GMAT Quantitative Review

5.0 Data Sufficiency

5.3 Sample Questions

Each data sufficiency problem consists of a question and two statements, labeled (1) and (2), which contain certain data. Using these data and your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of the word counterclockwise), decide whether the data given are sufficient for answering the question and then indicate one of the following answer choices:

1.  A Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

2.  B Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

3.  C BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

4.  D EACH statement ALONE is sufficient.

5.  E Statements (1) and (2) TOGETHER are not sufficient.

Note: In data sufficiency problems that ask for the value of a quantity, the data given in the statements are sufficient only when it is possible to determine exactly one numerical value for the quantity.

Example:

image

In ΔPQR, what is the value of x?

1.  (1) PQ = PR

2.  (2) y = 40

Explanation: According to statement (1) PQ = PR; therefore, ΔPQR is isosceles and y = z. Since x + y + z = 180, it follows that x + 2y = 180. Since statement (1) does not give a value for y, you cannot answer the question using statement (1) alone. According to statement (2), y = 40; therefore, x + z = 140. Since statement (2) does not give a value for z, you cannot answer the question using statement (2) alone. Using both statements together, since x + 2y= 180 and the value of y is given, you can find the value of x. Therefore, BOTH statements (1) and (2) TOGETHER are sufficient to answer the questions, but NEITHER statement ALONE is sufficient.

Numbers: All numbers used are real numbers.

Figures:

• Figures conform to the information given in the question, but will not necessarily conform to the additional information given in statements (1) and (2).

• Lines shown as straight are straight, and lines that appear jagged are also straight.

• The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero.

• All figures lie in a plane unless otherwise indicated.

1.           What is the tenths digit of the number d when it is written as a decimal?

1.  (1) image

2.  (2) 1,000d = 2,160

image

2.           A framed picture is shown above. The frame, shown shaded, is 6 inches wide and forms a border of uniform width around the picture. What are the dimensions of the viewable portion of the picture?

1.  (1) The area of the shaded region is 24 square inches.

2.  (2) The frame is 8 inches tall.

3.           What is the value of the integer x?

1.  (1) x rounded to the nearest hundred is 7,200.

2.  (2) The hundreds digit of x is 2.

4.           Is 2x > 2y?

1.  (1) x > y

2.  (2) 3x > 3y

5.           If p and q are positive, is pq less than 1?

1.  (1) p is less than 4.

2.  (2) q is less than 4.

6.           In a certain factory, hours worked by each employee in excess of 40 hours per week are overtime hours and are paid for at 1image times the employee’s regular hourly pay rate. If an employee worked a total of 42 hours last week, how much was the employee’s gross pay for the hours worked last week?

1.  (1) The employee’s gross pay for overtime hours worked last week was $30.

2.  (2) The employee’s gross pay for all hours worked last week was $30 more than for the previous week.

7.           If x > 0, what is the value of x5?

1.  (1) √x = 32

2.  (2) x2 = 220

8.           What is the value of the integer N?

1.  (1) 101 < N < 103

2.  (2) 202 < 2N < 206

9.           Is zw positive?

1.  (1) z + w3 = 20

2.  (2) z is positive.

image

10.       In the rectangular coordinate system above, if ΔOPQ and ΔQRS have equal area, what are the coordinates of point R?

1.  (1) The coordinates of point P are (0,12).

2.  (2) OP = OQ and QS = RS.

11.       If y is greater than 110 percent of x, is y greater than 75?

1.  (1) image

2.  (2) image

12.       What is the average (arithmetic mean) of x and y?

1.  (1) The average of x and 2y is 10.

2.  (2) The average of 2x and 7y is 32.

13.       What is the value of image?

1.  (1) image

2.  (2) image

image

14.       If image and w represent the length and width, respectively, of the rectangle above, what is the perimeter?

1.  (1) image

2.  (2) image

15.       For all x, the expression x∗ is defined to be ax + a, where a is a constant. What is the value of 2∗?

1.  (1) 3∗ = 2

2.  (2) 5∗ = 3

16.       Is k + m < 0?

1.  (1) k < 0

2.  (2) km > 0

17.       A retailer purchased a television set for x percent less than its list price, and then sold it for y percent less than its list price. What was the list price of the television set?

