ALGEBRA
Significant Figures
1.
Round off 0.003986 to three significant figures.
a.
0.00399
c.
0.0040
b.
0.004
d.
0.003986
2.
Perform the indicated operations and follow the rule in computing
with rounded numbers. (12.0*27.043)/(6.11*127.8)
a.
0.4156
c.
0.416
b.
0.41559
d.
0.42
Prime Numbers (Multiple and Divisor Definition)
3.
Which of the following is a prime number?
a.
203
c.
367
b.
301
d.
259
4.
Find the sum of all the prime numbers between 10 and 50.
a.
360
c.
311
b.
381
d.
1560
System of Real Numbers
Natural Numbers (N) and Whole Numbers (W)
5.
Given A = {1, 2, 3, 4, 5, 6}, B = {2, 4}, and C = {0, 1, 2, 3, 5, 8}.
Which of the following statement is false?
a.
B is a proper subset of A
c.
99 ∈ W
b.
C is a proper subset of
d.
B is a proper subset of C
W
6.
Given A = {1, 2, 3, 4, 5, 6}, B = {2,4}, and C = {0, 1, 2, 3, 5, 8}.
Which of the following statement is true?
a.
99 ∈ W
c.
2 is not an element of W
b.
0∈N
d.
C is a proper subset of N
Integers (Z), Rational (Q), Irrational (H) Numbers and Real Numbers (R)
7.
List the numbers in set A = {-2, 0, 5, √7, 12, 2/3, 4.5, √21, π, 0.75} that belong to Rational Numbers.
a.
-2, 0, 5, 12, 2/3, 4.5, c.
√7, √21, π
0.75
b.
0, 5, 12
d.
-2, 0, 5, 12
8.
Which of the following statement is false?
a.
N is a proper subset of Q
c.
H is a proper subset of Q
b.
W is a proper subset of Z
d.
Z is a proper subset of R
Algebraic Expressions Translation
x plus y
x decreased to
y
x added to y
x diminished
by y
x more than y
x reduced by y
x reduced to y
x greater than
y
x minus y
x less y
x less than y
x subtracted by
y
x subtracted to
x exceeds y
y
x decreased by
y
9.
Translate the phrase ‘Thrice a number, increased by 7’ into an
algebraic expression
a.
3n – 7
c.
7 + 3n
b.
7 – 3n
d.
3n + 7
10.
Translate the phrase ‘Four less than five times the width’ into an
algebraic expression
a.
4 + 5W
c.
4 – 5W
b.
5W + 4
d.
5W - 4
Inequalities
Solve the inequality,
11.
−2
1 5
𝑥+ ≤
3
2 6
a.
c.
x ≤ -2
x ≥ -1/2
b.
x ≥ -2
d.
x ≤ -1/2
Solve the inequality,
12.
|3𝑥 + 2|
≤1
4
2
a.
c.
𝑥 ≥ −2
−2 ≤ 𝑥 ≤
3
2
2
b.
d.
𝑥≤
𝑥=
3
3
Theory of Equations
Nature of Roots
13.
Find the sum of the roots of 𝑥3 + 6𝑥2 – 13x – 42 = 0.
a.
-6
c.
42
b.
6
d.
-42
Remainder Theorem
14.
If (5x + 𝑥4 – 14 𝑥2) is divided by (x + 4), compute the remainder.
a.
-50
c.
52
b.
-30
d.
12
15.
If 4𝑥3– 9x – 8𝑥2 + 7 is divided by (2x – 3), compute the remainder.
a.
11
c.
-15
b.
15
d.
-11
Binomial Expansion
16.
In the expansion of the expression below, what is the term
involving 𝑥2?
𝑎 10
3
(𝑥 + )
𝑥
a.
252 𝑎4 𝑥2
c.
210 𝑎5 𝑥2
b.
45 𝑎8 𝑥2
d.
120 𝑎7 𝑥2
Proportion/Variation
17.
At constant temperature, the resistance of a wire varies directly as
its length and inversely as the square of its diameter. If a piece of
wire 0.20 inch in diameter and 100 feet long has a resistance of
0.20 ohm, what is the resistance of a piece of wire of the same
material, 4000 feet long, 0.40 inch in diameter?
a.
1 ohm
c.
2 ohms
b.
4 ohms
d.
3 ohms
18.
Fifteen students from Laguna decided to live in Manila for the
review of CE licensure examination. To minimize their expenses,
they agreed to bring 21 sacks (1 sack = 50 kg) of rice that will last
for 5 months. After 5 months, 5 students went back to Laguna to
focus on self-review. Remaining students stayed in Manila for 1
month. How much more rice will the remaining students need for
the extended stay in Manila?
a.
150 kg
c.
130 kg
b.
140 kg
d.
110 kg
Number/Digit Problems
19.
The sum of the digits of a 2-digit number is 10. If the number is
divided by the unit digit, the quotient is 5 and the remainder is 2.
Find the number.
a.
28
c.
37
b.
19
d.
64
20.
The sum of the digits of a two-digit number is 12. If the digits are
reversed, the new number is 4/7 times the original number.
Determine the original number.
a.
84
c.
93
b.
75
d.
39
Age Problems
21.
In three years, a boy will be fifteen-nineteenths as old as his sister.
Three years ago, he was nine-thirteenths as old as she. How old
is the boy?
a.
12
c.
10
b.
16
d.
14
22.
A man is 41 years old and his son is 9. In how many years will the
father be three times as old as the son?
a.
8
c.
6
b.
5
d.
7
23.
Ten years ago Jane was four times as old as Bianca. Now she is
only twice as old as Bianca. Find the sum of their ages.
a.
30
c.
50
b.
15
d.
45
24.
Mary is 24 years old. Mary is twice as old as Anna was when Mary
was as old as Anna is now. How old is Anna?
a.
12
c.
14
b.
16
d.
18
Clock Problems
25.
Find the angle between the hands of the clock at 3:33 P.M.
a.
140.5°
c.
120.5°
b.
91.5°
d.
75°
26.
In how many minutes after 3 o’clock will the hands of the clock
extend in opposite directions for the first time?
a.
49.09 mins
c.
45.00 mins
b.
43.64 mins
d.
42.24 mins
27.
At what time after 7 o’clock will the second hand bisect the hour
and the minute hand for the first time?
a.
7:00:16.659
c.
7:00:17.659
b.
7:00:18.079
d.
7:00:17.233
Motion Problems
28.
I live 260 miles from a popular mountain retreat. On my way there
to do some mountain biking, my car had engine trouble – forcing
me to bike the rest of the way. If I drove 2 hours longer than I biked
and averaged 60 miles per hour driving and 10 miles per hour
biking, how many hours did I spend pedaling to the resort?
a.
4 hours
c.
3 hours
b.
2 hours
d.
1 hour
29.
At 9:00 AM, Linda leaves work on a business trip, gets on the
expressway, and sets her cruise control at 60 mph. At 9:30 A.M.,
Bruce notices she’s left her briefcase and cellphone, and
immediately starts her driving 75 mph. At what time will Bruce
catch up with Linda?
a.
11:45 AM
c.
12:05 AM
b.
11:00 AM
d.
11:30 AM
30.
Jeff had a job interview in a nearby city 72 miles away. On the first
leg of the trip he drove an average of 30 mph through a long
construction zone, but was able to drive 60 mph after passing
through this zone. If driving time for the trip was 1-1/2 hr, how long
was he driving in the construction zone?
a.
24 minutes
c.
36 minutes
b.
12 minutes
d.
42 minutes
31.
Luffy and Naruto can run around a circular mile track in 6 and 10
minutes respectively. If they start at the same instant from the
same place, in how many minutes will they pass each other if they
run around the track in the same direction?
a.
3.75 minutes
c.
10 minutes
b.
15 minutes
d.
8.75 minutes
32.
Luffy and Naruto can run around a circular mile track in 6 and 10
minutes respectively. If they start at the same instant from the
same place, in how many minutes will they pass each other if they
run around the track in opposite directions?
a.
3.75 minutes
c.
10 minutes
b.
15 minutes
d.
8.75 minutes
A boat, propelled to move at 25mi/hr in still water, travels 4.2mi
33.
against the river current in the same time that it can travel 5.8mi
with the current. Find the speed of the current.
a.
4 mi/hr
c.
6 mi/hr
b.
3 mi/hr
d.
5 mi/hr
34.
The boat travels with the water current in 2/3 time as it does
against. If the speed of the river current is 8 kph, determine the
speed of the boat in tranquil water.
a.
10 kph
c.
20 kph
b.
40 kph
d.
30 kph
35.
An airplane could travel a distance of 1000 miles with the wind in
the same time it could travel a distance of 800 miles against the
wind. If the wind velocity is 40 mph, what is the speed of the plane.
a.
360 mph
c.
420 mph
b.
180 mph
d.
240 mph
36.
An airplane flying with the wind, took two hours to travel 1000 km
and 2.5 hrs. in flying back, what was the wind velocity?
a.
60 kph
c.
70 kph
b.
50 kph
d.
40 kph
Mixture Problems
37.
As a nasal decongestant, doctors sometimes prescribe saline
solutions with n concentration between 6% and 20%. In “the old
days,” pharmacists had to create different mixtures, but only
needed to stock these concentrations, since any percentage in
between could be obtained using a mixture. An order comes in for
a 15% solution. How many milliliters (mL) of the 20% solution must
be mixed with 10 mL of the 6% solution to obtain the desired 15%
solution?
a.
18 mL
c.
20 mL
b.
15 mL
d.
17 mL
Situation 1
Give the total amount of the mix that results and the
percent concentration or worth of the mix.
38.
Two quarts of 100% orange juice are mixed with 2 quarts of water.
a.
4 quarts; 100%
c.
2 quarts; 75%
b.
4 quarts; 50%
d.
2 quarts; 25%
39.
Eight pounds of premium coffee beans worth $2.50 per pound are
mixed with 8 lb of standard beans worth $1.10 per pound.
a.
16 lb; $1.80/lb
c.
16 lb; $1.90/lb
b.
16 lb; $2.10/lb
d.
16 lb; $2.25/lb
40.
To help sell more of a lower grade meat, a butcher mixes some
premium ground beef worth $3.10/lb with 8 lb of lower grade
ground beef worth $2.05/lb. If the result was an intermediate grade
ground beef worth $2.68/lb, how much premium ground beef was
used?
a.
12 lb
c.
16 lb
b.
14 lb
d.
18 lb
41.
How many pounds of walnuts of 84 cents/lb should be mixed with
20 lb of pecans at $1.20/lb to give a mixture worth $1.04/lb?
a.
18 lb
c.
16 lb
b.
17 lb
d.
15 lb
42.
A 320-kg alloy containing 50% tin and 25% lead is to be added
with amounts of pure tin and pure lead to make an alloy which is
60% tin and 20% lead. Determine how much pure tin must be
added.
a.
60 kg
c.
40 kg
b.
80 kg
d.
100 kg
Work Problems
43.
Sakura can do a job in 3 days, and Nami can do the same job in 6
days. How long will it take them if they work together?
a.
2.5 days
c.
1.75 days
b.
2 days
d.
2.25 days
44.
A tank can be filled by three pipes separately in 20, 30, and 60
minutes respectively. In how many minutes can it be filled by the
three pipes acting together?
a.
10 minutes
c.
14 minutes
b.
12 minutes
d.
16 minutes
45.
A and B working together can complete a job in 6 days. A works
twice as fast as B. How many days would it take each of them,
working alone, to complete the job?
a.
10, 20 days
c.
8, 16 days
b.
7, 14 days
d.
9, 18 days
46.
Zorro’s rate of doing work is three times that of Sanji. On a given
day Zorro and Sanji work together for 4 hours; then Sanji is called
away and Zorro finishes the rest of the job in 2 hours. How long
would it take Sanji to do the complete job alone?
a.
20 hours
c.
24 hours
b.
22 hours
d.
26 hours
47.
A bathtub is filled through the faucet and then emptied through its
drain in a total of 2 hours. If water enters through the faucet and
simultaneously allowed to leave through its drain, the bathtub is
filled in 1-7/8 hours. How long will it take to fill the tub with the drain
closed?
a.
60 minutes
c.
30 minutes
b.
25 minutes
d.
45 minutes
48.
Twenty-eight men can finish the job in 60 days. At the start of the
16th day, 5 men were laid off and after the 45th day, 10 more men
