Math 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. Please refer to the directions before question 16 on how to enter
your answers in the grid. You may use scratch paper 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 ℓw.
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 ℓwh.
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 ℓwh.
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.
A painter will paint n walls with the same size and shape in a building using a
specific brand of paint. The painter’s fee can be calculated by the expression
nKlh, where n is the number of walls, K is a constant with units of dollars per
square foot, l is the length of each wall in feet, and h is the height of each wall
in feet. If the customer asks the painter to use a more expensive brand of paint,
which of the factors in the expression would change?
A. h
B. l
C. K
D. n
Explanation for question 1.
Question 2.
If
A.
B.
C.
D.
3r equals 18, what is the value of
6 r plus 3?
6
27
36
39
Explanation for question 2.
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Question 3.
Which of the following is equal to
values of a ?
a, raised to the power two thirds, for all
A.
1
a3
B.
a 3 the square root of a, cubed, end root
C.
D.
3
1
a2
3 2
a
the square root of a, raised to the power one third, end root
the cube root of a, raised to the power one half, end root
the cube root of a, squared, end root
Explanation for question 3.
Question 4.
The number of states that joined the United States between 1776 and 1849 is twice
the number of states that joined between 1850 and 1900. If 30 states joined the
United States between 1776 and 1849 and x states joined between 1850 and 1900,
which of the following equations is true?
A. 30 x  2 30x equals 2
B. 2 x  30 2x equals 30
x
 30 the fraction x over 2 equals 30
2
D. x  30  2 x plus 30 equals 2
C.
Explanation for question 4.
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Question 5.
5
15

, the fraction 5 over x equals the fraction whose numerator is 15 and
x x  20
x
whose denominator is x plus 20 end fraction, what is the value of
the fraction x
5
over 5 ?
If
A. 10
B. 5
C. 2
D.
1
one half
2
Explanation for question 5.
Question 6 refers to the following equations.
2 x  3 y  14 2x minus 3y equals negative 14
3 x  2 y  6 3x minus 2y equals negative 6
Question 6.
If  x , y  the ordered pair x comma y is a solution to the preceding system of
equations, what is the value of
x minus y ?
A. 20 negative 20
B. 8 negative 8
C. 4 negative 4
D. 8
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Explanation for question 6.
Question 7 refers to the following table.
x
0
2
4
5
f  x  f of x
3
1
0
−2
negative 2
Question 7.
The function f is defined by a polynomial. Some values of x and f  x  f of x are
shown in the preceding table. Which of the following must be a factor of f  x 
f of x ?
A.
B.
C.
D.
x  2 x minus 2
x  3 x minus 3
x  4 x minus 4
x  5 x minus 5
Explanation for question 7.
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Question 8.
The line
y  kx  4 y equals kx plus 4, where k is a constant, is graphed in the
xy-plane. If the line contains the point  c , d , with coordinates c comma d, where
c  0 c is not equal to 0 and
d  0, d is not equal to 0, what is the slope of the line
in terms of c and d ?
d 4
the fraction d minus 4, over c
c
c4
B.
the fraction c minus 4, over d
d
4d
C.
the fraction 4 minus d, over c
c
4c
D.
the fraction 4 minus c, over d
d
A.
Explanation for question 8.
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Question 9 refers to the following system of equations.
kx minus 3y equals 4
4x minus 5y equals 7
Question 9.
In the preceding system of equations, k is a constant and x and y are variables.
For what value of k will the system of equations have no solution?
12
the fraction 12 over 5
5
16
B.
the fraction 16 over 7
7
16
C. 
the negative fraction 16 over 7
7
12
D. 
the negative fraction 12 over 5
5
A.
Explanation for question 9.
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Question 10.
2
In the xy-plane, the parabola with equation y   x  11 y equals, parenthesis,
x minus 11, close parenthesis, squared, intersects the line with equation
y equals 25 at two points, A and B. What is the length of
A.
