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