ECE Integration SUMMARY
MODULE 1 SUB 1: Fundamentals in Algebra
System of Numbers
THEORY OF EQUATIONS
•Number of Roots of an Equation“Every rational integral equation f(x) of the nth degree has
exactly ‘n’ roots”
•Consider:
𝒇𝒙=𝟒𝒙𝟐+𝟏𝟖𝒙𝟐+𝟖𝒙−𝟓
𝒇𝒙=𝒙𝟗+𝟖𝒙𝟓−𝟓𝒙+𝟏𝟐
Problem 1
The polynomial x3+ 4x2–3x + 8 is divided by x –5. What is the remainder?
=281
Problem 2
When the polynomial (x + 3) (x –4) + 4 is divided by (x –k), the remainder is k. Find the value of
k.
=4 or -2
Problem 3
Find the value of k so that (x –3) is a factor of x4–k2x2–kx –39 = 0.
=–7/3
Problem 4
When a certain polynomial p(x) is divided by (x –1), the remainder is 12. When the same
polynomial is divided by (x –4), the remainder is 3. Find the remainder when the polynomial is
divided by (x –1)(x –4).
= –3x+15
THEORY OF EQUATIONS
Descartes’ Rule of Signs:
•The number of positive real roots of a polynomial f(x) is either equal to the number of
variations in sign of f(x)or less than that number by an even integer.
•The number of negative real roots of a polynomial f(x) is either equal to the number of
variations in sign of f(-x)or less than that number by an even integer.
Problem 5
Find the least possible number of positive real zeros of the polynomial P(x) = 3x6+ 4x5+ 3x3–x –
3.
=1
Problem 6
Given P(x) = 3x6+ 4x5+ 3x3–x –3. What is the maximum number of real zeros in P(x)?
=4
Laws of Exponent
Properties of Radicals
Problem 7
Simplify the expression:3x–3x-1–3x-2
=5×3x-2
Problem 8
Solve for 3xX5x+1= 6x+2
3^x X 5^x+1=6^x+2
=2.1544
LOGARITHMS
•The logarithm of a number to a given base is the power or exponent to which the base must
be raised in order to produce the number. Mathematically:𝐥𝐨𝐠𝒂𝑵=𝒙𝒂𝒙=𝑵
Properties of Logarithm
Problem 9
Solve for x:log6+𝑥log4=log4+log(32+4𝑥)
=3
Problem 10
Find logx27+logx3=2
=9
Problem 11
What is the natural logarithm of 𝑒to the xy power?
=xy
QUADRATIC EQUATIONS
•General Form:
𝑨𝒙𝟐+𝑩𝒙+𝑪=𝟎
•Solutions:
1.Factoring
2.Completing the Square
3.Quadratic Formula
4.By Calculator
QUADRATICE EQUATIONS
•Quadratic Formula:
Where: 𝑩𝟐−𝟒𝑨𝑪 is called the discriminant.
•Relationship between roots and coefficients
𝑨𝒙𝟐+𝑩𝒙+𝑪=𝟎
•Let r1and r2be the roots of the equation.
Problem 12
Given the equation: 3x2+ 12x –6 = 0, determine the sum of its roots.
=-4
Problem 13
Given the equation: 3x2+ 12x –6 = 0, determine the product of its roots.
=-13
Problem 14
The equation whose roots are reciprocals of the roots of 2x2–3x –5 = 0 is:
=5x2+ 3x –2 = 0
Problem 15
Find k so that 4x^2+ kx+ 1 = 0 will only have one real solution.
=4
Problem 16
The roots of the quadratic equation are 1/3 and 1/4. What is the equation?
=12x2–7x + 1 = 0
BINOMIAL EXPANSION
Problem 17
Find the 5thterm of the expansion (a –2y)20.
=77520a16y4
Problem 18
Find the coefficient of the 6th term of the expansion (1/2a -3)^16
=-66339/128
Problem 19
Find the term involving x13in the expansion of: (4𝑥^2+1/x)^14.
=524812288x13
Problem 20
Find the sum of the coefficients of(𝑎+2𝑏)^6.
=729
Problem 21
Find the sum of the coefficients of (3𝑥+1)^4.
=255
Problem 22
Find the sum of the exponents of (𝑥+𝑦)^6.
=42
Problem 23
Find the sum of the exponents of
(3𝑥^3+2𝑦^4)^10.
=385
Problem 24
Find the term involving x^2yz in the expansion of:
(2𝑥+𝑦+5𝑧)^4
=240
SUPPLEMENTARY M1S1
INSTRUCTION: Resolve and study the below problems.
1.If f(x)=x2+x+1, then f(x)-f(x-1)=
a.0
b.x
c.2x d.3
2.Solve for the simultaneous equations: 3x-y=6; 9x-y=12.
a.x=3; y=1
b.x=1; y=-3
c.x=2; y=1
d.x=4; y=2
3.Solve for w from the following
equations:
3x-2y+w=11;
x+5y-2w=-9;
and
2x+y-3w=-6
a.1
b.2
c.3
d.4
4.Find A and B such that 𝑥+10𝑥2−4=𝐴𝑥−2+𝐵𝑥+2
a.A=-3; B=2
b.A=-3; B=-2
c.A=3; B=-2
d.A=-3; B=2
5.The arithmetic mean of 80 numbers is 55. If two numbers namely 250 and 850
are removed, what is the arithmetic mean of the remaining numbers?
a.42.31
b.57.12
c.50
d.38.62
6.If 2x-3y=x+y, then x2:y2=
a.1:4
b.4:1
c.1:16
d.16:1
7.Find the value of x in (3^5)(9^6)=3^(2x). Note: The expression ^ means power.
a.8.5
b.9
c.9.5
d.8
8.What is the sum of the coefficients of the expansion of (2x-1)20?
a.1
b.0
c.15
d.225
9.The sum of the logarithms of two numbers is 1.748188 and the difference of
their logarithms is -0.0579919. One of the numbers is
a.9
b.6
c.8
d.5
10.The term involving x9 in the expansion of (x2+ 2/x)12is:
a.25434 x9
b.52344 x9
c.25344 x9
d.23544 x9
MODULE 1 SUB 2: Applications of Algebra
AGE PROBLEMS
Let X = your age now
Past
Present
Future
X-3
x
x+5
3 years ago
5 years hence
Problem 1
Six years ago, Jun was 4 times as old as John. In 4 years, he would be twice as old as John.
How old is Jun now?
=26
Problem 2
In 5 years, Jose would be twice the age of Ana. Five years ago, Jose was 4 times as old as
Ana. Find the sum of their present ages.
=35
Problem 3
Two times the father’s age is 8 more than six times his son’s age. Ten years ago, the sum of
their ages was 44. The age of the son is:
=15
Problem 4
Mary is 24 years old. Mary is twice as old as Ana was when Mary was as old as Ana is now.
How old is Ana?
=18
WORK PROBLEMS
Case 1: Different rates
Steps to solve:
1. Compute the rate of work of each
2. Setup the equation
Problem 5
One pipe can fill a tank in 5 hours and another can fill the same tank in 4 hours. A drainpipe
can empty the full content of the tank in 20 hours. With all the three pipes open, how long
will it take to fill the tank?
=2.5 hrs
Problem 6
Mr. Brown can wash his car in 15 minutes, while his son John takes twice as long to do the
same job. If they work together, how many minutes can they do the washing?
=10
Problem 7
Jun can finish an accounting work in 8 hrs. Leo can finish the same work in 6 hrs. After 2
hrsof working together Jun left for lunch and Leo finished the job. How long does it take Leo
to finish the job?
=2.5hrs
WORK PROBLEMS
Case 1: Same rates
Steps to solve:
Problem 8
If 10 bakers can make 7 cakes in 1.5 hours, then how many bakers are needed make 14
cakes within 2 hours?
=15
MOTION PROBLEMS
In current of water or air
Let:X = speed in boat/plane
Y = speed in water/air
Then:X + Y : downstream or
with the wind
X –Y : upstream or against the wind
General Formula:
Let:
d = distance
v = speed
t = time
Then:𝒗=𝒅/𝒕
Problem 9
John left Pikit to drive to Davao at 6:15 PM and arrived at 11:45 PM. If he averaged 30 mph
and stopped 1 hour for dinner, how far was Davao from Pikit?
=135
Problem 10
A man travels in a motorized bancaat the rate of 12 kphfrom his barrio to the poblacionand
come back to his barrio at the rate of 10 kph. If his total time of travel back and forth is 3
hours 10 mins, the distance from the barrio to población is:
=17.27
Problem 11
CLOCK PROBLEMS
General Formula:Let:X= distance travelled by the clock’s minute hand Then:X/12=
distance travelled by the clock’s hour hand
NOTE: 5 mins = 30 degrees
Problem 12
How many minutes after 10 o’clock will the hands of the clock be opposite each other for
the first time?
=21.81
Problem 13
At what time after 12:00 noon will the hour hand and the minute hand of a clock first form
an angle of 120 degrees?
= 12:21.81
MITURE PROBLEM
General Form:
Problem 14
2000 kg of steel containing 8% nickel is to be made by mixing a steel containing 14% and
another containing 6% nickel. How much of the 14% nickel is needed?
=500
Problem 15
A chemist of a distillery experimented on two alcohol solutions of different strength, 35%
and 50 %. How many cubic meters of 35% strength must he use to produce a mixture of 60
cubic meters that contain 40% alcohol.
=40
COIN PROBLEMS
Penny =$0.01
Nickel =$0.05
Dime =$0.10
Quarter =$0.25
Half Dollar =$0.50
Problem 16
A wallet contains the same number of pennies, nickels, and dimes. The coins total $1.44.
How many of nickels does the wallet contain?
=9
Problem 17
A collection of 33 coins, consisting of nickels, dimes, and quarters, has a value of $3.30. If
there are three times as many nickels as quarters, and one-half as many dimes as nickels,
how many quarters are there?
=6
MISCELLANEOUS PROBLEM
Problem 18
The sum of two numbers is 21 and one number is twice the other. Find the product of the
numbers.
=98
Problem 19
The sum of digits of a 3 digit number is 14. The hundreds digit being 4 times the units digit.
If 594 is subtracted from the number, the order of the digits will be reversed. Find the ten’s
digit of the number.
=4
SUPPLEMENTARY M1S2
1. Mary is 24 years old. Mary is twice as old as Ana was when Mary was as old as Ana is
now. How old is Ana?
a.16
b.18
c.19
d.20
2. The sum of the parent’s age is twice the sum of their children’s ages. Five years ago, the
sum of the parent’s ages is four times the sum of their children’s ages. In fifteen years, the
sum of the parent’s ages will be equal to the sum of their children’s ages. How many
children were in the family?
a.2
b.3
c.4
d.5
3. Peterand Paul can do a certain job in 3 hours. On a given day, they worked together for 1
hour then Paul left and Peter finishes the rest of the work in 8 more hours. How long will it take
for Peter to do the job alone?
a.10 hours
b.11 hours
c.12 hours
d.13 hours
4.Pedro can paint a fence 50% faster than Juan and 20% faster than Pilar and together they
can paint a given fence in 4 hours. How long will it take Pedro to paint the same fence if he
had to work alone?
a.10 hours
b.11 hours
c.13 hours
d.15 hours
5.Nonoy can finish a certain job in 10 days if Imelda will help for 6 days. The same work can
be done by Imelda in 12 days if Nonoy helps for 6 days. If they work together, how long will
it take for them to do the job?
a.8.9
b.8.4
c.9.2
d.8
6. The enrolment at College A and College B both grew up by 8% from 1980 to 1985. If the
enrolment in college A grew up by 800 and the enrolment in college B grew up by 840, the
enrolment at college B was how much greater than the enrolment in college A in 1985?
a.650
b.504
c.483
d.540
7.A merchant has 3 items on sale: namely, a radio for P50, a clock for P30, and a flashlight
for P1. At the end of the day, she has sold a total of 100 of the 3 items and has taken exactly
P1000 on the total sales. How many radios did he sell?
a.80
b.4
c.16
d.20
8. A deck of 52 playing cards is cut into two piles. The first pile contains 7 times s many
black cards as red cards. The second pile contains the number of red cards that is an exact
multiple as the number of black cards. How many cards are there in the first place?
a.14
b.15
c.16
d.17
9. A shoe store sells 10 different sizes of shoes, each in both high-cut and low-cut variety, each
either rubber or leather, and each with white or black color. How many different kinds of shoes
does he sell?
a.64
b.80
c.72
d.92
10. An engineer was told that a survey had been made on a certain rectangular field but the
dimensions had been 100 ft longer and 25 ft narrower, the area would have been increased by
2500 sq ft, and that if it had been 100 ft shorter and 50ft wider, the area would have been
decreased 5000 sq ft. What was the area of the field?
a.25,000 ft2
b.15,000 ft2
c.20,000 ft2
d.22,000 ft2
MODULE 1 SUB 3: Advanced Algebra
NUMBER PROGRESSIONS
ARITHMETIC PROGRESSION
•A sequence of numbers in which the difference of any two adjacent terms is constant.
•nth term:
•Sum of terms:
Problem 1
The sum of all even numbers from 0 to 420 is:
=44310
Problem 2
Find the 30thterm of the sequence:4, 7, 10, ...
=91
GEOMETRIC PROGRESSION
•A sequence of numbers in which the ratio of any two adjacent terms is constant.
•nth term:
•Sum of terms:
Problem 3
The 3rdterm of a geometric progression is 3 and the 6thterm is 64/9. What is the 5thterm?
=16/3
Problem 4
If the 3rdterm of a GP is 28 and the 5thterm is 112, find the sum of the first
10 terms.
=7161
•Sequence of numbers are in H.P. if their reciprocals form A.P.
Arithmetic Mean:
Geometric Mean:
Harmonic Mean:
Problem 5
The arithmetic mean of two numbers is 7.5 and their harmonic mean is 4.8. Find the geometric
mean.
=6
Problem 6
The geometric mean and harmonic mean of two numbers are 6 and 72/13 respectively. What is
the sum of the two numbers?
=13
DETERMINANTS
•Determinant can be solved by diagonal multiplication for a 3x3
Problem 7
Identify the determinant of matrix:
=-306
COMPLEX NUMBERS
•Complex or imaginary number –noted as “i” –is used to define squares of
negative numbers:
Problem 8
Evaluate i^100
=1
Problem 9
Multiply (1+𝑖2)(1−𝑖2)
=3
Problem 10
The value of x + y in the complex equation 3 + xi = y + 2i is:
=5
COUNTING TECHNIQUES
FUNDAMENTALS OF COUNTING
Suppose that two events occur in order. If the first can occur in mways and the second in nways
(after the first), then the two occur in mxn ways.
Problem 11
How many ways can ice cream be served if there are 3 cones and 4 flavors?
=12
Problem 12
How many 3-digit numbers can be formed from 0 to 9 if it should be odd without repetition.
=320
PERMUTATION
•The order in which the objects come into the group is important.
•AB is different from BA
Permutation of distinct objects:𝒏𝑷𝒓=𝒏!𝒏−𝒓!Permutation of objects of whichsome are
identical:𝒏!𝒏𝟏!𝒏𝟐!𝒏𝟑!...Circular Permutation:𝒏−𝟏!
Problem 13
In the long jump competition in Olympics, 10 athletes participated. How many ways can the
gold, silver, and bronze medals be awarded?
=720
Problem 14
How many arrangements can be done for the letters in the word STATISTICS?
=50,400
Problem 15
In how many ways can 8 persons be seated ata round table?
=5,040
COMBINATION
•The order in which the objects come into the group is NOT important.
•AB is the same as BA
Problem 16
Six men (coded as A, B, C, D, E and F) are qualified to run a machine that requires five operators
as a team. How many different teams can be formed?
=6
SUPPLEMENTARY M1S3
1.If the sum of the first 13 terms of two arithmetic progressions are in the ratio 7:3, find the
ratio of their corresponding 7th term.
a.3:7
b.1:3
c.7:3
d.6:7
2.The 1st, 4th, and 8thterms of an A.P. are themselves geometric progression. What is the
common ration of the geometric progression.
a.4/3
b.5/3
c.2
d.7/3
3.The sum of three numbers in arithmetic progression is 45. If 2 is added to the first number, 3
to the second, and 7 to the third, the new numbers will be in geometrical progression. Find the
common difference in A. P.
a.-5
b.10
c.6
d.5
4.In the recent Bosnia conflict, the NATO forces captured 6400 soldiers. The provisions on hand
will last for 216 meals while feeding 3 meals a day. The provision lasted 9 more days because of
daily deaths. AT an average, how many died per day?
a.15.2
b.17.8
c.18.3
d.19.4
5.Throw a fair coin five times. What is the probability of getting three heads and two tails?
a.5/32
b.5/16
c.1/32
d.7/16
6.The probability of getting a credit in an examination is 1/3. If three students are selected at
random, what is the probability that at least one of them got a credit?
a.19/27
b.8/27
c.2/3
d.1/3
7.Dennis Rodman sinks 50% of all his attempts. What is the probability that he will make exactly
3 of his next 10 attempts?
a.1/256
b.3/8
c.30/128
d.15/128
8.The UN forces for Bosnia uses a type of missile that hits the target with a probability of 0.3.
How many missiles should be fired so that there is at least an 80% probability of hitting the
target?
a.2
b.4
c.5
d.3
9.Write -4 + 3i in polar form.
a.5∠36.870
b.5∠216.870
c.5∠323.130
d.5∠143.130
10.Find the value of x in the complex equation: (x + yi)(1 -2i) = 7 –4i.
a.1
b.3
c.4
d.2
MODULE 2 SUB 1:
Circular and Trigonometric Functions
TRIGONOMETRY
The study of triangles applying the relationship between sides and angles•Word
origin:•“TRIGONON”meaning triangle•“METRIA”meaning measurement
BRANCHES OF TRIGONOMETRY
TYPES OF TRIANGLE (ANGLE)
ACUTE ,RIGHT, OBTUSE
PARTS OF TRIANGLES
Median
•Line joining a vertex to the midpoint of the opposite side.
•Intersection: Centroid
Altitude
•Line through a vertex and perpendicular to the opposite side.
