MathDB
Problems
Contests
National and Regional Contests
USA Contests
MAA AMC
AMC 12/AHSME
1955 AMC 12/AHSME
1955 AMC 12/AHSME
Part of
AMC 12/AHSME
Subcontests
(50)
50
1
Hide problems
Passing Cars
In order to pass
B
B
B
going
40
40
40
mph on a two-lane highway,
A
A
A
, going
50
50
50
mph, must gain
30
30
30
feet. Meantime,
C
C
C
,
210
210
210
feet from
A
A
A
, is headed toward him at
50
50
50
mph. If
B
B
B
and
C
C
C
maintain their speeds, then, in order to pass safely,
A
A
A
must increase his speed by:
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A
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30 mph
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(
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10 mph
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5 mph
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(
D
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15 mph
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3 mph
<span class='latex-bold'>(A)</span>\ \text{30 mph} \qquad <span class='latex-bold'>(B)</span>\ \text{10 mph} \qquad <span class='latex-bold'>(C)</span>\ \text{5 mph} \qquad <span class='latex-bold'>(D)</span>\ \text{15 mph} \qquad <span class='latex-bold'>(E)</span>\ \text{3 mph}
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30 mph
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10 mph
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5 mph
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15 mph
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(
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3 mph
49
1
Hide problems
Intersection of Graphs
The graphs of y\equal{}\frac{x^2\minus{}4}{x\minus{}2} and y\equal{}2x intersect in:
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(
A
)
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1 point whose abscissa is 2
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1 point whose abscissa is 0
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C
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no points
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(
D
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two distinct points
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(
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two identical points
<span class='latex-bold'>(A)</span>\ \text{1 point whose abscissa is 2} \qquad <span class='latex-bold'>(B)</span>\ \text{1 point whose abscissa is 0}\\ <span class='latex-bold'>(C)</span>\ \text{no points} \qquad <span class='latex-bold'>(D)</span>\ \text{two distinct points} \qquad <span class='latex-bold'>(E)</span>\ \text{two identical points}
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(
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1 point whose abscissa is 2
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(
B
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<
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1 point whose abscissa is 0
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(
C
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no points
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x
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>
(
D
)
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two distinct points
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(
E
)
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two identical points
48
1
Hide problems
Correct Statements?
Given triangle
A
B
C
ABC
A
BC
with medians
A
E
AE
A
E
,
B
F
BF
BF
,
C
D
CD
C
D
;
F
H
FH
F
H
parallel and equal to
A
E
AE
A
E
;
B
H
BH
B
H
and
H
E
HE
H
E
are drawn;
F
E
FE
FE
extended meets
B
H
BH
B
H
in
G
G
G
. Which one of the following statements is not necessarily correct?
(A)
\ AEHF \text{ is a parallelogram} \qquad
(B)
\ HE\equal{}HG \\
(C)
\ BH\equal{}DC \qquad
(D)
\ FG\equal{}\frac{3}{4}AB \qquad
(E)
\ FG\text{ is a median of triangle }BFH
47
1
Hide problems
Comparing Expressions
The expressions a\plus{}bc and (a\plus{}b)(a\plus{}c) are:
(A)
\ \text{always equal} \qquad
(B)
\ \text{never equal} \qquad
(C)
\ \text{equal whenever }a\plus{}b\plus{}c\equal{}1 \\
(D)
\ \text{equal when }a\plus{}b\plus{}c\equal{}0 \qquad
(E)
\ \text{equal only when }a\equal{}b\equal{}c\equal{}0
46
1
Hide problems
Intersection of Graphs
The graphs of 2x\plus{}3y\minus{}6\equal{}0, 4x\minus{}3y\minus{}6\equal{}0, x\equal{}2, and y\equal{}\frac{2}{3} intersect in:
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6 points
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1 point
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2 points
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(
D
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no points
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(
E
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an unlimited number of points
<span class='latex-bold'>(A)</span>\ \text{6 points} \qquad <span class='latex-bold'>(B)</span>\ \text{1 point} \qquad <span class='latex-bold'>(C)</span>\ \text{2 points} \qquad <span class='latex-bold'>(D)</span>\ \text{no points} \\ <span class='latex-bold'>(E)</span>\ \text{an unlimited number of points}
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(
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6 points
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(
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1 point
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C
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2 points
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−
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(
D
)
<
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no points
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>
(
E
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an unlimited number of points
45
1
Hide problems
Sum of Terms in Sequence
Given a geometric sequence with the first term
≠
0
\neq 0
=
0
and
r
≠
0
r \neq 0
r
=
0
and an arithmetic sequence with the first term \equal{}0. A third sequence
1
,
1
,
2
…
1,1,2\ldots
1
,
1
,
2
…
is formed by adding corresponding terms of the two given sequences. The sum of the first ten terms of the third sequence is:
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)
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978
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(
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)
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557
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(
C
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467
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(
D
)
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1068
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(
E
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not possible to determine from the information given
<span class='latex-bold'>(A)</span>\ 978 \qquad <span class='latex-bold'>(B)</span>\ 557 \qquad <span class='latex-bold'>(C)</span>\ 467 \qquad <span class='latex-bold'>(D)</span>\ 1068 \\ <span class='latex-bold'>(E)</span>\ \text{not possible to determine from the information given}
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978
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(
B
)
<
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>
557
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(
C
)
<
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467
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(
D
)
<
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>
1068
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>
(
E
)
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not possible to determine from the information given
44
1
Hide problems
Relationship Between Angles
In circle
O
O
O
chord
A
B
AB
A
B
is produced so that
B
C
BC
BC
equals a radius of the circle.
