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(B) There exists at least one x such that x is a
C.
(C) There exists at least one x such that x is a
Q.
(D) There are no x’s that are C’s.
(E) There are no x’s that are Q’s.
STOP
IF YOU FINISH BEFORE TIME IS CALLED,
YOU MAY CHECK YOUR WORK ON THIS
TEST ONLY. DO NOT WORK ON ANY
OTHER TEST IN THIS BOOK.

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Lesson 8
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■ SAT II Subject Tests
ANSWER KEY
1. D
2. D
3. E
4. D
5. B
6. A
7. D
8. C
9. A
10.
A
11.
C
12.
B
13.
C
14.
B
15.
B
16.
D
17.
B
18.
C
19.
B
20.
E
21.
D
22.
B
23.
C
24.
C
25.
E
26.
C
27.
A
28.
E
29.
D
30.
A
31.
D
32.
B
33.
E
34.
D
35.
C
36.
B
37.
E
38.
E
39.
C
40.
C
41.
C
42.
C
43.
D
44.
D
45.
C
46.
A
47.
B
48.
E
49.
D
50.
A
EXPLANATORY ANSWERS
Each of the following explanations contains a “Calculator Status Report”:
[+] Calculator is necessary for this problem.
[0] Calculator is not necessary for this problem but might be useful.
[–] Calculator is probably not useful for this problem.
1. The correct answer is (D). [0] Since x divided by 7 leaves a remainder of 4, x has the form 7m + 4,
where m is an integer. And since y divided by 7 leaves a remainder of 3, y has the form 7n + 3, again
where n is an integer. So xy has the form:
(7m + 4)(7n + 3) = 49mn + 21m + 28n + 12
The sum of the first three terms of that expression—49mn + 21m + 28n—is evenly divisible by 7.
So the remainder is determined by the last term. And 12 divided by 7 leaves a remainder of 5.
Alternatively, you can just use some numbers. Let x be 4 (4 divided by 7 leaves a remainder of 4),
and let y be 3 (3 divided by 7 leaves a remainder of 3). On those assumptions, xy = 12. And 12 divided
by 7 leaves a remainder of 5.
2. The correct answer is (D). [+] First calculate the radius of each sphere:
35 =
4
3
πr
3
24 =
4
3
πr
3
r
3
=35(
3
4
)/
π
r
3 
=24(
3
4
)
π
r
3
=8.3556
r
3
=5.7295
r =2.0292
r =1.7894
So the difference between the length of the radius of the larger sphere and that of the smaller is:
2.0292 – 1.78940 = 0.2398
3. The correct answer is (E). [–] If x is a member of S, then x – 1 is a perfect square:
(A)
8 – 1 = 7 Not a perfect square.
(B)
16 – 1 = 15 Not a perfect square.
(C)
24 – 1 = 23 Not a perfect square.
(D)
52 – 1 = 51 Not a perfect square.
(E)
65 – 1 = 64 and 8 
× 8 = 64

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Mathematics Level IC/IIC Subject Tests
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■ SAT II Subject Tests
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4. The correct answer is (D). [0] One of 4 seniors must be selected as the presidential candidate, and
one of 5 juniors must be selected as the vice–presidential candidate. So 13 people remain for the
position of treasurer:
4 
× 5 × 13 = 260
5. The correct answer is (B). [–] Since f (1) = 4:
(1)
2
+ b(1) + 1 = 4
1 + b + 1 = 4
b + 2 = 4
b = 2
Alternatively, you could test the answer choices:
(A)
(1)
2
+ 4(1) + 1 = 4
6 = 4
False.
(B)
(1)
2
+ 2(1) + 1 = 4
4 = 4
True.
(C)
(1)
2
+ 1(1) + 1 = 4
3 = 4
False.
(D)
(1)
2
+ 0(1) + 1 = 4
2 = 4
False.
(E)
(1)
2
+ –2(1) + 1 = 4
0 = 4
False.
6. The correct answer is (A). [+] Use the distance formula:
7. The correct answer is (D). [–] To find the train’s average speed for the first 2 hours, you need to
find how many kilometers are traveled during that time. You can do that by subtracting the kilometers
covered during the last 4 hours from 400:
400 – 4(t) = 400 – 4t
The train covered that many kilometers during the first two hours:

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Lesson 8
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ARCO
■ SAT II Subject Tests
Alternatively, you can just assume a value for t. Let us assume that the train traveled 100 kilometers
per hour for the last four hours, that is, t = 100. On that assumption, the train covered all 400 kilometers
during the last four hours of the trip and didn’t move at all during the first two hours. So if we substitute
100 for t into the formulas given as answer choices, the correct choice will generate the value 0: