Mathreviewer_1_.pdf

ALGEBRA 1.

Any number that can be expressed as a quotient of two integers (division of zero excluded) is called a. irrational number c. imaginary number b. rational number d. odd number

2.

In the expression a. power b. exponent

3.

Which of the following non – terminating decimals is rational? a. 3.14159265… c. 2.71828180… b. 2.470470… d. 1.141421356…

4.

The sum of the integers between 288 and 887 that are exactly divisible by 15 is: a. 23,700 c. 22,815 b. 21,800 d. 24,150

5.

Find the zeroes of the given polynomial (x2 – 4x + 3)(x2 + 3x – 4). a. 1, 2, 4 c. 1, 2, -4 b. 1, 3, 4 d. 1, 3, -4

6.

Ten liters of 25 % salt solution and 15 liters of 35% salt solution are poured into a drum originally containing 30 liters of 10 % salt solution. What is the percent concentration of salt in the mixture? a. 22.15% c. 25. 75% b. 27. 05% d. 19.55%

7.

A stack of bricks has 61 in the bottom layer, 58 bricks in the second layer, 55 bricks in the third later, and so on until there are 10 bricks in the last layer. How many bricks are there all together? a. 637 c. 640 b. 639 d. 638

8.

If f(x) = 2x2 + 2x + 4. What is f(2)? a. 16 c. 8 b. x2 + x + 2 d. 4x + 2

n

a , the letter n represents c. order d. radicand

9.

Once a month a man put some money into the cookie jar. Each month he puts 50 centavos more into the jar than the month before. After 12 years, he counted his money: he had P 5, 436. How much money did he put in the jar in the last month? a. P75.50 c. P72.50 b. P74.50 d. P73.50

10.

A boatman rows to a place 48 miles distant and back in 14 hours, but find that he rows 4 miles with the steam in the same time as 3 miles against the steam. Find the rate of the steam. a. 1 mile/hour c. 0.5 mile/hour b. 0.8 mile/hour d. 1.5 mile/hour

11.

A girl on a bicycle coasts down hill covering 4 ft in the 1st second, 12 ft in the 2nd second, and in general, 8 ft more each second than the previous second. If she reaches the bottom at the end of 14 seconds, how far did she coast? a. 782 ft c. 786 ft b. 780 ft d. 768 ft

12.

A jogger starts a course at a steady rate of 8 KPH. Five minutes later, a second jogger starts the same course at 10 KPH. How long will it take the second jogger to catch the first? a. 20 min c. 22 min b. 21 min d. 18 min

13.

The sum of Kim’s and Kevin’s ages in 18. In 3 years, Kim will be twice as old as Kevin. What are their ages? a. 5, 13 c. 6, 12 b. 7, 11 d. 4, 14

14.

Find the 10th term of 3, 6, 12, 24… a. 1563 c. 1653 b. 1356 d. 1536

15.

A bookstore purchased a best selling book at P200.00 per copy. At what price should this book be sold so that, giving a 20% discount, the profit is 30%. a. P 450 c. P 350 b. P 500 d. P 400

16.

A bookstore contracted to purchase a bestselling book at P250/copy. At what price should the bookstore retail this book so that, despite a 15% discount, the profit on each copy will be 30%? a. 375.66 c. 500 b. 413.22 d. 454.55

17.

Determine the SUM of the positive-valued solutions to the simultaneous equations: xy = 15, yz = 35, zx = 21: a. 13 c. 19 b. 17 d. 15

18.

If the polynomial x3 + 4x2 – 3x + 8 is divided by x – 5, determine the remainder. a. 45 c. 210 b. 42 d. 218

19.

The areas of two square differ by 7sq. ft and their perimeters differ by 4 ft. determine the SUM of their areas. a. 27.00 sq. ft c. 22.00 sq. ft b. 29.00 sq. ft d. 25.00 sq. ft

20.

Find the square root of 96 using binomial theorem. a. 9.79796 c. 9.81817 b. 9. 58584 d. 9. 67673

21.

In a certain community of 1200 people, 60 % are literate. Of the males, 50% are literate, and of the females, 70% are literate. What is the female population? a. 500 c. 600 b. 550 d. 850

22.

Gravity cause a body to fall 16.1 feet in the 1st second, 48.3 ft in the 2nd second, 80.5 ft in the 3rd second, and so on. How far did the body fall during the 10 th second? a. 273.7 ft. c. 241.5 ft. b. 338.1 ft. d. 305.9 ft.

23.

A and B can do a piece of work in 42 days, B and C in 31 days and C and A in 20 days. In how many days can all of them do the work together? a. 17 c. 21 b. 15 d. 19

24.

Find the 37th term of the arithmetic sequence 8, 11, 14. a. 114 c. 110 b. 112 d. 116

25.

Solve the inequality: x2 is less than 9. a. -3/2 is less than x is less than 3/2 b. -2 is less than x is less than 2 c. -4 is less than x is less than 4 d. -3 is less than x is less than 3

26.

A rubber ball is dropped from a height of 15 meters. On each rebound, it rises 2/3 of the heightt from which it last fell. Find the distance traversed by the ball before it comes to rest. The distance traversed by the ball before it comes to rest. The geometric progression occurs after the first rebound. a. 75 c. 80 b. 60 d. 85

27.

A student makes 100% of his first test and 80% on the second. On the third test he made 60% of the grade he made on the second, while on the fourth he made 80% of the grade he made on the third. What constant average rate of decrease would give the first and the last grades? a. 20.5 percent c. 20.1 percent b. 20.7 percent d. 20.9 percent

28.

A student has test scores of 75, 83 and 78. The final test counts half the total f=grade. What must be the minimum (integer) score on the final so that the average us 80? a. 81 c. 84 b. 82 d. 83

29.

Find the 12th term if the harmonic progression 1, 1/3, 1/5,... a. 1/9 c. 1/17 b. 1/23 d. 1/21

30.

Factor the following expression: x2 + 2xy – z2 – 2zy. a. (x – z)(x – 2y + z) c. (x – y)(x – 2y + z) b. (x – y)(x + 2y - z) d. (x – z)(x + 2y + z)

31.

Find the sum of the geometric series 3 + 3/2 + 3/4 + ... a. 8 c. 6 b. 4 d. 2

32.

Solve the inequality, expressing the solution in terms of interval: -4 is equal or less than (2x – 1)/3 is equal or less than 4. a. {x: -7/2 is less or equal to x is less or equal to 7/2} b. {x: -2/3 is less or equal to x is less or equal to 9/2} c. {x: -9/2 is less or equal to x is less or equal to 9/2} d. {x: -11/2 is less or equal to x is less or equal to 11/2}

33.

Find the dimensions of a rectangle whose perimeter is 40 inches and whose area is 96 square inches. a. 11, 9 c. 10, 9.6 b. 12, 8 d. 10, 10

34.

Find the harmonic mean of the numbers a and b by denoting h as the harmonic mean. a. h = ab/(a + b) c. h = 2ab/(a + b) b. h = ab/2(a + b) d. h = 3ab/(a + b)

35.

A particle is projected vertically upward from a point 112 ft above the ground with an initial velocity of 96 ft/sec., how fast is it moving when it is 240 ft above the ground? a. 36 ft per sec c. 34 ft per sec b. 32 ft per sec d. 30 ft per sec

36.

A box with an open top is to be made by taking rectangular piece of tin 8 x 10 inches and cutting a square of the same size out of each corner and folding up the sizes. If the area of the base is to be 24 square inches, what should the length of the sides of the square be? a. 2.0 inches c. 2.1 inches b. 2.2 inches d. 1.8 inches

37.

How many numbers between 10 and 200 are exactly divisible by 7? Find their sum. a. 27 numbers; S = 2835 c. 26 numbers; S = 2835 b. 26 numbers; S = 2830 d. 28 numbers; S = 2840

38.

A man buys a book for P200 and wishes to sell it. What price should he mark on it if he wishes a 40 percent discount while making 50 percent profit on the cost price? a. 667 c. 467 b. 567 d. 867

39.

When a bullet is fired into a sand bag, it will be assumed that its retardation is equal to the square root of its velocity on entering. For how long will it travel if the velocity on entering the bag is 144 ft/sec? a. 27 sec c. 25 sec b. 24 sec d. 26 sec

40.

If a dc generator has an emf of E volts and as an internal resistance of r ohms, what external resistance R will consume the most power? a. R = r c. R= 0.5 b. R = 0.5r d. R = 2r

41.

If (5x 3), (x + 2) and (3x – 11) form an arithmetic progression, find the fifteenth term. a. –86 c. -79 b. -81 d. -84

42.

A man on a wharf (pier) is pulling a rope tied to a raft at time rate of 0.60 m/sec if the hands of the man pulling the rope are 3.66m above the level of the water, how fast is the raft approaching the wharf there are 6.10 m of rope out? a. 0.75 m/sec c. 0.45 m/sec b. 0.55 m/sec d. -0.65 m/sec

43.

How many kg. of cream containing 25 percent butter fat should be added to 50 kg of milk containing one percent butter fat to produce milk containing 2 percent butter fat? a. 2.174 c. 4.170 b. 5.221 d. 3.318

44.

At what time will the hands of a clock be in a straight line between 7:00 and 8:00 in the morning? (Note: The hour hand is opposite that of the minute hand) a. 7:12.4545 A.M. c. 7:15.4545 A.M. b. 7:5.4545 A.M. d. 7:10.4545 A.M.

45.

A cask containing 20 gallons of wine was emptied on one-fifth of its content and then is filled with water. If this is done 6 times, how many gallons of wine remain in the cask? a. 5.121 c. 5.242 b. 5.010 d. 5.343

46.

Given a triangle of sides 10 cm and 15 cm with an included angle of 60 degrees. Find the area of the triangle in sq. cm. a. 65 c. 80 b. 72 d. 75

47.

Find the rational number equivalent to the repeating decimal 2.35242424… a. 23273/9900 c. 23289/9900 b. 23261/9900 d. 23264/9900

48.

Two vertical conical tanks are joined at the vertices by a pipe. Initially, the bigger tank is full of water. The pipe valve is opened to allow the water to flow to the smaller tank until it is full. At this instant, how deep is the water in the bigger tank? The bigger tank has a diameter of 6 ft and height of 10 ft, the smaller tank has a diameter of 6 ft and height of 8 ft. Neglect the volume of water in the pipeline. a. 25 exponent (1/5) c. 25 exponent (1/3) b. 200 exponent (1/3) d. 50 exponent (1/2)

49.

Find the most economical proportion for a box with an open top and a square base. a. b = h c. b = 3h b. b = 4h d. b = 2h

50.

The electric power which a transmission line can transmit is proportional to the product of its design voltage and current capacity, and inversely to the transmission distance. A 115 kilovolt line rated at 1,000 amperes can transmit 150 megawatts over 150 km. How much power, in megawatts, can a 230 kilovolt line rated at 1,500 amperes transmit over 100 km? a. 785 c. 675 b. 485 d. 595

51.

Determine the Greatest Common Divisor of the following numbers: 34, 58. a. 7 c. 2 b. 4 d. 17

52.

Find the values of x in the equation 24x2 + 5x – 1 = 0 a. (1/6, 1) c. (1/2, 1/5) b. (1/6, 1/5) d. (1/8, -1/3)

53.

54.

An arithmetic progression starts with 1, and 9 terms, and the middle term is 21. Determine the sum of the first 9 terms. a. 235 c. 112 b. 148 d. 189

 lnx3   If ln x = 3 and ln y = 4, determine  4   lny  a. 1.8750 c. 0.5625 b. 0.300 d. 1.000

55.

The hands of a tower clock are 4 ½ ft and 6 ft long respectively. How fast are the ends approaching at 4 o’clock in ft per minute? a. -0.246 c. -0.264 b. -0.203 d. -0.256

56.

Two men running at constant speeds along a circular track 1350 meters in circumference. Running in opposite directions, they meet each other every 3 minutes. Running in the same direction, they come abreast every 27 minutes. Determine the speed of the faster man, in kilometers per hour. a. 12 c. 15 b. 18 d. 21

57.

A geometric progression is 1 + z + z2 + …… + zn where z < 1. Determine the sum of the series as n approaches infinity. a. 1/(1 – 2z) c. 1/(1 – z) b. 1/(2 – z) d. 2/(1 – z)

58.

The vibration frequency of a string varies as the square root of the tension and the inversely as the product of the length and diameter of the string. If a string 3 feet long and 0.03 inch in diameter vibrates at 720 times per second under 90 pounds tension, at what frequency will a 2 feet long, 0.025 inch string vibrate under 2500 pounds tension? a. 5,645 c. 6,831 b. 7,514 d. 6,210

59.

Find the values of x and y from the equations: x – 4y + 2 = 0 2x + y – 4 = 0 a. 11/7, -6/7 c. 4/9, 8/9 b. 14/9, 8/9 d. 3/2, 5/3

60.

Ten liters of 25% salt solution and 15 liters of 35% salt solution are poured into a drum originally containing 30 liters of 10% salt solution. What is the percent concentration of salt in the mixture? a. 19.55% c. 27.05% b. 22.15% d. 25.72%

61.

If f(x) = 2x2 + 2x + 4. What is f(2)? a. 4x + 2 c. x2 + x +2 b. 16 d. 8

62.

The piston of an engine is connected by a 12-inch connecting rod to a point on a crank that rotates on a 4-inch radius about the crankshaft. If the crankshaft has an angular speed of 3,000 revolutions per minute, determine the rectilinear speed of the piston, in feet per minute, when the crank is 90 degrees to the motion of the piston. a. 1,319 c. 1,194 b. 1,257 d. 1,131

63.

An arithmetic progression starts with 1, has 9 terms, and the middle term is 21. Determine the sum of the first 9 terms. a. 235 c. 112 b. 148 d. 189

64.

A small line truck hauls poles from substation stockyard to pole sites along a proposed distribution line. The truck can handle only one pole at a time. The first pole site is 150 meters from the substation and the poles are to be 509 meters apart. Determine the total distance traveled by the truck, back and forth, after returning from delivering the 30th pole. a. 35 km c. 37.5 km b. 30 km d. 40 km.

65.

Horses sell for $25 and cows $26 a head. A rancher has $1,000 to spend and must spend it all with nothing left. If he buys the minimum number of horses, how many animals does he buy? a. 40 c. 26 b. 39 d. 28

66.

A man traveling 40 km finds that by traveling one more km per hour, he would made the journey in 2 hrs. less time. How many km per hour did he actually travel? a. 4 c. 18 b. 8 d. 6

67.

Two prime numbers which differ by 2 are called prime twins. Which of the following pairs of numbers are prime twins? a. (1,3) c. (7,9) b. (3,5) d. (9,11)

68.

If f(x)  a. 6 b. 5

x2 and g(y)  y  2 , then f[g(2)] equal: x2 b. 4 d. 3

69.

If x3 + 3x2 + (K + 5)x + 2 – K is divided by x + 1 and the remainder is 3, then the value of K is: a. -2 c. -4 b. -3 d. -5

70.

The value of k which will make 4x 2 – 4kx + 5k a perfect square trinomial is: a. 6 c. 4 b. 5 d. 3

71.

If the roots of ax2 + bx + c = 0, are a real and equal, then: a. b2 – 4ac > 0 c. b2 – 4ac = 0 b. b2 – 4ac < 0 d. none of the above

72.

The other form of loga N = b is: a. N = ab b. N = ba

73.

a b d. N = ab c. N 

Six times the middle of a three digit number is the sum of the other two. If the number is divided by the sum of the digits, the answer is 51 and the remainder is 11. If the digits are reversed, the number becomes smaller by 198. Find the number. Answer: 725

74.

Pedro is as old as Juan was when Juan is twice as old as Pedro was. When Pedro will be as old as Juan is now, the difference between their ages is 6 years. Find the age of each now. Answer: Juan is 24 years old, Pedro is 18 years old

75.

The sum of the areas of two unequal square lots is 5,200 square meters. If the lots were adjacent to each other, they would require 320 meters of fence to enclose the combined area formed by them. Find the dimensions of each lot. Answers: 60m and 40m 68m and 24m

76.

The area of a square field exceeds another square by 56 square meters. The perimeter of the larger field exceeds one half of the smaller by 26 meters. What are the sides of each field? Answers: 25 Larger field, 9m or m 3 11 Smaller field, 5m or m 3

77.

In an electric circuit, the voltage is 15 volts. If the current is increased by 2 amperes and the resistance is decreased by 1 ohm, the voltage is reduced by 1 volt. Find the original current and resistance. Answers: 5A, 3 ohms

78.

In an electric circuit A, the impressed voltage is 12 volts and the resistance is 3 ohms. In circuit B, the voltage is 20 volts and the resistance is 7 ohms. Additional batteries with a total voltage of 28 volts are to be added to these 2 circuits so that after the addition, the circuits in the two circuits are equal. How much voltage should be added to each circuit? Answers: 6V to A, 22V to B

79.

A number of two digits divided by the sum of the digits the quotient is 7 and the remainder is 6. If the digits of the number are interchanged, the resulting number exceeds three times the sum of the digits by 5. What is the number? Answer: 83

80.

Which of the following has no middle term? a. (x + y)3 c. (a - b)4 6 b. (u + v) d. (x - y)8

81.

Find the term containing x26 in the expansion of ( x-2 + x3): Answer: 66x26

82.

Maria was 36 years old; Maria was twice as old as Anna was when Maria was as old as Anna now. How old is Anna now? Answer: 24 years old

83.

Separate 132 into 2 parts such that the larger divided by the smaller the quotient is 6 and the remainder is 13. What are the parts? Answers: 17 and 115

84.

Find the number such that their sum multiplied by the sum of their squares is 65, and their difference multiplies by the difference of their squares is 5. Answer: 2 and 3

85.

Find three consecutive odd integers such that twice the sum of the first and the second integers plus four times the third is equal to 60. Answers: 5, 7, 9

86.

Three numbers are in ratio 2:5:8. If their sum is 60, find the numbers. Answers: 8, 20, 32

87.

The square of a number increased by 16 is the same as 10 times the number. Find the number. Answer: 8, 2

88.

The sum of the digits of a 3 – digit number is 12. The middle digit is equal to the sum of the other two digits and the number shall be increased by 198 if its digits are reversed. Find the number. Answer: 264

89.

How much water must be evaporated from 80 liters of 12% solution of salt in order to obtain a 20% solution of salt? Answer: 32 L

90.

A tank full of alcohol is emptied one third of its content and then filled up with water and mixed. If this is done six times, what fraction of the volume (original) of alcohol remains? 64 Answer: 729

91.

How many liters of water must be added to 45 liters of solution which is 90% alcohol in order to make the resulting solution 80 % alcohol? Answer: 5.63 L

92.

A 40 – gram solution of acid and water is 20% acid by weight. How much pure acid must be added to this solution to make it 30% acid? Answer: 5.71 grams

93.

Two numbers differ by 40 and their arithmetic mean exceeds their positive geometric mean by two. The numbers are: Answer: 81, 121

94.

A motorcycle messenger left the rear of a motorized troop 8 km long and rode to the front of the troop, returning at once to the rear. How far did he ride, if the troop traveled 15 km during this time and each traveled at a uniform rate? Answer: 25 kms

95.

September 1976. At the recent Olympic Games in Montreal, Canada, a team which participated in 1600 meters relay event had the following individual speed: First runner, 24 kph, second runner, 20 kph, third runner, 22 kph and fourth runner, 23 kph. What was the team’s speed? Answer: 22.149 kph

96.

A car running at 25 km per hour can cover a certain distance in 8 hours. By how many km per hour must its rate be increased in order to cover the same distance in three hours less? Answer: 15 km/hr

97.

A, B and C can do a piece of work in 10 days. A and B can do it in 12 days, A and C in 20 days. How many days would it take each to do the work alone? Answer: 30, 20, 60

98.

A one kilometer long caravan of men is walking at a constant rate. A man from the rear end walks towards the head and back to the rear at the instant when the caravan has covered a distance of one kilometer. Find the total distance traveled by the man. Answer: 2.414 kms

99.

A man receives a salary of Php36,000 per annum for the first year and a 10% rise every year for 10 years. What is his salary during the fifth year? Answer: Php52,707.60

100. A boat’s crew rowing at half their usual rate can negotiate 2 km down a river and back in one hour and 40 minutes. At their usual rate in still water, they would have gone over the same course in 40 minutes. Find their rate of rowing in still water. Answer: 6.4 km per hour 101. Two pipes running simultaneously can fill a swimming pool in 6 hours. If both pipes run for 3 hours and the first pipe is then shut off, it requires 4 hours more for the second pipe to fill the pool. How long does it take each pipe running separately to fill the pool? Answer: 8 and 24 102. Two brothers washed the family car in 24 minutes. Previously, when each had washed the car alone, the younger boy took 20 minutes longer to do the job than the older boy. How long did it take the older boy to wash the car alone? Answer: 40 minutes 103. A swimming pool holds 54 cubic meters of water. It can be drained at a rate of one cubic meter per minute faster than it can be filled. If it takes 9 mins. longer to fill it than to drain it, find the drainage rate. Answer: 3 m3/min. 104. How long will it be from the time the hour hand and the minute hand of a clock are together until they will be together again? Answer: 1 hr. and 5.45 minutes

105. At what time between 4 and 5 o’ clock do the hands of the clock coincide? Answer: 4:21.82 o’ clock 106. It is exactly 3 o’ clock. In how many seconds will the angle formed by the hour hand and the minute hand be twice the angle formed by the hour and the second hand? Answer: 22.4 sec. 107. It is now between 9 and 10 o’ clock. In 4 minutes, the hour hand will be exactly opposite the position occupied by the minute hand 3 minutes ago. What is the time now? Answer: 9:20 108. How many times in one complete day will the hour and the minute hands coincide with each other? Answer: 25 109. A man piles 150 logs in layers so that the top layer contains 3 logs and each lower layer has one more log than the layer above. How many logs are at the bottom? Answer: 17 logs 110. If log 6 + xlog 4 = log 4 + log (32 + 4X). Find x. Answer: 3 111. Find the number of terms of a geometric progression in which the first term is 48, the last term is 384 and the sum of the terms is 720. Answer: 4 terms 112. Evaluate log10 5. Answer: 2.321 113. If loga 10 = 0.25, find log10 a. Answer: 4 114. A man borrowed P100,000 at the interest rate of 12% per annum compounded quarterly. What is the effective rate? a. 3% c. 12% b. 13.2% d. 12.55%

115. A man purchased on monthly installment a P100,000 worth of land. The interest rate is 12% nominal and payable in twenty years. What is the monthly amortization? a. P 1,101.08 c. P 1,152.15 b. P 1,121.01 d. P 1,128.12 116. Once a month a man put some money into the cookie Each month he puts 50 centavos more into the jar than month before. After 21 years he counted his money; he P5436. How much money did he put in the jar in the month? a. P73.50 c. P74.50 b. P75.50 d. P72.50

jar. the has last

117. If equal spheres are piled in the form of a complete pyramid with an equilateral triangle as base, find the total number of spheres in the pile if each side of the base contains 4 spheres. a. 15 c. 21 b. 20 d. 18 118. A train, an hour after starting, meets with an accident which detains it an hour, after which it proceeds at 3/5 of its former rate and arrives three hours after time; but had the accident happened 50 miles farther on the line, it would have arrived one and one-half hour sooner. Find the length of the journey. a. 910/9 miles c. 920/9 miles b. 800/9 miles d. 850/9 miles 119. If n is any positive integer, then (n–1)(n–2)(n-3)…(3)(2)(1)…= a. e [exp(n-1)] c. n! b. (n - 1)! d. (n - 1) exp n 120. A runner and his trainer are standing together on a circular track of radius 100 meters. When the trainer gives a signal, the runner starts to run around the track at a speed of 10 m/s. How fast is the distance between the runner and the trainer increasing when the runner has run ¼ of the way around the track? a. 4 2 c. 6 2 b. 5 2 d. 3 2

121. A stack of bricks has 61 in the bottom layer, 58 bricks in the second layer, 55 bricks in the third layer, and so on until there are 10 bricks in the last layer. How many bricks are there all together? a. 638 c. 639 b. 637 d. 640 122. Multiply the following: (2x + 5y)(5x – 2y) a. 10 (x square) – 21 xy + 10 (y square) b. -10 (x square) + 21 xy + 10 (y square) c. 10 (x square) + 21 xy - 10 (y square) d. -10 (x square) – 21 xy -10 (y square) 123. The product of two positive numbers is 16. Find the number if the sum of one and the square of the other is least. a. 8, 2 c. 8, 5 b. 8, 4 d. 8, 0 124. The seventh term is 56 and the twelfth term is -1792 of a geometric progression. Find the common ratio and the first term. Assume the ratios are equal. a. -2, 5/8 c. -1, 7/8 b. -1, 5/8 d. -2, 7/8 125. There are two numbers whose sum is 50. Three times the first is 5 more than twice the second. What are the numbers? a. 23, 27 c. 21, 29 b. 20, 30 d. 23, 28 126. A chemist needs to dilute a 50% boric acid solution to a 10% solution. If it needs 25 liters of the 10% solution, how much of the 50% solution should it use? a. 7 c. 4 b. 6 d. 5 127. Mr. Tom purchases a selection of wrenches for his shop. His bill is $78. He buys the same number of $1.50 and $2.50 wrenches, and half that many of $4 wrenches. The number of $3 wrenches is one more than the $4 wrenches. How m any $2.50 wrenches did he purchase? a. 5 c. 8 b. 6 d. 10

128. Lita has ten bills in her wallet. She has a total of Php 40. If she has one more Php 5 bills than Php 10 bills, and two more Php 1 bill than Php 5 bills, how many Php 10 bill does she have? a. 5 c. 2 b. 8 d. 4 129. There is a number such that three times the number minus 6 is equal to 45. Find the number. a. 16 c. 20 b. 17 d. 19 130. A motorboat starting from 90 miles per Find the total a. 22.91 b. 25.43

with acceleration and deceleration of 4 ft/sec2 rest and reaches its maximum cruising speed at hour and maintained its speed for 15 minutes. distance traveled until it stops. c. 23.33 d. 27.56

