Five Points Each. Total 150 Points. Choose the best answer from (A) – (D). 1. How many of the following 5 statements are correct statements? statements? (1) If both a and b are irrational numbers, then ab is irrational. (2) If both a and b are irrational numbers, then a + b is irrational. 2 3 (3) If both a and a are rational numbers, then a is rational. (4) If a is rational and b is irrational, then ab is irrational. (5) if both a+b and a–b are rational numbers, then both a and b are rational numbers. (A) 4 (B) 3 (C) 2 (D) 1 2 2. Let f (t )＝－t ＋12t ＋13 be the function that represents the temperature of a desert area at a certain time t where 1 ≤ t ≤ 12. What is the difference between the highest and lowest temperatures temperatures during this period of time? time? (A) 49 (B) 36 (C) 25 (D) 16 3 7 x y . − 3. Let x and y be real numbers. If 47 = 8 and 376 = 128, find x y (A) –2 (B) –4 (C) –3 (D) –5 4. Let real numbers a1, a2, a3, a4, and a5 be an arithmetic sequence satisfying 1 < a1 < 3 and a3 = 5. Define bn = 2 a . Which one of the following fo llowing statements is correct ? (A) b1, b2, b3, b4, and b5 form an arithmetic sequence (B) b4 > b5 (C) (b1)( b5) = 512 (D) b2 > 8 5. Which one of the following statements about sets is correct ? (A) 4 ⊂ {2, 4, 6} (B) {2, 4, 6} and {1, 4, 9} have exactly one subset in common (C) Universal set U is a set that consists of positive integers 1 to 9. If A = {2, 4, 6} c c and B = {1, 4, 7, 9}, and A and B are the set complements of A and B, then c c A ∪B = {3, 5, 8}. (D) There are 8 sets A that satisfy {2, 4} ⊆ A ⊆ {2, 4, 6, 8, 9} 6. Let {A, B, C} be a partition for the sample space S. Suppose D ⊂ S and P(A) : 2 1 1 , P(D | B) B) = , and P(D | C) = , P(B) : P(C) = 2 : 5 : 3. If P(D | A) = 3 5 4 n
find P(C | D). (The conditional probability P(D|A) is defined to be (A)
9 37
(B)
7 37
(C)
16 37
(D)
P ( D ∩ A) P ( A)
)
12 37 2015 Final
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7. As shown in the figure on the right, use AC as diameter to make a half circle. Let B be a point on AC and connect B to a point D on the half circle so that BD is perpendicular to AC . a
−
If AB =
19 + 280 and BD = 3 and BC =
b where a and b are positive integers, what is the remainder when a is
divided by b? (A) 4
(B) 2
8. Compute (A) 2
4(
(C) 3
5+ 3 2 4
(B) 4
(D) 1
4
) – 16(
(C) 3
5+ 3 2 2
3
) – 17(
5+ 3 2 2
)
2
＋27(
5+ 3 2 2
) – 3.
(D) 5
9. The statements below describe the function
y = f ( x ) = 3
− x −1
+
2 . Which
statement is an incorrect description? (A) f ( x) ≥ 2 is true for all x. (B) The graph of f ( x)
can be obtained by moving the graph of y = 3
– x
horizontally to the right 1 unit and then up 2 units. (C) The graph of f ( x)
x–1
is symmetric to that of y = 3
+ 2 with respect to the
y–axis.
(D) f ( x) = x has exactly one real root and this root is between 2 and 3. 10. Suppose a62 × a89 = 1000 where {an} is a geometric sequence. Find
log10 ( a1 ) + log10 ( a2 ) + log10 ( a3 ) + L + log10 ( a150 ) (A) 450
(B) 300
(C) 375
(D) 225
11. Use 5 colors to color the 6 regions of the figure on the right. Each region can only use one color and neighboring regions must use different colors. Under these restrictions, how many ways can these regions be colored? (A) 840 (B) 960 (C) 1020 (D) 1260 12. A shop has three production lines A, B, and C all together producing 21000 nails. Production line A produces 6000 nails a day. B and C produce p and q nails a day, respectively. The proportion of bad nails produced by A, B, and C are 3%, 4%, and 2%, respectively. One nail is picked from the combined lot of 21000 nails. If 3 the probability of that nail is bad nail produced in line A is , find 3 p – 2q. 11 (A) 5000 2
2015 Final
(B) 10000
(C) 15000
(D) 20000
13. Let k be a real number. range of values for (A) 1
|x + k | + |x – 3| = 8
If
k is
(B) –17
a ≤ k ≤ b.
(C) 3
Find
has real solutions, then the 2a + b.
(D) –15
14. Suppose f ( x) is a fourth degree polynomial with rational coefficients.
If
f (1 + 3i ) = f (1 − 3 2 ) = 0 and f(7) = 243, find f (0).
