Mathematics Test—No Calculator
20 Questions
Turn to Section 3 of your answer sheet to answer the questions in this section.
Directions
For questions 1 through 15, solve each problem, choose the best answer from the choices
provided, and fill in the corresponding circle on your answer sheet. For questions
16 through 20, solve the problem and enter your answer in the grid on the answer sheet.
You may use any available space in your test booklet for scratch work.
Notes
1. The use of a calculator is not permitted.
2. All variables and expressions used represent real numbers unless otherwise indicated.
3. Figures provided in this test are drawn to scale unless otherwise indicated.
4. All figures lie in a plane unless otherwise indicated.
5. Unless otherwise indicated, the domain of a given function f is the set of all real
numbers x for which
f of x is a real number.
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Reference
Begin skippable figure descriptions.
The figure presents information for your reference in solving some of the problems.
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Reference figure 1 is a circle with radius r. Two equations are presented below reference
figure 1.
A equals pi times the square of r.
C equals 2 pi r.
Reference figure 2 is a rectangle with length
ℓ and width w. An equation is presented
below reference figure 2.
A equals ℓ w.
Reference figure 3 is a triangle with base b and height h. An equation is presented
below reference figure 3.
A equals one-half b h.
Reference figure 4 is a right triangle. The two sides that form the right angle are labeled
a and b, and the side opposite the right angle is labeled c. An equation is presented
below reference figure 4.
c squared equals a squared plus b squared.
Special Right Triangles
Reference figure 5 is a right triangle with a 30-degree angle and a 60-degree angle. The
side opposite the 30-degree angle is labeled x. The side opposite the 60-degree angle is
labeled x times the square root of 3. The side opposite the right angle is labeled 2 x.
Reference figure 6 is a right triangle with two 45-degree angles. Two sides are each
labeled s. The side opposite the right angle is labeled s times the square root of 2.
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Reference figure 7 is a rectangular solid whose base has length
ℓ and width w and
whose height is h. An equation is presented below reference figure 7.
V equals
ℓ wh.
Reference figure 8 is a right circular cylinder whose base has radius r and whose height
is h. An equation is presented below reference figure 8.
V equals pi times the square of r times h.
Reference figure 9 is a sphere with radius r. An equation is presented below reference
figure 9.
V equals four-thirds pi times the cube of r.
Reference figure 10 is a cone whose base has radius r and whose height is h.
An equation is presented below reference figure 10.
V equals one-third times pi times the square of r times h.
Reference figure 11 is an asymmetrical pyramid whose base has length
ℓ and width w
and whose height is h. An equation is presented below reference figure 11.
V equals one-third ℓ w h.
End skippable figure descriptions.
Additional Reference Information
The number of degrees of arc in a circle is 360.
The number of radians of arc in a circle is 2 pi.
The sum of the measures in degrees of the angles of a triangle is 180.
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Question 1.
If
the fraction whose numerator is x minus 1, and whose denominator is 3,
equals k, and
k equals 3, what is the value of x ?
A. 12
B. 14
C. 19
D. 10
Answer and explanation for question 1.
Question 2.
For
i equals the square root of negative 1, what is the sum
parenthesis, 7 plus 3i, close parenthesis, plus, parenthesis, negative
8 plus 9i, close parenthesis ?
A.
B.
C.
D.
negative 1 plus 12i
negative 1 minus 6 i
15 plus 12i
15 minus 6i
Answer and explanation for question 2.
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Question 3.
On Saturday afternoon, Armand sent m text messages each hour for 5 hours, and Tyrone
sent p text messages each hour for 4 hours. Which of the following represents the total
number of messages sent by Armand and Tyrone on Saturday afternoon?
A.
B.
C.
D.
9mp
20mp
5m plus 4p
4m plus 5p
Answer and explanation for question 3.
Question 4.
Kathy is a repair technician for a phone company. Each week, she receives a batch of
phones that need repairs. The number of phones that she has left to fix at the end of each
day can be estimated with the equation
P equals 108 minus 23 d, where
P is the number of phones left and d is the number of days she has worked that week.
What is the meaning of the value 108 in this equation?
A. Kathy will complete the repairs within 108 days.
B. Kathy starts each week with 108 phones to fix.
C. Kathy repairs phones at a rate of 108 per hour.
D. Kathy repairs phones at a rate of 108 per day.
Answer and explanation for question 4.
