value of (f D g)(8)?
[A] 16
[B] 1,280
[C] 80
2
3
−
[D] 8 10
1
2
134. If f ( x ) = x and g( x ) = 8 x , find ( f D g)( x )
and (f D g)(27).
[B]
135. If f and g are two functions defined by
f ( x ) = 3x + 5 and g( x ) = x 2 + 1, then g(f( x ))
is
[A] x 2 + 3x + 6
[B] 3x 2 + 8
[C] 9 x 2 + 26
[D] 9 x 2 + 30 x + 26
[C]
2
1
and g( x ) = , then (g D f)( x )
x+3
x
is equal to
136. If f ( x ) =
[A]
1 + 3x
2x
[B]
x+3
2x
[C]
x+3
2
[D]
2x
1 + 3x
[D]
137. If f ( x ) = x + 1 and g( x ) = x 2 − 1, the
expression (g D f)( x ) equals 0 when x is
equal to
COMPOSITIONS OF FUNCTIONS
130. If f ( x ) = −2 x + 7 and g( x ) = x 2 − 2, then
f(g(3)) is equal to
[A] -1
[B] -3
[C] -7
[D] 7
131. If f ( x ) = 5x 2 − 1 and g( x ) = 3x − 1, find
g(f(1)).
132. If f ( x ) = 2 x − 1 and g( x ) = x 2 − 1, determine
the value of (f D g)(3).
[A] 0, only
[B] 1 and -1
[C] 0 and -2
[D] -2, only
138. If f ( x ) = 2 x 2 + 4 and g( x ) = x − 3, which
number satisfies f ( x ) = ( f D g)( x ) ?
[A]
3
4
[B]
3
2
[C] 5
[D] 4
Math B Regents Exam Questions by Topic
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139. The accompanying graph is a sketch of the
function y = f ( x ) over the interval 0 ≤ x ≤ 7.
What is the value of ( f D f )(6) ?
[A] 1
[B] -2
[C] 0
[D] 2
140. A certain drug raises a patient's heart rate,
h( x ), in beats per minute, according to the
function h( x ) = 70 + 0.2 x , where x is the
bloodstream drug level, in milligrams. The
level of the drug in the patient's bloodstream
is a function of time, t, in hours, according to
the formula g(t ) = 300(0.8) t . Find the value
of h(g(4)), the patient's heart rate in beats per
minute, to the nearest whole number.
141. The temperature generated by an electrical
circuit is represented by t = f (m) = 0.3m2 ,
where m is the number of moving parts. The
resistance of the same circuit is represented
by r = g(t ) = 150 + 5t , where t is the
temperature. What is the resistance in a
circuit that has four moving parts?
[A] 8,670
[B] 174
[C] 156
143. The cost (C) of selling x calculators in a store
is modeled by the equation
3,200,000
C=
+ 60,000. The store profit (P)
x
for these sales is modeled by the equation
P = 500 x . What is the minimum number of
calculators that have to be sold for profit to be
greater than cost?
144. A company calculates its profit by finding the
difference between revenue and cost. The
cost function of producing x hammers is
C ( x ) = 4 x + 170. If each hammer is sold for
\$10, the revenue function for selling x
hammers is R( x ) = 10 x.
How many hammers must be sold to make a
profit?
How many hammers must be sold to make a
profit of \$100?
INVERSE OF FUNCTIONS
145. If a function is defined by the equation
y = 3x + 2 , which equation defines the
inverse of this function?
[A] y = −3x − 2
[B] y =
1
1
x+
3
2
1
2
x−
3
3
[D] x =
1
1
y+
3
2
[C] y =
[D] 51
OPERATIONS WITH FUNCTIONS
142. The revenue, R(x), from selling x units of a
product is represented by the equation
R( x ) = 35x , while the total cost, C(x), of
making x units of the product is represented
by the equation C ( x ) = 20 x + 500. The total
profit, P(x), is represented by the equation
P( x ) = R( x ) − C ( x ). For the values of R(x)
and C(x) given above, what is P(x)?
[A] 10x + 100
[B] 15x
[C] 15x - 500
[D] 15x + 500
146. A function is defined by the equation
y = 5x − 5. Which equation defines the
inverse of this function?
[A] y =
1
5x − 5
[C] y = 5x + 5
[B] x = 5 y − 5
[D] x =
1
5y − 5
Math B Regents Exam Questions by Topic
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147. A function is defined by the equation
1
3
y = x − . Which equation defines the
2
2
inverse of this function?
[A] y = 2 x −
3
2
[C] y = 2 x − 3
[B] y = 2 x +
150. Which graph represents the inverse of f(x) =
{(0,1),(1,4),(2,3)}?
[A]
3
2
[D] y = 2 x + 3
148. Given: f ( x ) = x 2 and g ( x ) = 2 x
a The inverse of g is a function, but the
inverse of f is not a function. Explain why
this statement is true.
b Find g −1 ( f (3)) to the nearest tenth.
149. If the point (a, b) lies on the graph y = f ( x ),
[B]
[C]
the graph of y = f −1 ( x ) must contain point
[A] (b,a)
[B] (0,b)
[C] (-a,-b)
[D] (a,0)
[D]
Math B Regents Exam Questions by Topic
Page 23
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151. Draw f ( x ) = 2 x 2 and f −1 ( x ) in the interval
0 ≤ x ≤ 2 on the accompanying set of axes.
State the coordinates of the points of
intersection.
154. The inverse of a function is a logarithmic
function in the form y = log b x. Which
equation represents the original function?
[A] y = bx
[B] x = b y
[C] by = x
[D] y = b x
SYSTEMS
WRITING LINEAR SYSTEMS
155. At the local video rental store, José rents two
movies and three games for a total of \$15.50.
At the same time, Meg rents three movies and
one game for a total of \$12.05. How much
money is needed to rent a combination of one
game and one movie?
152. The accompanying diagram shows the graph
1
of the line whose equation is y = − x + 2.
3
On the same set of axes, sketch the graph of
the inverse of this function.
State the coordinates of a point on the inverse
function.
153. What is the inverse of the function
y = log 4 x ?
[A] x 4 = y
[B] y 4 = x
[C] 4 y = x
[D] 4 x = y
156. The cost of a long-distance telephone call is
determined by a flat fee for the first 5 minutes
and a fixed amount for each additional
minute. If a 15-minute telephone call costs
\$3.25 and a 23-minute call costs \$5.17, find
the cost of a 30-minute call.
BREAK EVEN
157. A cellular telephone company has two plans.
Plan A charges \$11 a month and \$0.21 per
minute. Plan B charges \$20 a month and
\$0.10 per minute. After how much time, to
the nearest minute, will the cost of plan A be
equal to the cost of plan B?
[A] 81 hr 48 min
[B] 1 hr 36 min
[C] 81 hr 8 min
[D] 1 hr 22 m
Math B Regents Exam Questions by Topic
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158. Island Rent-a-Car charges a car rental fee of
\$40 plus \$5 per hour or fraction of an hour.
Wayne's Wheels charges a car rental fee of
\$25 plus \$7.50 per hour or fraction of an
hour. Under what conditions does it cost less
to rent from Island Rent-a-Car? [The use of
the accompanying grid is optional.]
SOLVING NONLINEAR SYSTEMS
159. Solve the following system of equations
algebraically:
9x2 + y2 = 9
3x − y = 3
160. What is the total number of points of
intersection for the graphs of the equations
y = x 2 and y = − x 2 ?
[A] 1
[B] 2
[C] 0
[D] 3
161. A pelican flying in the air over water drops a
crab from a height of 30 feet. The distance
the crab is from the water as it falls can be
represented by the function h(t ) = −16t 2 + 30,
where t is time, in seconds. To catch the crab
as it falls, a gull flies along a path represented
by the function g (t ) = −8t + 15. Can the gull
catch the crab before the crab hits the water?
accompanying grid is optional.]
Math B Regents Exam Questions by Topic
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162. The price of a stock, A(x), over a 12-month
period decreased and then increased
according to the equation
A( x ) = 0.75x 2 − 6 x + 20, where x equals the
number of months. The price of another
stock, B(x), increased according to the
equation B( x ) = 2.75x + 150
. over the same
12-month period. Graph and label both
equations on the accompanying grid. State all
prices, to the nearest dollar, when both stock
values were the same.
165. On the accompanying grid, sketch the graphs
of y = 2 x and 3 y = 7 x + 3 over the interval
− 3 ≤ x ≤ 4. Identify and state the coordinates
of all points of intersection.
166. On the accompanying grid, solve the
following system of equations graphically:
y = − x2 + 2x + 1
y = 2x
163. What is the total number of points of
intersection of the graphs of the equations
xy = 12 and y = − x 2 + 3?
[A] 1
[B] 2
[C] 4
[D] 3
164. The graphs of the equations y = 2 x and
y = −2 x + a intersect in Quadrant I for which
values of a?
[A] a > 1
[B] a < 1
[C] a ≥ 1
[D] 0 < a < 1
Math B Regents Exam Questions by Topic
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167. The path of a rocket is represented by the
equation y = 25 − x . The path of a missile
designed to intersect the path of the rocket is
3
represented by the equation x =
y . The
2
value of x at the point of intersection is 3.
What is the corresponding value of y?
2
[A] 2
[B] 4
[C] -4
[D] -2
168. A pair of figure skaters graphed part of their
routine on a grid. The male skater's path is
1
represented by the equation m( x ) = 3 sin x ,
2
and the female skater's path is represented by
the equation f ( x ) = −2 cos x . On the
accompanying grid, sketch both paths and
state how many times the paths of the skaters
intersect between x = 0 and x = 4π .
169. On a monitor, the graphs of two impulses are
recorded on the same screen, where
0° ≤ x < 360° . The impulses are given by the
following equations:
y = 2 sin 2 x
y = 1 − sin x
Find all values of x, in degrees, for which the
two impulses meet in the interval
0° ≤ x < 360° . [Only an algebraic solution
will be accepted.]
INEQUALITIES
ABSOLUTE VALUE INEQUALITIES
170. Which equation states that the temperature, t,
in a room is less than 3° from 68°?
[A] |68 + t| < 3
[B] |68 - t| < 3
[C] |3 - t| < 68
[D] |3 + t| < 68
171. The solution set of 3x + 2 < 1 contains
[A] both positive and negative real numbers
[B] only negative real numbers
[C] only positive real numbers
[D] no real numbers
172. What is the solution set of the inequality
3 − 2x ≥ 4 ?
[A] {x|−
1
7
≤x≤ }
2
2
[B] {x| x ≤ −
[C] {x| x ≤
1
7
or x ≥ }
2
2
7
1
or x ≥ }
2
2
7
1
[D] {x| ≤ x ≤ − }
2
2
Math B Regents Exam Questions by Topic
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173. What is the solution of the inequality
x + 3 ≤ 5?
[A] x ≤ −8 or x ≥ 2
[C] − 2 ≤ x ≤ 8
[B] − 8 ≤ x ≤ 2
[D] x ≤ −2 or x ≥ 8
174. The solution of 2 x − 3 < 5 is
[A] x < 4
[B] -1 < x < 4
[C] x > -1
[D] x < -1 or x > 4
175. What is the solution of the inequality
y + 8 > 3?
[A] -11 < y < -5
[B] y > -5 or y < -11
[C] -5 < y < 11
[D] y > -5
176. What is the solution set of the inequality
2x − 1 < 9 ?
[A] {x| x < −4}
[C] {x|−4 < x < 5}
[B] {x| x < −4 or x > 5}
[D] {x| x < 5}
177. Which graph represents the solution set of
2x − 1 < 7 ?
[A]
[B]
[C]
[D]
178. Which graph represents the solution set for
the expression 2 x + 3 > 7 ?
[A]
179. The solution set of which inequality is
represented by the accompanying graph?
[A] x − 2 < 7
[B] x − 2 > 7
[C] 2 − x > −7
[D] 2 − x < −7
180. The inequality 15
. C − 24 ≤ 30 represents the
range of monthly average temperatures, C, in
degrees Celsius, for Toledo, Ohio. Solve for
C.
181. The heights, h, of the students in the chorus at
Central Middle School satisfy the inequality
h − 57.5
≤ 3.25, when h is measured in
2
inches. Determine the interval in which these
nearest tenth of a foot. [Only an algebraic
182. A depth finder shows that the water in a
certain place is 620 feet deep. The difference
between d, the actual depth of the water, and
the reading is d − 620 and must be less than
or equal to 0.05d. Find the minimum and
maximum values of d, to the nearest tenth of
a foot.
183. Which graph represents the solution set of the
inequality x 2 − 4 x − 5 < 0 ?
[A]
[B]
[B]
[C]
[C]
[D]
[D]
Math B Regents Exam Questions by Topic
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184. Which graph represents the solution set of
x 2 − x − 12 < 0 ?
[A]
[B]
[C]
[D]
185. When a baseball is hit by a batter, the height
of the ball, h(t), at time t, t ≥ 0, is determined
by the equation h(t ) = −16t 2 + 64t + 4. For
which interval of time is the height of the ball
greater than or equal to 52 feet?
186. The height of a projectile is modeled by the
equation y = −2 x 2 + 38 x + 10, where x is
time, in seconds, and y is height, in feet.
During what interval of time, to the nearest
tenth of a second, is the projectile at least 125
feet above ground? [The use of the
accompanying grid is optional.]
187. The profit a coat manufacturer makes each
day is modeled by the equation
P( x ) = − x 2 + 120 x − 2000 , where P is the
profit and x is the price for each coat sold.
