__
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_
Choices B, C, and D are incorrect because
g
is a function and there can
only be one value of
(
g
ơ
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
[
(the number of weeks after July 1) when the two
prices were equal. The prices were equal when
E
=
F
; that is, when
[
[
. This last equation is equivalent to
[
, and so
[
=
= 4. Then to determine
E
, the price per
pound of beef, substitute 4 for
[
in
E [
, which gives
E GROODUVSHUSRXQG
Choice A is incorrect. It results from substituting the value 1, not 4, for
[
in
E [
. Choice B is incorrect. It
results from substituting
the value 2, not 4, for
[
in
E [
. Choice C is incorrect. It
UHVXOWVIURPVXEVWLWXWLQJWKHYDOXHQRWIRU
[
in
F [
.
QUESTION 12
Choice D is correct.
In the
[\
-plane, all lines that pass through
the origin are of the form
y
=
P[
, where
P
is the slope of the line.
_
1
Therefore, the equation of this line is
y
=
[
, or
[
= 7
y
. A point
7
with coordinates (
D
,
E
) will lie on the line if and only if
D
= 7
E2IWKH
JLYHQFKRLFHVRQO\FKRLFH'VDWLVƮHVWKLVFRQGLWLRQ
&KRLFH$LVLQFRUUHFWEHFDXVHWKHOLQHGHWHUPLQHGE\WKHRULJLQ
DQGLVWKHYHUWLFDOOLQHZLWKHTXDWLRQ
[ WKDWLVWKH
y
-axis.
_
The slope of the
y
D[LVLVXQGHƮQHGQRW
1
7
7KHUHIRUHWKHSRLQW
_
does not lie on the line that passes the origin and has slope 17.
Choices B and C are incorrect because neither of the ordered pairs has
_
a
y
-coordinate that is 17 the value of the corresponding
[
-coordinate.
QUESTION 13
1
Choice B is correct.
To
rewrite
—
, multiply
[
1
+ 2 +
1
—
—
[
(
[
+ 2)(
[
(
[
+ 2)(
[
by
—
. This results in the expression
, which
(
[
+ 2)(
[
(
[
+ 2) + (
[
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.
48(67,21
_
Choice A is correct.
2QHDSSURDFKLVWRH[SUHVV
8
[
so that the
2
y
numerator and denominator are expressed with the same base. Since
2 and 8 are both powers of 2, substituting 2
for 8 in the numerator
_
_
_
_
_
(2
)
[
of 8
[
gives
, which can be rewritten as 2
[
. Since the numerator
2
y
2
y
2
y
_
and denominator of 2
[
have a common base, this expression can be
2
y
rewritten as 2
[ơ\
,WLVJLYHQWKDW
[ơ
y
= 12, so one can substitute 12
_
for
WKHH[SRQHQW
[ơ
y
, given that the expression 8
[
is equal to 2
12
.
2
y
Choice B is incorrect. The expression 8
[
can be rewritten as 22
y
[
, or
2
y
2
[ơ\
. If the value of 2
[ơ\
is 4
4
, which can be rewritten as 28, then
2
[ơ\
= 2
8
,
ZKLFKUHVXOWVLQ
[ơ
y
= 8, not 12. Choice C is incorrect. If
_
the value of 8
[
is 8
2
, then 2
[ơ
y
= 8
2
ZKLFKUHVXOWVLQ
[ơ
y
= 6, not 12.
2
y
_
Choice D is incorrect because the value of 82
[
y
can be determined.
QUESTION 15
Choice D is correct.
2QHFDQƮQGWKHSRVVLEOHYDOXHVRI
D
and
E
in (
D[
+ 2) (
E[
+ 7)
by using the given equation
D
+
E
= 8 and
ƮQGLQJDQRWKHUHTXDWLRQWKDWUHODWHVWKHYDULDEOHV
D
and
E
. Since
(
D[
+ 2)(
E[
+ 7) = 15
[
2
+
F[
+ 14, one can expand the left side of the
equation to obtain
DE[
2
+ 7
D[
+ 2
E[
+ 14 = 15
[
2
+
F[
+ 14. Since
DE
is the
FRHƱFLHQWRI
[
2
RQWKHOHIWVLGHRIWKHHTXDWLRQDQGLVWKHFRHƱFLHQW
of
[
2
on the right side of the equation, it must be true that
DE
= 15. Since
D
+
E
= 8, it follows that
E ơ
D
. Thus,
DE
= 15 can be rewritten as
D
ơ
D
) = 15, which
in turn can be rewritten as
D
2
ơ
D )DFWRULQJ
gives (
Dơ
Dơ 7KXVHLWKHU
D DQG
E
= 5, or
D
= 5 and
E ,I
D DQG
E
= 5, then (
D[
+ 2)(
E[ [
+ 2)(5
[
+ 7) = 15
[
2
[
+ 14.
Thus, one of the possible values of
FLV,I
D
= 5 and
E WKHQ
(
D[
+ 2)(
E[
+ 7) = (5
[[
+ 7) = 15
[
2
+ 41
[
+ 14. Thus, another possible
value for
F
is 41. Therefore, the two possible values for
FDUHDQG
&KRLFH$LVLQFRUUHFWWKHQXPEHUVDQGDUHSRVVLEOHYDOXHVIRU
D
and
E
, but not possible values for
F
. Choice B is incorrect; if
D
= 5
and
E WKHQDQGDUHWKHFRHƱFLHQWVRI
[
when the expression
(5
[[
+ 7) is expanded as 15
[
2
[
+ 6
[
+ 14. However, when
WKHFRHƱFLHQWVRI
[DUHDQGWKHYDOXHRI
F
is 41 and not 6 and
&KRLFH&LVLQFRUUHFWLI
D DQG
E WKHQDQGDUHWKH
FRHƱFLHQWVRI
[ZKHQWKHH[SUHVVLRQ
[
+ 2)(5
[
+ 7) is expanded as
15
[
2
+ 21
[[+RZHYHUZKHQWKHFRHƱFLHQWVRI
[DUHDQG
21, the value of
FLVDQGQRWDQG
QUESTION 16
The correct answer is 2.
To solve for
t
, factor the left side of
t
2
ơ
giving (
t
ơ
t
7KHUHIRUHHLWKHU
t
ơ RU
t
,I
t
ơ WKHQ
t
= 2, and if
t
WKHQ
t
ơ6LQFHLWLVJLYHQWKDW
t
!WKHYDOXHRI
t
must be 2.
Another way to solve for
t
is to add 4 to both sides of
t
2
ơ JLYLQJ
t
2
= 4.
Then, taking the square root of the left and the right side of the equation
_
gives
t
= ±
Ƥ
4 = ± 2. Since it is given that
t
!WKHYDOXHRI
t
must be 2.
$16:(5(;3/$1$7,216
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6$73UDFWLFH7HVW
QUESTION 17
The correct answer is 1600.
It is given that
∠
$(%
and
∠
&'%
have
the same measure. Since
∠
$%(
and
∠
&%'
are vertical angles, they
have the same measure. Therefore,
triangle
($%
is similar to triangle
'&%
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
CD
=
(%
6XEVWLWXWLQJWKHJLYHQYDOXHVRIIRU
CD
,
[
_
_
%'
_
_
for
%'DQGIRU
(%
in
CD
=
(%
gives
=
. Therefore,
[
[
(
[
=
—
QUESTION 18
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