B.
C.
D.
y  25
AB line segment AB?
10
12
14
16
Explanation for question 10.
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Question 11 refers to the following figure.
Begin skippable figure description.
The figure presents three lines k, l, and m that intersect at a point. The three lines
form six angles at the point of intersection. The horizontal line is labeled m, the
line that slants upward and to the right is labeled l, and the line that slants
downward and to the right is labeled k. Of the 6 angles, 3 angles are above line m,
and 3 angles are below line m. Starting from the leftmost angle above line m and
going clockwise, the 6 angles are labeled x degrees, y degrees, z degrees,
t degrees, u degrees, and w degrees, respectively.
End skippable figure description.
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Question 11.
In the preceding figure, lines k,
l, and m intersect at a point. If x  y  u  w
x plus y equals u plus w, which of the following must be true?
I. x  z x equals z
II. y  w y equals w
III. z  t z equals t
A.
B.
C.
D.
I and II only
I and III only
II and III only
I, II, and III
Explanation for question 11.
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Question 12 refers to the following quadratic equation.
y  a  x  2  x  4  y equals a, parenthesis, x minus 2, close parenthesis, times,
parenthesis, x plus 4, close parenthesis
Question 12.
In the preceding quadratic equation, a is a nonzero constant. The graph of the
equation in the xy-plane is a parabola with vertex  c , d . with coordinates
c comma d. Which of the following is equal to d ?
A.
B.
C.
D.
9a
8a
5a
2a
negative 9 a
negative 8 a
negative 5 a
negative 2 a
Explanation for question 12.
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Question 13.
24 x 2  25 x  47
53
 8 x  3 
The equation
with the fraction whose
ax  2
ax  2
numerator is 24x squared, plus 25x minus 47, and whose denominator is a,x minus
2, equals, negative 8x minus 3, minus, the fraction whose numerator is 53, and
2
whose denominator is a,x minus 2, is true for all values of x  , x that do not
a
equal the fraction, 2 over a, where a is a constant. What is the value of a ?
A. 16 negative 16
B. 3 negative 3
C.
3
D. 16
Explanation for question 13.
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Question 14.
What are the solutions to 3 x 2  12 x  6  0 3x squared, plus 12x plus 6, equals 0 ?
A. x  2  2 x equals negative 2, plus or minus the square root of 2
B. x  2 
30
x equals negative 2, plus or minus, the fraction the square root of
3
30, end square root, over 3
C. x  6  2 x equals negative 6, plus or minus the square root of 2
D. x  6  6 2 x equals negative 6, plus or minus, 6 times the square root of 2
Explanation for question 14.
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Question 15 refers to the following equation.
C
5
 F  32  C equals five ninths, parenthesis F minus 32 close parenthesis
9
Question 15.
The preceding equation shows how a temperature F, measured in degrees
Fahrenheit, relates to a temperature C, measured in degrees Celsius. Based on the
equation, which of the following must be true?
I. A temperature increase of 1 degree Fahrenheit is equivalent to a temperature
5
increase of
five ninths degree Celsius.
9
II. A temperature increase of 1 degree Celsius is equivalent to a temperature
increase of 1.8 degrees Fahrenheit.
5
III. A temperature increase of
five ninths degree Fahrenheit is equivalent to a
9
temperature increase of 1 degree Celsius.
A.
B.
C.
D.
I only
II only
III only
I and II only
Explanation for question 15.
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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
seven slash two. (If
will be interpreted as
three and one half must be gridded as 3.5 or
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 16 refers to the following equation.


x 3 x 2  5  4 x x cubed, parenthesis, x squared minus 5, close parenthesis,
equals negative 4x
Question 16.
If x  0 x is greater than 0, what is one possible solution to the preceding
equation?
Explanation for question 16.
Question 17.
7
4
1 5
x  x   , seven ninths x, minus four ninths x, equals, one fourth plus
9
9
4 12
five twelfths what is the value of x ?