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Question 5.
parenthesis, x squared, y, minus
3y squared, plus 5xy squared, close parenthesis, minus, parenthesis, negative x squared, y,
plus 3xy squared, minus 3y squared, close parenthesis
Which of the following is equivalent to the preceding expression?
A.
4x squared, y squared
B.
C.
D.
8xy squared, minus 6y squared
2x squared, y, plus, 2xy squared
2x squared, y, plus 8xy squared, minus 6y squared
Answer and explanation for question 5.
Question 6.
h equals 3a, plus 28.6
A pediatrician uses the model above to estimate the height h of a boy, in inches, in terms
of the boy’s age a, in years, between the ages of 2 and 5. Based on the model, what is the
estimated increase, in inches, of a boy’s height each year?
A. 13
B. 15.7
C. 19.5
D. 14.3
Explanation for question 6.
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Question 7.
m equals an expression times P, where the expression is
the fraction whose numerator is parenthesis, the fraction r over 1,200, close parenthesis,
times parenthesis, 1 plus the fraction r over 1,200, close parenthesis, to the power N, and
whose denominator is parenthesis, 1 plus the fraction r over 1,200, close parenthesis, to
the power N, end power, minus 1, end fraction.
The preceding formula gives the monthly payment m needed to pay off a loan of
P dollars at r percent annual interest over N months. Which of the following gives
P in terms of m, r, and N ?
A.
P equals an expression times m, where the
expression is the fraction whose numerator is parenthesis, the fraction r over 1,200, close
parenthesis, times, parenthesis, 1 plus the fraction r over 1,200, close parenthesis, to the
power N, and whose denominator is parenthesis, 1 plus the fraction r over 1,200, close
parenthesis, to the power N, minus 1.
B.
P equals an expression times m, where the
expression is the fraction whose numerator is parenthesis, 1 plus the fraction r over
1,200, close parenthesis, to the power N, minus 1, and whose denominator is
parenthesis, the fraction r over 1,200, close parenthesis, times, parenthesis, 1 plus the
fraction r over 1,200, close parenthesis, to the power N.
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C.
P equals, parenthesis, the fraction r over 1,200, close parenthesis,
times m.
D.
P equals, parenthesis, the fraction 1,200 over r, close parenthesis,
times m.
Answer and explanation for question 7.
Question 8.
If
the fraction, a over b, equals 2, what is the value of
the fraction 4b
over a ?
A. 0
B. 1
C. 2
D. 4
Answer and explanation for question 8.
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Question 9.
3x plus 4y equals negative 23
2y minus x equals negative 19
What is the solution
system of equations?
A.
parenthesis, x comma y, close parenthesis, to the preceding
parenthesis, negative 5, comma negative 2, close parenthesis
B.
parenthesis, 3 comma, negative 8, close parenthesis
C.
parenthesis, 4 comma, negative 6, close parenthesis
D.
parenthesis, 9 comma, negative 6, close parenthesis
Answer and explanation for question 9.
Question 10.
g of x equals a, x squared, plus 24.
For the function g defined, a is a constant and
g of 4 equals 8. What is the
value of
g of negative 4?
A. 8
B. 0
C.
negative 1
D.
negative 8
Answer and explanation for question 10.
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Question 11.
b equals 2.35 plus 0.25x
c equals 1.75 plus 0.40x
In the preceding equations, b and c represent the price per pound, in dollars, of beef
and chicken, respectively, x weeks after July 1 during last summer. What was the price
per pound of beef when it was equal to the price per pound of chicken?
A. $2.60
B. $2.85
C. $2.95
D. $3.35
Answer and explanation for question 11.
Question 12.
A line in the xy-plane passes through the origin and has a slope of
one seventh. Which
of the following points lies on the line?
A.
parenthesis, 0 comma 7, close parenthesis
B.
parenthesis, 1 comma 7, close parenthesis
C.
parenthesis, 7 comma 7, close parenthesis
D.
parenthesis, 14 comma 2, close parenthesis
Answer and explanation for question 12.
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Question 13.