131. A support wire is anchored 12 m up from the base of a flagpole and the horizontal distance of the base of a flagpole from the other end of a wire is 16 ft., find the length of the supporting wire: a. 34 ft c. 20 ft b. 36 ft d. 22 ft 132. What are the roots of the quadratic equation if b2 – 4ac < 0? a. real and equal c. complex and equal b. real and unequal d. complex and imaginary 133. What is the value of x so that a. 0 < x < 1 b. -1 < x < 0

x will always be negative? (x  1)3 c. 1 < x < 2 d. -2 < x < 1

134. Solve the inequality 2x  3  1 . a. -1 < x < 2 b. 2 < x < 3

c. -2 < x < 1 d. 1 < x < 2

135. Solve for the particular solution if y(1) = 4. a. y = x4 c. y = x4 + 3x3 b. y = 3x2 + 4x + 2 d. y = 3x2 + x + 8

136. Mario bought two Php 1 stamps. How many 19-cent stamps did he purchase? a. 16 c. 12 b. 14 d. 11 137. In the afternoon, Pedro and Juan rode their bicycles 4 km more than three times the distance in kilometer they rode in the morning on a trip to the lake. If the entire trip was 112 km, how far did they ride in the morning? a. 27 c. 36 b. 28 d. 34 138. Given y = (x + 1)2 and y = (1 - x)2. Solve the equations simultaneously. a. -1, 0 c. 0, 1 b. no solution d. 1, 0 139. Two cars are headed for Las Vegas. One is 50 km ahead of the other on the same road. The one in front is traveling 60 kph while the second car is traveling 70 kph. What is the distance at which the second car will overtake first? a. 350 c. 340 b. 300 d. 400 140. Two people get in an elevator at the first floor. At the second floor, one person gets in. At the third floor, two people get off. At the fourth floor, the last person gets off. If each person weighs 150 lbs, and each floor is 12 ft high, find the total work done in ft-lb done by the elevator? a. 10,800 c. 5,400 b. 11,200 d. 12,600 141. A tank in the form of a frustum of a right circular cone is filled with oil weighing 50 pounds per cubic foot. If the height of the tank is 10 feet, the base radius, 6 feet, and the top radius, 4 feet, find the work required to pump the oil height 10 feet above the tank. a. 232 c. 195 b. 83 d. 312 142. Find the nth term of 6, 2, -2… a. -4n + 6 b. -2n + 6

c. 2n + 8 d. -4n + 10

143. A man inherited Php 2,000,000 which he invited in stocks and bonds. The stocks returned 6 percent and the bonds 8 percent. If the return on the bonds was Php 8,000 less than the return on the stocks, how much did he invest in the stock? a. Php 1,250,000 c. Php 1,400,000 b. Php 1,200,000 d. Php 1,500,000 144. Bill, Bob, and Barry are hired to paint signs. In 8 hours Bill can paint 2 signs, and Barry can paint 1 1/3 signs. They all come to work the first day, but Barry doesn’t like the job and quits after 3 hours. Bob works half an hour longer than Barry and quits. How long will it take Bill to finish the two signs they were supposed to paint? a. 2 hrs. c. 2 1/3 hrs. b. 1 1/2 hrs. d. 3 hrs. 145. Binoy, Boboy, and Bata are hired to paint signs. In 8 hours Binoy can paint 1 signs, Boboy can paint 2 signs, and Bata can paint 1 1/3 signs. They all come to work the first day, but Bata doesn’t like the job and quits after 3 hours. Boboy works half an hour longer than Bata and quits. How long will it take Binoy to finish the two signs they were supposed to paint? a. 1-3/4 hours c. 2-1/4 hours b. 1-1/2 hours d. 2-1/2 hours 146. The sum of three numbers in arithmetic progression is 33. If the numbers are increased by 2, 1 and 6, respectively, the new numbers will be in geometric progression. Find the product of the three numbers in arithmetic progression. a. 397 c. 792 b. 957 d. 872 147. The perimeter of an isosceles right triangle is 10.2426. Find the area of a triangle. a. 2 c. 4.5 b. 3 d. 4 148. Mr. Manuel makes a business trip from his house to Laguna in 2 hours. One hour later, he returns home in traffic at a rate of 20 kph less than his rate going. If Mr. Manuel is gone a total of 6 hours, what was his rate going to Laguna? a. 50 kph c. 40 kph b. 60 kph d. 30 kph

149. A stone is dropped into a pond causing water waves that form concentric circles, if after a few seconds the radius of the waves is r = 40t, where t is in seconds, r in cm, find the rate of change of area of the disturbed region increase with respect to t at t = 1. a. 2400 c. 6400 b. 3200 d. 1200 150. A clerk at the Dior Department Store receives $15 in change for her cash drawer at the start of each day. She receives twice as many dimes as fifty-cent pieces, and the same number of the quarters as dimes. She has twice as many nickels as dimes and a dollar’s worth of pennies. How many are dimes? a. 30 c. 40 b. 20 d. 10 151. Shelly and Karie go out to play. Shelly, who weighs 90 pounds, sits on one end of a 14-foot teater-tooter. Its balance point is at the center of the board. Karie, who weighs 120 pounds, climbs on the other end and slides towards the center until they balance. What is Karie’s distance from her end of the teater-tooter when they balance? a. 2-1/2 ft c. 1-3/4 ft b. 1-1/2 ft d. 5-1/4 ft 152. What are the values of n if (2n – 6) is greater than 1 but less than 14? a. 4, 5, 6, 7, 8, 9, 10 c. 3, 4, 5, 6, 7, 8 b. 4, 5, 6, 7, 8, 9 d. 2, 3, 4, 5, 6, 7 153. A collection of 36 coins consists of nickels, dimes and quarters. There are three fewer quarters than nickels and six more dimes than quarters. How many are quarters? a. 12 c. 9 b. 15 d. 6 154. A plane takes 1 ½ hours to fly from Los Angeles to San Francisco and 2 hours from San Francisco to Los Angeles. If the wind blows north on both trips at 24 mph, what is the speed of the plane in still air? a. 170 c. 120 b. 110 d. 150

155. A ball is thrown vertically upward with a velocity of 48 ft/sec at the edge of a cliff 432 ft above the ground. What is the acceleration in ft/s2? a. 32 c. -39.8 b. -32 d. 98 156. Terry bought some gum and some candy. The number of packages of gum was one more than the number of mints. The number of mints was three times the number of candy bars. If the gum was 24 cents per package, mints were 10 cents each, and candy bars were 35 cents each, how many gums did he get for $5.72? a. 6 c. 14 b. 9 d. 13 157. Without expanding, find the coefficient of a 10b5 in the expansion of (a2 + b)10. a. 252 c. 126 b. 210 d. 1260 158. The first term of a geometric series is 256 and the last term is 81, the sum is 781, what is the geometric ratio? a. 2/3 c. 3/5 b. 3/4 d. 5/6 159. A contactor has 50 men of the same capacity at work on a job in 30 days, the working day being 8 hours, but the contract expires in 20 days, how many workers should he add? a. 15 c. 25 b. 20 d. 30 160. Evaluate 12 + 13 + 22 + 23 +32 + 33 + 42 + 43 + … + 1002 + 1003. a. 28,485,240 c. 26,854,520 b. 25,840,850 d. 28,240,290 161. If equal spheres are piled in the form of a complete pyramid with a rectangular base, find the total number of spheres in the pile if there are 5 and 4 spheres in the long and short sides of the base, respectively. a. 36 c. 40 b. 39 d. 42

162. If equal spheres are piled in the form of a complete pyramid with a rectangular base, find the total number of spheres in the pile if there are 6 and 5 spheres in the long and short sides of the base, respectively. a. 70 c. 68 b. 74 d. 72 163. Pipes between stations as indicated have the following maximum flow capacities, in cubic meters per second: Between A and B 40.0, between B and C 30.0, between A and C 20.0. What is the maximum possible flow rate from A to C, in cubic meter per second, without exceeding this the above maximum flow capacities a. 60 c. 50 b. 30 d. 40 164. Solve for x in the equation: 3X + 9X = 27X a. 0.438 b. 0.460

c. 0.416 d. 0.482

165. Which of the following is a prime number? a. 91 c. 97 b. 119 d. 133 166. Three geometric means are to be inserted between 6 and 14,406. Determine their product. a. 74,088 c. 10,374,481 b. 1,452,729,852 d. 25,412,184 167. Which of the following is a prime number? a. 377 c. 357 b. 313 d. 333 168. A 500 lb body rest on the plane that is inclined 29. What is the force exerted perpendicular to the plane? Neglect friction. a. 430 lb c. 437 lb b. 431 lb d. 500 lb 169. What are the values of n if (2n – 6) is greater than 1 but less than 14? a. 4, 5, 6, 7, 8, 9, 10 c. 3, 4, 5, 6, 7, 8 b. 4, 5, 6, 7, 8, 9 d. 2, 3, 4, 5, 6, 7

170. Solve for one value of x in x3 – 8 = 0. a. 3 c. 1 b. -2 d. 2 171. A man invested Php 50,000. Part of it he put on an oil stock from which he hoped to receive 20 percent return per year. The rest he invested in a bank stock which was paying 6 percent per year. If he received Php 400 more the first year from the bank stock than from the oil stock, how much did he invest in the oil stock? a. Php 12,000 c. Php 10,000 b. Php 13,000 d. Php 11,000 172. A window in Mr. Jones’s house is stuck. He takes an 8-inch screwdriver to pry open the window. If the screwdriver rests on the sill (fulcrum) 3 inches from the window and Mr. Jones has to exert a force of 10 pounds on the other and to pry open the window, how much force was the window exerting? a. 18 lbs. c. 17 1/2 lbs. b. 16 2/3 lbs. d. 15.5 lbs. 173. A baseball diamond is a square whose sides are 90 ft long. If a batter hits a ball and runs to first base at the rate of 20 ft/sec, how fast is his distance from second base changing when he has run 50 ft? a. 70/ 70 c. 7 b. 90/ 90

d. 80/ 97

174. A cistern in the form of an inverted right circular cone 12 ft. diameter at the top and 20 ft. high is filled to a depth of 16 ft. with the liquid weighing 60 pfc. A ½ hp pump (that is, the engine can do the work at the rate of 16, 500 ft-lb per minute) is used to pump the liquid to a height of 10 ft. above the top of the cistern. Compute the number of minutes it will take the pump to empty the cistern. a. 36.50 min c. 25.57 min. b. 27.14 min d. 34.63 min. 175. The sum of two numbers is 41. The larger number is 1 less than twice the smaller number. Find the larger number. a. 26 c. 27 b. 30 d. 28

176. An anchor chain of a ship weighs 730 N per lineal meter while the anchor weight 8900 N. What is the work done in pulling up the anchor if 30 meters of chain are out, assuming that the left is vertical? a. 328.5 kJ c. 61.5 kJ b. 267 kJ d. 595.5 kJ 177. The average of six scores is 83. If the highest score is removed, the average of the remaining scores is 81.2. Find the highest score. a. 91 c. 93 b. 92 d. 94 178. If 4, 2, 5 and 18 are added respectively to an arithmetic progression, the resulting series is a geometric progression. What is the sum of A.P.? a. 48 c. 46 b. 49 d. 47 179. Evaluate 12 + 13 + 22 + 23 + 32 + 33 + ……… + 1502 + 1503. a. 128,348,358 c. 135,391,800 b. 129,391,900 d. 147,920,368 180. A bookstore contracted to purchase a best-selling book at P250.00 per copy. At what price should the bookstore retail this book so that, despite a 15% discount, the profit on each copy will be 30%? a. 375.66 c. 500 b. 413.22 d. 454.55 181. Find the inequality of 1 < 2x – 1 < 3. a. 2<x<3 c. 2<x<1 b. 1<x<2 d. 0<x<1 182. Solve the inequality (2x – 3) < 1. a. -1<x<2 c. -2<x<1 b. 2<x<3 d. 1<x<2 183. In a 3-digit number, the hundreds digit is 4 more than the units digit and the tens digit is twice the hundreds digit. If the sum of the digits is 12, find the units digit. a. 4 c. 8 b. 0 d. 2

184. In a 3-digit number, the hundred’s digit is 4 more than the unit’s digit. The ten’s digit is twice the hundred’s digit. If the sum of the digits is 28. Find the units digit. a. 4 c. 2 b. 1 d. 0 185. A box with rectangular base 2 ft by 4 ft and a height of 1 ft is full of water. Calculating the work done in ft-lb to pump water 2 ft above the top of a box. a. 1248 c. 1498 b. 1982 d. 2296 186. A farmer has 100 gallons of 70% pure disinfectant. He wishes to mix it with disinfectant which is 90% pure in order to obtain 75% pure disinfectant. How much of the 90% pure disinfectant must he use? a. 25 1/2 c. 33 b. 30 d. 33 1/3 187. Jose Ramirez had $50 to buy his groceries. He needed milk at $1.95 a carton, bread at $2.39 a loaf, breakfast cereal at $3.00 box and meat at $5.39 a pound. He bought twice as many cartons of milk as loaves of bread and one more package of cereal than loaves of bread. He also bough t the same number of pounds of meat as packages of cereal. How many pounds of meat did he purchase if he received $12.25 in change? a. 5 c. 7 b. 3 d. 10 188. The sum of three consecutive integers is 54. Find the largest integer. a. 17 c. 19 b. 18 d. 20 189. The linear density of a rod is the rate of change of its mass with respect to its length. A nonhomogeneous rod has a length of 9 feet and a total mass of 24 slugs. If the mass of a section of the rod of length x (measured from its leftmost end) is proportional to the square root of this length, compute the average density of the rod. a. 8/3 slugs/ft c. 4/3 slugs/ft b. 7/3 slugs/ft d. 7/6 slugs/ft

190. Robin flies to San Francisco from Santa Barbara in 3 hours. He flies back in 2 hours. If the wind was blowing from the north at the velocity of 40 mph going, but changed to 20 mph from the north returning, what was the airspeed of the plane? a. 140 mph c. 160 mph b. 150 mph d. 170 mph 191. Find the force on one side of the surface of an isosceles trapezoid of height 4 feet and bases 6 feet and 12 feet with the smaller base lying in the water surface. a. 5,000 lb c. 7,500 lb b. 6,000 lb d. 8,500 lb 192. A cable 100 feet long and weighing 3 pounds per foot hangs from a windlass. Find the work done in winding it up. a. 15 c. 22 ½ b. 7 ½ d. 3 ¾ 193. A store manager wishes to reduce the price on her fresh ground coffee by mixing two grades. If she has 50 pounds of coffee which sells for $10 per pound, how much coffee worth $6 per pound must the mix with it so that she can sell the final mixture for $8.50 per pound? a. 25 pounds c. 35 pounds b. 30 pounds d. 40 pounds 194. Mario bought Php 21.44 worth of stamps at the post office. He bought 10 more 4-cent stamps than 19-cent stamps. He also bought two Php 1 stamps. How many 19-cent stamps did he purchase? a. 16 c. 12 b. 14 d. 11 Ans. The number of 32-cent stamps was three times the number of 19-cent stamps 195. A florist wishes to make bouquets of mixed spring flowers. Each bouquet is to be made up of chrysanthemum (mums) at Php30 a bunch and roses at Php21 a bunch. How many bunches of mums should she use to make 15 bunches which she can sell for Php24 a bunch? a. 6 c. 4 b. 3 d. 5

196. Two cars are headed for La Union. One is 50 km ahead of the other on the same road. The one in front is traveling 60 kph while the second car is traveling 70 kph. What is the distance at which the second car will overtake the first? a. 350 c. 340 b. 300 d. 400 197. The water in 4 ft by 2 ft by 1 ft rectangular water tank is discharged at a point 2 ft above its surface level. Find the work done in lb ft. a. 1248 c. 380 b. 301 d. 1000 198. A basket contains mangoes, papayas and watermelons, 8 fruit in all. Mangoes cost P12 each, papayas cos P25 each and watermelons P50 each. If the total cost of all fruit is P198, determine the total cost of the mangoes. a. 48 c. 52 b. 44 d. 56 199. A bicycle travels along a straight road. A 1:00 it is 1 mile from the end of the road and at 4:00 it is 16 miles from the end of the road. Compute its average velocity from 1:00 to 4:00. a. 5 mph c. 3.4 mph b. 2.5 mph d. 6.8 mph 200. A 35-pound weight is 2 feet from the fulcrum, and a 75-pound weight on the same side is 10 feet from the fulcrum. If a weight on the other end 6 feet from the fulcrum balances the first two, how much does it weigh? a. 128 3/4 pounds c. 142 3/7 pounds b. 116 2/5 pounds d. 136 2/3 pounds 201. Jones can paint a car in 8 hours. Smith can paint the same car in 6 hours. They start to paint the car together. After 2 hours, Jones leaves for lunch and Smith finishes painting the car alone. How long does it take Smith to finish? a. 4 ½ hrs c. 3½ hrs b. 2½ hrs d. 5½ hrs 202. Find the tenth element of the given sequence 11, 4, -3, -10…. a. –52 c. 99 b. -106 d. 58

203. A plane takes 1 ½ hours to fly from Los Angeles to San Francisco and 2 hours from San Francisco to Los Angeles. If the wind blows north on both trips at 24 mph, what is the speed of the plane in still air? a. 170 c. 120 b. 110 d. 150 204. A right circular tank of depth 12 feet and radius 4 feet is half full of oil weighing 60 pounds per cubic foot. Find the work done in pumping the oil to a height 6 feet above the tank. a. 272 c. 109 b. 136 d. 164 205. Alex is 8 years older Cynthia. Twenty years ago Alex was three times as old as Cynthia. How old is Cynthia now? a. 18 c. 30 b. 20 d. 24 206. Find the nth term of 6, 2, -2,… a. -4n + 6 b. -2n + 6

c. 2n + 8 d. -4n + 10

207. Find the nth term of -5, -13, -21,… a. -7n + 4 c. -3n + 5 b. -8n + 3 d. -6n + 2 208. If an alloy containing 30% silver is mixed with a 55% silver alloy to get 800 pounds of 40% alloy, how much is a 30% silver alloy? a. 480 c. 450 b. 420 d. 460 209. Find the number which is greater to its square by a minimum difference. a. 1/2 c. 1/4 b. 1 d. 1/3 210. Tom, Dick, and Harry decided to fence a vacant lot adjoining their properties. If it would take Tom 4 days to build the fence. Dick 3 days, and Harry 6 days, how long would it take them working together? a. 1 -3/4 c. 1 -1/3 b. 2 -1/3 d. 2 -1/4

211. A circular water main 4 meter in diameter is closed by a bulkhead whose center is 40m below the surface of the mater reservoir. Find the force on the bulkhead. a. 4319 kN c. 3419 kN b. 4931 kN d. 5028 kN 212. If (5x-3), (x+2) and (3x-11) form arithmetic progression, find the 15th term. a. –86 c. -79 b. -81 d. –84 213. In a proportion of four quantities, the first and the fourth terms are referred to as the: a. means c. denominators b. extremes d. axiom

TRIGONOMETRY 214. A 100 kg weight rests on a 30 inclined plane. Neglecting friction how much pull must one exert to bring the weight up the plane? a. 86.67 kg c. 70.71 kg b. 100 kg d. 50 kg 215. If sin x cos x + sin 2x = 1, what are the values of x? a. 32.2°, 69.3° c. 20.90°, 69.1° b. -20.67°, 69.3° d. -32.2°, 69.3° 216. The two legs of a triangle are 300 and 150 each respectively. The angle opposite the 150 side is 26°. What is the third leg? a. 197.49 c. 341.78 b. 218.61 d. 282.15 Ans. A or C 217. From a hill 600 ft. high, the angles of depression to the bases in opposite directions are 42° and 19°23’, respectively. Find the length of the proposed tunnel through the bases. a. 2,589.15 ft c. 2,590.05 ft b. 2,371.74 ft d. 2,591.20 ft 218. Solve for G if csc (11G – 16 degrees) = sec (5G +26 degrees) a. 7 degrees c. 6 degrees b. 5 degrees d. 4 degrees 219. Perform the operation 4(cos 60° + i sin 60°) divided by 2(cos 30° + i sin 30°) in rectangular coordinates. a. (square root of 3) – 2i c. (square root of 3) + i b. (square root of 3) –i d. (square root of 3) + 2i 220. Evaluate sin 73. a. 0.8752 b. 0.9563

c. 0.5241 d. 0.7254

221. The slope of a line is 1/2. The slope of the second line is -2/3. The lines intercept at the point (3, 1). What is the acute angle between the lines? a. 27 degrees c. 60 degrees b. 50 degrees d. 80 degrees

222. A ship on a certain day is at latitude 20 degrees N and longitude 149 degrees E. After sailing for 150 hours at a uniform speed along a great circle route, it reaches a point at latitude 10 degrees S and longitude 170 degrees E. If the radius of the earth is 3959 miles, find the speed in miles per hour. a. 17.4 miles per hour c. 16.4 miles per hour b. 15.4 miles per hour d. 19.4 miles per hour 223. The horizontal angle from the ground to the top of a palm tree some unknown distance away is 46.18. at a point 40 m directly behind the first point, the horizontal angle to the top of the tree is 29.23. What is the distance from the palm tree to the first point? a. 42 m c. 51 m b. 46 m d. 61 m 224. Given that sin θ = 3/5 and θ is acute, find cos 2θ. a. 7/25 c. -4/5 b. -7/25 d. 4/5 225. In the curve y = tan 3x. What is the period?   a. c. 3 2 3  b. d. 2 4 226. Find the period of y = sin 3x. a. 33 b. 2pi/3

c. 1/3 d. 3/2pi

227. Two cities are 270 miles apart lie on the same meridian. What is the difference in latitude, if the radius of the earth is 3,960 miles? 2 4 a. rad c. rad 44 44 3 5 b. rad d. rad 44 44 228. Given the curve y = 4 sin 2x, find the amplitude. a. 4 c. 3 b. 2 d. 5

229. An airplane flies at a speed of 240 mph in still air S30W with a wind speed of 40 mph due west. What is the new bearing? a. S 32 W c. S 39 W b. S 35 W d. S 38 W 230. A telegraph pole is kept vertical by a guy wire which makes an angle of 25 with the pole and which exerts a pull of F = 300 Ib on the top. Find the horizontal component of the pull F. a. 140 lb c. 110 lb b. 135 lb d. 127 lb 231. Determine the simplified form of 2/(1 – cos2C) a. csc C c. sec C b. sec² C d. csc² C 232. Determine the simplified form of [cos 2A – (cos A)²]/(cos A)² a. –(sec A)² c. sec A b. –(tan A)² d. tan A 233. Given the curve y = 3 cos a. 3, pi/2 b. 3, 2pi

1 x, find the amplitude and period. 2 c. 3, 3pi/2 d. 3, 4pi

234. Given the curve y = 4 cos 2x, find the period. a. pi/4 c. pi b. 3pi d. 2 pi

1 235. Given the curve y = 3 sin   x, find the period. 2 a. 3/2 c. 3 b. 1/2 d. 2 236. In what quadrants will  be terminated if cos  is negative? a. 1, 2 c. 1, 3 b. 2, 3 d. 2, 4 237. In a triangle ABC, side AB = 12 cm angle A = 30, and angle B = 45. Find the length of the segment from vertex C and perpendicular to side AB. a. 5.4 cm c. 5.8 cm b. 4.4 cm d. 4.8 cm

238. A tree 120 ft tall casts a shadow 120 ft long. Find the angle of elevation of the sun in radian. a. pi/2 c. pi/3 b. pi/4 d. pi/6 239. Find the exact value of tan (5/6) a. 3 /2 c. b. - 3 /2

3 /3

d. - 3 /3

240. Convert 4 radians into degrees. a. 1540/ c. 90/ b. 180/ d. 720/ 241. The arc length is equal to the radius of a circle is called _______. a. 1 grad c. quarter arc b. 1 radian d. pi radians 242. A bicycle with 20-in wheels is mi/hr. Find the angular velocity minute. a. 190 b. 252

traveling down a road at 15 of the wheel is revolutions per c. 180 d. 342

243. Two buildings with flat roofs are 60 m apart. From the roof of the shorter building, 40 m in height, the angle of elevation to the edge of the roof of the taller building is 40. How high is the taller building? a. 60 m c. 80 m b. 70 m d. 90 m 244. Three ships are situated as follows: A is 225 mi due north of C, and B is 375 mi due east of C. What is the bearing of B from A? a. N 56 E c. N 59 E b. S 56 d. S 59 E 245. A support wire is anchored 12 m up from the base of a flagpole and the wire makes a 15 angle with the ground. How long is the wire? a. 12 m c. 46 m b. 92 m d. 24 m

246. 1 radian is equal to a. 3120/pi deg. b. 360/pi deg.

c. 180/pi deg. d. 170/pi deg.

247. 4 radians is equal to a. 720/pi deg. b. 360/pi deg.

c. 120/pi deg. d. 270/pi deg.

248. The two sides of a triangle are 3.2 km and 2.5 km. If the included angle is 143 degrees. Find the length of the third side. a. 5.41 c. 4.15 b. 6.54 d. 3.45 249. Given the curve y = 4sin 2x, find the amplitude. a. 4 c. 3 b. 2 d. 5 250. How many possible triangles can be formed in an angle A = 126 and sides a = 20 cm and b = 25 cm? a. 1 solution c. no solution b. 2 solutions d. infinite 251. Solve for F if TAN(8F + 1 degree) = COT(F + 17 degrees). a. 8° c. 9° b. 6° d. 7° 252. An airplane flew from Manila (14 degrees 36 minutes N, 121 degrees 5 minutes E) at an average speed of 300 miles per hour on a course S 32 degrees E. At what point will it cross the equator? The radius of the earth is 3959 miles. a. 128° 2’ E c. 134° 2’ E b. 130° 2’ E d. 132° 2’ E 253. In the triangle ABC, side a is 9 cm, side b is 12 cm and C = 500. Find angle B. a. 830 25’ c. 820 2’ b. 810 15’ d. 840 12’ 254. A spherical triangle where given parts are a=100 010.2’, b=4800.4’, and c=55036.8’. Find the vertex A. a. 121031.6’ c. 119041’ 0 b. 121 46.5’ d. 120040.2’

255. If tan (2D – 3) = 1/tan (5D – 9), determine D in degrees. a. 13.88° c. 14.57° b. 15.30° d. 16.97° 256. If 77° + 0.40x = Arctan (cot 0.25x), find x. a. 10° c. 20° b. 30° d. 40° 257. A certain angle has a supplement 5 times the compliment, find the angle. a. 67.5° c. 58.5° b. 30° d. 27° 258. Find the supplement of an angle whose compliment is 62 degrees: a. 30° c. 152° b. 28° d. 118° 259. Two angles whose sum is 360 degrees are said to be: a. supplementary c. elementary b. complimentary d. explementary 260. Sin (x + y) = 0.9659, sin x = 0.5. Find cos y. Answer: 0.707 261. The hypotenuse of a right triangle is 34 cm. Find the length of the two legs if one leg is 14 cm longer than the other. a. 18 and 32 cm c. 17 and 32 cm b. 15 and 29 cm d. 16 and 30 cm 262. sin (x + y) = 0.9659, sin x = 0.5. Find cos y. a. 0.816 c. 1.0 b. 0.707 d. 0.425 263. The piston of an engine is connected by a 12-inch connecting rod to a point on a crank that rotates on a 4-inch radius about the crank shaft. If the crankshaft has an angular speed of 3000 revolutions per minute, determine the rectilinear speed of the piston, in feet per minute, when the crank is 90° to the motion of the piston. a. 1, 319 c. 1, 194 b. 1, 257 d. 1, 131

264. From a hill 600 ft high, the angles of depression to the bases in the opposite direction are 42° and 19° 23 minutes, respectively. Find the length of the proposed tunnel through the bases. a. 2591. 10 ft c. 2591.20 ft b. 2590.05 ft d. 2589.15 ft 265. What value of F satisfy the equation tan (8F + 1) = cot (17) where all angles in degrees? a. 10 c. 7 b. 9 d. 8 266. Determine the simplified form of sin 2/ (1 – cos 2B). a. cot B c. sin B b. tan B d. cos B 267. Solve for x in the equation: arctan ( x + 1) + arctan (x – 1) = arctan (12): a. 1.34 c. 1.25 b. 1.20 d. 1.50 268. If sec (2A) = 1/sin (13A), determine the angle A in degrees. a. 6 degrees c. 8 degrees b. 7 degrees d. 5 degrees 269. An observer is 200 ft. from a building, observes that the top of the pole on top of the building makes an angle of elevation of 30. Assuming the height of the pole is 50 ft. and the height of the eyes of the observer is 5 ft. from the ground level. Find the height of the building in feet. a. 72.4 c. 80.1 b. 70.5 d. 65.8 270. Determine the simplified form of cos (2A) – cos2 (A)/sin (A). a. cos 2A c. cos A b. –sin A d. sin 2A 271. An observation made in Hongkong (Latitude 22 degrees 18 N) gave the altitude of the sun to be 43 20’ and its declination was 15 degrees 52’. Find the time of the day, if the observation is in the morning. One hour is 15 degrees. a. 9:12 A.M. c. 8:36 A.M. b. 8:52 A.M. d. 8:44 A.M.