If
Explanation for question 17.
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Question 18 refers to the following figure.
Begin skippable figure description.
The figure presents two triangles that have one vertex in common. The two
triangles, one on the left and one on the right, each have a vertical side opposite the
common vertex, and the other two sides in each triangle are marked showing that
both sides are equal in length. In the triangle on the left, the angle at the common
vertex is labeled y degrees, and in the triangle on the right, the angle at the
common vertex is labeled z degrees. The exterior angle of the triangle on the right
that is between one of the equal sides and the vertical line extended from the
vertical side is labeled x degrees.
End skippable figure description.
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Question 18.
Two isosceles triangles are shown in the preceding figure. If 180  z  2 y
180 minus z equals 2y and
y  75, y equals 75, what is the value of x ?
Explanation for question 18.
Question 19.
At a lunch stand, each hamburger has 50 more calories than each order of fries. If
2 hamburgers and 3 orders of fries have a total of 1700 calories, how many calories
does a hamburger have?
Explanation for question 19.
Question 20.
In triangle ABC, the measure of
B angle B is 90 90 degrees, BC  16 BC equals
16, and AC  20 AC equals 20. Triangle DEF is similar to triangle ABC, where
vertices D, E, and F correspond to vertices A, B, and C, respectively, and each side
1
of triangle DEF is
one third the length of the corresponding side of triangle
3
ABC. What is the value of sin F sine F?
Explanation for question 20.
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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 C is correct. The painter’s fee is given by
nKlh where n is the
number of walls, K is a constant with units of dollars per square foot, l is the
length of each wall in feet, and h is the height of each wall in feet. Examining this
equation shows that l and h will be used to determine the area of each wall. The
variable n is the number of walls, so n times the area of the walls will give the
amount of area that will need to be painted. The only remaining variable is K,
which represents the cost per square foot and is determined by the painter’s time
and the price of paint. Therefore, K is the only factor that will change if the
customer asks for a more expensive brand of paint.
Choice A is incorrect because a more expensive brand of paint would not cause the
height of each wall to change.
Choice B is incorrect because a more expensive brand of paint would not cause the
length of each wall to change.
Choice D is incorrect because a more expensive brand of paint would not cause the
number of walls to change.
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Explanation for question 2.
Choice D is correct. Dividing each side of the equation
3r equals 18 by 3
gives
r equals 6. Substituting 6 for r in the expression
6r plus 3
gives
6, parenthesis, 6, close parenthesis, plus 3, equals 39.
Alternatively, the expression
6r plus 3 can be rewritten as 2(3r) + 3
2, parenthesis, 3r, close parenthesis, plus 3. Substituting 18 for 3r 3r in the
expression 2(3r) + 3 2, parenthesis, 3r, close parenthesis, plus 3, yields 2(18) + 3 =
36 + 3 = 39 2, parenthesis 18 close parenthesis, plus 3, equals 36 plus 3, which
equals 39.
Choice A is incorrect because 6 is the value of r; however, the question asks for the
value of the expression 6r + 36 r plus 3.
Choices B and C are incorrect because if 6r + 3 6 r plus 3 were equal to either of
these values, then it would not be possible for 3r 3 r to be equal to 18, as stated in
the question.
Choices B and C are incorrect because if
6r plus 3 were equal to either of
these values, then it would not be possible for 3r to be equal to 18, as stated in the
question.
Explanation for question 3.
Choice D is correct. By definition,
a, raised to the power the fraction
m over n, equals the nth root of a, raised to the power m, for any positive integers
m and n. It follows, therefore, that
equals, the cube root of a, squared, end root.
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a, raised to the power two thirds,
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Choice A is incorrect. By definition,
a, raised to the power the fraction
one over n, equals the nth root of a for any positive integer n. Applying this
definition as well as the power property of exponents to the expression
square root of a, raised to the power one third, end root, yields
the
the square root of a, raised to the power one third, end root,
equals, parenthesis, a, raised to the power one third, close parenthesis, raised to the
power one half, which equals, a, raised to the power one sixth. Because
a, raised to the power one sixth, does not equal a, raised to the power
two thirds,
the square root of a, raised to the power one third, end root, is not
the correct answer.
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Choice B is incorrect. By definition,
a, raised to the power the fraction
one over n, equals the nth root of a, for any positive integer n. Applying this
definition as well as the power property of exponents to the expression
square root of a, cubed, end root, yields
the
the square root of a,
cubed, end root, equals parenthesis, a, cubed, close parenthesis, raised to the power
one half, which equals a, raised to the power three halves. Because
a, raised to the power three halves, does not equal a, raised to the power two thirds,
the square root of a, cubed, end root, is not the correct answer.
Choice C is incorrect. By definition,
a, raised to the power the fraction
one over n, equals the nth root of a, for any positive integer n. Applying this
definition as well as the power property of exponents to the expression
the
cube root of a, raised to the power one half, end root, yields
the cube root of a, raised to the power one half, end root, equals, parenthesis, a,
raised to the power one half, close parenthesis, raised to the power one third, which
equals a, raised to the power one sixth. Because
a, raised to the power
one sixth, does not equal a, raised to the power two thirds,
the cube root of
a, raised to the power one half, end root, is not the correct answer.
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Explanation for question 4.
Choice B is correct. To fit the scenario described, 30 must be twice as large as x.
This can be written as
2x equals 30.
Choices A, C, and D are incorrect. These equations do not correctly relate the
numbers and variables described in the stem. For example, the expression in C
states that 30 is half as large as x, not twice as large as x.
Explanation for question 5.
Choice C is correct. Multiplying each side of
the fraction 5 over x
equals, the fraction whose numerator is 15, and whose denominator is x plus 20,
end fraction by
x, parenthesis, x plus 20, close parenthesis, gives
15x equals, 5, parenthesis, x plus 20, close parenthesis.
Distributing the 5 over the values within the parentheses yields
15x equals 5x plus 100, and then subtracting 5x from each side gives
10x equals 100. Finally, dividing both sides by 10 gives
x equals 10.
Therefore, the value of
the fraction x over 5 is
the fraction 10 over 5,
equals 2.
Choice A is incorrect because it is the value of x, not
the fraction x over 5.
Choices B and D are incorrect and may be the result of errors in arithmetic
operations on the given equation.
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Explanation for question 6.
Choice C is correct. Multiplying each side of the equation
2x minus
3y equals negative 14, by 3 gives
6x minus 9y equals negative 42.
Multiplying each side of the equation
3x minus 2y equals negative 6
by 2 gives
6x minus 4y equals negative 12. Then, subtracting the
sides of
6x minus 4y equals negative 12 from the corresponding
sides of
6x minus 9y equals negative 42 gives
negative 5y equals negative 30. Dividing each side of the equation
negative 5y equals negative 30 by −5 negative 5 gives
y equals 6. Finally,
substituting 6 for y in
2x minus 3y equals negative 14 gives
2x minus 3, parenthesis, 6, close parenthesis, equals negative 14,
or
x equals 2. Therefore, the value of
x minus y is
2 minus
6 equals negative 4.
Alternatively, adding the corresponding sides of
2x minus 3y equals
negative 14 and
3x minus 2y equals negative 6 gives
5x minus 5y equals negative 20, from which it follows that
x minus y
equals negative 4.
Choices A, B, and D are incorrect and may be the result of an arithmetic error
when solving the system of equations.
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Explanation for question 7.
Choice C is correct. If
x minus b is a factor of f(x) f of x, then f(b) f of b must
equal 0. Based on the table,
f of 4 equals 0. Therefore,
x minus 4
must be a factor of f(x) f of x.
