ตัวอย่างข้อสอบ gmat
CHECKING EXTREME CASES
• When drawing a geometric figure or checking a given one, be sure to include drawings of extreme cases as well as ordinary ones.
| Example 1: In the figure to the right, AC is a chord and B is a point on the circle. What is the measure of angle x? |  |
Although in the drawing AC looks to be a diameter, that cannot be assumed. All we know is that AC is a chord. Hence, numerous cases are possible, three of which are illustrated below:
In Case I, x is greater than 45 degrees; in Case II, x equals 45 degrees; in Case III, x is less than 45 degrees. Hence, the given information is not sufficient to answer the question.
Example 2: Three rays emanate from a common point and form three angles with measures p, q, and r. What is the measure of q + r ?
It is natural to make the drawing symmetric as follows:
In this case, p = q = r = 120, so q + r = 240. However, there are other drawings possible. For example:
In this case, q + r = 180. Hence, the given information is not sufficient to answer the question.
Problems:
1. Suppose 3p + 4q = 11. Then what is the value of q?
(1) p is prime.
(2) q = -2p
Solution: (1) is insufficient. For example, if p = 3 and q = 1/2, then 3p + 4q = 3(3) + 4(1/2) = 11. However, if p = 5 and q = -1, then 3p + 4q = 3(5) + 4(-1) = 11. Since the value of q is not unique, (1) is insufficient.
Turning to (2), we now have a system of two equations in two unknowns. Hence, the system can be solved to determine the value of q. Thus, (2) is sufficient, and the answer is B.
2. What is the perimeter of triangle ABC above?
(1) The ratio of DE to BF is 1: 3.
(2) D and E are midpoints of sides AB and CB, respectively.
Solution: Since we do not even know whether BF is an altitude, nothing can be determined from (1). More importantly, there is no information telling us the absolute size of the triangle.
As to (2), although from geometry we know that DE = AC/2, this relationship holds for any size triangle. Hence, (2) is also insufficient.
Together, (1) and (2) are also insufficient since we still don't have information about the size of the triangle, so we can't determine the perimeter. The answer is E.
3. A dress was initially listed at a price that would have given the store a profit of 20 percent of the wholesale cost. What was the wholesale cost of the dress?
(1) After reducing the asking price by 10 percent, the dress sold for a net profit of 10 dollars.
(2) The dress sold for 50 dollars.
Solution: Consider just the question setup. Since the store would have made a profit of 20 percent on the wholesale cost, the original price P of the dress was 120 percent of the cost: P = 1.2C. Now, translating (1) into an equation yields:
P - .1P = C + 10
Simplifying gives
.9P = C + 10
Solving for P yields
P = (C + 10)/.9
Plugging this expression for P into P = 1.2C gives
(C + 10)/.9 = 1.2C
Since we now have only one equation involving the cost, we can determine the cost by solving for C. Hence, the answer is A or D.
(2) is insufficient since it does not relate the selling price to any other information. Note, the phrase "initially listed" implies that there was more than one asking price. If it wasn't for that phrase, (2) would be sufficient. The answer is A.
4. What is the value of the two-digit number x?
(1) The sum of its digits is 4.
(2) The difference of its digits is 4.
Solution: Considering (1) only, x must be 13, 22, 31, or 40. Hence, (1) is not sufficient to determine the value of x.
Considering (2) only, x must be 40, 51, 15, 62, 26, 73, 37, 84, 48, 95, or 59. Hence, (2) is not sufficient to determine the value of x.
Considering (1) and (2) together, we see that 40 and only 40 is common to the two sets of choices for x. Hence, x must be 40. Thus, together (1) and (2) are sufficient to uniquely determine the value of x. The answer is C.
5. If x and y do not equal 0, is x/y an integer?
(1) x is prime.
(2) y is even.
Solution: (1) is not sufficient since we don't know the value of y. Similarly, (2) is not sufficient. Furthermore, (1) and (2) together are still insufficient since there is an even prime number--2. For example, let x be the prime number 2, and let y be the even number 2 (don't forget that different variables can stand for the same number). Then x/y = 2/2 = 1, which is an integer. For all other values of x and y, x/y is not an integer. (Plug in a few values to verify this.) The answer is E.
6. Is 500 the average (arithmetic mean) score on the GMAT?
(1) Half of the people who take the GMAT score above 500 and half of the people score below 500.
(2) The highest GMAT score is 800 and the lowest score is 200.
Solution: Many students mistakenly think that (1) implies the average is 500. Suppose just 2 people take the test and one scores 700 (above 500) and the other scores 400 (below 500). Clearly, the average score for the two test-takers is not 500. (2) is less tempting. Knowing the highest and lowest scores tells us nothing about the other scores. Finally, (1) and (2) together do not determine the average since together they still don't tell us the distribution of most of the scores. The answer is E.
7. The set S of numbers has the following properties:
I) If x is in S, then 1/x is in S.
II) If both x and y are in S, then so is x + y.
Is 3 in S?
(1) 1/3 is in S.
(2) 1 is in S.
Solution: Consider (1) alone. Since 1/3 is in S, we know from Property I that 1/(1/3) = 3 is in S. Hence, (1) is sufficient.
Consider (2) alone. Since 1 is in S, we know from Property II that 1 + 1 = 2 (Note, nothing in Property II prevents x and y from standing for the same number. In this case both stand for 1.) is in S. Applying Property II again shows that 1 + 2 = 3 is in S. Hence, (2) is also sufficient. The answer is D.
8. What is the area of the triangle above?
(1) a = x, b = 2x, and c = 3x.
(2) The side opposite a is 4 and the side opposite b is 3.
Solution: From (1) we can determine the measures of the angles: a + b + c = x + 2x + 3x = 6x = 180
Dividing the last equation by 6 gives: x = 30
Hence, a = 30, b = 60, and c = 90. However, different size triangles can have these angle measures, as the diagram below illustrates:
Hence, (1) is not sufficient to determine the area of the triangle.
Turning to (2), be careful not to assume that c is a right angle. Although from the diagram c appears to be a right angle, it could be 91 degrees or 89 degrees--we can't tell. Hence, (2) is not sufficient to determine the area of the triangle.
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