PSAT Regular Multiple Choice questions.
1. Look at the answer choices before you begin to work on each question.
2. Read each question carefully, even if it looks like a question you don't think you can answer. Don't let the form of the question keep you
from trying to answer it.
3. If your answer isn't among the choices, try writing it in a different form. You may have the same answer in a different mathematical format.
If y = (x + 3)2, then (-2x - 6)2 must equal which
of the following?
2
(A) -4y
2
(B) -2y
(C) -4y
(D) 2y
(E) 4y
In the figure above, AD is a diameter of the
circle with center O and AO = 5. What is the
length of arc BCD ?
(A)
(B)
Explanation:
The expression (-2x - 6)2 can be rewritten as [-2(x + 3)]2, which
equals 4(x + 3]2.
Since y = (x + 3)2, it follows that (-2x - 6)2 = 4(x + 3)2 = 4y. The
correct answer is choice (E).
Explanation:
To solve this problem, it is helpful to draw segment OB in the figure.
Since OB and OD are both radii of the circle, they both equal 5.
Therefore, the angles opposite these congruent sides of
BOD are congruent and OBD = 36°. The third angle of the
triangle, BOD, equals
180°- 36°- 36° = 108°. Arc BCD is a fraction of the circumference of
the circle and more specifically equals
correct answer is choice (D).
, which equals
The
(C)
(D)
(E)
If 0 < a < b < c < d < e in the equation above,
then the greatest increase in S would result from
adding 1 to the value of which variable?
(A) a
Explanation:
When the denominator of a fraction is increased, the value of the fraction decreases.
Therefore, adding 1 go b, d, or e will decrease the sum S. Increasing one of the
numerators, either a or c, will increase S. Adding 1 to a changes
to
, thereby
(B)
b
c
(D) d
(E) e
increasing S by
(C)
. Adding 1 to c changes
to
, thereby increasing S by
. Since
b<d, then
. Therefore, adding 1 to a will result in the greatest increase in S. The
correct answer is (A).
If is defined for all positive numbers a and b by Explanation:
a b=
, then 10 2 =
Substituting 10 for a and 2 for b in the expression
The correct answer is (A).
yields
(A)
(B)
(C)
5
(D)
(E)
20
If m and p are positive integers and (m + p) x m
is even, which of the following must be true?
(A) If m is odd, then p is odd
(B) If m is odd, then p is even.
(C) If m is even, then p is even.
(D) If m is even, then p is odd.
(E) m must be even.
If xy = 2 and xy 2 = 8, what is the value of x ?
(A)
(B)
2
(C) 4
(D) 8
(E) 16
Explanation:
If m is even, then the expression (m + p) x m will always be even and it cannot be
determined whether p is even or odd. This eliminates choices (C) and (D). If m is odd,
then (m + p) x m will be even only when m + p is even and m + p will be even only
when p is odd. The correct answer is (A) since the truth of statement (A) also
eliminates choices (B) and (E).
Explanation:
Substituting xy = 2 into the equation xy2 = 8, you will obtain (xy)y = 2y = 8, thus y = 4.
To find x, substitute y = 4 into one of the two original equations to obtain x = . The
answer to this problem is (A).
Explanation:
Since the sum of the integers in each row, column, and diagonal is the same, it follows
that w - 2 + a = 3 + a - 3. Thus w - 2 = 0 so that w = 2. The answer to this problem is
(C).
In the incomplete table above, the sum of the
three integers in each row, column, and diagonal
is the same. If the numerical values in four of the
blocks are as shown, what is the value of w ?
(A) -6
(B) -5
(C) 2
(D) 5
(E) 8
If n is an odd integer, which of the following
must be an odd integer?
(A) n - 1
(B) n + 1
(C) 2n
(D) 3n + 1
(E) 4n + 1
Explanation:
If n is an odd integer, both one more and one less than n will be even integers,
eliminating choices (A) and (B). Any even multiple of n will be an even integer,
eliminating choice (C). However, 4n is even, making 4n +1 an odd integer. The
answer to this problem is (E). Note that 3n + 1 is even if n is odd and it is odd if n is
even. Since the question asks, "Which of the following MUST be an odd integer," (D)
cannot be the correct answer.
A 19-liter mixture consists by volume of 1 part
juice to 18 parts water. If x liters of juice and y
liters of water are added to this mixture to make
a 54-liter mixture consisting by volume of 1 part
juice to 2 parts water, what is the value of x ?