•Intersection: Orthocenter
Perpendicular Bisector
•Create a right angle at the midpoint of the segment
•Intersection: Circumcenter
Angle Bisector
•Split the internal angle into two congruent angles.
•Intersection: Incenter
EULER LINE
•the line joining the centroid, circumcenter, and the incenter.
FUNDAMENTAL PROPERTIES
Trigonometric Functions
Pythagorean Theorem
“In a right triangle, the sum of the squares of the sides is equal to the square of the
hypotenuse”
-Pythagoras, 500 BC
Coterminal Angles
•Two angles have the same initial side and the same terminal side, but different amounts of
rotation
•To find coterminal angles for any angle, add or subtract 2πor 360°to the given angle
Trigonometric Cofunctions
Trigonometric Functions for Special Triangles
Unit Circle
Positive sign convention for terminal sides falling in region:
A = all trigonometric functions are positive
C = cosine function is positive
T = tangent function is positive
S = sine function is positive
PROBLEM 1
If the value of cos A is a negative fraction, then angle A terminates in what quadrant?
2nd and 3rd
PROBLEM 2
The angle between 180 degrees and 270 degrees has:
Negative sine and cosine
PROBLEM 3
Simplify: (sinθ+ cosθtanθ)/(cosθ)
2tanθ
PROBLEM 4
If tan A = -3 and tan B = 2/3, find tan(A -B).
=11/3
PROBLEM 5
If tan 4x = cot 6y, then
2x + 3y = 450
PROBLEM 6
The measure of 2.25 revolutions counterclockwise is:
810°
M2 S1 SUPPLEMENTARY
1.If cos A = -15/17 and A is in quadrant III, find cos ½ A.
2.Simplify (sin 2x) / ( 1 + cos 2x)
3.Triangle ABC has sides a, b and c. If a = 75 m, b = 100 m and the angle opposite
side a is 32°, find the angle opposite side c.
4. If 82° + 0.35x = Arctan( cot 0.45x ), find x.
5. Evaluate cos( Arcsin 3/5 + Arctan 8/15 )
6.If sec A = -5/4, A in quadrant II,find tan 2A.
7.Express -4 -4√3 i in trigonometric form.
8.If the cosine of angle x is 3/5, then the value of the sine of x/2 is
9.If sin A = 3/5 and cos B = 5/13, find sin (A + B).
10.Find the product of (4cis120°)(2cis30°) in rectangular form.
MODULE 2 SUB 2:
Trigonometric Identities and Equations
POLAR COORDINATES
COMPLEX NUMBERS
OBLIQUE TRIANGLES
Law of Sines
Law of Cosines
PROBLEM 1
If Arccos( x –2 ) = π/3, find x
=5/2
PROBLEM 2
If sin x = ¼ , find the value of 4sin(x/2)cos(x/2)
=1/2
PROBLEM 3
The trigonometric expression ( 1 -tan²x ) / ( 1 + tan²x ) is equal to
cos 2x
PROBLEM 4
If tanθ= √3, θ in quadrant I, find the value of (1 + cosθ) / (1 –cosθ).
=3
PROBLEM 5
If cscθ= 2 and cosθ< 0, then ( secθ+ tanθ) / ( secθ–tanθ) =
=3
PROBLEM 6
If sin A + sin B = 1 and sin A –sin B = 1, find A.
90o
PROBLEM 7
Simplify( secA + cscA ) / ( 1 + tan A )
=csc A
PROBLEM 8
Three times the sine of an angle is equal to twice the square of the cosine of the
same angle. Find the angle.
=300
PROBLEM 9
Simplifysin(4x) / cos(2x)
=4sin(x)cos(x)
PROBLEM 10
Find the product of (4cis1200)(2cis300) in rectangular form.
=-4(√3 –i)
M2S2 SUPPLEMENTARY
1.Within what limits between 0 degrees and 360 degrees must the angle θ lie if
cos θ= -2/5 ?
2.If sin x + sin y = ½ and cos x –cos y = 1, find x.
3.If cot(80° -x/2) cot(2x/3) = 1, find x.
4.Given: sec2θ = √10 and 2θ in quadrant IV. Find : cos4θ
5.If tanθ = ½ and θ is in the 1stquadrant, find tan 4θ.
6.If cos A = -15/17 and A is in quadrant III, find cos ½ A.
7.If Arctan(2x) + Arctan(x) = π/4, find x.
8.If Arctanx + Arctan(1/3) = 45°, find x.
9.If cscθ = 2 and cosθ < 0, then ( secθ + tanθ ) / ( secθ –tanθ ) =
10.If sin A = -7/25 where 180° < A < 270°, find tan(A/2).
MODULE 2 SUB 3:
Triangles and Spherical Trigonometry
AREA OF TRIANGLE
𝐴=1/2𝑏xℎ
Given:2 sides & angle
𝐴=1/2 𝑎𝑏 sin𝜃
Given:3 sides
TRIANGLES AND CIRCLES
Triangle Inscribed in a circle
Triangle circumscribing a circle
𝐴=𝑟𝑠
PROBLEM 1
From the top of a 200-meter-high building, the angle of depression to the bottom
of a second building is 20 degrees. From the same point, the angle of elevation to
the top of the second building is 10 degrees. Calculate the height of the second
building.
296.9 m
PROBLEM 2
A 20-meter-high mast is placed on the top of the cliff whose height above sea
level is unknown. An observer at sea sees the top mast at an elevation of 46
degrees 42 mins, the foot at 38 degrees 23 mins. The height of the cliff is closest
to:
=59 m
PROBLEM 3
The hypotenuse of a right triangle is 34 cm. Find the length of the shortest leg if it
is 14 cm shorter than the other leg.
=16 cm
PROBLEM 4
Find the supplement of an angle whose complement is 62°
=152°
PROBLEM 5
A pole cast a shadow of 15 meters long when the angle of elevation of the sun is
61°If the pole has leaned 15°from the vertical directly toward the sun, what is the
length of the pole?
=54.23 m
PROBLEM 6
Two sides of a triangle measures 6 cm. and 8 cm. and their included angle is 40 0.
Find the third side.
=5.144 cm
PROBLEM 7
Given a triangle: C = 100°, a = 15, b = 20. Find c:
=27
PROBLEM 8
Points A and B 1000m apart are plotted on a straight highway running from East
and West. From A, the bearing of a tower C is 32deg W of N and from B the
bearing of C is 26 deg N of E. Approximate the shortest distance of tower C to the
highway.
𝑑=374𝑚
PROBLEM 9
A ship started sailing S 42°35’ W at the rate of 5kph. After two hours, ship B
started at the same port going N 46°20’ W at the rate of 7 kph. After how many
hours will the second ship be exactly north of ship A?
=4.03
PROBLEM 10
An aero lift airplane can fly at an air speed of 300 mph ,If there is a wind blowing
towards the cast at 50mph, what should be the plane’s compass heading in order
for its course to be 30°?
=21.7°
PROBLEM 11
An aero lift airplane can fly at an air speed of 300mph, If there is a wind blowing
towards the cast at 50mph, If its course is to be 30°? What will be the plane’s
ground speed if it flies in this course?
𝑉=321.8𝑚𝑝ℎ
By sine law
PROBLEM 12
The sides of a triangle are 195 , 157 , and 210, respectively. What is the area of
the triangle?
=14586
PROBLEM 13
The sides of a triangular lot are 130m , 180m , and 190m. The lot is to be divided
by a line bisecting the longest side and drawn from the opposite vertex. Find the
length of the line.
=125
SPHERICAL TRIANGLES
1. SOLUTION TO RIGHT TRIANGLE
2. SOLUTION TO OBLIQUE TRIANGLE
3. AREA OF THE SPHERICAL TRIANGLE
TERRESTRIAL SPHERE
NOTES for Terrestrial Sphere
1. Radius of Earth = 3659 miles
2. 1 minute of the great circle area on the surface of the earth = 1 NM
3. 1 NM (nautical mile) = 6080 feet
4. 1 statute mile = 5280 feet
PROBLEM 14
If Greenwich Mean Time (GMT) is 6AM, what is the time at a place located 30
degrees east longitude?
=8 AM
PROBLEM 15
If the longitude of Tokyo is 139 E and that of Manila is 121 E, what is the time
difference between Tokyo and Manila?
=1 hr12 mins
PROBLEM 16
One degree on the equator of the earth is equivalent to:
=4 MIN.
PROBLEM 17
A spherical triangle ABC has an angle C=90 degrees, and sides a=50 degrees, and
c=80 degrees. Find the value of “b” in degrees.
=74.33°
PROBLEM 18
Solve for the remaining side of the right spherical triangle whose given parts
A=B=80 degrees and a=b=89 degrees.
=168°31’
PROBLEM 19
Determine the spherical excess of a spherical triangle whose angles are all right
angles?
=90°
PROBLEM 20
The area of spherical triangle ABC whose parts are A=93°40’, B=64°12’, C=116°51’ and the
radius of the sphere is 100 mis:
𝐴=16531𝑠𝑞.𝑚
M2S3 SUPPLEMENTARY
1.Find the length of side AB in the figure below.
2.A pole stands on a plane which makes an angle of 15° with the horizontal. A wire from the top
of the pole is anchored on a point 8 m from the foot of the pole. If the angle between the wire
and the plane is 30 degrees, find the length of the wire.
3.What is the greatest distance on the surface of the earth that can be seen from the top of
Mayon volcano which is 2.4 kilometers high if the radius of the earth is 6370 km ?
4.A bicycle race follows a triangular course. The 3 legs of the race are in order, 2.3 km, 5.9 km,
and 6.2 km. Find the angle between the starting leg and the finishing leg, to the nearest degree.
5.The sides of a triangular lot are 130 m, 180 m, and 190 m. The lot is to be divided by a line
bisecting the longest side and drawn from the opposite vertex. Find the length of the line.
6.A 3.0-m ladder leans against a wall and makes an angle with the wall of 28°. What is the
height above the ground where the ladder makes contact with the wall?
7.Find the side opposite the given angle for a spherical triangle havingb = 60°, c = 30°, A = 45°
8.Find the side opposite the given angle for a spherical triangle havinga = 45°, c = 30°, B = 120°
9.Solve the remaining side of the spherical triangle whose given parts are A = B = 80° and a = b
= 89°.
10.A spherical triangle ABC has an angle C = 90° and sides a = 50° and c = 80°. Find angle B
FORMATIVES COMPILATION
The Logarithms of the quotient and the product of two numbers are 0.352162518 and
1.55630501, respectively, Find the first number
9
If (log10 𝑥)/ (1 − log10 2) = 2, what is the value of x?
25
The arithmetic mean of 80 numbers is 55. If two numbers namely 250 and 850 are removed,
what is the arithmetic mean of the remaining numbers?
42.31
There are seven arithmetic means between 3 and 35. Find the sum of all the terms.
171
If the roots of the quadratic equation ax^2 + bx + c = 0 are 3 and 2 and a, b, and c are all whole
numbers, find a + b + c.
2
If 3x=4y then 3x^2 / 4y^2 is equal to:
4/3
A Group consists of n boys and n girls. If two of the boys are replaced by two other girls then
49% of the group members will be boys. Find the value of n:
100
Pedro bought two cars, one for P600000.00 and the other for P400000.00. He sold the first at a
gain of 10% and the second at a loss of 12%. What was his total percentage gain or loss?
1.20%gain
A grocery owner raises the prices of his goods by 10%. Then he starts his Christmas sale by
offering the customers a 10% discount. How many percent of discount does the customers
actually get?
1%
Find the value of x in (3^5) (9^6) = 3^(2x). Note: the expression ^ means power.
8.5
9.5
8
9
Peter and Paul can do a certain job in 3 hours. On a given day, they work together for 1 hour
then Paul left and Peter finishes the rest work in 8 more hours. How long will it take for Peter to
do the job alone?
12hrs
A deck of 52 playing cards is cut into two piles. The first pile contains 7 times as many black
cards as red cards. The second pile contains the number of red cards that is an exact multiple as
the number of black cards. How many cards are there in the first pile.
16
The population of the country increases 5% each year. Find the percentage it will increase in 3
years
15.74%
A man travels in a motorized banca at rate of 12 kph from his barrio to the poblacion and come
back to his barrio at the rate of 10 kph. If his total time of travel back and forth is 3 hours and
10 minutes, the distance from the barrio to the poblacion is:
17.27
If (x)=x^2+x+1, then f(x)-f(x-1)=
2x
A man rows downstream at the rate of 5 mph and upstream at the rate of 2mph. How far
downstream should he go if he is to return 7/4 hours after leaving?
2.5
The sum of the logarithms of two numbers is 1.748188 and the difference of their logarithms is
---0.0579919. One of the numbers is
8
Find the value of log(a^a)^a?
a^2 log a
One pipe can fill a tank in 5 hours and another pipe can fill the same tank in 4 hours. A drain
pipe can empty the full content of the tank in 20 hours. With all the three pipes open, how long
will it take to fill the tank?
2.5
log([x^y)] / log[(y^x)] is equal to:
ylogx-xlogy
In what ratio must a peanut costing of P 240.00 per kg be mixed with a peanut costing P 340.00
per kg so that a profit of 20% is made by selling the mixture at P 360.00 per kg
2:3
Kim sold a watch for P3500.00 at a loss of 30% on the cost price. Find the corresponding loss or
gain if he sold it for P5050.00.
1%gain
A merchant has three items on sale, namely a radio for P50, a clock for P30 and a flashlight for
P1. At the end of the day, he sold a total of 100 of the three items and has taken exactly P1000
on the total sales.
16
Throw a fair coin five times what is the probability of getting 3 heads and 2 tails
5/32
There are 9 arithmetic means between 11 and 51. The sum of the progression is
341
Write -4+3i in polar form
5<143.13
The constant term in the expansion of (x+1/ (x ^ (3/2)) ^15 is
5005
Find the positive value of x if log 𝑥 36= 2.
6
A pump can pump out water from a tank in 11 hours. Another pump can pump out water from
the same tank in 20 hours. How long will it take both pumps to pump out water in the tank?
7 hours
Juan can walk from his home to his office at the rate of 5mph and back at the rate of 2mph.
What is the average speed in mph?
2.86
The reciprocal of 20 is:
0.05
Determine the sum of the progression if there are 7 arithmetic mean between 3 and 35.
171
Find two consecutive even integers such that the square of the larger is 44 greater than the
square of the smaller integer.
10 & 12
the time required for an elevator to lift a weight varies directly with the weight and the distance
through which it is lifted and inversely as the power of the motor. It takes 20 seconds for a 5 hp
motor list 50 lbs. through 40 feet, what weight can an 80-hp motor lift through a distance of 40
feet within 30 seconds
1200lbs
A 100-kg salt solution initially 4% be weight. Salt in water is boiled to reduce water content until
the concentration is 5% by weight. How much water evaporated?
20
𝑚
Which of the following is equivalent to √ 𝑛√𝑎
a^(1/mn)
the positive value of a so that 4x, 5x + 4, 3x^2 – 1 will be arithmetic progression is
3
if 1/4 and -7/2 are the roots of the quadratic equation ax^2+bx+C=0, what is the value of B?
26
solve for a in the equation a=64^x 4^y
4^(3x+y)
Twice the father's age is 8 more than 6 times the sons. Ten years ago, their ages had a total of
44. The age of the son is?
15
In one day (24 hours.) how many will the hour hand and minute hand of a continuously driven
clock be together?
22
At what time after 12:00 noon will the hour hand and minute hand of the clock first form an
angle of 120°?
12:21.818
At 2:00 pm, an airplane takes off at 340mph on an aircraft carrier. The aircraft carrier moves
due south at 25kph in the same direction as the plane. At 4:05 pm, the communication
between the plane and aircraft carrier was lost. Determine the communication range in miles
between the plane and the carrier.
656 miles
The enrollment at college A and college B both grew up by 8% from 1980 to 1985. If the
enrollment in college A grew up by 800 and the enrollment in college B grew up by 840, the
enrollment at college B was how much greater than the enrollment in college A in 1985?
540
The sum of coefficients in the expansion of (x+2y+z) ^4 (x+3y) ^5 is:
262144
Log of the nth root of x equals log of x the 1/n power and also equal to:
(log(x)^(1/2))/n
In two hours, the minute hand of a clock rotates through an angle of:
720
what is the sum of the coefficients of the expansion of (2x-1) ^20?
0
The value of x + y in the complex equation 3 + xi = y + 2i is:
5
A man rows downstream at the rate of 5 mph and upstream at the rate of 2mph. How far
downstream should he go if he is to return 7/4 hours after leaving?
2.5 mi
Geometric mean and the harmonic mean of two numbers are 12 and 36/5respectively.
What are the numbers?
36 and 4
If 33y = 1, what is the value of y/33?
0
sum of the coefficients in the expansion (x + y – z) ^8?
1
Two engineering students are solving a problem leading to a quadratic equation. One student
made A mistake in the coefficient of the first-degree term, got roots of 2 and -3. The other
student made a mistake in the coefficient of the constant term got roots of -1 and 4. What is
the correct equation?
X^2 – 3x – 6 = 0
How many 4-digit numbers can be formed without repeating any digit from the following digits:
1, 2, 3, 4 and 6?
120
Solve for x: -(1/-27) ^-2/3
-9
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?
20 min
Mr. Brown can wash his car in 15 minutes, while his son John takes twice as long as the same
job. If they work together, how many minutes can they do the washing?
10
if is an odd number and q is an even number, which of the following expressions must be even
pq
Delia can finish a job in 8 hours. Daisy can do it in 5 hours. If Delia worked for 3 hours and then
Daisy was asked to help her finish it, how long will Daisy have to work with Delia to finish the
job?
1.923 hours
If x varies directly as y and inversely as z, and x = 14 when y =7 and z = 2, find the value of x
when y = 16 and z = 4.
16
Solve for the simultaneous equations: 3x-y=6; 9x-y=12
X=1; y=-3
Find k so that the equation 4x^2+kx+1=0 will have just one real root.
4
In a two-digit number the unit digit is 3 greater than the tens digit. find the number if it is 4
times as large as the sum of its digit.
