GRE Math Practice Test 18

Test 18

GRE MATH

Complete GRE MATH Preparation Materialhttp://studymaterialcollection.blogspot.com/2015/12/complete-gre-math-preparation-material.html

Do You Want to Search the Specific:Fulbright Scholarships, JOB, Online Work or Online Job, Free Video Tutorial, Free VideoLectures for Different Subjects and Free Online Courses, GRE, GMAT, IELTS, TOEFL ThatAssist You in Your Professional Life or Field. Click below To Visit the SEARCH ENGINE’s Collection page http://searchenginecollectionpage.blogspot.com/2015/10/SearchEngineCollection.html

GRE Math Tests

346

Questions: 24 Time: 45 minutes

[Multiple-choice Question – Select One Choice Only] 1. Which one of the following is the solution of the system of equations given?

x + 2y = 7 x + y = 4

(A) x = 3, y = 2 (B) x = 2, y = 3 (C) x = 1, y = 3 (D) x = 3, y = 1 (E) x = 7, y = 1

[Multiple-choice Question – Select One or More Answer Choices] 2. The last digit of the positive even number n equals the last digit of n2. Which of the following could

be n ?

(A) 10 (B) 14 (C) 15 (D) 16 (E) 17

[Multiple-choice Question – Select One Answer Choice Only] 3. How many positive five-digit numbers can be formed with the digits 0, 3, and 5?

(A) 14 (B) 15 (C) 108 (D) 162 (E) 243

Test 18 — Questions

347

Quantitative Comparison Question] 4. Column A The sum of the positive integers

from 1 through n can be calculated by the formula

n n +1( )2

Column B

The sum of the multiples of 6 between 0 and 100

The sum of the multiples of 8 between 0 and 100

Quantitative Comparison Question] 5. Column A Column B The average of three numbers if

the greatest is 20 The average of three numbers if

the greatest is 2

[Multiple-choice Question – Select One Choice only] 6. The side length of a square inscribed in a circle is 2. What is the area of the circle?

(A) (B)

2 (C) 2 (D)

2 2 (E) 2

GRE Math Tests

348

[Multiple-choice Question – Select One Choice Only] 7. In the figure, P =

(A) 15° (B) 30° (C) 35° (D) 40° (E) 50°

[Multiple-choice Question – Select One or More Answer Choices] 8. In the figure, lines l and k are parallel. Which of the following must be true?

(A) a < b (B) a = 2b + 30 (C) a = b + 10 (D) a b (E) a > b

P D

C

A

B

105°

115°

b + 10°

(b + 20)°

a°

l

k

O


349

The following figure is for problems 9 and 10. Lines l and k are parallel and a is an acute angle, that is, a is less than 90 degrees in measure.

[Multiple-choice Question – Select One or More Answer Choices] 9. Which of the following must be true?

(A) b > 10 (B) b > 15 (C) b < 20 (D) b < 30 (E) b > 45

[Multiple-choice Question – Select One or More Answer Choices] 10. Which of the following could be true?

(A) b > 10 (B) b > 15 (C) b < 20 (D) b < 30 (E) b > 45

(b + 30)° l

k

O

b°

a°

GRE Math Tests

350

[Multiple-choice Question – Select One Answer Choice Only] 11. In a triangle with sides of lengths 3, 4, and 5, the smallest angle is 36.87°. In the figure, O is the

center of the circle of radius 5. A and B are two points on the circle, and the distance between the points is 6. What is the value of x ?

(A) 36.87 (B) 45 (C) 53.13 (D) 116.86 (E) 126.86

Quantitative Comparison Question] 12. Column A In the figure, ABC is inscribed

in the circle. The triangle does not contain the center of the circle O.

Column B

x 90

.O

x°

A

B C

A

B

O

x° 6


351

[Multiple-choice Question – Select One or More Answer Choices] 13. The slope of the line 2x + y = 3 is NOT the same as the slope of which of the following lines?

