1. In the figure below, the graph of y = kx2 intersects
triangle ABC at B. If AB = BC and the area of triangle ABC is 6, what is the value of k?
4. If (x − h)(x + k) = x2 − 16, what is the value of h + k?
(A) −8
(B) −4
y = kx2
A (–1,0)
(C)
0
(D)
4
(E)
8
B
7
6
5
4
3
2
1
C (3,0)
-3 -2 -1
-1
-2
2. For all n, let n∗ be defined as n∗ = n2 − n − 2. If a
and b are positive integers and a , b, which of the
following could be true?
1 2 3 4 5 6 7
5. A portion of the graph of the function h is shown in
the xy-plane above. If h(4) − h(s) = 4, which of the
following could be the value of s?
I. (a + b)∗ is even.
II. (a + b)∗ is odd.
III. (a − b)∗ is negative.
(A) −2
(B) 0
(C) 1
(D) 4
3. The graphs of two linear functions, f and g, are perpendicular. If f (2) = 4 and f (−4) = 1, which of the
following could define g?
(E) 7
1
(A) g(x) = − x + 4
2
(B) g(x) = −x + 1
(C) g(x) = −2x + 4
(D) g(x) = 2x − 3
(E) g(x) = x + 5
1
9. The value of an investment increased by 20 percent
to $800 by the end of the first week. To the nearest
dollar, what was the value of the investment at the
beginning of the week?
Q
O
7
P
6
5
y = g(x)
4
6. In the figure above, three identical circles of radius
8 with centers O, P and Q are tangent to each other.
What is the area of the shaded region bound by the
circles?
3
y = f (x)
2
1
-3
-2
-1
1
2
3
4
5
6
7
-1
7. The participants in a certain community service club
come from all four grade levels in Smithville High
School. There are equal numbers of sophomores
and juniors. The number of seniors is twenty more
than twice the number of sophomores and juniors
combined, and there are 20 freshmen. If the number
of sophomores, juniors and seniors combined
accounts for 80 percent of the club’s participants,
how many seniors are there in the club?
-2
10. The figure above shows the graphs of the functions f
and g. If g is defined in terms of a transformation of
h
f such that g(x) = f (x+h)+k, what is the value of ?
k
11. For all x and y, let the operation x _ y be equal to the
2x
whole number remainder obtained when finding .
y
If s is an even integer and the value of 7 _ s = 2,
8. For integers a and b where a > 0 and b ≥ 0, let the
ab
operation be defined as a b = . Which of the
a
following could be the value of a b?
what is a possible value of s?
I. ab · a−1
II. 0
III. 1
2
O
P
30º
A
S
B
E
R
Q
(A)
π
2
(B)
π
4
(C)
π
8
(D)
π
12
(E)
π
15
D
(–1,0)
12. In the figure above, OPQR is a square, and PS is the
arc of a circle with center O. If the length of arc PS
is π, the area of the shaded sector is what fraction of
the area of OPQR?
C
15. In the figure above, rectangle ABCD lies on the
coordinate plane with point D located at the origin.
If AB = AE = ED, what is the area of quadrilateral
BCDE?
B
105º
C
A
D
F
150º
E
Note: Figure not drawn to scale.
13. Jack drove to work in the morning at an average
speed of 45 miles per hour. He returned home in
the evening along the same route and averaged 30
miles per hour. If Jack spent a total of one hour
commuting to and from work, how many miles did
he drive to work in the morning?
16. In the figure above, triangle ABC and parallelogram
CDEF are constructed with B, C, and F along the
same line.
√ If the ratio of CF : BC = 1 : 2 and
DE = 2 3, what is the area of triangle ABC?
√
(A) 3 2
(B) 6
√
√
(C) 6 3 − 3 2
√
(D) 6 3
√
(E) 6 3 + 6
14. A student is weighing large and small marbles.
While weighing one large marble with 8 small marbles, the student discovers that the large marble’s
weight is 5 times the average (arithmetic mean) of
the weight of the small marbles. The large marble’s
weight is what fraction of the total weight of the 9
marbles?