were hired. How many days were they delayed in finishing the job?
a.
1.65 days
c.
2.67 days
b.
2.83 days
d.
2.27 days
Consecutive Integers
49.
The sum of three consecutive odd integers is 69. One of the
integers is
a.
27
c.
19
b.
29
d.
23
50.
Give one of the two consecutive even integers such that the sum
of twice the smaller integer plus the larger integer is one hundred
forty-six.
a.
50
c.
54
b.
52
d.
56
51.
Seven times the first of two consecutive odd integers is equal to
five times the second. Find the larger integer.
a.
5
c.
9
b.
7
d.
11
Arithmetic Progression
52.
Find the 16th term of the arithmetic sequence: 4,7,10,…
a.
52
c.
46
b.
49
d.
55
Determine the sum of the first 12 terms of the arithmetic sequence:
53.
3,8,13,…
a.
328
c.
432
b.
412
d.
366
54.
Find the 40th term and the sum of the first 40 terms of the
arithmetic sequence: 10,8,6,…
a.
-72, -1164
c.
-70, -1162
b.
-66, -1158
d.
-68, -1160
55.
Which term of the sequence 5, 14, 23,... is 239?
a.
27
c.
25
b.
26
d.
28
56.
Compute the sum of the first 100 positive integers exactly divisible
by 7.
a.
36057
c.
36771
b.
35350
d.
34650
57.
How many consecutive integers, beginning with 10, must be taken
for their sum to equal 2035?
a.
74
c.
65
b.
55
d.
34
58.
How long will it take to pay off a debt of $880 if $25 is paid the first
month, $27 the second month, $29 the third month, etc.?
a.
10 months
c.
20 months
b.
25 months
d.
15 months
59.
Determine the 20th term of the arithmetic sequence whose sum to
n terms is n2 +2n.
a.
39
c.
43
b.
37
d.
41
60.
Find the second number of three numbers in an arithmetic
sequence such that the sum of the first and third is 12 and the
product of the first and second is 24.
a.
8
c.
4
b.
10
d.
6
61.
Compute the sum of all integers between 100 and 800 that are
divisible by 3.
a.
104844
c.
104850
b.
104841
d.
104847
62.
There are 5 arithmetic means between 8 and 26. Which of the
following is not part of the sequence?
a.
11
c.
17
b.
15
d.
20
63.
Give the sum of all arithmetic means between 1 and 36 if the sum
of the resulting arithmetic sequence will be 148.
a.
111
c.
116
b.
106
d.
101
64.
There are 5 arithmetic means between 8 and 26. One is
a.
13
c.
19
b.
17
d.
22
Geometric Progression
65.
Find the 8th term of the sequence 4, 8, 16,…
a.
128
c.
256
b.
1024
d.
512
66.
Find the sum of the first seven terms of the sequence 9, -6, 4,…
a.
552/81
c.
463/81
b.
367/81
d.
243/81
The second term of a geometric sequence is 3 and the fifth term
67.
is 81/8. Find the eighth term.
a.
2257/64
c.
2635/64
b.
2187/64
d.
2517/64
68.
Find three numbers in a geometric sequence whose sum is 26
and whose product is 216.
a.
2, 4, 8
c.
2, 8, 32
b.
2, 10, 50
d.
2, 6, 18
69.
In a geometric sequence consisting of four terms in which the ratio
is positive, the sum of the first two terms is 8 and the sum of the
last two terms is 72. Find the 3rd term of sequence.
a.
2
c.
54
b.
6
d.
18
70.
It is estimated that the population of a certain town will increase
10% each year for four years. What is the percentage increase in
population after four years?
a.
21%
c.
46%
b.
61%
d.
33%
71.
From a tank filled with 240 gallons of alcohol, 60 gallons are drawn
off and the tank is filled up with water. Then 60 gallons of the
mixture are removed and replaced with water, etc. How many
gallons of alcohol remain in the tank after 5 drawings of 60 gallons
each are made?
a.
43 gal
c.
76 gal
b.
101 gal
d.
57 gal
There are five geometric means between 9 and 576. Which of the
72.
following cannot be a mean?
a.
288
c.
-36
b.
144
d.
-72
Infinite Geometric Series
73.
Find the sum of the infinite geometric series
1/3, -2/9, 4/27, -8/81…
a.
1/8
c.
1/5
b.
1/7
d.
1/6
74.
The distances passed over by a certain pendulum bob in
succeeding swings form the geometric sequence 16, 12, 9,…
inches respectively. Calculate the total distance traversed by the
bob before coming to rest.
a.
18 inches
c.
64 inches
b.
54 inches
d.
32 inches
Harmonic Progression
75.
Compute the 15th term of the harmonic sequence 1/4, 1/7, 1/10…
a.
1/40
c.
1/43
b.
1/49
d.
1/46
76.
Derive the formula for the harmonic mean, H, between two
numbers p and q.
a.
2(p+q)/(pq)
c.
(p+q)/(2pq)
b.
pq/[2(p+q)]
d.
2pq/(p+q)
77.
What is the harmonic mean between 3/8 and 4?
a.
34/25
c.
35/24
b.
25/34
d.
24/35
78.
There are three harmonic means between 10 and 20. One of these
is
a.
1/16
c.
7/80
b.
40/3
d.
1/16
79.
Identify the sequence -1, -4, 2…
a.
arithmetic
c.
harmonic
b.
geometric
d. None among the choices
Sequence and Series
80.
Find the nth term for the sequence ¼, 2/7, 3/10, 4/13…
a.
n/(2n + 1)
c.
n/(3n + 1)
b.
n/(n+3)
d.
n/(n + 2)
Exponents
81.
The number of population of a certain microorganism after t hours
is 3.86(105)(2t/4). How long will it take for the population to
quadruple?
a.
2 hours
c.
8 hours
b.
6 hours
d.
4 hours
Situation 1
A small business makes a new discovery and
begins an aggressive advertising campaign,
confident they can capture 66% of the market in a
short period of time. They anticipate their market
share will be modeled by the function
66
𝑀(𝑡) =
1 + 10𝑒−0.05𝑡
Where M(t) represents the percentage after t days.
82.
What was the company’s initial market share?
a.
8%
c.
6%
b.
14%
d.
12%
83.
What was their market share after 30 days later?
a.
33.1%
c.
20.4%
b.
42.6%
d.
18.6%
84.
How long will take to double the market share?
a.
12 days
c.
16 days
b.
18 days
d.
14 days
85.
Solve for y if 8𝑥 = 2𝑦+2 and 163𝑥−𝑦 = 4𝑦 .
a.
5
c.
2
b.
3
d.
4
Logarithms
86.
If loga10 = 0.50, what is the value of log10 a?
a.
6
c.
4
b.
5
d.
2
Determine the value of y in the equation
87.
𝑒𝑥
𝑦 = 𝑙𝑛 ( 𝑥−4)
𝑒
a.
5
c.
2
b.
6
d.
4
88.
If log 3 = x and log 2 = y, find log 2.4
a.
3xy – 1
c.
3x + y - 1
b.
3x + y
d.
x + 3y - 1
89.
Which of the following is the cologarithm of 256 to the base 10?
a.
-0.47
c.
2.41
b.
0.47
d.
-2.41
90.
Solve the logarithmic equation log(x – 1) – logx = log (x – 3).
a.
-3.732
c.
3.732
b.
-0.267
d.
0.267
x+1
2x
91.
Solve the exponential equation 5 = 6
a.
0.815
c.
0.811
b.
0.851
d.
0.855
92.
If log x = y where b is the base of logarithm, then x is equal to
a.
c.
y
by
b.
d.
b
yb
Find the value of logx146 if logx18 = 3.154.
93.
a.
5.44
c.
2.67
b.
2.50
d.
5.23
If the three positive numbers x, y, z are in geometric progression,
94.
which of the following is true?
a.
2 log y = log x + log z
c.
(log x)(log z ) = 2log y
b.
log (x+y) = log 2z
d.
log x – log y = log z
Partial Fraction
Find the value of A in the equation
95.
4𝑥 + 11
𝐴
𝐵
=
+
𝑥2 + 7𝑥 + 10 𝑥 + 5 𝑥 + 2
a.
2
c.
3
b.
4
d.
1
Find the value of C in the equation
96.
9
𝐴
𝐵
𝐶
=
+
+
(𝑥 + 5)(𝑥2 + 7𝑥 + 10) 𝑥 + 2 𝑥 + 5 (𝑥 + 5)2
a.
2
c.
-3
b.
4
d.
1
97.
Find the value of B in the equation
3𝑥2 − 𝑥 − 11
𝐴
𝐵𝑥 + 𝐶
=
+
(𝑥 − 3)(𝑥2 + 4) 𝑥 − 3 (𝑥2 + 4)
a.
1
c.
2
b.
3
d.
5
Imaginary Number
98.
The complex number 5 – 3i is divided by 2 – i. Find the quotient.
13 + i
13 + i
a.
c.
5
3
13 – i
13 – i
d.
b.
5
3
99.
Find the absolute value of 5 – 3i.
a.
5.30
c.
2.00
b.
2.58
d.
5.83
100.
Solve for x in the equation 3ix – 5 + 3i = (3 – i)y + i.
a.
-1/9
c.
5/3
b.
3/5
d.
1/9
101.
If you multiply the expression (-2)^(1/2) and (-1/8)^(1/2), what will
be the result?
a.
i
c.
-1/2
b.
-i
d.
1
PLANE TRIGONOMETRY
Complementary, Supplementary and Explementary/Conjugate Angles
1.
The supplement of an angle is thrice its complement. Determine
the angle.
a.
60°
c.
112.5°
b.
45°
d.
35°
2.
Give the explement of angle 22.5°.
a.
67.5°
c.
157.5°
b.
247.5°
d.
337.5°
Coterminal Angles
3.
Which of the following angles is not coterminal of 29°52’11”?
a.
389°52’11”
c.
-690°7’49”
b.
749°52’11”
d.
569°52’11”
Vertical Angles
4.
Find the measure of angle 2, given that lines m and n are parallel.
(3x + 2)° 1
m
2
3
n
4 (5x – 40)°
a.
120°
c.
130°
b.
115°
d.
105°
Similar Triangles
5.
Firefighters at the Monumento Fire Station need to measure the
height of the station flagpole. They find that at the instant when the
shadow of the station is 18 m long, the shadow of the flagpole is
99 ft long. The station is 10 m high. Find the height of the flagpole.
a.
55.0 m
c.
69.10 m
b.
16.8 m
d.
28.44 m
6.
A lifeguard located 20 yd from the water spots a swimmer in
distress. The swimmer is 30 yd from shore and 100 yd east of the
lifeguard. Suppose the lifeguard runs, then swims to the swimmer
in a direct line. How far east from his original position will he enter
the water?
a.
120 ft
c.
45 ft
b.
40 ft
d.
35 ft
Trigonometric Functions
7.
The terminal side of an angle θ in standard position passes
through the point (9, 15). Determine the cotangent of the angle.
a.
1.94
c.
1.67
b.
0.51
d.
0.60
8.
Identify the quadrant/s of an angle that satisfies the given condition
sin θ > 0, tan θ < 0.
a.
II, IV
c.
II
b.
I, II, IV
d.
I, II
9.
Identify the quadrant/s of an angle that satisfies the given condition
cos θ < 0, sec θ < 0.
a.
II, IV
c.
II
b.
II, III
d.
III
10.
Which of the following is possible?
a.
sec θ = 2/3
c.
sin θ = 2.5
b.
csc θ = -0.25
d.
tan θ = 110.47
11.
Suppose that angle θ is in quadrant II and sin θ = 2/3. Find the
value of cotangent function.
a.
c.
-2√5/2
√5/2
b.
d.
3/2
-√5/2
12.
Find tan θ, given that cos θ = -√3/4 and sin θ > 0.
a.
c.
√13/4
−√39/3
b.
d.
-√3/4
4/√13
13.
Determine the value of θ.
1
tan (3θ - 4°)=
cot (5θ - 8°)
a.
2
c.
5
b.
3
d.
0.03
14.
Determine the value of θ.
sin (4θ + 2°) csc(3θ+5°) = 1
a.
4°
c.
2°
b.
5°
d.
3°
15.
The haversine of an angle is 0.152. Determine the angle in degrees.
a.
32.01°
c.
31.02°
b.
45.89°
d.
54.98°
Cofunction Identities
16.
Find the solution of the equation
cos (θ + 4°) = sin(3θ+2°)
a.
24°
c.
21°
b.
12°
d.
32°
17.
If sin 3A = cos 6B, find the value of A + 2B.
a.
30°
c.
15°
b.
60°
d.
45°
Reference Angles
18.
Determine the reference angle of of 1387°.
a.
53°
c.
233°
b.
307°
d.
127°
Solution of Right Triangles
19.
Edgar knows that when he stands 123 ft from the base of a flagpole,
the angle of elevation to the top of the flagpole is 26°40’. If his eyes
are 5.30 ft above the ground, find the height of the flagpole.
a.
58.2 ft
c.
61.8 ft
b.
52.4 ft
d.
67.1 ft
20.
The length of the shadow of a building 34.09 m tall is 37.62 m. Find
the angle of elevation of the sun.
a.
36.58°
c.
29.33°
b.
42.18°
d.
19.12°
21.
A 13.5-m fire truck ladder is leaning against a wall. Find the
distance the ladder goes up to the wall (above the top of the fire
truck) if the ladder makes an angle of 43°50’ with the horizontal.
a.
8.35 m
c.
9.35 m
b.
7.35 m
d.
6.35 m
22.
From a window 30.0 ft above the street, the angle of elevation to the
top of the building across the street is 50.0° and the angle of