(A) 2x + y = 5 (B) x + y/2 = 3 (C) x = –y/2 – 3 (D) y = 7 – 2x (E) x + 2y = 9

[Multiple-choice Question – Select One or More Answer Choices] 14. If

x + x = 4 , then which of the following is odd?

(A) x2 + 3x (B) x2 + 3x + 2 (C) x2 + 4x (D) x2 + 4x + 2 (E) x2 + 4x + 3

Quantitative Comparison Question] 15. Column A (x – 2y)(x + 2y) = 5

(2x – y)(2x + y) = 35 Column B

2x2 – y2 x2 – 2y2

GRE Math Tests

352

[Multiple-choice Question – Select One Answer Choice Only] 16. In 2012, the arithmetic mean of the annual incomes of Jack and Jill was $3800. The arithmetic mean

of the annual incomes of Jill and Jess was $4800, and the arithmetic mean of the annual incomes of Jess and Jack was $5800. What is the arithmetic mean of the incomes of the three?

(A) $4000 (B) $4200 (C) $4400 (D) $4800 (E) $5000

Quantitative Comparison Question] 17. Column A The savings from a person’s

income is the difference between his income and his expenditure. The ratio of Marc's income to his boss's is 3 : 4. Their respective expenditures ratio is 1 : 2.

Column B

The savings from income of Marc

The savings from income of Marc’s Boss

Quantitative Comparison Question] 18. Column A Item A cost John $90 and item B

cost him $100. Column B

Percentage of profit that John earned by selling item A with a profit of $10

Percentage of profit that John earned by selling item B with a profit of $10


353

Quantitative Comparison Question] 19. Column A James purchased Medicine A for x

dollars, which included a sales tax of 5%. Kate was charged 5% for sales tax on x dollars that Medicine B costs.

Column B

Sales tax paid by James Sales tax paid by Kate

[Multiple-choice Question – Select One or More Answer Choices] 20. A driver plans that he should travel at 80 mph for the first half (either by distance or by time) of a trip

and at 40 mph for the second half of the trip. Which of the following could be the average speed of the car during the entire trip?

(A) 50 (B) 60 (C) 80 (D) 80/3 (E) 160/3

GRE Math Tests

354

[Multiple-choice Question – Select One Answer Choice Only] 21. For how many integers n between 5 and 20, inclusive, is the sum of 3n, 9n, and 11n greater than 200?

(A) 4 (B) 8 (C) 12 (D) 16 (E) 20

[Multiple-choice Question – Select One Answer Choice Only] 22. What is the probability that the product of two integers (not necessarily different integers) randomly

selected from the numbers 1 through 20, inclusive, is odd?

(A) 0 (B) 1/4 (C) 1/2 (D) 2/3 (E) 3/4

[Multiple-choice Question – Select One or More Answer Choices] 23. Removing which of the following numbers from the set S = {1, 2, 3, 4, 5, 6} would move the median

of the set S to the right on the number line?

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5 (F) 6


Do You Want to Search the Specific:Fulbright Scholarships, JOB, Online Work or Online Job, Free Video Tutorial, Free VideoLectures for Different Subjects and Free Online Courses, GRE, GMAT, IELTS, TOEFL ThatAssist You in Your Professional Life or Field. Click below To Visit the SEARCH ENGINE’s Collection page http://searchenginecollectionpage.blogspot.com/2015/10/SearchEngineCollection.html


355

[Multiple-choice Question – Select One Answer Choice Only] 24. On average, a sharpshooter hits the target once every 3 shots. What is the probability that he will hit

the target in 4 shots?

(A) 1 (B) 1/81 (C) 1/3 (D) 65/81 (E) 71/81

GRE Math Tests

356

Answers and Solutions Test 18:

Question Answer 1. C 2. A, D 3. D 4. A 5. D 6. C 7. D 8. E 9. D 10. A, B, C, D 11. C 12. A 13. E 14. E 15. A 16. D 17. D 18. A 19. B 20. B, E 21. C 22. B 23. A, B, C 24. D

If you got 18/24 correct on this test, you are likely to get 750+ on the actual GRE by the time you complete all the tests in the book. 1. The given system of equations is x + 2y = 7 and x + y = 4. Now, just substitute each answer-choice into the two equations and see which one works (start checking with the easier equation, x + y = 4):

Choice (A): x = 3, y = 2: Here, x + y = 3 + 2 = 5 4. Reject.

