Question 1: If x and y are integers, and 4x + 5y = 19, which of the following could be the value of x?
(a) 1
(b) 2
(c) 3
(d) 4
(e) 5
Solution: 4x + 5y = 19.
y = (19 – 4x)/5 is an integer, so 19 – 4x must be divisible by 5.
Out of the five choices given, the only value of x that satisfies this condition is x = 1:
y = (19 – 4)/5 = 3
(a) is the correct answer.
Question 2: Is y > 0 ?
(1) x2 > 5
(2) x + y = 32
(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:
The first inequality, x2 > 5 does not give is any information about y.
x can be greater than √5 or less then -√5. Statement (1) alone is not sufficient.
The second statement is not sufficient since it does not give any information about x.
y = 32 – x can be positive or negative. For x greater than 32, y is negative and for x less than 32 it is positive, so both statements together are not sufficient.
In conclusion (e) is the correct answer.
Question 3: What is the value of angle a in the figure below?
(a) 60o
(b) 70o
(c) 80o
(d) 90o
(e) 100o
Answer: Two of the angles of the triangle formed by the 3 lines are 180o -140o = 40o and 180o -120o = 60o
The sum of the angles of any triangle is 180o.
Anglea = 180o – 60o – 40o.
Anglea = 80o.
Question 4: When a coin is taken at random from a box with US, British and French coins, what is the probability that the coin is British?
1. There are twice as many American coins as there are British coins in the box.
2. There are 24 coins in the box.
(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:
From the first statement alone we cannot find the ratio between the number of British coins and the total number of coins. Statement (1) is not sufficient.
The same applies for statement (2).
From both statements together we have a system of 2 equations with 3 parameters:
x + y + z = 240,
y = 2x,
where x, y and z are the number of American, British and French coins.
The system has an infinity of solutions, so both statements are not sufficient.
Question 5: In triangle ABC, MN is parallel with BC and AM/AB = 2/3. What is the ratio between the area of triangle AMN and the area of triangle ABC?
(a) 2/3
(b) 2/9
(c) 4/5
(d) 4/9
(e) 5/9
Answer: Triangles ABC and AMN are similar, so the lengths of their corresponding sides are proportional. This means MN/BC = 2/3 and also the ratio between their altitudes is also 2/3.
The area of triangle ABC = (BC · H)/2 where H is the altitude of triangle ABC.
The area of triangle AMN = (MN · h)/2 where h is the altitude of triangle AMN.
AreaAMN / AreaABC = (2/3) · (2/3) = 4/9.
Question 6: Is the volume of cylinder C1 greater than the volume of cube C2?
(1) The radius of C1 is equal to the edge of C2
(2) The height of C1 is twice the edge of the C2
(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:
The volume of a cube is a function of the length of its edges and the volume of a cylinder is a function of its radius and height.
To determine the relation between the volumes of the cylinder and cube, we need both the ratios (height of the cylinder)/(edge of the cube) and (radius of the cylinder)/(edge of the cube).
Both statements together are sufficient, but neither statement alone is sufficient.
Question 7: 4 students go to the movies and seat on a row of 4 seats. How many different ways can the 4 students sit?
(a) 12
(b) 24
(c) 26
(d) 18
(e) 16
Answer: The number of arrangements of n distinct items in a row is n!.
4! = 4·3·2·1.
4! = 24.
Question #8: If the length of a 3-degree arc of a circle is 2 inches, what is the circumference of the circle?
(a) 200 inches
(b) 360 inches
(c) 240 inches
(d) 120 inches
(e) 100 inches
Answer: The circumference of the circle is 2 inches·(360o/3o) = 240 inches.
Question 9: For any real x, which of the following is equivalent to x6 + x5 + x4?
(a) x4(x3 + x + 1)
(b) x4(x2 + 1)
(c) x5(x2 + x + 1)
(d) x4(x2 + x + 1)
(e) x3(x2 + x + 1)
Answer:
x6 + x5 + x4 = x4(x2 + x + 1).
Question 10: Is the slope of line l positive?
(1) (0,5) is the intersection point of line l with the y axis
(2) Line l is perpendicular to line y = 5x + 9
(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) alone is not sufficient. Different lines, all passing through (0, 5) have different positive or negative slopes.
Statement (2) alone is sufficient. A line perpendicular to any line y = mx + n will have a slope of -1/m.
,br/>