3
17. The function f is defined as f (x) = (x − n)(x + m)
for all values of x. If the function g is defined as
g(x) = f (x) + 3, what is the y-intercept of the graph
of g in terms of n and m?
20. Triangle ABC has side lengths 3, 8, and a. Triangle
DEF has side lengths 7, 10, and d. If a and d are
both integers, what is the smallest possible value of
(d − a)?
(A) 3n + 3m
(A) −10
(B) 3n − 3m
(B) −8
(C) 3nm
(C) −6
(D) 3 − nm
nm
(E)
3
(D) 0
(E) 5
a, b, b − a, −a, . . .
21. An antique coin collector has a number of coins and
several display cases. If he puts one coin in each
case, there are four coins left out of the cases. If
instead he puts three coins in each case, all coins are
placed, and there are six empty cases left over. How
many coins does the collector have?
18. In the sequence above, each term after the first two
is obtained by finding the difference between the
two previous terms. If a = 3 and b = 4, what is the
sum of the first 200 terms of this sequence?
g(x) = x2 + c
h(x) = g(x − 5) + 3
A
19. The equations of two functions, g and h, are shown
above, where c is a constant integer. If the graph of
h has two solutions, what is the maximum possible
value of c?
Start
B
xº
(A) −5
(B) −4
Stop
(C) −3
(D) 1
22. A small circle, A, is rolled around the circumference
of a larger circle, B, as shown in the figure above.
Circle A starts at the position indicated “Start” and
makes two complete rotations, coming to rest at the
position marked “Stop”. If the ratio of the areas of A
to B is 1 to 25, what is the value of x?
(E) 3
4
23. When the positive integer w is divided by 6, the
remainder is 3. When the positive integer x is
divided by 7, the remainder is 5. How many possible
values of wx are there such that wx < 500?
27. In a store, the list price of one item is x dollars, and
the list price of another is y dollars. If a discount of p
percent is offered for the first item and a discount of
q percent is offered for the second item, which of the
following gives the total percent discount off of the
original total price when both items are purchased at
their respective discounted prices?
24. For their senior class trip, all seniors at Smithville
High School chose between going to an amusement
park and seeing a theater performance. If 20 more
than 40 percent of the seniors chose to go to the
amusement park and the remaining 160 students
chose to go to the theme park, how many total
seniors went on the senior class trip?
(A) x + y − (p + q) %
!
(1 − p)x + (1 − q)y
%
100
x + y p + q
(C)
−
%
100
100
(B) x + y −
(D)
25. Mr. Halyard has forgotten two of the numbers in the
5-digit zip code of his son’s mailing address. He
remembers that the digits 3-8-7 appear together, in
that order, somewhere within the zip code, and that
a zero does not appear in the remaining digits. If he
guesses at random, what is the probability that Mr.
Halyard will write the correct zip code on a package
he is mailing to his son?
px + qy
%
x+y
(E) xy −
pq
%
100
28. In Marksville Elementary School, 40 classrooms do
not have computers, and 25 percent of all rooms with
computers do not have a white board. If rooms with
computers represent two-thirds of all classrooms in
the school, and if 50 of the classrooms do not have
a white board, how many classrooms with a white
board do not have computers?
26. Let the operation j k be defined as “good” if the
set of all prime factors of j includes all numbers in
the set of prime factors of k. Which of the following
is NOT good?
(A) 80 100
(B) 60 15
(C) 12 4
(D) 11 21
(E) 10 20
5
32. If 40x + 7y = 3663 and x and y are positive integers,
which of the following is a possible value of xy?
(A) 500
(B) 620
y = h(x)
(C) 730
(D) 810
(E) 940
29. In the graph of the function h shown above, h(0) = 2.
If r and s are integers and h(r) = 2h(s), which of the
following are possible values of s?
A
I. 0
II. 1
III. 5
D
(A) I only
O
B
(B) II only
(C) III only
Note: Figure not drawn to scale.
(D) I and II only
33. In the figure above, points B and D lie on the circumference of the circle with center O, circumference c
and radius r. If the area of triangle ABO is equal to
the area of the circle, then in terms of c and r, what
is the area of triangle ABD?