272. Which is identically equal to (sec A + tan A)? a. 1/(sec A – tan A) c. 2/(1 – tan A) b. csc (A – 1) d. csc (A + 1) 273. If tan(2D – 3) = 1/tan(5D – 9), determine D in degrees. a. 13.88 degrees c. 14.57 degrees b. 15.30 degrees d. 16.97 degrees 274. Find the polar equation of a circle, if its center is at (4,0) and the radius is 4. a. r – 8 cos u = 0 c. r – 12 cos u = 0 b. r – 6 cos u = 0 d. r – 4 cos u = 0

GEOMETRY 275. A and B are points on the opposite banks of a certain body of water. Another point C is located such that AC is 600 m and BC is 500 m. Points A, B, and C from a triangle, whose vertex is (A), with an angle of 55 degrees. What is the width of the body of water in meters? a. 651.12 c. 630.21 b. 632.48 d. 648.33 Ans. 469.03 m or 252.25 m 276. A right circular cylinder is inscribed in a right circular cone of radius r. Find the radius R of the cylinder if its lateral area is a maximum. 1 2 a. R = r c. R = r 2 3 1 3 b. R = r d. R = r 3 2 277. Find the greatest area of the rectangle that can be cut from a semicircle of radius 6cm. a. 12 sq. cm. c. 24 sq. cm. b. 36 sq. cm. d. 72 sq. cm. 278. A regular hexagon is inscribed in a circle whose diameter is 20 meters. Find the area of the 6 segments of the circle formed by the sides of the hexagon. a. 42.47 c. 54.36 b. 50.21 d. 64.38 279. If the volume of a regular tetrahedron is 85.92 cm 3, compute its surface area. a. 110.30 cm2 c. 140.30 cm2 2 b. 120.30 cm d. 150.40 cm2 280. Given the square with 20 cm sides. Another square is to be inscribe in the given square such that the vertices of the former lies on the sides of the latter. Determine the area in sq. cm of the smallest inscribe square? a. 200 c. 180 b. 220 d. 160

281. Find the volume of a paraboloid having a radius of 8 cm and height of 16 cm. a. 512 cm3 c. 630 cm3 b. 569 cm3 d. 780 cm3 282. A right prism with a hexagonal base has a surface of 908.554 sq. cm. If the height of the prism is equal to 12 cm, find the base edge. a. 8 cm c. 6 cm b. 10 cm d. 5 cm 283. A right circular cone has a surface area of 15 sq. m. If the radius of the cone is 3 m, find the volume of the cone. a. 12 c. 14 b. 15 d. 16 284. Find the maximum area of a rectangle circumscribed about a fixed rectangle of length 6 and width 4. a. 50 c. 63 b. 72 d. 32 285. Find the maximum area of a rectangle circumscribed about a fixed rectangle with length 8 and width 4. a. 50 c. 32 b. 64 d. 72 286. How many diagonals does an octagon have? a. 20 c. 22 b. 18 d. 24 287. What is the perimeter of a regular 15-sided polygon inscribed in a circle with radius 10 cm? a. 63.77 cm2 c. 64.52 cm2 2 b. 62.37 cm d. 68.48 cm2 288. Seven regular hexagons each with 6-cm sides, are arranged so that they share some sides and the centers of six hexagons are equidistant from the seventh central hexagon. Determine the ratio of the total area of the hexagons to the total outer perimeter enclosing the hexagons. a. 0.6014 c. 0.7217 b. 1.0392 d. 0.8660

289. A rectangle is inscribed in an equilateral triangle with 10 cm sides, such that one sides of the rectangle rests on one side of the triangle. Determine the area in sq. cm of the largest possible inscribe rectangle. a. 21.65 c. 3.82 b. 22.73 d. 19.48 290. What is the sum of the interior angles of a 15-sided regular polygon? a. 2560 c. 2480 b. 2340 d. 2620 291. The longest diagonal of a cube is 15 cm, find the volume of the cube. a. 625.85 cm3 c. 1193.24 cm3 b. 649.52 cm3 d. 1295.36 cm3 292. A water tank is a horizontal circular cylinder 10 ft long and 10 ft in diameter. If the water inside is 7.5 ft deep determine the volume of water contained. a. 663.44 cu ft c. 631.85 cu ft b. 600.26 cu ft d. 568.67 cu ft 293. The perimeter of an isosceles right triangle is 10.2426. Find the area of a triangle. a. 2 c. 4.5 b. 3 d. 4 294. A wire is shaped to form a rectangle 15 cm in length, the rectangle has an area of 150 cm2. Then reshaped to form a square, what is the area of the square? a. 168.45 c. 156.25 b. 165.25 d. 152.65 295. Find the maximum area of a rectangle circumscribed about a fixed rectangle of length 6 and width 4. a. 50 c. 64 b. 72 d. 32 296. Find the two bases of a trapezoid if they are in the ratio 4:5. The altitude is 20 cm and the area is 360 sq. cm. a. 20, 25 c. 12, 15 b. 16, 20 d. 24, 30

297. The sum of the sides of two polygons is 9 and the sum of its diagonal is 7. Find the number of sides of its polygon. a. 2 and 3 c. 4 and 5 b. 3 and 6 d. 5 and 7 298. Suppose that a dam is shaped like a trapezoid with height 100 feet, 300 feet long at the top and 200 feet long at the bottom. When the water level behind the dam is level with its top, what is the total force that the water exerts on the dam? a. 42,600 tons c. 84,600 tons b. 36,400 tons d. 24,600 tons 299. Find the sum of the interior angle of a regular hexagon. a. 810 c. 720 b. 540 d. 630 300. The 3 sides of a triangle are a = 12 cm, b = 10 cm, and c = 8 cm. What is the sum of the 3 heights each perpendicular to the 3 sides. a. 23.25 cm c. 24.47 cm b. 22.03 cm d. 25.70 cm 301. The sum of the sides of 2 polygons is 12 and their diagonals is 19. Determine the number of sides of each polygon. a. 2 sides and 3 sides c. 4 sides and 5 sides b. 3 sides and 6 sides d. 5 sides and 7 sides 302. A rectangular hexagonal pyramid has a slant height of 4 cm and the length of each side of the base is 6 cm. Find the lateral area. a. 52 cm2 c. 72 cm2 2 b. 62 cm d. 82 cm2 303. A rectangle is inscribed in an equilateral triangle with 10 cm sides, such that one side of the rectangle rests on one side of the triangle. Determine the area in sq. cm. of the largest possible inscribed rectangle. a. 21.65 c. 23.82 b. 22.73 d. 19.48 304. Find the area of a square with a diagonal of 15 cm. a. 225 cm2 c. 112.5 cm2 2 b. 114.5 cm d. 121.5 cm2

305. A quadrilateral have sides equal to 12 m, 20 m, 8 m and 16.97 m, respectively. If the sum of the two opposite angles is equal to 225 degrees, find the area of the quadrilateral. a. 100 c. 124 b. 168 d. 158 306. A water tank is a horizontal circular cylinder 10 feet long and 10 ft in diameter. If the water inside is 7.5 feet deep determine the volume of water contained. a. 663.44 cu ft c. 631.85 cu ft b. 600.26 cu ft d. 586.67 cu ft 307. A solid has a circular base of radius r. Find the volume of the solid if every plane perpendicular to a given diameter is a square. a. 5r3 c. 16r3/3 b. 6r3 d. 19r3/3 308. A tetrahedron is a regular solid whose 4 equal faces (surfaces) are each an equivalent triangle. What is the volume of such a solid whose 6 edges are each equal to 10 cm? a. 67.21 cu. cm c. 83.33 cu. cm b. 91.67 cu. cm d. 73.94 cu. cm 309. Find the area of a regular five-pointed star that is inscribed in a circle. Note: Pentagon formed in a star has 10 cm on each side. a. 658.86 cm2 c. 549.75 cm2 b. 655.87 cm2 d. 556.76 cm2 310. Find the perimeter of a regular pentagon inscribed in a circle with a circumference of 100 cm. a. 93.55 cm c. 115.63 cm b. 125.68 cm d. 89.56 cm 311. A hole of radius 2 is drilled through the axis of a sphere of radius 3. Compute the volume of the remaining solid 40 3 20π 5 a. c. 3 2 25 3 26 5 b. d. 2 3

 radian, and the chord of the circle 4 subtended by this angle is 12 2 cm. Find the radius of the circle. a. 10 cm c. 14 cm b. 12 cm d. 16 cm

312. An inscribed angle is

313. Find the maximum area of the rectangle circumscribed about a fixed rectangle of length 6 and width 4. a. 50 c. 63 b. 72 d. 32 314. A diagonal of a cube is 6 cm. The total area of the cube is a. 36 2 sq. cm. c. 24 2 sq. cm. b. 72 sq. cm. d. 236 sq. cm. 315. Two rectangles one of length 4 less than the width and one of length 4 more than the width. The difference of the two areas is 64. What are the lengths of the two rectangles? a. 4, 12 c. 8, 4 b. 10, 16 d. 12, 16 316. A sphere of diameter 8 inches has a thickness of 1/16 inches. Find the volume by approximation. a. 2 pi c. 8 pi b. 4 pi d. 16 pi 317. Find the angle at which the arc length is always equal to its radius. a. 45 deg. c. 57.296 deg. b. 141.372 deg. d. 122.322 deg. 318. The sum of the sides of two polygons is 12 and the sum of the diagonals is 19. Find the number of sides of each polygon. a. 3 and 9 c. 4 and 8 b. 5 and 7 d. 6 and 6 319. Find the radius of the base maximum volume that could be 10 m. a. 13.33 b. 4.28

of a right circular cone of inscribed in a sphere of radius c. 9.43 d. 9.04

320. A piece of wire 36 cm long is cut into 2 parts. One part will be used to form into an equilateral triangle and the other into a rectangle whose length is twice its width. Find the length of the piece that was cut into a rectangle. a. 16.708 cm c. 6.431 cm b. 19.293 cm d. 27.846 cm 321. An open cylindrical tank is 3 ft in diameter with 4.5 ft in height is tilted so that one half if its bottom is exposed. Find how many cu ft of water remaining in the tank if the container was initially full of water? a. 6.75 c. 8.75 b. 5.06 d. 7.45 322. In a frustum of a cone of revolution the radius of the lower base is 11 in., the radius of the upper base is 5 in., and the altitude is 8 in. find the total area in square inches. a. 306 c. 226 b. 160 d. 80 323. Find the area of the largest isosceles triangle that can be inscribed in a circle of radius 6 inches. a. 24 3 sq. in c. 54 3 sq. in b. 12 3 sq. in

d. 27 3 sq. in

324. A tank with a cross-sectional shape of an equilateral triangle has dimensions of 4 m on all three sides. What is the water level if the tank is 50% full by volume? The tank vertex points up. a. 1.0 m from the bottom c. 1.7 m from the bottom b. 1.2 m from the bottom d. 2.2 m from the bottom 325. If (6 – x), (13 – x), and (14 –x) are the lengths of the sides of a right triangle, find the area of the triangle. a. 78 s.u. c. 32.5 s.u b. 30 s.u. . d. 60 s.u. 326. Two corridors respectively 2.5m and 1.0m wide intersect at right angels. Find the length in meters of the largest thin rod that will go horizontally around the corner. a. 3.97 c. 5.32 b. 4.79 d. 5.23

327. A side and a diagonal of a parallelogram are 12 inches and 19 inches, respectively. The angle between the diagonals, opposite the given side, is 124. Find the length of the other diagonal. a. 7.84 in c. 3.74 in b. 7.48 in d. 7.73 in 328. An equilateral triangle is circumscribed in a circle of radius 10 cm. Find the length of each side of the triangle. a. 34.69 cm c. 37.05 cm b. 32.09 cm d. 36.07 cm 329. A swimming pool is rectangular in shape of length 40 ft and width 18 ft. It has a sloping bottom and is 3 ft at one end and 12 ft at the other end. The water from a full cylindrical reservoir is 12 ft in diameter and 40 ft deep is emptied into the pool. Find the depth of the water at the deeper end. a. 10.91 c. 11.21 b. 12.01 d. 10.78 330. Given a triangle of sides 10 cm and 15 cm with an included angle of 60°. Find the area of the triangle in sq. cm. a. 72 c. 75 b. 80 d. 65 331. Find the dimensions of a rectangle whose perimeter is 40 inches and whose area is 96 square inches. a. 11, 9 c. 10, 9.6 b. 12, 8 d. 10, 10 332. A box with an open top is to be made by taking rectangular piece of tin 8x10 inches and cutting a square of the same size out of each corner and folding up the sides. If the area of the base is to be 24 square inches, what should be the length of the sides of the square be? a. 2 in. c. 2.1 in. b. 2.2 in. d. 1.8 in. 333. Given the triangle with sides 10 cm, 16 cm and 18 cm. Find the area of the triangle in sq. cm. a. 79.6 c. 80.5 b. 84 d. 81.2

334. A target with a black circular center and a white ring of uniform width is to be made. If the radius of the center is to be 3 cm, how wide should the ring be so that the area of the ring is the same as the area of the center? a. 1.232 cm c. 1.252 cm b. 1.263 cm d. 1.243 cm 335. Which of the following is not a property of a triangle: a. the sum of three angles of a triangle is equal to two right triangles. b. the sum of the two sides of a triangle is less than the third side. c. if two sides of the triangle are unequal, the angles opposite are equal. d. the altitude of a triangle meets in a point. 336. Which of the following is not a property of a circle: a. through 3 points not in the straight line one circle and only one can be drawn. b. a tangent to a circle is perpendicular to the radius at the point of tangency and conversely. c. an inscribed angle is measured by one half of the intercepted arc. d. the arcs of two circles subtended by equal central angles are equal. 337. The radius of the circle inscribed in a polygon, is called as: a. internal radius c. radius of gyration b. apothem d. hydraulic radius 338. An isosceles triangle has a 10 cm base and a 10 cm altitude. Determine the moment of inertia of the triangular area relative to a line parallel to the base and through the upper vertex, in cm4. a. 3025 c. 2273 b. 2500 d. 2750 339. If equal spheres are piled in the form of a complete pyramid with an equilateral triangle as base, find the total number of spheres in the pile of each side of the base contain 4 spheres. a. 20 c. 18 b. 21 d. 15

340. A polygon with 12 sides is called as: a. bidecagon c. dodecagon b. nonagon d. pentedecagon 341. All circles having the same center but with unequal radius are called as: a. eccentric circle c. inner circle b. concentric circle d. pythagorean circle 342. A triangle having three sides of unequal length is known as: a. equilateral triangle c. isosceles triangle b. scalene triangle d. equiangular triangle 343. The intersection of the sphere and the plane through the center is the: a. great circle c. small circle b. poles d. polar distance 344. The sides of a triangle are 195, 157 and 210 respectively. What is the area of the triangle? a. 10, 250 sq. unit c. 11, 260 sq. unit b. 14, 586 sq. unit d. 73, 250 sq. unit 345. Determine the total area of a regular six-star polygon if the inner regular hexagon has 10 cm sides. a. 467.64 sq. cm c. 493.62 sq. cm b. 519. 60 sq. cm d. 441. 66 sq. cm 346. A rhombus has diagonals of 32 and 20 inches. Determine its area. a. 360 sq. in c. 320 sq. in b. 400 sq. in. d. 280 sq. in 347. Given the triangle with sides 10 cm, 16 cm and 18 cm. Find the area of the triangle in sq. cm. a. 79.6 c. 80.5 b. 84.0 d. 81.2 348. A circle has a 20 cm diameter. Determine the moment of inertia of the circular area relative to the axis perpendicular to the area through the center of the circle, in cm 4. a. 14,280 c. 17,279 b. 15,708 d. 19,007

349. A target with a black circular center and a white ring of uniform width is to be made. If the radius of the center is to be 3 cm, how wide should the ring be so that the area of the ring is the same as the area of the center? a. 1.232 cm c. 1.252 cm b. 1.263 cm d. 1.243 cm 350. Three circles C1, C2 and C3 are externally tangent to each other. Center-to-center distances are 10 cm between C1, and C2, 8 cm between C2 and C3, and 6 cm between C3 and C1. Determine the total areas of the circles. a. 39.58 sq. cm. c. 43.98 sq. cm. b. 45.08 sq. cm. d. 46.18 sq. cm. 2 Ans. 175.93 cm 351. A regular pentagon has sides of 20 cm. An inner pentagon with sides of 10 cm is inside and concentric to the larger pentagon. Determine the area inside the larger pentagon but outside of the smaller pentagon. a. 430.70 sq. cm. c. 473.77 sq. cm. b. 573.26 sq. cm. d. 516.14 sq. cm. 352. A rhombus has diagonals of 32 and 20 inches. Determine its area. a. 280 sq. in. c. 400 sq. in. b. 360 sq. in. d. 320 sq. in. 353. A part of a circle is often called as: a. sector c. arc b. cord d. segment 354. Equal-sized spheres are contained in a regular tetrahedron (with equilateral triangle for each face), such that along the 6 sides of the tetrahedron, there are 4 spheres in line. How many total spheres are there? a. 14 c. 20 b. 22 d. 18 355. The sides of a triangle are 195, 157, and 210 respectively. What is the area of the triangle? a. 73,250 sq. unit c. 14,586 sq. unit b. 10,250 sq. unit d. 11,260 sq. unit

356. The surface S and the volume V of a sphere changes accordingly with radius r. There is a value of r when the rates of change in S and V are numerically equal. Determine the equal values of S and V. a. 20 c. 16 b. 18 d. 14 Ans. S = 16 s. u., V = 32/3 c. u. 357. Two pulleys, 10ft. between centers, are linked by a noncrossing belt. The larger pulley is 10ft in diameter and the smaller pulley is 5ft in diameter. Determine the circumference (total length) of the belt. a. 47.85 ft c. 46.88 ft b. 41.33 ft d. 43.50 ft Ans. 44.18 ft 358. Three circles are externally tangent to each other. The distances between their centers are 50 cm between circles A and B, 46 cm between circles B and C, and 40 cm between circles C and A. determine the diameters in cm, of each circle A, B, and C, respectively. a. 56 for A, 36 for B, 44 for C b. 40 for A, 50 for B, 46 for C c. 56 for A, 36 for B, 40 for C d. 44 for A, 56 for B, 36 for C 359. Two equilateral triangles, each with 12-cm sides, overlap each other to form a 6-point “Star of David”. Determine the overlapping area, in sq. Cm. a. 34.64 c. 28.87 b. 41.57 d. 49.88 360. A central circle has a 10-cm radius. Six equal smaller circles are to be arranged so that they are externally tangent to each other and the centers lie in the circumference of the central circle. What should be the radius in cm, of the small circle? a. 4.167 c. 3.472 b. 6.000 d. 5.000 361. The sum of the interior angles of a polygon is 540 degrees. Find the number of sides. a. 5 c. 8 b. 6 d. 11

362. A right circular cylinder has a 10 cm diameter and a 10 cm height. Determine the moment of inertia of the cylindrical volume relative to its center, in cm5. a. 14,399 c. 13,090 b. 11,900 d. 9,818 363. A cylindrical tin can has its height equal to the diameter of its base. Another cylindrical tin can with the same capacity has its height equal to twice the diameter of its base. Find the ratio of the amount of tin required for making the two cans with covers. Answer: 0.9524 364. The diameters of two spheres are in the ratio 2:3 and the sum of their volumes is 1,260 cubic meters. Find the volume of the larger sphere. Answer: 972 cu. Meter 365. Find the area of a regular 5 – pointed star that can be inscribed in a circle with radius of 10 cm. Answer: 112.257 cm2 366. Find the radius of the largest circle that can be inscribed in the triangle with sides: a = 8cm, b = 15cm, c = 17 cm. Answer: 3 367. Find the radius of the smallest circle that can circumscribed in the triangle in the previous problem. Answer: 8.5 368. A trapezoid gutter will be made from a sheet of metal 18” wide by bending up the edges at the one-third points. Find the width at the top for a maximum carrying capacity. Answer: 12” 369. The internal angle of a polygon is 150 degrees greater than its external angle. How many sides has the polygon had? Answer: 8 370. A circle of radius has 6 half its area removed by cutting off a border of uniform width. Find the width of the border. a. 22 c. 37.5 b. 13.5 d. 1.76

371. A circle is inscribed in a 3 – 4 – 5 right triangle. How long is the line segment joining the points of tangency of the “3 – side” and the “5 – side”? a. 1.28 c. 1.46 b. 1.35 d. 1.79 372. Let D be the set of vertices of a regular dodecagon. How many triangles may be constructed have d as vertices? a. 220 c. 240 b. 120 d. 180 373. A group of children playing with marbles placed 50 pieces of the marbles inside a cylindrical container with water filled to a height of 20 cm. If the diameter of each marble is 1.5 cm and that of the cylindrical container 6 cm, what would be the new height of water inside the cylindrical container after the marbles where placed inside? a. 23.125 c. 26.125 b. 24.125 d. 25.125 374. If a right circular has a base radius of 35cm and an altitude of 45 cm, solve for the total surface area in square cm. a. 10116 c. 11117 b. 10117 d. 12117 375. Find the maximum weight of a circular cylinder that can be cut from a spherical shot weighing 100 kg. a. 70.7 kg c. 92.6 kg b. 50 kg d. 57.7 kg 376. A circular sector is to have a perimeter of 16 cm. Find the radius that makes the area of the sector greatest. a. 4.8 c. 3 b. 5.2 d. 4 377. The side of a square is 16 inches. A second square is formed by joining, in the proper order, the midpoints of the sides of the first square. A third square is formed by joining the midpoints of the second square, and so on. Find the side of the eleventh square. a. 1/2 in. c. 1/4 in. b. 1/3 in. d. 1/5 in.

378. The length of the side of a square is 12 inches. A second square is inscribed by connecting the midpoints of the sides of the first square, a third by connecting the midpoints of the sides of the second, and so on. Find the sum of the areas of the infinitely many square formed, including the first. a. 72 sq. in. c. 288 sq. in. b. 576 sq. in. d. 144 sq. in 379. A silo of given volume is to be made in the form of a cylinder surmounted by a hemisphere. Find the proportions if the total cost of floor walls and roof all made of the same material. a. H = 2R c. H = R/2 b. H = 3R d. H= R 380. Equal-sized spheres are contained in a regular tetrahedron such (with equilateral triangle for each face) such that along each of the 6 sides of the tetrahedron, there are 4 spheres in line. How many total spheres are there? a. 14 c. 22 b. 20 d. 18 381. A right circular cylinder has a 10 cm diameter and a 10 cm height. Determine the moment of inertia of the cylindrical volume relative to its center, in centimeters. a. 14, 399 c. 13, 090 b. 11, 900 d. 10, 818

ANALYTICAL GEOMETRY 382. Find the major axis of the ellipse x 2 + 4y2 – 2x – 8y + 1 = 0. a. 2 c. 4 b. 10 d. 6 383. Find the location of the focus of the parabola y2 + 4y – 4x -8 = 0. a. (2.5, -2) c. (2, 2) b. (3, 1) d. (-2, -2) 384. What is the x-intercept of the line passing through (1, 4) and (4, 1)? a. 4.5 c. 6 b. 5 d. 4 385. Given the polar equation r = 5 sin . Determine the rectangular coordinates (x, y) of a point in the curve when  is 30. a. (2. 17, 1.25) c. (2.51, 4.12) b. (3.2, 1.5) d. (6, 3) 386. Find the area bounded by the line x – 2y + 10 = 0, the x-axis, the y-axis, and x = 10. a. 75 c. 100 b. 50 d. 25 387. If y = 4 cos x + sin 2x, what is the slope of the curve when x = 2? a. -2.21 c. -3.25 b. -4.94 d. 2.21 388. Find the distance of the line 3x + 4y = 5 from the origin. a. 4 c. 1 b. 2 d. 3 389. The center of a circle is at (1, 1) and one point on its circumference is (-1, -3). Find the other end of the diameter through (-1, -3). a. (2, 4) c. (3, 6) b. (3, 5) d. (1, 3)

390. Find the polar equation of the circle, if its center is at (4, 0) and the radius is 4. a. r – 8 cos u = 0 c. r – 12 cos u = 0 b. r – 6 cos u = 0 d. r – 4 cos u = 0 391. Find the area bounded by the parabolas y = 6x – x2 and y = x2 – 2x. Note: The parabolas intersect at points (0, 0) and (4, 8). a. 44/3 sq. units c. 74/3 sq. units b. 64/3 sq. units d. 54/3 sq. units 392. Locate the point of inflection of the curve y = f(x) = x 2ex a.  2  3 c.  2  2 b. 2  2

d. 2  3

393. Find the radius of the curvature of y = sin x at (, 1) a. 2 square root of 3 c. 1 b. 2 d. square root of 3 394. Find the equation of one of the medians of a triangle with vertices (0, 0), (6, 0) and (4, 4). a. 2x – y = 10 c. x + 10y = 4 b. 2x – 5y = 4 d. 2x – 5y = 0 395. Determine the nature of the surface 6x2 – 3y2 – 2z2 – 12x – 18y + 16z = 83 a. Ellipsoid c. Hyperboloid b. Sphere d. Elliptic Paraboloid 396. Find the distance between the line x + y = 2 and the given point (1/2, 1/3). 7 12 2 2 a. c. 7 12 5 6 2 2 b. d. 6 5 397. Find the equation of the normal to the circle x 2 + y2 = 25 at (3, -4). 4 4 a. y  x  6 c. y   x  8 3 3 2 4 x b. y   x  6 d. y  3 3