1.  (1) image

2.  (2) image

18.       If Ann saves x dollars each week and Beth saves y dollars each week, what is the total amount that they save per week?

1.  (1) Beth saves $5 more per week than Ann saves per week.

2.  (2) It takes Ann 6 weeks to save the same amount that Beth saves in 5 weeks.

19.       What is the total number of executives at Company P?

1.  (1) The number of male executives is image the number of female executives.

2.  (2) There are 4 more female executives than male executives.

20.       What is the ratio of c to d?

1.  (1) The ratio of 3c to 3d is 3 to 4.

2.  (2) The ratio of c + 3 to d + 3 is 4 to 5.

21.       A certain dealership has a number of cars to be sold by its salespeople. How many cars are to be sold?

1.  (1) If each of the salespeople sells 4 of the cars, 23 cars will remain unsold.

2.  (2) If each of the salespeople sells 6 of the cars, 5 cars will remain unsold.

22.       Committee member W wants to schedule a one-hour meeting on Thursday for himself and three other committee members, X, Y, and Z. Is there a one-hour period on Thursday that is open for all four members?

1.  (1) On Thursday W and X have an open period from 9:00 a.m. to 12:00 noon.

2.  (2) On Thursday Y has an open period from 10:00 a.m. to 1:00 p.m. and Z has an open period from 8:00 a.m. to 11:00 a.m.

23.       Some computers at a certain company are Brand X and the rest are Brand Y. If the ratio of the number of Brand Y computers to the number of Brand X computers at the company is 5 to 6, how many of the computers are Brand Y?

1.  (1) There are 80 more Brand X computers than Brand Y computers at the company.

2.  (2) There is a total of 880 computers at the company.

image

24.       In the figure shown, lines k and m are parallel to each other. Is x = z?

1.  (1) x = w

2.  (2) y = 180 − w

25.       When the wind speed is 9 miles per hour, the wind-chill factor w is given by

w = −17.366 + 1.19t,

where t is the temperature in degrees Fahrenheit. If at noon yesterday the wind speed was 9 miles per hour, was the wind-chill factor greater than 0?

1.  (1) The temperature at noon yesterday was greater than 10 degrees Fahrenheit.

2.  (2) The temperature at noon yesterday was less than 20 degrees Fahrenheit.

image

26.       What is the volume of the cube above?

1.  (1) The surface area of the cube is 600 square inches.

2.  (2) The length of diagonal AB is 10image inches.

27.       Of the 230 single-family homes built in City X last year, how many were occupied at the end of the year?

1.  (1) Of all single-family homes in City X, 90 percent were occupied at the end of last year.

2.  (2) A total of 7,200 single-family homes in City X were occupied at the end of last year.

28.       If J, S, and V are points on the number line, what is the distance between S and V?

1.  (1) The distance between J and S is 20.

2.  (2) The distance between J and V is 25.

29.       If x is a positive integer, what is the value of x?

1.  (1) image

2.  (2) image.

30.       During a certain bicycle ride, was Sherry’s average speed faster than 24 kilometers per hour? (1 kilometer = 1,000 meters)

1.  (1) Sherry’s average speed during the bicycle ride was faster than 7 meters per second.

2.  (2) Sherry’s average speed during the bicycle ride was slower than 8 meters per second.

31.       If x and y are integers, what is the value of x?

1.  (1) xy = 1

2.  (2) x ≠ −1

32.       If p, s, and t are positive, is |ps − pt| > p(s − t)?

1.  (1) p < s

2.  (2) s < t

33.       What were the gross revenues from ticket sales for a certain film during the second week in which it was shown?

1.  (1) Gross revenues during the second week were $1.5 million less than during the first week.

2.  (2) Gross revenues during the third week were $2.0 million less than during the first week.

34.       The total cost of an office dinner was shared equally by k of the n employees who attended the dinner. What was the total cost of the dinner?

1.  (1) Each of the k employees who shared the cost of the dinner paid $19.

2.  (2) If the total cost of the dinner had been shared equally by image of the n employees who attended the dinner, each of the image employees would have paid $18.

35.       For a recent play performance, the ticket prices were $25 per adult and $15 per child. A total of 500 tickets were sold for the performance. How many of the tickets sold were for adults?

1.  (1) Revenue from ticket sales for this performance totaled $10,500.