If
x is greater than 3, which of the following is equivalent to
the
fraction whose numerator is 1, and whose denominator is the sum of fraction, 1 over x
plus 2, end fraction, and the fraction 1 over x plus 3, end fraction, end expression?
A.
the fraction whose numerator is 2x plus 5, and whose denominator is
x squared, plus 5x, plus 6.
B.
the fraction whose numerator is x squared plus 5x plus 6, and whose
denominator is 2x plus 5.
C.
2x plus 5
D.
x squared plus 5x plus 6
Answer and explanation for question 13.
Question 14.
If
3x minus y equals 12, what is the value of
the fraction 8 to the
power x, over 2 to the power y?
A.
B.
2 to the power 12
4 to the power 4
C.
8 to the power 2
D. The value cannot be determined from the information given.
Answer and explanation for question 14.
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Question 15.
If
parenthesis, a, x plus 2, close parenthesis, times,
parenthesis, bx plus 7, close parenthesis, equals 15x squared, plus cx, plus 14, for all
values of x, and
a, plus b equals 8, what are the two possible values for c ?
A. 3 and 5
B. 6 and 35
C. 10 and 21
D. 31 and 41
Answer and explanation for question 15.
Question 16.
If
t is greater than 0 and
value of t ?
t squared minus 4 equals 0, what is the
Answer and explanation for question 16.
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Directions
For questions 16 through 20, solve the problem and enter your answer in the grid, as
described below, on the answer sheet.
1. Although not required, it is suggested that you write your answer in the boxes at the
top of the columns to help you fill in the circles accurately. You will receive credit
only if the circles are filled in correctly.
2. Mark no more than one circle in any column.
3. No question has a negative answer.
4. Some problems may have more than one correct answer. In such cases, grid only one
answer.
5. Mixed numbers such as
three and one half must be gridded as 3.5 or
seven slash two. (If
will be interpreted as
three, one, slash, two, is entered into the grid, it
thirty one halves, not
three and one half.)
6. Decimal answers: If you obtain a decimal answer with more digits than the grid can
accommodate, it may be either rounded or truncated, but it must fill the entire grid.
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The following are four examples of how to record your answer in the spaces provided.
Keep in mind that there are four spaces provided to record each answer.
Examples 1 and 2
Begin skippable figure description.
Example 1: If your answer is a fraction such as seven-twelfths, it should be recorded as
follows. Enter 7 in the first space, the fraction bar (a slash) in the second space, 1 in the
third space, and 2 in the fourth space. All four spaces would be used in this example.
Example 2: If your answer is a decimal value such as 2.5, it could be recorded as follows.
Enter 2 in the second space, the decimal point in the third space, and 5 in the fourth
space. Only three spaces would be used in this example.
End skippable figure description.
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Example 3
Begin skippable figure description.
Example 3: Acceptable ways to record two-thirds are: 2 slash 3, .666, and .667.
End skippable figure description.
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Example 4
Note: You may start your answers in any column, space permitting. Columns you don’t
need to use should be left blank.
Begin skippable figure description.
Example 4: It is not necessary to begin recording answers in the first space unless all four
spaces are needed. For example, if your answer is 201, you may record 2 in the second
space, 0 in the third space, and 1 in the fourth space. Alternatively, you may record 2 in
the first space, 0 in the second space, and 1 in the third space. Spaces not needed should
be left blank.
End skippable figure description.
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Question 17 is based on the following graphic.
Begin skippable figure description.
The figure presents the outline of a lake and some geometric figures with measurements.
In the figure, from point A, which is on the top side of the lake, to point E, which is on
the bottom side of the lake, the length of the lake, AE, is labeled x feet. To the right of
the lake, line segments AC and ED are drawn such that AC slants downward, ED slants
upward, and both line segments intersect at point B that is to the right of the lake. In
triangle AEB and triangle CDB, angle AEB and angle CDB are both marked with an
angle symbol.
End skippable figure description.
Question 17.
A summer camp counselor wants to find a length, x, in feet, across a lake as represented
in the preceding sketch. The lengths represented by AB, EB, BD, and CD on the sketch
were determined to be 1800 feet, 1400 feet, 700 feet, and 800 feet, respectively.
Segments AC and DE intersect at B, and
angle AEB and
angle CDB
have the same measure. What is the value of x ?
Answer and explanation for question 17.