63
A father is 3 times as old as his son. four years ago, he was four times as old as his son. How old
is his son?
12
If x to the ¾ power equals 8, then what is x?
16.
In a certain department store, the monthly salary of a saleslady is partly constant and partly
varies as the value of her sales for the month. When the value of her sales for the month is
P10,000, her salary for that month is P900, when her monthly sales goes up to P12,000, her
monthly salary goes up to P1,000. What must be the value of her sales for the month so that
her salary for that month would be P2,000?
P32,000
The ages of the mother and her daughter are 45 and 5 years, respectively. How many years will
the mother be three times as old as her daughter?
15
Find the value of x that will satisfy the following expressions √(𝑥 − 2) = −√𝑥 + 2
None
A shore sells 10 different sizes of shoes, each in both high-cut and low-cut variety, each either
rubber or leather, and each with white or black color. How many different kinds of shoes does
he sell?
80
The sum of the parents' ages is twice the sum of their children’s ages. Five years ago, the sum of
the parents' ages was four times the sum of their children’s ages. In fifteen years, the sum of
the parents' ages will be equal to the sum of their children’s ages. How many children were in
the family?
5
If log 8 𝑥 = −𝑛,then x is equal to
1/8^n
The denominator of a certain fraction is three more than twice the numerator. If 7 is added to
both terms of the fraction, the resulting fraction is 3/5. Find the original fraction.
5/13.
the term involving x^9 in the expansion of (x^ + 2/x) ^12 is:
25344 x^9
The only root of the equation x^2-6x+9=0
3
The constant term in the expansion of (x+1/(x^3/2)) ^15 is:
5005
What is the magnitude of the vector f= 2i + 5j +6k?
8.06
A bag contains 3 white and 5 black balls, if two balls are drawn in succession without
replacement, what is the probability that both balls are black?
5/14
In the recent Bosnia conflict, The NATO forces captured 6400 soldiers. The provisions on hand
will last for 216 meals while feeding 3 meals a day. The provisions lasted 9 more days because
of daily deaths. At an average, how many died per day?
17.8
Find the sum of the infinite geometric progression 6, -2, 2/3, . . .
9/2
Two fair dice are thrown. What is the probability that the sum shown on the dice is divisible by
5?
7/36
Nonoy can finish a certain job in 10 days if Imelda will help for 6 days. The same work can be
done by Imelda in 12 days if Nonoy helps for 6 days. If they work together, how long will it take
for them to do the job?
8.4
The sum of all numbers between 0 and 10000 which is exactly divisible by 77 is:
645645
There are 6 geometric means between 4 and 8748
13120
If log 5.2 1000 = 𝑥, what is the value of x?
4.19
By selling balut at P5 per dozen, a vendor gains 20%. The cost of the eggs rises by 12.5%. If he
sells at the same price as before, find his new gain %.
6.25%
Simplify the following: (√5-√3)/(√5+√3)
4 - √15
A boat travels downstream in 2/3 of the time as it goes going upstream. If the velocity of the
river’s current is 8 kph, determine the velocity of the boat in still water.
40
Solve for w from the following equations
3x-2y+w=11 x+5y-2w=-9 and 2x+y-3w=-6
3
Simplify (sin θ/ 1 – cos θ) – (1 + cos θ/ sin θ)
0
Of the 316 people watching a movie, there are 78 more children than women and 56 more
women than men. The number of men in the movie house is:
42
Factor x^4-y^2+y-x^2 as completely as possible
(x^2-y) (x^2+y-1)
In a two-digit number, the unit’s digit is 3 greater than the ten’s digit. Find the number if it is 4
times as large as the sum of its digits.
63
Simplify (sin θ + cosθtanθ)/(cosθ)
2tanθ
If sin 3A = cos 6B, then
A + 2B = 30°
Give the conjugate of 2+(square root of -25) in standard form.
2 + 5i
Find the value of sin (arccos 15/17)
8/17
it is an angular unit that is equal to 1/6400 of four right angles
mil
if 10^(ax+b) = P, what is the value of x?
(1/a) (logP-b)
The coreference angle of any angle A is the positive acute angle determined by the terminal
side of A and the y-axis. What is the coreference angle of 290 degrees?
20
In what quadrant do the secant and cosecant of an angle have the same algebraic sign?
I and III
Which of the following systems of angle measurements uses the degrees as the unit of
measure?
sexagesimal system
an angle between 90 degrees and 180 degrees has
Negative cotangent and cosecant
Relative to a right triangle ABC where C=90 degrees, which of the following is not true?
Csc A = secB
Which of the following is true?
Tan(180 degrees - θ) = -tan θ
Express 3i + 5 + (square root of -16) in the standard form
5+7i
The value of versθ is equal to
1-cosθ
Coversine A is equal to
1 – sin A
The point P(x,y) where x not equal to 0 and y > 0 is located in quadrant
I or II
It is an angular unit that is equal to 1/6400 of four right angles
Mil
It is defined as the angle subtended by a circular arc whose length is equal to the radius of the
circle
Radian
The cosecant of 960 degrees is equal to
-2 (sqrt of 3/3)
If sin x = 5/13, find sin 2x
120/169
If tan A = 2 and tab B = ½, find A+B
45 degrees
In what quadrant does an angle terminate if its cosine and tangent are both negative
Second
if sin θ = 3.5x and cos θ = 5.5x, find x
0.1534
A measure of 3200 mil is equal to
180
In what quadrant will the angle θ terminate, if sine θ is positive and sec θ is negative?
IV
To find the interior angles of a triangle whose sides are given use the law of
Cosine
in what quadrant can we locate the point (x, -4) if x is positive?
IV
If sin 40 degrees + sin 20 degrees = sin θ, find the value of θ
80
The expression (1-sinx) / (cosx) is equal to
(cosx) / (1+sinx)
For the trigometric function y = asin(bx+c), the absolute value of the ratio c/b is called
Phase shift
Sec A – cos A identically equal to
sin A tan A
In what quadrant can we locate the point (x, -4) if x is positive
IV
If tan θ = ½ and θ is in the 1st quadrant, find tan4 θ
-24/7
Evaluate tan (arcsec sqrt 5 – arccot 2)
¾
What is the period of y=3sin(x/2)?
4pi
If cos 3A + sin A= 0, find the value of A
45 degrees
If sin2x sin4x = cos2x, find the value of x
15
find θ if 2 tanθ = (1 tan^2 θ) cot 56 degrees
17
In what quadrant does an angle terminate if its cosine and tangent are both negative
Second
If tan A = 2 and tan B ½, find A+B
90
If sin 40 degrees+ sin20 degrees = sin θ, find the value of θ
80
The expression (1-sinx)/(cosx) is equal to
(cosx)/(1+sinx)
In what quadrant can we locate the point (x,-4) if x is positive?
IV
If cos 3A + sin A=0, find the value of A
45 degrees
If sin2xsin4x=cos2x cos4x, find the value of x
15
Which of the following relation is true for any angle θ?
sin (-θ) = sin θ
which of the following is an undirected distance?
The distance of a point from a line
If the product of cot 2 θ and cot 68 degrees is equal to unity, find θ
11 degrees
The secant is the cofunction of:
Cosecant
Which of the following is not true?
(secx/tanx) =(cosx/cotx)
If tan θ = 1/2 and θ is in the 1st quadrant, find tan 4 θ
-24/7
Simplify sin (4x)/cos(2x)
Tan(x)sec(x)
If sin + sin y = ½ and cos x – cos y = 1, find x
15
If the versin θ = x and 1 – sin θ = ½, find x if θ<90
0.134
If the tan θ = √3, θ quadrant III, find the value of (1+cos θ)/(1-cos θ)
1/3
Express sin (2arccosx) in terms of x
2x√1-x^2
Evaluate tan (arcsec√5 - Arccot 2)
¾
If arctan(2x) + Arctan(x) = π/4, find x
0.281
The rationalized value of (4 - 4√3 𝑖)(-2√3 – 2i is
-16√3 +16i
If tan x= ½ and tany = 1/3, find tan (x+y)
1
The trigonometric expression (1 - tan^2x)(1+ tan^2x) is equal to
Cos2x
How many solutions for 0< x < 2π solves the equation (2sin(x)-1)(tan(x)-1)=0?
4
If sinx + siny = ½ and cos x – cosy = 1, find x
15
Which of the following functions is positive if angle A terminates in the second quadrant?
Csc A
Relative to a right triangle ABC where C = 90 degrees, which of the following is not true
Csc A = sec B
Express 3i + 5+(square root of -16) in standard form
5+7i
The value of vers θ is equal to
1 – cos θ
If sin A= 5/13 and A in quadrant I, find cot A
12/5
The point p(x,y) where x not equal to 0 and y > 0 is located in quadrant
I or II
If the value of sin A is a negative fraction, then angle A terminate in
Quadrant II and IV
If tan A = 2 and tan B = ½, find A+B
90degrees
If arcsin (x-2)= π/6, find x
5/2
If tanθ =√3, θ in quadrant III, find the value of (1+cos θ) / (1-cosθ)
1/3
If sin x = ¼, find the value of 4sin(x/2) cos(x/2)
½
Twelve round holes are bored through a square piece of steel plate. their centers are equally
spaced on the circumference of a circle 18cm in diameter. find the distance between the
centers of two consecutive holes
4.66
Which of the following is a correct value of x for the equation sin(x)+sin(x/2) =0?
0
If 77 + (2x/5) = arccos (sin x/4), find x
20
If sin^2 + y = m and cos^2x+y = n, find y
(m + n – 1)/2
If cscθ = 2 and cosθ < 0, then (secθ + tanθ)/(secθ - tanθ) =
3
The angle between 90 and 180 degrees has
Negative secant and tangent
In what quadrants do the secant and cosecant of an angle have the same algebraic sign
I and III
Which of the following angles in standard position is a quadrantal angle?
540
In what quadrant does an angle terminate if its cosine and tangent are both negative
Second
A ladder leans against the wall of a building with its lower end 4m from the building. How long
is the ladder if it makes an angle 70 with the ground
11.7
If tan 35 = y, then (tan145 – tan125) / (1 + tan145tan125) =
(1 – y^2) /2y
Evaluate tan (Arccos(12/13) – Arcsin(4/5))
-33/56
A quadrilateral ABCD has its side AB perpendicular to side BC at B and its side AD perpendicular
to side CD at D. if angle BAD equals 60,AB = 10 m and AD = 12m find the distance (diagonal)
from A to C
12.86
Find the height of a tree if the angle of elevation of its top charges from 20 to 40 as the
observer advances 23m towards its base
14.78
The bearing of B from A is N20 E the bearing of C from B is S30 E and the bearing of A from C is
S40 W. if AB = 10. Find the area of the ABC
13.94
If sin A + sin B = 1 and sin A – sin B, find A
90
On the top of a cliff the farthest distance that can be seen on the surface of the earth is 60
miles. How high is the cliff if the radius of the earth is taken to be 4000 miles?
0.47mi
Which of the following proves the identity (1 + cos(x) + cos(2x) / (sin(x) + sin(2x))?
Cot(x)
If Arctan z = x/2, find cos x in terms of z
(1 – z^2) / (1+z^2)
Three times the sine of an angle is equal to twice the square of the cosine of the same angle.
Find the angle
30
A quadrilateral ABCD has its side AB and BC perpendicular to each other at B side AD makes an
angle of 45 with the vertical while side CD makes an angle of 70 with the horizontal. If AB = 15
and BC = 10, find the length of side CD
41.5
If two sides of a triangle are each equal to 8 units and included angle 70 find the third side
9.18
If Arctan z = x/2, find cos x in terms of z
(1 – z^2) / (1 + z^2)
A pole which leans to the sun by 10 15 from the vertical casts a shadow of 9.43m on the level
ground when the angle elevation of the sun is 54 50. The length of the pole is
18.3
Write (square root of 2) cis 135 in rectangular form
-1+i
The sum of the sines of two angles a and b is 3/2 while sum of the cosines of the angles is√3/2
find A
½
What is the greatest quadrant on the surface of the earth that can be seen from the top of
mayon volcano which is 3.4 kilimeters high if the radius of the earth is 6370 km?
174.8km
if cot theta square root of 3 and cos theta < 0, find csc theta
-1/2
if ysinx = a and ycosx = b, find y in terms of a and b
√𝑎^2 + b^2
If cot(80 - x/2)(cot(2x/3)=1, find x.
60
From the top of a tower 18m high, the angles of depression of two object situated in the
horizontal line with base of the tower and on the same side, are 30 and 45. Find the distance
between the two objects
13.18m
Two observers 100m apart and facing each other on a horizontal plane, observer at the same
angles of elevation of a balloon in them to be 58 and 44. Find the height of the balloon
60.23
A point P is at a distance of 4, 5 and 6 from the vertices of an equilateral triangle of side of x.
find x
8.5
find the angle which a 9 m will make with the ground if it is leaned against a window still 6m
high
51.8
two cars start at the same time from the same station and move along straight that roads that
form of 30, one car at the rate 30kph and the other at the rate of 40 kph. How far apart are the
cars at the end of half an hour
10.27
Two ships start from the same point one going south and the other north 28 east. If the speed
of the first is 12kph and the second is 16kph, find the distance between them after 45 minutes
20.4
In what quadrants do secant and cosecant of an angle have the same algebraic sign?
Ans. I and III
Coversine A is:
Ans. 1-sin A
Which of the following systems of angle measurements uses the degree as the unit of measure?
Ans. Sexagesimal system
In what quadrant does an angle terminate if its cosine and tangent are both negative?
Ans. Second
What limits between 0 degrees and 360 degrees must the angle θ lie if cos θ = -2/5
Ans. 90 and 270
The angle between 90 degrees and 180 degrees has:
Ans. Negative secant and tangent
3200 mils is equal to:
Ans. 180 degrees
The point P (x, y) where x not equal to 0 and y > 0 is located in quadrant:
Ans. I or II
Find θ if 2tan θ = (1 – tan^2 θ) cot 56
Ans. 17
If sin 3A = cos 6B, then:
Ans. A + 2B = 30 degrees
What is the period of y = 3 sin (x/2)?
Ans. 4 pi
For the trigonometric function y = a sin (bx +c), the absolute value of the ratio c/b is called
Ans. Phase Shift
Find the value of sin (Arc cos 15/17)
Ans. 8/17
If the product of cot 2θ and cot 68 degrees is equal to unity, find θ
Ans. 11 degrees
If sin 2x sin 4x = cos 2x cos 4x, find x
Ans. 15
Simplify (sin θ / 1 – cos θ) - (1 + cos θ / sin θ)
Ans. 0
If cos θ = 3/5 and θ in quadrant IV, find cos 2θ
Ans. -7/25
The expression (1 – sinx) / (cosx) is equal to
Ans. (cosx) /(1+ sinx)
The trigonometric expression (1 – tan^2 x) / (1 + tan^2 x) is equal to
Ans. cos2x
How many solutions for 0 ≤ x ≤ 2π solves the equation (2sin(x) - 1) (tan(x) - 1) = 0?
Ans. 4
If sin^2 x + y = m and cos^2 x + y = n, find y
Ans. (m + n – 1)/2
If Arctan(2x) + Arctan(x) = π/4, find x
Ans. 0.281
Express ½ (1 - √3 I) in trigonometric form
Ans. cis 300
If versine θ = x and 1 - sin θ = ½, find x if θ < 90
Ans. 0.134
If Arcsin (x – 2) = π/6, find x
Ans. 5/2
If tan35 = y, then (tan145 – tan125) / 91 = tan145*tan125) =
Ans. (1 – y^2) / 2y
It is defined as the angle subtended by a circular arc whose length is equal to the radius of the
circle
Ans. Radian
The secant is the cofunction of?
Ans. Cosecant
Which of the following is true?
Ans. tan(180 degrees - θ) = -tan θ
If csc θ = 2 and cos θ< 0, then (sec θ + tan θ) / (sec θ - tan θ) =
Ans. 3
The angles B and C of a triangle ABC are 50”30’ and 122”09’ respectively and BC = 9,
find the length of AB
Ans. 59.56
If versinθ = x and 1 - sinθ = ½, find x if θ < 90
Ans. 0.134
To find the interior angles of a triangle whose sides are given, use the law of:
Ans. Cosine
If sin x + sin y = ½ and cos x – cos y = 1, find x
Ans. 30
A ladder leans against the wall of a building with its lower end 4 m from the building. How long
Is the ladder if it makes an angle of 70 degrees with the ground?
Ans. 11.7
Two sides and the included angle of a triangle are measured to be 11 cm, 20 cm, and 112
degrees respectively. Find the length of the third side
Ans. 26.19 cm
Sec A – cos A is identically equal to
Ans. Sin A Tan A
Which of the following proves the identity tan^2 (x) - sin^2 (x)?
Ans. Tan^2 (x) csc^2 (x)
If Arctan z = x/2, find coz x in terms of z
Ans. (1 – z^2) / (1 + z^2)
Twelve round holes are bored through a square piece of steel plate. Their centers are equally
spaced on the circumference of a circle 18 cm in diameter. Find the distance between the
centers of two consecutive holes.
Ans. 4.66 cm
The sum of the sines of two angles A and B is 3/2 while the sum of the cosines of the angles is
√3 /2. Find cos (A – B)
Ans. ½
A flag stands on the top of a house 15m high from a point on the plane on which thee house
stands. The angles of elevation of the top and bottom of the flag staff are found to be 60 and 45
respectively. Find the height of the flag staff
10.98
The sides of triangle ABC are AB = 5, BC = 12 and AC = 10 find the length of the line segment
drawn from the vertex A and bisecting BC
5.15
Find the height of a tree if the angle of elevation of its top changes from 20 to 40 as the
observer advance 23m towards its base
14.78
A tower 28.65 m high is situated on the bank of a river. The angle of depression of an object on
the opposite bank of the river is 25.20. find the width of the river.
60.52m
if cos(x + y) = 0.17 and cosx = 0.50, find sin y.
