Question 1: A(2, 2) and B(-1, -2) are two points in the x,y plane.
| Column A | Column B |
| Distance between points A and B. | 4.5 |
(a) The quantity in Column A is greater.
(b) The quantity in Column B is greater.
(c) The two quantities are equal.
(d) The relationship cannot be determined from the information given.
Solution:
The distance between 2 points A(xa, ya) and B(xb, yb) can be found if we apply the Pythagorean theorem:
AB2 = (xa – xb)2 + (ya – yb)2
In our case,
AB2 = (2 + 1)2 + (2 + 2)2
AB2 = 32 + 42
AB2 = 25
AB = 5 and the quantity in Column A of the table is greater.
Question 2: For a given three digit positive integer, the tens digit is equal to the sum of the other two digits.
| Column A | Column B |
| tens digit | units digit |
(a) The quantity in Column A is greater.
(b) The quantity in Column B is greater.
(c) The two quantities are equal.
(d) The relationship cannot be determined from the information given.
Answer: If a, b and c are the hundreds digit, the tens digit and the units digit, we know that b = c + a and a≥1.
b > c, as any a≥1 added to c will result in a number greater than c, so (a) is the correct answer.
Question #3: What is the value of an angle of a regular polygon with six sides?
(a) 140o
(b) 110o
(c) 100o
(d) 120o
(e) 90o
Answer:
The sum of the angles of a regular polygon with n sides is (n – 2)·180o.
A polygon with six sides has the sum of the angles equal to (6 – 2)·180o = 720o.
The polygon is regular so all angles are equal to 720o/6 = 120o
Question #4: The circle with center O below has radius 5.
| Column A | Column B |
| Length of arc ACB | Radius of circle O |
(a) The quantity in Column A is greater.
(b) The quantity in Column B is greater.
(c) The two quantities are equal.
(d) The relationship cannot be determined from the information given.
Answer: The length of arc ACB is equal to (angleA/360o)·(2¶r)
We do not have any information about the value of angle a, so we cannot evaluate the length of arc ACB.
(d) is the correct answer.
Question #5: Rectangle ABCD below has AB = 4, AD = 6 and triangle EBC is isosceles, EB = EC. What is the perimeter of triangle EDC?
(a) 9
(b) 10
(c) 12
(d) 13
(e) 15
Answer: ED = 3, DC = 4 and we need to determine EC in order to find the perimeter of triangle EDC.
We apply the Pythagorean theorem:
EC2 = ED2 + DC2.
EC2 = 9 + 16.
EC = 5.
The perimeter of triangle EDC is 5 + 4 + 3 = 12.
Question #6: A particular stock is valued at $20 per share. If the value of a share increases 50%, then decreases 50% and then increases 20%, what is the final value of a share?
(a) $16
(b) $18
(c) $20
(d) $24
(e) $25
Answer: The final value of a share is $20(1 + .5)(1 – .5)(1 + .2) = $18.
Question #7: In the x,y plane, find the y-intercept of a line with slope equal to -2 and that passes through the point (3,2).
(a) 5
(b) 6
(c) 8
(d) 9
(e) 10
Answer: The equation of a line with intercept -2 is:
y = -2x + n.
If the line passes through point (3,2), 2 = (-2)3 + n, and n = 8.
The equation of the line is y = -2x + 8 and its the y-intercept is 8.
Question #8: If it takes John 6 days to paint the apartments of a building and it takes Kevin 8 days to do the same job, how long would it take to do the job if both painters worked simultaneously?
(a) 24/7 days
(b) 18/7 days
(c) 48/7 days
(d) 30/7 days
(e) 32/7 days
Answer: Since it takes John 6 days paint the apartments, John can paint (1/6) of the apartments in one day. Similarly, Kevin can paint (1/8) of the apartments in one day. And if we let x represent the number of days it would take for the painters working simultaneously to paint the apartments, then they would do the (1/x) of the job in 1 day.
1/6 + 1/8 = 1/x
x = 24/7 days.
Question #9: In a survey of 190 manufacturing companies, 103 hired operators, 67 hired technicians, and 49 hired both operators and technicians, as illustrated in the Venn diagram below:
(a) 1971
(b) 1972
(c) 1973
(d) 1974
(e) 1977
Answer:
The slope of the line between 1972 and 1973 is higher than all of the other slopes, so the Company A’s revenues increased the most from 1972 to 1973.