For what values of x does the company make
a profit? [The use of the accompanying grid
is optional.]
Math B Regents Exam Questions by Topic
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188. The profit, P, for manufacturing a wireless
device is given by the equation
P = −10 x 2 + 750 x − 9,000, where x is the
selling price, in dollars, for each wireless
device. What range of selling prices allows
the manufacturer to make a profit on this
wireless device? [The use of the grid is
optional.]
TRIGONOMETRIC INEQUALITIES
189. A building's temperature, T, varies with time
of day, t, during the course of 1 day, as
follows:
T = 8 cos t + 78
The air-conditioning operates when
T ≥ 80° F . Graph this function for 6 ≤ t < 17
and determine, to the nearest tenth of an hour,
the amount of time in 1 day that the airconditioning is on in the building.
Math B Regents Exam Questions by Topic
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190. The tide at a boat dock can be modeled by the
π
equation y = −2 cos( t ) + 8, where t is the
6
number of hours past noon and y is the height
of the tide, in feet. For how many hours
between t = 0 and t = 12 is the tide at least 7
feet? [The use of the grid is optional.]
192. A ball is thrown straight up at an initial
velocity of 54 feet per second. The height of
the ball t seconds after it is thrown is given by
the formula h(t ) = 54t − 12t 2 . How many
seconds after the ball is thrown will it return
to the ground?
[A] 9.2
[B] 4.5
[C] 4
[D] 6
193. If the equation x 2 − kx − 36 = 0 has x = 12 as
one root, what is the value of k?
[A] -3
[B] -9
[C] 3
[D] 9
194. For which equation is the sum of the roots
equal to the product of the roots?
191. On the accompanying set of axes, graph the
equations y = 4 cos x and y = 2 in the
domain − π ≤ x ≤ π .
Express, in terms of π , the interval for which
4 cos x ≥ 2.
[A] x 2 − 4 x + 4 = 0
[B] x 2 + 3x − 6 = 0
[C] x 2 + x + 1 = 0
[D] x 2 − 8 x − 4 = 0
Math B Regents Exam Questions by Topic
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197. What is the equation of a parabola that goes
through points (0,1), (-1,6), and (2,3)?
195. Which quadratic function is shown in the
accompanying graph?
[A] y = 2 x 2 − 3x + 1
[B] y = 2 x 2 + 1
[C] y = x 2 − 3x + 1
[D] y = x 2 + 1
198. For which quadratic equation is the axis of
symmetry x = 3?
[A] y = x 2 + x + 3
[B] y = − x 2 + 3x + 5
[C] y = − x 2 + 6 x + 2
[D] y = x 2 + 6 x + 3
199. The graph of y = ( x − 3) 2 is shifted left 4
units and down 2 units. What is the axis of
symmetry of the transformed graph?
1
[A] y = − x 2
2
[B] y = −2 x
[C] y = 2 x 2
[D] y =
2
1 2
x
2
196. Which equation represents the parabola
shown in the accompanying graph?
[A] f ( x ) = −( x − 3) 2 + 1
[B] f ( x ) = ( x + 1) 2 − 3
[C] f ( x ) = −( x + 3) 2 + 1
[D] f ( x ) = −( x − 3) 2 − 3
[A] x = -2
[B] x = 1
[C] x = -1
[D] x = 7
200. A small rocket is launched from a height of
72 feet. The height of the rocket in feet, h, is
represented by the equation
h(t ) = −16t 2 + 64t + 72, where t = time, in
seconds. Graph this equation on the
accompanying grid.
Use your graph to determine the number of
seconds that the rocket will remain at or
above 100 feet from the ground. [Only a
graphic solution can receive full credit.]
Math B Regents Exam Questions by Topic
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201. An acorn falls from the branch of a tree to the
ground 25 feet below. The distance, S, the
acorn is from the ground as it falls is
represented by the equation
S (t ) = −16t 2 + 25, where t represents time, in
seconds. Sketch a graph of this situation on
the accompanying grid.
Calculate, to the nearest hundredth of a
second, the time the acorn will take to reach
the ground.
204. An archer shoots an arrow into the air such
that its height at any time, t, is given by the
function h(t ) = −16t 2 + kt + 3. If the
maximum height of the arrow occurs at time
t = 4 , what is the value of k?
[A] 8
[B] 128
[C] 64
[D] 4
205. The height of an object, h(t), is determined by
the formula h(t ) = −16t 2 + 256t , where t is
time, in seconds. Will the object reach a
maximum or a minimum? Explain or show
206. Vanessa throws a tennis ball in the air. The
function h(t ) = −16t 2 + 45t + 7 represents the
distance, in feet, that the ball is from the
ground at any time t. At what time, to the
nearest tenth of a second, is the ball at its
maximum height?
MINIMUM AND MAXIMUM OF
202. What is the turning point, or vertex, of the
parabola whose equation is y = 3 x 2 + 6 x − 1?
[A] (-l,-4)
[B] (-3,8)
[C] (1,8)
[D] (3,44)
203. What is the minimum point of the graph of
the equation y = 2 x 2 + 8 x + 9 ?
[A] (2,33)
[B] (-2,1)
[C] (-2,-15)
[D] (2,17)
207. The height, h, in feet, a ball will reach when
thrown in the air is a function of time, t, in
seconds, given by the equation
h(t ) = −16t 2 + 30t + 6 . Find, to the nearest
tenth, the maximum height, in feet, the ball
will reach.
208. When a current, I, flows through a given
electrical circuit, the power, W, of the circuit
can be determined by the formula
W = 120 I − 12 I 2 . What amount of current, I,
supplies the maximum power, W?
209. The equation W = 120 I − 12 I 2 represents the
power (W), in watts, of a 120-volt circuit
having a resistance of 12 ohms when a current
(I) is flowing through the circuit. What is the
maximum power, in watts, that can be
delivered in this circuit?
Math B Regents Exam Questions by Topic
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210. A baseball player throws a ball from the
outfield toward home plate. The ball's height
above the ground is modeled by the equation
y = −16 x 2 + 48 x + 6 where y represents
height, in feet, and x represents time, in
seconds. The ball is initially thrown from a
height of 6 feet.
How many seconds after the ball is thrown
will it again be 6 feet above the ground?
What is the maximum height, in feet, that the
ball reaches? [The use of the accompanying
grid is optional.]
211. A rock is thrown vertically from the ground
with a velocity of 24 meters per second, and it
reaches a height of 2 + 24t − 4.9t 2 after t
seconds. How many seconds after the rock is
thrown will it reach maximum height, and
what is the maximum height the rock will
reach, in meters? How many seconds after
the rock is thrown will it hit the ground?
[Only an algebraic or graphic solution will be
accepted.]
212. If the sum of the roots of x 2 + 3x − 5 is added
to the product of its roots, the result is
[A] 15
[B] -15
[C] -2
[D] -8
Math B Regents Exam Questions by Topic
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213. Barb pulled the plug in her bathtub and it
started to drain. The amount of water in the
bathtub as it drains is represented by the
equation L = −5t 2 − 8t + 120, where L
represents the number of liters of water in the
bathtub and t represents the amount of time,
in minutes, since the plug was pulled.
How many liters of water were in the bathtub
when Barb pulled the plug? Show your
reasoning.
Determine, to the nearest tenth of a minute,
the amount of time it takes for all the water in
the bathtub to drain.
USING THE DISCRIMINANT
220. The roots of a quadratic equation are real,
rational, and equal when the discriminant is
[A] 0
[B] 2
[C] -2
221. Jacob is solving a quadratic equation. He
executes a program on his graphing calculator
and sees that the roots are real, rational, and
unequal. This information indicates to Jacob
that the discriminant is
[A] zero
[B] a perfect square
[C] not a perfect square
214. Matt’s rectangular patio measures 9 feet by
12 feet. He wants to increase the patio’s
dimensions so its area will be twice the area it
is now. He plans to increase both the length
and the width by the same amount, x. Find x,
to the nearest hundredth of a foot.
215. If 2 + 3i is one root of a quadratic equation
with real coefficients, what is the sum of the
roots of the equation?
216. Express, in simplest a + bi form, the roots of
the equation x 2 + 5 = 4 x.
[D] 4
[D] negative
222. The roots of the equation x 2 − 3x − 2 = 0 are
[A] imaginary
[B] real, rational, and equal
[C] real, rational, and unequal
[D] real, irrational, and unequal
223. The roots of the equation 2 x 2 − 8 x − 4 = 0 are
[A] real, irrational, and unequal
[B] real, rational, and unequal
[C] real, rational, and equal
217. Solve for x in simplest a + bi form:
x 2 + 8 x + 25 = 0
218. In physics class, Taras discovers that the
behavior of electrical power, x, in a particular
circuit can be represented by the function
f ( x ) = x 2 + 2 x + 7. If f ( x ) = 0, solve the
a + bi form.
219. Which quadratic equation has the roots 3 + i
and 3 − i ?
[A] x − 6 x − 8 = 0
[B] x + 6 x − 10 = 0
[C] x + 6 x + 8 = 0
[D] x − 6 x + 10 = 0
2
2
2
2
[D] imaginary
224. The roots of the equation 2 x 2 − x = 4 are
[A] real, rational, and unequal
[B] real and irrational
[C] imaginary
[D] real, rational, and equal
225. The roots of the equation 2 x 2 − 5 = 0 are
[A] real, rational, and unequal
[B] imaginary
[C] real and irrational
[D] real, rational, and equal
Math B Regents Exam Questions by Topic
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226. Which equation has imaginary roots?
[A] x 2 − 1 = 0
[B] x 2 − x − 1 = 0
[C] x 2 + x + 1 = 0
[D] x 2 − 2 = 0
227. Which equation has imaginary roots?
[A] (2x + l)(x - 3) = 7
[C] x(5 - x) = -3
[B] x(x + 6) = -10
[D] x(5 + x) = 8
234. If the roots of ax 2 + bx + c = 0 are real,
rational, and equal, what is true about the
graph of the function y = ax 2 + bx + c ?
[A] It lies entirely below the x-axis.
[B] It is tangent to the x-axis.
[C] It intersects the x-axis in two distinct
points.
[D] It lies entirely above the x-axis.
228. For which positive value of m will the
equation 4 x 2 + mx + 9 = 0 have roots that are
real, equal, and rational?
[A] 12
[B] 9
[C] 3
[D] 4
229. The roots of the equation ax 2 + 4 x = −2 are
real, rational, and equal when a has a value of
[A] 3
[B] 1
[C] 2
[D] 4
230. In the equation ax 2 + 6 x − 9 = 0, imaginary
roots will be generated if
[A] -1 < a < 1
[B] a < -1
[C] a > -1, only
[D] a < 1, only
231. The equation 2 x 2 + 8 x + n = 0 has imaginary
roots when n is equal to
[A] 6
[B] 8
[C] 10
[D] 4
232. Find all values of k such that the equation
3x 2 − 2 x + k = 0 has imaginary roots.
233. Which statement must be true if a parabola
represented by the equation y = ax 2 + bx + c
does not intersect the x-axis?
[A] b 2 − 4ac > 0, and b 2 − 4ac is not a perfect
square.
[B] b 2 − 4ac = 0
[C] b 2 − 4ac < 0
[D] b 2 − 4ac > 0, and b 2 − 4ac is a perfect
square.
235. Which is a true statement about the graph of
the equation y = x 2 − 7 x − 60?
[A] It intersects the x-axis in two distinct
points that have irrational coordinates.
[B] It does not intersect the x-axis.
[C] It is tangent to the x-axis.
[D] It intersects the x-axis in two distinct
points that have rational coordinates.
Math B Regents Exam Questions by Topic
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236. Which graph represents a quadratic function
with a negative discriminant?
[A]
OPERATIONS WITH POWERS
239. The product of (5ab) and ( −2a 2b) 3 is
[A] − 40a 6b 4
[B] − 30a 7b 4
[C] − 30a 6b 4
[D] − 40a 7b 4
(b 2 n +1 ) 3
240. The expression n 4 n + 3 is equivalent to
b ⋅b
[B]
[A] b −3n
[B]
bn
2
[C] b −3n +1
[D] b n
SCIENTIFIC NOTATION
241. Two objects are 2.4 × 1020 centimeters apart.
A message from one object travels to the
other at a rate of 12
. × 105 centimeters per
second. How many seconds does it take the
message to travel from one object to the
other?
[C]
[D]
[A] 2.0 × 104
[B] 2.0 × 1015
[C] 2.88 × 1025
[D] 12
. × 1015
EXPONENTIAL FUNCTIONS
242. Which equation models the data in the
accompanying table?
POWERS
ZERO AND NEGATIVE POWERS
237. If f ( x ) = 4 x 0 + (4 x ) −1 , what is the value of
f ( 4) ?
[A] − 12
[B] 0
238. Solve for x: x
−3
27
=
64
1
[C] 4
16
1
[D] 1
16
[A] y = 2 x
[B] y = 2 x
[C] y = 5(2 x )
[D] y = 2 x + 5
243. What is the domain of f ( x ) = 2 x ?
[A] x ≤ 0
[B] all real numbers
[C] x ≥ 0
[D] all integers
Math B Regents Exam Questions by Topic
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244. The height, f(x), of a bouncing ball after x
bounces is represented by f ( x ) = 80(0.5) x .
How many times higher is the first bounce
than the fourth bounce?
[A] 8
[B] 4
[C] 16
247. Which equation best represents the
accompanying graph?