0.3455
Evaluate [2(cos60 + isin60)]^3
-8
The angle of triangle ABC are in ratio 5:10:21 and the side opposite the smallest angle is 5. find
the side opposite the largest angle
11.43
a tower standing on level ground is due north of point A and due east of point B at A and B, the
angles of elevation of the top of the tower are 60 and 45 respectively. if AB = 20, find the height
of the tower.
17.32
If arcsin(2x) = 30 Degrees, find x
.25
Find the angle which 9-m ladder will make with the ground if it leaned against a window till 6m
high
51.8
station A and B 1000 m apart are plotted on a straight road running East and West. From A, the bearing
of a tower C is 32° W of N and from B the bearing of C is 26° N of E. Approximate the shortest distance of
tower C to the highway
373.81m
A spherical triangle ABC has an angle C=90 degrees, and sides a=50 degrees, and
c=80 degrees. Find angle A
51.07
from the top of a lighthouse 37m above sea level, the angle of depression of a
boat is 15 degrees. how far is the boat from the lighthouse?
138.1 m
a tree stands vertically on a hillside which makes an angle of 22 degrees with the horizontal.
from a point 60ft down the hill directly from the base of the tree, the angle of elevation of the
top of the tree is 55. how high is the tree?
56.97
two points a and b, 150m apart lie on the same side of a tower on a hill ang in a horizontal line
passing directly line passing directly under the tower, the angles of elevation of the top and
bottom of the tower viewed from B are 42 degrees and 34 degrees respectively and at A, the
angle of elevation of the bottom is 10 degrees. find the height of the tower.
8.3
In what quadrant will angle A terminate if sec A is positive and csc A is negative
IV
The sum of the sines of two angles a and b is 3/2 while sum of the cosines of the angles is√3/2
find A
30
two building with flat roofs are 15 m apart. from the edge of the roof the lower building, the
angle of elevation of the edge of the roof of the taller building is 32. how high is the taller
building of the lower building is 18m high?
27.4
The coreference angle of any angle A is the positive acute angle determined by the terminal
side of A and the y-axis. What is the coreference angle of 290 degrees?
20
if rcosxsiny = a, rcosxcosy = b and rsinx = c, find r
√𝑎2 + 𝑏2 + 𝑐^2
Which of the following is a correct value of x for the equation sin(x) + sin(x/2) = 0
All of them
1. A sequence of numbers such that the same quotient is obtained by dividing a term by the preceding
term is called
A.
B.
C.
D.
arithmetic progression
harmonic progression
infinite progression
geometric progression
2.If x is an irrational number not equal to zero and x 2 = N, then which of the following best describes N?
A. N is a natural number.
B. N is any rational number.
C. N is a positive rational number.
D. N is a positive integral number.
3. In the expression an , the number n is referred to as the
A. power
B. exponent
C. degree
D. index
4. The polynomial 2x3y + 8xyz4 – 3x2y3 has a degree of
A. 6
B. 3
C. 4
D. 8
5. The equations x + y = 2 and 2x + 2y = 8 are examples of equations which are
A. dependent
B. independent
C. conditional
D. inconsistent
6. A non-terminating and non-periodic decimal is
A. rational
B. prime
C. irrational
D. imaginary
7. The probability that an event is certain to occur is
A. greater than one
B. less than one
C. equal to one
D. equal to zero
8. Radicals can be added to form a single radical if they have the same radicand and the same
A. power
B. exponent
C. index
D. coefficient
9. A set of elements that is taken without regard to the order in which the elements are arranged is
called a
A. sequence
B. permutation
C. combination
D. progression
10. If b = 0, then the number a + bi is
A. complex
B. real
C. imaginary
D. irrational
11. How many prime numbers are there between 200 and 210?
A. one
B. three
C. none
D. two
12. In the expression
A. power
B. exponent
C. index
D. radicand
, the number n is called the
13. A harmonic progression is a sequence of numbers such that the reciprocals of the numbers will form
a
A. geometric progression
B. arithmetic progression
C. infinite progression
D. finite progression
14. If a, b and c are rational numbers and if b2 – 4ac is positive but not perfect square, then the roots of
the quadratic equation ax2 + bx + c = 0 are
A. real, irrational and unequal
B. real, rational and unequal
C. real, rational and equal
D. real, irrational and unequal
15. The equation xy = 0 implies that
A. x = 0 and y = 0
B. x = 0 or y = 0
C. x = 0 and y is not equal to zero
D. x = 0 or y is not equal to zero
16. Which of the following events are mutually exclusive?
A. event “Ace” and event “black card”
B. event “Queen” and event “heart”
C. event “Ten” and event “Spade”
D. event “diamond” and event “club”
17.Which of the following best describes (-3)1/2?
A. irrational number
B. pure imaginary number
C. natural number
D. complex number
18. It is a sequence of numbers such that the successive terms differ by a constant.
A. geometric progression
B. arithmetic progression
C. harmonic progression
D. infinite progression
19. If the discriminant of a quadratic equation is greater than zero, the roots of the equation are
A. real and equal
B. real and distinct
C. complex and unequal
D. imaginary and distinct
20. Which of the following terms is not rational in x?
A. 6x2
B. -4x
C.
x4
D.
21. In the theory of sets, the relation (A B)’ = A’ B’ expresses which of the following laws on ᴗ
operations?
ᴖ set
A. De Morgan’s Law
B. Involution Law
C. Complement Law
D. Identity Law
22. The set of odd integers is closed under the operation of
A. addition
B. subtraction
C. multiplicationD. division
23. Which of the following law states that the factors of a product may be grouped in any manner
without affecting the result?
A. commutative law
B. associative law
C. distributive law
D. inverse law
24. Which of the following terms has a degree of 4?
A. x4y
B. xy4
C. 4xy
D. xy3
25. The product of two conjugate complex numbers is
A. a real number
B. an imaginary number
C. zero
D. an irrational number
26. The statement “The examinees are not more than 30 years old.” implies that the examinees are
A. less than 30 years old
B. at least 30 years old
C. 30 years old or less
D. 30 years old or more
27. The closure property of numbers is not satisfied by the set of all integers under the operation of
A. addition
B. multiplication
C. subtractionD. division
28. The conditional probability of B given A is denoted symbolically by P(B/A). If P(B/A) = P(B), then the
events A and B are
A. dependent
B. independent
C. mutually inclusive
D. disjoint
29. What is the value of k that will make x2 – 28x + k a perfect square trinomial?
A. 100
B. 121
C. 144
D. 196
30. The roots of 6x2 + 7x + 34 = 0 are
A. real and equal
B. real and unequal
C. complex and unequal
D. pure imaginary
31. What is the conjugate of -6
A. 6
B. -6
C. 6i
D. -6i
32. Which of the following is true?
A.
B. (a + b)2 = a2 + b2 C. a /
(b – c) = a/b – a/c
D.
33. Which of the following cannot be a probability value?
A. (0.99)4
B. 88/100
C.
D. (0.5)-1
34. How many subsets has the set {c, u, t, e}?
A. 12
B. 14
C. 16
D. 18
35. Using the remainder theorem, find the remainder when x 6 – x + 1 is divided by x – 2.
A. 61
B. 62
C. 63
D. 64
36. What is the sum of the numerical coefficient of (2x – y)20?
A. zero
B. one
C. greater than one
D. less than one
37. How many subsets of one or more elements can be formed from a set containing 12 elements?
A. 4,096
B. 4,095
C. 4,094
D. 4,093
38. What is the product of
A. 6i
B. -6i
C. 6
and
?
D. -6
39. Which of the following is an irrational number?
A. (16)3/4
B. 0.0075
C. 1.36363636...
D. 3(5)1/2
40. Two prime numbers which differ by 2 are called prime twins. Which of the following pairs of
numbers are prime twins?
A. 1 and 3
B. 7 and 9
C. 17 and 19
D. 13 and 15
41. If A B C is not equal to zero, then which of the following notations refers to the set of ᴖ ᴖ elements
found in A and B but not in C?
A. A B C’ᴖ ᴗ
B. A B C’ᴗ ᴖ
C. A B C’ᴖ ᴖ
D. A B C’ᴗ ᴗ
42. Which of the following sequence is a geometric progression?
A. 16, 12, 8, ...
B. 16, 8, 2, ...
C. 16, 12, 9, ...
D. 16, 14, 12, ...
43. The point (x, y) where x = 2 and y = -x is in what quadrant?
A. first
B. second
C. third
D. fourth
44. Experiment:
Event:
A die is tossed.
A prime number results.
Which of the following is not an outcome of the event?
A. 1
B. 2
C. 3
D. 5
45. The logarithmic equation equivalent to 1/a = bc is
A. logc(1/a) = b
B. logb(1/a) = c
C. logc b = 1/a
D. logb c = 1/a
46. If P(A) = 0.78 and P(B) = 0.35, what is P(A’) + P(B’) ?
A. 0.83
B. 0.85
C. 0.87
D. 0.89
47. Which of the following is a polynomial in x ?
A. x -2 + x + 4
B.
+ 3x + 5
C. x3 + 2x + 3
D. 4/x + 3x + 1
48. If a set A has 1,024 subsets, how many elements does A contain?
A. 8
B. 9
C. 10
D. 11
49. Which of the terms in the expansion (y3 + y -1)10 will involve y2?
A. 6th term
B. 7th term
C. 8th term D. 9th term
50. P(A) = 0.60 and P(B’) = 0.30 while P(A B) = 0.15, find P(A B’) by using Venn Diagram. ᴖ ᴖ
A. 0.90
B. 0.30
C. 0.45
D. 0.75
51. Evaluate (i – 1)8.
A. 16
B. -16
C. 16i
D. -16i
52. If 16 is 4 more than 3x, then 2x – 5 =
A. 2
B. 3
C. 4
D. 5
53. In the series 2, -4, 8, -16, x, -64, ..., what is x?
A. -24
B. -32
C. 24
D. 32
54. If a, b, 2b, -a, ... is an arithmetic progression, find the next term.
A. 2b – 3a
B. 3b – 2a
C. 2b + 3a
D. 2b + a
55. In how many ways can 6 boys be seated in a row?
A. 520
B. 620
C. 720
D. 820
56.
is true only if
A. x > 2y
B. x = 2y
C. x <= 2y
D. x >= 2y
57. Find the fourth proportional to 3, 5 and 21.
A. 27
B. 56
C. 65
D. 35
58. Give the value of –(-1/27)-2/3
A. 9
B. -9
C. 1/9
D. -1/9
59. Simplify (a -1 + b -1) / (ab) -1
A. ab
B. b + a
C. 1/ab
D. a/b
60. If a die is thrown once, what is the probability of getting a prime number?
A. 1/3
B. ¼
C. ½
D. 1/6
61. Which of the following are similar radicalsA.
B.
and
C.
and
D.
and
and
62. Evaluate x = log 2 8
A. 4
B. 3
C. 2
D. 1
63. What is the greatest common factor (GCF) of 48 and 72 ?
A. 12
B. 24
C. 36
D. 42
64. If x, y and 5x are three consecutive terms of an arithmetic progression whose sum is 81, find x.
A. 9
B. 10
C. 11
D. 12
65. If f(x) = 2x3 – 3x + 1, then f(1) =
A. 0
B. 1
C. 2
D. 3
66. Find the sum to infinity of 3 -1, 3 -3, 3 -5, ...
A. 1/8
B. 3/8
C. 7/8
D. 5/8
67. Find the value of x if3 2=
10
1
x
A. 3
B. 4
C. 5
D. 6
68. Find the value of x in the series 1, 8, 27, x, 125, ...
A. 100
B. 81
C. 30
D. 64
69. Find the least common multiple (LCM) of 72x3y2, 108x2y3 and 9x2y
A. 108x3y3
B. 648x3y3
C. 972x3y3
D. 216x3y3
70. Evaluate (-1/27)-2/3 + (-1/32)-2/5
A. 6.25
B. 3.25
C. 9.25
D. 7.25
71. For what values of x will (x+3) < 2(2x+1)?
A. x=3
B. x>1/3
C. x<1/3
D. x=0
72. In how many ways can a man choose one or more of 7 ties?
A. 128
B. 127
C. 126
D. 125
73. If i =
, solve for x and y if x+2+4i = 5+(y-3)i
A. -3, 7
B. 3, -7
C. 3, 7
D. -3, -7
74. Determine the number of word of five different letters each that can be formed with the letters of
the word “VOLTAGE”.
A. 5,040
B. 2,520
C. 4,050
D. 2,520
75. If log A 10 = 25, find log10 A.
A. 3
B. 4
C. 5
D. 6
76. The set notation A-B is called the relative complement of B in A. It is equivalent to which of the
following
A. A
B. A
C. A’
D. A’
B’ᴗ
B’ᴖ
Bᴖ
Bᴗ
77. What is the sum of the first five prime numbers?
A. 17
B. 18
C. 28
D. 29
78. How many straight lines are determined by 8 points?
A. 28
B. 56
C. 36
D. 64
79. From a group of 10 men, in how many ways can we select a group of 6 men?
A. 120
B. 210
C. 200
D. 60
80. Find x if x = log2 (1/64)
A. -6
B. -5
C. -4
D. -3
81. If (x-2)i = y-3i, solve for x.
A. -4
B. -3
C. -2
D. -1
82. If 1/a, 1/b and 1/c are consecutive terms of an arithmetic progression, then b equals
A. 2ac/(a+c)
B. ac/(a+c)
C. (a+c)/2ac
D. (a+c)/ac
83. What is the third proportional to y/x and 1/x?
A. x/y
B. xy
C. y
D. 1/xy
84. (1/2) –x(8) –y is equal to
A. 23xy
B. 43y-x
C. 2x-3y
D. 4xy
85. A father is 27 years older than his son and 10 years from now, he will be twice as old as his son. How
old is his son now?
A. 15
B. 16
C. 17
D. 18
86. Find the 7th term of the geometric progression
,
,
, ...
A.
B.
C.
D.
87. If x = (2)^(log2 x), find the value of x.
A. 3
B. 4
C. 5
D. 6
88. In how many ways can 3 boys be seated in a room where there are 7 seats?
A. 200
B. 205
C. 210
D. 215
89. A card is drawn from a deck of 52 cards. What is the probability of drawing an ace?
A. 0.0763
B. 0.0765
C. 0.0767
D. 0.0769
90. If log10 x = -1/n, then x is equal to
A. 101/n
B. 10-1/n
C. 10-n
D. -10-n
91. The constant remainder when x30 – x + 5 is divided by x + 1 is
A. 7
B. 6
C. 8
D. 5
92. Find the mean proportional between
and
.
A. 6
B. 4
C. 8
D. 5
93. In how many ways can a poster be colored if there are 5 different colors available?
A. 30
B. 29
C. 28
D. 31
94. If 3x = 4y, then 3x2/4y2 is equal to
A. 16/9
B. 4/3
C. ¾
D. 27/64
95. Find the 8th term of 5x+1, 52x+1, 53x+1, ...
A. 56x+1
B. 57x+1
C. 58x+1
D. 59x+1
96. Find the larger of two numbers if their sum is 190 and the smaller number is 3/7 of the larger
number.
A. 132
B. 133
C. 134
D. 135
97. In how many ways can 6 boys be seated at a round table?
A. 120
B. 110
C. 100
D. 90
98. For what value of x is 2x+4 equal to 1/16
A. 6
B. -6
C. 8
D. -8
99. If x:6 = y:2 and x-y = 12, find y
A. 8
B. 2
C. 4
D. 6
100. Three balls are drawn from a bag containing 5 white balls and 4 red balls. What is the probability
that the balls drawn are all white?
A. 5/42
B. 3/42
C. 7/42
D. 9/42
101. If x3/4 = 8, then x =
A. 16
B. 14
C. 12
D. 10
102. How many 4-digit numbers can be made by using the digits from 1 to 9 if no digit is repeated in
each number?
A. 3,204
B. 3,024
C. 3,240
D. 3,402
103. Find the sum of the infinite geometric series 64 – 16 + 4 - ...
A. 256/5
B. 256/3
C. 256/2
D. 256/4
104. If 2x = 8x-1, solve for x.
A. ½
B. 3/2
C. 1
D. 2
105. A tank can be filled by one pipe in 6 hrs and by another in 8 hrs. If both pipes are open, how long
will it take them to fill the tank?
A. 2 hr
B. 2.5 hr
C. 3 hr
D. 3.5hr
106. The 100th term of the series 1.01, 1.00, 0.99, ... is
A. 0.0002
B. 0.002
C. 0.02D. 0.2
107. How many consecutive numbers beginning with 5 must be taken for their sum to be equal to 95?
A. 12
B. 11
C. 10
D. 9
108. If 2log2x + log2 4 = 1, find x. A.
B.
C.
D.
109. Find the next tem in the harmonic progression whose first three
terms are 1/3, 2/7 and 1/4.
A. 1/9
B. 2/9
C. 5/9
D. 4/9
110. A card is drawn from a deck of 52 cards. What is the probability of drawing a black king?
A. 1/25
B. 1/26
C. 1/27
D. 1/28
111. In how many ways can the position of President, Vice-President and Secretary be filled in a club of
12 members if no person is to hold more than one position?
A. 1,230
B. 1,320
C. 1,203
D. 1,302
112. How many arrangement can be made from the letters of the word “RESISTORS” when all are taken
at a time?
A. 30,240
B. 20,340
C. 40,320
D. 40,230
113. The odds are 13 to 8 in favor of winning the first prize of a lottery. What is the probability of
winning that prize?
A. 0.691
B. 0.617
C. 0.619
D. 0.671
114. For a geometric progression for which the first term is x+y and the common ratio is the reciprocal
of the first term, find the 10th term.
A. (x+y)-5
B. (x+y)-6
C. (x+y)-7
D. (x+y)-8
115. Twelve boys go out for tennis. How many matches are required if each boy is to play all the others
exactly once?
A. 64
B. 66
C. 68
D. 62
116. An urn contains 5 white balls, 4 black balls and 3 red balls. If 3 balls are drawn simultaneously, find
the probability that all are white balls.
A. 1/20
B. 1/21
C. 1/22
D. 1/23
117. How many committees can be formed from a group of 9 persons by taking any member at any
time?
A. 411
B. 511
C. 611
D. 711
118. A bag contains 4 black balls and 6 red balls. Two balls are drawn at random. What is the probability
that the balls drawn are both black?
