Question 1: If a, b and c are the sides of any triangle, which of the following inequalities is not true?
(a) ab > 0
(b) a + b > c
(c) a + c/2 >b
(d) b + c > a
(e) (a + b)·(b + c) > a·c
Solution: The first answer is true, since the product of 2 positive reals is positive.
The second and the fourth answers are also true, since the sum of 2 sides of a triangle is always higher than the third side.
The fifth answers is also true because it is just a multiplication of the second and fourth inequalities.
Answer three should be the one that is not true, and we can verify this result with an example: an isosceles triangle with a = 3, b= 3, c= 10 will not satisfy the inequality.
Question 2: If x is an odd integer, which of the following is also an odd integer?
(a) 2x + 6
(b) x2 + x + 3
(c) x2 + 2x + 1
(d) 4x
(e) x2 + 5
Answer:
2x + 6: even + even = even.
x2 + x + 3: odd + odd + 3 = odd.
x2 + 2x + 1: odd + even + 1 = even.
4x: even.
x2 + 5: odd + 5 = even.
The correct answer is (b).
Question 3: Does (x + a)2 = y2?
(1) x = y – a
(2) x = y + a
(a) Statement (1) ALONE is sufficient, but statement (2) is not sufficient.
(b) Statement (2) ALONE is sufficient, but statement (1) is not sufficient.
(c) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(d) EACH statement ALONE is sufficient.
(e) Statements (1) and (2) TOGETHER are NOT sufficient.
Answer: Statement (1): x = y – a.
x + a = y. We square this equation and (x + a)2 = y2.
Statement (1) alone is sufficient.
Statement (2): x = y + a
(y + 2a)2 = y2.
This equation has multiple (a, y) pair solutions so statement (2) is not sufficient.
(a) is the correct answer.
Question 4: A coin is tossed six times. What is the probability that exactly three heads will show?
(a) 9/16
(b) 1/2
(c) 7/16
(d) 5/16
(e) 3/16
Answer: Since on any given coin toss the probability of heads is 1/2, the probability of a specified sequence of heads in six tosses is (1/2)6 = 1/64.
The number of sequences with 3 heads out of 6 tosses is given by the formula, n!/[k!(n - k)!], where n = 6 is the number of tosses, and k = 3 is the number of times the result is heads.
6!/[3!(6 - 3)!] = 20.
The probability will be 20·(1/64) = 5/16.
Question 5: One member of an organization of 15 teachers and 25 students resigned. What fraction of the remaining members were teachers?
(1) 24/39 of the remaining members were students.
(2) The member that resigned was a teacher.
(a) Statement (1) ALONE is sufficient, but statement (2) is not sufficient.
(b) Statement (2) ALONE is sufficient, but statement (1) is not sufficient.
(c) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(d) EACH statement ALONE is sufficient.
(e) Statements (1) and (2) TOGETHER are NOT sufficient.
Answer:
Statement (1): 24/39 of the remaining members were students so (39 – 24)/39 = 15/39 was the fraction of remaining teachers. Statement (1) alone is sufficient.
Statement (2): If the member that resigned was a teacher, the fraction is (15 – 14)/39 = 14/39.
EACH statement ALONE is sufficient.
Question 6: If ABCD in the figure below is a square, what is the length of AE?
(1) The area of triangle AEB is 25.
(2) The length of AE is equal to the length of FC.
(a) Statement (1) ALONE is sufficient, but statement (2) is not sufficient.
(b) Statement (2) ALONE is sufficient, but statement (1) is not sufficient.
(c) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(d) EACH statement ALONE is sufficient.
(e) Statements (1) and (2) TOGETHER are NOT sufficient.
Answer:
Statement (1): The area of triangle AEB is 25 and AB = 10.
AreaAEB = (1/2)·AB·AE, which gives AE = 5.
Statement (2): The length of AE is equal to the length of FC does not determine the length of AE.
Any length AE between 0 and 10 will satisfy statement (2).
Question 7: If y = x2 + ax + b, y is minimum when x is:
(a) a/b
(b) -a/b
(c) -a/2
(d) -b/2
(e) b/a
Answer: Parabola px2 + qx + r = 0 is minimum for x = -(q/2p).
In our case y is minimum for x = -a/2.
Question #8: If y = mx + n is a line in the (x, y) plane, what is the slope m?
(1) y(2) = 2
(2) y(3) = -4
(a) Statement (1) ALONE is sufficient, but statement (2) is not sufficient.
(b) Statement (2) ALONE is sufficient, but statement (1) is not sufficient.
(c) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(d) EACH statement ALONE is sufficient.
(e) Statements (1) and (2) TOGETHER are NOT sufficient.
Answer:
A line in the standard rectangular plane is determined by 2 points.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Question 9: A box contains 11 tennis balls, la from 1 to 11. If 2 tennis balls are selected at random, what is the probability that both of them are numbered with odd numbers?
(a) 3/11
(b) 6/11
(c) 5/11
(d) 5/7
(e) 4/11
Answer:
The probability that the first tennis ball is an odd one is (number of ‘odd’ balls)/(total number of balls) = 6/11.
The probability that the second tennis ball is labeled with an odd number is (number of ‘odd’ balls left in the box)/(total number of balls left in the box) = 5/10.
The probability that both of them are labeled with odd numbers is the product (6/11)·(5/10) = 3/11.
Question 10: There are small books and large books on a desk. What is the total number of books on the desk?
(1) The ratio between the number of large books and small books is 3.
(2) There are 12 more large books than small books on the desk.
(a) Statement (1) ALONE is sufficient, but statement (2) is not sufficient.
(b) Statement (2) ALONE is sufficient, but statement (1) is not sufficient.
(c) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(d) EACH statement ALONE is sufficient.
(e) Statements (1) and (2) TOGETHER are NOT sufficient.
Answer:
If the number of large books is x and the number of small books is y, statement (a) can be written as x = 3y. We cannot find x + y from this equation alone. Statement (1) alone is not sufficient.
Statement (2) can be written as x = 12 + y.
Alone, this equation is not sufficient to find x + y.
The system:
x = 12 + y
x = 3y
is sufficient to find x + y.
(c) is the right answer.