[D] 2
245. The accompanying graph represents the value
of a bond over time.
Which type of function does this graph best
model?
[B] trigonometric
[C] exponential
[D] logarithmic
246. The strength of a medication over time is
represented by the equation y = 200(15
. )−x ,
where x represents the number of hours since
the medication was taken and y represents the
number of micrograms per millimeter left in
the blood. Which graph best represents this
relationship?
[A]
[B]
[C]
[D]
[A] y = 2 x
[B] y = 2 − x
[C] y = x 2 + 2
[D] y = −2 x
248. On January 1, 1999, the price of gasoline was
\$1.39 per gallon. If the price of gasoline
increased by 0.5% per month, what was the
cost of one gallon of gasoline, to the nearest
cent, on January 1 one year later?
249. A used car was purchased in July 1999 for
\$11,900. If the car depreciates 13% of its
value each year, what is the value of the car,
to the nearest hundred dollars, in July 2002?
250. The Franklins inherited \$3,500, which they
want to invest for their child's future college
expenses. If they invest it at 8.25% with
interest compounded monthly, determine the
value of the account, in dollars, after 5 years.
r
Use the formula A = P(1 + ) nt , where A =
n
value of the investment after t years,
P = principal invested, r = annual interest
rate, and n = number of times compounded
per year.
Math B Regents Exam Questions by Topic
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258. If log x = a, log y = b, and log z = c, then
x2 y
is equivalent to
log
z
PROPERTIES OF LOGARITHMS
251. If logb x = y , then x equals
[A]
y
b
[B] y ⋅ b
[C] y b
[D] b y
252. The function y = 2 x is equivalent to
[A] x = log 2 y
[B] y = x log 2
[C] x = y log 2
[D] y = log 2 x
253. For which value of x is y = log x undefined?
[A] 0
[B] 1483
.
[C] π
1
[A] 42a + b + c
2
1
[B] 2a + b − c
2
1
[C] 2ab − c
2
1
[D] a 2 + b − c
2
259. The expression log 10 x + 2 − log 10 x is
equivalent to
[A] 100
1
[D]
10
[B]
1
100
[C] − 2
[D] 2
260. If log a = x and log b = y , what is log a b ?
254. The expression log 3 (8 − x ) is defined for all
values of x such that
[A] x > 8
[B] x ≥ 8
[C] x < 8
[D] x ≤ 8
[B] 25a
[C] 10 + 2a
[D] 50a
256. Which expression is not equivalent to
logb 36 ?
[A] logb 9 + logb 4
[B] 2 logb 6
[C] 6 logb 2
[D] logb 72 − log b 2
257. If log a = 2 and log b = 3, what is the
a
numerical value of log 3 ?
b
[A] -25
[B] 25
[C] -8
x+ y
2
[C] x + 2 y
255. If log 5 = a, then log 250 can be expressed as
[A] 2a + 1
[A]
[D] 8
[B] 2 x + 2 y
[D] x +
y
2
261. The speed of sound, v, at temperature T, in
degrees Kelvin, is represented by the equation
T
v = 1087
. Which expression is
273
equivalent to log v?
1
1
[A] log 1087 + log T − log 273
2
2
1
1
[B] 1087( log T − log 273)
2
2
[C] log 1087 + 2 log(T + 273)
1
[D] 1087 + log T − log 273
2
Math B Regents Exam Questions by Topic
Page 39
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262. 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
2GM
formula, R = 2 .
c
Based on the laws of logarithms, log R can
be represented by
[A] log 2 + log G + log M − 2 log c
[B] 2 log G + log M − log 2c
[C] log 2G + log M − log 2c
[D] 2 log GM − 2 log c
264. A hotel finds that its total annual revenue and
the number of rooms occupied daily by
guests can best be modeled by the
function R = 3 log(n 2 + 10n), n > 0, where R
is the total annual revenue, in millions of
dollars, and n is the number of rooms
occupied daily by guests. The hotel needs an
annual revenue of \$12 million to be
profitable. Graph the function on the
accompanying grid over the interval
0 < n ≤ 100.
Calculate the minimum number of rooms that
must be occupied daily to be profitable.
GRAPHING LOGARITHMIC FUNCTIONS
263. The cells of a particular organism increase
logarithmically. If g represents cell growth
and h represents time, in hours, which graph
best represents the growth pattern of the cells
of this organism?
[A]
[B]
LOGARITHMIC EQUATIONS
265. Solve for x: log 4 ( x 2 + 3x ) − log 4 ( x + 5) = 1
[C]
[D]
266. In the equation log x 4 + log x 9 = 2, x is equal
to
[A]
13
[B] 6
[C] 18
[D] 6.5
267. If log5 x = 2, what is the value of
[A] 25
[B] 2
2
5
[C] 5
x?
[D]
268. Solve for x: log 2 ( x + 1) = 3
269. Solve for x: logb 36 − logb 2 = logb x
5
Math B Regents Exam Questions by Topic
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270. If log k = c log v + log p, k equals
276. Solve algebraically for x: 82 x = 4 6
[A] v c p
[B] (vp) c
[C] v + p
[D] cv + p
c
277. What is the value of x in the equation
81x + 2 = 275 x + 4 ?
271. The relationship between the relative size of
an earthquake, S, and the measure of the
earthquake on the Richter scale, R, is given by
the equation log S = R. If an earthquake
measured 3.2 on the Richter scale, what was
its relative size to the nearest hundredth?
272. The magnitude (R) of an earthquake is related
I
to its intensity (I) by R = log( ), where T is
T
the threshold below which the earthquake is
not noticed. If the intensity is doubled, its
magnitude can be represented by
[A] 2(log I - log T)
[B] 2 log I - log T
[C] log 2 + log I - log T
[D] log I - log T
273. The scientists in a laboratory company raise
amebas to sell to schools for use in biology
classes. They know that one ameba divides
into two amebas every hour and that the
formula t = log 2 N can be used to determine
how long in hours, t, it takes to produce a
certain number of amebas, N. Determine, to
the nearest tenth of an hour, how long it takes
one ameba.
EXPONENTIAL EQUATIONS
274. The solution set of 2 x
[A] {1}
[B] { }
2
+2 x
= 2 −1 is
[C] {-1}
[D] {-1, 1}
275. What is the value of b in the equation
4 2b − 3 = 81−b ?
[A]
10
7
[B]
−3
7
[C]
9
7
[D]
7
9
[A] −
3
2
[B] −
4
11
[C]
4
11
[D] −
2
11
278. Solve algebraically for x: 27 2 x +1 = 9 4 x
279. Solve for m:
3m+1 − 5 = 22
280. Determine the value of x and y if 2 y = 8 x and
3 y = 3x + 4 .
[A] x = -2, y = -6
[B] x = 6, y = 2
[C] x = 2, y = 6
[D] x = y
281. The growth of bacteria in a dish is modeled
t
by the function f (t ) = 2 3 . For which value
of t is f (t ) = 32 ?
[A] 8
[B] 16
[C] 15
[D] 2
282. Growth of a certain strain of bacteria is
modeled by the equation G = A(2.7) 0.584 t ,
where:
G = final number of bacteria
A = initial number of bacteria
t = time (in hours)
In approximately how many hours will 4
bacteria first increase to 2,500 bacteria?
Math B Regents Exam Questions by Topic
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283. Since January 1980, the population of the city
of Brownville has grown according to the
mathematical model y = 720,500(1022
. )x ,
where x is the number of years since January
1980.
Explain what the numbers 720,500 and 1.022
represent in this model.
If this trend continues, use this model to
predict the year during which the population
of Brownville will reach 1,548,800. [The use
of the grid is optional.]
284. After an oven is turned on, its temperature, T,
is represented by the equation
T = 400 − 350(3.2) −0.1m where m represents the
number of minutes after the oven is turned on
and T represents the temperature of the oven,
in degrees Fahrenheit.
How many minutes does it take for the oven's
temperature to reach 300°F? Round your
answer to the nearest minute. [The use of the
grid is optional.]
Math B Regents Exam Questions by Topic
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285. An amount of P dollars is deposited in an
account paying an annual interest rate r (as a
decimal) compounded n times per year. After
t years, the amount of money in the account,
in dollars, is given by the equation
r
A = P(1 + ) nt .
n
Rachel deposited \$1,000 at 2.8% annual
interest, compounded monthly. In how many
years, to the nearest tenth of a year, will she
have \$2,500 in the account? [The use of the
grid is optional.]
287. The equation for radioactive decay is
t
H
p = (0.5) , where p is the part of a substance
with half-life H remaining radioactive after a
period of time, t.
A given substance has a half-life of 6,000
years. After t years, one-fifth of the original
sample remains radioactive. Find t, to the
nearest thousand years.
288. An archaeologist can determine the
approximate age of certain ancient specimens
by measuring the amount of carbon-14, a
specimen. The formula used to determine the
−t
5760
age of a specimen is A = A0 2 , where A is
the amount of carbon-14 that a specimen
contains, A0 is the original amount of carbon14, t is time, in years, and 5760 is the half-life
of carbon-14.
A specimen that originally contained 120
milligrams of carbon-14 now contains 100
milligrams of this substance. What is the age
of the specimen, to the nearest hundred
years?
286. Sean invests \$10,000 at an annual rate of 5%
compounded continuously, according to the
formula A = Pe rt , where A is the amount, P is
the principal, e = 2.718, r is the rate of
interest, and t is time, in years.
Determine, to the nearest dollar, the amount
of money he will have after 2 years.
Determine how many years, to the nearest
year, it will take for his initial investment to
double.
289. Depreciation (the decline in cash value) on a
car can be determined by the formula
V = C (1 − r ) t , where V is the value of the car
after t years, C is the original cost, and r is the
rate of depreciation. If a car's cost, when
new, is \$15,000, the rate of depreciation is
30%, and the value of the car now is \$3,000,
how old is the car to the nearest tenth of a
year?
290. The amount A, in milligrams, of a 10milligram dose of a drug remaining in the
body after t hours is given by the formula
A = 10(0.8) t . Find, to the nearest tenth of an
hour, how long it takes for half of the drug
dose to be left in the body.
Math B Regents Exam Questions by Topic
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291. The current population of Little Pond, New
York, is 20,000. The population is
decreasing, as represented by the formula
P = A(13
. ) −0.234 t , where P = final population, t
= time, in years, and A = initial population.
What will the population be 3 years from
hundred people.
To the nearest tenth of a year, how many
years will it take for the population to reach
half the present population? [The use of the
grid is optional.]
295. What is the fourth term in the expansion of
(2 x − y ) 5 ?
296. What is the third term in the expansion of
bcos x + 3g ?
5
[A] 90 cos2 x
[B] 270 cos2 x
[C] 60 cos3 x
[D] 90 cos3 x
RATIONALIZING DENOMINATORS
297. Which expression is equivalent to
[A]
12 − 4 2
7
[B]
12 + 4 2
11
[C]
12 − 4 2
11
[D]
12 + 4 2
7
298. The expression
BINOMIAL EXPANSIONS
292. What is the last term in the expansion of
( x + 2 y)5 ?
[A] 2 y 5
[B] 10 y 5
[C] 32 y 5
[D] y 5
293. What is the middle term in the expansion of
( x + y)4 ?
[A] 2 x 2 y 2
[B] 4 x 2 y 2
[C] 6 x 2 y 2
[D] x 2 y 2
294. What is the fourth term in the expansion of
( y − 1) 7 ?
[A] 35 y 4
[B] − 35 y 3
[C] − 35 y 4
[D] 35 y 3
4
?
3+ 2
12
is equivalent to
3+ 3
[A] 6 − 2 3
[B] 4 − 2 3
[C] 2 + 3
[D] 12 − 3
299. The expression
7
is equivalent to
2− 3
[A] 14 − 7 3
[B]
14 + 3
7
[C] 14 + 7 3
[D]
2+ 3
7
Math B Regents Exam Questions by Topic
Page 44
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300. The expression
7
is equivalent to
3− 2
305. Which expression is equal to
[A]
3+ 2
7
[B] 3 + 2
[A]
[C]
21 + 2
7
[D] 3 − 2
[C] 7 + 4 3
301. The expression
1
5 − 13
is equivalent to
[B] 5 + 13
12
[A]
[C] 5 + 13
− 12
[D] 5 + 13
8
[C]
4
5 − 13
is equivalent to
[A]
2(5 − 13 )
19
[B]
2(5 + 13 )
19
[C]
5 + 13
3
[D]
5 − 13
3
303. The expression
[A]
− 3 +5
2
[C]
3 −5
2
304. The expression
[A]
[C]
11
is equivalent to
3 −5
[B]
3 +5
2
− 3 −5
[D]
2
5
is equivalent to
5 −1
5
4
[B]
5 5 −5
6
[D]
5 5 −5
4
5 5 +5
4
[B]
7+4 3
7
[D] 1 − 4 3
306. Which expression represents the sum of
1
1
+
?
3
2
[A] 5 + 13
−8
302. The expression
1− 4 3
7
2+ 3
?
2− 3
3+ 2
3
2
5
[B]
3+ 2
2
[D]
2 3+3 2
6
307. What is the domain of h( x ) = x 2 − 4 x − 5 ?
[A] {x − 1 ≤ x ≤ 5}
[B] {x − 5 ≤ x ≤ 1}
[C] {x x ≥ 1or x ≤ −5}
[D] {x x ≥ 5 or x ≤ −1}
308. Which statement is true for all real number
values of x?