398. Find the equation of a line perpendicular to y = 1 through A(-1, 1). a. x = 1 c. x + 1 = 0 b. x = -1 d. x – 2 = 0 399. Find the dimensions of the largest rectangular parallelepiped that can be inscribed in the ellipsoid 16x2 + 4y2 + 9z2 = 144. a. 8/ 3 , 4/ 3 , 12/ 3 c. 6/ 3 , 12/ 3 , 8/ 3 b. 8/ 3 , 6/ 3 , 16/ 3

d. 4/ 3 , 6/ 3 , 8/ 3

400. What are the vertical and non-vertical asymptotes for xy = x2 – ln x? a. y = x, x =0 c. y = -x, x = 0 b. x = 2y, y = 0 d. x = 4y, y = 0 401. What is the radius of a circle with the following equation? x2 – 6x + y2 – 4y -12 = 0 a. 3.5 c. 5.0 b. 4.0 d. 6.0 402. Classify the graph of the equation x 2 + xy + y2 – 6 = 0 as a a. circle c. ellipse b. parabola d. hyperbola 403. The graph of 3x2 – 6xy + 5y2 – x + 3y + 4 = 0 is a. circle c. parabola b. ellipse d. hyperbola 404. Find the length of the common chord of the curves x2 + y2 = 64 and x2 + y2 - 16x = 0 a. 13.86 c. 22.64 b. 15.53 d. 20.46 405. Find the rectangular coordinates of [3 (square root of 2), 45] a. (3, 3) c. (1, 1) b. (2, 2) d. (3, 2) 406. Given a parabola x2 = 4y, a line passes through point A(4, 4) and the focus of the parabola, find the length of the chord from A to B, where B is a point on the curve. a. 4.83 c. 5.96 b. 5.36 d. 6.25

407. Find the polar equation of the circle with radius a = 3/2 and the center in polar coordinates (3/2, ). 3 a. r = cos θ c. r = -3cos θ 2 1 b. r = cos θ d. r = -2cos θ 2 408. Find the area bounded by the curve y = 6x + x 2 – x3, x-axis and the 1st quadrant. a. 12 3/5 c. 10 2/3 b. 15 3/4 d. 13 1/2 409. Given the points of a triangle A(1, 0), B(9, 2) and C(3, 6). Find the intersection at which the median will meet.  13 8   8 13  a.  c.  , ,    3 3 5 5 

 8 13  b.  ,  3 3 

 13 8  d.  ,   5 5

410. Find the distance from point A(3, 4) and B(4, 3) along the arc of the circle x2 + y2 = 25. a. 1.33 c. 1.58 b. 1.42 d. 1.64 411. Change y = x from rectangular to polar form. a. theta = 2 or 3/2 c. theta =  /4 or 5 /4 b. theta = /3 or 4/3 d. theta =  or 3 412. Find the equation of one of the medians of a triangle with vertices (0, 0), (6, 0) and (4, 4). a. 2x – y = 10 c. x + 10y = 4 b. 2x – 5y = 4 d. x + 2y = 6 413. Find the equation of the bisector of the pair of acute angles formed by the line 4x + 2y = 9 and 2x – y = 8. a. y + 4x – 25 = 0 c. y – 8x – 25 = 0 b. y + 8x – 25 = 0 d. 8x - 25 = 0 414. Determine the equation of the line normal to the graph x2 + 3xy + y2 = 5 at the point (1, 1). a. x + y = 0 c. x + y = 1 b. x - y = 0 d. x - y = 1

415. Given the equation x2 – 2x + 3y2 + 6y = 0, find the length of the diameter whose slope is 1. a. 2 3 c. 3 2 b. 2 2

d. 3 3

416. Find the equation of the line, when the x-intercept is a = -3, and y-intercept b = 4. x y x y a. c.   1  1 4 3 3 4 x y x y b. d.  1  1 3 4 4 3 417. Three circles of radius 2, 4 and 6 are externally tangent to each other; find the radius of the circle that passes through the centers of the three circles. a. 3 c. 5 b. 4 d. 6 418. Given the hyperbola: xy = 1. Determine the new equation of this hyperbola if the x, y, axes are rotated about the origin by 45 degrees clockwise. a. y2 – x2 = 1 c. x2 – y2 = 1 b. x2 - y2 = 2 d. y2 – x2 = 2

 radian, and the chord of the circle 4 subtended by this angle is 12 2 cm. Find the radius of the circle. a. 10 cm c. 14 cm b. 12 cm d. 16 cm

419. An inscribed angle is

420. A regular n-sided polygon is inscribed in a circle of radius r, determine the ratio of the perimeter of the polygon to the diameter of the circle as n increases to infinity? a. 4.17 c. 1.57 b. 6.28 d. 3.14 421. A hemispherical bowl of radius depth of 5 cm. Find the volume a. 455 cm3 b. 655 cm3

10 cm is filled with water to a of the water. c. 434 cm3 d. 347 cm3

422. Find the length of the common external tangent of two circles of radii 5 cm and 12 cm, respectively, if the distance between their centers is 25 cm. a. 24 cm c. 26 cm b. 25 cm d. 27 cm 423. A chord is 36 cm long and its midpoint is 36 cm from the midpoint of the longer arc. Find the area of the circle. a. 1595 cm2 c. 1590 cm2 b. 1593 cm2 d. 1598 cm2 424. Given the parabola x2 + 4y, a line passes through point A(4,4) and the focus of the parabola, find the length of the chord from A to B, where B is a point on the curve. a. 6.43 c. 5.36 b. 5.9 d. 4.83 Ans. 6.25 425. An arch is in the form of an inverted parabola and has span of 12 feet at the base and a height of 12 feet. Determine the equation of the parabola and give the vertical clearance 4 feet from the vertical centerline. a. 7.33 ft. c. 5.33 ft b. 6.00 ft d. 6.67 ft 426. Determine the equation of the line through (3,4) which forms, with the positive y axes, the triangle with the last area. a. 4x + 5y =32 c. 4x +3y = 24 b. 3x + 4y = 25 d. 2x + 3y = 18 427. Determine the equation describing the locus of points P (x,y), such that the sum of the distances between P and (-5,0) and between P and (5,0) is constant at 20 units. a. (x/10)² + (y/8.66)² = 1 c. (x/8)² + (y/10)² = 1 b. (x/10)² + (y/8)² = 1 d. (x/8.66)² + (y/10)² = 1 428. Determine the distance between coordinates (8,9) and (-9,-8) a. 28.85 units c. 24.04 units b. 16.70 units d. 20.03 units 429. What is the slope of a curve y(x2) – 4 = 0 at (4, 4)? a. 8 c. -2 b. 4 d. –4

430. Two lines pass through (5,5) and separate tangents to the circles C: x² + y² = 9. Determine the distance between the x-intercepts of the two lines. a. 13.21 c. 12.01 b. 10.81 d. 14.41 431. A tetrahedron is a regular solid with equilateral triangles for each of the 3 surface. If each sides is 10 cm. What is the volume of the tetrahedron? a. 67.21 cu cm c. 83.33 cu cm b. 91.67 cu cm d. 73.94 cu cm Ans. 117.85 cu. cm, but choose (c). 432. Find the equation of the line that passes through (2, -3) and has a slope of 5. a. 5x + 3y = 13 c. 5x + y = 13 b. 5x - 3y = 13 d. 5x - y = 13 433. Given the line 2x = 5y + 9. Find its equation in x and y intercept form. x y x y  1   1 a. c. (9/2) (9/5) (9/2) (9/5) x y x y  1  1 b. d. (9/2) ( 9/5) (9/2) (9/5) 434. Find the equation of the line in slope-intercept form, if slope is -3 and y intercept is 4. a. y = 3x – 4 c. y = -3 + 4 b. y = -3x - 4 d. y = 4x - 3 435. The radii of the two circles that are tangent externally are 8 and 3 m., respectively. What is the distance between the point of tangency of one of their common external tangents? a. 6.90 c. 8.90 b. 7.80 d. 9.80 436. Three circles C1, C2 and C3 are externally tangent to each other. Center-to-center distances are 15 cm between C1 and C2, 12 cm between C2 and C3, and 9 cm between C3 and C1. Determine the total areas of the circles. a. 98.86 sq. cm c. 103.96 sq. cm b. 99.98 sq. cm d. 100.76 sq. cm

437. Find the distance of the directrix from the center of an ellipse if its major axis is 10 and its minor axis is 8. a. 8.1 c. 8.5 b. 8.3 d. 8.7 438. Two vertices of triangle are (2, 4) and (-2, 3) and the area is 2 sq. units, the locus of the third vertex is a. 4x – y = 14 c. x + 4y = 12 b. 4x + 4y = 14 d. x – 4y = -10 439. Find the curl of F = i(x2 + yz) + j(y2 + zx) + k(z2 + xy). a. xi – yj c. 1 b. 0 d. 2xi + yj + zk 440. Find the distance of the centroid from the y-axis, bounded by x = 10, y = x, and y = -x. a. 6.67 c. 5.51 b. 6.06 d. 7.33 441. Find the x-intercept of a line tangent to y= x ln x at x = e. a. 1.500 c. 1.0 b. 1.750 d. 1.359 442. Find the greatest area of a rectangle inscribed in a given parabola y = 16 – x2 and the x-axis. a. 24.63 s.u. c. 98.53 s.u. b. 49.27 s.u. d. 46.87 s.u. 443. A trapezoidal area has the following vertices on the x-y plane: A(6.0, 1.5), B(10.0, 2.50), C(10.0, -2.50) and D(6.0, -11.5). With all coordinates in cm. If this area is rotated about the yaxis, determine the generated volume in cu. cm. a. 746 c. 821 b. 903 d. 578 444. Find the equation of the pair of acute angles formed by the line 4x + 2y = 9 and 2x – y = 8. a. y + 4x – 25 = 0 c. y - 8x – 25 = 0 b. y + 8x – 25 = 0 d. 8x – 25 = 0 445. Find the equation of a line parallel to y= 1 through (-1, 1). a. y = 1 c. y = 2 b. y = -1 d. y = -2

446. Two circles of different radii are concentric. If the length of the chord of the larger circle that is tangent to the smaller circle is 40 cm., find the different in area of the two circles. a. 350 sq. cm c. 500 sq. cm. b. 400 sq. cm. d. 550 sq. cm. 447. The towers of a 60 meter parabolic suspension bridge are 12 m high and the lowest point of the cable is 3 m above the roadway. Find the vertical distance from the roadway to the cable at 15 m from the center. a. 3 m c. 6 m b. 5 m d. 8 m 448. From a point outside and equilateral triangle the distances of the vertices are 10 m, 18 m, and 10 m respectively. Find the length of the side of the triangle. a. 16.95 m c. 18.95 m b. 17.95 m d. 19.95 m 449. Two lines pass through (5, 5) and separate tangents to the circle C: x2 + y2 = 9. Determine the distance between the x-intercepts of the two lines. a. 13.21 c. 12.01 b. 10.81 d. 14.41 450. Find the equation of a line whose x-intercept a = 2, and yintercept b = 3. x y x y a. c.  1  1 3 2 3 2 x y x y b. d.  1  1 2 3 2 3 451. Determine the equation of the line through (3, 4) which forms, with the positive x and positive y axes, the triangle with the least area. a. 4x + 5y = 32 c. 4x + 3y = 24 b. 3x + 4y = 25 d. 2x = 3y = 18 452. What is the length of the arc intercepted by a central angle of 1/3 radian on a circle of radius 30 cm? a. 5cm c. 8.32cm b. 10cm d. 12.44cm

453. Find the center of curvature x cube + y cube = 4xy at the point (2, 2). a. (1/3, 3/4) c. (-1/3, -1/4) b. (7/4, 7/4) d. (1/3, 1/4) 454. What is the slope of the equation 2x – 8y – 5 = 0. a. 1/4 c. -1/4 b. 4 d. -4 455. Find approximately the difference between the areas of two spheres whose radii are 4 feet and 4.5 feet. a. 1.4  sq. ft. c. 1.2 sq. ft. b. 1.6  sq. ft. d. 1.8 sq. ft. 456. Find the point where the normal to y  x  x crosses the y-axis. a. y = 23 c. y = 11 b. y = 5.75 d. y = 9.2

at (4, 6)

457. Find the equation of a line perpendicular to y = 1 through A(-1, 1). a. x = 1 c. y – 1 = 0 b. x = -1 d. x – 2 = 0 458. The points A (0, 0), B(5, 1), C(1, 3) are vertices of a parallelogram. Find the coordinates of the fourth vertex if BC is the diagonal. a. (6, 3) c. (5, 4) b. (6, 5) d. (6, 4) 459. Find the slope of the tangent line of yx2 – 4 = 0 passing thru (4, 4). a. 4 c. 8 b. -4 d. -2 460. Find the equation of the normal to the circle x 2 + y2 = 25 at (3, -4). 4 4 a. y = x – 6 c. y = - x + 8 3 3 2 4 b. y =  x - 6 d. y = - x 3 3

461. Given: Radius = 30 cm Central angle = 1/3 rad Determine the length of the arc. a. 5 cm c. 20 cm b. 10 cm d. 12 cm 462. What is the slope of the tangent line of x2y + sin y = 12pi at P(2 3pi)? a. -4 pi c. -pi b. -3 pi d. -2 pi 463. What is the slope of the line 2x – 8y = 6. a. 1/2 c. 1/6 b. 1/4 d. 1/3 464. A circle is tangent to the line 3x – 4y – 4 = 0 at the point (-4, -4) and the center is on the line x + y + 7 = 0. Find the equation of the circle. a. x2 + y2 – 6y + 4 = 0 c. x2 + y2 + 14y + 24 = 0 b. x2 + y2 + 4x + 16 = 0 d. x2 + y2 – 6y + 14 = 0 465. Find the equation of the circle x2 + y2 – 2x – 6y + 4 = 0 if the origin is moved to O’(2, 3). a. x2 + y2 – 6y + 4 = 0 c. x2 + y2 + 2x - 5 = 0 b. x2 + y2 – 2x + 10 = 0 d. x2 + y2 – 6y - 5 = 0 466. Find the point to which the origin must be translated so that the transformed equation of 2x 2 + 20x +y2 – 4y -12 = 0 will have no first-degree terms. a. (5, -2) c. (-5, 2) b. (-2, 5) d. (2, -5) 467. What is the circumference of the circle with the following equation? x2 – 6x + y2 – 4y -12 = 0 a. 5 c. 15 b. 10 d. 20 468. Identify the generated conic 2xy – x + y + 6 = 0. a. hyperbola c. parabola b. circle d. ellipse

469. Given the equation x2 – 2x + 3y2 + 6y = 0, find the length of the diameter whose slope is 1. a. 2 3 c.3 2 b. 2 2

d. 3 3

470. A mirror for a reflecting telescope has the shape of a (finite) paraboloid of diameter 8 inches and depth 1 inch. How far from the center of the mirror will the incoming light collect? a. 2 in c. 16 in b. 8 in d. 4 in 471. Find the equation of the line, when the x-intercept a = -3, and y-intercept b = 4. x y x y a. c.  1  1 4 3 3 4 x y x y b. d.  1  1 3 4 4 3 472. Find the distance between the line x + y = 2 and a given point (1/2, 1/3).  12 7 a. c. 2 2 12 7 5 6 b. d. 2 2 6 5 473. Find the locus of a point which moves so that its distance from (4,0) is equal to two-thirds its distance from the line x=9. a. 6x2 + 10y2 = 200 c. 4x2 + 6y2 = 120 b. 3x2 + 6y2 = 150 d. 5x2 + 9y2 = 180 474. Find the distance from point A (3, 4) and B(4, 3) along the arc of the circle x2 + y2 = 25. a. 1.33 c. 1.58 b. 1.42 d. 1.64 475. Find the equation of the plain passing through the points P(2, -3, 1), P’(5, -3, -5) and perpendicular to the plane x – 2y + 5z + 20 = 0. a. x – 2y + 55z = 0 c. x – 2y + 5z + 15 = 0 b. 4x + 7y + 2z + 11 = 0 d. 4x + 7y + 2z – 11 = 0

476. Determine the nature of the surface whose equation is 2x2 – 3y2 + z2 + 8x + 16y – 2z = 30. a. paraboloid c. ellipsoid b. hyperboloid d. sphere 477. Find the equation of the line through (13, 5) which makes an angle of 45° with the line 2x + y = 12. a. 3x – y = 34 c. x + 3y = 4 b. 3x + y =8 d. x + 3y = 6 478. The line through the points (4,3) and (-6,0) intersects the line thru (0,0) and (-1,5). Find the angle of intersection. a. 84°, 94° c. 87°, 92° b. 85°, 95° d. 80°, 90° 479. Find the equation of the circle whose center is on the x-axis and which passes through the points (1,3) and (4,6). a. x2 + y2 – 12x + 4 = 0 c. x2 + y2 – 12x + 14 = 0 2 2 b. x + y – 14x + 4 = 0 d. x2 + y2 – 14x + 8 = 0 480. Find the minimum distance from the point (4,2) to the parabola y2 =8x. a. 2 3 units c. 4 2 units b. 2 2 units

d. 3 3 units

481. A curve passing through the origin has a slope of 2x at any point of the curve. The equation of the curve is a. x2 + y2 = 4 c. y2 = 2x b. y = x2 d. y = 2x + c 482. Two lines pass through (5,5) and separate tangents to the circle C: x2 + y2 = 9. Determine the distance between the xintercepts of the two lines. a. 13.21 c. 12.01 b. 10.81 d. 14.41 483. Find the equation of the hyperbola which has a center at (0, 0), transverse axis along the x-axis, a locus at (5, 0) and a transverse axis of 6. a. x2/9 – y2/16 = 1 c. x2/16 – y2/25 = 1 b. x2/9 – y2/25 = 1 d. x2/16 – y2/9 = 1

484. Find the equation of the circle whose center is on the x-axis and which passes through the points (1, 3) and (4, 6). a. x2 + y2 – 12x + 4 = 0 c. x2 + y2 – 12x + 14 = 0 b. x2 + y2 – 14x + 4 = 0 d. x2 + y2 – 14x + 8 = 0 485. A line passes thru (1, -3) and (4, -2). Write the equation of the line in slope-intercept form. a. y = -x – 2 c. y – 2 = x b. y = x – 4 d. y – 4 = x 486. If y = 4 cos x + sin 2x, what is the slope of the curve when x = 2? a. -4.94 c. 2.21 b. -3.25 d. -2.21 487. Given the line L: 7x + 5y – 35 = 0. Determine the line M parallel to L and 7 units distant from L. From these, determine the algebraic sm of the intercepts of M. a. 8.65 c. 7.78 b. -8.65 d. -7.78 488. If the line connecting coordinates (x, 7) and (10, y) is bisected at (8, 2), determine x and y. a. x=7, y= -2 c. x=6, y= -3 b. x=7, y=2 d. x=6, y=3 489. Find the locus of a point which moves so that its distance from (4, 0) is equal to two-thirds its distance from the line x = 9. a. 6x2 + 10y2 = 200 c. 4x2 + 6y2 = 120 b. 3x2 + 6y2 = 150 d. 5x2 + 9y2 = 180 490. A curve passing through the origin has a slope of 2x at any point of the curve. The equation of the curve is a. x2 + y2 = 4 c. y2 = 2x b. y = x2 d. y = 2x + C 491. Find the point of division of the line segment joining A(4, 5) and B(-3, -2) if it is divided into two segments one of which is three times as long as the other. a. (9/4, 12/14) c. (9/4, 13/4) b. (3/4, 9/4) d. (3/4, 10/4)

492. Determine the nature of the surface whose equation is 2x2 3y2 + z2 + 8x + 16y a. Sphere c. Hyperboloid b. Paraboloid d. Ellipsoid 493. Find the equation of the line through (13, 5) which makes an angle of 45 with the line 2x + y = 12. a. x + 3y = 6 c. 3x + y = 8 b. 3x – y = 34 d. x + 3y = 4 494. Determine the point of division of the line segment from A(5, 6) to B(-3, -2) that divides this line segment starting from A, into two parts in the ratio 1:3. a. (-1, 1) c. (1, 3) b. (3, 4) d. (0, 2) 495. A circle has its center at (0, 0) and its radius is 10 units. Determine the equations of the lines through (15, 15) and tangent to the circle. a. x – 2.897y + 34.450 = 0 and x – 0.303y – 10.450 = 0 b. x – 3.297y + 34.450 = 0 and x – 0.303y – 10.450= 0 c. x – 2.897y + 34.450 = 0 and x – 0.353y – 10.450 = 0 d. x – 2.897y + 34.450 = 0 and x – 0.353y – 10.450 = 0 496. Determine the tangent to the curve 3y 2 = x3 at (3, 3) and calculate the area of the triangle bounded by the tangent line the x-axis, and the line x = 3. a. 2.50 sq. units c. 4.00 sq. units b. 3.50 sq. units d. 3.00 sq. units 497. Determine the radius of the sphere whose equation is: x2 + y2 + z2 – 2x + 8y + 16z + 65 = 0 a. 5 units c. 3 units b. 4 units d. 6 units 498. Through the point (3, 3), determine the equation of the line making an angle of 30°with the y-axis. a. y – 3 = 1.1.92 (x – 3) c. y – 3 = 1.732 (x – 3) b. y – 3 = 1.428 ( x – 3) d. y – 3 = 1.000 (x – 3) 499. Points that lie on the same plane are said to be: a. collinear c. dihedral b. coplanar d. parallel

500. If y =

sinx , what is the slope of the curve when x is 2 x2  1

radians? a. -0.51 b. 6.15 Ans. -0.23

c. 2.1 d. -0.53

501. Determine the tangent to the curve 3y 2 = x3 at (3,3) and calculate the area of the triangle bounded by the tangent line, the x-axis, and the line x = 3. a. 2.50 sq. units c. 4.00 sq. units b. 3.50 sq. units d. 3.00 sq. units 502. A hyperbola has the equation: xy = 1 Determine the equation of this hyperbola if the x, y-axes are rotated 45 degrees counterclockwise. a. 0.25 x’2 – 0.25 y’2 = 1 c. 1.00 x’2 – 1.00 y’2 = 1 b. 2.00 x’2 – 2.00 y’2 = 1 d. 0.50 x’2 – 0.50 y’2 = 1 503. A circle is circumscribed about a triangle with vertices at (-2, 3), (5, 2), and (6, -1). Determine the equation of the circle. a. x2 + y2 – 2x + 2y + 23 = 0 b. x2 + y2 + 2x + 2y + 23 = 0 c. x2 + y2 – 2x + 2y - 23 = 0 d. x2 + y2 + 2x - 2y - 23 = 0 504. If the line connecting coordinates (x, 7) and (10, y) is bisected at (8, 2), determine x and y. a. x = 6, y = 3 c. x = 7, y = 2 b. x = 7, y = -2 d. x = 6, y = -3 505. Through the point (3, 3), determine the equation of the line making an angle of 30 degrees with the y-axis. a. y – 3 = 1.192(x – 3) c. y – 3 = 1.732(x – 3) b. y – 3 = 1.428(x – 3) d. y – 3 = 1.000(x – 3) 506. A curve, in rectangular coordinates, is to have a slope equal to the ratio x/y at any of its point. If this curve must pass through (1, 0), determine the equation of the curve. a. x = y + 1 c. y2 – x2 = 1 b. x = y – 1 d. x2 – y2 = 1

DIFFERENTIAL CALCULUS 507. dy = x2dx. What is the equation of y in terms of x if the curve passes through (1, 1)? a. x2 – 3y + 3 = 0 c. x3 + 3y2 + 2 = 0 b. x3 – 3y + 2 = 0 d. 2y + x + 2 = 0 508. Evaluate

d 1   dx  x6 

x 6x6 1 b.  6x7 6 Ans. 7 x

a.

1 6x5 1 d. 6x7

c.

509. Differentiate y = sec (x2 + 2) a. 2x cos (x2 + 2) b. –cos (x2 + 2) cot (x2 + 2) c. 2x sec (x2 + 2) tan (x2 + 2) d. cos (x2 + 2) 510. Differentiate y = log10(x2 + 1)2 a. log10 e(x) (x2 + 1) b. 4x(x2 + 1) 511. Differentiate (x2 + 2)1/2 (x 2  2)1/2 a. 2 x b. 2 (x  2)1/2

log10 e (x 2  1) d. 2x(x2 + 1) c. 4x

c.