C
O
CO
CO
is drawn and extended to
D
D
D
.
A
O
AO
A
O
is drawn. Which of the following expresses the relationship between
x
x
x
and
y
y
y
?[asy]size(200);defaultpen(linewidth(0.7)+fontsize(10)); pair O=origin, D=dir(195), A=dir(150), B=dir(30), C=B+1*dir(0); draw(O--A--C--D); dot(A^^B^^C^^D^^O); pair point=O; label("
A
A
A
", A, dir(point--A)); label("
B
B
B
", B, dir(point--B)); label("
C
C
C
", C, dir(point--C)); label("
D
D
D
", D, dir(point--D)); label("
O
O
O
", O, dir(285)); label("
x
x
x
", O+0.1*dir(172.5), dir(172.5)); label("
y
y
y
", C+0.4*dir(187.5), dir(187.5)); draw(Circle(O,1)); [/asy]
(A)
\ x\equal{}3y \\
(B)
\ x\equal{}2y \\
(C)
\ x\equal{}60^\circ \\
(D)
\ \text{there is no special relationship between }x\text{ and }y \\
(E)
\ x\equal{}2y \text{ or }x\equal{}3y\text{, depending upon the length of }AB
43
1
Hide problems
Solutions of Simultaneous Equations
The pairs of values of
x
x
x
and
y
y
y
that are the common solutions of the equations y\equal{}(x\plus{}1)^2 and xy\plus{}y\equal{}1 are:
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(
A
)
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3 real pairs
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(
B
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4 real pairs
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(
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4 imaginary pairs
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(
D
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2 real and 2 imaginary pairs
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(
E
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1 real and 2 imaginary pairs
<span class='latex-bold'>(A)</span>\ \text{3 real pairs} \qquad <span class='latex-bold'>(B)</span>\ \text{4 real pairs} \qquad <span class='latex-bold'>(C)</span>\ \text{4 imaginary pairs} \\ <span class='latex-bold'>(D)</span>\ \text{2 real and 2 imaginary pairs} \qquad <span class='latex-bold'>(E)</span>\ \text{1 real and 2 imaginary pairs}
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(
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)
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3 real pairs
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(
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4 real pairs
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>
(
C
)
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4 imaginary pairs
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>
(
D
)
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2 real and 2 imaginary pairs
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(
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)
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1 real and 2 imaginary pairs
42
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Equivalent Radical Expressions
If
a
a
a
,
b
b
b
, and
c
c
c
are positive integers, the radicals \sqrt{a\plus{}\frac{b}{c}} and
a
b
c
a\sqrt{\frac{b}{c}}
a
c
b
are equal when and only when:
(A)
\ a\equal{}b\equal{}c\equal{}1 \qquad
(B)
\ a\equal{}b\text{ and }c\equal{}a\equal{}1 \qquad
(C)
\ c\equal{}\frac{b(a^2\minus{}1)}{2} \\
(D)
\ a\equal{}b \text{ and }c\text{ is any value} \qquad
(E)
\ a\equal{}b \text{ and }c\equal{}a\minus{}1
41
1
Hide problems
Aytown to Beetown
A train traveling from Aytown to Beetown meets with an accident after
1
1
1
hr. It is stopped for
1
2
\frac{1}{2}
2
1
hr., after which it proceeds at four-fifths of its usual rate, arriving at Beetown
2
2
2
hr. late. If the train had covered
80
80
80
miles more before the accident, it would have been just
1
1
1
hr. late. The usual rate of the train is:
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(
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20 mph
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>
(
B
)
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>
30 mph
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>
(
C
)
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/
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40 mph
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s
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o
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>
(
D
)
<
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50 mph
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60 mph
<span class='latex-bold'>(A)</span>\ \text{20 mph} \qquad <span class='latex-bold'>(B)</span>\ \text{30 mph} \qquad <span class='latex-bold'>(C)</span>\ \text{40 mph} \qquad <span class='latex-bold'>(D)</span>\ \text{50 mph} \qquad <span class='latex-bold'>(E)</span>\ \text{60 mph}
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20 mph
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30 mph
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40 mph
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50 mph
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60 mph
40
1
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Unequal Fractions
The fractions \frac{ax\plus{}b}{cx\plus{}d} and
b
d