[A]
x2 = x
[B] |x - 1| > (x - 1)
[C]
x2 = x
[D] |x - 1| > 0
309. What is the axis of symmetry of the graph of
the equation x = y 2 ?
[A] line y = -x
[B] y-axis
[C] line y = x
[D] x-axis
310. If
2 x − 1 + 2 = 5, then x is equal to
[A] 4
[B] 2
[C] 5
[D] 1
Math B Regents Exam Questions by Topic
Page 45
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311. What is the solution of the equation
2x − 3 − 3 = 6?
[A] 39
[B] 3
[C] 6
312. The solution set of the equation
[A] {-2,3}
[B] {3}
[D] 42
x + 6 = x is
[C] { }
[D] {-2}
313. What is the solution set of the equation
9 x + 10 = x
[A] {9}
[B] {-1}
[C] {10}
[D] {10, -1}
314. What is the solution set of the equation
x = 2 2x − 3 ?
[A] {2}
[B] { }
[C] {2,6}
[D] {6}
315. Solve for all values of q that satisfy the
equation 3q + 7 = q + 3.
316. Solve algebraically:
x + 5 +1= x
317. Solve algebraically for x:
3x + 1 + 1 = x
318. A wrecking ball suspended from a chain is a
type of pendulum. The relationship between
the rate of speed of the ball, R, the mass of the
ball, m, the length of the chain, L, and the
mL
force, F, is R = 2π
. Determine the
F
force, F, to the nearest hundredth, when L =
12, m = 50, and R = 0.6.
319. The lateral surface area of a right circular
cone, s, is represented by the equation
s = π r r 2 + h 2 , where r is the radius of the
circular base and h is the height of the cone.
If the lateral surface area of a large funnel is
236.64 square centimeters and its radius is
4.75 centimeters, find its height, to the
nearest hundredth of a centimeter.
320. The equation V = 20 C + 273 relates speed
of sound, V, in meters per second, to air
temperature, C, in degrees Celsius. What is
the temperature, in degrees Celsius, when the
speed of sound is 320 meters per second?
[The use of the accompanying grid is
optional.]
Math B Regents Exam Questions by Topic
Page 46
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321. The number of people, y, involved in
recycling in a community is modeled by the
function y = 90 3x + 400 , where x is the
number of months the recycling plant has
been open.
Construct a table of values, sketch the
function on the grid, and find the number of
people involved in recycling exactly 3 months
after the plant opened.
After how many months will 940 people be
involved in recycling?
324. The value of
(
30
27
[A]
1
9
[B] −
2
3
) −1
1
9
is
[D] − 9
[C] 9
1
325. If x is a positive integer, 4 x 2 is equivalent to
[A]
2
x
[B] 4
1
x
−
[C] 2 x
[D] 4 x
3
2
326. The expression b , b > 0, is equivalent to
[B] − ( b ) 3
1
[A]
(3 b ) 2
[C] (3 b ) 2
1
[D]
( b )3
327. The expression 4 16a 6b 4 is equivalent to
3
[A] 4a 2 b
[B] 2a 2b
[C] 4a 2b
[D] 2a 2 b
3
3
3
[B] 4
323. The expression
[C] 16
3
3
[A]
3
[D] 8 2
1
3
−
[B]
[A]
6
m5
[B]
4
m3
[C]
5
m −4
[D]
3
m −2
is equivalent to
2
3
3
1
3
−
1
2
328. When simplified, the expression ( m )(m )
is equivalent to
1
322. The expression 4 2 ⋅ 2 3 is equal to
[A] 4 2
4
3
[C] 3
[D] 1
329. Find the value of ( x + 2) 0 + ( x + 1)
x = 7.
−
−
2
3
when
3
2
1
330. If f ( x ) = x , then f ( ) is equal to
4
[A] − 2
[B] − 4
[C] −
1
8
[D] 8
Math B Regents Exam Questions by Topic
Page 47
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2
331. If (a x ) 3 =
[A] -1
1
, what is the value of x?
a2
[B] 2
[C] -3
[D] 1
332. Meteorologists can determine how long a
storm lasts by using the function
3
2
t (d ) = 0.07d , where d is the diameter of the
storm, in miles, and t is the time, in hours. If
the storm lasts 4.75 hours, find its diameter,
to the nearest tenth of a mile.
RATIONALS
333. A rectangular prism has a length of
2 x 2 + 2 x − 24
x2 + x − 6
a
width
of
,
, and a
x+4
4x2 + x
8x 2 + 2 x
height of
. For all values of x for
x2 − 9
which it is defined, express, in terms of x, the
volume of the prism in simplest form.
334. If the length of a rectangular garden is
x2 + 2x
represented by 2
and its width is
x + 2 x − 15
2x − 6
represented by
, which expression
2x + 4
represents the area of the garden?
[C]
x
x +5
x2 − 9
5x − x 2
x−4
• 2
÷ 2
2
x − 5x x − x − 12 x − 8 x + 16
RATIONALS
[B] x + 5
[D]
x2 + 2x
2( x + 5)
335. Express in simplest form:
4x + 8 2 − x
x2 − 4
•
÷ 2
x + 1 3x − 15 2 x − 8 x − 10
1
1
+
x x+3
337. Express in simplest form:
338. What is the sum of
MULTIPLICATION AND DIVISION OF
RATIONALS
[A] x
336. Perform the indicated operations and simplify
completely:
[A] 1
3
x
and
?
x−3
3− x
[B] 0
[C] − 1
339. What is the sum of ( y − 5) +
[D]
x+3
x−3
3
?
y+2
[A] y − 5
[B] y 2 − 3 y − 7
y+2
[C] y − 2
y+2
[D] y 2 − 7
y+2
SOLVING RATIONALS
340. Solve for all values of x:
9
9
+
= 12
x x−2
4
3
−
=7
x x +1
342. What is the solution set of the equation
1
28
x
−
= 2
?
x − 4 x + 3 x − x − 12
[A] {4,-6}
[B] { }
[C] {-6}
[D] {4}
Math B Regents Exam Questions by Topic
Page 48
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343. A rectangle is said to have a golden ratio
w
h
when =
, where w represents width
h w−h
and h represents height. When w = 3,
between which two consecutive integers will
h lie?
344. Working by herself, Mary requires 16
minutes more than Antoine to solve a
mathematics problem. Working together,
Mary and Antoine can solve the problem in 6
minutes. If this situation is represented by the
6
6
equation +
= 1, where t represents the
t t + 16
number of minutes Antoine works alone to
solve the problem, how many minutes will it
take Antoine to solve the problem if he works
by himself?
RATIONAL EXPRESSIONS
346. Written in simplest form, the expression
x2 y2 − 9
is equivalent to
3 − xy
[B]
[C] − 1
[D] 3 + xy
347. Written in simplest form, the expression
x2 − 9x
is equivalent to
45x − 5x 2
[B] −
[A] 5
348. The expression
345. Electrical circuits can be connected in series,
one after another, or in parallel circuits that
branch off a main line. If circuits are hooked
up in parallel, the reciprocal of the total
resistance in the series is found by adding the
reciprocals of each resistance, as shown in the
accompanying diagram.
1
3 + xy
[A] − (3 + xy )
[A] −
9
4
1
5
[C] − 5
[D]
1
5
3 y 2 − 12 y
is equivalent to
4 y2 − y3
[B] −
3
y
[C]
3 12
−
4 y2
[D]
3
y
349. Express the following rational expression in
simplest form:
9 − x2
10 x 2 − 28 x − 6
350. For all values of x for which the expression is
2x + x2
defined, 2
is equivalent to
x + 5x + 6
If R1 = x , R2 = x + 3, and the total resistance,
RT , is 2.25 ohms, find the positive value of
R1 to the nearest tenth of an ohm.
[A]
1
x+2
[B]
x
x+2
[C]
x
x+3
[D]
1
x+3
Math B Regents Exam Questions by Topic
Page 49
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354. Which graph shows that soil permeability
varies inversely to runoff?
RATIONAL FUNCTIONS
351. Which function is symmetrical with respect to
the origin?
[A] y = x + 5
[C] y = −
5
x
[A]
[B]
[C]
[D]
[B] y = 5 − x
[D] y = 5x
352. Which equation represents a hyperbola?
16
x
[A] y = 16 − x 2
[B] y =
[C] y 2 = 16 − x 2
[D] y = 16 x 2
353. The accompanying graph shows the
relationship between a person's weight and
the distance that the person must sit from the
center of a seesaw to make it balanced.
355. Which graph represents an inverse variation
between stream velocity and the distance
from the center of the stream?
[A]
[B]
[C]
[D]
356. What is the domain of the function
2x2
f ( x) = 2
?
x −9
[A] all real numbers except 0
Which equation best represents this graph?
[A] y = 12 x 2
[B] y = −120 x
[C] y = 2 log x
[D] y =
120
x
[B] all real numbers
[C] all real numbers except 3 and -3
[D] all real numbers except 3
357. What is the domain of the function
3x 2
f ( x) = 2
?
x − 49
[A] {x | x ∈ real numbers, x ≠ 7}
[B] {x | x ∈ real numbers, x ≠ 0}
[C] {x | x ∈ real numbers, x ≠ ±7}
[D] {x | x ∈ real numbers}
Math B Regents Exam Questions by Topic
Page 50
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1
, the domain of f ( x ) is
2x − 4
358. If f ( x ) =
[A] x < 2
[B] x ≥ 2
[C] x = 2
[D] x > 2
1
+
x
364. The expression
1
−
x2
COMPLEX FRACTIONS
a b
−
359. The expression b a is equivalent to
1 1
+
a b
[A]
a −b
ab
[B] a + b
x2 y
1+ y
[C] 2 x
2 xy
1+ y
[D]
[B] y − x
[C]
xy
y−x
[D]
[A]
y−x
xy
[B] x − y
[C]
x− y
xy
[D] y − x
x +1
x+3
[C]
3x + 3
x+3
[A]
2x
x+2
[B]
2x
x +4
2
[C]
2
x
[D]
x
2
367. When simplified, the complex fraction
1
1+
x , x ≠ 0, is equivalent to
1
−x
x
[A] 1
1 1
+
3
3x is equivalent to
362. The expression
1 1
+
x 3
[B]
xy
x− y
1 1
−
r
366. Express in simplest form: 2 s
r
−1
s2
1
1
− 2
2
x
y
361. In simplest form,
is equal to
1 1
+
y x
[A] 2
y−x
xy
365. Which expression is equivalent to the
x
complex fraction x + 2 ?
x
1−
x+2
x
+x
y
360. The fraction
is equivalent to
1
+1
y
[B]
1
y
is equivalent to
1
y2
[A]
[D] a − b
[C] ab
[A] x
x 4
−
363. Express in simplest form: 4 x
4
1−
x
[D]
1
3
[B] -1
[C]
1
x −1
1− m
m
368. Simplify completely:
1
m−
m
[D]
1
1− x
Math B Regents Exam Questions by Topic
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369. Simplify for all values of a for which the
2
1−
a
expression is defined:
4
−1
a2
370. In a science experiment, when resistor A and
resistor B are connected in a parallel circuit,
1
. This complex
the total resistance is
1 1
+
A B
fraction is equivalent to
[A] A + B
[B]
AB
A+ B
[D] AB
[C] 1
INVERSE VARIATION
371. Explain how a person can determine if a set
of data represents inverse variation and give
an example using a table of values.
372. For a rectangular garden with a fixed area, the
length of the garden varies inversely with the
width. Which equation represents this
situation for an area of 36 square units?
[A] y =
36
x
[B] x − y = 36
[C] x + y = 36
[D] y = 36 x
373. If R varies inversely as S, when S is doubled,
R is multiplied by
[A]
1
4
[B]
1
2
[C] 4
375. The speed of a laundry truck varies inversely
with the time it takes to reach its destination.
If the truck takes 3 hours to reach its
destination traveling at a constant speed of 50
miles per hour, how long will it take to reach
the same location when it travels at a constant
speed of 60 miles per hour?
[A] 2 hours
[C] 2
1
hours
3
[B] 2
2
hours
3
[D] 2
1
hours
2
376. 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?
377. When air is pumped into an automobile tire,
the pressure is inversely proportional to the
volume. If the pressure is 35 pounds when
the volume is 120 cubic inches, what is the
pressure, in pounds, when the volume is 140
cubic inches?
378. Boyle's Law states that the pressure of
compressed gas is inversely proportional to its
volume. The pressure of a certain sample of a
gas is 16 kilopascals when its volume is 1,800
liters. What is the pressure, in kilopascals,
when its volume is 900 liters?
[D] 2
374. In a given rectangle, the length varies
inversely as the width. If the length is
doubled, the width will
[A] increase by 2
[B] be multiplied by 2
[C] be divided by 2
[D] remain the same
379. 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?
[A] 50 lb / in 2
[B] 25 lb / in 2
[C] 16 lb / in 2
[D] 40 lb / in 2
Math B Regents Exam Questions by Topic
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380. Camisha is paying a band \$330 to play at her
graduation party. The amount each member
earns, d, varies inversely as the number of
members who play, n. The graph of the
equation that represents the relationship
between d and n is an example of
[A] an ellipse
[B] a hyperbola
[C] a line
[D] a parabola
381. The price per person to rent a limousine for a
prom varies inversely as the number of
passengers. If five people rent the limousine,
the cost is \$70 each. How many people are
renting the limousine when the cost per
couple is \$87.50?