2x (x  2)1/2 2

d. (x 2  2)3/2

512. A poster is to contain 300 cm2 of printed matter with margins of 10 cm at the top and bottom and 5 cm at each side. Find the overall dimensions if the total area of the poster is minimum. a. 27.76, 47.8 cm c. 22.24, 44.5 cm b. 20.45, 35.6 cm d. 25.55, 46.7 cm

513. The cost of fuel in running a locomotive is proportional to t he square of the speed and is $25 per hour for a speed of 25 miles per hour. Other costs amount to $10 per hour, regardless of the speed. What is the speed which will make the cost per mile a minimum? a. 40 c. 50 b. 55 d. 45 514. The coordinates (x, y) in ft. of a moving particle P are given by x = cos t - 1 and y = 2 sin t + 1, where t is the time in seconds. At what extreme rates in ft per sec is P moving along the curve? a. 3 and 2 ft per sec c. 2 and 0.5 ft per sec b. 3 and 1 ft per sec d. 2 and 1 ft per sec 515. The rotating beacon of a lighthouse makes 0.2 revolutions per second. The nearest at point P along the straight shoreline from the beacon is 200 ft. What is the rate of change, that the ray of light makes along the shore at a point 100 ft from P? a. 125 c. 200 b. 100 d. 150 516. If the x intercept of the tangent to the curve y = e-x is increasing at a rater of 4 units per second, find the rate of change of the y intercept when the x intercept is 6 units. a. -0.135 c. -0.248 b. -0.05 d. -0.368 517. The dimensions of a rectangle are continuously changing. The width increases at the rate of 3 in/sec while the length decreases at the rate of 2 in/sec. At one instant the rectangle is a 20-inch square. How fast is its area changing 3 seconds later? a. –15 c. -11 b. -16 d. -8 518. A trough filled with water is 2 m long and has a cross section in the shape of an isosceles trapezoid 30 cm wide at the bottom, 60 cm wide at the top, and a height of 50 cm., if the trough leaks water at the rate of 2000 cm 3/min, how fast is the water level falling when the water is 20 cm deep? a. -5/21 c. -4/26 b. -7/20 d. -8/21

x2 is discontinuous? x 1 c. -1 d. 2

519. At what value of x does the equation a. 0 b. 1 520. A bicycle with 20-in wheels is mi/hr. Find the angular velocity minute. a. 190 b. 252

traveling down a road at 15 of the wheel in revolutions per c. 180 d. 342

521. A rectangular metal sheet of 12 inches wide is used to make a rain gutter. It needs to fold up equal widths along the edges perpendicularly to form a rain gutter. What a dimension should be folded up at each side to yield a maximum carrying capacity? 1 a. 3 c. 2 2 3 b. 2 d. 3 4 522. A function is given. What value of x maximizes y? y2 + y + x2 – 2x =5 a. –1 c. 1 b. 1/2 d. 5 523. Find the maximum area of a rectangle inscribed on the curve y= if the base of a rectangle lies on the x-axis and the two vertices on the curve. a. a squared c. 4 (a squared) b. 3 (a squared) d. 2 (a squared) 524. A balloon was released from the ground level at a point 160 m from an observer on ground level. If the balloon goes straight up at the rate of 4 m/s, find the rate of change of velocity of separation after 30 sec. a. 0.0512 c. 0.0482 b. 0.0613 d. 0.0385 525. Find the limit of (-1)n(2)-n+ as n approaches infinity. a. 1 c. indeterminate b. 0 d. infinity

526. A kite is flying 100 ft above the ground moving in a strictly horizontal direction at a rate of 10ft/sec. How fast is the angle between the string and the horizontal changing when there is 300 ft of string out? a. -1/90 rad/s c. -1/80 rad/s b. -1/60 rad/s d. -1/70 rad/s 527. A long piece of galvanized iron 60 cm wide is to be made into a trough by bending up two sides. Find the width of the base if the carrying capacity is maximum? a. 30 c. 40 b. 20 d. 50 528. Two poles support an electrical line at the height. The left pole is 50 m tall while the right pole is 20 m tall with the ground sloping upward. The line is described by the equation  x  y =  x  100  50 . The x and y coordinates are set such  50  that the origin lies at the foot of the left pole. What is the minimum vertical distance of the line from the ground? a. 14.59 c. 19.38 b. 23.26 d. 24.56 529. Find the limit of a. 1/3 b. 1/2

x3  2x  5 as x approaches infinity. 2x3  7 c. 1/4 d. 1/5

530. The rotating beacon of a lighthouse makes 1 revolution in 15 seconds. The nearest at point P along the straight shoreline from the beacon is 200 ft. What is the rate of change, in ft/min, that the ray of light makes along the shore at a point 400 ft from P? a. 23,155 c. 21,533 b. 25,133 d. 21,355 531. The height (in feet) at any time t (in seconds) of a projectile thrown vertically is h(t) = 16t2 + 256t. What is the rate of h changing when t = 6? a. 960 c. 160 b. 128 d. 64

532. A spherical balloon is being inflated with r = 3 3 t as t is greater than zero and t is less than or equal to 10. Find the rate of change of volume in cubic cm at t = 8. a. 12 c. 36 b. 48 d. 24 533. Determine (1 – n)1/n as n approaches zero n non-negative. a. 0 c. 1 b. 0.368 d. infinity 534. Determine the csc (n x ) divided by (n x ) as n approaches to zero. a. 0 c. infinity b. 1 d. indeterminate 535. An area in the x, y-plane is bounded by the following lines: x = 0 (y-axis) x + 4y = 30 y = 0 (x-axis) 4x + y = 30 The linear function z = x + y attains its maximum value within the bounded area only at one of the vertices (intersections of the above lines). Determine the maximum value of z. a. 10 c. 6 b. 12 d. 8 536. A point travels as described by the following parametric equations, x = 10t + 10cos (t), y = 10t + 10sin (t), z = 10t, where x, y, z are in meters, t in seconds, all angles in radians. The vector locating the body at any time is r = ix + jy + kz. Determine the magnitude of the velocity of the body in meters per second at a time t = 0.25 second. a. 33.07 c. 35.87 b. 34.57 d. 33.85 537. With the x, y region defined as follows: x  0, y  0, (2x + y)  4, (x + 2y)  4 Determine the maximum value of the function f(x, y) = 3x + 4y. a. 11.333 c. 15.333 b. 7.333 d. 9.333 538. Determine (1-n)(1/n) as n approaches zero n non-negative. a. 0 c. 1 b. 0.368 d. infinity

539. A projectile has an initial upward speed of 1,000 feet per second which is steadily decreased by gravity at the rate of 32.2 feet per second per second, so that the upward speed at a subsequent time id dy/dt = 1000 – 32.2t ft/sec. If the initial position of the projectile is 200 ft above ground, what is the maximum height that the projectile will attain? a. 18,874 ft c. 22,648 ft b. 15,728 ft d. 13,107 ft 540. Maximize z= (x/5) + (y/3) subject to (x/5)² + (y/3)² = 1 a. Z = 1.27 c. Z = 1.54 b. Z =1.69 d. Z = 1.40 541. A right triangle has a fixed hypotenuse of 30 cm. And the other two sides allowed to vary. Determine the largest possible area of the triangle. a. 225 sq. cm c. 243 sq. cm b. 234 sq. cm d. 216 sq. cm 542. What is the limiting value of the following function as n approaches infinity? y = [1 + (1/n)]ⁿ a. indeterminate c. 0 b. 2.7183 d. infinity 543. A body move according the parametric equations: x = (10-t) sin (2t/10), y = (10 – t) cos (t/10), z = (10 –t) where x, y, z are in meters, t in seconds, all angles are in radians. The vector locating the body at any time t is r = ix + jy + kz. Determine the magnitude of the velocity of the body, in meters per second at time t = 5 seconds, by computing for the vector dr/dt. a. 15.76 m/sec c. 13.14 m/sec b. 18.92 m/sec d. 10.95 m/sec

Ans. 2.1136 544. The velocity of a particle in three dimensional motion is defined by v = m/s where t is in seconds. Determine the magnitude of the position vector r at t = 6 seconds, if initially x0 = 0, y0 = -200 m and z0 = -100 m. a. 3 c. 1 b. -2 d. 2

545. Determine (1 + n)(1/n) as n approaches zero n non-negative. a. 1 c. infinity b. 0 d. 2.7183 546. A particle’s position (in inches) along the x axis after t seconds of travel is given by the equation x = 24t 2 – t3 + 10. What is the particle’s average velocity during the first 3 seconds of travel? a. 72 c. 32 b. 48 d. 63 547. Find the rate of change of the area of a square with respect to its side, when x = 5. a. 12 c. 10 b. 14 d. 8 548. A fencing material has a limited length of 60 ft. What is the largest triangular area than can be fenced in? a. 190.53 sq. ft. c. 173.21 sq. ft. b. 181.87 sq. ft. d. 164.74 sq. ft. n

1  549. Determine the limiting value of 1   as n approaches zero. n  a. 1 c. 2.721 b. 0 d. infinity

 180  550. Evaluate lim n sin  . n   n  a. 2 b. infinity

c. 3.14 d. indeterminate

551. A ladder 4.5 m long leans against a vertical wall. If the top slides down at 0.6 m/s, how fast is the angle of elevation of the ladder decreasing, when the lower end is 3.8 m from the wall? a. -0.14 rad/sec c. -0.16 rad/sec b. -0.15 rad/sec d. -0.17 rad/sec 552. Find the limit of (-1 raised to n)(2 raise to –n) as n approaches to infinity. a. 1 c. indeterminate b. 0 d. none of the above

553. A funnel in the form of cone is 10 cm across the top and 8 cm deep. Water is flowing into the tunnel at the rate of 12 cm3/sec and out at the rate of 4 cm 3/sec. How fast is the surface of the water rising when it is 5 cm deep. a. 0.26 cm/sec c. 0.14 cm/sec b. 0.32 cm/sec d. 0.40 cm/sec 554. Pipes between stations as indicated have the following maximum flow capacities, in cubic meters per second: Between A and B 40.0; Between B and C 30.0; Between A and C 20.0 What is the maximum possible flow rate from A to C, in cubic meters per second, without exceeding the above maximum flow capacities? a. 60 c. 50 b. 30 d. 40 555. A light is placed on the ground 32 feet from a building. A man 6 feet tall walks from the light toward the building at a rate of 6 feet per second. Find the rate at which his shadow on the building is decreasing when he is 16 feet from the building. a. 4 ½ ft/sec c. 4 ft/sec b. 5½ ft/sec d. 5 ft/sec 556. The base diameter and the altitude of a right circular cone are observed at a certain instant to be 10 and 20 inches, respectively. If the lateral area is constant and the base diameter is increasing at a rate of 1 inch per minute, find the rate at which the altitude is decreasing. a. 3.75 in/min c. 1.25 in/min b. 2.25 in/min d. 4.75 in/min 557. A rocket is launched vertically and is tracked by an observing station located on the ground 100 m from the launch pad. Suppose that the elevation angle  of the line of the sight to the rocket is increasing 15 per second when  = 60. What is the velocity of the rocket at this instant? 100 76 a.  c.  3 3 80 112 b.  d.  3 3

558. If 24 mango trees are planted per hectare, each tree will produce 600 mangoes per year. For each additional tree planted, the number of mangoes produced per tree diminishes by 12. What is the number of trees per hectare for maximum harvest? a. 45 c. 37 b. 28 d. 38 559. A piece of wire 36 cm long is cut to make and equilateral triangle and a rectangle with length is twice its width. Find the length of the rectangle so that the sum of the area of a triangle and a rectangle is minimum. a. 3.16 cm c. 5.57 cm b. 4.27 cm d. 5.28 cm 560. The dimensions of a rectangle are continuously changing. The width increases at the rate of 3in/sec while the length decreases at the rate of 2in/sec. At one instant the rectangle is a 20-inch square. How fast is its area changing 3 seconds later? a. -18 c. -16 b. 20 d. 24 561. The dimensions of a rectangle are continuously changing. The width increases at the rate of 3 in/sec while the length decreases at the rate of 2 in/sec. At one instant the rectangle is a 20-inch square. How fast is its area changing 3 seconds later? a. -15 c. -11 b. -16 d. -8 562. A right circular cylinder is inscribed in a right circular cone of radius r. Find the radius R of the cylinder if its lateral area is maximum. 1 2 a. R = r c. R = r 2 3 1 3 b. R = r d. R = r 3 2 563. Find the second derivative of y = x + x(exp -2). a. 1-6x-4 c. 6x4 b. 1 – 2x3 d. 6x-4

564. If the x-intercept of the tangent to the curve y = e -x is increasing at a rate of 4 units per second, find the rate of change of the y-intercept when the x-intercept is 6 units. a. -0.135 c. -0.248 b. -0.05 d. -0.368 565. Find the rate of which the volume of a right circular cylinder of constant altitude 10 feet changes with respect to its diameter when the radius is 5 feet. a. 25  cu. ft/ft c. 100 cu. ft/ft b. 50 cu. ft/ft d. 200 cu. ft/ft x2 is discontinuous? x 1 c. -1 d. 2

566. At what value of x does the equation a. 0 b. 1

567. The diameter of a right circular cone increases at 1 inch per min. Find the rate at which its altitude is changing at the instant its diameter is 10 in while its altitude is 20 inches. The lateral area remains constant. a. -1.25 c. -4.75 b. -2.25 d. -3.45 568. A lot has the form of a right triangle, with perpendicular sides, 90 m and 120 m long. Find the area of the largest rectangular building that can be erected facing the hypotenuse of the triangle. a. 36 m by 75 m c. 30 m by 80 m b. 45 m by 60 m d. 40 m by 80 m 569. The acceleration of a moving body is a = 0.6 s. If the initial velocity was 0.90 m/sec, determine the velocity after moving 2.00 m. a. 1.79 m/sec c. 0.90 m/sec b. 3.21 m/sec d. 1.34 m/sec 570. A balloon is rising vertically over a point A on the ground at the rate of 20m/sec. A point B on the ground is level with and 30m from A. When the balloon is 40m from A, at what rate is its distance from B changing? a. 20 c. 12 b. 16 d. 14

571. When a bullet is fired into a sand bag, it will be assumed that its retardation is equal to the square root of its velocity on entering. For how long will it travel of the velocity on entering the bag is 144 ft/sec? a. 24 sec c. 26 sec b. 25 sec d. 27 sec 572. The cost C of the product is a function of the quantity x, of the product. C (x) = x2 – 400x + 50. Find the quantity for which the cost is minimum. a. 3000 c. 2000 b. 1500 d. 1000 573. Find the y” for the equation x3  3xy  y3  1 21  x   2 xy a. c. 2 2 y  x3 1  y  21  y   4xy b. d. 3 2 1  x 2 y x









574. A particle is projected vertically upward from a point 112 ft. above the ground with an initial velocity of 96 ft./sec. How fast is it moving when it is 240 ft. above the ground? a. 36 ft/sec c. 34 ft/sec b. 32 ft/sec d. 30 ft/sec 575. Find the most economical proportion for a box with an open top and a square base. a. b = h c. b = 3h b. b = 4h d. b = 2h 576. An elliptical plot of garden has a semi-major axis of 10m and semi-minor axis of 7.5m. If they are increased 0.25m each, find by differentials the increase in area of garden in square meters. a. 14.74 c. 16.74 b. 13.74 d. 1.74 577. The second derivative of the function f (x) is – f (x). What is the characteristic of this function? a. hyperbolic c. trigonometric b. exponential d. logarithmic

578. Pipes between stations as indicated have the following maximum flow capabilities, in cubic meters per second: between A and B 40, between B and C 30, between A and C 20. What is the maximum possible flow rate from A to C in cubic meters per second, without exceeding the above maximum flow capabilities? a. 60 c. 50 b. 30 d. 40 579. A window consists of a rectangle surmounted by an equilateral triangle. For a given perimeter, what must be the ratio of the total height of its breadth so that the ventilation is a maximum? Answer: H = 3W/2 580. Find the most economical proportions of a cylindrical cup. Answer: r = h 581. A rectangular box with a square base contains 540 cubic inches. If the cost $0.30/ sq. in of material, the bottom $0.20 and the sides $0.10, find the dimension of the box so that the cost is minimum. Answer: 6” x 15” 582. At what rate is the shadow of a 6-ft tall man shortening as he walks at 8 ft/sec on a level path toward the street light that is 20 ft above the pavement? Answer: 24/7 583. Find the area of the largest rectangle with lower base on the x-axis and upper vertices on the curve y = 12 – x2. Answer: 32 sq. units 584. A 5m picture hung on the wall so that its location is 4m above an observer eye. How far should the observer stand from the wall so that the angle subtended by the picture at the eye shall be maximized? Answer: 6m 585. Differentiate y = ex cos x2 a. ex (cos x2 – 2x sin x2) b. ex cos x2 – 2x sin x2

c. -2xex sin x d. –ex sin x2

586. A poster is to contain 300 cm2 of printed matter with margins of 10 cm at the top and bottom and 5 cm at each side. Find the overall dimensions of the total area of the poster is a minimum. a. 27.76, 47.8 cm c. 22.24, 44.5 cm b. 20.45, 35.6 cm d. 25.55, 46.7 cm 587. A farmer has enough money to build only 100 meters of fence. What are the dimensions of the field he can enclose the maximum area? a. 15 m x 35 m c. 22.5 m x 27.5 b. 20 m x 30 m d. 25 m x 25 m 588. A train is traveling at a speed of 100 kph. The locomotive has traction steel wheels of 1.2 meters diameter on level steel rails. Determine the maximum rectilinear speed of a point on the circumference of the traction wheel. a. 4000 m per min c. 3667 m per min b. 3000 m per min d. 3333 m per min 589. A triangle has variable sides x, y, and z subject to the constraint that the perimeter P is 18 cm. What is the maximum possible area for the triangle? a. 18.71 sq. cm c. 17.15 sq. cm b. 14.03 sq. cm d. 15.59 sq. cm 590. Find the second derivative of y = x + x -2. a. 1 – 6x-4 c. 6x4 b. 1 – 2x-3 d. 6x-4 591. What is the limit value of y = (x 3 + x) / (x2 + x) as x approaches 0? a. indeterminate c. 3 b. 0 d. 1 592. A train is traveling at a speed of 100 km per hr. The locomotive has traction steel wheels of 1.2 meters diameter on level steel rails. Determine the maximum rectilinear speed of a point on the circumference of the traction wheel. a. 3,333 m per min c. 3,000 m per min b. 4,000 m per min d. 3,667 m per min

593. A fencing material is limited to 20 ft in length. What is the maximum rectangular area that can be fenced in, using the two perpendicular corner sides of an existing wall? a. 100 c. 90 b. 140 d. 120 594. Find the y’’ for the equation x3 - 3xy + y3 = 1 a. 2(1 + x)/(1 + y)2 c. -2 xy/(y2 + x3) 2 3 b. -4xy/(y – x) d. 2(1 + y)/(1 + x)2 595. Two corridors respectively 2.5 m and 1.0 m wide intersect at right angles. Find the length in meters of the largest thin rod that will go horizontally around the corner. a. 3.97 c. 5.32 b. 4.79 d. 5.23 596. Find the minimum distance from the point (4, 2) to the parabola y2 = 8x. a. 2 3 units c. 4 2 units b. 2 2 units

d. 3 3 units

597. The cost C of a product is a function of the quantity x, of the product: C(x) = x2 – 400x + 50. Find the quantity for which the cost is minimum. a. 1,000 c. 1,500 b. 3,000 d. 2,000 598. An elliptical plot of garden has a semi-major axis of 10 m and semi-minor axis of 7.5 m. If they are increased 0.25 m each, find by differentials the increase in area of garden in sq. m. a. 14.74 c. 16.74 b. 13.74 d. 1.74 599. A triangle has variable sides x, y, and z subject to the constraint that the perimeter P is 18 cm. What is the maximum possible area for the triangle? a. 15.59 cm2 c. 14.03 cm2 2 b. 18.71 cm d. 17.15 cm2 600. Differentiate y = ex cos x2 a. –ex sin x2 b. ex (cos x2 – 2x sin x2)

c. ex cos x2 – 2x sin x2 d. -2xex sin x

601. An area in the x, y-plane is bounded by the following lines: x = 0 (y-axis) x + 4y = 20 y = 0 (x-axis) 4x + y = 20 The linear function z = 5x + 5y attains its maximum value within the bounded area only at one of the vertices (intersections of the above lines). Determine the maximum value of z a. 40.0 c. 50.0 b. 25.0 d. 45.0 602. What is the limit value of y = (x 3 + x)/(x2 + x) as x approaches 0? a. 1 c. 0 b. indeterminate d. 3 603. A fencing material is limited to 20 ft in length. What is the maximum rectangular area that can be fenced in, using the two perpendicular corner sides of an existing wall? a. 120 c. 140 b. 100 d. 90 604. Determine the limiting value of the following expression as n approaches infinity x = n sin (180/n) where the angle is in degrees. a. Infinity c.  b. e d. indeterminate 605. If y = (t2 + 2)2 and t = x1/2, determine dy/dx. a. 2x3/2 + x c. 2(x + 2) 5/2 1/2 b. 2x + x d. (2x2 + 2x)/3

INTEGRAL CALCULUS 606. Find the area bounded by the curve y = x 2 + 2, and the lines x = 0, y = 0 and x = 4. a. 88/3 c. 54/4 b. 64/3 d. 64/5 607. Evaluate the integral e2x over all (4 + 3e2x)dx, from -1 to 2? a. 15.32 c. 1.25 b. 28.51 d. 0.61 Ans. 0.61 608. Evaluate the integral, two log e to base 10, over x times, dx from x = 1 to 10. a. 2.0 c. 3.0 b. 49.7 d. 5.12 609. Find the integral of (ex – 1) divided by (ex + 1)dx. a. ln (e exp x – 1) square + x + C b. ln (e exp x + a) – x + C c. ln (e exp x + a) – x + C d. ln (e exp x – 1) square + x + C 610. Find the area which is inside r2 = 2cos20 and outside r = 1. a. 2 + pi/3 c. 3 - pi/3 b. 3 + pi/3

d.

2 - pi/3

611. Find the volume of the pentagon with given vertices at (1, 0), (2, 2), (0, 4), (-2, 2), (-1, 0) if it is revolve about the x axis. 98 104 a.  c.  5 3 b. 36 d. 72 612. Suppose that a motorboat is moving at 40 feet per second when its motor suddenly quits, and that 10 seconds later the boat has slowed to 20 feet per second. Assume that the resistance it encounters while coasting is proportional to its velocity v, so that dv/dt + -kv. How far will the motorboat coast in all? a. 400/ln 2 c. 420/ln 2 b. 380/ln 2 d. 440/ln 2

613. Compute the volume of the solid obtained by rotating the region bounded by y = x2, y = 8 – x2 and the y axis about the x axis. 250 a.  c. 56 3 256 b.  d. 50 3 614. A rectangle of sides a and b. Find the volume of the solid generated when rectangle is rotated on side b.  a.  a2b c. a2b 2  b. ab2 d. ab2 2 615. Find the area bounded by the curve r = 4sin 2θ cosθ. a. /4 c. /3 b.  /2 d. 24 616. Find the centroid of the volume generated by revolving about the y-axis the area bounded by the curve y 2 = 4ax in the first quadrant and the line x = a. 3 4 a. c. a a 5 5 5 3 b. d. a a 6 4 617. Find the volume of the solid generated by revolving the area bounded by the curve y = 1- x2 and the x-axis about x = 1. a. 10/3 c. 8 /3 b. 9/2 d. 11/3 618. Find the area of the region bounded by the curve y = x 3 and the line y = 8. a. 12 c. 11 b. 13 d. 10 619. A circle with a radius of 10 cm is revolved about a line tangent to it. Find the volume generated. a. 19,739 cm3 c. 15,250 cm3 b. 17,834 cm3 d. 18,235 cm3

620. Determine the area of the region bounded by the parabola y = 9 – x2 and the line x + y = 7. a. 7/2 c. 10/3 b. 9/2 d. 7/6 621. Find the volume generated by revolving about the x-axis the area bounded by y = 2x + 1, y = 0, x = 1, and x = 2. 42 36 a.  c.  3 5 49 89 b.  d.  3 4 622. Find the radius of the curvature of y = sin x at (/2, 1). a. 2 square root of 3 c. 1 b. 2 d. square root of 3 623. What is the area bounded by the parabola x2 = 4ay, its latus rectum and the y-axis? 2 2 3 a. a c. a2 3 4 4 2 3 b. a d. a2 3 2 624. Find the area bounded by the curve y 2 – 3x + 3 =0 and x = 4. a. 12 c. 16 b. 9 d. 8 625. A right circular cone with 10 cm circular base and height of 10 cm, find the volume of moment of inertia along the axis along the tip of the cone and perpendicular to the base in cm 5. a. 8,760 c. 9,801 b. 8,450 d. 7,854 626. Find the integral of (ex – 1) divided by (ex +1) dx. a. ln (e exp x – 1) square + x + C b. ln (e exp x – 1) - x + C c. ln (e exp x – 1) + x + C d. ln (e exp x – 1) square - x + C 627. Find the area of one loop of r2 = 4sin2. a. 2 c. 3 b. 4 d. 5

628. Determine the area bounded by y = 8 – x3, the x-axis and y-axis. a. 14 c. 16 b. 10 d. 12 629. An ellipse with major axis 16 and minor axis 8 is revolved about its minor axis. Find the volume of the solid generated. 512 1024 a.  c.  3 3 1125 1058 b.  d.  3 3 630. Evaluate the integral of xy respect to y and then to x, for the limit x = 0 to x = 1, and the limits from y = 1 to y = 2. a. 7/9 c. 3/4 b. 1/2 d. 5/8 631. A plane area is bounded by the lines: y = x, y = -x, x = 10 By integration, determine the distance of the centroid of the area from the y-axis. a. 7.33 c. 6.06 b. 6.67 d. 5.51 632. Determine the area bounded by the x-axis and the curve y = 1/(x²) from x = 1 to x = infinity. a. 1.00 c. indeterminate b. infinity d. 2.00 633. Find the volume obtained if the region bounded by y = x 2 and y = 2x is rotated about the x axis. a. 7pi c. 5pi 64 b. pi d. 3pi 15 634. Find the volume of the solid of revolution formed by rotating the region bounded by the parabola y = x2 and the lines y = 0 and x = 2 about the x axis. 32 a. 15pi c. pi 5 35 b. pi d. 20pi 2

635. Find the area of the region bounded by the parabola x = y 2 and the line y = x – 2. a. 12/7 c. 7/6 b. 10/3 d. 9/2 636. Find the area of the curve r = a(1 – sin u). 4 2 1 a. a c. a2 3 2 3 2 b. a2 d. a2 2 3 2 1

637. Evaluate

  xydxdy . 0 0

3 4 3 b. 8

a.

c.

7 9

d. 1

638. A trapezoid has two equal slanting sides a 6cm base and a 3 cm top parallel to and 5 cm above the base. Determine the moment of inertia of the trapezoidal area relative to the base, in cm4. a. 142.05 c. 129.13 b. 171.88 d. 156.25 639. Determine the area bounded by y = 8 – x3, the x-axis and the y-axis. a. 14 c. 16 b. 10 d. 12 1

640. Evaluate

 sinh xdx . 2

0

a. 0.40672 b. 0.50678

c. -0.40672 d. 0.25085

641. Determine the tangent to the curve 3y 2 = x3 at (3, 3) and calculate the area of the triangle bounded by the tangent line, the x-axis, and the line x= 3. a. 2.50 s.u. c. 4.00 s.u. b. 3.50 s.u d. 3.00 s.u.

e(expx)  1 dx. e(expx)  1 ln (e exp x – 1)square + x + C ln (e exp x – 1) + x + C ln (e exp x + 1) - x + C ln (e exp x – 1)square - x + C

642. Evaluate the integral of a. b. c. d.

643. Find the area bounded by r = 4 cos2θ . a. 4 c. 12 b. 8 d. 16 644. Find the area of the region bounded by y = x2 – 5x + 6, the x-axis, and the vertical lines x = 0 and x = 4. a. 19/6 c. 17/3 b. 14/3 d. 16/3

dP  2(sqr.rt.P), find P. dQ a. P = 2(Q sqr.) + C b. P = 4(Q sqr.) + QC

645. If

c. P = (Q sqr.) + C d. P = (Q sqr.) + 2QC + C2

646. Find the area bounded by the curve y = 4 over the square root of (1 - 2x) and the lines y = 0, x = -4 and x = 0. a. –4 c. -2 b. 8 d. 4 647. Considering the volume of a spherical shell as an increment of volume of a sphere, find approximately the volume of a spherical shell whose outer diameter is 8 inches and whose 1 thickness is inch. 16 a.  cu. in c. 3 cu. in b. 2 cu. in d. 4 cu. in 648. Compute the volume of the solid obtained by rotating the region bounded by y = x2, y = 8 – x2 and the y axis about the x axis. 250  a. c. 56 3 256  b. d. 50 3

649. Find the centroid of the areas bounded by 2x + y = 6, x = 0, y = 0. a. (1, 3) c. (1, 2) b. (1, 5/2) d. (1, 3/2) 650. Find the area enclosed by the curve r = 8sin 2 c. 24 d. 20

a. 12 b. 6

1 θ. 2



651. Evaluate  sin 2 cos 3 cosh4d . 

a. 3 b. 5

c. 2/3 d. 0

652. Find the area bounded by the curve y = 2 over (x – 3) and the lines y = 0, x = 4, and x = 5. 1 a. 2 ln4 c. ln4 2 b. 4 ln2 d. ln4 653. A reversed curve on a railroad track consists of two circular arcs. The central angle of one is 20 with radius 2500 ft and the central angle of the other is 25 with radius 3000 ft. Find the total length of the two arcs. a. 2812 ft c. 2482 ft b. 2821 ft d. 2848 ft 654. Find how far an airplane will move in landing, if in t seconds after touching the ground its speed in feet per second is given by the equation v = 180 – 18t. a. 400 ft c. 900 ft b. 450 ft d. 1,800 ft 655. A pendulum is brought to rest by air resistance, each swing being 11/12 as much as the preceding one> if the lower end of the pendulum describes an arc 60 cm long in the first swing, what will be the total length of the path which the pendulum describes before it comes to rest? a. 390 cm c. 360 cm b. 720 cm d. 1,440 cm

656. Find the volume of the pentagon with given vertices at (1, 0), (2, 2), (0, 4), (-2, 2), (-1, 0) if it is to revolve about the x-axis. 98 104 a.  c.  5 3 b. 36 d. 72 657. Find the volume of the solid generated by revolving the region bounded by y = x2 and y2 = x, about x = -1. a. 29/15 c. 29/60 b. 29/14 d. 29 /30 658. Find the area which is inside r2 = 2cos 2 and outside r = 1. a. 2 + pi/3 c. 3 - pi/3 b. 3 + pi/3

d.