\frac{b}{d}
d
b
are unequal if:
(A)
\ a\equal{}c\equal{}1, x\neq 0 \qquad
(B)
\ a\equal{}b\equal{}0 \qquad
(C)
\ a\equal{}c\equal{}0 \\
(D)
\ x\equal{}0 \qquad
(E)
\ ad\equal{}bc
39
1
Hide problems
Least Possible Value
If y\equal{}x^2\plus{}px\plus{}q, then if the least possible value of
y
y
y
is zero
q
q
q
is equal to:
(A)
\ 0 \qquad
(B)
\ \frac{p^2}{4} \qquad
(C)
\ \frac{p}{2} \qquad
(D)
\ \minus{}\frac{p}{2} \qquad
(E)
\ \frac{p^2}{4}\minus{}q
38
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Averages and Numbers
Four positive integers are given. Select any three of these integers, find their arithmetic average, and add this result to the fourth integer. Thus the numbers
29
29
29
,
23
23
23
,
21
21
21
, and
17
17
17
are obtained. One of the original integers is:
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19
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21
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23
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29
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17
<span class='latex-bold'>(A)</span>\ 19 \qquad <span class='latex-bold'>(B)</span>\ 21 \qquad <span class='latex-bold'>(C)</span>\ 23 \qquad <span class='latex-bold'>(D)</span>\ 29 \qquad <span class='latex-bold'>(E)</span>\ 17
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19
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21
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23
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29
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17
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Digits of Number
A three-digit number has, from left to right, the digits
h
h
h
,
t
t
t
, and
u
u
u
, with
h
>
u
h>u
h
>
u
. When the number with the digits reversed is subtracted from the original number, the units' digit in the difference of
r
r
r
. The next two digits, from right to left, are:
<
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>
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A
)
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5 and 9
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(
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9 and 5
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impossible to tell
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5 and 4
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4 and 5
<span class='latex-bold'>(A)</span>\ \text{5 and 9} \qquad <span class='latex-bold'>(B)</span>\ \text{9 and 5} \qquad <span class='latex-bold'>(C)</span>\ \text{impossible to tell} \qquad <span class='latex-bold'>(D)</span>\ \text{5 and 4} \qquad <span class='latex-bold'>(E)</span>\ \text{4 and 5}
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5 and 9
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9 and 5
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impossible to tell
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5 and 4
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4 and 5
36
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Oil Tank
A cylindrical oil tank, lying horizontally, has an interior length of
10
10
10
feet and an interior diameter of
6
6
6
feet. If the rectangular surface of the oil has an area of
40
40
40
square feet, the depth of the oil is:
(A)
\ \sqrt{5} \qquad
(B)
\ 2\sqrt{5} \qquad
(C)
\ 3\minus{}\sqrt{5} \qquad
(D)
\ 3\plus{}\sqrt{5} \\
(E)
\ \text{either }3\minus{}\sqrt{5}\text{ or }3\plus{}\sqrt{5}
35
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Dividing Marbles
Three boys agree to divide a bag of marbles in the following manner. The first boy takes one more than half the marbles. The second takes a third of the number remaining. The third boy finds that he is left with twice as many marbles as the second boy. The original number of marbles:
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is none of the following
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cannot be determined from the given data
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is 20 or 26
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is 14 or 32
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is 8 or 38
<span class='latex-bold'>(A)</span>\ \text{is none of the following} \qquad <span class='latex-bold'>(B)</span>\ \text{cannot be determined from the given data}\\ <span class='latex-bold'>(C)</span>\ \text{is 20 or 26} \qquad <span class='latex-bold'>(D)</span>\ \text{is 14 or 32} \qquad <span class='latex-bold'>(E)</span>\ \text{is 8 or 38}
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is none of the following
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cannot be determined from the given data
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is 20 or 26
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is 14 or 32
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is 8 or 38
34
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Wire around Poles
A
6
6
6
-inch and
18
18
18
-inch diameter pole are placed together and bound together with wire. The length of the shortest wire that will go around them is:
(A)
\ 12\sqrt{3}\plus{}16\pi \qquad
(B)
\ 12\sqrt{3}\plus{}7\pi \qquad
(C)
\ 12\sqrt{3}\plus{}14\pi \\
(D)
\ 12\plus{}15\pi \qquad
(E)
\ 24\pi
33
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Hands of a Clock
Henry starts a trip when the hands of the clock are together between
8
8
8
a.m. and
9
9
9
a.m. He arrives at his destination between
2
2
2
p.m. and
3
3
3
p.m. when the hands of the clock are exactly
18
0
∘
180^\circ
18
0
∘
apart. The trip takes:
(A)
\ \text{6 hr.} \qquad
(B)
\ \text{6 hr. 43\minus{}7/11 min.} \qquad
(C)
\ \text{5 hr. 16\minus{}4/11 min.} \qquad
(D)
\ \text{6 hr. 30 min.} \qquad
(E)
\ \text{none of these}
32
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Discriminant of Quadratic
If the discriminant of ax^2\plus{}2bx\plus{}c\equal{}0 is zero, then another true statement about
a
a
a
,
b
b
b
, and
c
c
c
is that:
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they form an arithmetic progression
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(
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they form a geometric progression
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C
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they are unequal
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they are all negative numbers
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(
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only b is negative and a and c are positive
<span class='latex-bold'>(A)</span>\ \text{they form an arithmetic progression} \\ <span class='latex-bold'>(B)</span>\ \text{they form a geometric progression} \\ <span class='latex-bold'>(C)</span>\ \text{they are unequal} \\ <span class='latex-bold'>(D)</span>\ \text{they are all negative numbers} \\ <span class='latex-bold'>(E)</span>\ \text{only b is negative and a and c are positive}
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)
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they form an arithmetic progression
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(
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)
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they form a geometric progression
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they are unequal
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they are all negative numbers
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(
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)
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only b is negative and a and c are positive
31
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Length of Median of Trapezoid
An equilateral triangle whose side is
2
2
2
is divided into a triangle and a trapezoid by a line drawn parallel to one of its sides. If the area of the trapezoid equals one-half of the area of the original triangle, the length of the median of the trapezoid is:
(A)
\ \frac{\sqrt{6}}{2} \qquad
(B)
\ \sqrt{2} \qquad
(C)
\ 2\plus{}\sqrt{2} \qquad
(D)
\ \frac{2\plus{}\sqrt{2}}{2} \qquad
(E)
\ \frac{2\sqrt{3}\minus{}\sqrt{6}}{2}
30
1
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Type of Roots
Each of the equations 3x^2\minus{}2\equal{}25, (2x\minus{}1)^2\equal{}(x\minus{}1)^2, \sqrt{x^2\minus{}7}\equal{}\sqrt{x\minus{}1} has:
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(
A
)
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two integral roots
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(
B
)
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no root greater than 3
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no root zero
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(
D
)
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only one root
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(
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one negative root and one positive root
<span class='latex-bold'>(A)</span>\ \text{two integral roots} \qquad <span class='latex-bold'>(B)</span>\ \text{no root greater than 3} \qquad <span class='latex-bold'>(C)</span>\ \text{no root zero} \\ <span class='latex-bold'>(D)</span>\ \text{only one root} \qquad <span class='latex-bold'>(E)</span>\ \text{one negative root and one positive root}
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)
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two integral roots
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(
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)
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no root greater than 3
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(
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)
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no root zero
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(
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)
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only one root
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>
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one negative root and one positive root
29