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Question 18 is based on the following system of equations.
x plus y equals negative 9
x plus 2y equals negative 25
Question 18.
According to the preceding system of equations, what is the value of x ?
Answer and explanation for question 18.
Question 19.
In a right triangle, one angle measures
x degrees, where
x degrees equals four fifths. What is
sine of
cosine of, parenthesis, 90 degrees
minus x degrees, close parenthesis ?
Answer and explanation for question 19.
Question 20.
If
a, equals 5 times the square root of 2 and
root of 2x, what is the value of x ?
2a, equals the square
Answer and explanation for question 20.
Stop.
If you finish before time is called, you may check your work on this section only. Do
not turn to any other section.
Answers and explanations for questions 1 through 20 are provided in the
next section of this document.
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Answers and Explanations for Questions 1
through 20.
Explanation for question 1.
Choice D is correct. Since
k equals 3, one can substitute 3 for k in the equation
the fraction whose numerator is x minus 1, and whose denominator is 3,
equals k which gives
the fraction whose numerator is x minus 1, and whose
denominator is 3, equals 3. Multiplying both sides of
the fraction whose
numerator is x minus 1 and whose denominator is 3, equals 3, by 3 gives
x minus 1, equals 9 and then adding 1 to both sides of
x minus 1, equals 9
gives
x equals 10.
Choices A, B, and C are incorrect because the result of subtracting 1 from the value and
dividing by 3 is not the given value of k, which is 3.
Explanation for question 2.
Choice A is correct. To calculate
parenthesis, 7 plus 3i, close
parenthesis, plus, parenthesis, negative 8 plus 9i, close parenthesis, add the real parts of
each complex number,
7 plus, parenthesis, negative 8, close parenthesis,
equals negative 1, and then add the imaginary parts,
The result is
negative 1 plus 12i.
3i plus 9i, equals 12i.
Choices B, C, and D are incorrect and likely result from common errors that arise when
adding complex numbers. For example, Choice B is the result of adding 3i and
negative 9i Choice C is the result of adding 7 and 8.
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Explanation for question 3.
Choice C is correct. The total number of messages sent by Armand is the 5 hours he spent
texting multiplied by his rate of texting:
m texts
per hour, times 5 hours, equals 5 m texts. Similarly, the total number of messages sent by
Tyrone is the 4 hours he spent texting multiplied by his rate of texting:
p texts per hour, times 4 hours, equals 4 p texts.
The total number of messages sent by Armand and Tyrone is the sum of the total number
of messages sent by Armand and the total number of messages sent by Tyrone:
5m plus 4p.
Choice A is incorrect and arises from adding the coefficients and multiplying the
variables of 5m and 4p. Choice B is incorrect and is the result of multiplying 5m
and 4p. The total number of messages sent by Armand and Tyrone should be the sum of
5m and 4p, not the product of these terms. Choice D is incorrect because it multiplies
Armand’s number of hours spent texting by Tyrone’s rate of texting, and vice versa. This
mix-up results in an expression that does not equal the total number of messages sent by
Armand and Tyrone.
Explanation for question 4.
Choice B is correct. The value 108 in the equation is the value of P in
P equals 108 minus 23d, when
d equals 0. When
d equals 0, Kathy has
worked 0 days that week. In other words, 108 is the number of phones left before Kathy
has started work for the week. Therefore, the meaning of the value 108 in the equation is
that Kathy starts each week with 108 phones to fix because she has worked 0 days and
has 108 phones left to fix.
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Choice A is incorrect, because Kathy will complete the repairs when
Since
P equals 108 minus 23d, this will occur when
0 equals 108 minus 23d, or when
P equals 0.
d equals the fraction 108 over 23, not when
d equals 108. Therefore, the value 108 in the equation does not represent the
number of days it will take Kathy to complete the repairs. Choices C and D are incorrect
because the number 23 in
P equals 108 minus 23, P equals 108
indicates that the number of phones left will decrease by 23 for each increase in the value
of d by 1; in other words, that Kathy is repairing phones at a rate of 23 per day, not
108 per hour (choice C) or 108 per day (choice D).
Explanation for question 5.