382. To balance a seesaw, the distance, in feet, a
person is from the fulcrum is inversely
proportional to the person's weight, in
pounds. Bill, who weighs 150 pounds, is
sitting 4 feet away from the fulcrum. If Dan
weighs 120 pounds, how far from the fulcrum
should he sit to balance the seesaw?
[A] 3.5 ft
[B] 5 ft
[C] 3 ft
[D] 4.5 ft
383. A pulley that has a diameter of 8 inches is
belted to a pulley that has a diameter of 12
inches. The 8-inch-diameter pulley is running
at 1,548 revolutions per minute. If the speeds
of the pulleys vary inversely to their
diameters, how many revolutions per minute
does the larger pulley make?
ANGLES
UNIT CIRCLE
384. Which angle is coterminal with an angle of
125°?
[A] 235°
[B] -235°
[C] -125°
[D] 425°
385. Expressed as a function of a positive acute
angle, sin (-230°) is equal to
[A] -cos 50°
[B] -sin 50°
[C] sin 50°
[D] cos 50°
386. If θ is an angle in standard position and its
terminal side passes through the point
1 3
( , ) on a unit circle, a possible value of
2 2
θ is
[A] 120°
[B] 30°
[C] 150°
[D] 60°
387. In the accompanying diagram, point
P(0.6,−0.8) is on unit circle O. What is the
value of θ , to the nearest degree?
Math B Regents Exam Questions by Topic
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388. In the accompanying diagram of a unit circle,
3 1
the ordered pair ( −
,− ) represents the
2
2
point where the terminal side of θ intersects
the unit circle.
3
and the angle is
5
not in Quadrant I, what is the value of the
cosine of the angle?
390. If the sine of an angle is
391. In the accompanying diagram, PR is tangent
to circle O at R, QS⊥ OR, and PR⊥ OR.
What is m∠θ ?
[A] 233
[B] 240
[C] 210
[D] 225
389. In the unit circle shown in the accompanying
diagram, what are the coordinates of ( x , y ) ?
Which measure represents sin θ ?
[A] QS
[B] PR
[C] SO
[D] RO
392. If x is a positive acute angle and cos x =
what is the exact value of sin x ?
[A]
[A] ( −30,−210)
[B] ( −
2
2
,−
)
2
2
3
)
2
[D] ( −
3
,−0.5)
2
[C] ( −0.5,−
3
5
[B]
13
4
[C]
4
5
[D]
3
5
3
,
4
Math B Regents Exam Questions by Topic
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393. The accompanying diagram shows unit circle
O, with radius OB = 1.
398. If sin θ is negative and cosθ is negative, in
which quadrant does the terminal side of θ
lie?
[A] I
[B] III
[C] IV
[D] II
399. If tan θ = 2.7 and csc θ < 0, in which
[A] IV
[B] II
[C] I
[D] III
400. If θ is an obtuse angle and sin θ = b, then it
can be concluded that
[A] cosθ > b
[B] cos2θ > b
[C] tan θ > b
[D] sin 2θ < b
1
sin 2 x the same expression as sin x ?
2
401. Is
Which line segment has a length equivalent to
cosθ ?
[A] OC
[B] AB
[C] OA
[D] CD
394. Two straight roads intersect at an angle whose
measure is 125°. Which expression is
equivalent to the cosine of this angle?
[A] cos 35°
[B] -cos 55°
[C] cos 55°
[D] -cos 35°
4
5
[B]
4
5
[C]
3
5
[D] −
3
5
396. If sin θ > 0 and sec θ < 0, in which quadrant
does the terminal side of angle θ lie?
[A] IV
[B] III
[C] II
[D] I
397. If the tangent of an angle is negative and its
secant is positive, in which quadrant does the
angle terminate?
[A] I
[B] IV
[C] II
402. What is the number of degrees in an angle
7π
?
12
403. Through how many radians does the minute
hand of a clock turn in 24 minutes?
395. If θ is an angle in standard position and
P( −3,4) is a point on the terminal side of θ ,
what is the value of sin θ ?
[A] −
[D] III
[A] 0.6π
[B] 0.2π
[C] 0.4π
[D] 0.8π
404. What is the radian measure of the angle
formed by the hands of a clock at 2:00 p.m.?
[A]
π
4
[B]
π
6
[C]
π
2
[D]
π
3
405. 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?
Math B Regents Exam Questions by Topic
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406. A wedge-shaped piece is cut from a circular
pizza. The radius of the pizza is 6 inches.
The rounded edge of the crust of the piece
measures 4.2 inches. To the nearest tenth, the
angle of the pointed end of the piece of pizza,
[A] 1.4
[B] 0.7
[C] 7.0
[D] 25.2
408. Kristine is riding in car 4 of the Ferris wheel
represented in the accompanying diagram.
The Ferris wheel is rotating in the direction
indicated by the arrows. The eight cars are
equally spaced around the circular wheel.
Express, in radians, the measure of the
smallest angle through which she will travel
to reach the bottom of the Ferris wheel.
407. 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.
What is the measure of angle θ between the
garage and the fence, in radians?
[A] 0.36
[B] 3.14
[C] 2.79
[D] 160
409. An arc of a circle that is 6 centimeters in
length intercepts a central angle of 1.5
radians. Find the number of centimeters in
410. The pendulum of a clock swings through an
angle of 2.5 radians as its tip travels through
an arc of 50 centimeters. Find the length of
the pendulum, in centimeters.
TRIGONOMETRIC IDENTITIES
411. The expression
[A] sin x
1 − cos2 x
is equivalent to
sin 2 x
[B] 1
[C] cos x
[D] − 1
Math B Regents Exam Questions by Topic
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412. The expression (1 + cos x)(1 - cos x) is
equivalent to
[A] sec 2 x
[B] 1
[C] csc 2 x
[D] sin 2 x
2 − 2 sin 2 x
413. Express in simplest terms:
cos x
414. If θ is a positive acute angle and sin θ = a ,
which expression represents cosθ in terms of
a?
[A]
1
a
[C]
1− a
[B]
2
415. The expression
[D]
1
1− a2
a
tan θ
is equivalent to
sec θ
[A] cosθ
[B] sin θ
sin θ
[C]
cos2 θ
cos2 θ
[D]
sin θ
416. The expression
[A]
sin θ
cosθ
[C] sin θ
sec θ
is equivalent to
csc θ
[B]
cosθ
sin θ
[D] cosθ
417. The expression
2 cosθ
is equivalent to
sin 2θ
[A] sin θ
[B] sec θ
[C] cot θ
[D] csc θ
418. A crate weighing w pounds sits on a ramp
positioned at an angle of θ with the
horizontal. The forces acting on this crate are
modeled by the equation Mw cosθ = w sin θ ,
where M is the coefficient of friction. What is
an expression for M in terms of θ ?
[A] M = cot θ
[B] M = csc θ
[C] M = sec θ
[D] M = tan θ
DOUBLE ANGLE AND ANGLE SUM AND
DIFFERENCE IDENTITIES
419. If A is a positive acute angle and sin A =
5
,
3
what is cos 2A?
[A]
1
3
[B] −
1
9
[C] −
1
3
[D]
420. If x is an acute angle and sin x =
1
9
12
, then
13
cos2 x equals
[A] −
[C]
119
169
[B] −
25
169
421. If sin θ =
[A] −
1
3
25
169
119
169
[D]
5
, then cos2θ equals
3
[B]
1
3
[C]
1
9
[D] −
1
9
422. If θ is an acute angle such that sin θ =
what is the value of sin 2θ ?
[A]
12
13
[B]
60
169
[C]
10
26
[D]
5
,
13
120
169
Math B Regents Exam Questions by Topic
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423. If θ is a positive acute angle and
3
sin 2θ =
, then (cosθ + sin θ ) 2 equals
2
12
3
, cos y = , and x and y are acute
13
5
angles, the value of cos( x − y ) is
429. If sin x =
[D] 1 +
3
2
424. If x is a positive acute angle and sin x =
1
,
2
[A] 30°
[B] 60°
[C] 1
what is sin 2 x ?
[A] −
1
2
[B]
425. The expression
1
2
[C]
3
2
3
2
[D] −
sin 2θ
is equivalent to
sin 2 θ
[A] 2 cot θ
[B] 2 tan θ
[C] 2 cosθ
[D]
[A] −
33
65
[B]
21
65
[C]
63
65
[D] −
14
65
430. The expression cos 40° cos10°+ sin 40° sin 10°
is equivalent to
[A] sin 30°
[B] cos50°
[C] cos30°
[D] sin50°
SOLVING TRIGONOMETRIC EQUATIONS
431. A solution set of the equation 5 sin θ + 3 = 3
contains all multiples of
[A] 45°
2
sin θ
[B] 90°
[C] 180°
[D] 135°
432. Solve the following equation algebraically for
all values of θ in the interval 0° ≤ θ ≤ 180° .
2 sin θ − 1 = 0
4
, where 0° < x < 90°, find the
5
value of cos (x + 180°).
426. If sin x =
427. If A and B are positive acute angles,
5
4
sin A = , and cos B = , what is the value
13
5
of sin( A + B) ?
[A]
63
65
[B]
33
65
[C]
56
65
[D] −
16
65
4
5
, tan B = , and angles A and B
5
12
are in Quadrant I, what is the value of
sin( A + B) ?
428. If sin A =
[A]
63
65
[B] −
33
65
[C]
33
65
[D] −
63
65
433. An architect is using a computer program to
design the entrance of a railroad tunnel. The
outline of the opening is modeled by the
function f ( x ) = 8 sin x + 2, in the interval
0 ≤ x ≤ π , where x is expressed in radians.
Solve algebraically for all values of x in the
interval 0 ≤ x ≤ π , where the height of the
terms of π .
If the x-axis represents the base of the tunnel,
what is the maximum height of the entrance
of the tunnel?
434. What value of x in the interval 0° ≤ x ≤ 180°
satisfies the equation 3 tan x + 1 = 0 ?
[A] 150°
[B] -30°
[C] 30°
[D] 60°
Math B Regents Exam Questions by Topic
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435. Solve algebraically for all values of θ in the
interval 0° ≤ θ ≤ 360° that satisfy the equation
sin 2 θ
= 1.
1 + cosθ
436. In the interval 0° ≤ A ≤ 360° , solve for all
values of A in the equation
cos 2 A = −3 sin A − 1.
437. Navigators aboard ships and airplanes use
nautical miles to measure distance. The
length of a nautical mile varies with latitude.
The length of a nautical mile, L, in feet, on
the latitude line θ is given by the formula
L = 6,077 − 31cos 2θ .
Find, to the nearest degree, the angle
θ , 0 ≤ θ ≤ 90° , at which the length of a
nautical mile is approximately 6,076 feet.
438. Find, to the nearest degree, all values of θ in
the interval 0° < θ < 360° that satisfy the
equation 3 cos 2θ + sin θ − 1 = 0.
439. If (sec x − 2)(2 sec x − 1) = 0, then x
terminates in
[A] Quadrants I and II, only
[B] Quadrants I and IV, only
442. If sin 6A = cos 9A, then m∠A is equal to
[A] 36
[D] Quadrants I, II, III, and IV
440. Find, to the nearest degree, all values of θ in
the interval 0° ≤ θ ≤ 180° that satisfy the
equation 8 cos2 θ − 2 cosθ − 1 = 0.
[A] 30°
[B] 45°
[C] 60°
[D] 90°
[D] 1
1
2
443. What is the period of the function
y = 5 sin 3x ?
[A] 5
[B] 3
[C]
2π
5
[D]
2π
3
444. What is the period of the graph of the
1
equation y = 2 sin x ?
3
[A]
3π
2
[B] 2π
[C]
2
π
3
[D] 6π
445. A sound wave is modeled by the curve
y = 3 sin 4 x. What is the period of this curve?
[A] 4
[B]
π
2
[C] π
[D] 3
446. A certain radio wave travels in a path
represented by the equation y = 5 sin 2 x.
What is the period of this wave?
[B] π
[C] 2
[D] 5
447. A modulated laser heats a diamond. Its
variable temperature, in degrees Celsius, is
given by f (t ) = T sin at . What is the period
of the curve?
[A]
441. What is a positive value of x for which
1
9 − cos x = ?
3
[C] 54
TRIGONOMETRIC GRAPHS
[A] 2π
[B] 6
2π
a
[B]
1
a
[C] T
[D]
2 aπ
a
448. The brightness of the star MIRA over time is
given by the equation y = 2 sin
π
x + 6,
4
where x represents time and y represents
brightness. What is the period of this
Math B Regents Exam Questions by Topic
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449. 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?
[A] 3
[B] 2π
[C]
π
4
454. The shaded portion of the accompanying map
indicates areas of night, and the unshaded
portion indicates areas of daylight at a
particular moment in time.
[D] π
450. What is the amplitude of the function shown
in the accompanying graph?
Which type of function best represents the
curve that divides the area of night from the
area of daylight?
[A] cosine
[B] logarithmic
[C] tangent
455. Which transformation could be used to make
the graph of the equation y = sin x coincide
with the graph of the equation y = cos x ?
[A] 2
[B] 1.5
[C] 12
[D] 6
451. What is the amplitude of the function
2
y = sin 4 x ?
3
[A] 4
[B] 3π
[C]
π
2
[D]
2
3
452. A monitor displays the graph y = 3 sin 5x.
What will be the amplitude after a dilation of
2?
[A] 6
[B] 5
[C] 7
[D] 10
453. 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?
[A] 27
[B] 57
[C] 13
[D] 30
[A] dilation
[B] rotation
[C] translation
[D] point reflection
456. The graphs below show the average annual
precipitation received at different latitudes on
Earth. Which graph is a translated cosine
curve?
