# Graduate School - Verbal And Quantitative Practice, Gre

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## Graduate School - Verbal And Quantitative Practice, Gre

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The sample questions that follow are organized by content category and represent the types of questions included in the General Test. The purpose of these questions is to provide some indication of the range of topics covered in the test as well as to provide some addi- tional questions for practice purposes. These questions do not represent either the length of the actual test or the proportion of actual test questions within each of the content categories.

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## Nội dung Text: Graduate School - Verbal And Quantitative Practice, Gre

1. Graduate Record Examinations® Preparing for the Verbal and Quantitative Sections of the GRE General Test Sample Questions with Explanations Copyright © 2002 by Educational Testing Service. All rights reserved. EDUCATIONAL TESTING SERVICE, ETS, the ETS logos, GRADUATE RECORD EXAMINATIONS, and GRE ® are registered trademarks of Educational Testing Service.
7. ALGEBRA the xy-plane. For example, the graph of the linear equation (including coordinate geometry) y = − 3 x − 2 is a line with a slope of − 3 and a y-intercept of 5 5 Questions that test algebra include those involving the follow- –2, as shown below. ing topics: rules of exponents, factoring and simplifying algebraic expressions, concepts of relations and functions, equations and inequalities, and coordinate geometry (including slope, intercepts, and graphs of equations and inequalities). The skills required include the ability to solve linear and qua- dratic equations and inequalities, and simultaneous equations; the ability to read a word problem and set up the necessary equations or inequalities to solve it; and the ability to apply basic algebraic skills to solve problems. Some facts about algebra that may be helpful If ab = 0, then a = 0 or b = 0; for example, if (x 1) (x + 2) = 0, it follows that either x 1 = 0 or x + 2 = 0; therefore, x = 1 or x = 2. Adding a number to or subtracting a number from both sides of an equation preserves the equality. Similarly, multiplying or GEOMETRY dividing both sides of an equation by a nonzero number preserves the equality. Similar rules apply to inequalities, except that multi- Questions that test geometry include those involving the plying or dividing both sides of an inequality by a negative number following topics: properties associated with parallel lines, reverses the inequality. For example, multiplying the inequality circles, triangles (including isosceles, equilateral, and 3x 4 > 5 by 4 yields the inequality 12x 16 > 20; however, mul- 30˚ 60˚ 90˚ triangles), rectangles, other polygons, area, tiplying that same inequality by 4 yields 12x + 16 < 20. perimeter, volume, the Pythagorean Theorem, and angle mea- The following rules for exponents may be useful. If r, s, x, and y sure in degrees. The ability to construct proofs is not measured. are positive numbers, then 1 1 5 – 3 = 53 = 125 1 (a) x–r = ; for example, Some facts about geometry that may be helpful xr If two lines intersect, then the opposite angles (called vertical (b) (x )(x ) = x r+s; r s for example, (32 )(34 ) = 36 = 729 angles) are equal; for example, in the figure below, x = y. (c) (x r )(yr ) = (xy)r; for example, (34 )(24 ) = 64 = 1,296 (d) (x r )s = x rs; for example, (23 )4 = 212 = 4,096 x r = x r–s 42 1 1 y˚ x˚ (e) xs ; for example, = 42–5 = 4–3 = 3 = 45 4 64 If two parallel lines are intersected by a third line, certain angles The rectangular coordinate plane, or xy-plane, is shown below. that are formed are equal. As shown in the figure below, if , then x = y = z. z˚ x˚ y˚ The sum of the degree measures of the angles of a triangle is 180. The square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the two legs (Pythagorean Theorem). The sides of a 45˚– 45˚– 90˚ triangle are in the ratio 1: 1: 2, and the sides of a 30˚– 60˚– 90˚ triangle are in the ratio 1 : 3 : 2. Drawing in lines that are not shown in a figure can sometimes be helpful in solving a geometry problem; for example, by drawing the dashed lines in the pentagon below, The x-axis and y-axis intersect at the origin O, and they partition the plane into four quadrants, as shown. Each point in the plane has coordinates (x, y) that give its location with respect to the axes; for example, the point P(2, –8) is located 2 units to the right of the y- the total number of degrees in the angles of the pentagon can be axis and 8 units below the x-axis. The units on the x-axis are the found by adding the number of degrees in each of the three same length as the units on the y-axis, unless otherwise noted. triangles: 180 + 180 + 180 = 540. Equations involving the variables x and y can be graphed in 7
9. Here are some more examples: Column A Column B Column A Column B Correct Answer 1. 9.8 100 Examples 4-6 refer to PQR. 100 denotes 10, the positive square root of 100. (For any posi- tive number x, x denotes the positive number whose square is x.) Q Since 10 is greater than 9.8, the best answer is (B). It is important not to confuse this question with a comparison of 9.8 and x where x x2 = 100. The latter comparison would yield (D) as the correct y answer because x2 = 100 implies that either x = 10 or x = 10, and there would be no way to determine which value x would w z actually have. P N R 2. ( 6) 4 ( 6)5 Example 4: PN NR A B C D E Since ( 6)4 is the product of four negative factors, and the (since equal measures product of an even number of negative numbers is positive, cannot be assumed, ( 6)4 is positive. Since the product of an odd number of negative even though PN and numbers is negative, ( 6)5 is negative. Therefore, ( 6)4 is greater NR appear to be equal) than ( 6)5 since any positive number is greater than any negative Example 5: x y A B C D E number. The best answer is (A). It is not necessary to calculate (since N is between that ( 6)4 = 1,296 and that ( 6)5 = 7,776 in order to make P and R) the comparison. 