depression to the base of this building is 20.0°. Find the height of
the building across the street
a.
102 ft
c.
133 ft
b.
128 ft
d.
119 ft
23.
To determine the diameter of the sun, an astronomer might sight
with a transit first to one edge of the sun and then the other,
estimating that the included angle equals 32’. Assuming that the
distance from Earth to the sun is 92,919,800 mi, approximate the
diameter of the sun.
a.
864943 mi
c.
834694 mi
b.
846439 mi
d.
894634 mi
The length of the base of an isosceles triangle is 42.36 in. Each
24.
base angle is 38.12°. Find the length of each of the two equal sides
of the triangle.
a.
22.92 in
c.
33.12 in
b.
31.23 in
d.
26.92 in
Find the altitude of an isosceles triangle having base 184.2 cm if the
25.
angle opposite the base is 68°44’.
a.
143.6 cm
c.
123.2 cm
b.
134.7 cm
d.
152.5 cm
A pyramid has a square base with sides 700 ft long and its height is
26.
200 ft. Find the angle of elevation of the pyramid’s edge.
a.
26°
c.
28°
b.
22°
d.
20°
Edmundo needs to know the height of a tree. From a given point on
27.
the ground, he finds that the angle of elevation of the top of the tree
is 36.7°. He then moves back 50 ft. From the second point, the
angle of elevation to the top of the tree is 22.2°. Find the height of
the tree.
a.
40 ft
c.
45 ft
b.
35 ft
d.
50 ft
28.
From a point on a level ground, the angles of elevation of the top
and bottom of a flagpole situated on the top of a hill are measured
as 47°54’ and 39°45’. Find the height of the hill if the height of the
flagpole is 115.5 ft.
a.
439.3 ft
c.
349.3 ft
b.
644.8 ft
d.
464.8 ft
29.
From the top of a lighthouse, 175 ft above the water, the angle of
depression of a boat due south is 18°50’. Calculate the speed of the
boat if, after it moves due west for 2 min, the angle of depression is
14°20’.
a.
312 ft/min
c.
203 ft/min
b.
227 ft/min
d.
355 ft/min
30.
A circular log, 5 ft in diameter rolls up an incline of 18°20’. What is
the height of the center of the log above the base of the incline
when the wheel has rolled 5 ft up the incline?
a.
2.67 ft
c.
3.45 ft
b.
3.95 ft
d.
2.95 ft
31.
Determine the shortest distance a lizard can travel from upper
corner of the room with dimension 3 m x 3 m x 3 m to the lower
corner. Line connecting the upper corner and lower corner is the
diagonal of the cube.
a.
6.71 m
c.
7.24 m
b.
9.00 m
d.
5.20 m
32.
In what direction should the ship cruise in order to sail due south at
40 kph if the ocean current is moving due east at 8 kph?
a.
N 11°19’ E
c.
S 11°19’ E
b.
N 11°19’ W
d.
S 11°19’ W
Unit of Angles
33.
Convert 9π/4 to degrees. Express answer in sexagesimal system.
a.
7200 mil
c.
1.125 circle
b.
450 grad
d.
405°
34.
What is the equivalent of 405° in mils?
a.
6400 mils
c.
5400 mils
b.
7200 mils
d.
3600 mils
Radian Measure Applications
35.
A circle has radius 18.20 cm. Find the length of the arc intercepted
by a central angle 144°.
a.
35.84 cm
c.
54.34 cm
b.
45.74 cm
d.
63.45 cm
36.
The latitude of Reno is 40°N, while that of Los Angeles is 34°N. The
radius of Earth is 6400 km. Find the north-distance between two
cities.
a.
590 km
c.
620 km
b.
670 km
d.
540 km
37.
Two gears are adjusted so that the smaller gear (r = 2.5 cm) drives
the larger one (R = 4.8 cm). If the smaller gear rotates through an
angle of 225°, through how many degrees will the larger gear
rotate?
a.
147°
c.
117°
b.
137°
d.
127°
Graphs of the Circular Function
38.
Determine the period of the function f(x) = sin x
a.
π
c.
3/2 π
b.
2π
d.
π/2
39.
Function f(x) = cos x has x-intercepts in the form (where n is an
integer)
a.
(1 + n) π/2
c.
(1 + 2n) 2π
b.
(2n + 1) π/2
d.
(1 + n) 2π
The voltage E in an electrical circuit is modeled by
40.
𝐸 = 5 cos 120𝜋𝑡
where t is time measured in seconds.
Find the amplitude.
a.
24
c.
5
b.
12
d.
120
41.
Give the phase shift of the function y = sin (x – π/3).
a.
π/3 left
c.
π/3 down
b.
π/3 right
d.
π/3 up
42.
Give the phase shift of the function y = –2 cos (3x + π).
a.
π/3 left
c.
π/3 down
b.
π/3 right
d.
π/3 up
43.
Determine the vertical translation of the function
y = 3 – 2 cos 3x
a.
3 up
c.
2 up
b.
3 down
d.
2 down
44.
Determine the period of the function
y = – 1 + 2 sin (4x + π)
a.
2π
c.
π/2
b.
π/8
d.
π/4
Determine the period of the function
45.
y = tan x
a.
2π
c.
π
b.
π/4
d.
π/2
46.
Given the function,
y = cot x
one of its vertical asymptote is
a.
π/8
c.
π
b.
π/4
d.
π/2
47.
Given the function,
y = – 2 – cot (x – π/4)
determine the vertical translation.
a.
2 left
c.
2 down
b.
2 right
d.
2 up
48.
Determine the period of the function
y = 2 sec (1/2)x.
a.
4π
c.
π
b.
2π
d.
π /2
Harmonic Motion
Situation 1
Suppose that an object is attached to a coiled spring. It is
pulled down a distance of 5 in from its equilibrium
position, and then released. The time for one complete
oscillation is 4 sec.
49.
Give the equation that models the position of the object at time t.
𝜋
a.
c.
𝑠(𝑡) = 5 sin 𝜋 𝑡
𝑠(𝑡) = −5 cos 𝑡
2
𝜋
b.
d.
𝑠(𝑡) = 5 cos 𝜋 𝑡
𝑠(𝑡) = −5 sin 𝑡
2
50.
Determine the position at t = 1.5 sec.
a.
4.65 in
c.
-0.35 in
b.
3.54 in
d.
-2.34 in
51.
Find the frequency.
a.
0.75
c.
0.50
b.
1
d.
0.25
52.
Suppose that an object oscillates according to the model
𝑠(𝑡) = 8 sin 3𝑡
Where t is in seconds and s(t) is in feet. Determine the frequency of
the oscillation.
a.
0.64
c.
0.33
b.
0.48
d.
0.51
2.
Trigonometric Identities
53.
Given a trigonometric expression,
cot 𝜃 + 1
Determine its identity.
a.
csc 𝜃 (cos 𝜃 + sin 𝜃)
c.
b.
csc 𝜃 (cos 2𝜃)
d.
Given a trigonometric expression,
54.
tan 𝑡 − cot 𝑡
sin 𝑡 cos 𝑡
Determine its identity.
a.
sec2 𝑡 − csc2 𝑡
c.
b.
sin2 𝑡 − cos2 𝑡
d.
55.
Which of the following is not an identity?
cos 𝑥
1 + sin 𝑥
=
a.
c.
1 − sin 𝑥
cos 𝑥
sec 𝛼 + tan 𝛼
sec 𝛼 − tan 𝛼
b.
d.
1 + 2 sin 𝛼 + sin2 𝛼
=
cos2 𝛼
Trigonometric Equations
56.
Solve
cos-1 x = sin-1
csc 𝜃 (sin 2𝜃)
sec 𝜃 (cos 𝜃 + sin 𝜃)
csc2 𝑡 − cot2 𝑡
sec2 𝑡 − tan2 𝑡
cos 𝜃 + 1
= cot 𝜃
sin 𝜃 + tan 𝜃
tan 𝑥 − cot 𝑥
= 2sin2 𝑥
tan 𝑥 + cot 𝑥
1
2
a.
c.
√3/4
√2/3
b.
√2/4
d.
√3/2
57.
Solve 2 sin θ + 1 = 0
a.
210°, 330°
c.
330°, 120°
b.
120°, 310°
d.
210°, 310°
58.
Solve tan x + √3 = sec x
a.
7 π /6
c.
5 π /6
b.
2 π /6
d.
11 π /6
59.
Determine the value of tan A + tan B + tan C when (tan A)(tan
B)(tan C) = 4.98 and A + B + C = 180°.
a.
4.98
c.
3.12
b.
0.20
d.
0.86
Solution of Oblique Triangles
60.
Two ranger stations are on an east-west line 110 miles apart. A
forest fire is located on a bearing of N 42° E from the western
station at A and a bearing of N 15° E from the eastern station at B.
How far is the fire from the western station?
a.
180.06 mi
c.
133.12 mi
b.
234.04 mi
d.
206.83 mi
61.
In order to determine the elevation of point Y on top of a cliff, a
transit was set over a point X, the elevation of which was known to
be +100.345 m and the height of instrument from the ground was 5
m. The recorded angle of elevation of Y from point X was 17°38’.
Point Z was next located at a horizontal distance of 200 m from
point X. Then the transit was set next at a point Z and the angle of
elevation of Y from this point was 25°34’. Finally, with the telescope
horizontal, the reading on the leveling rod on point X was 7.80 m.
Determine the elevation of Y.
a.
313.15 m
c.
289.24 m
b.
227.33 m
d.
355.71 m
62.
A tower 125 ft high is on a cliff on the bank of a river. From the top
of the tower, the angle of depression of a point on the opposite
shore is 28°40’, and from the base of the tower, the angle of
depression of the same point is 18°20’. Find the width of the river.
a.
365 ft
c.
580 ft
b.
412 ft
d.
192 ft
63.
From A a pilot flies 125 km in the direction N38°20’W and turns
back. Through an error, the pilot then flies 125 km in the direction
S51°40’E. In what direction must the pilot now fly to reach the
intended destination A?
a.
S 49°20’ W
c.
S 45°20’ W
b.
S 35°13’ W
d.
S 38°15’ W
64.
To determine the height of a distant mountain, two observation
points were set up at A and B, which is 290 m closer to but 25 m
lower in elevation than A. The angle of elevations from both points
are 24.45° and 35.45°, respectively. If the elevation at A is 550.45
m, find the elevation of the top of the mountain.
a.
959.48 m
c.
970.22 m
b.
967.11 m
d.
981.23 m
65.
Two insects fly from the same point but towards different directions.
One was flying at a speed of 12.2 m/min while the other was flying
at 17.4 m/min. the angle between their flight direction is 84.1°. How
many meters are they far apart after 2.1 minutes?
a.
37.65 m
c.
55.11 m
b.
42.42 m
d.
29.47 m
66.
A mast leans 15° from the vertical towards the sun. If it casts a
shadow 10 m long when the angle of elevation of the sun is 35°,
calculate the length of the mast.
a.
6.98 m
c.
7.86 m
b.
6.03 m
d.
8.92 m
67.
A pilot wishes a course 15°0’ against a wind of 25 mi/h from
160°30’. Find the groundspeed when the airspeed is 175 mph.
a.
185 mph
c.
175 mph
b.
195 mph
d.
165 mph
PLANE GEOMETRY
Plane Figures
Triangle
1.
Two sides of a triangle measure 30 m and 35 m. One possible
dimension of the third side is
a.
65 m
c.
66 m
b.
64 m
d.
67 m
A triangle has sides 36 m, 32 m, and 60 m. Determine the area.
a.
367.1 m2
c.
478.9 m2
b.
106.7 m2
d.
59.9 m2
Two sides of a triangle measure 201 m and 150 m. If its area is1.26
3.
ha, calculate the perimeter.
a.
625.78 m
c.
660.84 m
b.
592.34 m
d.
548.23 m
4.
Triangle ABC has courses AB = 160 km, BC = 190 km and CA =
190 km. Station D is along course AB and AD is 100 km. Station E
is along course CA. Calculate the length of course AE if the area of
triangle ADE formed is 3/5 the area of triangle ABC.
a.
167.3 km
c.
159.4 km
b.
182.4 km
d.
175.9 km
5.
The angles of a triangle are in the ratio of 1:4:5. Determine the area
of triangle if the longest side measures 120 m.
a.
2161.05 m2
c.
2116.03 m2
2
b.
2210.21 m
d.
2301.09 m2
6.
The sides of a triangle measure 42 cm, 54 cm, and 72 cm. A circle
is drawn such that its center lies on the shortest side and tangent to
the other sides. Calculate the radius of the circle.
a.
15.9 cm
c.
16.7 cm
b.
16.1 cm
d.
17.9 cm
7.
Two of the interior angles of a triangle are 107°31’ and 52°20’. If its
area is 2400 m2, determine the shortest side.
a.
43.6 cm
c.
52.1 cm
b.
72.6 cm
d.
46.8 cm
8.
The ratio of the angles of a triangle is 2:3:7. If the area of the
triangle is 10 hectares, calculate the perimeter of the triangle.
a.
1662.96 m
c.
1626.66 m
b.
1266.69 m
d.
1696.99 m
9.
The hypotenuse of a right triangle is 95 m. If one of its leg is 52 m,
find its area.
a.
2067.12 m2
c.
2167.11 m2
b.
2076. 21 m2
d.
2760.22 m2
Quadrilateral

Parallelogram
10.
One pair of parallel sides of a parallelogram measure 35 m and the
distance between them is 28 m. Calculate the area of the
parallelogram.
a.
490 m2
c.
693 m2
b.
980 m2
d.
890 m2
11.
The diagonals of a parallelogram measure 31 cm and 45 cm. If they
intersect at an angle of 48°, determine the area of the
parallelogram.
a.
518.34 cm2
c.
337.89 cm2
b.
429.65 cm2
d.
629.11 cm2
12.
Sides of a parallelogram measure 150 and 85 mm. If one of the
diagonals is 135 mm long, determine the length of the other
diagonal.
a.
178 mm
c.
203 mm
b.
245 mm
d.
187 mm

Square
13.
Sides of a square measure 65 m. Calculate its diagonal.
a.
86.75 m
c.
79.29 m
b.
91.92 m
d.
81.92 m

Rhombus
14.
The diagonals of a rhombus measure 260 cm and 500 cm.
Calculate its area.
a.
55,000 cm2
c.
45,000 cm2
b.
60,000 cm2
d.
65,000 cm2
15.
Sides of a rhombus measure 56 mm. If one of its interior angles
measure 129°, calculate the area of the rhombus.
a.
3247.41 mm2
c.
2437.14 mm2
b.
4732.41 mm2
d.
2347.41 mm2
16.
The diagonals of a rhombus measure 30 mm and 20 mm. Calculate
the area of the circle inscribed in it.
a.
217.49 mm2
c.
311.29 mm2
b.
223.94 mm2
d.
324.67 mm2

Trapezoid
17.
Base angles of a trapezoid measure 79° and 36°, respectively. If
parallel sides of the trapezoid measure 46 m and 31 m, determine
the area of the trapezoid.
a.
367.66 m2
c.
429.01 m2
b.
483.56 m2
d.
356.67 m2

General Quadrilateral: Trapezium
18.
Given the sides of a quadrilateral: 36 mm, 24 mm, 50.91 mm, and
60 mm. The sum of two opposite interior angles is 225°. Calculate
the area of the quadrilateral.
a.
1545.43 mm2
c.
1511.97 mm2
b.
1533.26 mm2
d.
1587.79 mm2

Cyclic Quadrilateral: Trapezium
19.
Three sides of a cyclic quadrilateral measure 252 m, 368 m and 280
m, consecutively. If the angle measurement from the first to second
side is 107°, calculate the area of the quadrilateral.
a.
23.11 ha
c.
21.13 ha
b.
13.22 ha
d.
11.23 ha
20.
A quadrilateral having an area of 260 m2 is inscribed in a circle.
Three of its sides measure 24 m, 16 m, and 10 m, consecutively.
Compute for the length of the fourth side.
a.
14.56 m
c.
18.22 m
b.
14.23 m
d.
17.67 m
21.
The sides of a cyclic quadrilateral measure 20.097 m, 13.893 m,
27.5385 m, and 21.516 m, consecutively. Give the value of the
product of its diagonals.
a.
852.363 m2
c.
784.986 m2
b.
258.645 m2
d.
542.979 m2
Relationships Within Triangles

Mid-Segments of a Triangle
22.
The sides of a triangle ABC measure AB = 31.5 m, BC = 49.5 m,
and CA = 54.0 m. Find the length of the mid-segment parallel to the
side CA.
a.
24
c.
25
b.
27
d.
28
23.
The sides of the triangle ABC measure AB = 25.74 m, BC = 14.30
m, and CA = 31.46 m. Calculate the area of the given triangle
formed by the three mid-segments.
a.
48.19 m2
c.
45.37 m2
2
b.
54.87 m
52.44 m2

Medians of a Triangle
24.
The sides of a triangle measure 33.48 m, 44.64 m, and 66.96 m.
Determine the length of the median to the longest side of the
triangle.
a.
19.67 m
c.
21.43 m
b.
20.88 m
d.
23.21 m
25.
The sides of the triangle ABC measure AB = 67.83 m, BC = 33.25
m, and CA = 62.51 m. Compute the distance from the intersection
of the medians to the midpoint of side CA.
a.
18.67 m
c.
12.86 m
b.
14.44 m
d.
13.23 m
26.
The sides of the triangle ABC measure AB = 199.92 cm, BC =
344.96 cm, and CA = 411.60 cm. How far is the intersection of the
medians to side BC?
a.
77.11 cm
c.
66.45 cm
b.
80.43 cm
d.
58.92 cm

Angle Bisectors of a Triangle
27.
In triangle ABC, AB = 74.80 cm, BC = 112.20 cm, and CA = 149.60
cm. Compute the length of the bisector Angle ABC.
a.
45.79 cm
c.
81.21 cm
b.
54.97 cm
d.
72.27 cm
28.
In triangle ABC, AB = 25.05 m, BC = 30.06 m, and CA = 40.08 m.
Compute the distance from the point of intersection of the angle
bisectors from vertex C.
a.
32.45 m
c.
44.87 m
b.
23.89 m
d.
38.12 m
29.
The sides of the triangle ABC are AB = 19.35 cm, BC = 23.22 cm,
and CA = 30.96 cm. Determine the distance from the point of
intersection of the angular bisectors to the side AB.
a.
7.11 cm
c.
6.10 cm
b.
6.39 cm
d.
5.96 cm

Perpendicular Bisectors of a Triangle
30.
The sides of a triangle measure 42.90 mm, 64.35 mm and 72.93
mm. Calculate the distance from the intersection of the
perpendicular bisectors to the vertices
a.
48.93 mm
c.
36.73 mm
b.
58.46 mm
d.
63.37 mm
31.
A triangle has the following sides: 170.4 m, 198.8 m and 227.2 m.
Determine the distance from the intersection of the perpendicular
bisector of all sides of the triangle to the shortest side.
a.
80.66 m
c.
85.12 m
b.
90.33 m
d.
92.56 m

Altitudes of a Triangle
32.
A triangle ABC has sides AB = 185.6 mm, BC = 232.0 mm and CA
= 278.4 mm. Calculate the altitude to the longest side of the
triangle.
a.
145.53 mm
c.
135.54 mm
b.
163.52 mm
d.
153.45 mm
33.
The courses of a triangular lot ABC were measured by a tape. The
following were the results: AB = 66.3 m, BC = 114.4 m and CA =
136.5 m. Calculate the distance from the point of intersection of
medians to course BC.
a.
32.40 m
c.
18.65 m
b.
22.04 m
d.
41.32 m
Escribed Circles of a Triangle
34.
The sides of a triangle measure 573.24 m, 632.25 m, and 649.11 m.
Compute for the radius of the circle which is escribed outside the
triangle if it is tangent to the shortest side.
a.
463.65 m
c.
433.98 m
b.
448.25 m
d.
472.67 m
35.
The sides of a triangle measure 111.3 m, 143.1 m and 174.9 m.
Determine the radius of the circle which is escribed outside triangle
if it is tangent to the 143.1-m side.
a.
132.33 m
c.
121.67 m
b.
106.54 m
d.
111.02 m
Sector of a Circle
36.
Calculate the central angle of the sector if its area is 32% of the
circle.
a.
115.2°
c.
105.3°
b.
123.5°
d.
132.1°
37.
The area and the perimeter of a circular sector are 250 m2 and 66
m, respectively. Calculate the radius of the sector.
a.
21.22 m
c.
18.11 m
b.
17.83 m
d.
20.33 m
38.
A circle of radius 15 cm is inscribed in a sector of radius 75 cm.
Calculate the area of the sector.
a.
1421 cm2
c.
1142 cm2
b.
1241 cm2
d.
1412 cm2
Segment of a Circle
39.
A fish pond is constructed in the shape of two intersecting identical
circles having a radius of 10 m. The distance between their centers
is 10 m. Calculate the area of the artificial fish pond.
a.
809.12 m2
c.
414.16 m2
b.
505.48 m2
d.
705.12 m2
40.
Two circles with radii of 20 m and 32 m are placed on a plane so
that they intersect at right angles. Compute for the area common to
the two circles.
a.
425.32 m2
c.
528.15 m2
b.
336.88 m2
d.
260.65 m2
Theorems on Circles

Peripheral Angle Theorem
41.
Two chords of a circle AB and BC intersect at an angle 52°.
Calculate the central angle of arc AC.
a.
65°
c.
26°
b.
104°
d.
84.5°
42.
Two chords of a circle AB and BC measure 20 m and 8 m,
respectively. If the angle between the chords is 127°, determine the
radius of the circle.
a.
14.71 m
c.
17.42 m
b.
18.71 m
d.
19.23 m

Cross-Chord Theorem
43.
Two chords of a circle AC and BD intersect at point O inside the
circle. If OA = 42 cm, OC = 84 m and OD = 63 m, determine the
measure of OB.
a.
42 m
c.
36 m
b.
56 m
d.
84 m
44.
From the figure shown, determine the
area of the shaded region.
76.97 m2
82.08 m2

Secant-Secant Theorem
45.
From the figure shown,
determine the area of
quadrilateral ABDC. Given:
OA = 156 m, AB = 60 m,
CD = 108 m, and θ = 35°.
a.
b.
a.
b.
c.
d.
70.65 m2
54.65 m2
85.93 are
c.
75.67 are
90.53 are
d.
84.25 are

Tangent-Tangent Theorem
46.
Two tangents OA and OB to a circle intersect at point O. If the area
of the smaller sector intercepted by the arc AB is 35% the area of
the circle, determine the measure of ∠BOA.
a.
39°
c.
54°
b.
62°
d.
48°
Inscribed Figures

Incircle of a Triangle
47.
The sides of a triangle measure 99.96 cm, 158.27 cm, and 191.59
cm. Determine the radius of the circle inscribed in the triangle.
a.
47.91 cm
c.
35.12 cm
b.
60.51 cm
d.
52.87 cm

Circumcircle of a Triangle
48.
Determine the diameter of the circle circumscribed about the
triangle for which A = 43° and a = 65 m.
a.
87.81 cm
c.
85.12 cm
b.
90.71 cm
d.
95.31 cm
49.
The sides of a triangle measure 33.48 m, 44.64 m, and 66.96 m.
Determine the radius of the circle circumscribed about the triangle.
a.
39.45 m
c.
51.65 m
b.
37.67 m
d.
42.67 m
Ellipse
50.
The major and minor axis of an ellipse measure 56 m and 48 m,
respectively. Calculate the area of the ellipse.
a.
2314.31 m2
c.
2421.34 m2
b.
2213.25 m2
d.
2111.15 m2
Parabolic Segment
51.
A parabolic segment has a base width of 20 cm and an altitude of
32 cm. A dividing line 12 cm long is drawn parallel to the base.
Calculate the divided smaller area.
a.
58.93 m2
c.
42.14 m2
b.
81.67 m2
d.
92.16 m2
Polygon

Lines, Sides and Diagonals of Polygon
52.
Determine the number of lines that can be drawn from the vertices
of an octagon.
a.
32
c.
30
b.
26
d.
28
53.
How many sides are there in an icosagon?
a.
10
c.
30
b.
20
d.
50
54.
How many diagonals are there in a tetracontakaipentagon?
a.
1430
c.
945
b.
2015
d.
560
55.
The sum of the number of lines drawn from the vertices of two
polygons is 214 and the sum of the number of diagonals is 184.
One of the polygon is
a.
dodecagon
c.
enneadecagon
b.
tridecagon
d.
decagon