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Measure of Angle
In the figure,
P
A
PA
P
A
is tangent to semicircle
S
A
R
SAR
S
A
R
;
P
B
PB
PB
is tangent to semicircle
R
B
T
RBT
RBT
;
S
R
T
SRT
SRT
is a straight line; the arcs are indicated in the figure. Angle
A
P
B
APB
A
PB
is measured by: [asy]unitsize(1.2cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=3;pair O1=(0,0), O2=(3,0), Sp=(-2,0), R=(2,0), T=(4,0); pair A=O1+2*dir(60), B=O2+dir(85); pair Pa=rotate(90,A)*O1, Pb=rotate(-90,B)*O2; pair P=extension(A,Pa,B,Pb); pair[] dots={Sp,R,T,A,B,P};draw(P--P+5*(A-P)); draw(P--P+5*(B-P)); clip((-2,0)--(-2,2.5)--(4,2.5)--(4,0)--cycle); draw(Arc(O1,2,0,180)--cycle); draw(Arc(O2,1,0,180)--cycle);dot(dots); label("
S
S
S
",Sp,S); label("
R
R
R
",R,S); label("
T
T
T
",T,S); label("
A
A
A
",A,NE); label("
B
B
B
",B,N); label("
P
P
P
",P,NNE); label("
a
a
a
",midpoint(Arc(O1,2,0,60)),SW); label("
b
b
b
",midpoint(Arc(O2,1,85,180)),SE); label("
c
c
c
",midpoint(Arc(O1,2,60,180)),SE); label("
d
d
d
",midpoint(Arc(O2,1,0,85)),SW);[/asy]
(A)
\ \frac {1}{2}(a \minus{} b) \qquad
(B)
\ \frac {1}{2}(a \plus{} b) \qquad
(C)
\ (c \minus{} a) \minus{} (d \minus{} b) \qquad
(D)
\ a \minus{} b \qquad
(E)
\ a \plus{} b
28
1
Hide problems
Two Graphs' Intersection
On the same set of axes are drawn the graph of y\equal{}ax^2\plus{}bx\plus{}c and the graph of the equation obtained by replacing
x
x
x
by \minus{}x in the given equation. If
b
≠
0
b \neq 0
b
=
0
and
c
≠
0
c \neq 0
c
=
0
these two graphs intersect:
(A)
\ \text{in two points, one on the x\minus{}axis and one on the y\minus{}axis}\\
(B)
\ \text{in one point located on neither axis} \\
(C)
\ \text{only at the origin} \\
(D)
\ \text{in one point on the x\minus{}axis} \\
(E)
\ \text{in one point on the y\minus{}axis}
27
1
Hide problems
Roots of Quadratic
If
r
r
r
and
s
s
s
are the roots of x^2\minus{}px\plus{}q\equal{}0, then r^2\plus{}s^2 equals:
(A)
\ p^2\plus{}2q \qquad
(B)
\ p^2\minus{}2q \qquad
(C)
\ p^2\plus{}q^2 \qquad
(D)
\ p^2\minus{}q^2 \qquad
(E)
\ p^2
26
1
Hide problems
Buying and Selling a House
Mr. A owns a house worth
$
10000
\$10000
$10000
. He sells it to Mr. B at
10
%
10 \%
10%
profit. Mr. B sells the house back to Mr. A at a
10
%
10 \%
10%
loss. Then:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
Mr. A comes out even
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
Mr. A makes
$
100
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
Mr. A makes
$
1000
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
Mr. B loses
$
100
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
none of the above is correct
<span class='latex-bold'>(A)</span>\ \text{Mr. A comes out even} \qquad <span class='latex-bold'>(B)</span>\ \text{Mr. A makes }\$100 \qquad <span class='latex-bold'>(C)</span>\ \text{Mr. A makes }\$1000 \\ <span class='latex-bold'>(D)</span>\ \text{Mr. B loses }\$100 \qquad <span class='latex-bold'>(E)</span>\ \text{none of the above is correct}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
Mr. A comes out even
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
Mr. A makes
$100
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
Mr. A makes
$1000
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
Mr. B loses
$100
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
none of the above is correct
25
1
Hide problems
Factors of Polynomial
One of the factors of x^4\plus{}2x^2\plus{}9 is:
(A)
\ x^2\plus{}3 \qquad
(B)
\ x\plus{}1 \qquad
(C)
\ x^2\minus{}3 \qquad
(D)
\ x^2\minus{}2x\minus{}3 \qquad
(E)
\ \text{none of these}
24
1
Hide problems
Behavior of Function
The function 4x^2\minus{}12x\minus{}1:
(A)
\ \text{always increases as }x\text{ increases}\\
(B)
\ \text{always decreases as }x\text{ decreases to 1} \\
(C)
\ \text{cannot equal 0} \\
(D)
\ \text{has a maximum value when }x\text{ is negative} \\
(E)
\ \text{has a minimum value of \minus{}10}
23
1
Hide problems
Clerk Counts Incorrectly
In checking the petty cash a clerk counts
q
q
q
quarters,
d
d
d
dimes,
n
n
n
nickels, and
c
c
c
cents. Later he discovers that
x
x
x
of the nickels were counted as quarters and
x
x
x
of the dimes were counted as cents. To correct the total obtained the clerk must:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
make no correction
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
subtract 11 cents
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
subtract 11
x
cents
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
add 11
x
cents
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
add
x
cents
<span class='latex-bold'>(A)</span>\ \text{make no correction} \qquad <span class='latex-bold'>(B)</span>\ \text{subtract 11 cents} \qquad <span class='latex-bold'>(C)</span>\ \text{subtract 11}x\text{ cents} \\ <span class='latex-bold'>(D)</span>\ \text{add 11}x\text{ cents} \qquad <span class='latex-bold'>(E)</span>\ \text{add }x\text{ cents}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
make no correction
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
subtract 11 cents
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