2.  (2) The average (arithmetic mean) price per ticket sold was $21.

36.       What is the value of x?

1.  (1) image

2.  (2) image

37.       If x and y are positive integers, what is the remainder when image is divided by 3?

1.  (1) image

2.  (2) image

38.       What was the amount of money donated to a certain charity?

1.  (1) Of the amount donated, 40 percent came from corporate donations.

2.  (2) Of the amount donated, $1.5 million came from noncorporate donations.

39.       What is the value of the positive integer n?

1.  (1) image

2.  (2) image

40.       If the set S consists of five consecutive positive integers, what is the sum of these five integers?

1.  (1) The integer 11 is in S, but 10 is not in S.

2.  (2) The sum of the even integers in S is 26.

41.       A total of 20 amounts are entered on a spreadsheet that has 5 rows and 4 columns; each of the 20 positions in the spreadsheet contains one amount. The average (arithmetic mean) of the amounts in row i is Ri (1 ≤ i ≤ 5). The average of the amounts in column j isCj (1 ≤ j ≤ 4). What is the average of all 20 amounts on the spreadsheet?

1.  (1) R1 + R2 + R3 + R4 + R5 = 550

2.  (2) C1 + C2 + C3 + C4 = 440

42.       Was the range of the amounts of money that Company Y budgeted for its projects last year equal to the range of the amounts of money that it budgeted for its projects this year?

1.  (1) Both last year and this year, Company Y budgeted money for 12 projects and the least amount of money that it budgeted for a project was $400.

2.  (2) Both last year and this year, the average (arithmetic mean) amount of money that Company Y budgeted per project was $2,000.

image

43.       If a, b, c, and d are numbers on the number line shown and if the tick marks are equally spaced, what is the value of a + c?

1.  (1) a + b = −8

2.  (2) a + d = 0

image

44.       In the triangle above, does image?

1.  (1) image

2.  (2) image

45.       If image, is image?

1.  (1) image

2.  (2) image

46.       If x, y, and z are three integers, are they consecutive integers?

1.  (1) image

2.  (2) image

47.       A collection of 36 cards consists of 4 sets of 9 cards each. The 9 cards in each set are numbered 1 through 9. If one card has been removed from the collection, what is the number on that card?

1.  (1) The units digit of the sum of the numbers on the remaining 35 cards is 6.

2.  (2) The sum of the numbers on the remaining 35 cards is 176.

48.       In the xy-plane, point (r, s) lies on a circle with center at the origin. What is the value of image?

1.  (1) The circle has radius 2.

2.  (2) The point image lies on the circle.

49.       If r, s, and t are nonzero integers, is r 5s 3t 4 negative?

1.  (1) rt is negative.

2.  (2) s is negative.

50.       If x and y are integers, what is the value of y?

1.  (1) image

2.  (2) image

51.       How many newspapers were sold at a certain newsstand today?

1.  (1) A total of 100 newspapers were sold at the newsstand yesterday, 10 fewer than twice the number sold today.

2.  (2) The number of newspapers sold at the newsstand yesterday was 45 more than the number sold today.

52.       John took a test that had 60 questions numbered from 1 to 60. How many of the questions did he answer correctly?

1.  (1) The number of questions he answered correctly in the first half of the test was 7 more than the number he answered correctly in the second half of the test.

2.  (2) He answered image of the odd-numbered questions correctly and image of the even-numbered questions correctly.

53.       If x is a positive integer, is image an integer?

1.  (1) image is an integer.

2.  (2) image is not an integer.

54.       Is the value of n closer to 50 than to 75?

1.  (1) image

2.  (2) image

55.       Last year, if Elena spent a total of $720 on newspapers, magazines, and books, what amount did she spend on newspapers?

1.  (1) Last year, the amount that Elena spent on magazines was 80 percent of the amount that she spent on books.

2.  (2) Last year, the amount that Elena spent on newspapers was 60 percent of the total amount that she spent on magazines and books.

56.       If p, q, x, y, and z are different positive integers, which of the five integers is the median?

1.  (1) image

2.  (2) image

57.       If image, what is the value of wz?

1.  (1) w and z are positive integers.

2.  (2) w and z are consecutive odd integers.