A. 0.131
B. 0.133
C. 0.134
D. 0.135
119. How many liters of pure alcohol must be added to 10 liters of 15 percent alcohol solution in order
to obtain a mixture of 25 percent alcohol?
A. 1/3
B. ½
C. ¼
D. 1/5
120. What is the probability of getting a sum of 5 by throwing two dice once?
A. 0.111
B. 0.112
C. 0.113
D. 0.115
121. The first term of a geometric progression is 3 and the last term is 48. If each term is twice the
previous term, find the sum of the geometric progression.
A. 93
B. 92
C. 91
D. 90
122. How many numbers of two different digits each can be formed by using the digits 1,3,5,7,9?
A. 18
B. 20
C. 22
D. 24
123. What is the sum of the coefficients of (x + y + z)5 ?
A. 240
B. 241
C. 242
D. 243
124. In a single throw of a pair of dice, what is the probability of obtaining a total of 9?
A. 1/8
B. 1/9
C. 1/7
D. 1/6
125. Evaluate i113 + 4i84 + i3
A. 4
B. -4
C. 4 + 2iD. 4 – 2i
126. What is the number of permutation of the letters in the word “CHACHA”?
A. 80
B. 85
C. 90
D. 95
127. What is the sum of the first one hundred positive odd integers?
A. 13,000
B. 12,000
C. 11,000
D. 10,000
128. In the arithmetic progression -9, -2, 5, ... which term is 131?
A. 20
B. 21
C. 22
D. 23
129. Fnd the sum of the first fifty positive multiples of 12.
A. 15,300
B. 15,200
C. 15,100
D. 15,000
130. In a geometric progression, if the first term is x 2 and the common ratio is x4, which term is x18?
A. 4th
B. 5th
C. 6th
D. 7th
131. When f(x) = (x+3(x-4)+4 is divided by x-k, the remainder is k. The values of k are
A. 2 and -4
B. -2 and 4
C. 3 and 4
D. -3 and 4
132. From a group of 6 men and 5 women, in how many ways can we select a group of 4 men and 3
women?
A. 25
B. 39
C. 150
D. 420
133. In a single throw of pair of dice, what is the probability of obtaining a total greater than 9?
A. 1/6
B. 1/3
C. 2/3D. ½
134. A man can finish a certain job in 10 days. A boy can finish the same job in 15 days. If the man and
the boy plus the girl can finish the job in 5 days, how long will it take the girl to finish the job alone?
A. 30
B. 45
C. 15
D. 35
135. In how many ways can 4 boys and 4 girls be seated at a round table if each girl is to sit between two
boys?
A. 256
B. 16
C. 144
D. 96
136. If f(x) = x – 1 and f(g(x)) = 4, find g(x).
A. 4
B. 5
C. 6
D. 7
137. If John is 10 percent taller than Peter and Peter is 10 percent taller than May, then John is taller
than May by
A. 18%
B. 20%
C. 21%
D. 22%
138. A bag contains 4 white balls and 5 black balls. Four balls are drawn in succession and not replaced.
Find the probability that the first two balls are white and the last two balls are black.
A. 3/63
B. 4/63
C. 5/63
D. 6/63
139. If xy = 12, xz = 15 and yz = 20, find the value of xyz.
A. 60
B. 55
C. 50
D. 45
140. A committee of 4 is to be selected by lot from a group of 6 men and 4 women. What is the
probability that it will consist exactly of 2 men?
A. 2/7
B. 1/7
C. 3/7
D. 4/7
141. Solve for x in the equation (x+3):10 = (3x-2):8
A. 1
B. 2
C. 3
D. 4
142. If
, find the value if x3y.
A. 8
B. 32
C. 64
D. 128
143. If four electricians can earn P465.80 in 7 days, how much can 14 carpenters paid at the same rate
earn in 12 days?
A. P2,749.80
B. P2,974.80
C. P2,479.80
D. P2,749.80
144. How many four-digit even numbers can be written by using the digits 1 up to 9?
A. 1,144
B. 1,244
C. 1,344
D. 1,444
145. If 1 + x + x2 + ... = ¾, find the value of x.
A. -1/2
B. -1/3
C. -1/4
D. -1/5
146. If it takes A twice as it takes B to do a piece of work and if working together, they can do the work
in 6 days, how long would it take B to do it alone?
A. 8
B. 9
C. 7
D. 6
147. What is the probability that a coin will turn up heads twice in 6 tosses of the coin?
A. 15/64
B. 14/64
C. 13/64
D. 12/64
148. Simplify 3n – 3n-1 – 3n-2.
A. 5(3n-2)
B. 3(3n-2)
C. 33n-2
D. 33-n
149. If the odds are 5:3 that Juan will receive P5,000 in a Math contest, find his mathematical
expectation.
A. P2,135
B. P2,315
C. P3,215
D. P3,125
150. Find the sum of all integers between 90 and 190 if each integer is exactly divisible by 17?
A. 847
B. 857
C. 867
D. 887
151. If P(n,3) = 60(n,5), find n.
A. 7
B. 6
C. 5
D. 8
152. Solve for x in the equation
A. 2
B. 4
C. 8
D. 7
153. Two balls are drawn from a bag containing 9 balls numbered from 1 to 9. Find the probability that
both balls drawn are numbered even.
A. ¼
B. 1/5
C. 1/6
D. 1/7
154. What is the coefficient of the term containing x4 in the expansion of (2x+x -1)8 ?
A. 19,270
B. 19,720
C. 17,920
D. 17,290
155. The bob of a pendulum swings through an arc of 24cm long on its first swing. If each successive
swing is approximately 5/6 the length of the preceding swing, find the approximate total distance it
travels before coming to rest.
A. 121 cm
B. 114 cm
C. 144 cm
D. 169 cm
156. If 6x2 + 36x + k = 6(x+a)2, what is the value of k?
A. 12
B. 18
C. 54
D. 36
157. Two dice are rolled. Find the probability that the sum of the two dice is greater than 10.
A. 1/11
B. 1/12
C. 1/13
D. 1/14
158. If z = 2 + i and w = i – 2, find (z-w)/(z+w).
A. 2i
B. -2i
C. i
D. –i
159. A card is chosen from a deck of 52 cards. In how many ways can a spade or a ten be chosen?
A. 14
B. 15
C. 16
D. 17
160. Simplify
.
A. a7/8
B. a1/8
C. a2/3
D. a5/6
161. Determine the 7th term of the arithmetic progression 3xy – y, 2xy, xy +y, ...
A. 5y – 3xy
B. 5y + 3xy
C. 5x – xy
D. 5x + xy
162. How many arithmetic means must be inserted between 1 and 36 so that the sum of the resulting
arithmetic progression will be 148?
A. 5
B. 6
C. 7
D. 8
163. Transform the logarithmic equation 4(log x)2 + 9(log y)2 = 12 (log x) (log y) to its equivalent cartesion
form.
A. x3 = y2
B. x2 = y3
C. 3x = 2y
D. 2x = 3y
164. Given 3 dots and 3 dashes. How many code words of exactly 6 symbols can be formed?
A. 18
B. 20
C. 22
D. 24
165. Given1 2
6= 25, find x. x -
21
0 1 -1
A. 1
B. 2
C. 3
D. 4
166. A card is drawn from a deck of 52 cards. What is the probability of drawing an ace or a spade?
A. 17/52
B. 16/52
C. 15/52
D. 14/52
167. A man invested P4,000 at a certain rate of interest and P7,200 at 2% less than the first rate.
The yearly income from both investments is P640. Find the rate of interest for P4,000
A. 5%
B. 6%
C. 8%
D. 7%
168. Rationalize (2+i)/(3-i)
A. i/2
B. (5+i)/2
C. (1+i)/2D. (1-i)/2
169. If the first term and third term of a harmonic progression are 5/21 and 5/23 respectively, find the
6th term.
A. 26/5
B. 27/5
C. 24/5
D. 23/5
170. A boat travels 12km downstream and 16km upstream in 12 hours. If the rate of the current is 3kph,
what is the rate of the boat in still water?
A. 7 kph
B. 8 kph
C. 9 kph
D. 10 kph
171. Two people are selected randomly from a group of 4 men and 4 women. The probability that a man
and a woman are selected is
A. 4/7
B. 2/7
C. ¼
D. 3/7
172. Find the 5th term of (x2 – 3y)5 without expanding.
A. 403x2 y4
B. 402x2 y4
C. 404x2 y4
D. 405x2 y4
173. The equation whose roots are the reciprocals of the roots of 3x 2 – 7x – 20 = 0 is
A. 20x2 + 7x + 3 = 0
B. 20x2 – 7x + 3 = 0
C. 20x2 + 7x – 3 = 0
D. 20x2 - 7x – 3 = 0
174. Find the 50th term of 1 + i, 2 + 4i, 3 + 7i, ...
A. 47 + 148i
B. 48 + 148i
C. 49 + 148i
D. 50 + 148i
175. Fifty liters of acid solution contains 22% water. How many liters of water must be added to the
solution so that the resulting mixture will be 60% acid?
A. 15
B. 10
C. 20
D. 12
176. From the letters a, e, i, o, r, s, t, how many arrangements of 5 different letters each can be formed
if each arrangement involves 2 consonants and 3 vowels?
A. 100
B. 121
C. 144
D. 169
177. Six coins are tossed. What is the probability that exactly two of them are heads?
A. 0.423
B. 0.342
C. 0.234
D. 0.243
178. There are 0 defective per 1000 items of product in a long run. What is the probability that there is
one and only one defective in a random lot of 100?
A. 0.2770
B. 0.2707
C. 0.2077
D. 0.2207
179. The value of k which will make 8x2 + 8kx + 3k + 2 a perfect square trinomial is
A. 5
B. 6
C. 3
D. 2
180. The 3rd term of an arithmetic progression is 4 and the 9th term is -14. Find the 5th term. A. -2
B. -3
C. -4
D. -5
181. If x:y:z = 4:-3:2 and 2x + 4y – 3z = 20, find x.
A. -8
B. -6
C. -4
D. -2
182. Two brothers are respectively 5 and 8 years old. In how many years will the ratio of their ages be
3:4?
A. 3
B. 4
C. 5
D. 6
183. If f(x) = (x+2)/(x-2) and G(y) = y+2, find g(f(3)).
A. 4
B. 5
C. 6
D. 7
184. How many consecutive even integers beginning with 4 must be taken for their sum to equal 648?
A. 20
B. 22
C. 24
D. 26
185. Determine the value of the given determinant
1
1
0
0
1
1
1
0
0
1
1
1
0
0
1
1
A. 1
B. 0
C. -2
D. -1
186. At what time after 3 o’clock will the minute hand of a clock be as far in front of the 5 o’clock mark
as the hour hand is behind that mark?
A. 32.21
B. 32.31
C. 32.41
D. 32.51
187. If 2x = 4y and 8y = 16z, find x/z.
A. 1/3
B. 2/3
C. 8/3
D. 5/3
188. If n is a perfect square, what is the next larger perfect square?
A. n2 + 2n + 1
B. n2 + n + 1
C. n2 + 1
D. n +
+1
189. If 2 men can repair 6 machines in 4 hours, how many men are needed to repair 18 machines in 8
hours?
A. 6
B. 5
C. 4
D. 3
190. In how many ways can 9 books be arranged in a shelf so that 4 books are always together?
A. 2,880
B. 3,024
C. 14,400
D. 17,280
191. If the roots of (2k +2)x2 + (4 – 4k)x + k – 2 = 0 are reciprocals to each other, find the value of k.
A. -2
B. -3
C. -4
D. -5
192. The 3rd term of a geometric progression is 5 and the 6th term is -40. Find the 8th term.
A. -140
B. -150
C. -160
D. -170
193. The equation x2 – 4kx + 10 – 6k = 0 will have two equal roots if the value of k is
A. 5/3
B. -5/2
C. -5/3D. 5/2
194. The arithmetic mean of a set of 50 numbers is 38. Two numbers of the set, namely 45 and 55, are
discarded. What will be the arithmetic mean of the remaining set of numbers?
A. 35.5
B. 36.5
C. 37.5
D. 38.5
195. The sum of two numbers is 37. If the larger is divided by the smaller, the quotient is 3 and the
remainder is 5. Find the smaller number.
A. 6
B. 7
C. 8
D. 9
196. In how many ways can 6 boys be seated at a round table if two particular boys must always sit
together?
A. 42
B. 44
C. 46
D. 48
197. Maria is twice as old as Ana was when Maria was as old as Ana is now. If Maria is 24 years old now,
how old is Ana now?
A. 18
B. 17
C. 16
D. 15
198. If 1/x = a + b and 1/y = a – b, then x – y is equal to
A. 2b/(b2 – a2)
B. 2a/(b2 – a2)
C. 2b/(a2 – b2)
D. 2a/(a2 – b2)
199. The 4th term of a geometric progression is 81 and the 7th term is 9. What is the 10th term?
A. 1
B. 2
C. 3
D. 4
200. If log 2 = x and log 3 = y, find log(1.2)
A. 2x + y -1
B. 2x + y +1
C. 2x – y + 1D. 2x – y – 1
201. How many consecutive terms must be taken from the sequence 3, -6, 12, -24, ... for the sum to
equal 8,193?
A. 11
B. 12
C. 13
D. 14
202. At what time between 7 and 8AM will the minute hand and the hour hand of a clock be opposite
one another?
A. 7:05:23
B. 7:05:25
C. 7:05:27D. 7:05:29
203. A man can do a job with his son in 30 days. If after working together for 12 days, the son worked
alone and finished the job in 24 more days, how long will it take the son to do the job alone?
A. 40 days
B. 45 days
C. 50 days
D. 55 days
204. How many arrangements can be made from the letters of the word “TRANSIENTS” when all are
taken at a time?
A. 435,600
B. 453,600
C. 436,500
D. 463,500
205. Find the middle term of (x1/3 – y1/3)12 without expanding.
A. -924x2 y2
B. 924x2 y2
C. -492x2 y2 D. 492x2 y2
206. A bag contains 6 red balls, 4 white balls and 8 black balls. If 3 balls are drawn at random, determine
the probability that 2 balls are white and one is red.
A. 0.0414
B. 0.0441
C. 0.0144
D. 0.0141
207. If 102x = 4 find 106x-1.
A. 4.4
B. 5.4
C. 6.4
D. 7.4
208. Find the sum of the first n terms of 2, 8, 14, ...
A. n(3n+1)
B. n(2n+3)
C. n(n+3)
D. n(3n-1)
209. In how many ways can 4 men be selected out of 12 men if 2 of the men are to be excluded from
every selection?
A. 66
B. 45
C. 210
D. 495
210. Three drawn from a pack of 52 cards. Determine the probability that all cards drawn are of the
same suit.
A. 0.0516
B. 0.0518
C. 0.0520
D. 0.0522
211. If log x = 2(1 – log 2), find x.
A. 5
B. 10
C. 15
D. 25
212. Mary is three times as old as Ricky. Three years ago, she was four times as old Ricky was then. Find
the sum of their ages now.
A. 32
B. 24
C. 38
D. 36
213. An air plane went 360 miles in 2 hours with the wind and flying back the same route, took 3 hours
and 36 minutes against the wind. What was its speed in still air?
A. 60 mph
B. 120 mph
C. 140 mphD. 160 mph
214. A ball is dropped from a height of 28 cm. If it always rebounds ½ of the height from which it falls,
how far does it travel after the fifth bounce?
A. 372 cm
B. 374 cm
C. 376 cmD. 378 cm
215. A tank can be filled by one pipe in 16 minutes, by a second pipe in 24 minutes and can be drained
by a third pipe in 48 minutes. If all pipes are open, in how many minutes can the tank be filled?
A. 10
B. 12
C. 14
D. 16
216. If 3x 3y = 27 and 2x + y = 5, find x.
A. 2
B. 3
C. 4
D. 5
217. The amount of P300,000 is divided into 3 parts in the ratio 2:5:8 and these parts are invested at 2%,
4% and 6% respectively. Find the income from the 6% investment.
A. P6,600
B. P7,600
C. P8,600
D. P9,600
218. Find the sum of the first 12 terms of an arithmetic progression whose 7 th term is 5/3 and with a
common difference of -2/3
A. 22
B. 24
C. 26
D. 28
219. By use of 3 different red flags and 4 different green flags, how many signals can formed by flying all
the flags from seven positions on a pole if the same color are to be consecutive?
A. 144
B. 288
C. 70
D. 84
220. In a throw of two dice, the probability of obtaining a total of 10 or 12 is
A. 1/16
B. 1/12
C. 1/9
D. 1/18
221. Find x so that x-1, x+2 and x+8 are the first three terms of a geometric progression.
A. 4
B. 3
C. 5
D. 2
222. If
, find x.
A. 5/9
B. 9/5
C. 7/9
D. 9/7
223. Find the sum of the infinity of 1 – ½ + ¼ - 1/8 + ...
A. 1/3
B. 2/3
C. ¼
D. ¾
224. If log 2 = x and log 3 = y, find log4 9
A. x/y
B. xy
C. y/xD. xy
225. A man drives a certain distance at 50kph and a second man drives the same distance in 20 minutes
less time at 60kph. Find the distance traveled.
A. 130 km
B. 120 km
C. 110 km
D. 100 km
226. How many numbers between 200 and 500 can be formed by using the digits 0, 1, 2, 3, 4, 5 if each
digit must not be repeated in any number?
A. 120
B. 80
C. 60
D. 100
227. A bag contains 10 white balls and 5 black balls. If 3 balls are drawn in succession without
replacement, find the probability that the balls are drawn in the order black, black and white.
A. 0.0733
B. 0.0723
C. 0.0743
D. 0.0713
228. Find the term free of x in the expansion of (x 2 – x -1)9 .
A. 64
B. 84
C. 96
D. 48
229. How many different signals; each consisting of 6 flags hung in a vertical line can be formed from 4
identical red flags and 2 identical blue flags?
A. 12
B. 13
C. 15
D. 16
230. Juan can do a job in 6 days and Pedro can do the same job in 10 days. If Juan worked for 2 days and
Pedro joined him, in how many days more will the two boys finished the job together?