[A]
[B]
[C]
[D]
Math B Regents Exam Questions by Topic
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457. Which type of symmetry does the equation
y = cos x have?
[A] line symmetry with respect to the x-axis
the top of a 50-foot tower. The wave is
represented by the accompanying graph.
[B] point symmetry with respect to the origin
π
[C] point symmetry with respect to ( ,0)
2
[D] line symmetry with respect to y = x
458. In physics class, Eva noticed the pattern
shown in the accompanying diagram on an
oscilloscope.
What is the equation of this radio wave?
[A] y = sin x
[B] y = 2 sin x
[C] y = 15
. sin x
[D] y = sin 15
. x
460. The accompanying diagram shows a section
of a sound wave as displayed on an
oscilloscope.
Which equation best represents the pattern
shown on this oscilloscope?
1
[A] y = 2 sin( − x ) + 1
2
[B] y = sin x + 1
[D] y = 2 sin x + 1
1
[C] y = sin( x ) + 1
2
Which equation could represent this graph?
[A] y =
1
cos 2 x
2
[C] y = 2 cos
x
2
[B] y =
1
π
sin x
2
2
[D] y = 2 sin
x
2
Math B Regents Exam Questions by Topic
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461. A student attaches one end of a rope to a wall
at a fixed point 3 feet above the ground, as
shown in the accompanying diagram, and
moves the other end of the rope up and down,
producing a wave described by the equation
y = a sin bx + c. The range of the rope's
height above the ground is between 1 and 5
feet. The period of the wave is 4π . Write
the equation that represents this wave.
462. The times of average monthly sunrise, as
shown in the accompanying diagram, over the
course of a 12-month interval can be modeled
by the equation y = A cos( Bx ) + D.
Determine the values of A, B, and D, and
explain how you arrived at your values.
TRIANGLES
PYTHAGORAS
463. The accompanying diagram shows a
semicircular arch over a street that has a
radius of 14 feet. A banner is attached to the
arch at points A and B, such that AE = EB = 5
feet. How many feet above the ground are
these points of attachment for the banner?
PERIMETER AND AREA OF TRIANGLES
464. If the perimeter of an equilateral triangle is
18, the length of the altitude of this triangle is
[A] 3
[B] 3 3
[C] 6
[D] 6 3
465. A garden in the shape of an equilateral
triangle has sides whose lengths are 10
meters. What is the area of the garden?
[A] 50 3 m2
[B] 50 m2
[C] 25 3 m2
[D] 25 m2
Math B Regents Exam Questions by Topic
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466. The accompanying diagram shows two cables
of equal length supporting a pole. Both cables
are 14 meters long, and they are anchored to
points in the ground that are 14 meters apart.
USING TRIGONOMETRY TO FIND AREA
469. The accompanying diagram shows the floor
plan for a kitchen. The owners plan to carpet
all of the kitchen except the "work space,"
which is represented by scalene triangle ABC.
Find the area of this work space to the nearest
tenth of a square foot.
What is the exact height of the pole, in
meters?
[B] 14
[A] 7 2
[C] 7 3
[D] 7
TRIANGLE INEQUALITIES
467. A box contains one 2-inch rod, one 3-inch
rod, one 4-inch rod, and one 5-inch rod.
What is the maximum number of different
triangles that can be made using these rods as
sides?
[A] 3
[B] 1
[C] 2
[D] 4
BASIC TRIGONOMETRIC RATIOS
468. At Mogul's Ski Resort, the beginner's slope is
inclined at an angle of 12.3°, while the
advanced slope is inclined at an angle of
26.4°. If Rudy skis 1,000 meters down the
advanced slope while Valerie skis the same
distance on the beginner's slope, how much
longer was the horizontal distance that
Valerie covered?
[A] 81.3 m
[B] 977.0 m
[C] 895.7 m
[D] 231.6 m
470. Two sides of a triangular-shaped pool
measure 16 feet and 21 feet, and the included
angle measures 58°. What is the area, to the
nearest tenth of a square foot, of a nylon
cover that would exactly cover the surface of
the pool?
471. The triangular top of a table has two sides of
14 inches and 16 inches, and the angle
between the sides is 30°. Find the area of the
tabletop, in square inches.
472. A landscape architect is designing a triangular
garden to fit in the corner of a lot. The corner
of the lot forms an angle of 70°, and the sides
of the garden including this angle are to be 11
feet and 13 feet, respectively. Find, to the
nearest integer, the number of square feet in
the area of the garden.
Math B Regents Exam Questions by Topic
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473. In ΔABC , AC = 18, BC = 10, and
1
cos C = . Find the area of ΔABC to the
2
nearest tenth of a square unit.
474. The accompanying diagram shows a
triangular plot of land that is part of Fran's
garden. She needs to change the dimensions
of this part of the garden, but she wants the
area to stay the same. She increases the
length of side AC to 22.5 feet. If angle A
remains the same, by how many feet should
side AB be decreased to make the area of the
new triangular plot of land the same as the
current one?
LAW OF COSINES
476. Two straight roads, Elm Street and Pine
Street, intersect creating a 40° angle, as
shown in the accompanying diagram. John's
house (J) is on Elm Street and is 3.2 miles
from the point of intersection. Mary's house
(M) is on Pine Street and is 5.6 miles from the
intersection. Find, to the nearest tenth of a
mile, the direct distance between the two
houses.
477. A ship at sea is 70 miles from one radio
transmitter and 130 miles from another. The
angle between the signals sent to the ship by
the transmitters is 117.4°. Find the distance
between the two transmitters, to the nearest
mile.
475. Gregory wants to build a garden in the shape
of an isosceles triangle with one of the
congruent sides equal to 12 yards. If the area
of his garden will be 55 square yards, find, to
the nearest tenth of a degree, the three angles
of the triangle.
478. The Vietnam Veterans Memorial in
Washington, D.C., is made up of two walls,
each 246.75 feet long, that meet at an angle of
125.2°. Find, to the nearest foot, the distance
between the ends of the walls that do not
meet.
479. To measure the distance through a mountain
for a proposed tunnel, surveyors chose points
A and B at each end of the proposed tunnel
and a point C near the mountain. They
determined that AC = 3,800 meters, BC =
2,900 meters, and m∠ACB = 110. Draw a
diagram to illustrate this situation and find the
length of the tunnel, to the nearest meter.
Math B Regents Exam Questions by Topic
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480. A wooden frame is to be constructed in the
form of an isosceles trapezoid, with diagonals
acting as braces to strengthen the frame. The
sides of the frame each measure 5.30 feet, and
the longer base measures 12.70 feet. If the
angles between the sides and the longer base
each measure 68.4°, find the length of one
brace to the nearest tenth of a foot.
481. Kieran is traveling from city A to city B. As
the accompanying map indicates, Kieran
could drive directly from A to B along County
Route 21 at an average speed of 55 miles per
hour or travel on the interstates, 45 miles
along I-85 and 20 miles along I-64. The two
interstates intersect at an angle of 150° at C
and have a speed limit of 65 miles per hour.
How much time will Kieran save by traveling
along the interstates at an average speed of 65
miles per hour?
482. A surveyor is mapping a triangular plot of
land. He measures two of the sides and the
angle formed by these two sides and finds that
the lengths are 400 yards and 200 yards and
the included angle is 50°.
What is the measure of the third side of the
plot of land, to the nearest yard?
What is the area of this plot of land, to the
nearest square yard?
483. A triangular plot of land has sides that
measure 5 meters, 7 meters, and 10 meters.
What is the area of this plot of land, to the
nearest tenth of a square meter?
484. A farmer has determined that a crop of
strawberries yields a yearly profit of \$1.50 per
square yard. If strawberries are planted on a
triangular piece of land whose sides are 50
yards, 75 yards, and 100 yards, how much
profit, to the nearest hundred dollars, would
the farmer expect to make from this piece of
land during the next harvest?
LAW OF SINES
485. The accompanying diagram shows the
approximate linear distances traveled by a
sailboat during a race. The sailboat started at
point S, traveled to points E and A,
respectively, and ended at point S.
Based on the measures shown in the diagram,
which equation can be used to find x, the
distance from point A to point S?
[A]
sin 65° sin 75°
=
x
32
[B]
sin 75°
x
=
sin 65°
32
[C]
65 32
=
x 75
[D]
x 32
=
65 75
Math B Regents Exam Questions by Topic
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486. In ΔABC , a = 19, c = 10, and m∠A = 111.
Which statement can be used to find the value
of ∠C ?
[A] sin C =
19 sin 69°
10
[B] sin C =
10 sin 21°
19
[C] sin C =
10
19
[D] sin C =
10 sin 69°
19
487. In ΔABC , m∠A = 53, m∠B = 14, and
a = 10. Find b to the nearest integer.
488. In ΔABC , m∠A = 33, a = 12, and b = 15.
What is m∠B to the nearest degree?
[A] 41
[B] 43
[C] 48
[D] 44
489. A ski lift begins at ground level 0.75 mile
from the base of a mountain whose face has a
50° angle of elevation, as shown in the
accompanying diagram. The ski lift ascends
in a straight line at an angle of 20°. Find the
length of the ski lift from the beginning of the
ski lift to the top of the mountain, to the
nearest hundredth of a mile.
490. In the accompanying diagram of ΔABC ,
m∠A = 30, m∠C = 50, and AC = 13.
What is the length of side AB to the nearest
tenth?
[A] 11.5
[B] 10.1
[C] 12.0
[D] 6.6
491. As shown in the accompanying diagram, two
tracking stations, A and B, are on an east-west
line 110 miles apart. A forest fire is located
at F, on a bearing 42° northeast of station A
and 15° northeast of station B. How far, to
the nearest mile, is the fire from station A?
492. In the accompanying diagram of ΔABC ,
m∠A = 65, m∠B = 70, and the side opposite
vertex B is 7. Find the length of the side
opposite vertex A, and find the area of ΔABC.
Math B Regents Exam Questions by Topic
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493. Carmen and Jamal are standing 5,280 feet
apart on a straight, horizontal road. They
observe a hot-air balloon between them
directly above the road. The angle of
elevation from Carmen is 60° and from Jamal
is 75°. Draw a diagram to illustrate this
situation and find the height of the balloon to
the nearest foot.
494. A ship at sea heads directly toward a cliff on
the shoreline. The accompanying diagram
shows the top of the cliff, D, sighted from two
locations, A and B, separated by distance S. If
m∠DAC = 30, m∠DBC = 45, and S = 30
feet, what is the height of the cliff, to the
nearest foot?
495. The accompanying diagram shows the plans
for a cell-phone tower that is to be built near a
busy highway. Find the height of the tower,
to the nearest foot.
497. While sailing a boat offshore, Donna sees a
lighthouse and calculates that the angle of
elevation to the top of the lighthouse is 3°, as
shown in the accompanying diagram. When
she sails her boat 700 feet closer to the
lighthouse, she finds that the angle of
elevation is now 5°. How tall, to the nearest
tenth of a foot, is the lighthouse?
498. A sign 46 feet high is placed on top of an
office building. From a point on the sidewalk
level with the base of the building, the angle
of elevation to the top of the sign and the
angle of elevation to the bottom of the sign
are 40° and 32°, respectively. Sketch a
diagram to represent the building, the sign,
and the two angles, and find the height of the
building to the nearest foot.
USING TRIGONOMETRY TO SOLVE
TRIANGLE INEQUALITIES
499. How many distinct triangles can be formed if
m∠A = 30, side b = 12, and side a = 8?
[A] 3
[B] 2
[C] 0
[D] 1
500. What is the total number of distinct triangles
that can be constructed if AC = 13, BC = 8,
and m∠A = 36?
[A] 2
496. A ship captain at sea uses a sextant to sight an
angle of elevation of 37° to the top of a
lighthouse. After the ship travels 250 feet
directly toward the lighthouse, another
sighting is made, and the new angle of
elevation is 50°. The ship's charts show that
there are dangerous rocks 100 feet from the
base of the lighthouse. Find, to the nearest
foot, how close to the rocks the ship is at the
time of the second sighting.
[B] 3
[C] 0
[D] 1
501. An architect commissions a contractor to
produce a triangular window. The architect
describes the window as ΔABC , where
m∠A = 50, BC = 10 inches, and
AB = 12 inches. How many distinct triangles
can the contractor construct using these
dimensions?
[A] more than 2
[B] 1
[C] 2
[D] 0
Math B Regents Exam Questions by Topic
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502. Sam is designing a triangular piece for a
metal sculpture. He tells Martha that two of
the sides of the piece are 40 inches and 15
inches, and the angle opposite the 40-inch
side measures 120°. Martha decides to sketch
the piece that Sam described. How many
different triangles can she sketch that match
Sam's description?
[A] 1
[B] 0
[C] 3
[D] 2
503. Sam needs to cut a triangle out of a sheet of
paper. The only requirements that Sam must
follow are that one of the angles must be 60°,
the side opposite the 60° angle must be 40
centimeters, and one of the other sides must
be 15 centimeters. How many different
triangles can Sam make?
[A] 1
[B] 3
[C] 2
506. In ΔABC , if AC = 12, BC = 11, and
m∠A = 30, angle C could be
[A] an acute angle, only
[B] a right angle, only
[C] an obtuse angle, only
[D] either an obtuse angle or an acute angle
507. In ΔABC , m∠A = 30, a = 14, and b = 20.
Which type of angle is ∠B ?