Interior and Exterior Angles of a Polygon
56.
Each interior angle of a regular chiliagon measures
a.
168°26’30”
c.
167°44’15”
b.
158°33’29”
d.
179°38’24”
57.
Each exterior angle of a regular icosagon measures
a.
18°
c.
22°
b.
20°
d.
24°
58.
The interior angle of a regular polygon is 144° greater than its
exterior angle. The regular polygon is
a.
tetracontagon
c.
heptadecagon
b.
octadecgon
d.
icosagon

Area of Regular Polygon
59.
A circle is inscribed in a pentagon. If the area of the circle is
[7]
153.938 m2, determine the area of pentagon.
a.
158 m2
c.
168 m2
b.
178 m2
d.
188 m2
A nonagon-shaped outline is to be formed from an 18-m length
60.
adhesive tape for a wall design. If 5% of the tape was not utilized,
compute for the area of the nonagon.
a.
22.32 m2
c.
18.99 m2
b.
19.67 m2
d.
21.83 m2
61.
Two hexagons each with 15 m sides overlapped each other such
that the overlapping area is a regular dodecagon. Compute the area
of the dodecagon.
a.
385.22 m2
c.
245.87 m2
b.
467.56 m2
d.
542.60 m2
Find the difference between the radii of two circles if one is
62.
circumscribed while the other is inscribed in a regular heptagon
whose side is 35 cm.
a.
4.99 cm
c.
2.99 cm
b.
3.99 cm
d.
5.99 cm
Polygram
63.
Determine the area of the hexagram inscribed in a circle of radius
21 m.
a.
582.21 m2
c.
673.38 m2
b.
598.12 m2
d.
763.83 m2
SOLID GEOMETRY
Prism
Polygonal Bases
1.
Compute the weight of water that can be filled in a waterbed
mattress that is 7 ft by 4 ft by 1 ft.
a.
0.90 kip
c.
1.75 kip
b.
1.23 kip
d.
1.65 kip
2.
Each edge of the large cube is 12 inches long. The cube is painted
on the outside, and then cut into 216 smaller cubes. How many
cubes are painted?
a.
132
c.
144
b.
138
d.
152
3.
A hexagonal pencil is a hexagonal prism. A base edge has length 4
mm. The pencil (without eraser) has height 170 mm. How much
surface area of a hexagonal pencil gets painted?
a.
5280 mm2
c.
4650 mm2
b.
4080 mm2
d.
3980 mm2
4.
The sides of the base of a pentagonal right prism measure 3 cm, 5
cm, 6 cm, 8 cm and 9 cm. If its lateral area is 200 cm2, find the
altitude of the prism.
a.
4.65 cm
c.
5.65 cm
b.
6.45 cm
d.
3.95 cm
Cylinder
5.
A standard drinking straw is 19.5 cm long and has a diameter of 0.6
cm. Calculate the area of plastic used in one straw.
a.
33.4 cm2
c.
42.1 cm2
b.
29.7 cm2
d.
36.8 cm2
6.
A metal washer 1 inch in diameter is pierced by ½ inch hole. What
is the volume of the washer if it is 1/8 inch thick?
a.
0.062 in3
c.
0.052 in3
b.
0.042 in3
d.
0.074 in3
Truncated Prism
7.
The base of a truncated prism is a triangle with sides 8 cm, 12 cm
and 10 cm. If the heights are 20 cm, 18 cm and 12 cm respectively,
compute the volume.
a.
661.43 cm3
c.
456.23 cm3
b.
521.54 cm3
d.
583.88 cm3
8.
A right cylinder 5 cm in diameter was cut by a plane at an angle of
45° with its axis. If the average altitude of the truncated cylinder is
19 cm, determine the total surface area.
a.
345.85 cm2
c.
365.85 cm2
b.
355.85 cm2
d.
335.85 cm2
9.
A truncated prism having a square base has a volume of 1000 cubic
meters. The height of the prism at each corner is respectively 7m,
7m, 10m, and 10m. What is the area of the base?
a.
132.85 m2
c.
120.45 m2
b.
109.95 m2
d.
117.65 m2
10.
A triangular prism has a horizontal triangular base ABC with AB =
10 cm, BC = 12 cm, and CA = 8 cm. The vertical edges through A,
B, and C are 20 cm, 12 cm, and 18 cm, respectively. Calculate the
volume of the prism.
a.
674.92 cm2
c.
661.44 cm2
b.
649.11 cm2
d.
658.32 cm2
11.
The base of a truncated prism is a rectangle with length twice its
width. The corner edges have heights of 12 m, 12 m, 16 m, and 16
m respectively. If the volume of the prism is 8,200 m3, determine the
length of its base.
a.
36.12 cm
c.
35.59 cm
b.
34.23 cm
d.
37.10 cm
Pyramid

Polygonal Bases
12.
A regular triangular pyramid has an altitude of 9 m and a volume of
187.06 m3. Determine the base edge.
a.
15 m
c.
12 m
b.
11 m
d.
14 m
13.
How far from a vertex is the opposite face of a tetrahedron if an
edge is 50 cm long?
a.
40.83 cm
c.
39.12 cm
b.
38.60 cm
d.
41.67 cm
14.
The surface area of a regular tetrahedron is 173.2 square
centimeters. What is its altitude?
a.
9.33 cm
c.
8.16 cm
b.
8.56 cm
d.
9.10 cm
15.
The altitude of the great Pyramid of Cheops in Egypt originally was
480 ft and its square base was 764 ft on an edge. It is said to have
cost $10 a cubic yard and $3 more for each square yard of lateral
surface (considered as planes). What was its cost?
a.
$33,885,000
c.
$34,845,000
b.
$34,901,000
d.
$32,905,000
16.
Find the area of the base of a regular square pyramid whose lateral
faces are equilateral triangles and whose altitude is 8 in.
a.
107 in2
c.
128 in2
b.
119 in2
d.
135 in2

Cone
17.
A cone was formed by rolling a thin sheet of metal in the form of a
sector of a circle 72 cm in diameter with a central angle of 210°.
What is the volume of the cone?
a.
14503 cm3
c.
13504 cm3
b.
10543 cm3
d.
15304 cm3
18.
The lateral area of a right circular cone of radius 4 cm is 100.53
cm2. Determine the slant height.
a.
8 cm
c.
9 cm
b.
7 cm
d.
10 cm
The base diameter of a cone is 18 cm and its axis is inclined 60°
19.
with the base. If the axis is 20 cm long, what is the volume of the
cone?
a.
1478.12 cm3
c.
1455.23 cm3
b.
1459.56 cm3
d.
1469.13 cm3
20.
The lateral area of a right circular cone is 386 square meters. If its
diameter is one-half its altitude, determine its altitude in meters.
a.
20.98 cm
c.
23.09 cm
b.
21.65 cm
d.
22.51 cm
Similar Figures
21.
If the edge of a cube is increased by 30%, by how much is the
surface area increased?
a.
30%
c.
63%
b.
9%
d.
69%
22.
A right circular cone with an altitude of 9 m is divided into two
segments. One is smaller circular cone having the same vertex with
an altitude of 6m. Find the ratio of the volume of the two cones.
a.
2:3
c.
8:27
b.
2:5
d.
1:3
23.
Two identical cones with vertical axis, one inverted and the other is
upright has base radius of 1.2 m and height of 4.8 m. Each cone
contains equal volume of oil having specific gravity of 0.8. If the
depth of oil in the inverted cone is 2.4 m, what is the depth of oil in
the upright cone?
a.
0.21 m
c.
0.42 m
b.
0.30 m
d.
0.51 m
24.
The sides of the base of a pentagonal right prism measure 3 cm, 5
cm, 6 cm, 8 cm, and 9 cm, consecutively. Another prism similar to
the first has its shortest side measuring 2 cm and a lateral area of
144 cm2. Determine the altitude of the larger prism.
a.
11.5 cm
c.
9.5 cm
b.
8.5 cm
d.
10.5 cm

Frustum
25.
The volume of a frustum of a regular triangular pyramid is 135 m3.
The lower base is an equilateral triangle with an edge of 9 m. The
upper base is 8 m above the lower base. What is the upper base
edge?
a.
2m
c.
2.5 m
b.
3m
d.
3.25 m
26.
The edges of the bases of a frustum of a regular square pyramid
are 10 m and 20 m, respectively and the altitude is 7 m. Calculate
its lateral surface area.
a.
516.14 m2
c.
502.76 m2
b.
498.25 m2
d.
480.34 m2
27.
A frustum of a cone has an upper base whose radius is 5 m and a
lower base whose radius is 7 m. Its altitude is 11 m. Determine the
total surface area.
a.
421.50 m2
c.
653.97 m2
2
b.
569.34 m
d.
645.21 m2
Sphere
28.
The corners of a cubical block touch the closed spherical shell that
encloses it. The volume of the box is 2744 cm3. Determine the
volume inside the shell is not occupied by the block?
a.
4471 cm3
c.
4371 cm3
b.
4721 cm3
d.
4234 cm3
29.
A solid spherical steel ball 20 cm in diameter is placed into a tall
vertical cylinder containing water causing the water level to rise by
10 cm. What is the radius of the cylinder?
a.
12.2 cm
c.
11.4 cm
b.
11.6 cm
d.
12.6 cm
30.
The volume of a sphere is 65.45 cubic meters. Determine its
surface area.
a.
88.34 cm2
c.
78.54 cm2
b.
73.64 cm2
d.
68.34 cm2

Spherical Segment
31.
Determine the area of the zone of a spherical segment having a
volume of 1470.265 m3, if the diameter of the sphere is 30 m?
a.
88.34 cm2
c.
78.54 cm2
b.
73.64 cm2
d.
68.34 cm2
32.
A mixture compounded from equal parts of two liquids, one white
and the other black was placed in a hemispherical bowl. The total
depth of the two liquids is 6”. After standing for a short time the
mixture separated the white liquid settling below the black. If the
thickness of the segment of the black is 2”, find the total volume of
the two liquids.
a.
603.15 in3
c.
628.35 in3
b.
593.25 in3
d.
543.55 in3

Spherical Wedge/Lune
33.
Find the radius of the spherical wedge whose volume is 12 m3 with
a central angle of 1.8 radians.
a.
2.45 m
c.
2.63 m
b.
2.33 m
d.
2.15 m