(E) I, II, and III
3x − ay = 11
x + 4y = 6
30. For which of the following values of a will the system of equations above have no solution?
(A) −12
(B) −6
(C) 0
(A)
1
cr
4
(B)
1
cr
2
(C) cr
(D) 6
(D) 2cr
(E) 12
(E) 4cr
31. Both a and b are positive integers less than 30. If the
sum of a and b is even, what is the smallest possible
a
value of ?
b
6
34. For a certain exam, the “base score” is calculated
as the difference between the number of questions
answered correctly, c, and one-third the number of
questions answered incorrectly. If there are q questions on an exam and Lynn left b questions unanswered, which of the following expressions gives the
base score for Lynn’s exam?
36. The warehouse at Paper Supply Unlimited must keep
an inventory that ranges from 5000 to 7500 reams of
paper. If r is the number of reams of paper in the
warehouse, which of the following must be true?
(A) r − 5000 ≤ 6250
(B) r + 7500 ≤ 1250
1
(A) c − (q − b)
3
(C) r − 6000 ≤ 6250
1
(B) c − (q − b − c)
3
(D) r − 7500 ≤ 1500
1
(C) c + b − (q − c)
3
(E) r − 6250 ≤ 1250
1
(D) c + b − (q + b − c)
3
37. A drawer contains only blue pens. When 3 red pens
are added, the fraction of blue pens in the drawer
4
decreases to . How many red pens must be added
5
to the drawer so that blue pens account for one-third
the final number of all pens in the drawer?
1
(E) c − (c + b − q)
3
A
A
O
B
m
C
B
35. The graph above shows circle O with center at
38. In the figure above, three-fourths of a circle’s arc
is drawn from A and C,
√ and right triangle ABC is
isosceles. If AC = 10 2, what is the perimeter of
the figure outlined by the solid line?
3
(1, −2). Line m has equation y = − x + 6, and is
2
tangent to the circle at point A. If point B is directly
across from A and has coordinates (x, y), what is the
y
value of ?
x
7
D
42. What is the maximum number of intersection points
of a circle with radius r and a square with side length
2r?
C
(A) 2
(B) 4
(C) 5
B
(D) 6
(E) 8
A
43. If 4s − 6w = 2t + 18 and 3w − 2s = 5 + t, what is the
value of t?
39. The figure above shows a small square with diagonal BC within a large square with diagonal AD,
and A, B, C, and D are collinear. If the ratio of
AB : BC : CD is 1 : 2 : 1, what is the ratio of the
area of the small square to the area of the shaded
region?
(A) −7
(B) −3
(C)
0
(D)
4
(E) 10
3x − y = y − 2
x>0
y>7
44. The average of a set of five positive integers is a.
When the number 9 is added to the set, the average
of the numbers increases by 1. What is the value of
a?
40. If the statements above are true and y is a prime
integer, what is the smallest possible integer value
of x?
(A) 12
41. The quadratic function h is given by h(x) = ax2 ,
where a is a positive constant. If h(m) = h(n) for
some values of m and n, which of the following could
be true?
I. m = n
II. m = −n
III. −h(m) = h(−n)
(B)
9
(C)
8
(D)
3
(E)
1
45. A cylindrical tank with a base radius of 3 feet and
height of 12 feet is filled with water to two-thirds
of its capacity. The water is then transferred to a
larger cylindrical container with a base radius of 2
feet. What is the height of the water, in feet, in the
the larger container?
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I, II, and III
8
r
49. To make a cran-apple juice, 3 parts of cranberry juice
are mixed with 2 parts of apple juice. To make
an apple-pomegranate juice, 2 parts of pomegranate
juice are mixed with 1 part of apple juice. If equal
amounts of cran-apple and apple-pomegranate juice
are mixed, what fraction of the new mixture is apple
juice?
t
Note: Figure not drawn to scale.
46. In the figure above, a circular track is shown. The
outside of the track is a distance r meters from the
center, and the inside of the track is t meters from
the center. If the area of the shaded portion of the
3
track shown is 28π square meters and t = r, what
4
is the value of t?