Choice (B): x = 2, y = 3: Here, x + y = 2 + 3 = 5 4. Reject.

Choice (C): x = 1, y = 3: Here, x + y = 1 + 3 = 4 = 4, and x + 2y = 1 + 2(3) = 7. Correct.

Choice (D): x = 3, y = 1: Here, x + y = 3 + 1 = 4, but x + 2y = 3 + 2(1) = 5 7. Reject.

Choice (E): x = 7, y = 1: Here, x + y = 7 + 1 = 8 4. Reject. The answer is (C).

Test 18 — Solutions

357

Method II (without substitution): In the system of equations, subtracting the bottom equation from the top one yields (x + 2y) – (x + y) = 7 – 4, or y = 3. Substituting this result in the bottom equation yields x + 3 = 4. Solving the equation for x yields x = 1. The answer is (C). 2. Numbers ending with the digits 0, 1, 5, or 6 will have their squares also ending with the same digit. For example,

10 ends with 0, and 102 (= 100) also ends with 0. 11 ends with 1, and 112 (= 121) also ends with 1. 15 ends with 5, and 152 (= 225) also ends with 5. 16 ends with 6, and 162 (= 256) also ends with 6.

Among the four numbers 0, 1, 5, or 6, even numbers only end with 0 or 6. Choices (A) and (D) have such numbers. The answer is (A) and (D). 3. Let the digits of the five-digit positive number be represented by 5 compartments:

Each of the last four compartments can be filled in 3 ways (with any one of the numbers 0, 3 and 5). The first compartment can be filled in only 2 ways (with only 3 and 5, not with 0, because placing 0 in the first compartment would yield a number with fewer than 5 digits).

3 5

0 3 5

0 3 5

0 3 5

0 3 5

Hence, the total number of ways the five compartments can be filled in is 2 3 3 3 3 = 162. The answer is (D). 4. The sum of the multiples of 6 between 0 and 100 equals

6 + 12 + 18 + ... + 96 = 6(1) + 6(2) + 6(3) + ... + 6(16) = 6(1 + 2 + 3 + ... + 16) =

616 16+1( )

2

= 3 16 17.

The sum of multiples of 8 between 0 and 100 equals

8 + 16 + 24 + ... + 96 = 8(1) + 8(2) + 8(3) + ... + 8(12) = 8(1 + 2 + 3 + ... + 12) =

812 12+1( )

2= 4 12 13

Since 3 16 17 (= 48 17) is clearly greater than 4 12 13 (= 48 13), Column A is greater than Column B and the answer is (A).

GRE Math Tests

358

5. At first glance, Column A appears larger than Column B. However, the problem does not exclude negative numbers. Suppose the three numbers in Column A are –20, 0, and 20 and that the three numbers

in Column B are 0, 1, and 2. Then the average for Column A would be

20+ 0+ 203

=03

= 0 , and the

average for Column B would be

0+1+ 23

=33

= 1. In this case, Column B is larger. Clearly, there are also

numbers for which Column A would be larger. Hence, the answer is (D). 6. The diagonal of a square inscribed in a circle is a diameter of the circle. The formula for the diagonal of a square is

2 side. Hence, the diameter of the circle inscribing the square of side length 2 is 2 2 =

2 2 . Since radius = diameter/2, the radius of the circle is 2 22

= 2 . Hence, the area of the circle is

radius2 =

2( )2 = 2 . The answer is (C).