Spherical Cone
34.
What is the volume of a spherical sector with angle of 60° in a
sphere of radius 30 cm?
a.
7657 cm3
c.
7756 cm3
b.
7576 cm3
d.
7567 cm3
Prismatoid
35.
A railway embankment across a valley has the following
dimensions: width at top, 24 ft; width at base, 66 ft; height, 14 ft;
length at top, 286 ft; length at base, 210 ft. Find its volume.
a.
5484.3 yd3
c.
5864.6 yd3
b.
5744.1 yd3
d.
5648.7 yd3
36.
A solid has a circular base of base radius 20 cm. Find the volume of
the solid if every plane section perpendicular to a certain diameter is
an isosceles right triangle with one leg in the plane of the base.
a.
22333.33 cm3
c.
23333.33 cm3
b.
21333.33 cm3
d.
20333.33 cm3
37.
A cylinder of radius 6 m has its axis along the X-axis. A second
cylinder of the same radius has its axis along the Y-axis. Find the
volume, in the first octant, common to the two cylinders.
a.
144 m3
c.
111 m3
b.
133 m3
d.
123 m3
ANALYTIC GEOMETRY
Plane Analytic Geometry
Distance Between Two Points
1.
Find the distance between P1 = (1, 4) and P2 = (-3, 2).
a.
4.47
c.
2.97
b.
3.87
d.
5.01
If P1 = (x, 0), P 2 = (2, 5), and ̅P=15√2,
find x.
2.
2
a.
-2
c.
-3
b.
5
d.
6
3.
The following are vertices of a rectangle: (1,2), (4,7), (-6,13). Find
the fourth vertex.
a.
(-4, 3)
c.
(-8, 9)
b.
(-7, 8
d.
(-9, 8)
Division of Line Segment (Point-of-Division)
4.
Find the point one-third of the way from A = (2, 5) to B = (8, -1).
a.
(4, 3)
c.
(-3, 4)
b.
(3, 4)
d.
(4, -3)
5.
Given the segment AB, where A = (-3, 1) and B = (2, 5), is extended
beyond A to a point P twice as far from B as A is; find P.
a.
(-3, 8)
c.
(-8, 3)
b.
(8, -3)
d.
(-8, -3)
6.
If P = (4, -1) is the midpoint of the segment AB, where A = (2, 5),
find B.
a.
(7, -6)
c.
(6, 7)
b.
(6, -7)
d.
(-7, 6)
7.
Find the point of intersection of medians of a triangle with vertices
(5, 2), (0, 4) and (-1, -1).
a.
3/5, 4/3
c.
4/3, 5/3
b.
5/3, 4/3
d.
5/3, 3/4
Inclination and Slope
Find the slope of the line containing P1 = (1,5) and P2 = (7, -7).
8.
a.
-2
c.
1/2
b.
2
d.
-1/2
Parallel and Perpendicular Lines
9.
If the line through (x, 1) and (0, y) is coincident with the line through
(1, 4) and (2, -3), find x and y.
a.
7/10, 11
c.
11, 10/7
b.
10/7, 11
d.
11, 7/10
10.
If the line through (x, -3) and (3, 1) is perpendicular to the line
through (x, -3) and (-1, -2), find x.
a.
-1
c.
2
b.
1
d.
-2
Angle from One Line to Another
11.
If two lines have slopes 3 and -2 respectively, find the angle
between them.
a.
60°
c.
45°
b.
40°
d.
30°
12.
Find the angle between the vertical line and another line which has
slope equal to ½.
a.
87°
c.
117°
b.
62°
d.
98°
13.
Find the slope of the line bisecting the angle from l1, with slope 7, to
l2, with slope 1.
a.
-1/2
c.
1/2
b.
-2
d.
2
14.
Find the slope of the line bisecting the angle from l1, with slope 2, to
l2, with no slope.
a.
2.44
c.
1.44
b.
4.24
d.
-0.24
Equation of a Locus
15.
Find an equation for the set of all points in the xy plane which are
equidistant from (1, 3) and (-2, 5).
a.
6x + 4y + 19 = 0
c.
6x – 4y + 19 = 0
b.
4x – 6y + 19 = 0
d.
4x – 6y – 19 = 0
16.
Find an equation for the set of all points (x, y) such that the sum of
its distances from (3, 0) and (-3, 0) is 8.
a.
7x2 – 16y2 = 112
c.
7x2 + 16y2 = 112
b.
16x2 – 7y2 = 112
d.
16x2 + 7y2 = 112
17.
Find an equation for the set of all points (x, y) such that are
equidistant from (-2, 4) and the y axis
a.
y2 + 4x – 8y + 20 = 0
c.
x2 – 4x + 8y – 20 = 0
b.
y2 – 4x + 8y – 20 = 0
d.
x2 + 4x – 8y – 20 = 0
Point-Slope Form of a Line
18.
Find an equation of the line through (-2, -3) with slope ½.
a.
x – 2y – 4 = 0
c.
x + 2y – 4 = 0
b.
x – 2y + 4 = 0
d.
x + 2y + 4 = 0
19.
Find an equation of the vertical line through (3, -2).
a.
x–3=0
c.
y–3=0
b.
x+3=0
d.
y+3=0
Two-Point Form of a line
20.
Find an equation of the line through (4, 1) and (-2, 3).
a.
x – 3y + 7 = 0
c.
x + 3y + 7 = 0
b.
x + 3y – 7 = 0
d.
x – 3y – 7 = 0
21.
Find the perpendicular bisector of the line segment joining (5, -3)
and (1, 7).
a.
2x + 5y + 4 = 0
c.
2x – 5y – 4 = 0
b.
2x – 5y + 4 = 0
d.
2x + 5y – 4 = 0
Slope-Intercept Form a Line
22.
Find an equation of the line with slope 2 and y intercept 5.
a.
2x + y – 4 = 0
c.
2x – y + 4 = 0
b.
2x + y + 4 = 0
d.
2x – y – 4 = 0
23.
Find an equation of the line that is parallel to 3x + 2y – 5 = 0 and
contains the point (3, 1).
a.
3x + 2y – 11 = 0
c.
3x – 2y – 11 = 0
b.
3x + 2y + 11 = 0
d.
3x – 2y + 11 = 0
24.
Find an equation of the line that is perpendicular to 3x + 2y – 5 = 0
and contains the point (3, 1).
a.
2x + 3y – 3 = 0
c.
2x – 3y – 3 = 0
b.
2x + 3y + 3 = 0
d.
2x – 3y + 3 = 0
Intercept Form
25.
Find an equation of the line with x and y intercepts 5 and -2.
a.
2x + 5y – 10 = 0
c.
2x – 5y – 10 = 0
b.
2x – 5y + 10 = 0
d.
2x + 5y + 10 = 0
Normal Form of a Line
26.
The equation of the line is given as x + y – 8 = 0. Compute the value
ρ if it is expressed in the normal form x cos β + y sin β – ρ = 0.
a.
c.
√2
3√2
b.
d.
2√2
4√2
Distance from a Point to a Line
27.
For what value of m is the line y – 1 = m(x + 3) at a distance 3 from
the origin?
a.
-4/3
c.
4/3
b.
-3/4
d.
3/4
28.
Find an equation of the line bisecting the angle from 3x – 4y – 3 = 0
to 5x + 12y + 1 = 0.
a.
32x + 4y – 17 = 0
c.
32x + 4y + 17 = 0
b.
32x – 4y – 17 = 0
d.
32x – 4y + 17 = 0
Distance Between Two Parallel Lines
29.
Find the distance between the parallel lines: 2x – 5y – 10 = 0 and 2x
– 5y + 4 = 0.
a.
4.6
c.
2.6
b.
1.6
d.
3.6
Circle: Standard Form for an Equation
30.
Give an equation for the circle with center (3, -5) and radius 2.
a.
(x + 3)2 + (y + 5)2 = 4
c.
(x – 3)2 + (y – 5)2 = 4
b.
(x – 3)2 + (y + 5)2 = 4
d.
(x + 3)2 + (y – 5)2 = 4
Determine the area enclosed by the curve 2x2 + 2y2 – 2x + 6y – 3 =
31.
0.
a.
13.62
c.
13.79
b.
12.57
d.
12.88
Conditions to Determine a Circle
32.
Find an equation of the circle through points (1, 5), (-2, 3) and (2, 1).
a.
5x2+5y2-9x-19y+26=0
2
33.
c.
5x2+5y2+9x-19y-26=0
b.
5x +5y -9x+19y-26=0
d.
5x2+5y2-9x-19y-26=0
Find an equation of the circle of radius 4 with center on the line 4x
+ 3y + 7 = 0 and tangent to 3x + 4y + 34 = 0.
a.
x2 + y2 - 4x - 10y - 13 =0
c.
x2 + y2 - 4x - 10y + 13 =0
b.
2
x2 + y2 + 4x - 10y - 13 =0
d.
x2 + y2 - 4x + 10y + 13 =0
Parabola
34.
Where is the vertex of the parabola x2 – 2x – 6y + 13 = 0?
a.
(-1, -2)
c.
(1, 2)
b.
(-2, -1)
d.
(2, 1)
35.
Determine the length of the latus rectum of the curve y2 = 8x.
a.
4
c.
8
b.
10
d.
6
36.
Give the equation of the directrix of the curve x2 = -12y
a.
x–3=0
c.
y–3=0
b.
x+3=0
d.
y+3=0
Conditions to Determine a Parabola
37.
Find an equation of the parabola with vertex at the origin and focus
(-4, 0)
a.
x2 = -16y
c.
y2 = -16x
b.
x2 = 16y
d.
y2 = 16x
A parabola whose vertex is at the origin and contains (2, 3) and (38.
2, 3). Find its equation.
a.
y2 = 4y/3
c.
x2 = 4y/3
b.
y2 = -4y/3
d.
x2 = -4y/3
39.
Determine the equation of a parabola whose axis is parallel to the
y-axis, contains (1, 1), (2, 2) and (-1, 5).
a.
x2 – 2x – y + 2 = 0
c.
x2 + 2x + y + 2 = 0
b.
x2 – 2x + y + 2 = 0
d.
x2 – 2x + y – 2 = 0
Ellipse: Standard Form for an Equation
40.
Determine the area enclosed by the curve 4x2 + y2 – 8x + 4y – 8 =
0.
a.
25.13 sq. units
c.
28.32 sq. units
b.
27.21 sq. units
d.
17.33 sq. units
41.
Determine the latera recta length of the curve 9x2 + 25y2 – 225 =
0.
a.
4.2
c.
7.2
b.
3.6
d.
5.1
42.
Give the point of intersection of the major and minor axis of the
2
2
curve 25x + 16y = 400
a.
(0, 0)
c.
(±4, 0)
b.
(0, ±5)
d.
(0, ±3)
Geometric Conditions to Determine an Ellipse
43.
Find an equation of the ellipse with vertices (0, ±8) and foci (0, ±5).
a.
8x2 + 13y2 = 832
c.
64x2 + 39y2 = 2496
b.
13x2 + 8y2 = 832
d.
39x2 + 64y2 = 2496
44.
Find an equation of the ellipse with foci (±6, 0) and eccentricity, e
= 3/5.
a.
25x2 + 16y2 = 1060
c.
16x2 + 25y2 = 1600
b.
16x2 + 25y2 = 1060
d.
25x2 + 16y2 = 1600
45.
Find an equation of the ellipse with vertices (±6, 0) and latus
rectum’s length 3.
a.
4x2 + y2 = 36
c.
12x2 + 3y2 = 36
b.
x2 + 4y2 = 36
d.
3x2 + 12y2 = 36
Hyperbola
46.
Determine the length of the latus rectum of the curve 16x2 – 9y2 +
144 = 0
a.
4.5
c.
2.5
b.
3.5
d.
2
47.
Find the equation of one of the asymptotes of the hyperbola 16x2
– 9y2 + 144 = 0.
a.
12x + 9y = 0
c.
5x + 3y = 0
b.
4x – 3y = 0
d.
3x – 4y = 0
48.
Determine the coordinates of one of the foci of the hyperbola 9x2
– 4y2 – 90x + 189 = 0.
a.
(1.39, 0)
c.
(0, -1.39)
b.
(0, 1.39)
d.
(-1.39, 0)
49.
Find the distance from the center to directrices of x2 – 2y2 + 4x +
4y + 4 = 0.
a.
2.828
c.
1.732
b.
2.000
d.
0.577
Geometric Conditions to Determine a Hyperbola
50.
Find an equation of the hyperbola with vertices (±2, 0) and focus
(-4, 0).
a.
x2 – 3y2 – 12 = 0
c.
x2 – 3y2 + 12 = 0
b.
3x2 – y2 – 12 = 0
d.
3x2 – y2 + 12 = 0
Find an equation of the hyperbola with directrices x = ±9/5 and
51.
eccentricity 5/3.
a.
9x2 – 16y2 + 144 = 0
c.
16x2 – 9y2 – 144 = 0
b.
9x2 – 16y2 – 144 = 0
d.
16x2 – 9y2 + 144 = 0
Find an equation of the hyperbola with asymptotes y = ±4x/3 and
52.
contains (3√2, 4).
a.
9x2 – 16y2 + 144 = 0
c.
16x2 – 9y2 – 144 = 0
b.
9x2 – 16y2 – 144 = 0
d.
16x2 – 9y2 + 144 = 0
DIFFERENTIAL CALCULUS
Differentials
1.
If y = x4/5, determine the approximate change from 4 to 4.02.
a.
0.01213
c.
0.01123
b.
0.21231
d.
0.31232
Approximation and Errors
2.
Two sides of a triangle change from 7 m and 15 m to 7.15 m and
15.25 m, respectively. If the angle between the sides is 35°,
determine the approximate change in area.
a.
c.
1.15 m2
1.35 m2
b.
1.25 m2
d.
1.45 m2
3.
The radius and altitude of a right circular cone are 4 m and 15 m,
respectively. Determine the maximum percentage error in the
volume when these dimensions are in the error of ± 0.40 m.
a.
13.33%
c.
16.67%
b.
22.67%
d.
26.33%
Derivatives of Powers of a Variable
4.
Determine the derivative of (4x2 – 3x)2 with respect to x.
a.
c.
16x3 – 24x2 + 9x
16x3 + 24x2 – 9x
3
2
b.
64x – 72x + 18x
d.
64x3 + 72x2 – 18x
5.
Determine the second derivative of y = x3 + 6x2 – 9x + 8 at x = 0.
a.
8
c.
12
b.
6
d.
10
Derivatives of Products and Quotients
6.
Determine the derivative of (x2 – 1)(2x2 – 3) with respect to x.
a.
c.
4x(4x2 – 5)
2x(4x2 + 5)
2
b.
2x(4x – 5)
d.
4x(4x2 + 5)
7.
Determine the derivative of (2x – 1)/(4x + 3) with respect to x.
8
–8x+6
a.
c.
(4x+3)2
(4x+3)
8x – 6
10
b.
d.
(4x+3)
(4x+3)2
Derivative of Trigonometric Functions
8.
Find the first derivative of 2 cos(2+x3).
a.
6x2 sin(2+x3)
c.
2 sin(2+x3)
b.
–6x2 sin(2+x3)
d.
–2 sin(2+x3)
Derivative of Inverse Trigonometric Functions