2 - pi/3

659. Given the curves y = x3, x = 1, x = 2 and y = 0 is rotated about the x-axis. Find the moment of inertia rotated about the axis of revolution. a. 4 pi/1269 c. pi/1200 b. pi/1300 d. 3 pi/1300 660. Find the entire area enclosed by the curve r = 2sin3. a. /3 c. /2 b. /4 d.  661. Determine the area of the region bounded by the parabola y = 9 – x2 and the line x + y = 7. a. 7/2 c. 10/3 b. 9/2 d. 7/6 662. Find the maximum area of the rectangle inscribed on the 8a3 curve y = 2 if the base of a rectangle lies on the x-axis x  4a2 and the two vertices on the curve. a. a squared c. 4(a squared) b. 3(a squared) d. 2(a squared) 663. Find the area bounded by the curve y = 6x + x 2 – x3, x-axis and the 1st quadrant. a. 12 3/5 c. 10 2/3 b. 15 3/4 d. 13 ½

664. A rectangle has sides of a and b. Find the volume of the solid generated when the rectangle is rotated on side a.  a. a2b c. a2b 2  b.  ab2 d. ab2 2 665. Find the centroid of the volume generated by revolving about the y-axis the area bounded by the curve y2 = 4ax in the first quadrant and line x = a 3 3 a. a c. a 5 5 3 5 b. a d. a 6 4 666. Determine the area of the region bounded by the curve y = x3 – 4x2 + 3x and the x-axis, 0  x  3 . a. 37/12 c. 33/12 b. 135/12 d. 39/12 667. An ellipse with major axis 16 and minor axis 8 is revolved about its minor axis. Find the volume of the solid generated. 512 1024 a.  c.  3 3 1125 1058 b.  d.  3 3 668. Find the polar moment of inertia of the area of a circle of radius 2 cm with respect to their center. a. 6pi c. 4pi b. 8pi d. 2pi 669. Find the area of the curve r = a(1 sinu). a. 4/3 ∏a2 c. ´ ∏a2 b. 3/2 ∏a2 d. 2/36 ∏a2 670. Find the moment of inertia with respect to the axis of the volumes of a sphere generated b revolving circle of radius r about a fixed diameter. a. 4/15 ∏r5 c. 2/15 ∏r5 b. 8/15 ∏r5 d. 7/15 ∏r5

671. Find the area bounded by the curve (y curve) 3x + 3 = 0 and then x = 4. a. 12 c. 16 b. 9 d. 8 672. Evaluate the integral sinh3 xcosh2 x dx a. cosh4 x – cosh2 x + c b. ¼ cosh4 x – ½ cosh2 x + c c. 1/5 cosh5 x – 1/3 cosh3 x + c d. cosh5 x – cosh3 x + c 673. A trapezoidal area has the following vertices on the x-y plane: A(6,1.5), B(10,2.5), C(10,-2.5) and D(6,-11.5). With all coordinates in cm, if this area is rotated about the y-axis, determine the generated volume in cm 3. a. 746 c. 821 b. 903 d. 578 674. Evaluate the integral of

  ln e   x  C 2

a. ln e x 1  x  C b.

x 1

e  1dx . e  1 x

x

  lne 

c. ln e x 1  x  C d.

x 1 2

 x C

675. Evaluate the integral of x dx, and the limits are 1 and e. a. 1 c. e b. 0 d. infinity 676. By integration, determine the area bounded by the curves: y = 6x – x2 and y = x2 – 2x. a. 25.60 sq. units c. 21.33 sq. units b. 17.78 sq. units d. 30.72 sq. units 677. Evaluate the integral (cos x – x sin x) dx from x = 1 to 2. a. 0.72 c. -0.53 b. 0.48 d. -1.37 678. Determine the integral of zr2 sin θ with respect to z, then r, and then 0, from the limits from z = 0 to z = 2, from r = 0 to r = 1, and from θ = 0 to θ = Π/2. a. 4/5 c. 2/3 b. 3/4 d. 1/2

679. Integrate 1/(3x +4) with respect to x and evaluate the result from x = 0 to x = 2. a. 0.336 c. 0.305 b. 0.252 d. 0.278 680. Evaluate the integral of tan (x/2) dx, the limits of which are 0 and Π. a. 2 + Π/2 c. 2 + Π /4 b. 2 – Π /2 d. 2 – Π /4 681. Find the area bounded by the curve y 2 – 3x + 3 = 0 and then x = 4. a. 12 c. 16 b. 9 d. 8 682. Find the area of the curve r = a(1 – sin u). a. (4/3)a2 c. (1/2)a2 b. (3/2) a2 d. (2/36)a2 683. Evaluate the double integral of xy dxdy when the limits of x = 0 and 1, and the limits of y = 1 and 2. a. 1 c. 0 b. 1/4 d. 3/4 684. Evaluate the integral of tan(x/2)dx, the limits of which are 0 and . a. 2 - /4 c. 2 - /2 b. 2 + /2 d. 2 + /4 Ans. 3.6 (By approximation) 685. Evaluate the triple integral of zr2 sinu dz dr du, where the limits of z is from 0 to 2, the limits of r is from 0 to 1, and the limits of u is from 0 to /2. a. 2/3 c. 5/3 b. 4/3 d. 1/3 686. Evaluate the triple integral of xyzdzdydx for the following limits: z from 0 to (2 – x), y from 0 to (1 – x), and x from 0 to 1. a. 85/30 c. 89/30 b. 87/30 d. 81/30 Ans. 13/240

687. Determine the integral of zr2 sin with respect to z, then r, and then θ, for the limits from z = 0 to z = 2, from r = 0 to r = 1 a. 1/2 c. 3/4 b. 4/5 d. 2/3 688. Integrate 1/(3x + 4) with respect to x and evaluate the result from x = 0 to x = 2. a. 0.278 c. 0.252 b. 0.336 d. 0.305 689. Evaluate the integral of r sin  with respect to r and then to , for the limits from r = 0 to r = cos , and from  = 0 to  = . a. 1/4 c. 1/2 b. 1/3 d. 1/6 690. By integration, determine the area bounded by the curves: y = 6x – x2 and y = x2 – 2x a. 25.60 sq. units c. 21.33 sq. units b. 17.78 sq. units d. 30.72 sq. units 691. Evaluate the integral (cos x – xsin x) dx from x = 1 to 2. a. -0.35 c. 0.48 b. 0.72 d. -0.53 Ans. -1.37 692. A circle has a 20 cm diameter. Determine the moment of inertia of the circular area relative to the axis perpendicular to the area through the center of the circle, in cm 4. a. 15, 708 c. 19, 007 b. 17, 279 d. 14, 280 693. Compute the moment of inertia of a rectangle 8 cm by 24 cm with respect to a line through its center of gravity and parallel to the short side. a. 8,734 cm4 c. 9,074 cm4 b. 8,576 cm4 d. 9,216 cm4 694. Find the moment of inertia with respect to the axis of the volume of a sphere generated by revolving a circle of radius r about a fixed diameter. a. (4/15)r5 c. (2/15)r5 5 b. (8/15) r d. (7/15)r5

695. Find the polar moment of inertia of the area of a circle of radius 2 cm with respect to its center. a. 2 c. 8 b. 6 d. 4

NUMERICAL METHODS 696. In the sequence 1, 1, 1/2, 1/6, 1/24…..an determine the 6th term. a. 1/74 c. 1/120 b. 1/100 d. 1/80 697. Find the sixth term of the series 1+ 1 + 1  1  1 …. 2 6 24 120 a. 1/240 c. 1/720 b. 1/360 d. 1/960 698. Find the power series y=  cn xn, satisfying the conditions: y=2 when x=0; y’=1 when x= 0 and y” +2y=0. a. y=2 – x + x2 + 2/3 x3 b. y=2 + x - x2 + 2/3 x3 c. y=x - x2 - 2/3 x3 d. none of the above 699. Using power series, evaluate the integral of x -1 sin x dx from zero to one limits. a. 0.966 c. 0.666 b. 0.946 d. 0.996 700. Given f(x) = 4y” + axy’ + b, if x  0 what kind of equation is the given? a. Laplace c. Bernoulli’s b. Mclaurin d. Euler 701. f(x) = sin x, find the first four terms of the Mclaurin series to find f(46), if the Mclaurin series expansion of 3 5 x x sinx  x   … 3! 5! a. 0.7931 c. 0.7139 b. 0.7193 d. 0.7319 702. Determine the sum of the infinite series: s = 1/3 + 1/9 + 1/27 +…+ (1/3)” +… a. 4/5 c. 2/3 b. 3/4 d. ½

703. Determine the sum of the infinite series: S = (0.8) + (0.8)² + (0.8)³ + … + (0.8)ⁿ + … a. 3 c. 2 b. 5 d. 4 704. Determine the sum of the infinite series: S = (0.9) + (0.9)² + (0.9)³ + … + (0.9)ⁿ + ... a. 7 c. 6 b. 9 d. 8 705. Evaluate (0.7)20 + (0.7)19(0.3)2 C20 + (0.7)17(0.3)3 C20 3 . 2 a. 0.0107… c. 8.54 b. 0.107… d. 7.01… 706. The sum of geometric series is: S = 1 + z + z² + z³ + ……zⁿ, where z is less than 1.0. What is S as n approaches infinity? a. 2/(1-z) c. 1/(2-z) b. 1/(1-z) d. 1/(1-2z)

MATRIX ALGEBRA

4 707. If A = 6 1

5 7 2

equal to? 4 0 a. 0 0

7 0

0 0 5

0 b. 0 1

0 7 0

0 0 0

0 3 5

1 and B = 0 0

3 708. Transpose the matrix  2 0 1 0 a. 2

3 b. 1 2

2 1 1

2 1 0

0 2 3

0 2 1

1 1 2

0 1 0

0 0 , what is A times B 1

6 c. 8 2

7 9 3

0 4 6

4 d. 6 1

5 7 2

0 3 5

2 0. 1 3 0 c. 2

1 2 1

2 1 0

1 d.  1 2

3 2 2

2 0 1

709. Solve the equations: 2x – y + 3z = -3; 3x + 3y –z = 10; -x – y + z = -4 by Cramer’s Rule. a. (2, 1, -1) c. (1, 2, -1) b. (2, -1, -1) d. (-1, -2, 1) 710. Matrix

2 1 1 1 + 2 Matrix equals 1 3 1 1

a.

2 4 2 2

c.

2 1 1 3

b.

1 2 1 1

d.

0 1

5 5

1 2 3 711. Evaluate the determinant  2  1  2 . 3 1 4 a. 4 b. 2

c. 5 d. 0

2 14 3 1 1 5 1 3 712. Evaluate the determinant . 1 2 2 3 3 4 3 4 a. 489 c. 326 b. 389 d. 452 713. Given the matrices A = a.

2 8 3 2

b.

 13  7 7 7

1 1 5 5 and B = , find 3A – 2B. 1 1 2 2  10  3 c. 5 1 13  7 d. 7 0

714. Determine the product of the following matrices: 2  1 1  2  5    1 0  x 3 4 0  3 4 

 1  a.  1  9  1  b.  1  9

 8  10  2 5  22 15   8  10 2 5  22 15 

0 0  10   c. 1  2  5  0 21 15  0 0  10   d. 1  2  5  0 21  15

  1  8  10    Ans.  1  2  5  , but choose (a). 15 10  15 

715. Find the value of y in the given equation: 4x – 2y + 6z = 2 2x + 3y – 2z = 10 x+y–z=2 a. 13/5 c. 15 b. 47/5 d. 14 Ans. y = 6 716. Given the equations: x+y+z=2 3x – y – 2z = 4 5x – 2y + 3z = -7 Solve for y by determinants. a. 1 b. -2

c. 3 d. 0

717. If the rows in the first determinant is the same as the columns of another determinants. a. equal c. indeterminate b. not equal d. zero 718. What are the values of B1 and B2? 9 7 B1 2  1 3 B2 1 a. B1 = -1/20, B2 = 7/20 b. B1 = 7/20, B2 = -1/20

c. B1 = -1/20, B2 = -7/20 d. B1 = 10, B2 = 20

719. Solve for x by determinants: 3x + 2y= 1, x – y = -8. a. 3 c. -2 b. -3 d. 4 720. Find the solution to the system of equations by using the inverse of the matrix method X = A(exp-1)D. The equations are 3x–z = 3, -3x + y + z = 2, and -5x + 2z = 4. a. (5,10,27) c. (10,5,27) b. (10,5,25) d. (5,10,25) 721. Solve the equations: x – y + 3z = -3; 3x + 3y – z = 10; -x – y + z = -4 by Cramer’s rule, a. (2, -1, -1) c. (-1, -2, 1) b. (1, 2, -1) d. (2, 1, -1)

722. The inverse of matrix a. Matrix

1 0

0 1 1 1 0 1

b. Matrix

1 0

1 is : 1 c. Matrix

0 1

d. Matrix

1 0

1 1 1 1

723. A rectangular array of numbers forming m rows and n columns are called as a. determinants c. pascal’s triangle b. elements d. none of the above 724. Find the solution to the system of equations by using the inverse of the matrix method X = A-1D. The equations are 3x – z = 3, -3x + y + z = 2, and -5x + 2z = 4. a. (5, 10, 27) c. (10, 5, 27) b. b. (10, 5, 25) d. (5, 10, 25) 725. Which of the following matrices has an inverse? a. matrix

6 9

2 3

c. matrix

b. matrix

3 6

1 2

d. matrix

726. If matrix

x matrix y z a. 3 b. 1

1 1 2 2 1 3 0 1 1

4 2

multiply by matrix

2 1

5 2

2 1

x y z

= 0, then

=? c. 0 d. -2

727. Solve for x by determinants: 3x + 2y = 1, x – y = -8. a. 4 c. -3 b. 3 d. -2

1 728. If matrix 4

4 x x multiply by matrix = 0, then is =? 1 y y

a. 8 b. 1

c. -4 d. 0

729. Solve for y by determinants of the second order 2y = 3x – 4 and x – 3y + 5 = 0. a. 22/7 c. 24/7 b. 17/7 d. 19/7 730. Determine the eigen values of the following matrix: 5 3

0 2 a. 1, 6 b. 4, 1

c. 3, 4 d. 2, 5

731. If a 3 x 3 matrix and its inverse are multiplied together write the product. 1 0 0 0 0 1 a. 0 0

1 0

0 1

c. 0 1

0 b. 0 0

0 0 0

0 0 0

1 1 1 d. 1 1 1 1 1 1

3 732. If A =  2 0

1 1 2

 2 1 0 2

0 0

2 0 , what is the cofactor of the first row, 1

second column element? 3 2 a. 0 1 b.

1 0

3 2 0 1 2 0 d. 0 1 c.

2 733. If A =  1 0

3 2 5

1 4 what is the cofactor with the second row, 7

third column element? 2 3 a. 0 5 b. -

2 0

3 5

734. Solve by determinants: 3x + 2y + z = 4 2x + y – 3z = -3 4x + 3y + 4z = 9 a. x = -1, y = -3, z = 6 b. x = -1, y = 2, z = 4

1 7 2 0 3 1 d. 0 7 c. -

c. x = 4, y = -5, z = 2 d. x = 2, y = -5, z = 4

DIFFERENTIAL EQUATIONS 735. Solve the differential equation dy – xdx = 0 if the curve passes through (1, 0). a. 3x2 + 2y – 3 = 0 c. x2 – 2y + 1 = 0 b. 2y + x2 – 1 = 0 d. 2x2 + 2y – 2 = 0 2 Ans. x – 2y – 1 = 0 736. If e = 100 sin (t + 30) – 50 cos 3t + 25 sin (5t + 150) and i = sin (t + 40) + 10 sin (3t + 30) – 5 sin (5t + 50). Calculate the power in watts. a. 1177 c. 1043 b. 937.6 d. 1224 Ans. P = 918.54 watts 737. Radium decomposes at a rate proportional to the amount at any instant. In 100 years, 100 mg. of radium decomposes to 96 mg. How many mg. will be left after 100 years? a. 88.60 c. 92.16 b. 95.32 d. 90.72 738. In an LC circuit. Where R = 0 in RLC ckt. Under resonance condition express frequency in terms of L & C. Where E (t) = E sin t. a. 2/ LC c. 1/LC b. 2/LC

d. 1/ LC

739. Given y = emx, What values of m(-infinity to +infinity) will satisfy the relationship 6y” – y’ – y = 0. a. -1/3, 1/2 c. -1/3, -1/2 b. 1/3, -1/2 d. 1/3, 1/2 740. The rate at which a tablet of vitamin C begins to dissolve depends on the surface area of a tablet. One brand of tablet is 2 centimeters long and is in the shape of a cylinder with hemispheres of diameter 0.5 centimeter attached to both ends. A second brand of tablet is to be manufactured in the shape of a right circular cylinder of altitude 0.5 centimeter. Find the volume of the tablet. a. /16 c. 9/5 b.  /8 d. 9/4

741. The rate at which a tablet of vitamin C begins to dissolve depends on the surface area of a tablet. One brand of tablet is 2 centimeters long and is in the shape of a cylinder with hemispheres of diameter 0.5 centimeter attached to both ends. A second brand of tablet is to be manufactured in the shape of a right circular cylinder of altitude 0.5 centimeter. Find the diameter of the second tablet so that its surface area is equal to that of the first tablet. a. 1 cm c. 1/2 cm b. 2 cm d. 1/4 cm 742. The electric potential at any point (x, y, z) is given by V = x2 + 4y2 + 9z2. Find the rate of change of potential at point P(2, -1, 3) towards the origin. a. 164/ 15 c. -178/ 14 b. 178/ 14 743. Solve the particular solution of a. y = x4 b. y = 3x2 + 4x + 2

d. -164/ 15 dy 3y   x3 if y(1) = 4. dx x c. y = x4 + 3x3 d. y = 3x2 + x +8

744. A point travels as described by the following parametric equations x = 10t + 10cos 3.14t, y = 20t + 10sin 3.14t and z = 30t, where x, y, z are in meters, t in seconds, all angles in radians. The vector locating the body at any time is r = ix + jy + kz. Determine the magnitude of the velocity of the body in meters per second at a time t = 0.75 second. a. 35.72 c. 32.47 b. 33.41 d. 38.08 745. Two currents described as I1 = 20sin 377t and I2 = 30cos 377t, what is the instantaneous current at t = 0.002 sec? a. 35.56 c. 38.07 b. 37.07 d. 39.08 746. Solve the differential equation d²x/dt² + 4x = 0 With initial condition x(0) = 10, x(0) = 0 a. x(t) = 10 cos 2t b. x(t) = 10 cos t + 10 sin t c. x(t) = 10 cos 2t + 10 sin 2t d. x(t) = 10 sin 2t

747. A capacitor of 0.001 farad is connected in series with a 10  resistor. A voltage e = 100sin 377t is impressed in the circuit. Find the maximum amplitude of the current I(t). a. 8.36 amperes c. 9.39 amperes b. 10.34 amperes d. 15.84 amperes 748. Two electric bulbs B1 (100 watts, 230 volts) and B2 (50 watts, 230 volts) are in series across a 220-volt source so that the same current passes through them and their voltages sum up to equal the source voltage. For each bulb, its resistance directly varies with the current and further characterized as follows: For bulb B1, R1 = 484 ohms when I = 5/11 ampere For bulb B2, R2 = 484 ohms when I = 5/22 ampere Determine the power in watts consumed by B1, P1 = I²R1 a. 21.2 c. 18.2 b. 19.3 d. 20.3 749. Solve for the general solution of the differential equation (D + 3)(D2 + 3D + 2)y = 0, where D is the differential operator d/dx, where x = 1. Determine y in terms of the constant of integration. a. 2.718C1 + 7.389C2 + 20.086C3 b. 0.368C1 + 0.135C2 + 0.05C3 c. 0.368C1 + 7.389C2 + 20.086C3 d. 0.368C1 + 0.135C2 + 20.086C3 750. Obtain general solution of the following differential equation and determine y (0.06545) in terms of the constant of integration: (D² + 144)(D + 12) y(x) = 0, where D = d/dx a. y = 0.7071 C1 + 0.7071 C2 + 0.4559 C3 b. y = 0.5815 C1 + 0.5815 C2 + 0.5815 C3 c. y = 0.5815 C1 + 0.5815 C2 + 0.4559 C3 d. y = 0.4559 C1 + 0.4559 C2 + 0.7071 C3 751. Solve for the general solution of the differential equation: (D2 + 100)(D + 10)y = 0 Where D is the differential operator d/dx and x = 0.05. a. 0.479 C1 + 0.878 C2 + 0.368 C3 b. 0.479 C1 + 0.878 C2 + 0.368 C3 c. 0.479 C1 + 0.878 C2 + 0.368 C3 d. 0.479 C1 + 0.878 C2 + 0.368 C3

752. A spherical snowball is melting in such a way that its surface area decreases at the rate of 1cm2/min. How fast is its radius shrinking when it is 3 cm? a. -1/(24 pi) c. -1/(32 pi) b. -1/(36 pi) d. -1/(48 pi) 753. Solve the differential equation: x(x +1) dx + (x² - 1) dy = 0 If y =2 when x = 2, determine y when x = 4. a. 0.376 c. 0.282 b. 0.311 d. 0.342 754. Which of the following is the solution of ylll – 3yll + 3yl – y = 0 l. (e to the x) ll. x(e to the x) lll. (e to the –x) a. l and ll c. l only b. lll only d. ll only dy  4 divided by x(y – 3). dx a. x3y4 = Cey c. x4y2 = Cey b. x4y3 = Cey d. x3y2 = Cey

755. Solve:

756. Solve for x and y in the equation 2(exp x) + (1/9)i = 8 + log to the base 3 of y exp i. a. x = 2, y = 3/9 c. x = 2, y = 3[exp (1/9)] b. x = 3, y = 3[exp (1/9)] d. x = 3, y = 3/9 757. Determine the solution to the differential equation: (xy + y2)dx – x2dy = 0 if y = 1 when x = 1 a. x = exp (1 – y/x) c. y = exp (1 – y/x) b. x = exp (1 – x/y) d. y = exp (1 – x/y) 758. Suppose that a motorboat is moving at 40 feet per second when its motor suddenly quits, and that 10 seconds later the boat has slowed to 20 feet per second. Assume that the resistance it encounters while coasting is proportional to its velocity v, so that dv/dt = -kv. How far will the motorboat coast in all? a. 400/ln 2 c. 420/ln 2 b. 380/ln 2 d. 440/ln 2

759. A bacteria has a growth rate constant of 0.02. If initially the number of bacteria is 1000, what is the time before it reaches a population of 100,000? a. 208.36 c. 230.26 b. 205.87 d. 209.22 760. Solve the particular solution a. x3 = et b. x2 = et

dx x if x(0)=1.  dt 2 c. x = et d. x4 = tet

761. The solution of the differential equation condition y = 1 when t = 0, is a. y = ln kt b. y = ekt

dy  ky , with initial dt

c. y = sin kt d. y2 = 2t +k

762. The charge in coulombs that passes through a wire after t seconds is given by the equation Q(t) = t 3 – 2t2 + 5t + 2. Determine the average current during the first two seconds. a. 9 A c. 5 A b. 8 A d. 6 A 763. The electric potential at any point (x, y, z) is given by V = x2 + 4y2 + 9z2. Find the rate of change of potential at point P(2, -1, 3) toward the origin. a. 164/ 15 c. -178/ 14 b. 178/ 14

d. -164/ 15

764. dx/dt = x/2, x(0) = 1. Find the particular solution. a. -1/2 et/2 c. et/2 b. -3/2 et/2 d. 1/2 et/2 765. Solve the particular solution of a. y = x4 b. y = 3x2 + 4x + 2 766. For

the differential y=Cx2 + 1. a. xy’ = 2 (y – 1) b. x’ = 2 (y – 1)

dx 3y   x3 if y(1) = 4. dy x c. y = x4 + 3x3 d. y = 3x2 + x + 8

equation

whose

general

c. x = y’ – 1 d. xy’ = 2 (1 – y)

solution

is

767. Solve (x + y) dy = (x – y) dx. a. x2 + 2xy + y2 = c b. x2 – 2xy – y2 = c

c. x2 – 2xy + y2 = c d. x2 + y2 = c

768. Given: V = 70.7 2 sin(t + 30 degrees) – 35.35 2 sin(3t + 60 degrees) I = 4.35 2 sin(3t + 99.8 degrees). Determine the active power in watts if V is in volts and I is in amperes. a. 243 c. 238 b. 257 d. 264 769. Solve: dy/dx = 4y divided by x(y – 3) a. x3 y4 = Cey c. x4y3 = Cey 4 2 y b. x y = Ce d. x3y2 = Cey 770. Solve: ydy – 4xdx = 0. a. y2 + x2 = c b. y2 = 4x2 + c

c. y2 + 4x2 = c d. –y2 + 4x2 = c

771. Solve the differential equation d2x/dt2 + 4x = 0. With the initial conditional x(0) = 10, x 2 (0) = 0. a. x(t) = 10cos2t c. x(t) = 10cos2t + 10sin2t b. x(t) = 10cost + 10sint d. x(t) = 10sin2t 772. Determine the solution to the differential (xy+y2)dx-x2dy = 0, if y=1 when x=1. a. x = exp(1-y/x) c. y = exp(1-y/x) b. x = exp(1-x/y) d. y = exp(1-x/y) 773. Solve: ydy – 4xdx = 0. a. x2 + y2 = C b. y2 = 4x2 + C

equation:

c. y2 + 4x2 = C d. none of the above

774. For the differential equation whose general solution is y = Cx 2 + 1. a. xy’ = 2(1 – y) c. x’ = 2(y – 1) b. xy’ = 2(y – 1) d. x = y’ – 1 775. Solve: xy’(2y – 1) = y(1 – x). a. ln(xy) = 2(x – y) + C b. ln(xy) = 2y - x + C

c. ln(xy) = x – 2y + C d. ln(xy) = x + 2y + C

776. Under certain conditions cane sugar with water is converted into dextrose at a rate which is proportional to the amount unconverted at anytime. If 75 grams at a time t = 0.8 grams are converted during the first 30 minutes, find the amount converted in 1.5 hours, in grams. a. 22.6 c. 20 b. 19.6 d. 21.5 777. A generator has a field winding with an inductance L = 10 henrys and a resistance Rt = 0.1 ohm. To break an initial field current of 1000 amperes, the field breaker inserts a field discharge resistance Rd across the field terminals before its main contacts open. As a result, the field current decays to zero according to the differential equation. L di/dt + Ri = 0, where R = Rf + Rd Preventing the sudden decrease of I to zero, and a resulting high inductive voltage due to L. Solve the differential equation and determine the value of Rd that will limit the initial voltage across it to 1,000 volts. a. 0.90 ohm c. 0.85 ohm b. 0.80 ohm d. 0.95 ohm 778. A voltage waveform e(t) is described as follows, for the first half-cycle: e(t) = 60 volts at t = 0 sec. increases linearly to 100 volts at t = 0.005 sec decreases linearly to 60 volts at t = 00.01 sec Determine the root-mean-square or RMS value of this voltage, noting that, at least for a half-cycle the RMS procedure involves squaring the voltage, taking the mean of the square , and taking the square root of the mean. a. 0.90 ohm c. 0.85 ohm b. 0.80 ohm d. 0.95 ohm 779. A 10 ohm resistance R and 1.0 Henry inductance L are in series. An AC voltage e(t) = 100 sin 377t is applied across the series circuit. The applicable differential equation is: Ri + L(di/dt) = e(t) Solve for the particular solution (without the complimentary solution) to the differential equation, and determine the amplitude of the resulting sinusoidal current i(t). a. 0.265 ampere c. 0.292 ampere b. 0.321 ampere d. 0.241 ampere