A. 3.5
B. 3
C. 2.5D. 2
231. What must be the value of x in the arithmetic progression x-7, x-2, x+3, ... so that its 10th term will
be 40?
A. 4
B. 3
C. 2
D. 1
232. The sum of the digits of a two-digit number is 12. If the digits are reversed, the new number is 4/7
of the original number, Determine the original number.
A. 57
B. 75
C. 93
D. 84
233. Find the probability that a couple with three children have exactly two boys.
A. 0.375
B. 0.365
C. 0.345
D. 0.335
234. The sum of three consecutive odd integers is 75. Find the largest integer.
A. 25
B. 27
C. 24
D. 29
235. In how many ways can a man choose 2 or more of 5 ties?
A. 31
B. 26
C. 20
D. 10
236. A boat can travel 10 kph in still water. It can travel 60 km downstream in the same time that it can
travel 40 km upstream. What is the rate of the current?
A. 2 kph
B. 2.5 kph
C. 3 kph
D. 3.5 kph
237. The first term of a geometric progression is 160 and the common ratio is 2/3. How many
consecutive terms must be taken to give a sum of 2,110?
A. 7
B. 6
C. 5
D. 4
238. At what time after 4PM will the 6 o’clock mark bisect the angle formed by the minute and the hour
hand?
A. 4:36.62
B. 4:36.72
C. 4:36.82
D. 4:36.92
239. If the product of two positive numbers is 14 and their sum is 6, find the sum of their reciprocals.
A. 3/7
B. 5/7
C. 4/7
D. 2/7
240. In how many ways can 6 plus signs and 4 minus signs be written on a straight line?
A. 24
B. 744
C. 360
D. 210
241. John and Jack can do a job in 4 hours and the working rate of John is twice that of Jack. How many
hours would it take John to work alone?
A. 5
B. 6
C. 7
D. 10
242. How many arrangements can be made out of the letters of the word “CONSTITUTION”?
A. 9,979,200
B. 9,799,200
C. 7,999,200
D. 2,979,900
243. A fair coin is tossed 3 times. Find the probability of getting either 3 heads or 3 tails.
A. 1/8
B. ½
C. 3/8
D. ¼
244. In how many ways can 4 boys and 3 girls be seated in a row of 7 seats with the girls always in
consecutive seats?
A. 620
B. 720
C. 820
D. 920
245. A tank can be filled by one pipe in 16 minutes; by another pipe in 24 minutes and can be drained by
a third pipe in 48 minutes. If all pipes are open at the same time, in how many minutes can the tank be
filled?
A. 12
B. 14
C. 10
D. 16
246. In how many ways can a panel of 5 judges make a majority decisions?
A. 18
B. 25
C. 14
D. 16
247. A man drove from station A to station B, 60 km away, at an average speed of 30kph and return to A
at an average speed of 20 kph. What was the average speed for the whole journey?
A. 22 kph
B. 23 kph
C. 24 kphD. 25 kph
248. A man can do a job in 8 days. After the man has worked for 3 days, his son joins him and together
they complete the job in 3 more days. How long will it take the son to do the job alone?
A. 12
B. 11
C. 13
D. 10
249. A tunnel is one kilometer long. A train 250 meters long is passing through the tunnel at 25kph. How
long will it take the train to completely pass the tunnel?
A. 6 min
B. 4 min
C. 5 min
D. 3 min
250. If two dice are cast, what is the probability that the sum will be less than 6?
A. 1/15
B. 1/16
C. 5/18
D. 2/19
251. If (3)^(log3 x) = 4x3, find the value of x.
A. 1/3
B. ½
C. ¼
D. 1/5
252. The probability that both Vic and Ric can solve a certain puzzle is 0.95. The probability that Vic
alone can solve the same puzzle is 0.98. What is the probability that Ric can solve the puzzle given the
Vic does?
A. 0.9361
B. 0.9694
C. 0.9310
D. 0.9496
253. In how many ways can 8 students be divided into 4 groups of 2?
A. 2,250
B. 2,502
C. 2,205
D. 2,520
254. A lottery has a first prize of P10,000,000. Suppose only 8,000,000 tickets are sold and you have
bought 40 tickets, what is your mathematical expectation?
A. P30
B. P40
C. P50D. P60
255. Thrice the sum of two numbers is 30 and the sum of their squares is 52. Find the product of the
numbers.
A. 22
B. 24
C. 26
D. 28
256. What is the harmonic mean between 3/8 and 4?
A. 24/35
B. 23/35
C. 22/35
D. 21/35
257. Find the third proportional to 4n2 and 2mn2.
A. mn
B. (mn)2
C. 2/m
D. m/2
258. One basket contains 5 apples and 2 oranges and a second basket contains 4 apples and 3 oranges.
If a fruit is taken from one of the two baskets at random, what is the probability that it is an orange?
A. 0.4133
B. 0.4143
C. 0.4153
D. 0.4163
259. If
, find the value of x.
A. 3/14
B. 3/15
C. 3/16
D. 3/17
260. The sum of the first three terms of an arithmetic progression is -3 while the sum of the first five
terms of the same arithmetic progression is 10. Find the first term.
A. -5
B. -4
C. -3
D. -2
261. The product of n P n-r and r P 1 is equal to
A. n Pn-r+1
B. nPn+r-1
C. nPn-1 D. nPr-1
262. In how many ways can 12 books be divided among four students so that each student receive 3
books?
A. 390,660
B. 396,600
C. 366,900
D. 369,600
263. Ann is eleven times as old as Beth. In a certain number of years, Ann will be five times as old as
Beth and five years after that, Ann will be three times as old as Beth. How old is Beth now?
A. 20
B. 21
C. 22
D. 23
264. Juan after working on a job for 2 hours was helped by Jose and it took 3 hours more for them
working together to finish the job. Had they worked together from the start, it would only require 4
hours to finish the job. How long would it take Juan to finish the job alone?
A. 6 hr
B. 7 hr
C. 8 hr
D. 9 hr
265. Find the common ratio of a geometric progression whose first term is 1 and for which the sum of
the first 6 terms is 28 times the sum of the first 3 terms.
A. 2
B. 3
C. 4
D. 5
266. If x:y:z = 2:5:7 and 4x – y + 2z = 51, find z.
A. 21
B. 6
C. 15
D. 8
267. A bag contains an assortment of red and blue balls. If two balls are drawn from it at random, the
probability that 2 red balls are drawn is 5 times the probability that 2 blue balls are draw. Furthermore,
the probability that one ball of each color is drawn is 6 times the probability that 2 blue balls are drawn.
How many red balls are there in the bag?
A. 3
B. 4
C. 5
D. 6
268. Solve for x in the equation x + 3x + 5x + 7x + ... + 49x = 625.
A. ¼
B. ½
C. 1
D. 2
269. The sum of the digits of a three-digit number is 14. The units digit is half the tens digit. If the digits
are reversed, the resulting number is 198 more than the original number. Find the original number.
A. 563
B. 842
C. 284
D. 921
270. Te arithmetic mean of two positive numbers exceeds their geometric mean by 2. Find the smaller
number if it is 40 less than the larger number.
A. 90
B. 101
C. 121
D. 81
271. Juan’s age on his birthday in 1989 is equal to the sum of the digits of the year 19Juan’s age on his
birthday in 1989 is equal to the sum of the digits of the year 19XY in which he was born. If X and Y satisfy
the equation X – Y – 6 = 0, find the age of Juan in 1990.
A. 18 yr
B. 19 yr
C. 20 yr
D. 21 yr
272. A group of neighbors plan to pay equal amount in order to buy a small power mower which costs
P4,800. If the adding 2 more neigbor to the original group, cost to each is reduced by P120. Find the
number of neighbor in the original group.
A. 4
B. 6
C. 8
D. 10
273. A chemist mixed 40 ml of 8% hydrochloric acid with 60 ml of 12% hydrochloric acid solution. She
used a portion of this solution and replaced it with distilled water. If the new solution tested 5.2%
hydrochloric acid, how much of the original solution did she used?
A. 50 ml
B. 40 ml
C. 70 ml
D. 60 ml
274. There are two copies each of 3 different books. In how many ways can they be arranged on a shelf?
A. 80
B. 85
C. 90
D. 95
275. A coin is tossed 6 times. What are the odds in favor of getting at least 3 head?
A. 18:11
B. 19:11
C. 20:11
D. 21:11
276. A bag contains 10 red balls. 30 white balls, 20 black balls and 15 yellow balls. If 2 balls are drawn,
replacement being made after each drawing, find the probability that only one is red.
A. 0.2211
B. 0.2311
C. 0.2411
D. 0.2511
277. In how many ways can 9 people cross a river riding 3 bancas whose maximum capacity is 2,4 and 5
respectively?
A. 5,346
B. 3,654
C. 6,453
D. 4,536
278. A realtor bought a group of lots for S90,000. He then sells them at a gain of S3,750 per lot and has a
total profit equal to the amount he received for the last 4 lots sold. How many lots were originally in the
group?
A. 10
B. 11
C. 12
D. 13
279. A man left Manila for Baguio City at past 9AM. Between 4 to 5 hours, he arrived at Baguio and
noticed the minute and hour hands of his wrist watch interchanged in position. At what time did the
man arrived at Baguio?
A. 1:45.53 PM
B. 1:45.63 PM
C. 1:45.73 PM
D. 1:45.83 PM
280. A and B can do a piece of work in 42 days, B and C in 31 and A and C in 20 days. In how many days
can all of them do the work together?
A. 18.86 days
B. 18.76 days
C. 18.66 days
D. 18.56 days
281. Using the relation nCr-1 + nCr = n+1Cr , find the value of y given that 89C63 –88 C63 =x Cy
A. 60
B. 61
C. 62
D. 63
282. A class of 40 students took examination in Mathematics and English. If 30 passed in English, 36
passed in Mathematics and 2 fails in both subjects, the number of students who passed both subjects is
A. 26
B. 28
C. 29
D. 30
283. There is a 30% chance of rain today. If it does not rain today, there is a 20% chance of rain
tomorrow. If it rains today, there is a 50% chance of rain tomorrow. What is the probability that it rains
tomorrow?
A. 0.27
B. 0.28
C. 0.29
D. 0.26
284. A group consists of n boys and n girls. If two of the boys are replaced by two other girls, then 49%
of the group members will be boys. Find the value of n.
A. 49
B. 51
C. 98
D. 100
285. If P(n,r) = 840 and C(n,r) = 35, find the value of r.
A. 2
B. 3
C. 4
D. 5
286. 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. 0.24
B. 0.21
C. 0.30
D. 0.36
287. A passenger train x times as fast as a freight train takes x times as long to pass when overtaking the
freight train as it takes to pass when two trains are going in opposite directions. What is the value of x?
A. 2.31
B. 2.41
C. 2.51
D. 2.61
288. Three card are drawn from a deck of 52 cards without replacement. Find the probability that all are
of the same color.
A. 0.2353
B. 0.3523
C. 0.3323
D. 0.2335
289. The simplest form of [(n+1)!]2 / n!(n-1)! is
A. n2
B. n(n+1)
C. n+1
D. n(n+1)2
290. How many products can be formed from the numbers 2,3,4,5,6,7 by taking two or more numbers
at a time?
A. 57
B. 64
C. 59
D. 69
291. Juan is thrice as old as Jose was when Juan was as old as Jose is now. When Jose becomes twice as
old as Juan is now, together they will be 78 years. How old is Juan now?
A. 12
B. 14
C. 16
D. 18
292. If z2 = 24 + 10i, find z.
A. 5+i or 5-i
B. 5+i or -5-i
C. 5-i or -5+iD. 5-i or -5-i
293. There are three candidates A, B and C for mayor in a town. If the odds that candidate A will win are
7:5 that of B are 1:3, what is the probability that candidate C will win?
A. 0.25
B. 0.13
C. 0.17
D. 0.21
294. The first term of an arithmetic progression is 6 and the tenth term is 3 times the second term. What
is the common difference?
A. 1
B. 2
C. 3
D. 4
295. A basket contains 3 red balls and 2 white balls while a second basket contains 2 red balls and 5
white balls. A man selected a basket at random and picked a ball and placed it on the other basket. Then
another ball is drawn from the second basket. Find the probability that both balls he picked are of the
same color.
A. 601/1680
B. 701/1680
C. 801/1680
D. 901/1680
296. If 7 coins are tossed once, find the probability of tossing at most 6 heads.
A. 0.9911
B. 0.9922
C. 0.9933
D. 0.9944
297. A businessman travelled 1,110km to attend a company conference. He drove his car 60 km to an
airport and flew the rest of the way. His plane speed is 10 times that of his car. If he flew 45 minutes
longer than he drove, how long did he fly?
A. 1.25 hr
B. 1.45 hr
C. 1.75 hrD. 1.50 hr
298. Maria, Ana, Cora, Cely, Jose, Juan and Pedro are participating in elections for four student officers:
President, Vice-President, Secretary and Treasurer. What is the probability that a girl becomes a
President and a boy Vice-President?
A. 0.8257
B. 0.2857
C. 0.5827
D. 0.7285
299. A tank can be filled separately in 10 and 15 minutes respectively by tow pipes. When a third pipe
was used simultaneously with the first two pipes, the tank can be filled in 4 minutes. How long would it
take the third pipe alone to fill the tank?
A. 12 min
B. 11 min
C. 10 minD. 9 min
300. In a single toss of a pair of dice, find the probability of tossing a total of at most 5.
A. 0.278
B. 0.268
C. 0.258
D. 0.248
301. In how many ways can 9 different books be divided among three boys A, B and C so that they
receive 4, 3 an 2 books respectively?
A. 1,600
B. 3,600
C. 1,260
D. 2,460
302. A family budget provides an expenditures of P5,100 per month for food. If the amount alloted for
meat is P300 more than that of milk and if the allotment for other food is twice as much as that for meat
and milk, find the amount alloted for milk.
A. P600
B. P650
C. P700
D. P750
303. Find the probability of throwing 11 each time in 3 tosses of 2 dice.
A. 0.00017
B. 0.00170
C. 0.01700
D. 0.17000
304. A 2.5-liter container has a mixture of 25% alcohol. How many liters of the mixture must be drained
out and replaced with pure alcohol in order to obtain a mixture containing 40% alcohol?
A. 0.35
B. 0.40
C. 0.45
D. 0.50
305. If the sum of two numbers is 1 and their product is also 1, find the sum of their cubes.
A. -3
B. -1
C. -2
D. -4
306. On a trip, a man noticed that his car averaged 21 km per liter of gasoline except for the days he
used the air conditioning and then it averaged only 17 km per liter. If he used 91 liters of gasoline to
drive 1,751 km, on how many of these kilometers did he used the air conditioning? A. 480 km
B. 580 km
C. 680 km
D. 780 km
307. The 2nd, 4th and 8th terms of an arithmetic progression are themselves in geometric progression.
Find the common ratio of the geometric progression.
A. 1
B. 2
C. 3
D. 4
308. The 4th term of a geometric progression is 343 and the 6th term is 16,807. Find the 8th term.
A. 853,243
B. 835,432
C. 824,533
D. 823,543
309. Mary is twice as old as Ann was when Mary was as old as Ann is now. If Mary is 20 years old now,
how old is Ann now?
A. 15
B. 16
C. 17
D. 18
310. The tens digit of a two-digit number is one third of the units digit. When the digits are reversed, the
new number exceeds twice the original number by 2 more than the sums of the digits. Find the units
digit.
A. 5
B. 6
C. 2
D. 3
311. Determine the common difference of an arithmetic progression whose sum to n terms is 2n 2 + 3n.
A. 7
B. 5
C. 4
D. 6
312. A man sold a book at 105% of the marked price instead of discounting the marked price by 5%. If he
sold the book at P4.20, what was the discounted price for which he should have sold the book?
A. P2.80
B. P3.80
C. P3.00
D. P2.50
313. If logb y = 2x + logb x, find y.
A. y = 2xbx
B. y = xbx
C. y = xb2x
D. y = x2 bx
314. The probability of Juan’s winning whenever he plays a certain game is 1/3. If he plays 4 times, find
the probability that he wins at most twice.
A. 0.86
B. 0.94
C. 0.89
D. 0.79
315. A tank can be filled by pipe A in half the time that pipe B can empty the same tank. When both
pipes are operating, the tank can be filled in 1 hour and 12 minutes. Find the time for pipe A to fill the
tank alone.
A. 0.60 hr
B. 0.50 hr
C. 0.40 hr
D. 0.30 hr
316. How many terms of the arithmetic progression 9,11,13,... must be added in order that the sum
should equal the sum of the first nine terms of the geometric progression 3,-6,12,-24,...?
A. 18
B. 19
C. 20
D. 21
317. How many arithmetic means must be inserted between 1 and 36 so that the sum of all numbers in
the resulting progression will be 148?
A. 4
B. 3
C. 5
D. 6
318. The head of a fish measures 22cm long. The tail is as long as the head and half the body and the
body is as long as the head and tail. How long is the fish?
A. 172 cm
B. 174 cm
C. 176 cmD. 178 cm
319. From a 2-liter vessel containing water a certain amount was drained and replaced with pure
alcohol. Later from the mixture, the same amount was drained and again replaced with pure alcohol.
What amount was removed each time if the resulting mixture has 36% alcohol?
A. 0.50 L
B. 0.40 L
C.0.30 L
D. 0.20 L
320. An urn contains 4 red marbles and 8 black marbles. A marble is drawn from the urn and a marble of
the other color is then put into the urn. A second marble is drawn from the urn. Find the probability that
the 2nd marble is red.
A. 5/18
B. 7/24
C. 13/36
D. 1/12
321. If a man was left with 10 hectares fewer than 40% of his land after selling 6 hectares more than
70% of his land, how many hectares of land did he initially own?
A. 30
B. 40
C. 50
D. 60
322. The 3rd term of an arithmetic progression is 4 and the 9th term is -14. Find the sum of the first six
terms.
A. 21
B. 19
C. 17
D. 15
323. A number of 5 different digits is written at random by use of the digits 1,2,3,4,5,6,8. Find the
probability that the number will have even digits at each end.