58.       What is the value of image?

1.  (1) image

2.  (2) image

59.       Machine X runs at a constant rate and produces a lot consisting of 100 cans in 2 hours. How much less time would it take to produce the lot of cans if both Machines X and Y were run simultaneously?

1.  (1) Both Machines X and Y produce the same number of cans per hour.

2.  (2) It takes Machine X twice as long to produce the lot of cans as it takes Machines X and Y running simultaneously to produce the lot.

60.       Can the positive integer p be expressed as the product of two integers, each of which is greater than 1?

1.  (1) image

2.  (2) p is odd.

61.       Is image?

1.  (1) image

2.  (2) image

62.       If image, is image?

1.  (1) image

2.  (2) image

63.       A certain list consists of 400 different numbers. Is the average (arithmetic mean) of the numbers in the list greater than the median of the numbers in the list?

1.  (1) Of the numbers in the list, 280 are less than the average.

2.  (2) Of the numbers in the list, 30 percent are greater than or equal to the average.

64.       What is the area of rectangular region R?

1.  (1) Each diagonal of R has length 5.

2.  (2) The perimeter of R is 14.

65.       If Q is an integer between 10 and 100, what is the value of Q?

1.  (1) One of Q’s digits is 3 more than the other, and the sum of its digits is 9.

2.  (2) image

66.       If p and q are positive integers and image, what is the value of p?

1.  (1) image is an integer.

2.  (2) image is an integer.

67.       What is the value of image?

1.  (1) image

2.  (2) image

68.       How many integers n are there such that image?

1.  (1) image

2.  (2) r and s are not integers.

69.       If the total price of n equally priced shares of a certain stock was $12,000, what was the price per share of the stock?

1.  (1) If the price per share of the stock had been $1 more, the total price of the n shares would have been $300 more.

2.  (2) If the price per share of the stock had been $2 less, the total price of the n shares would have been 5 percent less.

70.       If n is positive, is image?

1.  (1) image

2.  (2) image

71.       Is image?

1.  (1) image and image.

2.  (2) image

72.       In Year X, 8.7 percent of the men in the labor force were unemployed in June compared with 8.4 percent in May. If the number of men in the labor force was the same for both months, how many men were unemployed in June of that year?

1.  (1) In May of Year X, the number of unemployed men in the labor force was 3.36 million.

2.  (2) In Year X, 120,000 more men in the labor force were unemployed in June than in May.

73.       If image, what is the value of image?

1.  (1) image

2.  (2) image

74.       On Monday morning a certain machine ran continuously at a uniform rate to fill a production order. At what time did it completely fill the order that morning?

1.  (1) The machine began filling the order at 9:30 a.m.

2.  (2) The machine had filled image of the order by 10:30 a.m. and image of the order by 11:10 a.m.

75.       If image, is image?

1.  (1) image

2.  (2) image

image

76.       What is the radius of the circle above with center O?

1.  (1) The ratio of OP to PQ is 1 to 2.

2.  (2) P is the midpoint of chord AB.

77.       What is the number of 360-degree rotations that a bicycle wheel made while rolling 100 meters in a straight line without slipping?

1.  (1) The diameter of the bicycle wheel, including the tire, was 0.5 meter.

2.  (2) The wheel made twenty 360-degree rotations per minute.

78.       If image, what is the value of t?

1.  (1) image

2.  (2) image

79.       In the equation imagex is a variable and b is a constant. What is the value of b?

1.  (1) image is a factor of image.

2.  (2) 4 is a root of the equation image.

80.       A Town T has 20,000 residents, 60 percent of whom are female. What percent of the residents were born in Town T?

1.  (1) The number of female residents who were born in Town T is twice the number of male residents who were not born in Town T.

2.  (2) The number of female residents who were not born in Town T is twice the number of female residents who were born in Town T.

81.       In image, what is the length of YZ?

1.  (1) The length of XY is 3.

2.  (2) The length of XZ is 5.

82.       If the average (arithmetic mean) of n consecutive odd integers is 10, what is the least of the integers?

1.  (1) The range of the n integers is 14.

2.  (2) The greatest of the n integers is 17.

83.       What was the ratio of the number of cars to the number of trucks produced by Company X last year?

1.  (1) Last year, if the number of cars produced by Company X had been 8 percent greater, the number of cars produced would have been 150 percent of the number of trucks produced by Company X.