9.
Find y' = if y = arcsin x.
1
a.
√1 – x2 c.
1 – x2
1
1+x
b.
d.
√1 – x2
√1 – x2
10.
Find the first derivative of y = arcos 4x.
4
4
–
a.
c.
2
√
√1 – 16x
1 – 4x2
4
4
b.
–
d.
√1 – 16x2
√1 – 4x2
11.
Find the first derivative of y = arcsin 3x.
3
3
a.
c.
√1 – 9x2
√1 – 3x2
3
3
–
–
d.
b.
2
√1 – 9x
√1 – 3x2
12.
Find the derivative of arccsc x.
– dx
dx
a.
c.
√1 – x2
x√x2 – 1
– dx
dx
b.
d.
x√x2 – 1
√1 – x2
Derivative of Logarithmic Functions
13.
Find the derivative of logau with respect to x.
loga e du
du
∙
c.
a.
log u∙
u
dx
dx
u du
du
b.
∙
d.
log
a∙
ln a dx
dx
14.
Find y' if y = ln x.
1
1
a.
c.
ln x
x
2
b.
ln x
d.
x ln x
Derivative of Exponential Functions
u
15.
Find y' if y = a
au
a.
au ln a
c.
ln a
b.
u ln a
d.
a ln u
16.
Find the derivative of h with respect to u if h = π2u
a.
π2u
c.
2π2u ln π
b.
2u ln π
d.
2π2u
17.
Find y' if y = 33x
c.
a.
33x ln 3
3x+1 ln 3
3x ln 3
d.
b.
33x+1 ln 3
Derivative of Hyperbolic Functions
18.
Find y' if y = sinh x.
a.
csch x
c.
sech x
b.
cosh x
d.
tanh x
Total Derivatives
19.
Find f'(xyz) if f(xyz) = 5x2 y4 z.
a.
10xy4 z dx + 20x2 y3 z dy + 5x2 y4dz
b.
20x2 y3z dx + 10xy4 dy + 5x2 y4 dz
c.
10xy4 z dx + 5x2 y4 dy + 20x2 y3 z dz
d.
20x2 y3z dx + 5x2 y4 dy + 10xy4 dz
b.
d.
𝑦 = 𝐶1 + 𝐶2𝑒−2𝑥
𝑦 = 𝐶1 + 𝐶2𝑒2𝑥
Implici t Differentiation
20.
Find y' if 3xy + 4y2= 10.
– 3y
a.
3x – 8y
c.
3y
3x + 8y
1.
Express 1296x12 - 4320x9y2 + 5400x6y4 - 3000x3y6 + 625y8 in the form (a + b)n.
a. (𝟔𝒙𝟑 − 𝟓𝒚𝟐)𝟒
c. (𝟔𝒙𝟑 − 𝟏𝟓𝒚𝟑)𝟒
𝟑
b. (−𝟒𝒙𝟒 + 𝟏𝟓𝒚𝟐)
d. (𝟒𝒙𝟑 − 𝟐𝟓𝒚𝟐)𝟒
2.
Translate the statement into algebraic expression: 69 divided into a
number
a. 6/9
c. 69/n
b. 9/6
d. n/69
After how many seconds from 3:05 – before the second hand completes
one rotation – the hour hand is exactly between minute and second
hands, i.e. hour hand bisects the angle between the minute and second
hand.
a. 29.589 sec
c. 25.479 sec
b. 23.836 sec
d. 28.236 sec
3.
4.
Jayson is filling his bathtub but he forgot to put the plug in. The
volume of water for a bath is 0.350 m3 and the tap is flowing at 1.32
L/min and the drain is running at 0.32 L/min. Assuming Jayson shuts
off the water when the tub is full and does not flood the house, how
much water will be wasted?
a. 89 L
c. 77 L
b. 103 L
d. 112 L
5.
Find the two-digit number satisfying the following two conditions.
(1) Four times the units digit is six less than twice the tens digit.
(2) The number is nine less than three times the number obtained by
reversing the digits.
a. 81
c. 63
b. 56
d. 72
6.
If 8 men take 12 days to assemble 16 machines, how many days will it
take 15 men to assemble 50 machines?
a. 10 days
c. 20 days
b. 12 days
d. 15 days
Solution Y is 30% liquid X and 70% water. If 2 kg of water evaporate
from 8 kg of solutions Y and 2 kg of solution Y are added to the
remaining 6 kg of liquid, what percent of this new liquid solution is
liquid X?
a. 30%
c. 37.5%
b. 33.33%
d. 40%
7.
8.
Find the 3rd term of the 7 numbers, in arithmetical progression, such
that the sum of the first and sixth shall be 14, and the product of
the third and fifth shall be 60.
a. 8
c. 9
b. 6
d. 1
9.
Buggy and Shanks are running in opposite directions in a 400-m circular
track and meet each other every 35 seconds. If Buggy can complete one
track in 1 minute and 5 seconds, how long does it take for Shanks to
finish 400 m?
a. 75.8 s
c. 79.1 s
b. 69.4 s
d. 63.8 s
10.
A pump is attached to a container for the purpose of creating a vacuum.
For each stroke of the pump, 1/4 of the air that remains in the
container is removed. How much of the air remains in the container
after 15 strokes?
a. 3.12%
c. 2.65%
b. 0.15%
d. 1.34%
11.
12.
13.
A building 38 m x 45 m is to be laid out with a 50-m long metallic
tape. If during standardization the tape is found to be only 49.950
m. Which of the following most nearly gives the error in area?
a. 2.34 m2
c. 4.32 m2
b. 2.43 m2
d. 3.42 m2
The lengths of the radii of circles form an infinite geometric series.
The length of the first circle is 10 cm. The length of the radius of
each of the circles is 4/5 of the length of the previous circle.
Determine the total area of all the circles formed in this way.
a. 2500/9 π
c. 750 π
b. 500 π
d. 1500/9 π
Let x and y be positive real numbers such that
𝒍𝒐𝒈𝟓(𝒙 + 𝒚) + 𝒍𝒐𝒈𝟓(𝒙 − 𝒚) = 𝟑
And
𝒍𝒐𝒈𝟐𝒚 − 𝒍𝒐𝒈𝟐𝒙 = 𝟏 − 𝒍𝒐𝒈𝟐𝟑
What is the product of x and y? Hint: System
substitution with 2 unknowns
a. 150
c. 25
b. 100
d. 250
of
Equations
by
14.
If 3x – y = 4, determine the value of
𝟐𝟕𝒙
𝟑𝒚
a. 9
c. 81
b. 27
d. 243
15.
Find: 𝒊𝟏𝟕 + 𝒊𝟏𝟗
a. 0
c. -1 + i
b. 1 + i
d. -i
When the suns altitude increases from 30° to 60°, the length of the
shadow of a tower decrease by 5 m. Determine the height of the tower.
a. 4.33 m
c. 5.67 m
b. 5.11 m
d. 6.23 m
16.
17.
18.
19.
20.
21.
From the foot of the hill, angle of
After climbing 1000 m at the slope of
top becomes 60°. Calculate the height
a. 1277.08 m
c.
b. 1298.54 m
d.
elevation of its top is 45°.
30°, angle of elevation of the
of the hill.
1366.03 m
1372.55 m
Find sec 𝜃, given that cot 𝜃 = -√𝟑/𝟒 and csc 𝜃 > 0.
a.
c.
-√𝟏𝟑/4
√𝟓𝟕/3
b.
d.
√𝟑𝟗/4
-√𝟓𝟕/3
Which of the following is not true?
a. -1 < cos 𝜃 < 1
c.
b. -∞ < cot < ∞
d.
csc 𝜃 ≥ 1 and csc 𝜃 ≤ -1
none among the choices
The conjugate of an angle is 30° less than
complement.
a. 130°
c. 60°
b. 15°
d. 45°
Find the period of the trigonometric equation
y = 3 sin x – 4 sin3 x
a. 60°
c. 120°
b. 90°
d. 150°
five
times
of its
22.
A statue 3.20 m high is standing on a base 6 m high. If an observer’s
eye is 1.50 m above the ground, how far should he stand from the base
in order that the angle subtended by the statue is 12°?
a. 19.82 m
c. 12.22 m
b. 24.07 m
d. 15.61 m
23.
The latitude of the town of Exeter, New Hampshire, is about 43° N.
How far does Earth’s (R = 3960 miles) rotation carry the citizens of
Exeter during a single day?
a. 18,197.11 miles
c. 20,834.67 miles
b. 24,881.41 miles
d. 19,005.33 miles
24.
In triangle ABC,
3sin A + 4cos B = 6 and 4sin B + 3cos A = 1.
Find the measure of angle C. Hint: Take the square of the equations
and add.
a. 150°
c. 60°
b. 30°
d. 45°
25.
A plane travels in a direction of N. 30° W at an air speed of 600 kph
with the intention to travel 2000 km from its initial position to the
same direction. However, it did not consider the direction of the
prevailing winds, which has a s peed of 80 kph travelling at N 40° E.
The pilot noticed that he is off course after 1 and half hour. If he
immediately changed his direction and was able to travel to the correct
direction considering wind speed with a resultant speed of 700 kph,
what is the total time of travel of the plane?
a. 3.52 hours
c. 3.02 hours
b. 1.52 hours
d. 2.52 hours
The angle of elevation of the top of a tower from a point A due south
is 35° and from a point B due east of the tower is 45°. If AB = 200
m, then the height of the tower is
a. 114.72 m
c. 129.11 m
b. 123.56 m
d. 136.50 m
26.
27.
An object is observed from three points A, B, C in the same horizontal
line. Passing through the base of the object, the angle of elevation
at B is twice and at C thrice that at A. If AB = 38 m, BC = 25 m,
compute for the height of the object.
a. 32.07 m
c. 41.08 m
b. 36.69 m
d. 39.33 m
28.
A tree is broken by wind, its upper part touches the ground at a point
10 m from the foot of the tree and makes an angle of 60° with the
ground, the entire length of the tree is
a. 29.41 m
c. 37.32 m
b. 34.81 m
d. 25.33 m
29.
The hands of a clock show 11:20. Express the smaller obtuse angle
formed by the hour hand and minute hands in standard unit of angle.
a. 140°
c. 1400/9 grad
b. 7𝝅/9
d. 22400/9 mils
30.
A right triangle has a hypotenuse equal to 10 m and an altitude to
the hypotenuse equal to 6. Calculate the area of the triangle.
a. 20 sq units
c. 24 sq units
b. 30 sq units
d. None of the above
Mathematics 2 for FEU Alabang
1. In a family of five children, what is the probability that there are 3 boys and 2 girls?
A. 7/16
B. 3/8
C. 5/16
D. 3/16
2. A basketball player averages 65% in a free-throw line. What is the probability of missing one for two freethrows?
A. 0.523
B. 0.455
C. 0.574
D. 0.486
3. The corners of a cubical block touch the closed spherical shell that encloses it. The volume of the box is 27000
cc. What volume in cc, inside the shell is not occupied by the block?
A. 46461
B. 32987
C. 54873
D. 28652
4. A regular hexagon is inscribed in a circle having an area of 158 square cm. Find the area of the circle not
covered by hexagon.
A. 32.34
B. 27.34
C. 22.98
D. 45.63
5. If sin3A=cos6B, find A+2B in radians.
A. π/6
B. π/3
C. π/2
D. π/4
6. For the sequence 1, 1, 1/2, 1/6, 1/24, …, what is the 6th term?
A. 1/120
B. 1/60
C. 1/48
D. 1/240
7. The fourth term of an arithmetic progression is -8 and the eighth term is -16. What is the 6th term?
A. -14
B. -12
C. -10
D. -18
8. A can finish a job in 2 days, B can finish it in 3 days and C can finish it in 6 days. Working together, how many
days can they finish the job?
A. 2
B. 3.5
C. 2.5
D. 1
9. Find the angle in degrees made by the tangent line of the curve y=e^(4x) at x=0 with x-axis.
A. 45
B. 0
C. 76
D. 63
10. The base of a truncated prism is a rectangle with length twice its width. The corner edges have heights of 12 m,
12 m, 16 m, and 16 m respectively. If the volume of the prism is 8,200 cu m, find the length of its base.
A. 17.11
B. 34.23
C. 28.63
D. 14.31
11. The upper base of a frustum if a pyramid is 2.5 m by 4 m and the lower base is 5 m by 8m. Find its volume if the
distance between the bases is 6 m.
A. 140
B. 160
C. 130
D. 120
12. Find the distance of the line 8x-5y-121=0 to point (1,1).
A. 10.47
B. 11.36
C. 12.51
D. 13.87
13. Find the radius of a spherical wedge whose volume is 12 cu. m with a central angle of 1.8 radians.
A. 2.36 m
B. 2.73 m
C. 2.52 m
D. 2.15 m
14. X is 12% older than Y while Y is 11% older than Z. By how much, in percent, is X older than Z.
A. 25.43
B. 21.78
C. 24.32
D. 23.66
15. How many diagonals have an undecagon?
A. 44
B. 55
C. 54
D. 65
16. A cylindrical tank open at the top is made of metal sheet having a total area of 49.48 square meters. If the
height of the tank is 1.5 times its base diameter, determine the diameter.
A. 3.5 m
B. 2.5 m
C. 3 m
D. 3.2 m
17. Determine the sum of numerical coefficients in the expansion of (a+b)^8.
A. 128
B. 256
C. 512
18. What is the value of y in the progression: 1, 1/5, 1/9, 1/y, …?
A. 10
B. 12
C. 13
D. 1024
D. 11
19. A triangular piece of land has one side measuring 12 km. The land is to be divided into two equal areas by a