subtract 11
x
cents
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
add 11
x
cents
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
add
x
cents
22
1
Hide problems
Three Successive Discounts
On a
$
10000
\$10000
$10000
order a merchant has a choice between three successive discounts of
20
%
20 \%
20%
,
20
%
20 \%
20%
, and
10
%
10\%
10%
and three successive discounts of
40
%
40 \%
40%
,
5
%
5 \%
5%
, and
5
%
5 \%
5%
. By choosing the better offer, he can save:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
nothing at all
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
$
440
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
$
330
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
$
345
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
$
360
<span class='latex-bold'>(A)</span>\ \text{nothing at all} \qquad <span class='latex-bold'>(B)</span>\ \$440 \qquad <span class='latex-bold'>(C)</span>\ \$330 \qquad <span class='latex-bold'>(D)</span>\ \$345 \qquad <span class='latex-bold'>(E)</span>\ \$360
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
nothing at all
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
$440
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
$330
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
$345
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
$360
21
1
Hide problems
Altitude on Hypotenuse
Represent the hypotenuse of a right triangle by
c
c
c
and the area by
A
A
A
. The atltidue on the hypotenuse is:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
A
c
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
2
A
c
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
A
2
c
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
A
2
c
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
A
c
2
<span class='latex-bold'>(A)</span>\ \frac{A}{c} \qquad <span class='latex-bold'>(B)</span>\ \frac{2A}{c} \qquad <span class='latex-bold'>(C)</span>\ \frac{A}{2c} \qquad <span class='latex-bold'>(D)</span>\ \frac{A^2}{c} \qquad <span class='latex-bold'>(E)</span>\ \frac{A}{c^2}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
c
A
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
c
2
A
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2
c
A
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
c
A
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
c
2
A
20
1
Hide problems
Expression Equals Zero
The expression \sqrt{25\minus{}t^2}\plus{}5 equals zero for:
(A)
\ \text{no real or imaginary values of }t \qquad
(B)
\ \text{no real values of }t\text{ only} \\
(C)
\ \text{no imaginary values of }t\text{ only} \qquad
(D)
\ t\equal{}0 \qquad
(E)
\ t\equal{}\pm 5
19
1
Hide problems
Roots of Equation
Two numbers whose sum is
6
6
6
and the absolute value of whose difference is
8
8
8
are roots of the equation:
(A)
\ x^2\minus{}6x\plus{}7\equal{}0 \qquad
(B)
\ x^2\minus{}6x\minus{}7\equal{}0 \qquad
(C)
\ x^2\plus{}6x\minus{}8\equal{}0 \\
(D)
\ x^2\minus{}6x\plus{}8\equal{}0 \qquad
(E)
\ x^2\plus{}6x\minus{}7\equal{}0
18
1
Hide problems
Discriminant of Equation
The discriminant of the equation x^2\plus{}2x\sqrt{3}\plus{}3\equal{}0 is zero. Hence, its roots are:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
real and equal
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
rational and equal
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
rational and unequal
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
irrational and unequal
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
imaginary
<span class='latex-bold'>(A)</span>\ \text{real and equal} \qquad <span class='latex-bold'>(B)</span>\ \text{rational and equal} \qquad <span class='latex-bold'>(C)</span>\ \text{rational and unequal} \\ <span class='latex-bold'>(D)</span>\ \text{irrational and unequal} \qquad <span class='latex-bold'>(E)</span>\ \text{imaginary}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
real and equal
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
rational and equal
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
rational and unequal
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
irrational and unequal
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
imaginary
17
1
Hide problems
Logarithmic Equation
If \log x\minus{}5 \log 3\equal{}\minus{}2, then
x
x
x
equals:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
1.25
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
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(
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0.81
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2.43
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either 0.8 or 1.25
<span class='latex-bold'>(A)</span>\ 1.25 \qquad <span class='latex-bold'>(B)</span>\ 0.81 \qquad <span class='latex-bold'>(C)</span>\ 2.43 \qquad <span class='latex-bold'>(D)</span>\ 0.8 \qquad <span class='latex-bold'>(E)</span>\ \text{either 0.8 or 1.25}
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2.43
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either 0.8 or 1.25
16
1