(A) 17
(B) 18
(C) 27
(D) 35
(E) 36
Explanation:
It is given that the 19-liter mixture consists by volume of 1 part juice to 18 parts water, so that there is 1
If a and b are integers greater than 100 such that
Explanation:
liter of juice and 18 liters of water in the mixture. Since the ratio
and x liters of juice and y
liters of water are added to make a mixture consisting by volume of 1 part juice to 2 parts water, then
The new mixture is 54 liters; therefore, x + y = 54 - 19 = 35. The two simultaneous
equations to be solved are
and x + y = 35. Since the question askes for the value of x,
substitute y = 35 - x into the fractional equation obtaining
+ 35 - x or 3x = 51 so x = 17. The answer to this problem is (A).
It follows that 2 + 2x = 18
a + b = 300, which of the following could be the
exact ratio of a to b ?
(A) 9 to 1
(B) 5 to 2
(C) 5 to 3
(D) 4 to 1
(E) 3 to 2
To solve this question, you need to look at the answer choices. For any of the answer choices to be the
ratio of a to b, some multiple of the sum of the two numbers must evenly divide 300. For example, if the
ratio of a to b equaled 9 to 1, then a would equal 9x and b would equal x for some number x.
Furthermore, 9x + x would have to equal 300. This is possible since 10x = 300 yields an integer solution,
namely x = 30. However, if x = 30, then a would equal 270 and b would equal 30. Although the sum of
these numbers equals 300, they do not satisfy the other condition in the problem. That is, both of these
numbers are not greater than 100. Therefore, choice (A) can be eliminated.
Answer choices (B) and (C) can be eliminated since neither the sum of the two numbers in (B) nor the
sum of the two numbers in (C) evenly divided 300. (5x + 2x = 300 does not yield an integer solution, nor
does 5x + 3x = 300.)
Although answer choices (D) and (E) are possible ratios of a to b (both 4x + x = 300 and
3x + 2x = 300 yield integer solutions), (D) results in a = 240 and b = 60 and can be eliminated since 60 is
not greater than 100.
Only choice (E) gives a correct ratio of a to b that satisfies all of the conditions in the problem. For (E), a
= 180 and b = 120, and both integers are greater than 100.
Explanation:
The sum of the areas of the shaded regions is the area of the circle minus the area of the
In the figure above, a square is inscribed in a
circle with diameter d. What is the sum of the
areas of the shaded regions, in terms of d ?
(A)
(B)
square. The area of the circle is
where
Therefore, this area is
. To find
the area of the square, you need the length of one of the sides. Since the diagonal of the
square is d, each side equals
. Therefore, the area of the square is
sum of the areas of the shaded regions is
correct answer is (A).
which equals
. The
. The
(C)
(D)
(E)
Explanation:
Since the students are to be assigned to the lockers shown, each "assignment" is the
pairing of a student with a specific locker (locker #46, #47, or #48), not the pairing of a
student with another student. The conditions of the problem allow you to deduce which
students will share a locker, but they are not enough to allow you to deduce the specific
Two seniors, Abby and Ben, and two juniors, Cathy and
Dave, are to be assigned to the 3 lockers shown above
according to the following rules.
All 3 lockers are to be assigned.
Abby and Ben cannot share a locker with each other.
A senior cannot share a locker with a junior.
The locker assignments of all four students can be
determined from the assignments of which of the
following pairs?
I. Abby and Ben
II. Ben and Cathy
III. Cathy and Dave
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I, II, and III
locker assignments. For example, knowing that Cathy and Dave will share a locker
does not tell you to which locker they will be assigned.
First consider what knowing the assignments of Abby and Ben will tell you about the
locker assignments of the remaining two students. Since Abby and Ben are seniors and
they cannot share a locker with each other or with any juniors, you know that Cathy
and Dave must share the third locker. Since you know the specific locker assignments
of all four students, (I) is correct.
If you know the assignments of Ben and Cathy, you know that Abby is in the third
locker and Dave must share Cathy’s locker. Therefore, (II) is correct.
If you know the assignments of Cathy and Dave (they must share the same locker), you
only know to whom one of the lockers is assigned. You will not know specifically to
which lockers Abby and Ben are assigned — you will only know that they do not share
a locker. The correct answer is choice (D).
For positive integers a and b, let a b be defined
as ab +1. If x and y are positive integers and x
y = 16, which of the following could be a value
of y?
I. 1
II. 2
III. 3
(A) I only
(B) II only
(C) I and III only
(D) II and III only
(E) I, II, and III
Explanation:
For this question, you are given that x y = 16 where x y is
defined as xy +1. You are asked which of three values are possible for
y when xy +1 = 16.
The value of y could be 1 if x = 4, since 41+1 = 42 = 16. So I is
correct. The value of y could be 3 if x = 2, since 23+1 = 24 = 16. So
III is correct. Since there is no integer that can be raised to the
(2 + 1) or 3rd power to obtain 16, II is not correct. The correct
answer is (C).