Choice C is correct. Only like terms, with the same variables and exponents, can be
combined to determine the answer as shown here:
Parenthesis, x squared, y, minus 3y squared, plus 5xy squared, close parenthesis, minus,
parenthesis, negative x squared, y, plus 3xy squared, minus 3y squared, close parenthesis.
Equals, parenthesis, x squared, y, minus, parenthesis, negative x squared, y, close double
parentheses, plus, parenthesis, negative 3y squared, minus, parenthesis, negative 3y
squared, close double parentheses, plus, parenthesis, 5xy squared, minus 3xy squared,
close parenthesis. Equals, 2x squared, y, plus 0, plus 2xy squared. Equals, 2x squared, y,
plus 2xy squared
Choices A, B, and D are incorrect and are the result of common calculation errors or of
incorrectly combining like and unlike terms.
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Explanation for question 6.
Choice A is correct. In the equation
of the boy, increases by 1, then h becomes
h equals 3a, plus 28.6 if a, the age
h equals 3, parenthesis, a,
plus 1, close parenthesis, plus 28.6, equals, 3a, plus 3, plus 28.6, equals, parenthesis, 3a,
plus 28.6, close parenthesis, plus 3. Therefore, the model estimates that the boy’s height
increases by 3 inches each year.
Alternatively: The height, h, is a linear function of the age, a, of the boy. The
coefficient 3 can be interpreted as the rate of change of the function; in this case, the rate
of change can be described as a change of 3 inches in height for every additional year
in age.
Choices B, C, and D are incorrect and are likely to result from common errors in
calculating the value of h or in calculating the difference between the values of h for
different values of a.
Explanation for question 7.
Choice B is correct. Since the right hand side of the equation is P times the expression,
The fraction whose numerator is, parenthesis, the fraction r
over 1,200, close parenthesis, times, parenthesis, 1 plus the fraction r over 1,200, close
parenthesis, to the power N, and whose denominator is, parenthesis, 1 plus the fraction r
over 1,200, close parenthesis, to the power N, minus 1, end fraction.
multiplying both sides of the equation by the reciprocal of this expression results in
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The reciprocal of the expression times m equals P,
where the reciprocal of the expression is the fraction whose numerator is, parenthesis, 1
plus the fraction r over 1,200, close parenthesis, to the power N, minus 1, and whose
denominator is, parenthesis, the fraction r over 1,200, close parenthesis, times,
parenthesis, 1 plus the fraction r over 1,200, close parenthesis, to the power N.
Choices A, C, and D are incorrect and are likely the result of conceptual or computation
errors while trying to solve for P.
Explanation for question 8.
Choice C is correct. Since
the fraction a, over b equals 2, it follows that
the fraction b over a, equals one half. Multiplying both sides of the equation by 4 gives
4 times, parenthesis, the fraction b over a, close parenthesis, equals the
fraction 4b over a, equals 2.
Choice A is incorrect because if
the fraction 4b over a, equals 0, then
fraction a over b would be undefined. Choice B is incorrect because if
fraction 4b over a equals 1, then
incorrect because if
the
the
the fraction a, over b equals 4. Choice D is
the fraction 4b over a, equals 4, then
the fraction
a over b equals 1.
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Explanation for question 9.
Choice B is correct. Adding x and 19 to both sides of
2y minus x equals
negative 19 gives
x equals 2y plus 19. Then substituting
2y plus
19 for x in
3x plus 4y equals negative 23, gives
3 times, parenthesis, 2y plus 19, close parenthesis, plus 4y
equals negative 23. This last equation is equivalent to
10y plus 57
equals negative 23. Solving
10y plus 57 equals negative 23 gives
y equals negative 8. Finally, substituting
negative 8 for y in
2y minus x equals negative 19, gives
2 times, parenthesis, negative 8,
close parenthesis, minus x, equals negative 19, or
x equals 3. Therefore, the
solution
parenthesis, x comma y, close parenthesis, to the given system of
equations is
parenthesis, 3 comma negative 8, close parenthesis.
Choices A, C, and D are incorrect because when the given values of x and y are
substituted in
2y minus x equals negative 19, the value of the left side of
the equation does not equal
negative 19.
Explanation for question 10.
Choice A is correct. Since g is an even function,
g of negative 4,
equals, g of 4, equals 8.