780. A series circuit has R = 10 ohms, L = 0.1 henry and C = 0.0001 farad. An AC voltage e = 100 sin 377t is applied across the series circuit and the applicable differential equation is: L(d 2i / dt 2 )  R(di / dt)  (1 / C)i  de / dt Solve for the particular solution (without the complimentary solution), and determine the amplitude of the resulting sinusoidal current i(t). a. 5.51 amperes c. 7.34 amperes b. 6.67 amperes d. 6.06 amperes 781. A 20 ohms resistance R and a 0.001 farad capacitance C are in series. A direct current voltage E of 100 volts is applied across the series circuit at t = 0 and the initial current is i(0) = 5 amperes. The applicable differential equation is: R(di/dt) + i/C = 0 Solve the differential equation and determine the resulting current i(t) at t = 0.01 sec. a. 3.34 amperes c. 2.78 amperes b. 3.67 amperes d. 3.03 amperes 782. Radium decomposes at a rate proportional to the amount at any instant. In 10 years, 100 mg of radium decomposes to 96 mg. How many will be left after 100 years? a. 95.32 c. 90. 72 b. 92.16 d. 88.60 783. Solve: dy/dx = 4y divided by x(y - 3). a. x3y4 = Cey c. x4y3 = Cey b. b. x4y2 = Cey d. x3y2 = Cey

LAPLACE TRANSORMATIONS 784. The Laplace transform of [1 – e-at]/a is a. 1/s(s + a) c. 1/(s2 + a2) b. 1/a(s + a) d. 1/(s + a)2 785. The Laplace transform of cos wt is a. w / [(s square) + (w square)] b. w / (s + w) c. s/ (s + w) d. s / [(s square) + (w square)] 786. k divided by [(s square) + (k square)] is the laplace transform of: a. sin kt c. 1.0 b. e (exp kt) d. cos kt 787. Find the Laplace transform of e-ax a. 1/s – a c. – 1/s2 b. 1/s2 d. 1/s + a 788. Find the Laplace transform of x3 e4x a. 6 (s + 4) -4 c. 6 (s – 2)-2 -4 b. 6 (s – 4) d. none of the above 789. Find the Laplace transform of te -4t a. (s + 4)2 c. (s – 4)-2 b. (s + 4)-2 d. none of the above 790. k divided by [s2 + k2] is the Laplace transform of… a. cos kt c. e (exp kt) b. sin kt d. 1.0 791. Find the Laplace transform f(t) = t3e-2t. a. 6/(s + 2)4 c. 4/(s + 2)3 b. 6/(s - 2)4 d. 4/(s - 2)3 792. If the Laplace Transform of the function [1 – e-t] = 1/[s(s + 1)], then the final value of this function can be determined by the final value theorem as a. -1 c. 1 b. infinity d. 0

793. Using Laplace transform technique, find the transient response of the system describe by the differential equation [d(square)y/dx(square)] = 3 dy/dx + 2y = 1, with the initial condition y(0) = 1, dy/dx = 1 when t = 0. a. 2e-t – 3/2e-2t c. e-t – 3e-2t -t b. 2e + 3/2e d. e-t + 3e-2t

1 , then ss  1 the final values of this function can be determined by the final value theorem as a. –1 c. 1 b. infinity d. 0

794. If the Laplace Transform of the function 1  et 

795. Using Laplace Transform technique, find the transient response of the system describe by the differential equation dy2/dx2 + 3dy/dx + 2y = 1, with the initial condition y(0) = 1, dy/dx = 1 when t = 0. a. 2e-t – (3/2)e-2t c. e-t – 3e-2t -t -2t b. 2e + (3/2)e d. e-t + 3e-2t 796. Evaluate Laplace transform of cos2t. s2 s a. 2 c. 3 s 4 s 4 2 s 2 s2  2 b. d. s(s 2  4) s3  4 797. Evaluate Laplace transform of t2. a. 2/s c. 2/s2 2 b. 1/s d. 1/s 798. What is the Laplace transform of f(t) = cos at? a. a/(s2 + a2) c. a/(1 – a2) b. s/(s2 + a2) d. s/(1 – s2)

 s  1 799. Find f(t) if L {f(t)} = ln  .  s  1 2sinht 2sinht a. c. t t2 2cosht 2cosht b. d. t t2

800. Evaluate L(sinh at cos at). a(s 2  2a2 ) a. s 4  4a4 s2  2a2 b. 3 s  2a3

s2  2a2 s4  4a4 s2  2a2 d. 4 s  2a4 c.

100 . s s2  102 c. i(t) = t – 0.1sin 10t d. i(t) = t2 – cos 10t

801. Determine the inverse Laplace transform I(s) = a. i(t) = t – cos 10t b. i(t) = t2 – 0.1sin 10t

802. Find the Inverse Laplace transform of I(s) = find i(t) when t = 0.1 sec. a. 19.07 b. 17.56

2





10(2s  5) and s2  3s  2

c. 20.4 d. 21.23

803. Evaluate the inverse Laplace transform of 6 over (s 2 + 4). a. 3cos 2t c. 3cosh 2t b. 3sinh 2t d. 3sinh 2t 804. Determine the inverse Laplace transform of F(s) = (s + 2)(e-s)/(s2 + 4). a. cos 2(t - 1) – sin 2(t - 1) c. cos 2(t -1) + sin 2(t - 1) b. -cos 2(t - 1) + sin 2(t - 1) d. -cos 2(t - 1) – sin 2(t - 1) 805. Find the inverse Laplace transform of 6/ (s – 9)4 a. t6 e9t c. t5 e-9t b. t3 e9t d. none of the above 806. Find the inverse Laplace transform of (2s-18) / (s2 + 9) a. 2 cos 3x – 6 sin 3x c. 6 sin 3x + 2 cos 3x b. 2 cos 3x + 6 sin x d. 6 sin 3x+ 2 cos 3x 807. The inverse Laplace transform of s/(s2 + w2) is a. sin wt c. e (exp wt) b. w d. cos wt 808. Determine the inverse Laplace transform of: I(s) = 200/(s + 50s + 10625) a. i(t) = 2.0t-25t sin 100t c. i(t) = 2.0-25t cos 100t -25t b. i(t) = 2.0 sin 100t d. i(t) = 2.0-25t cos 100t

809. Determine the inverse Laplace transform of: I(s) = 100/[(s + 10)(s + 20)] a. i(t) = 10 c. 16.74 b. 13.74 d. 1.74 810. Determine the inverse Laplace Transform of: (s + a)/[(s + a)2 + w2] a. exp(-at) sint c. t exp(-at) cost b. t sint d. exp(-at) cos t 811. Find the inverse Laplace transform of [2/(s + 1)] – [4/(s + 3)] a. [2 e (exp –t) - 4 e (exp –3t)] b. [e (exp –2t) + e (exp –3t)] c. [e (exp –2t) – e (exp –3t)] d. [2 e (exp –t) - 2 e(exp –2t)] 812. Evaluate the inverse Laplace transform of a. 10e-5t b. 10e-t

10 . s  50

c. 10e-50t d. 10e-10t

813. Determine the inverse Laplace transform of: I (s) = 100/(s + 20)² a. i (t) = 100 – 10 cos 10t c. i (t) = 1 - .1 sin 10t b. i (t) = 100 – 100 cos 10t d. i (t) = 1 – 10cos 10t Ans. 100t e-20t

STATISTICS AND PROBABILITIES 814. If the sum of the squares of 10 numbers is 645 and their standard deviation is 2.87, find their arithmetic mean. a. 6.5 c. 8.5 b. 7.5 d. 9.5 815. The lotto uses numbers 1-42. A winning number consists six(6) different numbers in any order. What are your chances of winning it? a. 5,245,786 c. 10,127,420 b. 8,437,224 d. 2,546,725 816. There are four balls of four different colors. The two balls are taken at a time and arranged in a definite order. For example, if the white and the red balls are taken, one definite arrangement is white first, red second, and another arrangement is red first, white second. How many such arrangement is possible? a. 24 c. 12 b. 6 d. 36 817. A group of 3 people enter a theater after the lights had dimmed. They are shown to correct group of 3 seats by the usher. Each person holds a number stub. What is the probability that each is in the correct seat according to the numbers on the seat and stub? a. 1/6 c. 1/2 b. 1/4 d. 1/8 818. Four different colored flags can be hung in a row to make coded signal. How many signals can be made if a signal consists of the display of one or more flags? a. 64 c. 68 b. 66 d. 62 819. American was tested by their blood samples A, B, AB and O. The proportion of the blood type samples of the Caucasians are 0.41, 0.1, 0.04 and 0.45 respectively. Find the probability that the Caucasian that he or she is either A or AB. a. 0.43 c. 0.55 b. 0.51 d. 0.45

820. A toothpaste firm claims that in a survey of 54 people, they were using either Colgate, Hapee or Close-up brand. The following statistics were found: 6 people used all the three brands, 5 used only Hapee and Close-up, 18 used Hapee or Close-up, 2 used Hapee, 2 used only Hapee and Colgate, 1 used Close-up and Colgate, and 20 used only Colgate. Is the survey worth paying for? a. neither yes or no c. no b. yes d. either yes or no 821. The probability that a married man watches a certain television show is 0.4 and the probability that a married woman watches the show is 0.5. the probability that a man watches the show, given that his wife does, is 0.7. Find the probability that a wife watches the show given that her husband does. a. 0.635 c. 0.875 b. 0.925 d. 0.745 822. A box contains 2 blue socks and 2 white socks. Picking randomly, what is the probability that you will pick 2 socks of the same color? a. 1/6 c. 1/2 b. 1/3 d. 1/4 823. In a fuel economy study, each of 3 race cars is tested using 5 different brands of gasoline at 7 test sites located in different regions of the country. If 2 drivers are used in the study, the test runs are made once under each distinct set conditions, how many test runs are needed? a. 210 c. 10800 b. 420 d. 1400 824. In how many ways can two numbers whose sum is even be chosen from the numbers 1, 2, 3, 8, 9, 10, and 11? a. 8 c. 7 b. 10 d. 9 825. What is the probability of drawing 2 cards; both are spade in a standard deck of 52 cards? a. 3/52 c. 4/51 b. 3/51 d. /52

826. A random sample of 200 adults is classified below according to sex and the level of education attained. Education Male Female Elementary 38 45 Secondary 28 50 College 22 17 a. 14/39 c. 39/100 b. 7/50 d. 0.857 827. Find the probability of getting a spade and a face card (Jack, Queen, and King) in an ordinary deck of 52 cards. a. 3/39 c. 1/52 b. 3/52 d. 1/49 828. In how many ways can 6 persons line up to buy a ticket? a. 720 c. 120 b. 480 d. 540 829. The probability that a patient recovers from a delicate heart operation is 0.9. What is the probability that exactly 5 out of 7 patients will survive? a. 0.148 c. 0.128 b. 0.1240 d. 0.240 830. John, Peter and Charlie are suitors of Susan. The probability that Susan will say yes to John is equal to that of Peter, if the probability that Susan will say yes to Charlie is twice of either of the two. What is the chance that Charlie will win to Susan? a. 1/2 c. 1/4 b. 1/3 d. 2/3 831. Determine the value of c so that f(x, y) = c x y represents joint probability distributions of the random variables X and Y, if the random numbers are x = -2, 3 and y = 2, 3. a. 1/5 c. 1/15 b. 1/8 d. 1/3 832. Given a well-balanced coin, what is the probability of getting a head or a tail in the long run? a. 70 : 50 c. 40 : 60 b. 50 : 50 d. 30 : 50

833. A certain college campus, 250 of the 3,500 coed enrolled are over 5 ft, 6 inches in height. Find the probability that a coed chosen at random from the group of 3,500 has a height of less than 5 ft, 6 inches. a. 11/14 c. 1/14 b. 13/14 d. 3/14 834. A box contains 10 yellow balls, 7 green balls and 4 red balls. What is the probability of drawing either red or green ball in a single draw? a. 0.5444 c. 0.5238 b. 0.0635 d. 0.0667 835. A man bought 5 tickets in a lottery for a prize of P2,000. If there are a total of 400 tickets, what is the mathematical expectation? a. P25 c. P30 b. P20 d. P35 836. A survey of 500 television viewers produced the following results: 285 watch football games 195 watch hockey games 115 watch basketball games 45 watch football and basketball 70 watch football and hockey 50 watch hockey and basketball 50 do not watch any of the three games How many watch the football games only? a. 230 c. 160 b. 200 d. 190 837. Assuming that an examinee answered randomly each of 50 examination questions from 4 given answers 1 of which is correct. What is the probability that he answered correctly half of the examination questions? a. 72% c. 25% b. 50% d. 36% 838. With a throw of 3 dice, what is the probability of getting a 9 or an 11? a. 50/216 c. 54/215 b. 52/216 d. 56/216

839. A question was lost in which 600 persons had voted; the same persons having voted again on the same question, it was carried by twice as many as it was before lost by; and the new majority was to the former as 8:7. How many changed their minds? a. 200 c. 150 b. 250 d. 125 840. An association has 15 officials. How many committees can be formed from these officials? a. 1,365 c. 1,638 b. 1,966 d. 1,138

4-member

841. A drawer contains 10 white and 6 black balls. What is the possibility of random drawing of two white balls? a. 0.450 c. 0.313 b. 0.375 d. 0.260 842. A bank’s password each consists of any two letters of the English alphabet plus two digit from 0 to 9. How many different password are possible a. 100880 c. 26000 b. 175760 d. 67600 843. A basketball coach has a total ways can he field a team of 5 included? a. 42 b. 70

of 10 players. In how many if the captain ball is always c. 126 d. 25

844. A number between 1 and 69 inclusive is selected at random. What is the probability that it is prime? a. 21/69 c. 20/69 b. 19/69 d. 17/69 845. The RMS of the set 1, 3, 4, 5, and 7 is a. 20 c. 4.47 b. 5 d. 6.54 846. In a 5 horses race, Aubrey picked two horses at random. What is the probability of winning? a. 2/10 c. 2/5 b. 1/10 d. 1/5

847. A clinical record gives the following information on body types: Body types Endomorph Ectomorph Mesomorph Male 72 54 36 Female 62 64 38 How many subjects are either female or endomorphs? a. 298 c. 238 b. 296 d. 282 Ans. 236, but choose (c). 848. In Jones family each daughter has many brothers as sisters and each son has three times as many sisters as brothers. How many daughters and sons are there in the Jones family? a. 3, 2 c. 5, 2 b. 4, 2 d. 6, 3 849. A’s probability of hitting a target is 1/3 while B has a probability of 1/5 of hitting the same target. What is the probability that one of them hits the target? a. 7/15 c. 2/5 b. 3/5 d. 8/15 850. Given the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and the following sets M = {1, 4, 7, 10}, N = {1, 2, 3, 4, 5} and L = {2, 4, 6, 8}. Determine the set NOT (M AND N AND L) by enumerating its members. a. {1, 2, 3, 4, 6, 8, 10} b. {1, 2, 3, 4, 5, 6, 7, 8, 10} c. {1, 2, 3, 5, 6, 7, 8, 9, 10} d. {1, 2, 3, 4} 851. In throwing a pair of dice, what is the possible outcome of getting 10? 1 5 a. c. 36 36 1 1 b. d. 12 18 852. A box contains 10 yellow balls, 7 green balls and 4 red balls. What is the probability of drawing either red or green ball in a single draw. a. 0.5444 c. 0.5238 b. 0.0635 d. 0.0667

853. A question was lost which 600 persons had voted; the same persons having voted again on the same question, it was carried by twice as many as it was before lost by; and the new majority was to the former as 8:7. How many changed their minds? a. 200 c. 150 b. 250 d. 125 854. A man bought 5 tickets in a lottery for a prize of P2,000. If there are a total of 400 tickets, what is his mathematical expectation? a. P25 c. P30 b. P20 d. P35 855. A fair die is tossed twice. Find the probability of getting a 4, 5, or 6 on the first toss and a 1, 2, 3 or 4 on the second toss. 1 1 a. c. 2 4 1 1 b. d. 3 5 856. A real estate agent has 8master keys to open several new homes. Only one master key will open any given house. If 40% of these homes are usually left unlocked, what is the probability that the real estate agent can get into a specific home if the agent selects 3 master keys at random before leaving the office? a. 1/2 c. 5/8 b. 3/4 d. 3/8 857. A’s probability of hitting a target is 1/3 while B has a probability of 1/5 of hitting the same target. What is the probability that they both hit the target? a. 2/3 c. 8/15 b. 1/15 d. 7/15 858. Chemistry magazine stakes 30% of failure due to operator error. What is the probability of no more than 4 out of 20 is not an operator failure? a. 0.000352 c. 0.02375 b. 0.0000056 d. 0.00026

859. If 4 out 20 truck tires blowout of a cargo company. Find the probability that 3 to 6 trucks will have a blow out tire. a. 1/2 c. 1/4 b. 1/3 d. 1/5 860. A bag contains 20 balls numbered 1 to 20. If one ball is removed from the bag, what is the probability that the ball is even or is less than 5? a. 0.5 c. 0.65 b. 0.75 d. 0.6 861. There were 104,830 people who attended a rock festival. If there were 8110 more boys than girls, and 24, 810 fewer adults over 50 years of age than there were girls, how many girls attended the festival? a. 40,510 c. 48,620 b. 53,175 d. 15,700 862. A box contains 2 blue socks and 2 white socks. Picking randomly, what is the probability that you will pick 2 sock of the same color? a. 1/6 c. 1/2 b. 1/3 d. 1/4 863. A drug for the relief of asthma can be purchased from 5 different manufacturers in liquid, tablet, or capsule form, all of which come in regular and extra strength. In how many different ways can a doctor prescribe the drug for a patient suffering from asthma? a. 15 c. 30 b. 60 d. 40 864. One bag contains 4 white balls and 3 black balls, and a second bag contains 3 white balls and 5 black balls. One ball is drawn at random from the second bag and is placed unseen in the first bag. What is the probability that a bal now drawn from the first bag is white? a. 5/64 c. 35/64 b. 20/64 d. 15/64 865. In how many ways can 6 persons can line up to buy a ticket? a. 720 c. 120 b. 480 d. 540

866. In a deck of 52 cards, a poker hand consists of 5 cards, find the probability of holding 2 aces and 3 jacks. a. 0.7 x 10-5 c. 0.3 x 10-5 b. 0.9 x 10-5 d. 0.5 x 10-5 867. A shipment of 12 television sets contains 3 defective sets. In how many ways can a hotel purchase 5 of these sets and receive at least 2 of the defective sets? a. 0.264 c. 0.496 b. 0.659 d. 0.343 868. The probability that A hits a target is ½ while the probability that B hits the target is 1/3. Find the probability that one hits the target. a. 3/4 c. 2/3 b. 1/6 d. 5/6 869. A coin is biased so that a head is twice as like to occur as a tail. If the coin is tossed 3 times, what is the probability of getting 2 tails and 1 head? a. 1/3 c. 2/9 b. 2/27 d. 5/8 870. A and B are subsets of Q. (4, 7, 9); B = (4, 5, 9, 10); Q = (4, 5, 6, 7, 8, 9, 10) What is A  B? a. (4, 5, 6, 7, 8, 9, 10) c. (4, 5, 6, 8, 9, 10) b. (4, 5, 7, 9, 10) d. (5, 10)

A

=

871. A certain college campus, 250 of the 3, 500 women enrolled and are over 5 ft., 6 inches in height. Find the probability that a women chosen at random from the group of 3, 500 exceeds 5 ft.-6 in. in height. a. 3/7 c. 2/7 b. d. 1/16 872. A student makes 100% of his first test and 80% on the second. On the third test, he made 60% of the grade he made on the second, while on the fourth, he made 80% of the grade he made on the third. What constant average rate of decrease would give the first and last grades? a. 20.5% c. 20.1% b. 20.7% d. 20.9%

873. Determine the value of c so that f(x, y) = c x y represents joint probability distributions of the random variables X and Y, if the random numbers are x = -2, 3 and y = 2, -3. a. 1/17 c. 1/15 b. 1/8 d. 1/3 874. Given n = 5 with measurement 2, 1, 1, 3, 5. What is the sample variance? a. 1.496 c. 2.8 b. 2.24 d. 2.4 875. Two dice are tossed. How many sample events are in the same sample space? a. 24 c. 18 b. 36 d. 12 876. Find the probability of drawing a heart and a spade in standard deck of 52 cards. a. 13/102 c. 1/51 b. 1/26 d. 2/103 877. If the sum of the squares of 10 numbers is 645 and their standard deviation is 2.87, find their arithmetic mean? a. 6.5 c. 8.5 b. 7.5 d. 9.5 878. A and B are two independent events. The probability that A can occur is p and that for both A and B to occur is q. What is the probability that B will occur? a. p - 1 c. p q b. q / p d. p + q 879. A student has test scores of 75, 83, 78. the final test counts half the total grade. What must be the minimum (integer) score on the final so that the average is 80? a. 81 c. 84 b. 82 d. 83 880. If A= (1,2,3,4,5) and B= (2,3,4,5,6) the set A intersect of set B is a. (2,3,4,5,6) c. (2,3,4,5) b. (1,2,3,4,5) d. (2,4,6)

881. There are 5 main roads between the cities A and B, and four between B and C. In how many ways can a person drive from A to C and return, going through B on both trips, without driving on the same road twice? a. 260 c. 120 b. 240 d. 160 882. Suppose that 30% of the employees in a large factory are smokers. What is the probability that there will be exactly two smokers in a randomly chosen-five-person work group? a. 0.2557 c. 0.3671 b. 0.3267 d. 0.3087 883. In how many ways can a set of 6 distinct books be arranged in a bookshelf? a. 5040 c. 720 b. 120 d. 24 884. In a mathematics examination, a student may select 7 problems from a set of 10 problems. In how many ways can he make his choice? a. 120 c. 720 b. 530 d. 320 885. A and B are independent events. The probability that event A will occur is p(A) and the probability that A and B will occur is p(AB). From these two statements, what is the probability that the event B will occur? a. p(A) – p(B) c. p(A)p(B) b. p(B) – p(A) d. p(AB)/p(A) 886. The probability of getting at least two heads when a pair of coin is tossed four times is… a. 11/16 c. 1/4 b. 13/16 d. 3/8 887. How many license plates can be first two places and any of the last three? a. 38, 358 b. 35, 283

made using two letters for the numbers 0 through 9 for the c. 252, 000 d. 676, 000

888. If a card is drawn from a deck of 52 cards of 4 suits, what is the probability that it will be a jack, a queen or a king? a. 4/52 c. 8/52 b. 12/52 d. 16/52 889. Two dice are rolled. Find the probability that the sum of two dice is greater than 10. Answer: 1/12 890. Nine tickets, numbered 1 to 9 are in a box. If two tickets are drawn at random, determine the probability that both are odd. Answer: 5/18 891. How many permutations can be formed from the letters of the word “constitution”. Answer: 9, 979, 200 892. How many four-place numbers can be written using the digits from 1 to 9? Answer: 3, 204 893. A committee of three is to be chosen from a group of 5 men and four women. If the selection is made at random, find the probability that two are men. Answer: 10/21 894. 3 balls are drawn from box containing 5 red, 8 black, and 4 white balls. Determine the probability that all are white. Answer: 1/170 895. A bag contains 9 balls numbered 1 to 9. Two balls are drawn at random. Find the probability that one is even and the other is odd. Answer: 5/9 896. a) How many ways can 5 people be lined up to pay their electric bills? b) If two particular persons refuse to follow each other, how many ways are possible? Answer: a) 120, b) 72 897. In how many different ways can a ten-question true-false examination to be answered? Answer: 1024 ways

898. An engineering freshman must take a chemistry course, a humanities course, and a mathematics course. Is he may be selected in any of 2 chemistry courses, how many ways can he arrange his program? Answer: 24 ways 899. How many distinct permutations can be made from the letters of the word “mathematics”? Answer: 4, 989, 600 900. How many ways can the first five players in a basketball team be filled with twelve men who can play any position? Answer: 95, 040 ways 901. How many three-digit numbers can be formed from the digits 0, 1, 2, 3, 4 and 5? a. if each digit is used only once in a given number? b. if digits may be repeated in a given number? c. how many in (a) are odd numbers? d. how many in (a) are even numbers? e. how many in (b) are even numbers? f. how many are less than 330? g. how many are greater than 330? Answers: a) 100; b) 180; c) 48; d) 52; e) 90; f) 52; g) 89 902. A contractor wishes to build 5 houses, each different in design. In how many ways can he place these homes on a street if two lots are on one side and three lots are on the opposite side? Answer: 120 ways 903. In how many ways can four boys and three girls sit in a row if the boys and girls must alternately seated? Answer: 288 ways 904. In how many ways can seven trees be planted in a circle? Answer: 720 ways 905. In how many ways can two mango trees, three chico trees and two avocado trees be arranged in a straight line if one does not distinguish between trees of the same kind? Answer: 210 ways

906. A college team plays eight basketball games during an intramural. In how many ways can the team end games with four wins, three losses and one tie? Answer: 280 ways 907. Nine people will be shooting the rapids of Pagsanjan in three bancas that will hold 2, 4, and 5 passengers, respectively. How many ways is it possible to transport the nine people to the falls? Answer: 4, 536 ways 908. From a group of three men and seven women, how many committees of five people are possible? a. with no restrictions? b. with two men and three women? c. with one man and four women if a certain woman must be on the committee? Answers: a) 252; b)105; c) 60 909. From three red, four green and five yellow bubble gums, how many selections consisting of five bubble gums are possible if two of each color are to be selected? Answer: 390 910. A shipment of 10 Sony Betamax video recorders contains 3 defective sets. In how many ways can a hotel purchase 4 of these sets and receive at least 2 of the defective sets? Answer: 70 911. A bag contains four blue, five red and six yellow plastic chips. a) If two chips are drawn, find the probability that both are yellow b) If six chips are drawn, find the probability that there will be two balls of each color. c) If nine chips are drawn, find the probability that two will be red, five yellow and two blue . Answer: a) 1/7 b) 180/1001 c) 72/1001 912. In a single throw of two dice, what is the probability of throwing not more than 5? Answer: 5/18