7. Since the angle in a line is 180°, we have PAD + DAB = 180, or PAD + 105 = 180 (from the figure, DAB = 105°). Solving for PAD yields PAD = 75. Since the angle in a line is 180°, we have ADP + ADC = 180, or ADP + 115 = 180 (from the figure,

ADC = 115°). Solving for ADP yields ADP = 65. Now, summing the angles of the triangle PAD to 180° yields

P + PAD + ADP = 180

P + 75 + 65 = 180

P = 40 The answer is (D).

diameter


359

8. Draw line m passing through O and parallel to both line l and line k.

Now, observe that angle x is only part of angle a, and x = b since they are alternate interior angles. Since x is only part of angle a, a > x and a > b. The answer is (E). 9. Draw a line parallel to both of the lines l and k and passing through O.

We are given that a is an acute angle. Hence, a < 90. Since angles p and b + 30 are alternate interior angles, they are equal. Hence, p = b + 30. Similarly, angles q and b are alternate interior angles. Hence, q = b. Since angle a is the sum of its sub-angles p and q, a = p + q = (b + 30) + b = 2b + 30. Solving this equation for b yields b = (a – 30)/2 = a/2 – 15. Now, dividing both sides of the inequality a < 90 by 2 yields a/2 < 45. Also, subtracting 15 from both sides of the inequality yields a/2 – 15 < 30. Since a/2 – 15 = b, we have b < 30. The answer is (D). 10. From the previous question, all the choices could be true except (E). We know that b < 30. The answer is (A), (B), (C), and (D).

b°

(b + 30)°

x°

l

k

O (a – x)° m a°

b°

(b + 30)°

q°

l

k

O p° m

GRE Math Tests

360

11. We are given that the smallest angle of any triangle of side lengths 3, 4, and 5 is 36.87°. The smallest angle is the angle opposite the smallest side, which measures 3. Now, in AOB, OA = OB = radius of the circle = 5 (given). Hence, the angles opposite the two sides in the triangle are equal and therefore the triangle is isosceles. Just as in any isosceles triangle, the altitude on the third side AB must divide the side equally. Say, the altitude cuts AB at J. Then we have AJ = JB, and both equal AB/2 = 3.

Since the altitude is a perpendicular, applying The Pythagorean Theorem to AOJ yields AJ2 + JO2 = OA2; 32 + JO2 = 52; JO2 = 52 – 32 = (25 – 9) = 16 = 42. Square rooting both sides yields JO = 4. Hence, in AOJ, AJ = 3, JO = 4, and AO = 5. Hence, the smallest angle, the angle opposite the smallest side, AOJ equals 36.87°. Summing the angles of AOJ to 180° yields x + AJO + JOA = 180; x + 90 + 36.87 = 180. Solving this equation yields x = 180 – (90 + 36.87) = 180 – 126.87 = 53.13. The answer is (C). 12. A chord makes an acute angle on the circle to the side containing the center of the circle and makes an obtuse angle to the other side. In the figure, BC is a chord and does not have a center to the side of point A. Hence, BC makes an obtuse angle at point A on the circle. Hence, A, which equals x°, is obtuse and therefore is greater than 90°. Hence, Column A is greater than Column B, and the answer is (A). 13. The slope of a line expressed as y = mx + b is m. Expressing the given line 2x + y = 3 in that format yields y = –2x + 3. Hence, the slope is –2, the coefficient of x. Let’s express each line in the form y = mx + b and pick the line whose slope is not –2.

Choice (A): 2x + y = 5; y = –2x + 5, slope is –2. Reject. Choice (B): x + y/2 = 3; y = –2x + 6, slope is –2. Reject. Choice (C): x = –y/2 – 3; y = –2x – 6, slope is –2. Reject. Choice (D): y = 7 – 2x; y = – 2x + 7, slope is –2. Reject.

Choice (E): x + 2y = 9;

y =12x +

92

, slope is

12

2. Accept the choice.

The answer is (E).

