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1. If 2x < 100 and x is an integer, how many of the 2x + 2 integers will be divisible by 3 and by 2?
(a) 1
(b) 2
(c) 3
(d) 4
(d) 5
Answer:
26 = 64 and 27 = 128. If 2x < 100, then the highest value of x is 6.
Possible values for x: 0 , 1 , 2 , 3 , 4, , 5 , 6. 2x + 2 can take the values 3 , 4 , 6 , 10 , 18 , 34 , 66.
Out of these values, only 6 , 18 and 66 are divisible by 3 and by 2. The correct answer is (c).
2.sin(x)cos(x)(1 + tan2(x)) =
(a) tan(x) + 1
(b) cos(x)
(c) sin(x)
(d) tan(x)
(e) sin(x) + cos(x)
Answer:
sin(x)cos(x)(1 + tan2(x)) =sin(x)cos(x)(1 + sin2(x)/cos2(x)) = sin(x)cos(x)[(cos2(x) + sin2(x))/cos2(x)] = sin(x)cos(x)[1/cos2(x)] = sin(x)/cos(x) = tan(x)
3. If 5xy = 210, and x and y are positive integers, each of the following could be the value of x + y except:
(a) 13
(b) 17
(c) 23
(d) 15
(e) 43
Answer:
5xy = 210 so xy = 42. 42 = 2 · 3 · 7.
The following products of integers equal to 42: (6 · 7) or (2 · 21) or (14 · 3) or (42 · 1). 6 +7 = 13, 2 + 21 = 23, 14 + 3 = 17 42 + 1 = 43.
The only answer that is NOT 43, 13, 23 or 17 is (d), 15.
4.The average of the integers 24, 6, 12, x and y is 11. What is the value of the sum x + y?
(a) 11
(b) 17
(c) 13
(d) 15
(e) 9
Answer:
The average of the 5 numbers is (24 + 6 + 12 + x + y)/5. (24 + 6 + 12 + x + y)/5 = 11 24 + 6 + 12 + x + y = 11 · 5 24 + 6 + 12 + x + y = 55 x + y = 13
5.The inequality |2x - 1| > 5 must be true in which one of the following cases? I. x < -5 II. x > 7 III. x > 0
(a) II only
(b) I, II and II
(c) I and II only
(d) I and III only
(e) I only
Answer:
|2x - 1| > 5, -5 > 2x - 1 or 2x - 1 > 5 -4 > 2x or 2x > 6 -4 > 2x results in x < -2 2x > 6 results in x > 3
I answer is true, II answer is also true, but III answer is false, so the correct answer is (c)