913. Find the probability that all five cards drawn from a deck are all hearts. Answer: 4.95x10-4 914. A team of 5 students is to be chosen for a math contest. If there were ten male and eight female students to choose from, what is the probability that the three members will be male and two will be female? Answer: 20/51 915. A bag contains five pairs of socks. If four socks are chosen, what is the probability that there is no complete pair taken? Answer: 8/21 916. In the game “spin-a-win” the rim of the wheel is divided into 30 equal parts which is marked P10, P20, ….., P300. The win is indicated by a fixed pointer at the top. If the wheel is spun, what is the probability that the three digit number will be the players take home winning? Answer: 7/10 917. If eight different books are arranged at random in a shelf, what is the probability that a certain pair of books (a) will be beside each other? (b) will not be together? Answer: a) ¼ b) ¾ 918. Joey prepares 3 cards for his 3 girlfriends. He addresses three corresponding envelopes, a brown-out suddenly occurred and he hurriedly placed the cards in the envelope at random. What is the probability that (a) each card was sent to its proper addressee (b) no card is sent in the proper addressee Answer: a) 5/16 b) 9/16 919. A box containing 15 red eggs, 20 white eggs. If 12 eggs are taken at a random, what is the probability that this will have an equal number of red and white eggs? Answer: 209/899 920. During a fund raising lottery, 250 tickets were sold to the freshmen. Of which 3 are winners. Marissa, a freshman has 2 tickets. What is her probability of winning something? Answer: 248/10,375

921. If the probability that Nini will go to UP for a certain semester is 1/3 and the probability that she will go to UST that semester is ¼, find the probability that she will go to college in one of the two schools? Answer: 7/12 922. If the probabilities that Ginebra, Alaska and Shell will win the PBA Open conference championship are 1/5, 1/6 and 1/10 respectively. Find the probability that one of them will win the contest? Answer: 7/15 923. The probability that Joseph Estrada will be nominated to run for President is ¼ and the probability of his election if nominated is 1/3. Find the probability of (a) his being elected as President (b) of his being nominated and not elected. Answer: a) 1/12 b) 1/6 924. Find the probability of obtaining a 4 in each of two successive tosses of a pair of dice. Answer: 1/1296 925. A box containing 5 black and 3 white handkerchiefs and another seven black and five white handkerchief is drawn from each box, find the probability that both will be (a) black, (b) white and (c) the same color Answer: a) 35/96 b) 5/32 c) 25/48 926. The probabilities that Marita will win the preliminary, semifinal and final contest in singing are 3/8, 1/6 and 1/12 respectively. Failure in any contest prohibits participation in the following one. Find the probability that she will (a) reach the final contest (b) win the final contest. Answer: a) 1/16 b) 1/192 927. Six Algebra books, four Physics books and two Chemistry books are on the table. If a book is removed and replaced, then another is removed and replaced, and so on until six removals and replacement have been made. Fin the probability that an Algebra book was removed and replaced a) three times b) at least three times. Answers: a) 5/16 b) 21/32

928. Three Physics books, five Algebra books and two Chemistry books are on a shelf. Judd decides to take the two books and selects them at random. Find the probability that the first book drawn will be Physics and the second is Chemistry. Answer: 1/15 929. Find the probability of throwing in three tosses of a dice, (a) exactly two 4’s, (b) at least two 4’s. Answer: a) 5/72 b) 2/27 930. A bag contains three white, four red, and five green candies. Five withdrawal of one candy each are made, and the candy replaced after each. Find the probability that all will be red. Answer: 1/243 931. If the probability that Alaska basketball team will win the PBA Conference Championship is 2/3, find the probability that it will win exactly three championship that it will win exactly three championship in 5 years. Answer: 80/243 932. If the probability that Imelda will be elected to be office is 2/3, find the probability that she will be elected for four consecutive terms and then defeated on the fifth term. Answer: 16/243 933. The probability of an event happening exactly twice in four trials is 18 times the probability of it happening exactly five times in six trials. Find the probability that it will happen in one trial. Answer: 1/3 934. The probability of an event will happen exactly three times in ive trials is equal to the probability that it will happen exactly two times in six trials, find the probability that it will occur in one trial. Answer: 0.451 935. How many number of 5 different digits each number to contain 3 odd and 2 even digits can be formed from the digits 1,2,3,4,5,6,7,8 & 9? Answer: 7, 200

936. How many permutations can be formed from the letters of the word “constitution”? Answer: 9, 979, 000 937. How many four place numbers can be written using the digits from 1 to 9? Answer: 3, 024 938. Find n if P (n,3) = 6C (n,5) Answer: n=8 939. Two dice are rolled. Find the probability that the sum of the two dice is greater than 10. Answer: 1/12 940. A pair of dice is thrown. Find the probability of having 7 or 11. Answer: 2/9 941. A pair o dice is thrown. If it is known that one die shoes a 4, what is the probability that the other die shown a 5? Answer: 2/11 942. Nine tickets, numbered 1 to 9, are in a box. If two tickets are drawn at random, determine the probability that both are odd. Answer: 5/18 943. A committee of three is to be chosen from a group of 5 men and 4 women. If the selection is made at random, determine the probability that two are men. Answer: 10/21 944. 3 balls are drawn from box containing 5 red, 8 black and 4 white balls. Determine the probability that all are white. Answer: 1/170 945. A bag contains 9 balls numbered 1 to 9. Two balls are drawn at random. Find the probability that one is even and the other one is odd. Answer: 5/9 946. From a bag containing 3 white, 4 black and 5 red balls, a ball is drawn. Find the probability that it is not red. Answer: 7/12

947. How many cars can be given license plates having 5 digit numbers using the digits 1,2,3,4 and 5 with no digit repeated in any license plate? Answer: 120 948. A committee of 4 is selected by lot from a group of six men and 4 women. What will be the probability that will consist of exactly 2 men and 2 men? Answer: 3/7 949. A box contains 25 electric bulbs are drawn at random from the box. What is te probability that both electric bulbs are good? Answer: 7/20 950. There are 52 tickets in the lottery in which there is a first and a second prize/ What is the probability of a man drawing a prize if he owns 5 tickets? Answer: 0.18367 951. There are 3 candidates for A, B and C for mayor of a certain town. If the odds are 7:5 that candidate A will win and those of B are 1:3, what is the probability that candidate C will win? Answer: 1/6 952. A coin is biased so that the head is twice as likely to occur as tail. If the coin is tossed 3 times, what is the probability of getting: (a) 2 tails and 1 head (b) at least 2 heads? Answers: a) 2/9 b) 20/27 953. Three men are running for public office. Candidates A and B are given twice the chance of either A and B. Find the probability that: a) C wins b) A does not win Answers: a) ¾ b) ¾ 954. A player sinks 50% of all his shots. What is the probability that he will make exactly 3 of his next 4 shots? Answer: 25% 955. In a poker game consisting of 5 cards, what is the probability of holding: a) 2 aces and 2 kings b) 5 spades Answers: a) 1584/2598960 b) 1287/2598960

956. Find the harmonic mean 7, 1, 5, 2, 6 and 3. Answer: 2.56 957. Determine the number of permutations of 8 distinct objects, taken 3 at a time. a. 504 c. 120 b. 210 d. 336 958. A drawer contains 10 white and 6 black balls. What is the probability of randomly drawing a white and a black ball. a. 0.3 c. 0.36 b. 0.25 d. 0.208 959. If three sticks are drawn from 5 sticks whose lengths are 1, 3, 5, 7, and 9, what is the probability that they will form a triangle? a. 0.25 c. 0.35 b. 0.30 d. 0.40 960. There are four balls of different colors. Two balls at a time are taken and arrange in any way. How many such combination is a. 36 c. 6 b. 3 d. 12 961. A manufacturer of outboard motors received a shipment of shearpin to be used in assembly of its motors. A random sample of 10 pins is selected and tested to determine the amount of pressure required to cause the pin to break. When tested, the required pressures to the nearest pound are 19, 23, 27, 19, 23, 28, 27, 29 and 27. What is the measure of the mean? a. 27 c. 25 b. 24 d. 28 962. A set of elements that is taken without regard to the order in which the elements are arranged is called: a. combination c. sequence b. permutation d. series 963. What is the number of permutations of the letters in the word BANANA? a. 60 c. 42 b. 52 d. 36

964. How many different committees can be formed by choosing 4 men from an organization that has a membership of 15 men. a. 1240 c. 1365 b. 1435 d. 1390 965. A group of 3 people enter a theater after the lights had dimmed. They are shown to the correct group of 3 seats by the usher. Each person holds a number stub. What is the probability that each is in the correct seat according to the numbers on seat and stub? a. 1/4 c. 1/8 b. 1/2 d. 1/6 966. A number between 1 and 10, 000 (inclusive) is randomly selected. What is the probability what it will be divisible both by 4 and by 5? a. 0.20 c. 0.05 b. 0.25 d. 0.10 967. A student has a periodic test scores 75, 83, 78. The final test has weighted equal to 3 periodic tests. What should the student strive for minimum final test score so that he gets a passing minimum average of 80? a. 81 c. 82 b. 83 d. 80 968. In a certain college campus, 250 of the 3,500 women enrolled are over 5 ft, 6 inches in height. Find the probability that a woman chosen at random from the group of 3,500 exceeds 5 ft 6 inches in height. a. 3/7 c. 2/7 b. 1/14 d. 1/16 969. If A = (1, 2, 3, 4, 5) and B = (2, 3, 4, 5, 6) the set A intersect of set B is a. (2, 3, 4, 5, 6) c. (2, 3, 4, 5) b. (1, 2, 3, 4, 5) d. (2, 4, 6) 970. Suppose that 30% or the employees in a large factory are smokers. What is the probability that there will be exactly two smokers in a randomly chosen five-person work group? a. 0.2557 c. 0.3671 b. 0.3267 d. 0.3087

971. There are 5 main roads between the cities A and B, and four between B and C. in how many ways can a person drive from A to C and return, going through B on both trips, without driving on the same road twice? a. 260 c. 120 b. 240 d. 160 972. Electrical loads are arranged on horizontal x, y-axes as follows: Load No. X-coordinates Y-coordinates Kilowatts Load 1 0 2 100 2 1 1 180 3 1 3 200 4 2 0 120 5 2 4 150 6 3 1 200 7 3 3 180 8 4 2 100 Determine the coordinates of the center of load a. x = 2.000, y = 2.049 c. x = 2.163, y = 2.195 b. x = 1.854, y = 2.211 d. x = 2.146, y = 1.902 973. In a certain community of 1,200 people, 60% are literate. Of the males, 50% are literate, and of the females 70% are literate. What is the female population? a. 850 c. 550 b. 500 d. 600 974. In a commercial survey involving 1,000 persons on brand preference, 120 were found to prefer brand X only, 200 prefer brand Y only, 150 prefer brand Z only, 370 prefer either brand X or Y but not Z, 450 prefer brand Y or Z but not X, and 420 prefer either brand Z or X but not Y. How many persons have no brand preference, satisfied with any of the 3 brands? a. 80 c. 180 b. 230 d. 130 975. Given the sets A = {1, 2, 3, 4} and B = {3, 4, 5, 6, 7}, determine the intersection AB a. {6, 7} c. {1, 2, 3, 4, 5, 6, 7} b. {3, 4} d. {5, 6}

976. A number between 1 and 10,000 (inclusive) is randomly selected. What is the probability that it will be divisible both by 4 and by 5? a. 0.20 c. 0.05 b. 0.25 d. 0.10 977. There are four balls of different colors. Two balls at a time are taken and arranged any way. How many such combination is possible? a. 36 c. 6 b. 3 d. 12

ADVANCE MATH 978. This is the method used to represent a periodic function by a series of sinusoids to any desired degree of accuracy. a. Maclaurin’s Series c. Taylor’s series b. Fourier Series d. Laplace transforms 979. One term of a Fourier series in cosine form is 10 cos 40t. Write it in exponential form. a. 5ej40t c. 10 ej40t b. 5ej40t + 5e-j40t d. -10 ej40t 980. Given the Fourier series in cosine form. a. 12 c. -12 b. 3.25 d. 8 981. Given the Fourier series in Cosine form, f(t) = 5 cos 40t + cos 60t. What is the frequency of the fundamental? a. 10 c. 20 b. 40 d. 30 982. Given the Fourier series in cosine form, f(t) = 5 cos 20t + 2 cos 40t + cos 80t. What is the fundamental frequency? a. 20 c. 10 b. 40 d. 60 983. Evaluate the terms of a Fourier series 2e j10t + 2e-j10t at t = 1. a. 2 + j c. 4 b. 2 d. 2 + j2 984. Evaluate the Fourier series 2ej10Πt + 2e-j10Πt at t = 1. a. 2 c. 2 + j2 b. 4 d. 2 + j 985. The 5 vectors: 10cm at 72k degrees, k= 0, 1, 2, 3, 4 encompass the sides of a regular pentagon. Determine the magnitude of the vector cross-product: 2.5{(10/ at 144 deg) x (10/at 216 deg)}. a. 198.1 c. 285.2 b. 237.7 d. 165.1

986. The 3 vectors describe by: 10 cm/ at 120k degrees, k = 0,1,2 encompass the sides of an equilateral triangle. Determine the magnitude of the vector cross product: 0.5[(10/at 0 deg) x (10/at 120 deg)]. a. 86.6 c. 50.0 b. 25.0 d. 43.3 987. Convert the polar to rectangular coordinates x, y, and z. If  = 20.6,  = 60,  = 60 a. (10.3, 15.45, 8.92) c. (15.45, 8.92, 10.3) b. (8.92, 15.45, 10.3) d. (8.92, 10.3, 15.45) 988. Find the rectangular coordinates for the point cylindrical coordinates are (8, 30, 5). a. (5, 6.93, 4) c. (4, 5, 6.93) b. (6.93, 4, 5) d. (4, 6.93, 5)

whose

989. Given the vector V = i + 2j + k, what is the angle between V and the x-axis? a. 22 c. 66 b. 24 d. 80 990. Change y = x from rectangular to polar form. a. theta = 2 or 3/2 c. theta = /4 or 5/4 b. theta = /3 or 4/3 d. theta =  or 3 991. Find the polar equation of the circle with radius a = 3/2 and the center in polar coordinates (3/2, ). 3 a. r = cos(theta) c. r = -3cos(theta) 2 1 b. r = cos(theta) d. r = -2cos(theta) 2 992. Convert the rectangular coordinates (0, 180). a. (0, 1) b. (1, 1) 993. Find the rectangular r= 6sin2usecu a. x (x2+y2) = 6y2 b. x3 (x2+y2) = 6y2

coordinate

system

the

polar

c. (0, 0) d. (1, 0)

equation

of

the

polar

c. x (x2+y2) = 6y2 d. x2 (x2+y2) = 6y2

equation

994. What is the cosine of the angle between the planes of 2x – y – 2z – 5 = 0 and 6x – 2y +3z + 8 = 0? a. 3/21 c. 67.6 b. 8/21 d. 0 995. The acute angle between the two planes 3x + 4y = 0 and 4x -7y + 4z -6 = 0 is a. 70.5 c. 64.8 b. 69.2 d. 82.5 996. Convert the spherical-coordinate equation  = 60 degrees to its rectangular-coordinates equation. a. x2 + y2 = z3/3 c. x2 + y2 = z4/3 b. x2 + y2 = z5/3 d. x2 + y2 = 3z2 997. G-numbers are generated recursively as follows: G(0) = 0, G(1) = 1 for n = 0, 1 G(n) = 2G(n – 2) + G(n – 1) for n = 2, 3, 4, 5,.. Determine G(6). a. 5 c. 11 b. 21 d. 43 998. Determine the gradient of the function f(x, y, z)  x 2  y 2  z 2 at point (1, 2, 3). Give the magnitude of the gradient of f . a. 7.21 units c. 6.00 units b. 8.25 units d. 7.48 units 999. Determine the Divergence of the vector: V = i(x2y) + j(-xy) + k(xyz) at coordinates (3, 2, 1) a. 15.00 c. 7.00 b. 9.00 d. 11.00 1000. Find the angle in degrees between the vectors joining the origin to point P(1, 2, 3) and P’(2, -3, -1). Use space vector. a. 120 c. 180 b. 60 d. 240 1001.

Find the sum of infinite geometric series



 (0.2) . i 2

a. 1 b. 1/15

c. 1/45 d. 1/30

i

1002.

Given the 3-dimensional vectors A = i(xy) + j(2yz) + k(3zx) B = i(yz) + j(2zx) + k(3xy) Determine the MAGNITUDE of the vector sum (A + B) at coordinates (3, 2, 1). a. 32.92 c. 27.20 b. 29.92 d. 24.73

1003.

1004.

1005.

For z = x2y2 + e2xy3, find z a. xy2 + 2ex b. 2xy2 + 2e2xy3

. x c. 2xy + 2exy3 d. 2x + y3

Evaluate  (6) / 2  (3) : a. 10 b. 20

c. 30 d. none of the above

Evaluate  (5 / 2) /  (1 / 2) : a. ¼ b. ¾

c. ½ d. 1

1006.

Given the following numbers x(1) to x(10): 11, 13, 9, 5, 20, 15, 1, 17, 25 These numbers are processed by the following algorithm. 1. nos. = 0 2. for i = 1 to 10 3. number = number + x(i) * x(i) 4. next i 5. number = square – root of (number/10) What is the final value of the number? a. 15.04 c. 25.36 b. 14.08 d. 11.05

1007.

Given the following numbers x(1) to x(10): 11, 13, 9, 5, 20, 15, 1, 7, 17, 25 These numbers are processed by the following algorithm. 1. nos. = 0 2. for i = 1 to 10 3. number = number + x(i) 4. next I What is the final value of the number? a. 57 c. 123 b. 112 d. 88

COMPLEX NUMBER 1008. A number of the form a + bi with a and b as real constants an i is the square root of negative one is called a. imaginary number c. radical b. complex number d. compound number 1009.

If A = -2 – i3, and B = 3 + i4, what is A/B? a. (18 – i)/25 c. (-18 + i)/25 b. (-18 – i)/25 d. (18 + i)/25

1010.

Simplify

2  3 i5  i . (3  2 i)2

221  91i 169 21  52i b. 13

a.

1011.

 7  17i 13  90  220i d. 169

c.

What is 4i cube times 2i square? a. -8i c. -8 b. 8i d. -8i2

1012. What is the simplified complex expression of (4.33 + j2.5)2? a. 12.5 + j21.65 c. 15 + j20 b. 20 + j20 d. 21.65 + j12.5 1013.

Simplify: i (exp 29) + i (exp 21) + i. a. 3i c. 1 + i b. 1 – i d. 2i

1014.

Write in the form a + bi the i(exp 3219) – i(exp 427) + a. i b. –i

1015.

Evaluate (cos 15 + isin 15)3. 2 2  i a. c. 2  2i 2 2 2 2  2i b. d. 2 5

expression i(exp 18). c. -1 d. 1

1016.

1017.

1018.

1019.

If i  a. i2

 1 , what is the value of (i)i? c. -1

b. e2i

d. e

Evaluate i raised to 96. a. –1 b. 0

c. 1 d. -i

π 2

Express the power (l + i)8 in rectangular form. a. 16 c. 1 - i b. 8i d. -6 Express

5 as a product of i and real number.

a. 5i

c. i 5

b. -5i

d. -i 5

1020.

What is the angle between -2.5 + j4.33 and 4.33 – j2.5? a. 30° c. 150° b. 120° d. 0°

1021.

Which equation has the roots 2i? a. x2 – 4x + 4 = 0 c. x2 – 4 = 0 2 b. x + 4x - 1 = 0 d. x2 + 4 = 0

1022.

Which of the following is not a root of the equation x4 + x2 + 1 = 0? a. 1/120° c. 1/135° b. 1/240° d. 1/300°

1023. For a high-voltage transmission line, we have the vector relation: Er = DEs + BIs If the per phase sending-end voltage and current are: Es = 70,000 volts at 0 degree Is = 100 amperes at 30 degrees Determine the receiving-end per phase voltage Er if D = 0.95 at 0 degree and B = 100 ohms at 90 degrees. a. 68,317 volts c. 59,001 volts b. 62,107 volts d. 65,212 volts

1024.

Evaluate ln (50/at 70 degrees) a. 4.73 + j1.48 c. 3.56 + j1.34 b. 4.30 +j1.11 d. 3.91 + j1.22

1025.

Evaluate (1 + i) raised to 10th power. a. -32i c. 2i b. -16i d. 32i

1026.

Determine the principal value of j  j . a. 127.8 c. 112.5 b. 111.3 d. 142.5

1027.

What is the value of a.

 10 x

70 i

b. - 70

7? c. - 70 i d.

70

1028. Determine the cube roots of the complex number 8 at 120. a. 2 at 0, 2 at 120, 2 at 240 b. 2 at 40, 2 at 220, 2 at 300 c. 2 at 40, 2 at 160, 2 at 280 d. 2 at 20, 2 at 140, 2 at 260 1029.

1030.

1031.

Evaluate sinh(5 + j5). a. 23.15 – j78.28 b. 21.05 – j71.16

c. 25.47 – j64.69 d. 19.14 – j86.11

4 + 8i is in what form? a. polar form b. logarithmic form

c. exponential form d. rectangular form

Evaluate cosh (0.942 + j0.429) a. 1.435 + j0.532 c. 1.435 – j0.532 b. 1.345 + j0.452 d. 1.345 – j0.452

1032. a. b. c. d.

Convert from rectangular to polar form the vector 3 + j2. 4.583/at 39.63 degrees 4.165/ at 56.31 degrees 3.068/at 41.13 degrees 3.608/ at 33.69 degrees

1033.

Find the 12th term of (asubi) = (1 – i)cubed. a. -1331 c. -1311 b. 1331 d. 1311

1034.

Determine the magnitude of the vector cross-product: (10/90°)(10/180°) a. 100 c. 80 b. 90 d. 70.6

1035.

Find the x and y such that 2x – yi = 4 + 3i. a. x = 2, y = 3 c. x = 2, y = -3 b. x = 3, y = -2 d. x = -2, y = 3

1036.

Find the rectangular coordinate of (0, 180 degrees). a. 1, 0 c. 0, 0 b. 0, 1 d. 1, 1

1037.

Express the quotient (1 – 2i)/(1 + 2i) in rectangular form. a. 0.6 – 0.8i c. 0.6 + 0.8i b. -0.6 + 0.8i d. -0.6 – 0.8i

1038.

Evaluate sin h(0.942 + j0.429). a. 0.991 + j0.614 c. -0.991 + j0.614 b. 0.991 – j0.614 d. -0.991 – j0.614

1039.

Find the values of x and y when x and y are real numbers: (2x – 4) + 9i = 8 + 3yi a. x = 6, y = 3 c. x = -6, y = 3 b. x = 3, y = 6 d. x = -3, y = 6

1040. Express the product of the imaginary number i. a. -120i b. 120i

 64x  225 using the concept of

c. 120 d. -120

1041.

Solve or z of the equation iz/2 = 3 – 4i. a. z = -4 – 3i c. z = -4 – 4i b. z = -8 – 6i d. z = 4 – 3i

1042.

Evaluate (4 + 8i)/i 3. a. -8 + 4i b. -8 - 4i

c. 8 + 4i d. 8 - 4i

1043.

Find the rectangular coordinates of [3(square root of 2) /45]. a. (3, 3) c. (1,1) b. (2, 2) d. (3, 2)

1044. What is the simplified expression of the complex number (6 + j2.5)/(3 + j4)? a. 1.12 + j0.66 c. -1.75 + j1 b. 0.32 – j0.66 d. -0.32 + j0.66 1045. Find the coordinate of the trisection points of the line segment joining P and Q where P = -6i + 3j and Q = 3i + 6j a. (-3,3), (0,-5) c. (-3,4), (0,-5) b. (-3,3), (0,5) d. (-3,4), (0,5) 1046.

Solve for x and y in the equation 2x +j(1/9) = 8 + j log y to the base of 3i a. x=2, y=3/9 c. x=2, y=31/9 1/9 b. x=3, y=3 d. x=3, y=3/9

1047.

Evaluate ln (3 + j4) a. 1.46 + j0.102 b. 1.61 + j0.927

1048.

What is the simplified expression of (4.33 + j2.5) 2 a. 20 + j20 c. 21.65 + j12.5 b. 15 + j20 d. 12.5 + j21.65

c. 1.77 + j0.843 d. 1.95 + j0.112

1049. Solve for x and y in the equation 2(exp x) + (1/9)i = 8 + log to the base 3 of y exp i. a. x = 2, y = 3/9 c. x = 2, y = 3[exp (1/9)] b. x = 3, y = 3[exp (1/9) d. x = 3, y = 3/9 1050. Find the rectangular equation of the polar equation r = 6(sin u2) sec u. a. x[x2 + y2]2 = 6y2 c. x[x2 + y2] = 6y2 b. x3[x2 + y2] = 6y2 d. x2[x2 + y2] = 6y2 1051. If A = 40ej120°, B = 20-40°, and c = 26.46 + j0. Solve for A + B + C. a. 27.7 angle 45° c. 30.8 angle 45° b. 35.1 angle 45° d. 33.4 angle 45°

1052. If a is the unit vector at 120 deg angle, determine the vector sum (1 – a + a2) in polar form a. 1.732 at -30 deg angle c. 2.000 at -60 deg angle b. 2.000 at 60 deg angle d. 1.732 at 60 deg angle 1053.

Find the principal 5th root of [50 (cos 150 + i sin150)] a. (1.9 + j1.1) c. (2.87 + j2.1) b. (3.26 – j2.1) d. (2.25 – j1.2)

1054.

Rationalize

4  3i . 2i

a. 1 + 2i b. 1055.

1056.

11  10 i 5

c.

5  2i 5

d. 2 + 2i

Evaluate cosh (i/4). a. 1.414214/at 270 degrees b. 0.707107/at 0 degree

c. 1.414214/at 180 degrees d. 0.707107/at 90 degree

Evaluate tanh (i/3) a. 1.414214 at 180 degrees b. 0.707107 at 0 degree

c. 1.7321 at 90 degrees d. 0.8660 at -90 degrees

1057.

Simplify (3 – i)2 - 7(3 – i) + 10; a. – (3 + i) c. (3 - i) b. (3 + i) d. - (3 - i)

1058.

What is the simplified expression of the complex number (6 + j2.5)/(3 + j4). a. -0.32 + j0.66 c. 0.32 - j0.66 b. 1.12 - j0.66 d. -1.75 + j1

1059.

Find the principal 5th root of [50 (cos 150° + j sin 150°)] a. (3.26 – j2.1) c. (2.25 – j1.2) b. (2.87 + j2.1) d. (1.9 + j1.1)

1060.

What is 4i3 times 2i 2? a. 8i b. -8

c. -8i2 d. -8i

1061. Three vectors A, B, and C are related as follows: A/B = 2 at 180 degrees; A + C = -5 + j15, C is conjugate of B. Find B. a. -10 + j10 c. -15 + j15 b. 10 – j10 d. 5 – j5 1062. Loads are tapped along a single-phase primary distribution line as follows: Distance from sending end Load at unity power factor

1.0 km 2.5 km 4.0 km

6.0 km

20 kW

10 kW

15 kW

15 kW

Determine the equivalent length of line, with a load at the end equal to the total load that will give the same total moments of loads. Using the equivalent line with an impedance of (0 + j10) ohms per km, determine the sendingend voltage V, if the line end voltage V, is 4000 volts. Vectorially, Vs = Vr + IZ and I = P/V at unity power factor. a. 4,424 volts c. 4,625 volts b. 4,021 volts d. 4,222 volts