(A) –51 (B) –45 (C) 45 (D) 51 15. If 0 < x < 1 < y < 100 < z and satisfy the system of equations log 2 xyz = 103 1 1 1 1 , + + = log 2 x log 2 y log 2 z 103 find xyz( x + y + z) – xy – yz – zx. (A) 2
103
+1
(B) 2
206
30
16. Suppose 2
∑ k
2
103
+1
(C) 2
(D) 2
60
= 9455,
k ＝1
2
∑ k
2
k ＝1
2
(B) 19
206
– 1
30
∑ k
3
= 73810, and 2
1×2 + 2×3 + 3×4 + … + 30×31 . (A) 17
–1
(C) 16
= 216225 and let
P =
k ＝1
Find the sum of all digits of P. (D) 18
17. Define a "decreasing" number be a positive integer whose digits are in strictly decreasing order from left to right, such as the number 54321. 54331 is NOT a "decreasing" number. Consider all the "decreasing" 5–digit numbers and order them based on their sizes in descending (decreasing) order. What is the 175th number in this group? (A) 86430
(B) 86421
(C) 86321
(D) 86310
18. Let a "bare" candidate be a student who has not prepared for an examination. Given an exam paper that consists of n "true or false" questions, what is the smallest n such that the probability that a “bare” candidate will not have a zero score is more than 99.9999%? (A) 19
(B) 21
(C) 17
(D) 20
19. Suppose real numbers x and y satisfy xy = 162 and x + 8 y has its smallest value m when ( x, y) = (a, b). Find
(A) 36
(B) 45
m – a + 2b.
(C) 54
(D) 63 4
20. If the integer coefficient equation x distinct positive integer roots, find (A) 13
(B) 9
(C) –9
3
2
＋ax ＋bx ＋cx＋56＝0
has four
4 a – b – c.
(D) –13 2015 Final
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2
21. Let f ( x)＝log4( x
2
－2 x＋9)－log4( x ＋2 x＋9).
f ( x) is M and the minimum value is m.
(A) 4
(B) 8
(C) 3
Find
Suppose the maximum value for 7 M – m.
(D) 6
a1 = 2 3a − 1 22. Suppose the sequence {an} has a recursive definition of an 1 = n 4an − 1 +
where n is positive integer. Find its 15th term a15. 46 16 44 15 (A) (B) (C) (D) 95 31 85 29 23. For any list of letters, underline those identical letters that are grouped together to form a "chain."
A "chained" number is the number of chains in a particular
arrangement of a list. For example, arrange 3 P's and 5 Q's together to form QPPQPQQQ
and mark it Q PP Q P QQQ . This one gives a "chained"
number of 5.
If 5 P's and 3 Q's and 1 R are arranged, how many different
arrangements that would give a "chained" number of 4? (A) 18
(B) 36
(C) 24
(D) 42
24. According to the gymnastic competition scoring rules, among the scores assigned to a gymnast by 7 judges, both the highest and lowest scores are deleted. The average of the remaining 5 scores is the score assigned to the gymnast. Suppose the 7 scores were shown on the overhead monitor after a gymnast completed his round.
Because of the short time period those scores shown on
the monitor, only the first 6 scores assigned by judges were seen and they were 8.9, 8.7, 8.8, 8.9, 9.1, and 8.6. this gymnast which was 8.8.
The next screen showed the score received by What would be the score assigned by the 7th
judge? (A) 8.7
(B) 8.6
25. If a 4 digit number find
aabb
(D) 8.8
is the square of an integer where a and b are digits,
2a – b.
(A) 7
4
(C) 8.9
2015 Final
(B) 13
(C) 4
(D) 10
2
4
3
26. Suppose a and b are integers and x + x + a is a factor of f ( x) = 3 x + 11 x + 2
bx + 13 x – 6.
If the range of values of x that satisfy f ( x) < 0 is
c < x < d ,
which one of the followings is correct ? (A) a = –2
(B) b = 12
(C) c = –3
(D) c – b > d – a
27. An internationally accepted way to measure the strength of an earthquake is to use the Richter Scale. A Richter Scale measurement of R means that an earthquake released E ergs of energy and the relation between E and R is: log10( E ) = 1.6 R + 11.7.
Recently, Nepal had a 7.8 earthquake and Japan had an
8.3 earthquake. How many times larger was the energy released from the Japan earthquake than that released by the Nepal earthquake? (Note: 10 (A) 6.4
(B) 6.3
(C) 6.2
0.4
= 2.51)
(D) 6.1
28. Given a geometric sequence {2, 4, 8, 16, …}. If the number 1 is inserted in front of 2 in the sequence, two 1's are inserted between 2 and 4, and three 1's are inserted between 4 and 8, and so on by following the pattern that k 1's are st
th
inserted between this sequence's original ( k– 1) and k term (k ≥2). After all the insertions, the new series would look like {1, 2, 1, 1, 4, 1, 1, 1, 8, 1, 1, 1, 1, 16, …}. What is the sum of the first 99 terms of this new series? (A) 8276
(B) 16468
(C) 8277
n 29. Let C k be the binomial coefficient n
(D) 16469 n!
k !( n − k )!
.
Given that
＋……＋nC n ＝11264
where n is a natural number.
(A) 11
(C) 9
(B) 12
Find n.
(D) 10
30. Suppose a bag has 6 black balls and n (n≥2) red balls. from the bag.
C 1n ＋2C 2n ＋3C n3
Randomly take 3 balls
If the probability of taking any ball is the same and Pn is the
probability of taking 1 black ball and 2 red balls, what is the largest possible value for Pn? 46 35 (B) (A) 85 68
(C)
44 85
(D)
33 68
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