A

B

O

x°

5

3

3 5 J


361

14. We are given that

x + x = 4 . If x is negative or zero, then

x equals –x and

x + x equals –x + x = 0. This conflicts with the given equation, so x is not negative nor 0. Hence, x is positive and therefore

x equals x. Putting this in the given equation yields

x + x = 4 x + x = 4 2x = 4 x = 2

Now, select the answer-choice that results in an odd number when x = 2.

Choice (A): x2 + 3x = 22 + 3(2) = 4 + 6 = 10, an even number. Reject. Choice (B): x2 + 3x + 2 = 22 + 3(2) + 2 = 4 + 6 + 2 = 12, an even number. Reject. Choice (C): x2 + 4x = 22 + 4(2) = 4 + 8 = 12, an even number. Reject. Choice (D): x2 + 4x + 2 = 22 + 4(2) + 2 = 4 + 8 + 2 = 14, an even number. Reject. Choice (E): x2 + 4x + 3 = 22 + 4(2) + 3 = 4 + 8 + 3 = 15, an odd number. Accept.

The answer is (E). 15. We are given the two equations:

(x – 2y)(x + 2y) = 5 (2x – y)(2x + y) = 35

Applying the Difference of Squares formula, (a + b)(a – b) = a2 – b2, to the left-hand sides of each equation yields

x2 – (2y)2 = 5 (2x)2 – y2 = 35

Simplifying these two equations yields

x2 – 4y2 = 5 4x2 – y2 = 35

Subtracting the bottom equation from the top one yields

(x2 – 4y2) – (4x2 – y2) = 5 – 35 –3x2 – 3y2 = –30 –3(x2 + y2) = –30 x2 + y2 = –30/–3 = 10

Now, Column A – Column B = (2x2 – y2) – (x2 – 2y2) = x2 + y2 = 10. Hence, Column A is 10 units greater than Column B. The answer is (A).

GRE Math Tests

362

16. Let a, b, and c be the annual incomes of Jack, Jill, and Jess, respectively. Now, we are given that The arithmetic mean of the annual incomes of Jack and Jill was $3800. Hence, (a + b)/2 = 3800. Multiplying by 2 yields a + b = 2 3800 = 7600. The arithmetic mean of the annual incomes of Jill and Jess was $4800. Hence, (b + c)/2 = 4800. Multiplying by 2 yields b + c = 2 4800 = 9600. The arithmetic mean of the annual incomes of Jess and Jack was $5800. Hence, (c + a)/2 = 5800. Multiplying by 2 yields c + a = 2 5800 = 11,600. Summing these three equations yields

(a + b) + (b + c) + (c + a) = 7600 + 9600 + 11,600

2a + 2b + 2c = 28,800

a + b + c = 14,400 The average of the incomes of the three equals the sum of the incomes divided by 3:

(a + b + c)/3 = 14,400/3 =

4800 The answer is (D). 17. Suppose Marc’s income is $300 and his boss’s income is $400 (incomes match the given ratio, 3 : 4). Also, suppose Marc’s expenditure is $100 and his boss’s expenditure is $200 (expenditures match the given ratio, 1 : 2). Then the savings from income of Marc is 300 – 100 = 200 and that of his boss is 400 – 200 = 200. In this case, Marc’s savings equals his boss’s savings. Now, suppose the expenditures are different: Marc’s expenditure is $150 and his boss’s expenditure is $300 (expenditures match the given ratio, 1 : 2). In this case, Marc’s savings is 300 – 150 = 150, and his boss’s savings is 400 – 300 = 100 (savings are not equal). Hence, we have a double case, and the answer is (D).

18. The formula for the profit percentage is

ProfitCost

100 .

Hence, Column A equals 10/90 100 = 100/9% = 11.1%, and Column B equals 10/100 100 = 10%. Hence, Column A is greater than Column B, and the answer is (A). 19. Let the original cost of Medicine A be a dollars. 5% tax on this equals (5/100)a = a/20. So, the total cost including sales tax is a + a/20 = 21a/20. This equals x dollars (what medicine A cost James). Hence,

we have 21a/20 = x. Solving for a yields a = 20x/21. Sales tax on A is

a20

=

20x2120

=x21

= Column A.