Hide problems
Evaluating Expression
The value of \frac{3}{a\plus{}b} when a\equal{}4 and b\equal{}\minus{}4 is:
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3
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3
8
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any finite number
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meaningless
<span class='latex-bold'>(A)</span>\ 3 \qquad <span class='latex-bold'>(B)</span>\ \frac{3}{8} \qquad <span class='latex-bold'>(C)</span>\ 0 \qquad <span class='latex-bold'>(D)</span>\ \text{any finite number} \qquad <span class='latex-bold'>(E)</span>\ \text{meaningless}
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8
3
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any finite number
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meaningless
15
1
Hide problems
Difference Between Radii
The ratio of the areas of two concentric circles is
1
:
3
1: 3
1
:
3
. If the radius of the smaller is
r
r
r
, then the difference between the radii is best approximated by:
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0.41
r
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0.73
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0.75
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0.73
r
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0.75
r
<span class='latex-bold'>(A)</span>\ 0.41r \qquad <span class='latex-bold'>(B)</span>\ 0.73 \qquad <span class='latex-bold'>(C)</span>\ 0.75 \qquad <span class='latex-bold'>(D)</span>\ 0.73r \qquad <span class='latex-bold'>(E)</span>\ 0.75r
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0.41
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0.73
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0.73
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0.75
r
14
1
Hide problems
Ratio of Areas
The length of rectangle R is
10
10
10
percent more than the side of square S. The width of the rectangle is
10
10
10
percent less than the side of the square. The ratio of the areas, R:S, is:
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99
:
100
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101
:
100
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1
:
1
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199
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201
:
200
<span class='latex-bold'>(A)</span>\ 99: 100 \qquad <span class='latex-bold'>(B)</span>\ 101: 100 \qquad <span class='latex-bold'>(C)</span>\ 1: 1 \qquad <span class='latex-bold'>(D)</span>\ 199: 200 \qquad <span class='latex-bold'>(E)</span>\ 201: 200
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201
:
200
13
1
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Simplifying A Fraction
The fraction \frac{a^{\minus{}4}\minus{}b^{\minus{}4}}{a^{\minus{}2}\minus{}b^{\minus{}2}} is equal to:
(A)
\ a^{\minus{}6}\minus{}b^{\minus{}6} \qquad
(B)
\ a^{\minus{}2}\minus{}b^{\minus{}2} \qquad
(C)
\ a^{\minus{}2}\plus{}b^{\minus{}2} \\
(D)
\ a^2\plus{}b^2 \qquad
(E)
\ a^2\minus{}b^2
12
1
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Solving Equation with Radicals
The solution of \sqrt{5x\minus{}1}\plus{}\sqrt{x\minus{}1}\equal{}2 is:
(A)
\ x\equal{}2,x\equal{}1 \qquad
(B)
\ x\equal{}\frac{2}{3} \qquad
(C)
\ x\equal{}2 \qquad
(D)
\ x\equal{}1 \qquad
(E)
\ x\equal{}0
11
1
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Negation of Statement
The negation of the statement "No slow learners attend this school" is:
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All slow learners attend this school
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All slow learners do not attend this school
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Some slow learners attend this school
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Some slow learners do not attend this school
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No slow learners do not attend this school
<span class='latex-bold'>(A)</span>\ \text{All slow learners attend this school} \\ <span class='latex-bold'>(B)</span>\ \text{All slow learners do not attend this school} \\ <span class='latex-bold'>(C)</span>\ \text{Some slow learners attend this school} \\ <span class='latex-bold'>(D)</span>\ \text{Some slow learners do not attend this school} \\ <span class='latex-bold'>(E)</span>\ \text{No slow learners do not attend this school}
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All slow learners attend this school
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All slow learners do not attend this school
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Some slow learners attend this school
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Some slow learners do not attend this school
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No slow learners do not attend this school
10
1
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Time Train Takes
How many hours does it take a train traveling at an average rate of
40
40
40
mph between stops to travel
a
a
a
miles it makes
n
n
n
stops of
m
m
m
minutes each?