47. Of 5 members of a high school club, 3 are to be assigned to the finance committee and 2 are to be assigned to the activities committee. If 3 of the members are seniors and 2 are juniors, and if those assigned to the finance committee are to be chosen at
random, what is the probability that the finance committee assignments will be given to 2 of the seniors
and 1 of the juniors?
2
5
(C)
1
2
(D)
3
5
(E)
2
3
3
16
(B)
1
4
(C)
11
30
(D)
3
8
(E)
7
12
50. For a poll, 72 people were asked about two models
of a car, A and B. Of the people polled, 38 have
owned model A, 42 have owned model B, and 10
have owned neither model. How many of the people
polled have owned both models?
51. In choosing a color scheme for her living room,
Julia must pick 2 colors from a list of 6 available
paint colors. How many different two-color schemes
are available for Julia to use for painting her living
room?
1
(A)
3
(B)
(A)
2
48. If x 3 = y6 and x8 = yt , what is the value of t?
9
xº
P
20º
O
m
P
Q
52. In the figure above, the circle has center O and
radius r, and triangle OPQ is equilateral. If the area
3
of the shaded region is 120π and PQ = r, what is
2
the area of triangle OPQ?
Note: Figure not drawn to scale.
54. In the figure above, line m is tangent to a vertex of a
regular octagon at point P. What is the value of x?
53. There are blue and red shirts on a clothing display. If
x
15 of the shirts are blue and of the shirts are red
7
where x is an integer, what is the minimum number
of shirts that could be on display?
(A) 16
55. If the function g is defined as g(x) = 2x2 − 3x and
g(t + 1) = 0, what is the value of g(t + 1.5) if t > 0?
56. If a is inversely proportional to the square of b, and
a = 4 when b = 8, what is the value of b when
a = 16?
(B) 18
(C) 19
(D) 20
(A) 2
(E) 21
(B) 4
(C) 8
(D) 16
(E) 32
57. If x, y, and z are positive integers and xz · yz = 225,
what is the value of xyz?
(A) 90
(B) 60
(C) 45
(D) 30
(E) 15
10
A
C
60. Some of William’s friends arranged to contribute $6
each to buy him a birthday cake. Two of the friends
then decided to use their money to buy a different
present, and the remaining friends had to contribute
an extra $3 each to make up the difference for the
cost of the cake. How many friends were originally
going to contribute to the cost of the cake?
B
58. The cube above has a side length of x. Point A is at
the center of the top face of the cube, and points B
and C are midpoints of the bottom edges of the cube.
In terms of x, what is the perimeter of 4ABC?
√
(A) x + x 2
√
(B) x + x 3
√
(C) x + x 5
√
(D) x + 2x 2
(E) 3x
4π
B
1
of
4
1
the book’s pages. On the second day, he reads of
4
the pages that remain, leaving 189 pages left to read.
61. On his first day reading a book, Henrik reads
What is the total number of pages in Henrik’s book?
62. On the xy-coordinate plane, lines ` and m are
perpendicular and intersect at (−2, 0). If b is the yintercept of line ` and c is the y-intercept of line m,
what is the value of bc?
(A)
4
(B)
2
(C)
1
(D) −1
A
120º
(E) −4
O
−2, 4, −8, 16, . . .
63. In the sequence above, every term after the first is
obtained by multiplying the preceding term by −2.
The sum of which pair of terms in the sequence is
equal to 249 ?
59. In the figure above, 4ABO is inscribed in the circle
with center O.
If arc AB measures 4π and
∠AOB = 120◦ , what is the length of AB ?
(A) 24th and 25th
(B) 25th and 26th
(C) 48th and 49th
(D) 49th and 50th
(E) 50th and 51st
11
C
B
x
64
16
D
A
67. Given the table of x and y values above, y could be
inversely proportional to which of the following?
E
F
y
27
216
(A) x
√
(B) x
64. A semicircular window is divided into four equal
panes, as shown in the figure above. A straight line
drawn between points B and D (not shown) is 16 feet
long. What is the perimeter of one of the panes?