[A] It must be a right angle.
[B] It must be an acute angle.
[C] It must be an obtuse angle.
[D] It may be either an acute angle or an
obtuse angle.
[D] 0
VECTORS
504. A landscape designer is designing a triangular
garden with two sides that are 4 feet and 6
feet, respectively. The angle opposite the 4foot side is 30°. How many distinct triangular
gardens can the designer make using these
measurements?
505. Main Street and Central Avenue intersect,
making an angle measuring 34°. Angela lives
at the intersection of the two roads, and
Caitlin lives on Central Avenue 10 miles from
the intersection. If Leticia lives 7 miles from
Caitlin, which conclusion is valid?
[A] Leticia can live at one of two locations on
Main Street.
[B] Leticia cannot live on Main Street.
[C] Leticia can live at only one location on
Main Street.
[D] Leticia can live at one of three locations
on Main Street.
508. Two tow trucks try to pull a car out of a ditch.
One tow truck applies a force of 1,500 pounds
while the other truck applies a force of 2,000
pounds. The resultant force is 3,000 pounds.
Find the angle between the two applied
forces, rounded to the nearest degree.
509. One force of 20 pounds and one force of 15
pounds act on a body at the same point so that
the resultant force is 19 pounds. Find, to the
nearest degree, the angle between the two
original forces.
510. Two equal forces act on a body at an angle of
80°. If the resultant force is 100 newtons,
find the value of one of the two equal forces,
to the nearest hundredth of a newton.
Math B Regents Exam Questions by Topic
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511. Two forces of 40 pounds and 20 pounds,
respectively, act simultaneously on an object.
The angle between the two forces is 40°.
Find the magnitude of the resultant, to the
nearest tenth of a pound.
Find the measure of the angle, to the nearest
degree, between the resultant and the larger
force.
515. 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.
OTHER POLYGONS
PERIMETER AND AREA OF OTHER
POLYGONS
512. 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.
513. Chad had a garden that was in the shape of a
rectangle. Its length was twice its width. He
decided to make a new garden that was 2 feet
longer and 2 feet wider than his first garden.
If x represents the original width of the
garden, which expression represents the
difference between the area of his new garden
and the area of the original garden?
[A] x 2 + 3x + 2
[B] 6 x + 4
[C] 8
[D] 2 x 2
514. A small, open-top packing box, similar to a
shoebox without a lid, is three times as long
as it is wide, and half as high as it is long.
Each square inch of the bottom of the box
costs \$0.008 to produce, while each square
inch of any side costs \$0.003 to produce.
Write a function for the cost of the box
described above.
Using this function, determine the dimensions
of a box that would cost \$0.69 to produce.
CONICS
CIRCUMFERENCE AND AREA
516. Every time the pedals go through a 360°
rotation on a certain bicycle, the tires rotate
three times. If the tires are 24 inches in
diameter, what is the minimum number of
complete rotations of the pedals needed for
the bicycle to travel at least 1 mile?
[A] 12
[B] 5,280
[C] 561
[D] 281
517. Ileana buys a large circular pizza that is
divided into eight equal slices. She measures
along the outer edge of the crust from one
1
piece and finds it to be 5 inches. What is
2
the diameter of the pizza to the nearest inch?
[A] 7
[B] 8
[C] 4
[D] 14
Math B Regents Exam Questions by Topic
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518. A ball is rolling in a circular path that has a
radius of 10 inches, as shown in the
accompanying diagram. What distance has
the ball rolled when the subtended arc is 54°?
of an inch.
519. Cities H and K are located on the same line of
longitude and the difference in the latitude of
these cities is 9°, as shown in the
accompanying diagram. If Earth's radius is
3,954 miles, how many miles north of city K
to the nearest tenth of a mile.
520. The accompanying diagram shows the path of
a cart traveling on a circular track of radius
2.40 meters. The cart starts at point A and
stops at point B, moving in a
counterclockwise direction. What is the
length of minor arc AB, over which the cart
traveled, to the nearest tenth of a meter?
521. Kathy and Tami are at point A on a circular
track that has a radius of 150 feet, as shown in
the accompanying diagram. They run
counterclockwise along the track from A to S,
a distance of 247 feet. Find, to the nearest
degree, the measure of minor arc AS.
522. As shown in the accompanying diagram, a
dial in the shape of a semicircle has a radius
of 4 centimeters. Find the measure of θ , in
radians, when the pointer rotates to form an
arc whose length is 1.38 centimeters.
Math B Regents Exam Questions by Topic
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523. 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?
[A] 25%
[B] 10%
[C] 31%
526. Which equation represents the circle shown in
the accompanying graph?
[D] 21%
EQUATIONS OF CIRCLES
524. The center of a circular sunflower with a
diameter of 4 centimeters is (-2,1). Which
equation represents the sunflower?
[A] ( x − 2) 2 + ( y + 1) 2 = 2
[B] ( x + 2) 2 + ( y − 1) 2 = 4
[A] ( x − 1) 2 + ( y + 2) 2 = 9
[C] ( x − 2) 2 + ( y − 1) 2 = 4
[B] ( x − 1) 2 − ( y + 2) 2 = 9
[D] ( x + 2) 2 + ( y − 1) 2 = 2
[C] ( x + 1) 2 − ( y − 2) 2 = 9
525. What is the equation of a circle with center
[A] ( x + 3) 2 + ( y − 1) 2 = 49
[B] ( x − 3) 2 + ( y + 1) 2 = 7
[C] ( x + 3) 2 + ( y − 1) 2 = 7
[D] ( x − 3) 2 + ( y + 1) 2 = 49
[D] ( x + 1) 2 + ( y − 2) 2 = 9
527. For a carnival game, John is painting two
circles, V and M, on a square dartboard.
a On the accompanying grid, draw and label
circle V, represented by the equation
x 2 + y 2 = 25 , and circle M, represented by the
equation ( x − 8) 2 + ( y + 6) 2 = 4 .
b A point, (x,y), is randomly selected such
that − 10 ≤ x ≤ 10 and − 10 ≤ y ≤ 10 . What is
the probability that point (x,y) lies outside
both circle V and circle M?
Math B Regents Exam Questions by Topic
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528. A circle has the equation
( x + 1) 2 + ( y − 3) 2 = 16. What are the
coordinates of its center and the length of its
[A] (-1,3) and 16
[B] (1,-3) and 4
[C] (-1,3) and 4
[D] (1,-3) and 16
532. The accompanying diagram shows the
elliptical orbit of a planet. The foci of the
elliptical orbit are F1 and F2 .
529. What are the coordinates of the center of the
circle represented by the equation
( x + 3) 2 + ( y − 4) 2 = 25 ?
[A] (3,4)
[B] (-3,4)
[C] (-3,-4)
[D] (3,-4)
530. The center of a circle represented by the
equation ( x − 2) 2 + ( y + 3) 2 = 100 is located
[A] IV
[B] III
[C] I
[D] II
EQUATIONS OF ELLIPSES
531. An object orbiting a planet travels in a path
represented by the equation
3( y + 1) 2 + 5( x + 4) 2 = 15. In which type of
pattern does the object travel?
If a, b, and c are all positive and a ≠ b ≠ c,
which equation could represent the path of the
planet?
[A] ax 2 − by 2 = c 2
[B] y = ax 2 + c 2
[C] x 2 + y 2 = c 2
[D] ax 2 + by 2 = c 2
533. Which equation, when graphed on a Cartesian
coordinate plane, would best represent an
elliptical racetrack?
[A] 3x 2 − 10 y 2 = 288,000
[B] 30 xy = 288,000
[A] circle
[B] parabola
[C] 3x 2 + 10 y 2 = 288,000
[C] ellipse
[D] hyperbola
[D] 3x + 10 y = 288,000
534. A designer who is planning to install an
elliptical mirror is laying out the design on a
coordinate grid. Which equation could
represent the elliptical mirror?
[A] x 2 + 4 y 2 = 144
[B] y = 4 y 2 + 144
[C] x 2 + y 2 = 144
[D] x 2 = 144 + 36 y 2
Math B Regents Exam Questions by Topic
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535. The accompanying diagram represents the
elliptical path of a ride at an amusement park.
537. An architect is designing a building to include
an arch in the shape of a semi-ellipse (half an
ellipse), such that the width of the arch is 20
feet and the height of the arch is 8 feet, as
shown in the accompanying diagram.
Which equation represents this path?
[A] x 2 + y 2 = 300
[C]
[B]
x2
y2
−
=1
1502 502
x2
y2
+
=1
1502 502
[D] y = x 2 + 100 x + 300
536. A commercial artist plans to include an
ellipse in a design and wants the length of the
horizontal axis to equal 10 and the length of
the vertical axis to equal 6. Which equation
could represent this ellipse?
[A] 3 y = 20 x 2
[C] 9 x 2 + 25 y 2 = 225
[D] 9 x 2 − 25 y 2 = 225
[B] x 2 + y 2 = 100
Which equation models this arch?
x2
y2
[A]
+
=1
400 64
[C]
x2
y2
+
=1
64 400
x2 y2
[B]
+
=1
100 64
[D]
x2 y2
+
=1
64 100
Math B Regents Exam Questions by Topic
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538. The accompanying diagram shows the
construction of a model of an elliptical orbit
of a planet traveling around a star. Point P
and the center of the star represent the foci of
the orbit.
Which equation could represent the relation
shown?
2
2
2
CHORDS SECANTS AND TANGENTS
539. Kimi wants to determine the radius of a
circular pool without getting wet. She is
located at point K, which is 4 feet from the
pool and 12 feet from the point of tangency,
as shown in the accompanying diagram.
What is the radius of the pool?
[A] 20 ft
[B] 4 10 ft
[C] 32 ft
[D] 16 ft
2
[A]
x
y
+
=1
225 81
[B]
x
y
+
=1
15 9
[C]
x2 y2
+
=1
81 225
[D]
x2 y2
−
=1
15 9
540. The accompanying diagram represents
circular pond O with docks located at points A
and B. From a cabin located at C, two
sightings are taken that determine an angle of
JJJG
JJJG
30° for tangents CA and CB.
What is m∠CAB ?
[A] 30
[B] 60
[C] 75
[D] 150
Math B Regents Exam Questions by Topic
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541. A small fragment of something brittle, such as
pottery, is called a shard. The accompanying
diagram represents the outline of a shard from
a small round plate that was found at an
archaeological dig.
543. An overhead view of a revolving door is
shown in the accompanying diagram. Each
panel is 1.5 meters wide.
JJJG
If BC is a tangent to p
AC at B and
m∠ABC = 45, what is the measure of p
AC ,
the outside edge of the shard?
[A] 1.50 m
[B] 1.73 m
[A] 225°
[C] 3.00 m
[D] 2.12 m
[B] 90°
[C] 135°
[D] 45°
542. The accompanying diagram shows a child's
spin toy that is constructed from two chords
intersecting in a circle. The curved edge of
the larger shaded section is one-quarter of the
circumference of the circle, and the curved
edge of the smaller shaded section is one-fifth
of the circumference of the circle.
What is the measure of angle x?
[A] 108°
[B] 72°
[C] 40°
[D] 81°
What is the approximate width of d, the
opening from B to C?
544. The accompanying diagram shows a
revolving door with three panels, each of
which is 4 feet long. What is the width, w, of
the opening between x and y, to the nearest
tenth of a foot?
Math B Regents Exam Questions by Topic
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545. A toy truck is located within a circular play
area. Alex and Dominic are sitting on
opposite endpoints of a chord that contains
the truck. Alex is 4 feet from the truck, and
Dominic is 3 feet from the truck. Meira and
Tamara are sitting on opposite endpoints of
another chord containing the truck. Meira is 8
feet from the truck. How many feet, to the
nearest tenth of a foot, is Tamara from the
truck? Draw a diagram to support your
547. In the accompanying diagram of circle O,
chord AY is parallel to diameter DOE , AD
is drawn, and m p
546. In the accompanying diagram, the length of
3π
AB C is
2
What is m∠DAY ?
[A] 110
[B] 90
[C] 150
[D] 130
548. The new corporate logo created by the design
engineers at Magic Motors is shown in the
accompanying diagram.
What is m∠ABC ?
[A] 45
[B] 36
[C] 53
[D] 72
If chords BA and BC are congruent and
mBC = 140, what is m∠B ?
[A] 40
[B] 140
[C] 80
[D] 280
Math B Regents Exam Questions by Topic
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549. A machine part consists of a circular wheel
with an inscribed triangular plate, as shown in
the accompanying diagram. If SE ≅ EA,
p = 140, find the length of
SE = 10, and mSE
SA to the nearest tenth.
552. Point P lies outside circle O, which has a
diameter of AOC . The angle formed by
tangent PA and secant PBC measures 30°.
Sketch the conditions given above and find
the number of degrees in the measure of
minor arc CB.
553. In the accompanying diagram, cabins B and G
are located on the shore of a circular lake, and
cabin L is located near the lake. Point D is a
dock on the lake shore and is collinear with
cabins B and L. The road between cabins G
and L is 8 miles long and is tangent to the
lake. The path between cabin L and dock D is
4 miles long.
550. A regular hexagon is inscribed in a circle.
What is the ratio of the length of a side of the
hexagon to the minor arc that it intercepts?
[A]
6
π
[B]
π
6
[C]
3
6
[D]
3
π
551. In the accompanying diagram of circle O,
diameter AOB is extended through B to
external point P, tangent PC is drawn to
p = 7 : 2.
point C on the circle, and m p
AC : mBC
Find m∠CPA.