Choice A is incorrect because
f of 2 does not equal 0, choice B is
incorrect because no information is given about the value of f(3) f of 3, so
x minus 3 may or may not be a factor of f(x) f of x, and choice D is incorrect
because
f of 5 does not equal 0.
Explanation for question 8.
Choice A is correct. The linear equation
y equals kx plus 4 is in
slope-intercept form, and so the slope of the line is k. Since the line contains the
point (c, d) with coordinates c comma d, the coordinates of this point satisfy the
equation
y equals kx plus 4: d equals kc plus 4. Solving this
equation for the slope, k, gives
k equals the fraction d minus 4, over c.
Choices B, C, and D are incorrect and may be the result of errors in substituting
the coordinates of (c, d) c comma d in
y equals kx plus 4 or of errors in
solving for k in the resulting equation.
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Explanation for question 9.
Choice A is correct. If a system of two linear equations has no solution, then the
lines represented by the equations in the coordinate plane are parallel. The equation
kx minus 3y equals 4 can be rewritten as
y equals the
fraction k over 3, end fraction, times x, minus, the fraction 4 over 3, where
fraction k over 3 is the slope of the line, and the equation
equals 7 can be rewritten as
minus the fraction 7 over 5, where
the
4x minus 5y
y equals the fraction 4 over 5, times x,
the fraction 4 over 5 is the slope of the line. If
two lines are parallel, then the slopes of the line are equal. Therefore,
fraction 4 over 5 equals, the fraction k over 3, or
the
k equals the fraction 12
over 5. (Since the y-intercepts of the lines represented by the equations are
negative fraction 4 over 3 and
the
the negative fraction 7 over 5, the lines are
parallel, not identical.)
Choices B, C, and D are incorrect and may be the result of a computational error
when rewriting the equations or solving the equation representing the equality of
the slopes for k.
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Explanation for question 10.
Choice A is correct. Substituting 25 for y in the equation
y equals
parenthesis, x minus 11, close parenthesis, squared gives
25 equals
parenthesis, x minus 11, close parenthesis, squared. It follows that
x minus 11 equals 5 or
x minus 11 equals negative 5, so the
x-coordinates of the two points of intersection are
x equals 16 and
x equals 6, respectively. Since both points of intersection have a y-coordinate
of 25, it follows that the two points are (16, 25) 16 comma 25 and (6, 25) 6
comma 25. Since these points lie on the horizontal line y = 25 y equals 25, the
distance between these points is the positive difference of the x-coordinates:
16 minus 6 equals 10.
Choices B, C, and D are incorrect and may be the result of an error in solving
the quadratic equation that results when substituting 25 for y in the given
quadratic equation.
Explanation for question 11.
Choice B is correct. Since the angles marked
y degrees and
u degrees are
vertical angles,
y equals u. Subtracting the sides of
from the
corresponding sides of
x plus y equals, u plus w gives
x equals w. Since the angles marked
w degrees and
z degrees are vertical
angles,
w equals z. Therefore,
x equals z, and so statement I must be
true.
The equation in statement II need not be true. For example, if
x equals w, which equals z which equals t, which equals 70, and
y equals u, which equals 40, then all three pairs of vertical angles in the figure have
equal measure and the given condition
x plus y equals u plus w holds.
But it is not true in this case that y is equal to w. Therefore, statement II need not
be true.
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Since the top three angles in the figure form a straight angle, it follows that
x plus y plus z equals 180. Similarly,
w plus u plus t
equals 180, and so
x plus y plus z, equals, w plus u plus t.
Subtracting the sides of the given equation
x plus y equals u plus w
from the corresponding sides of
x plus y plus z, equals, w plus
u plus t gives
z equals t. Therefore, statement III must be true. Since
statements I and III only must be true, the correct answer is B.
Choices A, C, and D are incorrect because each of these choices includes
statement II, which need not be true.
Explanation for question 12.