(A)
\ \frac{3a\plus{}2mn}{120} \qquad
(B)
\ 3a\plus{}2mn \qquad
(C)
\ \frac{3a\plus{}2mn}{12} \qquad
(D)
\ \frac{a\plus{}mn}{40} \qquad
(E)
\ \frac{a\plus{}40mn}{40}
9
1
Hide problems
Radius of Circle
A circle is inscribed in a triangle with sides
8
8
8
,
15
15
15
, and
17
17
17
. The radius of the circle is:
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5
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3
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7
<span class='latex-bold'>(A)</span>\ 6 \qquad <span class='latex-bold'>(B)</span>\ 2 \qquad <span class='latex-bold'>(C)</span>\ 5 \qquad <span class='latex-bold'>(D)</span>\ 3 \qquad <span class='latex-bold'>(E)</span>\ 7
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6
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5
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7
8
1
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Graph of Equation
The graph of x^2\minus{}4y^2\equal{}0:
(A)
\ \text{is a hyperbola intersecting only the }x\text{ \minus{}axis} \\
(B)
\ \text{is a hyperbola intersecting only the }y\text{ \minus{}axis} \\
(C)
\ \text{is a hyperbola intersecting neither axis} \\
(D)
\ \text{is a pair of straight lines} \\
(E)
\ \text{does not exist}
7
1
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Cut in Wages
If a worker receives a
20
20
20
percent cut in wages, he may regain his original pay exactly by obtaining a raise of:
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20 percent
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25 percent
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22
1
2
percent
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)
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$
20
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$
25
<span class='latex-bold'>(A)</span>\ \text{20 percent} \qquad <span class='latex-bold'>(B)</span>\ \text{25 percent} \qquad <span class='latex-bold'>(C)</span>\ 22\frac{1}{2} \text{ percent} \qquad <span class='latex-bold'>(D)</span>\ \$20 \qquad <span class='latex-bold'>(E)</span>\ \$25
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20 percent
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25 percent
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22
2
1
percent
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$20
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$25
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Merchant Selling Oranges
A merchant buys a number of oranges at
3
3
3
for
10
10
10
cents and an equal number at
5
5
5
for
20
20
20
cents. To "break even" he must sell all at:
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8 for 30 cents
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(
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3 for 11 cents
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5 for 18 cents
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11 for 40 cents
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13 for 50 cents
<span class='latex-bold'>(A)</span>\ \text{8 for 30 cents} \qquad <span class='latex-bold'>(B)</span>\ \text{3 for 11 cents} \qquad <span class='latex-bold'>(C)</span>\ \text{5 for 18 cents} \\ <span class='latex-bold'>(D)</span>\ \text{11 for 40 cents} \qquad <span class='latex-bold'>(E)</span>\ \text{13 for 50 cents}
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8 for 30 cents
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3 for 11 cents
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5 for 18 cents
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11 for 40 cents
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13 for 50 cents
5
1
Hide problems
Varies Inversely
y
y
y
varies inversely as the square of
x
x
x
. When y\equal{}16, x\equal{}1. When x\equal{}8,
y
y
y
equals:
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2
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128
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64
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1
4
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1024
<span class='latex-bold'>(A)</span>\ 2 \qquad <span class='latex-bold'>(B)</span>\ 128 \qquad <span class='latex-bold'>(C)</span>\ 64 \qquad <span class='latex-bold'>(D)</span>\ \frac{1}{4} \qquad <span class='latex-bold'>(E)</span>\ 1024
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128
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64
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4
1
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1024
4
1
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Equation
The equality \frac{1}{x\minus{}1}\equal{}\frac{2}{x\minus{}2} is satisfied by:
(A)
\ \text{no real values of }x \qquad
(B)
\ \text{either }x\equal{}1 \text{ or }x\equal{}2 \qquad
(C)
\ \text{only }x\equal{}1 \\
(D)
\ \text{only }x\equal{}2 \qquad
(E)
\ \text{only }x\equal{}0
3
1
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Average of Ten Numbers
If each number in a set of ten numbers is increased by
20
20
20
, the arithmetic mean (average) of the original ten numbers:
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is increased by 20
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is increased by 200
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is increased by 10
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<span class='latex-bold'>(A)</span>\ \text{remains the same} \qquad <span class='latex-bold'>(B)</span>\ \text{is increased by 20} \qquad <span class='latex-bold'>(C)</span>\ \text{is increased by 200} \\ <span class='latex-bold'>(D)</span>\ \text{is increased by 10} \qquad <span class='latex-bold'>(E)</span>\ \text{is increased by 2}
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is increased by 20
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is increased by 200
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is increased by 10
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is increased by 2
2
1
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Angle Formed by Clock's Hands
The smaller angle between the hands of a clock at
12
:
25
12: 25
12
:
25
p.m. is:
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13
7
∘
<span class='latex-bold'>(A)</span>\ 132^\circ 30' \qquad <span class='latex-bold'>(B)</span>\ 137^\circ 30' \qquad <span class='latex-bold'>(C)</span>\ 150^\circ \qquad <span class='latex-bold'>(D)</span>\ 137^\circ 32' \qquad <span class='latex-bold'>(E)</span>\ 137^\circ
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1
1
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Equivalent Expressions
Which one of the following is not equivalent to
0.000000375
0.000000375
0.000000375
?
(A)
\ 3.75 \times 10^{\minus{}7} \qquad
(B)
\ 3 \frac{3}{4} \times 10^{\minus{}7} \qquad
(C)
\ 375 \times 10^{\minus{}9} \\
(D)
\ \frac{3}{8} \times 10^{\minus{}7} \qquad
(E)
\ \frac{3}{80000000}