2.  (2) Last year Company X produced 565,000 cars and 406,800 trucks.

84.       If x, y, and z are positive numbers, is image?

1.  (1) image

2.  (2) image

85.       K is a set of numbers such that

1.  (i) if x is in K, then −x is in K, and

2.  (ii) if each of x and y is in K, then xy is in K.

Is 12 in K?

3.  (1) 2 is in K.

4.  (2) 3 is in K.

86.       If image, what is the value of image?

1.  (1) image

2.  (2) image

87.       If x, y, and z are numbers, is image?

1.  (1) The average (arithmetic mean) of x, y, and z is 6.

2.  (2) image

88.       After winning 50 percent of the first 20 games it played, Team A won all of the remaining games it played. What was the total number of games that Team A won?

1.  (1) Team A played 25 games altogether.

2.  (2) Team A won 60 percent of all the games it played.

89.       Is x between 0 and 1?

1.  (1) x2 is less than x.

2.  (2) x3 is positive.

90.       Is p2 an odd integer?

1.  (1) p is an odd integer.

2.  (2) image is an odd integer.

91.       If m and n are nonzero integers, is mn an integer?

1.  (1) nm is positive.

2.  (2) nm is an integer.

92.       What is the value of xy?

1.  (1) image

2.  (2) image

93.       Is x2 greater than x?

1.  (1) x2 is greater than 1.

2.  (2) x is greater than −1.

94.       Michael arranged all his books in a bookcase with 10 books on each shelf and no books left over. After Michael acquired 10 additional books, he arranged all his books in a new bookcase with 12 books on each shelf and no books left over. How many books did Michael have before he acquired the 10 additional books?

1.  (1) Before Michael acquired the 10 additional books, he had fewer than 96 books.

2.  (2) Before Michael acquired the 10 additional books, he had more than 24 books.

95.       If image, does image?

1.  (1) image

2.  (2) image

96.       Last year in a group of 30 businesses, 21 reported a net profit and 15 had investments in foreign markets. How many of the businesses did not report a net profit nor invest in foreign markets last year?

1.  (1) Last year 12 of the 30 businesses reported a net profit and had investments in foreign markets.

2.  (2) Last year 24 of the 30 businesses reported a net profit or invested in foreign markets, or both.

97.       If k and n are integers, is n divisible by 7?

1.  (1) image

2.  (2) image is divisible by 7.

98.       Is the perimeter of square S greater than the perimeter of equilateral triangle T?

1.  (1) The ratio of the length of a side of S to the length of a side of T is 4:5.

2.  (2) The sum of the lengths of a side of S and a side of T is 18.

99.       If image, is image?

1.  (1) image

2.  (2) image

100.   Can the positive integer n be written as the sum of two different positive prime numbers?

1.  (1) n is greater than 3.

2.  (2) n is odd.

image

101.   In the figure above, segments RS and TU represent two positions of the same ladder leaning against the side SV of a wall. The length of TV is how much greater than the length of RV?

1.  (1) The length of TU is 10 meters.

2.  (2) The length of RV is 5 meters.

Cancellation Fees

Days Prior to Departure

Percent of Package Price

46 or more

10%

45−31

35%

30−16

50%

15−5

65%

4 or fewer

100%

102.   The table above shows the cancellation fee schedule that a travel agency uses to determine the fee charged to a tourist who cancels a trip prior to departure. If a tourist canceled a trip with a package price of $1,700 and a departure date of September 4, on what day was the trip canceled?

1.  (1) The cancellation fee was $595.

2.  (2) If the trip had been canceled one day later, the cancellation fee would have been $255 more.

103.   If P and Q are each circular regions, what is the radius of the larger of these regions?

1.  (1) The area of P plus the area of Q is equal to image.

2.  (2) The larger circular region has a radius that is 3 times the radius of the smaller circular region.

104.   For all zimage denotes the least integer greater than or equal to z. Is image?

1.  (1) image

2.  (2) image

image

105.   The circular base of an above-ground swimming pool lies in a level yard and just touches two straight sides of a fence at points A and B, as shown in the figure above. Point C is on the ground where the two sides of the fence meet. How far from the center of the pool’s base is point A?

1.  (1) The base has area 250 square feet.