Alternatively: First find the value of a, and then find
g of 4 equals 8, substituting 4 for x and 8 for
g of negative 4. Since
g of x gives
8 equals a, parenthesis, 4, close parenthesis, squared, plus
24, equals 16a, plus 24. Solving this last equation gives
Thus
a, equals negative 1.
g of x equals negative x squared, plus 24, from which it
follows that
g of negative 4 equals, negative, parenthesis,
negative 4, close parenthesis, squared, plus 24;
equals, negative 16, plus 24, and
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g of negative 4
g of negative 4 equals 8. The other choices
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are incorrect because g is a function and there can only be one value of
negative 4.
g of
Explanation for question 11.
Choice D is correct. To determine the price per pound of beef when it was equal to the
price per pound of chicken, determine the value of x (the number of weeks after July 1)
when the two prices were equal. The prices were equal when
b equals c; that is,
when
2.35 plus 0.25x, equals, 1.75 plus 0.40 x. This last
equation is equivalent to
0.60 equals 0.15 x, and so
x equals, the fraction whose numerator is 0.60 and whose denominator is 0.15, equals 4.
Then to determine b, the price per pound of beef, substitute 4 for x in
b equals 2.35 plus 0.25x, which gives
b equals 2.35 plus 0.25, parenthesis, 4, close parenthesis, equals, 3.35 dollars per pound.
Choice A is incorrect. It results from using the value 1, not 4, for x in
b equals 2.35 plus 0.25x. Choice B is incorrect. It results from using the value 2, not 4,
for x in
b equals 2.35 plus 0.25x. Choice C is incorrect. It results
from using the value 3, not 4, for x in
c equals 1.75 plus 0.40x.
Explanation for question 12.
Choice D is correct. Determine the equation of the line to find the relationship between
the x- and y-coordinates of points on the line. All lines through the origin are of the form
y equals mx, so the equation is
y equals one seventh x. A point lies on
the line if and only if its y-coordinate is
one seventh of its x-coordinate. Of the given
choices, only choice D,
this condition:
parenthesis, 14 comma 2, close parenthesis, satisfies
2 equals one seventh times, parenthesis, 14, close
parenthesis.
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Choice A is incorrect because the line determined by the origin
0 comma 0, close parenthesis, and
the vertical line with equation
y-axis is undefined, not
parenthesis,
parenthesis, 0 comma 7, close parenthesis, is
x equals 0; that is, the y-axis. The slope of the
one seventh. Therefore, the point
parenthesis, 0
comma 7, close parenthesis, does not lie on the line that passes the origin and has slope
one seventh. None of the other coordinate pairs satisfy the equation
y equals mx.
Explanation for question 13.
Choice B is correct. To rewrite
the expression which is a fraction whose
numerator is 1, and whose denominator is the sum of the fraction, 1 over x plus 2, and the
fraction 1 over x plus 3, end expression, multiply by
the fraction whose
numerator is, parenthesis, x plus 2, close parenthesis, times, parenthesis, x plus 3, close
parenthesis, and whose denominator is, parenthesis, x plus 2, close parenthesis, times,
parenthesis, x plus 3, close parenthesis. This results in the expression
the fraction whose numerator is, parenthesis, x plus 2, close parenthesis, times,
parenthesis, x plus 3, close parenthesis, and whose denominator is, parenthesis, x plus 3,
close parenthesis, plus, parenthesis, x plus 2, close parenthesis, which is equivalent to the
expression in choice B.
Choices A, C, and D are incorrect and could be the result of common algebraic errors that
arise while manipulating a complex fraction.
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Explanation for question 14.
Choice A is correct. One approach is to express
the fraction 8 to the power x, over
2 to the power y so that the numerator and denominator are expressed with the same base.
Since 2 and 8 are both powers of 2, substituting
2 cubed for 8 in the numerator of
the fraction 8 to the power x, over 2 to the power y gives
the fraction
whose numerator is, parenthesis, 2 cubed, close parenthesis, to the power x, and whose
denominator is 2 to the power y which can be rewritten as
the fraction 2 to the
power 3x, over 2 to the power y. Since the numerator and denominator of
the
fraction 2 to the power 3x, over 2 to the power y have a common base, this expression
can be rewritten as
2 to the 3x minus y power. It is given that
3x minus y equals 12 so one can substitute 12 for the exponent,
giving that the expression
equal to
3x minus y,
the fraction 8 to the power x, over 2 to the power y, is
2 to the power 12.