What is the length, in miles, of BD ?
[A] 24
[B] 12
[C] 8
[D] 4
Math B Regents Exam Questions by Topic
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554. The accompanying diagram shows a circular
machine part that has rods PT and PAR
attached at points T, A, and R, which are
located on the circle;
o : m AR
p : m RT
p = 1: 3 : 5; RA = 12
mTA
centimeters; and PA = 5 centimeters.
556. An architect is designing a park with an
entrance represented by point C and a circular
garden with center O, as shown in the
accompanying diagram. The architect plans
to connect three points on the circumference
of the garden, A, B, and D, to the park
entrance, C, with walkways so that walkways
CA and CB are tangent to the garden,
walkway DOEC is a path through the center
of the garden, m q
AEB = 3 : 2, BC = 60
meters, and EC = 43.6 meters.
Find the measure of the angle between
walkways CA and CB.
Find the diameter of the circular garden, to
the nearest meter.
Find the measure of ∠P, in degrees, and find
the length of rod PT, to the nearest tenth of a
centimeter.
555. In the accompanying diagram, PA is tangent
to circle O at A, secant PBC is drawn,
PB = 4, and BC = 12. Find PA.
557. Given circle O with diameter GOAL; secants
HUG and HTAM intersect at point H;
p : mML
p : m LT
o = 7 : 3 : 2; and chord GU ≅
mGM
chord UT . Find the ratio of m∠UGL to
m∠ H .
Math B Regents Exam Questions by Topic
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558. In the accompanying diagram, circle O has
radius OD, diameter BOHF, secant CBA,
and chords DHG and BD; CE is tangent to
p = 80; and
circle O at D; mDF
p : mp
p = 3 : 2 :1.
mBA
AG : mGF
p , and m∠BHD, m∠BDG ,
Find mGF
m∠GDE , m∠C , and m∠BOD.
SOLIDS AND
SIMILARITY
SIMILARITY
559. The accompanying diagram shows a 24-foot
ladder leaning against a building. A steel
brace extends from the ladder to the point
where the building meets the ground. The
brace forms a right angle with the ladder.
If the steel brace is connected to the ladder at
a point that is 10 feet from the foot of the
ladder, which equation can be used to find the
length, x, of the steel brace?
[A]
10 x
=
x 24
[C] 102 + x 2 = 24 2
[B] 102 + x 2 = 14 2
[D]
10 x
=
x 14
VOLUME
560. A rectangular piece of cardboard is to be
formed into an uncovered box. The piece of
cardboard is 2 centimeters longer than it is
wide. A square that measures 3 centimeters
on a side is cut from each corner. When the
sides are turned up to form the box, its
volume is 765 cubic centimeters. Find the
dimensions, in centimeters, of the original
piece of cardboard.
Math B Regents Exam Questions by Topic
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TRANSFORMATIONS
DILATIONS
564. Which transformation represents a dilation?
IDENTIFYING TRANSFORMATIONS
561. Which transformation of the graph of y = x 2
[A] (8,4) → (4,2)
[B] (8,4) → ( −4,−8)
[C] (8,4) → (11,7)
[D] (8,4) → ( −8,4)
would result in the graph of y = x + 2 ?
2
[A] ry = 2
[B] D2
[C] T0, 2
[D] R0 ,90
TRANSLATIONS
565. In which quadrant would the image of point
(5, − 3) fall after a dilation using a factor of
− 3?
[A] III
562. The image of the origin under a certain
translation is the point (2,-6). The image of
point ( −3,−2) under the same translation is
the point
[A] (-1,-8)
[B] (-6,12)
3 1
[C] ( − , )
2 3
[D] (-5,4)
563. Two parabolic arches are to be built. The
equation of the first arch can be expressed as
y = − x 2 + 9, with a range of 0 ≤ y ≤ 9, and
the second arch is created by the
transformation T7 ,0 . On the accompanying
set of axes, graph the equations of the two
arches. Graph the line of symmetry formed
by the parabola and its transformation and
label it with the proper equation.
[B] IV
[C] I
[D] II
566. The graph of the function g(x) is shown on the
accompanying set of axes. On the same set of
axes, sketch the image of g(x) under the
transformation D2 .
Math B Regents Exam Questions by Topic
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567. In the accompanying graph, the shaded region
represents set A of all points (x,y) such that
x 2 + y 2 ≤ 1. The transformation T maps point
(x, y) to point (2x, 4y).
Which graph shows the mapping of set A by
the transformation T?
[A]
REFLECTIONS
568. What are the coordinates of point P, the
image of point (3,-4) after a reflection in the
line y = x?
[A] (-4,3)
[B] (4,-3)
[C] (-3,4)
[D] (3,4)
569. Which transformation best describes the
relationship between the functions f ( x ) = 2 x
1
and g( x ) = ( ) x ?
2
[A] reflection in the origin
[B] reflection in the line y = x
[C] reflection in the x-axis
[D] reflection in the y-axis
[B]
570. In the accompanying diagram of square
ABCD, F is the midpoint of AB, G is the
midpoint of BC, H is the midpoint of CD,
and E is the midpoint of DA.
[C]
[D]
Find the image of ΔEOA after it is reflected
in line A.
Is this isometry direct or opposite? Explain
Math B Regents Exam Questions by Topic
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571. The graph below represents f(x).
577. Which transformation does not preserve
orientation?
[A] dilation
[B] reflection in the y-axis
[C] translation
[D] rotation
Which graph best represents f(-x)?
ROTATIONS
[B]
[A]
578. Point P ′ is the image of point P(-3,4) after a
translation defined by T( 7 , −1) . Which other
[C]
transformation on P would also produce P ′ ?
[D]
[A] R− 90°
[B] ry =− x
[C] ry − axis
[D] R90°
ISOMETRIES
572. Which transformation is not an isometry?
[A] T3,6
[B] ry = x
[C] R0,90°
[D] D2
573. Which transformation is not an isometry?
[A] line reflection
[B] rotation
[C] dilation
[D] translation
574. Which transformation is a direct isometry?
[A] D− 2
[B] T2 ,5
[C] D2
[D] ry − axis
575. Which transformation is an opposite
isometry?
[A] line reflection
[B] rotation of 90°
[C] translation
[D] dilation
COMPOSITIONS OF TRANSFORMATIONS
579. If the coordinates of point A are (-2,3), what
is the image of A under ry − axis D D3 ?
[A] (-6,-9)
[B] (6,9)
[C] (9,-6)
[D] (5,6)
580. What is the image of point (1,1) under
rx − axis D R0,90° ?
[A] (-1,1)
[B] (1,-1)
[C] (-1,-1)
[D] (1,1)
581. What are the coordinates of point A', the
image of point A(-4,l) after the composite
transformation R90° D ry = x where the origin is
the center of rotation?
576. Which transformation is an example of an
opposite isometry?
[A] (x,y) → (x + 3,y - 6)
[B] (x,y) → (3x,3y)
[C] (x,y) → (y,x)
[D] (x,y) → (y,-x)
[A] (-1,-4)
[B] (1,4)
[C] (4,1)
[D] (-4,-1)
Math B Regents Exam Questions by Topic
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582. If f(x) = cos x, which graph represents f(x)
under the composition ry − axis D rx − axis ?
[A]
[B]
[C]
[D]
584. The accompanying graph represents the
figure .
Which graph represents after a
transformation defined by ry = x D R90° ?
583. The graph of f(x) is shown in the
accompanying diagram.
[A]
[B]
[C]
[D]
Which graph represents f ( x ) rx − axis Dry − axis ?
[A]
[B]
[C]
[D]
585. a On the accompanying grid, graph the
equation 2 y = 2 x 2 − 4 in the interval
− 3 ≤ x ≤ 3 and label it a.
b On the same grid, sketch the image of a
under T5, −2 D rx − axis and label it b.
Math B Regents Exam Questions by Topic
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586. Graph and label the following equations, a
and b, on the accompanying set of coordinate
axes.
a: y = x 2
588. Given point A(-2,3). State the coordinates of
the image of A under the composition
T− 3, −4 D rx − axis . [The use of the accompanying
grid is optional.]
b: y = − ( x − 4) + 3
Describe the composition of transformations
performed on a to get b.
2
LOGIC
PROOFS
587. On the accompanying grid, graph and label
AB, where A is (0,5) and B is (2,0). Under
the transformation rx − axis D ry − axis ( AB), A maps
to A′′ and B maps to B ′′. Graph and label
A′′B ′′. What single transformation would
map AB to A′′B ′′ ?
589. Given: A(-2,2), B(6,5), C(4,0), D(-4,-3)
Prove: ABCD is a parallelogram but not a
rectangle. [The use of the grid is optional.]
Math B Regents Exam Questions by Topic
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590. The coordinates of quadrilateral ABCD are
A(-1,-5), B(8,2), C(11,13), and D(2,6). Using
ABCD is a rhombus. [The use of the grid is
optional.]
591. Jim is experimenting with a new drawing
program on his computer. He created
T( −2,3), E( −5,−4), A(2,−1), and M (5,6).
Jim believes that he has created a rhombus
but not a square. Prove that Jim is correct.
[The use of the grid is optional.]
592. Given: A(1,6), B(7,9), C(13,6), and D(3,1)
Prove: ABCD is a trapezoid. [The use of the
accompanying grid is optional.]
593. Quadrilateral KATE has vertices K(1,5),
A(4,7), T(7,3), and E(1,-1).
a Prove that KATE is a trapezoid. [The use of
the grid is optional.]
b Prove that KATE is not an isosceles
trapezoid.
Math B Regents Exam Questions by Topic
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594. The coordinates of quadrilateral JKLM are
J(1,-2), K(13,4), L(6,8), and M(-2,4). Prove
that quadrilateral JKLM is a trapezoid but not
an isosceles trapezoid. [The use of the grid is
optional.]
596. In the accompanying diagram of ΔABC,
1
1
AB ≅ AC , BD = BA, and CE = CA.
3
3
Triangle EBC can be proved congruent to
triangle DCB by
595. In the accompanying diagram of ABCD,
where a ≠ b, prove ABCD is an isosceles
trapezoid.
[A] SSS ≅ SSS
[B] HL ≅ HL
[C] ASA ≅ ASA
[D] SAS ≅ SAS
597. In the accompanying diagram, CA ⊥ AB,
ED ⊥ DF , ED
AB, CE ≅ BF , AB ≅ ED
and m∠CAB = m∠FDE = 90.
Which statement would not be used to prove
ΔABC ≅ ΔDEF ?
[A] SAS ≅ SAS
[B] AAS ≅ AAS
[C] HL ≅ HL
[D] SSS ≅ SSS
Math B Regents Exam Questions by Topic
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598. In the accompanying diagram of
parallelogram ABCD, DE ≅ BF .
601. Which statements could be used to prove that
ΔABC and ΔA′B ′C ′ are congruent?
[A] ∠A ≅ ∠A′, AC ≅ A′C ′, and BC ≅ B ′C ′
[B] AB ≅ A′B ′, ∠A ≅ ∠A′, and ∠C ≅ ∠C′
[C] ∠A ≅ ∠A′, ∠B ≅ ∠B ′, and ∠C ≅ ∠C′
[D] AB ≅ A′B ′, BC ≅ B ′C ′, and ∠A ≅ ∠A′
Triangle EGC can be proved congruent to
triangle FGA by
[A] SSA ≅ SSA
[B] AAA ≅ AAA
[C] AAS ≅ AAS
[D] HL ≅ HL
602. Given: parallelogram FLSH, diagonal
FGAS, LG⊥ FS , HA⊥ FS
599. In the accompanying diagram, HK bisects
IL and ∠H ≅ ∠K.
Prove: ΔLGS ≅ ΔHAF
603. 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
What is the most direct method of proof that
could be used to prove ΔHIJ ≅ ΔKLJ ?
[A] ASA ≅ ASA
[B] SAS ≅ SAS
[C] HL ≅ HL
[D] AAS ≅ AAS
600. Which condition does not prove that two
triangles are congruent?
[A] SAS ≅ SAS
[B] SSA ≅ SSA
[C] SSS ≅ SSS
[D] ASA ≅ ASA
Prove: ΔABE ≅ ΔCAB
Math B Regents Exam Questions by Topic
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604.
606. Given: chords AB and CD of circle O
intersect at E, an interior point of circle O;
chords AD and CB are drawn.
Prove: (AE)(EB) = (CE)(ED)
607. Prove that the diagonals of a parallelogram
bisect each other.
608. In ΔABC, D is a point on AC such that BD
is a median. Which statement must be true?
p = 70,
605. In the accompanying diagram, mBR
p = 70, and BOD is the diameter of circle
mYD
O. Write an explanation or a proof that shows
ΔRBD and ΔYDB are congruent.
[B] ∠ABD ≅ ∠CBD
[C] ΔABD ≅ ΔCBD
[D] BD⊥ AC
609. In the accompanying diagram, ΔABC is not
isosceles. Prove that if altitude BD were
drawn, it would not bisect AC.
Math B Regents Exam Questions by Topic
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610. Given: parallelogram ABCD, diagonal AC,
and ABE
Prove: m∠1 > m∠2
611. Given: ΔABT , CBTD, and AB⊥ CD
Write an indirect proof to show that AT is
not perpendicular to CD.
612. In the accompanying diagram of circle O, PA
is drawn tangent to the circle at A. Place B on
PA anywhere between P and A and draw
OA, OP, and OB. Prove that OB is not
perpendicular to PA.
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