A. 0.2578
B. 0.2785
C. 0.2857
D. 0.2587
324. Without expanding (4x -2 – (x/2) )7, find the term involving x -2
A. 130x-2
B. 140x-2
C. 150x-2
D. 160x-2
325. Juan and Pedro can do a job together in 4 days. If the working rate of Juan is twice that of Pedro,
how long would it take for Pedro to do the job alone?
A. 12 days
B. 10 days
C. 8 days
D. 6 days
326. A man decided to build a wire fence along one straight side of his property. He planned to place the
posts 6 feet apart but after he bought the posts and the wire, he found that he had miscalculated. He
had 5 posts too few. However, he discovered that he could do with the posts he had by placing them 8
feet apart. How ling was the side of the plot?
A. 100 ft
B. 110 ft
C. 120 ft
D. 125 ft
327. Solve the given system for z: x – y + 6z = -15
3y – 2z = 18
5x + 2z = -8
A. 5
B. -1
C. ½
D. -3/2
328. The positive value of x so that x, x2 – 5 and 2x will be in harmonic progression is
A. 6
B. 5
C. 4
D. 3
329. If a number of 6 different digits is written at random by using the digits 1,2,3,4,5,6,7, find the
probability that the number will be even.
A. 1/7
B. 2/7
C. 3/7
D. 4/7
330. Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 4 blue balls if each
selection consists of 3 balss of each color.
A. 600
B. 700
C. 800
D. 900
331. What is the probability of drawing a face card on the first selection from a deck of 52 cards without
replacement drawing an ace on the second selection?
A. 0.0181
B. 0.0118
C. 0.0811
D. 0.0188
332. A boy is two years more than twice as old as his brother. The two boys together are 17 years older
than their sister who is 7 years younger than the older boy. How old is the sister?
A. 10
B. 15
C. 18
D. 22
333. The 3rd term of a harmonic progression is 15 and the 9th term is 6. Find the 11th term.
A. 4
B. 8
C. 5
D. 7
334. It takes 10 second for two trains to pass each other when moving in opposite directions. If they
move in the same direction, the faster train could completely pass the slower train in 2 minutes. If the
faster train is 120 meters long and the slower train is 130 meters long, find the rate of the faster train.
A. 11 m/s
B. 12 m/s
C. 13 m/sD. 14 m/s
335. A bag contains 9 balls numbered 1 to 9. Two balls are drawn at random. Find the probability that
one is odd and the other is even.
A. 7/9
B. 5/9
C. 6/9
D. 4/9
336. Three boys A, B and C working together can do a job in a certain number of hours. If A can do the
job in 1 day more and B alone can do the same job in 6 days more while C alone can do the job twice as
much time, in how many days would the three boys finish the job?
A. 2
B. 3
C. 4
D. 5
337. A man drove 156 km at a constant rate of speed. If he had driven 9 km more per hour, he would
have made the trip in 45 minutes less time. What was his initial speed?
A. 28 kph
B. 39 kph
C. 42 kph
D. 50 kph
338. Find the positive value of x so that 4x, 5x+4 and 3x 2-1 will be in arithmetic progression.
A. 5
B. 4
C. 3
D. 2
339. In how many ways can 3 men be selected out of 15 men if 2 of the men are to be excluded from
every selection?
A. 284
B. 286
C. 288
D. 282
340. A bag contains 5 red balls and 6 white balls. If we draw 4 balls, find the probability that at least 3
balls are white.
A. 0.2578
B. 0.4385
C. 0.3485
D. 0.3845
341. The sum of the squares of two consecutive even integers is 340. Find the larger even integer.
A. 12
B. 14
C. 16
D. 18
342. Find the 10th term in an arithmetic progression where the first term is 3 and whose 1 st, 4th and 13th
terms form a geometric progression.
A. 21
B. 22
C. 23
D. 24
343. Town A is 11 kilometers from town B.A boy walks from A to B at the rate of 3kph and a man starting
at the same time walks from B to A at 4kph. When will they be 2kilometer apart after meeting each
other along the way?
A. 1.66 hr
B. 1.76 hr
C. 1.86 hrD. 1.96 hr
344. A man can do a job in 8 hours, a boy can do the same job in 12 hours and a girl can do it in 16
hours. How long will it take them to do the job if the man and the boy work together for one hour and
then the boy and the girl finish the job?
A. 6.54 hr
B. 6.45 hr
C. 6.34 hr
D. 6.43 hr
345. Three groups of men A, B and C assemble 96 machines. Group A assembles 3 machines, B 4
machines and C 5 machines per day. If B works twice as many days as A and if C works 1/3 as many days
as both A and B together, how many days does group A work?
A. 6
B. 8
C. 10
D. 12
346. The probability that Juan will win a game of chess whenever he plays is ¼. If he plays twice, what
are the odds that he wins the 1st game and loses the 2nd game?
A. 0.2108
B. 0.2208
C. 0.2308
D. 0.2408
347. A bag contains 3 black balls, 7 white balls, 6 black cubes and 14 white cubes. Find the probability of
drawing a ball and a black object.
A. 0.10
B. 0.15
C. 0.20
D. 0.25
348. A man is 4 years older than his wife and 5 times as old as his son. When the son was born,
the age of the wife was 6/7 the age of her husband. Find the age of the son now/
A. 6 yr
B. 7 yr
C. 8 yr
D. 9 yr
349. Equal volumes of different liquids evaporate at different but constant rates. If the first is totally
evaporated in 6 weeks and the second is 5 weeks, when will the second be ½ the volume of the first?
A. 27/7
B. 29/7
C. 33/7
D. 30/7
350. A piece of work can be done by women in 11 days and 30 men in 7 days. In how many days can the
work be done by 22 women and 21 men?
A. 4
B. 5
C. 3
D. 2
351. It is a measure of dispersion which depends only on two scores in the entire distribution.
A. mean deviation
B. quartile deviation
C. variance
D. range
352. The most numerous or most common value in a series of values is called the
A. range
B. mode
C. mean
D. median
353. Which of the following numbers has two significant figure?
A. 0.039
B. 3,009
C. 39.00
D. 30.09
354. What do you call each possible outcome of an experiment?
A. trial
B. sample
C. event
D. variate
355. Which of the following is a measure of central tendency whose magnitude depends directly on the
size of the scores in the group?
A. arithmetic mean
B. harmonic mean
C. median
D. mode
356. An arrangement of raw numerical data in ascending or descending order of magnitude is called
A. data
B. array
C. frequency
D. category
357. It is halfway between the upper limit and lower limit of a class interval.
A. average
B. class mark
C. mean
D. boundary
358. A tabulation of data showing the number of times a score or group of scores appears is called
A. normal distribution
B. Poisson distribution
C. frequency distribution
D. probability distribution
359. The probability of an impossible event is
A. zero
B. unity
C. infinity
D. undefined
360. It is a variable that can only assume designated values.
A. continuous variable
B. parameter
C. discrete variable
D. variate
361. If two events have no outcome in common, then they are said to be
A. dependent
B. independent
C. mutually exclusive
D. mutually inclusive
362. The class interval 50 – 52 theoretically includes all measurement from 49.5 and 52.5 and these end
points are called the
A. class limits
B. class boundaries
C. class marks
D. class widths
363. The total area under a probability curve is always
A. equal to unity
B. greater than one
C. between zero and one
D. less than one
364. The range is a measure of
A. variability
B. deviation
C. central tendency
D. distribution
365. When 130 numbers are arranged in an array, the median corresponds to the
A. 65th number in the array
B. the mid number of the array
C. the mean of the 65th and the 66th numbers in the array
D. the average of the sum of all numbers in the array
366. It is a measure of dispersion which depends upon the deviation of all scores from the mean.
A. average deviation
B. quartile deviation
C. mean deviation
D. standard deviation
367. Which of the following is not a continuous variable?
A. weight of a body
B. height of a body
C. temperature of an object
D. number of girls
368. How many significant figures has the number 10.01?
A. one
B. two
C. three
D. four
369. It represents that point in the data where one half of the scores fall below that point and one half
falls above it.
A. average
B. median
C. midscore
D. mean
370. Which of the following events are mutually exclusive?
A. events “ace or black card”
B. events “king or red card”
C. events “queen or face card”
D. events “diamond or black Jack”
371. If H is the event of getting a head by tossing a coin and N is the event of getting a prime number by
tossing a die, then which of the following means “The probability of getting a tail given that a prime
number has come up on the die.”?
A. p(H’ and N)
B. p(H’ or N)
C. p(H’/N)D. p(N/H’)
372. In probability theory, what do you call the set of all possible outcomes of an experiment?
A. event
B. trial
C. sample spaceD. variate
373. If A and B are two events and p(A and B) = p(A)p(B), then the two events are said to be
A. complementary
B. independent
C. disjoint
D. dependent
374. The highest score in a distribution minus the lowest score is called the
A. class mark
B. median
C. range
D. standard score
375. Which of the following is commonly used to illustrate the government income and
expenditure? A. pie chart
B. frequency polygon
C. histogram
D. scattergram
376. It refers to the number of times a score occurs in a sample.
A. outcome
B. mode
C. average
D. frequency
377. It refers to the facts and figures collected on some characteristics of a sample.
A. array
B. data
C. population
D. histogram
378. In set theory, the sum A + B is denoted by
A. A Bᴗ
B. A Bᴖ
C. A = B
D. A ↔ B
379. The most widely used measure of dispersion is the
A. mean deviation
B. standard deviation
C. quartile deviation
D. average deviation
380. The number of favorable outcomes divided by the total amount of outcomes is called
A. permutation
B. certainty
C. probability
D. frequency
381. The sum of the squared deviation about the mean is called
A. variate
B. variable
C. variance
D. value
382. In statistics, which of the following is a qualitative variable?
A. number grade in a card
B. letter grade in a card
C. number of people
D. salary of a teacher
383. The positive square root of the variance is equal to
A. quartile deviation
B. mean deviation
C. standard deviation
D. average deviation
384.
Which of the following is true?A.
B.
C.
D.
It is equal to the absolute difference between the observations
in a sample and the mean divided by the total number of observations in
the sample.
A. arithmetic mean
B. root mean square
C. quartile deviation
D. mean deviation
385.
386. When the sample is large and the variable is quantitative which of the following measures of
central tendency has a distinct advantage in terms of accuracy?
A. geometric mean
B. arithmetic mean
C. median
D. mode
387. When a coin is tossed 8 times in succession, head appeared 3 times and tail 5 times in the following
order HTTTHHTT. In how many other orders could they have appeared?
A. 53
B. 54
C. 55
D. 56
388. In a single toss of a pair of dice, the probability of obtaining a sum of 6 is
A. 5/36
B. 7/36
C. 4/36
D. 6/36
389. A point in the distribution of scores at which 50% of the score fall below and 50% fall above.
A. mode
B. mean
C. median
D. range
390. If a coin is tossed 100 times, find the theoretical standard deviation.
A. 4
B. 2
C. 3
D. 5
391. If a die is thrown 3 times, what is the probability that all throws show 6?
A. 1/8
B. ¼
C. 3/8D. ¾
392. If A and B are two independent events and P(A) = 0.9 and P(not B) = 0.2, find P(not A or B).
A. 0.8
B. 0.9
C. 0.7
D. 0.6
393. A boy has an average of 85 in four subjects. What grade must he make in the fifth subject so that
his average will be 87?
A. 93
B. 94
C. 95
D. 96
394. A Poisson distribution is given by p(X) = [(0.7) X e-0.7] / X!. Find p(2).
A. 0.1172
B. 0.1217
C. 0.1127
D. 0.1721
395. If the probability of a defective bolt is 0.20, how many bolts are expected to be defective if there
are a total of 600 bolts?
A. 100
B. 105
C. 115
D. 120
396. When a test was given, the probability of getting a score of 85 was 0.70. If 40 students took the
test, what is the expected number of students who will get a score of 85?
A. 28
B. 27
C. 26
D. 25
397. Consider two independent events A and B. If P(A) = 0.85 and P(B’) = 0.35, find P(A’ and B).
A. 0.0795
B. 0.0975
C. 0.0597
D. 0.0759
398. The grades of an examinee in a board examination in three subjects A, B and C were 70, 76 and 82
respectively. If the weights accorded to these grades are 25, 35 and 45 respectively, what is the mean
grade of the examinee?
A. 80
B. 79
C. 81
D. 78
399. The ages of 8 people are 17, 50, 19, 43, 20, 36, 21 and 29. Find the median
A. 24
B. 26
C. 27
D. 25
400. Find the mode for the following numbers: 16,29,19,27,18,20,27,24,19,27.
A. 19
B. 27
C. 18
D. 24
401. Find the mean of 67,53,50,76,66,81,69,77,91.
A. 73
B. 72
C. 71
D. 70
402. The odds that a new product will succeed are estimated as being 5:3. Find the probability that the
product will succeed.
A. 0.625
B. 0.652
C. 0.562
D. 0.626
403. Determine the root mean square (RMS) of the numbers 2.7, 3.2, 3.8, 4.3.
A. 1.55
B. 2.55
C. 3.55
D. 4.55
404. If the variable x assumes that values 1, 3 and 5 while those of the variable y are 2, 4 and 6, calculate
the value of
.
A. 184
B. 188
C. 187
D. 186
405. If P(A) = 0.25 and P(B) = 0.35 and if A and B are not mutually exclusive events, find P(A or B).
A. 0.0875
B. 0.0714
C. 0.6000
D. 0.5125
406. The number of minutes, a girl spent in making 6 phone calls was 3, 8, 9, 11, 15 and 20 minutes. Find
the mean number of calls.
A. 10 min
B. 11 min
C. 12 min
D. 13 min
407. In a basketball game, Jawo is given two free throws. Based on his previous record, the probability
that his first free throw will be successful is 0.75 and the probability that he will be successful on both
throws is 0.55. If Jawo is successful on the first throw, what is the probability that he makes the second
throw?
A. 0.71
B. 0.73
C. 0.74
D. 0.75
408. For a sample which consists of the values 45, 50, 55, 60 and 65, the average deviation is
A. 5
B. 4
C. 6
D. 7
409. If
and
, find
.
A. 18
B. 16
C. 15
D. 17
410. Find the geometric mean of 2, 3, 3, 5, 7 and 8.
A. 4.121
B. 4.131
C. 4.141
D. 4.151
411. Of 300 students, 100 are currently enrolled in mathematics and 80 are currently enrolled in Physics.
These enrolment figures include 30 students who are enrolled in both subjects. What is the probability
that a randomly chosen student will be enrolled in either Mathematic or Physics?
A. 0.45
B. 0.50
C. 0.55
D. 0.60
412. On a single roll of a die, what are the odds of rolling either an even number or a 5?
A. 2:1
B. 3:1
C. 4:1
D. 5:1
413. Find the standard deviation of 4, 7, 8, 9 and 12.
A. 2.61
B. 2.81
C. 2.41
D. 2.31
414. If the median (Md) is 57.22 and the mean (M) is 55.78, find the mode (Mo) by using the empirical
formula M-Mo = 3(M-Md).
A. 30.10
B. 40.10
C. 50.10
D. 60.10
415. Evaluate
by using the summation formulas.
A. 10
B. 11
C. 12
D. 13
416. Calculate the harmonic mean of the numbers 2, 4, 5 and 7.
A. 3.55
B. 3.66
C. 3.77
D. 3.88
417. In an electric company, the probability of passing an IQ test is 0.75. If ten applicants took the test,
what is the theoretical standard deviation of the group?
A. 1.57
B. 1.47
C. 1.37
D. 1.27
418. A student has test scores of 75, 83 and 78. The final test counts half the total grade. What must be
the minimum(integer) score of the final test so that the average is 80?
A. 83
B. 82
C. 84
D. 81
419. Out of 10,000 men, the probability that a man picked at random weighs over 86 kg is 0.25 and the
probability that the man weighs less than 61 kg is 0.15. What is the probability that a man picked at
random weighs between 61 kg and 86 kg?
A. 0.55
B. 0.60
C. 0.65
D. 0.70
420. The amount X of money a certain author earns is shown in the following probability function:
X:
P(X):
P1,000
0.20
P1,200
0.22
P1,600
0.24
P2,000
0.21
P2,400
0.13
What is the probability that the author will earn more than P1,500?
A. 0.66
B. 0.34
C. 0.58
D. 0.80
421. If the probability of a defective bolt is 0.10, find the standard deviation of defective bolts in a total
of 500 bolts.
A. 6.71
B. 7.61
C. 5.71
D. 6.17
422. An urn contains 3 white balls and 2 black balls. If two balls are drawn at random, what is the
probability that the two balls drawn are of different colors?
A. 4/5
B. 2/5
C. 1/5
D. 3/5
423. On the final examination on Algebra, Juan was informed that he received a standard score of 1.4. If
the standard deviation of the examination grades is 10 and the mean is 72, find the examination grade if
Juan.
A. 85
B. 86
C. 84
D. 83
424. A die is tossed 6 times. Using the binomial probability formula, determine the probability of rolling
the number 5 four times.
A. 0.00804
B. 0.00480
C. 0.08004
D. 0.00840
425. For the probability distribution given below, find the mean.
X:
P(X):
-10
1 /5
-20
3 /10
30
1 /2
A. 6
B. 7
C. 5
D. 8
426. The probability that a man will be alive in 20 years is 0.68 and the probability that his wife will be
alive in 20 years is 0.45. What is the probability than both will be alive in 20 years?
A. 0.360
B. 0.306
C. 0.630
D. 0.603
427. In problem 426, find the probability that at least one of them will be alive in 20 years.
A. 0.428
B. 0.482
C. 0.824
D. 0.842
428. If a pack of 52 cards is cut, what is the probability that it shows a king, a jack, a spade or an ace?
A. 0.3421
B. 0.2431
C. 0.1432
D. 0.4231
429. A box contains 4 red marbles, 8 white marbles and 12 blue marbles. If 3 marbles are drawn, what is
the probability that one of each color is drawn?