Choice A is correct. The parabola with equation
y equals a,
parenthesis, x minus 2, close parenthesis, times, parenthesis, x plus 4, close
parenthesis, crosses the x-axis at the points (−4, 0) negative 4 comma 0 and (2, 0)
2 comma 0. The x-coordinate of the vertex of the parabola is halfway between the
x-coordinates of (−4, 0) negative 4 comma 0 and (2, 0) 2 comma 0. Thus, the
x-coordinate of the vertex is
the fraction with numerator negative 4
plus 2, and denominator 2, which equals negative 1. This is the value of c. To find
the y-coordinate of the vertex, substitute −1 negative 1 for x in
y equals a, parenthesis, x minus 2, close parenthesis, times,
parenthesis, x plus 4, close parenthesis:
y equals a, parenthesis x
minus 2 close parenthesis, times, parenthesis, x plus 4, close parenthesis, which
equals, a, parenthesis, negative 1 minus 2, close parenthesis, times, parenthesis,
negative 1 plus 4, close parenthesis, which equals a, parenthesis negative 3, close
parenthesis, times, parenthesis, 3, close parenthesis, which equals negative 9a
Therefore, the value of d is −9a negative 9a.
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Choice B is incorrect because the value of the constant term in the equation is not
the y-coordinate of the vertex, unless there were no linear term in the quadratic.
Choice C is incorrect and may be the result of a sign error in finding the
x-coordinate of the vertex.
Choice D is incorrect because the negative of the coefficient of the linear term in
the quadratic is not the y-coordinate of the vertex.
Explanation for question 13.
Choice B is correct. Since
by
ax minus 2 is equal to
negative 53 it is true that
24x squared plus 25x minus 47 divided
negative 8x minus 3 with remainder
parenthesis,
negative 8x minus 3, close parenthesis, times, parenthesis, a, x minus 2, close
parenthesis, minus 53, equals 24x squared plus 25x minus 47. (This can be seen by
multiplying each side of the given equation by
ax minus 2) This can be
rewritten as
negative 8ax squared, plus
16x, minus 3ax, equals 24x squared plus 25x minus 47. Since the coefficients of
the x squared -term have to be equal on both sides of the equation,
negative 8a, equals 24, or
a, equals negative 3.
Choices A, C, and D are incorrect and may be the result of either a conceptual
misunderstanding or a computational error when trying to solve for the value of a.
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Explanation for question 14.
Choice A is correct. Dividing each side of the given equation by 3 gives the
equivalent equation
x squared plus 4x plus 2, equals 0. Then using
quadratic formula,
the fraction whose numerator is negative b
plus or minus, the square root of b squared minus 4ac, end square root, and whose
denominator is 2 a with
a, equals 1,
b equals 4, and
c equals 2,
gives the solutions
of 2.
x equals negative 2 plus or minus the square root
Choices B, C, and D are incorrect and may be the result of errors when applying
the quadratic formula.
Explanation for question 15.
Choice D is correct. If C is graphed against F, the slope of the graph is equal to
five ninths degrees Celsius/degrees Fahrenheit, which means that for an increase of
1 degree Fahrenheit, the increase is
five ninths of 1 degree Celsius. Thus,
statement I is true. This is the equivalent to saying that an increase of 1 degree
Celsius is equal to an increase of
nine fifths degrees Fahrenheit. Since
nine fifths equals 1.8, statement II is true. On the other hand, statement III is not
true, since a temperature increase of
nine fifths degrees Fahrenheit, not
five
ninths degree Fahrenheit, is equal to a temperature increase of 1 degree Celsius.
Choice A, B, and C are incorrect because each of these choices omits a true
statement or includes a false statement.
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Explanation for question 16.