2
(C) x 3
3
(D) x 2
(E)
(A) 16
2
3x
(B) 32
√
(C) (16 + 2π) 2
68. For all numbers x, y, and z, let (x, y, z) be defined
as (x, y, z) = xy + yz + x + z. For all numbers a, b,
and c where b = −1 and a > c > 0, for how many
ordered pairs (a, c) will (a, b, c) equal zero?
(D) 48
√
(E) (32 + 2π) 2
(A) None
65. When the square of a number is raised to 10 times
the number, the result is equal to 4 raised to 5 times
the number. What is the square of the number?
(B) One
(C) Two
(D) Three
(E) More than three
D
Q (0, b)
B
x
150º
A
P (3, 2)
m
y
C
n
69. In the figure above, m k n, and 4ABC has two side
lengths x and y. If AB bisects ∠DBC, what is the
x
ratio of ?
y
√
(A) 2 : 2
√
(B) 2 : 1
√
(C) 3 : 2
√
(D) 3 : 1
O
66. In the figure above, 4OPQ is shown on the xy-plane.
What is the value of b?
(E)
12
2:1
70. For all integers n and m, let n m be defined as the
sum of all integers from n to m, inclusive. What is
the value of (2 100) − (1 99)?
B
A
C
D
73. In the figure above, there are four squares each with
a perimeter of 8. The four points indicated are at the
centers of their respective squares. What is the area
of the square formed by connecting the four points
(not shown)?
E
(A) 4
71. In the figure above, square ABDE is inscribed in a
circle, and 4ACE has vertices on the square. What
is the ratio of the area of the circle to the area of
triangle ABDE?
(A)
(B) 8
(C) 16
(D) 20
4
π
3
(E) 32
(B) π
n
(C)
2
π
3
(D)
π
2
m
C
B
π
(E)
4
E
72. On a high school track team, there are initially 60
students, and 35 percent of the team members are
girls. After several more girls join the team, the percentage of girls on the team increases to 40 percent.
The number of boys on the team is how much greater
than the number of girls?
A
D
(A) 5
(B) 7
74. In the figure above, m k n, AB = DE = 2, and
CD = 3. What is the length of BD?
(C) 11
(D) 13
(E) 18
13
k2 (a − b)2 = 3a2 − 6ab + 3b2
75. The equation above is true for all values of a and b.
If k is a constant greater than zero, what is the value
of k?
O
(A) 0
√
(B) 3
√
(C) 6
k
(D) 3
P
(E) 6
79. The figure above shows the graph of a circle with
center O tangent to the x-axis at point P. Line k
passes through the origin and the center of the circle. If the radius of the circle has length 2 and the
π
area of the shaded region is , what is the slope of
3
line k?
1
(A)
2
1
(B) √
2
1
(C)
3
1
(D) √
3
1
(E)
4
Q
P
R
S
xº
O
76. In the figure above, QR and PS are arcs of circles
3
with the same center O. If segment OS is of OR,
5
_ 4
PQ = 8, and 5.0ptPS = π, what is the value of x ?
3
77. Four friends are arranged on a set of steps. Samuel
stands on the middle step, Trisha stands 5 steps
above him, David is 2 steps below Trisha, and
Martin is 9 steps above David. How many steps are
there in total?
PRICES AT ABBEY’S SANDWICH SHOP
ORDER
Sandwich & Salad
Salad & Soup
Sandwich & Soup
78. If 4x3 + 2x2 = 16x + 8, and x is a non-zero integer,
what is the value of x2 ?
PRICE
$16.00
$10.00
$18.00
80. The table above shows the prices of lunch combination orders at Abbey’s Sandwich Shop for the month
of May. The orders include different combinations
of three items: a sandwich, a soup, and a salad.
What is the average price, in dollars, of these items?
14
81. Five guitars, labeled V, W, X, Y, and Z, are to be
arranged in a line in a display case. Guitar X must
be positioned furthest to the right, and guitars V
and Y must be positioned next to one another. What
fraction of the total possible arrangements will result
in guitars W and Z being positioned next to one
another?
B
y = 2x3!
y = 2x!