3. The area of The area of Example 6: w+z 180 A B C D E an equilateral a right triangle (since PR is a straight line) triangle with with legs 3 side 6 and 9 A machine was in operation for t minutes. The area of a triangle is one half the product of the lengths of the base and the altitude. In Column A, the length of the altitude must Example 7: The number 60t A B C D E first be determined. A sketch of the triangle may be helpful. of seconds that the machine was in operation 6 6 h A farmer has two plots of land that are equal in area. The first 3 3 6 plot is divided into 16 parcels with m acres in each parcel, and the The altitude h divides the base of an equilateral triangle into two second plot is divided into 20 par- equal parts. From the Pythagorean Theorem, h 2 + 32 = 6 2, or cels with n acres in each parcel. h 3 3. Therefore, the area of the triangle in Column A is Example 8: m n ( 1 )(6)(3 A B C D E 3) 9 3. In Column B, the base and the altitude of the 2 Directions: Each of the sample questions consists of two right triangle are the two legs; therefore, the area is quantities, one in Column A and one in Column B. There may be additional information, centered above the two col- umns, that concerns one or both of the quantities. A symbol ( 1 )(9)( 2 3) 9 3 2 . Since 9 3 is greater than 9 3 2 , the best that appears in both columns represents the same thing in answer is (A). Column A as it does in Column B. You are to compare the quantity in Column A with the quantity in Column B and decide whether: x2 = y 2 + 1 (A) The quantity in Column A is greater. 4. x y (B) The quantity in Column B is greater. (C) The two quantities are equal. From the given equation, it can be determined that x2 > y2; however, (D) The relationship cannot be determined from the relative sizes of x and y cannot be determined. For example, if the information given. y = 0, then x could be 1 or 1 and, since there is no way to tell Note: Since there are only four choices, NEVER MARK (E). which number x is, the best answer is (D). 9
11. y Some examples of problem solving questions involving data analysis, with explanations, follow. Questions 11-13 refer to the following table. x O PERCENT CHANGE IN DOLLAR AMOUNT OF SALES IN CERTAIN RETAIL STORES FROM 1977 TO 1979 9. If the equation y = 3x – 18 were graphed on the coordinate Percent Change axes above, the graph would cross the y-axis at the point (x, y) where From 1977 From 1978 Store to 1978 to 1979 (A) x 0 and y 18 (B) x 0 and y 18 P 10 10 (C) x 0 and y 6 Q 20 9 (D) x 6 and y 0 R 5 12 (E) x 6 and y 0 S 7 15 T 17 8 A graph crosses the y-axis at a point (x, y) where x 0. In the given equation, when x 0, y 3(0) 18 18. Therefore, the graph would cross the y-axis at the point (0, 18), and the 11. In 1979, for which of the stores was the dollar amount of best answer is (B). sales greater than that of any of the others shown? (A) P (B) Q (C) R (D) S 10. The operation denoted by the symbol is defined for (E) It cannot be determined from the information given. all real numbers p and r as follows. Since the only information given in the table is the percent p r pr p r change from year to year, there is no way to compare the dollar amount of sales for the stores in 1979 or in any other year. The What is the value of ( 4) 5? best answer is (E). (A) 9 (B) 11 12. In store T, the dollar amount of sales for 1978 was approxi- (C) 19 mately what percent of the dollar amount of sales for 1979? (D) 19 (A) 86% (B) 92% (C) 109% (D) 117% (E) 122% (E) 21 If A is the amount of sales for store T in 1978, then 0.08A is the amount of decrease and A 0.08A = 0.92A is the amount of sales By the definition, for 1979. Therefore, the desired result can be obtained by dividing ( 4) 5 ( 4)(5) ( 4) 5 20 4 5 11, 1 A by 0.92A, which equals , or approximately 109%. The best 0. 92 and therefore the best answer is (B). answer is (C). Some problem solving questions involve data analysis; many 13. If the dollar amount of sales in store P was $800,000 in 1977, of these occur in sets of two to five questions that share com- what was the dollar amount of sales in that store in 1979? mon data in the form of tables, graphs, etc. In questions that (A)$727,200 (B) $792,000 (C)$800,000 involve data analysis, graphs are drawn as accurately as pos- (D) $880,000 (E)$968,000 sible. Therefore, you can read or estimate data values from the graphs (whether or not there is a note that the graphs are drawn If sales in store P were $800,000 in 1977, then in 1978 they were to scale). 110 percent of that, i.e.,$880,000. In 1979 sales were 90 percent of The following strategies may help in answering problem $880,000, i.e.,$792,000. Note that an increase of 10 percent in one solving questions that involve data analysis. year and a decrease of 10 percent in the following year does not Scan the data briefly to see what it is about, but do not result in the same dollar amount as the original dollar amount of attempt to analyze it in too much detail before reading the sales because the base used in computing the percents changes from questions. Focus on those aspects of the data that are neces- $800,000 to$880,000. The best answer is (B). sary to answer the questions. Be sure to read all notes related to the data. When possible, try to make visual comparisons of the data given in a graph and estimate products and quotients rather than perform involved computations. Remember that these questions are to be answered only on the basis of the data given, everyday facts (such as the number of days in a year), and your knowledge of mathematics. Do not make use of specific information you recall that may seem to relate to the particular situation on which the questions are based unless that information can be derived from the data provided. 11