2.  (2) The center of the base is 20 feet from point C.

106.   If image, what is the value of image?

1.  (1) image

2.  (2) image

107.   If the average (arithmetic mean) of 4 numbers is 50, how many of the numbers are greater than 50?

1.  (1) None of the four numbers is equal to 50.

2.  (2) Two of the numbers are equal to 25.

108.   [y] denotes the greatest integer less than or equal to y. Is image?

1.  (1) image

2.  (2) image

109.   If m is a positive integer, then m3 has how many digits?

1.  (1) m has 3 digits.

2.  (2) m2 has 5 digits.

110.   For each landscaping job that takes more than 4 hours, a certain contractor charges a total of r dollars for the first 4 hours plus 0.2r dollars for each additional hour or fraction of an hour, where r > 100. Did a particular landscaping job take more than 10 hours?

1.  (1) The contractor charged a total of $288 for the job.

2.  (2) The contractor charged a total of 2.4r dollars for the job.

111.   The sequence s1s2s3, . . . , sn, . . . is such that sn image for all integers image. If k is a positive integer, is the sum of the first k terms of the sequence greater than image?

1.  (1) image

2.  (2) image

112.   If x and y are nonzero integers, is image?

1.  (1) image

2.  (2) image

113.   In the sequence S of numbers, each term after the first two terms is the sum of the two immediately preceding terms. What is the 5th term of S?

1.  (1) The 6th term of S minus the 4th term equals 5.

2.  (2) The 6th term of S plus the 7th term equals 21.

114.   If d is a positive integer, is image an integer?

1.  (1) d is the square of an integer.

2.  (2) image is the square of an integer.

115.   Is the positive integer n a multiple of 24?

1.  (1) n is a multiple of 4.

2.  (2) n is a multiple of 6.

116.   If 75 percent of the guests at a certain banquet ordered dessert, what percent of the guests ordered coffee?

1.  (1) 60 percent of the guests who ordered dessert also ordered coffee.

2.  (2) 90 percent of the guests who ordered coffee also ordered dessert.

117.   A tank containing water started to leak. Did the tank contain more than 30 gallons of water when it started to leak? (Note: 1 gallon = 128 ounces)

1.  (1) The water leaked from the tank at a constant rate of 6.4 ounces per minute.

2.  (2) The tank became empty less than 12 hours after it started to leak.

118.   If x is an integer, is y an integer?

1.  (1) The average (arithmetic mean) of x, y, and image is x.

2.  (2) The average (arithmetic mean) of x and y is not an integer.

119.   If 2 different representatives are to be selected at random from a group of 10 employees and if p is the probability that both representatives selected will be women, is image?

1.  (1) More than image of the 10 employees are women.

2.  (2) The probability that both representatives selected will be men is less than image.

120.   In the xy-plane, lines k and  intersect at the point (1,1). Is the y-intercept of k greater than the y-intercept of ?

1.  (1) The slope of k is less than the slope of .

2.  (2) The slope of  is positive.

121.   Each of the 45 books on a shelf is written either in English or in Spanish, and each of the books is either a hardcover book or a paperback. If a book is to be selected at random from the books on the shelf, is the probability less than image that the book selected will be a paperback written in Spanish?

1.  (1) Of the books on the shelf, 30 are paperbacks.

2.  (2) Of the books on the shelf, 15 are written in Spanish.

122.   If S is a set of four numbers w, x, y, and z, is the range of the numbers in S greater than 2?

1.  (1) image

2.  (2) z is the least number in S.

123.   Stations X and Y are connected by two separate, straight, parallel rail lines that are 250 miles long. Train P and train Q simultaneously left Station X and Station Y, respectively, and each train traveled to the other’s point of departure. The two trains passed each other after traveling for 2 hours. When the two trains passed, which train was nearer to its destination?

1.  (1) At the time when the two trains passed, train P had averaged a speed of 70 miles per hour.

2.  (2) Train Q averaged a speed of 55 miles per hour for the entire trip.

image

124.   In the xy-plane shown, the shaded region consists of all points that lie above the graph of y = x2 − 4x and below the x-axis. Does the point (a, b) (not shown) lie in the shaded region if b < 0?

1.  (1) 0 < a < 4

2.  (2) a2 − 4a < b