Now, Kate was charged sales tax on Medicine B. The charge was 5% of the cost of the medicine (x). 5% of x is (5/100)x = x/20 = Column B. Since 1/20 > 1/21, x/20 > x/21 and Column B > Column A. The answer is (B).


363

20. Case I (Half by time). Let t be the entire time of the trip. We have that the car traveled at 80 mph for t/2 hours and at 40 mph for the remaining t/2 hours. Remember that Distance = Speed Time. Hence, the net distance traveled during the two periods equals 80 t/2 + 40 t/2. Now, remember that

Average Speed =

Net DistanceTime Taken

=

80 t2

+ 40 t2

t=

80 12

+ 40 12

=

40 + 20 =

60

Select choice (B). Case II (Half by distance). Let d be the entire time of the trip. We have that the car traveled at 80 mph for d/2 hours and at 40 mph for the remaining d/2 hours. Remember that Distance = Speed Time. Hence, the net time taken to travel the two half distances equals (d/2)/80 + (d/2)/ 40. Now, the average speed

Average Speed = Total distance / Total time = d / [(d/2)/80 + (d/2)/ 40] = 1 / [(1/2)/80 + (1/2)/ 40] =

2 / [2(1/2)/80 + 2(1/2)/ 40] = 2 / [1/80 + 1/ 40] =

2 80 / [80 1/80 + 80 1/ 40] = 160 / [1 + 2] =

160 / 3

Select choice (E). The answers to select are (B) and (E).

21. The sum of 3n, 9n, and 11n is 23n. Since this is to be greater than 200, we get the inequality 23n > 200. From this, we get n > 200/23 8.7. Since n is an integer, n > 8. Now, we are given that 5 n 20. Hence, the values for n are 9 through 20, a total of 12 numbers. The answer is (C). 22. The product of two integers is odd when both integers are themselves odd. Hence, the probability of the product being odd equals the probability of both numbers being odd. Since there is one odd number in every two numbers (there are 10 odd numbers in the 20 numbers 1 through 20, inclusive), the probability of a number being odd is 1/2. The probability of both numbers being odd (independent case) is

1/2 1/2 = 1/4 The answer is (B).

GRE Math Tests

364

23. The median of the S = set {1, 2, 3, 4, 5, 6} is half the sum of the middle numbers:

(3 + 4)/2 =

7/2 =

3.5 To move the median to the right on the number line, remove at least one of the elements on the left of the median on the number line. For example, if we remove 3 from the set S = set {1, 2, 4, 5, 6}, then median is 4, the median increased. The correct answers are the numbers to the left of 3.5 on the number line, the choices (A), (B), and (C). 24. The sharpshooter hits the target once in 3 shots. Hence, the probability of hitting the target is 1/3. The probability of not hitting the target is 1 – 1/3 = 2/3. Now, (the probability of not hitting the target even once in 4 shots) + (the probability of hitting at least once in 4 shots) equals 1, because these are the only possible cases. Hence, the probability of hitting the target at least once in 4 shots is

1 – (the probability of not hitting even once in 4 shots)

The probability of not hitting in the 4 chances is

23232323

=1681

. Now, 1 – 16/81 = 65/81. The answer is

(D). This methodology is similar to Model 2. You might try analyzing why. Clue: The numerators of

23232323

=1681

are the number of ways of doing the specific jobs, and the denominators are the number

of ways of doing all possible jobs.


Do You Want to Search the Specific:Fulbright Scholarships, JOB, Online Work or Online Job, Free Video Tutorial, Free Video Lectures forDifferent Subjects and Free Online Courses, GRE, GMAT, IELTS, TOEFL That Assist You in YourProfessional Life or Field. Click below To Visit the SEARCH ENGINE’s Collection page http://searchenginecollectionpage.blogspot.com/2015/10/SearchEngineCollection.html

Documents

GRE Math Practice Test 18