Choices B and C are incorrect because they are not equal to
Choice D is incorrect because the value of
2 to the power 12.
the fraction 8 to the power x, over 2 to
the power y, can be determined.
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Explanation for question 15.
Choice D is correct. One can find the possible values of a and b in
parenthesis, a, x plus 2, close parenthesis, times, parenthesis, bx plus 7, close parenthesis,
by using the given equation
a, plus b equals 8, and finding another equation
that relates the variables a and b. Since
parenthesis, a, x plus 2, close parenthesis, times, parenthesis, bx plus 7, close parenthesis,
equals, 15x squared, plus cx plus 14, one can expand the left side of the equation to
obtain
a, bx squared, plus 7a, x, plus 2bx,
plus 14, equals, 15x squared, plus cx, plus 14. Since ab is the coefficient of
,
x squared on the left side of the equation and 15 is the coefficient of
x squared on the
right side of the equation, it must be true that
a, b equals 15. Since
a, plus b equals 8, it follows that
b equals 8 minus a. Thus
a, b equals 15 can be rewritten as
a, times, parenthesis, 8 minus a, close
parenthesis, equals 15, which in turn can be rewritten as
minus 8a, plus 15, equals 0. Factoring gives
a, squared
parenthesis, a, minus 3,
close parenthesis, times, parenthesis, a, minus 5, close parenthesis, equals 0. Thus, either
a, equals 3 and
b equals 5, or
a, equals 5 and
b equals 3. If
a, equals 3 and
b equals 5, then
parenthesis, a, x plus 2,
close parenthesis, times, parenthesis, bx plus 7, close parenthesis, equals, parenthesis 3x
plus 2, close parenthesis, times, parenthesis, 5x plus 7, close parenthesis, equals,
15x squared, plus 31x, plus 14. Thus, one of the possible values of c is 31. If
a, equals 5 and
b equals 3, then
parenthesis, a, x plus 2,
close parenthesis, times, parenthesis, bx plus 7, close parenthesis, equals, parenthesis, 5x
plus 2, close parenthesis, times, parenthesis, 3x plus 7, close parenthesis, equals
15x squared, plus 41x, plus 14. Thus another possible value for c is 41. Therefore, the
two possible values for c are 31 and 41.
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Choice A is incorrect; the numbers 3 and 5 are possible values for a and b, but not
possible values for c. Choice B is incorrect; if
a, equals 5 and
b equals 3,
then 6 and 35 are the coefficients of x when the expression
parenthesis, 5x plus 2, close parenthesis, times, parenthesis, 3x plus 7, close parenthesis,
is expanded as
15x squared, plus 35x, plus 6x, plus 14.
However, when the coefficients of x are 6 and 35, the value of c is 41 and not 6 and 35.
Choice C is incorrect; if
a, equals 3 and
b equals 5, then 10 and 21 are the
coefficients of x when the expression
parenthesis, 3x plus 2, close
parenthesis, times, parenthesis, 5x plus 7, close parenthesis, is expanded as
15x squared, plus 21x, plus 10x, plus 14. However, when the
coefficients of x are 10 and 21, the value of c is 31 and not 10 and 21.
Explanation for question 16.
The correct answer is 2. To solve for t, factor the left side of
minus 4 equals 0, giving
t squared
parenthesis, t minus 2, close parenthesis,
parenthesis, t plus 2, close parenthesis, equals 0. Therefore, either
t minus 2
equals 0, or
t plus 2 equals 0. If
t minus 2 equals 0, then
t equals 2, and if
t plus 2 equals 0, then
t equals negative 2. Since it is
given that
t is greater than 0, the value of t must be 2.
Another way to solve for t is to add 4 to both sides of
equals 0, giving
t squared minus 4
t squared equals 4. Then taking the square root of the left and
right side of the equation gives
t equals plus or minus the square root of
4, equals, plus or minus 2. Since it is given that
t is greater than 0, the value of
t must be 2.
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Explanation for question 17.