A
84. In the figure above, the graphs of y = 2x3 and y = 2x
intersect at points A and B. If the diameter of a circle
(not shown) is formed by AB, what is the area of the
circle?
r
O
(A) 8π
(B) 6π
(C) 5π
(D) 4π
82. In the figure above, a small circle with radius r is
inscribed in a square (shaded), and the square is inscribed in a larger circle. What is the ratio of the area
of the larger circle to the area of the smaller circle?
(E) 2π
PETE'S AUTOMOTIVE REVENUES!
TIRE ROTATIONS!
11%!
(A) 2 : 1
√
(B) 2 2 : 1
(C) 3 : 1
√
(D) 3 2 : 1
OIL CHANGES!
19%!
(E) 4 : 1
INSPECTIONS!
6%!
83. If f and g are functions such that f (a) = 2g(2a) and
g(m) = 6m + 10, what is the value of f (5) ?
(A) 10
(B) 20
(B) $11,000
(D) 130
(C) $19,000
(E) 140
(D) $40,000
(E) $260,000
15
BRAKE TUNE-UPS!
22%!
85. The circle graph above shows the revenues for Pete’s
Automotive last year. During the year, transmissions
and inspections revenues totaled $240,000. How
much more in revenue did Pete’s Automotive earn
through oil changes than through tire rotations?
(A) $8,000
(C) 100
TRANSMISSION
REPAIRS!
42%!
88. There are 42 marbles in a bag, and each marble is
colored one of three colors. If more than one-third
of the marbles are blue, what is the greatest possible
fraction of marbles in the bag that could be one of
the other colors?
C
cº
B
D
bº
dº
A aº
2
3
(B)
9
14
(C)
13
21
(D)
11
20
(E)
1
2
E
gº
G
(A)
fº
F
86. In the figure above, ∠DEF is a right angle. What is
the average of a, b, c, d, f , and g?
87. A positive number greater than 1 is decreased by p
percent. The resulting number must be increased by
n percent to obtain the original value. If n and p are
positive numbers such that n < 100 and p < 100,
which of the following expresses n as a percent in
terms of p?
(A)
100 − p
%
p
(B)
p
%
100 − p
8
150º
A
6
B
89. The figure above is symmetric about line AB. What
is the area of the figure?
√
(A) 48 + 9 3
p
%
(C)
100(p − 1)
(D)
100(p − 1)
%
100 − p
(B) 48
100p
%
100 − p
(D) 44
(E)
√
(C) 44 + 9 3
(E) There is not enough information to determine
the area
16
C
X
92. Let the function r be defined by r(x) = −3(x+4)2 +2.
If line ` has a y-intercept at (0, 5) and intersects the
graph of the function r at the maximum value of r,
what is the slope of `?
D
B
W
Y
E
A
Z
F
Q
90. In the figure above, quadrilateral WXYZ is inscribed
in hexagon ABCDEF. What is the sum of the
marked angles?
6
P
O
93. The figure above shows two circles with the same
center O, and PQ = 6. If the area of the shaded
region is 60π, what is the ratio of OP : OQ ?
A
(A) 1 : 2
y = g(x)
(B) 1 : 3
(C) 1 : 4
B
0
C
(D) 1 : 5
E
(E) 1 : 6
D
91. The figure above shows the graph of the function g.
For how many of the points marked on the function
is g(x) − g(x) = 0 ?
94. A large cube with volume V greater than 1 has sides
of length x. The sides of a smaller cube are onethird the length of the sides of the large cube. If the
volume of the smaller cube is an integer value, what
is the smallest possible value of V?
(A) 4
(B) 3
(C) 2
(D) 1
(E) None
17
x
4
y
95. On the number line above, the marks are evenly
spaced. If the difference between x and y is six more
than the average of x and y, what is the value of x?
P
O
Q
30º
R
96. In the figure above, square OPQR lies partially in the
circle with center O, and has vertices P and R on the
circumference of the circle. If the area of the shaded
region is 24π, what is the perimeter of the square?
c Evan Wessler and Method Test Prep,
All problems 2013. No part of this document may be reproduced, distributed, or otherwise replicated in any form without permission from the above parties.
18