The correct answer is either 1 or 2. The given equation can be rewritten as
x raised to the power 5, end power, minus 5x cubed, plus 4x
equals 0. Since the polynomial expression on the left has no constant term, it has
x as a factor:
x, parenthesis, x raised to the power 4, end
power, minus 5x squared plus 4, close parenthesis, equals 0. The expression in
parentheses is a quadratic equation in
x squared that can be factored, giving
x, parenthesis, x squared minus 1, close parenthesis, times,
parenthesis, x squared minus 4, close parenthesis, equals 0. This further factors as
x, parenthesis, x minus 1, close parenthesis, times,
parenthesis, x plus 1, close parenthesis, times, parenthesis, x minus 2, close
parenthesis, times, parenthesis, x plus 2, close parenthesis, equals 0. The solutions
for x are
x equals 0,
x equals 1,
x equals negative 1,
x equals 2, and
x equals negative 2. Since it is given that
x is greater
than 0, the possible values of x are
x equals 1 and
x equals 2. Either
1 or 2 may be gridded as the correct answer.
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Explanation for question 17.
The correct answer is 2. First, clear the fractions from the given equation by
multiplying each side of the equation by 36 (the least common multiple of 4, 9, and
12). The equation becomes
28x minus 16x equals, 9 plus 15.
Combining like terms on each side of the equation yields
12x equals 24.
Finally, dividing both sides of the equation by 12 yields
x equals 2.
Alternatively, since
seven ninths x, minus four ninths x, equals
three ninths x, which equals one third x, and
one fourth
plus five twelfths, equals, three twelfths plus five twelfths, equals, eight twelfths,
which equals two thirds, the given equation simplifies to
two thirds. Multiplying each side of
yields
one third x equals
one third x equals two thirds by 3
x equals 2.
Explanation for question 18.
The correct answer is 105. Since
180 minus z equals 2y, and
y equals 75, it follows that
180 minus z equals 150, and so
z equals 30. Thus, each of the base angles of the isosceles triangle on the right has
measure
the fraction 180 degrees minus 30 degrees, over 2,
equals 75 degrees. Therefore, the measure of the angle marked
x degrees is
180 degrees minus 75 degrees, equals 105 degrees, and so the
value of x is 105.
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Explanation for question 19.
The correct answer is 370. A system of equations can be used, where h represents
the number of calories in a hamburger and f represents the number of calories in
an order of fries. The equation
2h plus 3f equals 1,700 represents
the fact that 2 hamburgers and 3 orders of fries contain a total of 1700 calories, and
the equation
h equals f plus 50 represents the fact that one hamburger
contains 50 more calories than an order of fries. Substituting
f plus 50 for
h in
2h plus 3f equals 1,700 gives
2, parenthesis, f plus 50, close parenthesis, plus 3 f, equals 1,700. This equation can
be solved as follows:
2f plus 100 plus 3f equals 1,700
5f plus 100 equals 1,700
5f equals 1,600
f equals 320
The number of calories in an order of fries is 320, so the number of calories in a
hamburger is 50 more than 320, or 370.
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Explanation for question 20.
The correct answer is
three fifths or .6. Triangle ABC is a right triangle with its
right angle at B. Thus,
line segment AC is the hypotenuse of right triangle
ABC, and
line segment AB and
line segment BC are the legs of right
triangle ABC. By the Pythagorean theorem,
AB equals, the square root of 20
squared minus 16 squared, end square root, which equals, the square root of 400
minus 256, end square root, which equals the square root of 144, which equals 12.
Since triangle DEF is similar to triangle ABC, with vertex F corresponding to
vertex C, the measure of angle F equals the measure of angle C. Thus,
sine F equals sine C. From the side lengths of triangle ABC,
sine C equals the fraction opposite side
over hypotenuse, which equals the fraction AB over AC, which equals the fraction
12 over 20, which equals three fifths. Therefore,
fifths. Either
sine F equals three
three fifths or its decimal equivalent, .6, may be gridded as the
correct answer
Stop. This is the end of the answers and explanations for questions 1
through 20.
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