The correct answer is 1600. It is given that
angle AEB and
angle CDB
have the same measure. Since
angle AEB and
angle CDB are vertical
angles, they have the same measure. Therefore, triangle EAB is similar to triangle DCB
because the triangles have two pairs of congruent corresponding angles (angle-angle
criterion for similarity of triangles). Since the triangles are similar, the corresponding
sides are in the same proportion; thus
the fraction CD over x equals the
fraction BD over EB. Substituting the given values of 800 for CD, 700 for BD, and 1400
for EB in
the fraction CD over x equals the fraction BD over EB. gives
the fraction 800 over x, equals, the fraction 700 over 1,400. Therefore,
x equals the fraction whose numerator is, parenthesis, 800,
close parenthesis, times, parenthesis, 1,400, close parenthesis, and whose denominator is
700, equals 1600
Explanation for question 18.
The correct answer is 7. Subtracting the left and right sides of
x plus y equals
negative 9 from the corresponding sides of
x plus 2y equals negative 25,
gives
parenthesis, x plus 2y, close parenthesis,
minus, parenthesis, x plus y, close parenthesis, equals, negative 25, minus, parenthesis,
negative 9, close parenthesis, which is equivalent to
y equals negative 16.
Substituting
negative 16 for y in
x plus y equals negative 9, gives
x plus, parenthesis, negative 16, close parenthesis, equals, negative 9,
which is equivalent to
x equals negative 9, minus, parenthesis,
negative 16, close parenthesis, equals 7.
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Explanation for question 19.
The correct answer is
four fifths or 0.8. By the complementary angle relationship for
sine and cosine,
sine, parenthesis, x degrees, close
parenthesis, equals, cosine, parenthesis, 90 degrees minus x degrees, close parenthesis.
Therefore,
cosine, parenthesis, 90 degrees minus x degrees, close
parenthesis, equals four fifths. Either the fraction
four fifths or its decimal equivalent,
0.8, may be gridded as the correct answer.
Alternatively, one can construct a right triangle that has an angle of measure
such that
x degrees
sine, parenthesis, x degrees, close parenthesis, equals four fifths,
as shown in the following figure, where
sine, parenthesis, x degrees, close
parenthesis, is equal to the ratio of the opposite side to the hypotenuse, or
four fifths.
Begin skippable figure description.
The figure presents a triangle with a 90 degree angle. The other two angles of the triangle
have degree measures of x and 90 minus x degrees. The side opposite the x degree
angle is labeled 4, and the side opposite the 90 degree angle is labeled 5.
End skippable figure description.
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Since two of the angles of the triangle are of measure
x degrees and
90 degrees,
the third angle must have the measure
180 degrees minus
90 degrees, minus x degrees, equals, 90 degrees minus x degrees. From the figure,
cosine, parenthesis, 90 degrees minus x degrees, close parenthesis,
which is equal to the ratio of the adjacent side to the hypotenuse, is also
four fifths.
Explanation for question 20.
The correct answer is 100. Since
can substitute
a, equals 5 times the square root of 2, one
5 times the square root of 2 for a in
square root of 2x, giving
2a equals the
10 times the square root of 2, equals, the
square root of 2x. Squaring each side of
10 times the square root of 2,
equals, the square root of 2x gives
parenthesis, 10 times the
square root of 2, close parenthesis, squared, equals, parenthesis, the square root of 2x,
close parenthesis, squared, which simplifies to
parenthesis, 10, close parenthesis, squared, times, parenthesis, the square root of 2, close
parenthesis, squared, equals, parenthesis, the square root of 2x, close parenthesis,
squared, or
200 equals 2x. This gives
x equals 100. Checking
x equals 100 in the original equation gives
2 times,
parenthesis, 5 times the square root of 2, close parenthesis, equals, the square root of,
parenthesis, 2, close parenthesis, times, parenthesis, 100, close parenthesis, end
square root which is true since
2 times, parenthesis, 5 times the
square root of 2, close parenthesis, equals, 10 times the square root of 2, and
the square root of, parenthesis, 2, close
parenthesis, times, parenthesis, 100, close parenthesis, end square root, equals,
parenthesis, the square root of 2, close parenthesis, times, parenthesis, the square root of
100, close parenthesis, equals, 10 times the square root of 2.
Stop. This is the end of the answers and explanations for questions 1 through 20.
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