dividing line parallel to the given side. What is the length of the dividing line?
A. 6
B. 8.485
C. 7.623
D. 8
Mathematics 2 for FEU Alabang
20. A car is travelling at a uniform speed 80 kph. The driver saw a roadblock ahead and stepped on the brake
causing the car to decelerate uniformly at 10 m/s2. The distance from the roadblock to the point where the car
stopped is 12 m. How many meters away from the roadblock was the car when the driver saw it if the
perception-reaction time is 3 s.
A. 123.32
B. 95.43
C. 103.36
D. 87.65
21. An observer wishes to determine the height of the tower. He observes the top of the tower from A and got an
angle of elevation 30°. He then walked 25 m closer to point B and observed the angle of elevation as 40°. Points
A and B are at the same elevation, and on a direct line with the tower. How high is the tower?
A. 43.21 m
B. 57.32 m
C. 46.27 m
D. 76.32 m
22. Steve Nash records the all-time highest percentage in the free throw line in NBA with 90.42%. If he is to shoot 3
free throws, what is the probability that he will miss one?
A. 0.15
B. 0.18
C. 0.23
D. 0.11
23. A particle moves according to the parametric equations of x and y in terms of time t: y = t^2 + 2lnt and x = t +
1/t. What is the acceleration when t=5?
A. 1.92
B. 0.016
C. 0.018
D. 1.93
24. In average, 11 examinees per room assignment pass the licensure examination bi-annually. What is the
probability that at least 6 in the room pass the examination bi-annually? Use Poisson’s distribution.
A. 0.079
B. 0.921
C. 0.038
D. 0.962
25. Squares ABCD and ADEF are perpendicular planes. If AB=4, what is the length of the line joining F and C?
A. 6.93
B. 5.66
C. 4.33
D. 4.98
26. The probability distribution of your winnings at a casino’s card game is shown below.
X
P0
P5
P10 P25
P(X) 0.1 0.4 0.3
0.2
How much should you expect to win if you play the game once?
A. P5
B. P7.5
C. P10
D. P12.5
27. The graphs of x^2 + y = 12 and x + y = 12 intersect at two points. What is the distance between these points?
A. 1
B. 1.4
C. 1.6
D. 14.87
SIT 1: P36,000 is divided among Rico, Rennie and Ray such that their shares in the same order, form an arithmetic
progression. Rennie’s share is three times that of Rico’s.
28. How much is Rico’s share?
A. P12,000
B. P4,000
C. P3,000
D. P9,000
29. How much is Rennie’s share?
A. P12,000
C. P9,000
C. P15,000
D. P4,000
30. What is the ratio of Ray’s share to that of Rico’s?
A. 3
B. 5
C. 8000
D. 9000
SIT 2: Evaluate the following limits:
31. (x^2-1)/(x-1) as x approaches 1.
A. 4
B. 1/4
C. 1/2
D. 2
32. (x^3-2x^2+7)/(2x^3+x-3) as x approaches infinity.
A. 2
B. 4
C. 1/2
D. ¼
33. x^3/(x^2-1) as x approaches 1.
A. 3
B. 0
C. 1
D. Does not exist
SIT 3: Given the equation 1 – 15x + 70x^2 – 120x^3 + 64x^4 = y.
34. What is the maximum ordinate of the graph of the curve?
A. 1.469
B. -1.469
C. 0.8358
D. 0.1593
35. What is the minimum ordinate of the graph of the curve?
A. 1.469
B. -1.469
C. 0.8358
D. 0.1593
36. Which of the following is a point of inflection of the curve?
A. 0.275
B. 0.356
C. 0.666
D. 0.671
Mathematics 2 for FEU Alabang
SIT 4: Given the data of a closed traverse:
Lines
Latitude
AB
446.56
BC
y
CD
-58.328
DA
-2.090
37. Find the value of x.
A. 135.667
B. 123.12
38. Find the value of y.
A. 15.762
B. 22.561
39. Find the value of z.
A. -88.698
B. 44.512
Departure
30.731
75.451
DMD
2A
1372.324
2158.023
-8668.766
z
x
148.621
-42.439
C. 136.913
D. 145.69
C. 32.987
D. 67.120
C. -44.512
D. 99.012
SIT 5: In the two peg test of a dumpy level, the following observations are taken:
Rod reading
Instrument set near X
Instrument set near Y
Station A
0.296
1.563
Station B
0.910
2.410
Point X is equidistant from both A and B, while Y is 2.50 m away from A along the extension of line AB and 79.27 m
from B.
40. Determine the true difference in elevation between A and B.
A. 0.555
B. 0.614
C. 0.656
D. 0.765
41. Determine the correct reading at A for a level sight if the instrument is at point Y.
A. 1.540
B. 1.564
C. 1.460
D. 1.555
42. Determine the correct reading at B for a level sight if the instrument is at point Y.
A. 2.170
B. 2.222
C. 2.238
D. 2.156
SIT 6: A 50 m steel tape is of standard length under a pull of 5.5 kg and a temperature of 20ᵒC when supported
throughout its entire length. The tape weighs 0.05 kg/m, has a cross-sectional area of 0.04 square cm and a
modulus of elasticity of 2.1 x 106 kg/cm2. This tape was used to measure a distance that was determined to be
458.650 m. At the time the measurement was made, the constant pull applied was 8 kg with the tape supported
only at end points. During the measurement, the temperature was observed to be at an average of 18ᵒC.
43. Determine the correction due to tension.
A. 0.016
B. 0.014
C. 0.011
D. 0.018
44. Determine the correction due to sag.
A. 1.828
B. 1.832
C. 1.622
D. 1.523
45. Determine the correct length of the line.
A. 456.825
B. 456.555
C. 456.821
D. 456.231
SIT 7: Answer the following money problems.
46. The cost of producing a certain commodity consists of P 45.00 per unit for labor and material cost and P 15.00
per unit for other variable cost. The fixed cost per month amounts P 450,000. If the commodity is sold at P
250.00 each, what is the break even quantity?
A. 2014
B. 2178
C. 2589
D. 2368
47. P 1,000,000 was invested to an account earning 8% compounded continuously. What us the amount after 20
years?
A. P 4,452,796.32
B. P 4,953,032.42
C. P 4,075,458.90
D. P 3,337,090.21
48. A man made a year-end payment of P 100,000 to an account earning 8% annually for 10 years. How much is
the account after 20 years?
A. P 3,127,540.18
B. P 4,075,458.99
C. P 3,327,452.88
D. P 3,247,112.92
49. A car is travelling at 60 kph applies a break and stopped at a distance of 30 m. The coefficient of friction
between the tires and the road is 0.50. What is the grade of the road?
A. -4.3%
B. 4.3%
C. -2.8%
D. 2.8%
50. The ground makes a uniform slope of 4.8% from STA 12 + 180 to STA 12 + 250. At STA 12 + 180, the center
height of the roadway is 1.20 m fill. At the other station, the center height is 3.11 m cut. Find the grade of the
finished road.
A. -4.214%
B. -2.149%
C. -1.427%
D. -1.357%
Mathematics 2 for FEU Alabang
51. A parabolic curve AB 400 m long connects two tangent grades of +6.5% and -3%. If the elevation of the summit
is 123.256 m, what is the elevation of point B?
A. 122.111 m
B. 119.625
C. 121.361
D. 120.542
52. A car is travelling at 80 kph on 4% upgrade suddenly applies a brake. If the coefficient of friction between the
tires and the pavement is 0.30, how far will the car travel after applying the brake?
A. 45 m
B. 87 m
C. 74 m
D. 61 m
53. Determine the equal payment series future worth factor of an annuity of P 15,000 per year for 25 years at 18%
interest annually.
A. 342.60
B. 133.41
C. 121.45
D. 33.1
54. Which of the following has the greatest effective interest rate?
A. 12.31% compounded quarterly
C. 12.35% compounded annualy
B. 12.20% compounded monthly
D. 12.32% compounded semi-annually
55. The impact factor of a road with radius 500m is 0.15. What maximum velocity can the car travel round the
curve? Neglect friction between the road and the wheel.
A. 97.65 kph
B. 124.78 kph
C. 63.25 kph
D. 81.12 kph
56. A man deposits P 6,000 every end of three months for his retirement. If the interest rate is 10% annually
compounded quarterly, what lump sum value can he expect after 20 years?
A. P 528,000
B. P 785,500
C. P 1,245,278.32
D. P 1,490,296.28
57. Determine the traffic flow in a certain highway if the traffic density is 4 vehicles per kilometer with space mean
speed of 30 mph.
A. 147.6 veh/hr
B. 127.2 veh/hr
C. 217.3 veh/hr
D. 193.2 veh/hr
SIT 8: Given the following data for a certain mode of payment:
Periodic payment: P 14,000
Term: 16 years
Interest rate: 10% compounded quarterly
Payment interval: 3 months
58. What is the sum of the payments after 16 years using ordinary annuity?
A. 2,213,657
B. 2,159,665
C. 2,315,678
D. 2,177,789
59. What is the sum of the payments after 16 years using annuity due?
A. 2,213,657
B. 2,159,665
C. 2,315,678
D. 2,177,789
“The good Lord in his infinite wisdom did not create us all equal when it comes to size, strength, appearance or various aptitudes. But
success is not being better than someone else, success is the peace of mind that is a direct result of self-satisfaction in knowing that you gave
your best effort to become the best of which you are capable.” –Coach John Wooden
Correlation 3
a.
b.
Quiz 2A – Mathematics
1. Which of the following is true these three coordinates:
A(2,3) B(5,6) C(0,-2)?
a. AB is parallel to BC
b. AB is perpendicular to BC
c. The three coordinates are not collinear
d. The three coordinates are collinear
2. Identify the conic section represented by the equation
4x2 – 2xy + 2y2 = 1
a. hyperbola
b. parabola
c. ellipse
d. circle
3. Evaluate:
a.
b.
infinity
0
c.
d.
-1
undefined
1500
1000
c.
d.
2000
3000
5. Water is pouring into a swimming pool. After t hours,
there are t +sqrt(t) gallons in the pool. At what rate is the
water pouring into the pool when t=9 hours?
a.
b.
6/5 gph
5/4 gph
c.
d.
8/7 gph
7/6 gph
6. What positive number added to its reciprocal gives the
minimum sum?
a.
b.
c.
d.
24°18’
32°25’
8. The lateral area of a right circular cone is 386 square
centimeters. If its diameter is one-half its altitude,
determine its altitude in centimeters.
a.
22.5
c.
17.4
b.
23.8
d.
18.5
9. The ratio of the volume to the lateral area of a right
circular cone is 2:1. If the altitude is 15 cm, what is the
ratio of the slant height to the radius?
a.
b.
5:1
4:1
c.
d.
5:2
4:2
10. Find the area bounded by the curve y=4sinx and the
x-axis from x=π/3 to x=π
4. The cost C of a product is a function of the quantity x
of the product: C(x)=x2 – 4000x + 50. Find the quantity for
which the cost is minimum.
a.
b.
42°36’
36°42’
1/2
3
c.
d.
2
1
7. The figure represents a rectangular parallelepiped;
AD=20 in., AB=10 in., AE=15 in. Find the number of
degrees in the angle BFO.
a.
b.
6.0 sq.units
8.4 sq.units
c.
d.
6.2 sq.units
7.2 sq.units
11. The probability that a patient recovers from a
delicate heart operation is 0.8. What is the probability
that exactly 2 of the next 3 patients who have this
operation survive?
a.
b.
0.51
0.27
c.
d.
0.38
0.44
12. Two cards are selected at random from 10 cards
numbered 1 to 10. Find the probability that the sum is
odd if the two cards are drawn one after the other
without replacement.
a.
b.
5/9
4/9
c.
d.
1/2
1/3