A. 0.1897
B. 0.1987
C. 0.1798
D. 0.1879
430. A lottery has one prize of P100,000, two prizes of P50,000, five prizes of P25,000 and ten prices of
P10,000. If there are 100,000 ticket sold, what is the expected value of a ticket?
A. P4.00
B. P4.25
C. P4.50
D. P4.75
431. Out of 800 families with 4 children each, how many of these families would have at least one boy?
A. 600
B. 650
C. 700
D. 750
432. What is the probability of obtaining a sum of 11 when 3 dice are tossed?
A. 0.145
B. 0.135
C. 0.125D. 0.115
433. From 5 men and 6 women, a committee consisting of 3 men and 2 women is to be formed. How
many different committees can be formed if 2 men must be on the committee?
A. 35
B. 40
C. 45
D. 50
434. Two students A and B were informed that they received standard scores of 2.6 and -0.8
respectively on the final examinations in Physics. If their examination grades were 83 and 62
respectively, find the standard deviation of the examination grades.
A. 15
B. 13
C. 11
D. 9
435. In how many ways can 30 boys be selected out of 100 boys? Hint: Use Stirling’s approximation to n!
A. 24
B. 25
C. 26
D. 27
436. Find the probability of winning the first prize of a state lottery in which one is required to choose six
of the numbers 1, 2, 3, ..., 45 in any order.
A. 1.52 x 10-7
B. 1.42 x 10-7
C. 1.32 x 10-7
D. 1.22 x 10-7
437. If 3 percent of the electric bulbs manufactured by a company are defective, find the probability that
in a sample of 100 bulbs, 5 will be defective by using Poisson distribution.
A. 0.105
B. 0.103
C. 0.101
D. 0.107
438. A bag contains 3 white balls and 4 red balls. Each of three boys A, B and C, named in that order,
draws a ball without replacement. The first to draw a red ball receives P70. Determine the mathematical
expectation of C.
A. P6.00
B. P8.00
C. P10.00D. P7.00
439. Find the probability of getting between 2 and 5 heads inclusive in 8 tosses of a fair coin.
A. 0.8203
B. 0.8302
C. 0.8230
D. 0.8032
440. Five sealed envelopes are placed in a box, three of them containing P50 bill each and two of them
containing P100 bill each. Another box has ten sealed envelopes, six of them containing P50 bill each
and four of them containing P100 bill each. If a box is selected at random and an envelope is drawn from
it, what is the probability that it contains a P100 bill?
A. 3/5
B. ¾
C. 2/3
D. 2/5
441. A die is tossed 8 times. What is the probability of tossing 5 and 6 twice?
A. 0.044
B. 0.064
C. 0.054
D. 0.034
442. Given the probability distribution
X:
8
P(X):
1 /4
2
Find the expected value of x or E(x2)
15
1 /3
16
3 /8
24
1 /6
A. 283
B. 273
C. 263
D. 253
443. If a man buys a lottery ticket, he can win first prize of P30,000,000 or a second prize of P20,000
with probabilities of 1.9 x 10-7 and 4.1 x 10-5 respectively. What should be a fair price to pay the ticket?
A. P5.52
B. P6.52
C. P7.52
D. P8.52
444. A box contains 3 red balls and 7 black balls. A person selects a ball at random and the color is
noted. Then the ball is replaced. After shaking the box, a second ball is drawn and followed by the same
procedure until five drawings were made. What is the probability that of the 5 balls drawn, 2 were red?
A. 0.3078
B. 0.3708
C. 0.3087
D. 0.3807
445. Three towns A, B and C are equidistant from each other. A car travels from A to B at 40kph, from B
to C at 50 kph and from C to A at 60 kph. Determine the average speed for the entire trip. (Hint: The
average speed is equal to the harmonic mean of the given speeds.)
A. 44.65 kph
B. 46.65 kph
C. 48.65 kphD. 59.65 kph
446. An airplane travels distances of 1,500 mi, 2,000 mi and 3,200 mi at speeds of 120 mph, 150 mph
and 200 mph respectively. Find the average speed of the plane.
A. 160 mph
B. 150 mph
C. 140 mph
D. 130 mph
447 . Given the following frequency distribution
Class Interval
5– 7
8 – 10
11 – 13
14 – 16
17 – 19
Find the arithmetic mean.
A. 10.95
B. 11.95
C. 12.95
D. 13.95
448. In problem 447, find the median.
:
Frequency
8
14
18
11
9
A. 11.73
B. 11.63
C. 11.83
D. 11.53
449. In problem 447, find the standard deviation.
A. 3.73
B. 3.63
C. 3.53
D. 3.43
450. In problem 447, find the coefficient of variance.
A. 31.21 %
B. 41.21 %
C. 51.21 %
D. 61.21 %
451. Out of 50 numbers, 8 were 10’s, 12 were 7’s, 15 were 16’s, 10 were 9’s and the remainder were
15’s. Find the mean.
A. 13.38
B. 10.38
C. 11.38
D. 12.38
452. The probability that a man will be alive in 25 years is 3/5 and the probability that his wife will be
alive in 25 years is 2/3. Find the probability that one of them will be alive in 25 years.
A. 4/15
B. 1/5
C. 2/5
D. 7/15
453. Three marbles are drawn without replacement from an urn containing 4 red marbles and 6 white
marbles. If X is a random variable that denotes the total number of red marbles, construct a table
showing the probability distribution and find the variance of the distribution.
A. 0.54
B. 0.56
C. 0.58
D. 0.52
454. Three teachers in mathematics reported mean examination grades of 2.45, 2.25 and 1.85 in their
classes which consisted of 35, 28 and 20 students respectively. Determine the mean grade of the
classes.
A. 2.22
B. 2.24
C. 2.26
D. 2.28
455. A fair die is tossed 6 times. Find the probability that one 2, two 3’s and three 4’s turn up.
A. 0.0013
B. 0.0015
C. 0.0017
D. 0.0019
456. If it rains, an umbrella salesman can earn P780 per day. If it is fair, he can lose P156 per day. What
is his mathematical expectation if the probability of rain is 0.30?
A. P120.80
B. P122.80
C. P124.80D. P126.80
457. A continuous random variable X that can be assume values only between X = 2 and X = 8 inclusive
has a density function p(X) = a(X + 3) where a is a constant. Find the value of a.
A. 1/45
B. 1/46
C. 1/47
D. 1/48
458. In problem 457, find P(X-4).
A. ¾
B. 3/5
C. 3/7
D. 3/8
459. A factory supervisor finds that 20 percent of the bolts produced by a machine will be defective. If 5
bolts are chosen at random, find the probability that at most 2 bolts will be defective.
A. 0.9214
B. 0.9421
C. 0.9124
D. 0.9412
460. A box contains 5 white balls, 3 red balls and 2 black balls. A ball selected at random from the box,
its color noted and then the ball is replaced. Find the probability that out of 5 balls selected in this
manner, 2 are white balls, 2 are red balls and 1 is a white ball.
A. 0.115
B. 0.125
C. 0.135
D. 0.145
461. Compute the standard deviation for a binomial distribution in which out of 60 bolts, 42 bolts are
found to be defective.
A. 3.5496
B. 3.6549
C. 3.4596
D. 3.9546
462. Joey took examinations in algebra, physics, chemistry and english and scored 84, 79, 88 and
93 respectively. If the mean grade in algebra is 80, in physics 75, in chemistry 85 and in english 90 and if
the standard deviation are 8, 6, 4 and 5 in algebra, physics, chemistry and english respectively, in which
subject was his relative standing higher? Hint: Calculate the standard grade corresponding to each
subject and compare.
A. algebra
B. physics
C. chemistryD. english
463. A bag contains 8 one-centavo coins, 6 ten-centavo coins, 4 twenty five-centavo coins and 2 onepeso coins. The coins are placed one each in uniform boxes. What is the mathematical expectation of a
person drawing a box at random?
A. 11.65
B. 12.65
C. 13.65
D. 14.65
464. If the variance of a sample is 29 and its arithmetic mean is 11, find the root mean square.
A. 11.26
B. 12.25
C. 13.24
D. 10.36
465. In a company, the mean earnings per hour is P180. If the mean earning paid to male nd female
employees were P200 and P150 respectively, determine the percentage of male employed by the
company.
A. 50%
B. 55%
C. 60%D. 65%
466. A box contains 10 red balls, 15 orange balls, 20 blue balls and 30 green balls. Two balls are drawn in
succession replacement being made after each drawing. Find the probability that at least one ball is blue
A. 103/225
B. 104/225
C. 105/225
D. 106/225
467. A bag contains 1 red marble and 7 white marbles. A marble is drawn from the bag. After its color
has been noted, it is put back into the bag and another marble is drawn from the bag. Using Poisson
approximation, find the probability that in 8 such drawings, a red ball is selected 3 times.
A. 0.0631
B. 0.0541
C. 0.0451
D. 0.0316
468. A bag contains 9 tickets numbered from 1 to 9 inclusive. If 3 tickets are drawn from the box one at
a time, find the probability that they are drawn in the order odd, odd, even or even, eve, odd.
A. 7/18
B. 5/18
C. 4/18
D. 3/18
469. Between 1 and 3 pm, the average number of phone calls per minute coming into the switch board
of a company is 2. Using Poisson approximation, find the probability that during one particular minute
there will be 4 phone calls.
A. 0.0702
B. 0.0802
C. 0.0902
D. 0.0602
470. A box contains a very large number of red, white, blue and yellow balls in the ratio 1:2:3:4. Find the
probability that in 10 drawing, 9 yellow balls and 1 red ball will be drawn
A. 0.00026
B. 0.00036
C. 0.00046
D. 0.00056
471. In how many ways can 8 persons be seated at a round table if a certain 2 persons are not to sit next
to each other?
A. 3,600
B. 4,600
C. 5,600
D. 6,600
472. How many sums of money each consisting 3 or more coins can be formed from 6 different kinds of
coins?
A. 40
B. 41
C. 42
D. 43
473. There are 5 different chemistry books, 4 different physics books and 2 different history books to be
placed on a shelf with the books of each subject kept together. Find the number of ways in which the
books can be placed.
A. 54,360
B. 64,350
C. 34,560
D. 45,630
474. Evaluate
A. ¾
B. ¼
C. 2/3
D. 1/3
475. Find the area bounded by the curve y = 2x – x2 and the x-axis.
A. 1/3
B. 2/3
C. 4/3
D. ¾
476. The integral of secn y tan y dy is
A. (secn+1 y)/(n+1)
B. (secn y)/n + C
C. tan y + C
D. (sec2n y)/(2n) + C
477. Use the Wallis’ formula to evalute
A. 8/693
B. 9/693
C. 10/693D. 11/693
476. If f(x) = x + 3 and g(x) = (x+1) 2, find
A. 2.4139
B. 2.4319
C. 2.3491
D. 2.1943
477. If the integral of
dx from x = 0 to x = y is equal to 14/3, find y.
A. 1
B. 2
C. 3
D. 4
478. Find the integral of 2dx / x3 from x = 0 tp x = infinity.
A. ½
B. 1/3
C. ¼
D. 1/5
479. The arc of the curve
from x=0 to x=1 is revolved about the x-axis. Find the area of the
surface generated.
A. 3.33
B. 4.33
C. 5.33
D. 6.33
480. If
, find k.
A. 0
B. 1
C. 2
D. 3
481. A 30-m long cable weighing 15N/m is to be wound about a windlass. Find the work done. A. 6750
joules
B. 7650 joules
C. 6507 joules
D. 5760 joules
482. The area bounded by 4x2 + 9y2 = 36 is revolved about the line y = 6 – x. Use Pappus’ theorem to find
the volume of the solid generated.
A. 501.4
B. 502.5
C. 503.6
D. 504.7
483. Evaluate
A. 0.271
B. 0.371
C. 0.471
D. 0.571
484. A particle moves along a straight line with velocity v given at time t by v = 12 t 2 m/s. Find the
distance traveled by the particle in the first 5 seconds.
A. 300 m
B. 400 m
C. 500 m
D. 600 m
485. The value of
is equal to
A. 0
B. 1
C. -1D. 2
486. If the area bounded by y = x2, x=k (k>0) and the x-axis is equal to 8/3, find k. A. -1
B. 1
C. 2
D. -3
487. Evaluate
.
A. [(4x2+1)3/2]/20 + C
B. [(4x2+1)3/2]/8 + C
C. [(4x2+1)5/2]/20 + C
D. [(4x2+1)5/2]/8 + C
488. The length of the arc of the curve y = ln sec x from x = 0 to x = pi/3 is
A. 1.4170
B. 1.3170
C. 1.2170
D. 1.1170
489. If
evaluate
A. 3
B. 7
C. 6
D. 5
and
,
490. Find the area bounded by y=x2-1 and y=3.
A. 31/3
B. 32/3
C. 35/3
D. 37/3
491.
Integrate
A.
B.
C.
D.
Find the moment of inertia with respect to the x-axis of
the area bounded by y2 = 4x, y = 4 and x = 0. A. 21.2
B. 31.2
C. 41.2
D. 51.2
492.
493. Find the y-coordinate (ŷ) of the centroid of the first-quadrant area under the curve y = ex between x
= 0 and x = 1.
A. 0.91
B. 0.93
C. 0.95
D. 0.97
494. Evaluate
A. 1.7726
B. 1.7627
C. 1.6772
D. 1.6727
495. Find the integral of
from x = 0 to x = 1.
A. pi/6
B. pi/7
C. pi/8
D. pi.9
496. Find the area bounded by y2 = 1 – x, y = x -2, y=1 and y=-1.
A. 7/3
B. 8/3
C. 10/3
D. 11/3
497. If the second-degree equation Ax2 + Bxy + Cy2 +Dx + Ey + F = 0 represents a real conic and B 2 –
4AC is positive, then it is
a. ellipse
b. circle
c. parabola
d. hyperbola
498. If the slopes of two lines are equal and their y-intercepts are different, then the lines are a.
intersecting
b. parallel
c. coincident
d. perpendicular
499. A line with inclination between 0° and 90° has
a. zero slope
b. no slope
c. positive slope
d. negative slope
500. The parabola x2 – 4x + 2y – 6 = 0 opens
a. downward
b. upward
c. to the right
d. to the left
501. The locus of a point on a circle which rolls without slipping on a straight line is called a.
strophoid
b. trochoid
c. astroid
d. cycloid
502. If b2 – 4ac < 0, then the graph of y = ax2 + bx + c
a. crosses the x-axis once
b. crosses the x-axis twice
c. does not cross the x-axis
d. touches the x-axis once
503. The point (4,y) where y < 0 lies in quadrant
a. I
b. II
c. III
d. IV
504. The slope of a vertical line is
a. zero
b. one
c. 90°
d. undefined
505. The graph of y2 – 1 = 0 is
a. a pair of parallel lines
b. a pair of intersecting lines
c. a parabola
d. a point
506. The curve y = x3 is symmetric with respect to
a. the z-axis
b. the y-axis
c. the origin
d. both axes
507. The polar equation of the line parallel to the polar axis and 4 units above it is
a. r = 4cscθ
b. r = 4secθ
c. r = 4sinθ
d. r = 4cosθ
508. The equation y2 + 12y + 36 = 0 represents
a. two parallel lines
b. two intersecting lines
c. a point
d. a straight line
509. If C = 0, then the graph of the line Ax + By + C = 0
a.
b.
c.
d.
is parallel to the x-axis
is parallel to the y-axis
crosses the positive x-axis
passes through the origin
510. If the inclination θ of a line is an obtuse angle, then the tangent of θ is
a. positive
b. negative
c. zero
d. infinity
511. Which of the following as no graph?
a. x2 + y2 – 9 = 0
b. x2 + y2 + 9 = 0
c. x2 – y2 – 9 = 0
d. x2 – y2 + 9 = 0
512. The ellipse is symmetric with respect to
a. the x-axis only
b. the y-axis only
c. the origin only
d. both axes and the origin
513. The circle x2 + y2 = 100 has a radius of
a. 25
b. 30
c. 10
d. 50
514. If the eccentricity of a conic is 3/5, then it is
a. an ellipse
b. a circle
c. a parabola
d. a hyperbola
515. The graph of the polar equation r(2 + 4sinθ) = 3 is
a. a circle
b. a hyperbola
c. a parabola
d. an ellipse
516. if a line slants downward to the right, then it has
a. negative slope
b. positive slope
c. no slope
d. zero slope
517. the equation of the directrix of the parabola x2 =16y is
a. x + 4 = 0
b. x – 4 = 0
c. y – 4 = 0
d. y + 4 = 0
518. the locus of a point such that its radius vector is proportional to its vectorial angle is called the
a. Conchoid of Nicomedes
b. Spiral of Archimedes
c. Cissoid of Diocles
d. Folium of Descartes
519. If A = 0 and B∙C ≠ 0, then the line Ax + By + C = 0 is
a. parallel to the x-axis
b. parallel to the y-axis
c. perpendicular to the x-axis
d. coincident with the y-axis
520. The graph of the equation 4y2 = 8 – x2 is
a. a circle
b. an ellipse
c. a parabola
d. a hyperbola
521. If the directed distance from a point to the line is negative, then which of the following is true?
a. The point and the origin are not on the side of the line.
b. The point and the origin are on the opposite sides of the line.
c. The point is below the line.
d. The point is above the line.
522. It is the locus of a point which moves in a plane so that the sum of its distance from two fixed
points is constant.
a. a circle
b. a parabola
c. an ellipse
d. a hyperbola
523. If M is a point that is 1/3 of the distance from point A to point B, then M divides the line
segment AB in the ratio
a.
b.
c.
d.
1:3
1:2
2:3
1:4
524. Which of the following is the polar equation of a limacon?
a. r = 1 + sinθ
b. r = 2(1 – sinθ)
c. r = 2 – sinθ
d. r = 2sinθ
525. A line will have a positive slope under which of the following conditions?
a. positive x-intercept and positive y-intercept
b. negative x-intercept and positive y-intercept
c. negative x-intercept and negative y-intercept
d. both b and c
526. If two lines with slopes m1 and m2 are perpendicular to each other, then which of the following
relations is true?
a. m1 = m2
b. m1m2 = -1
c. m1/m2 = -1
d. m1 – m2 = 1
527. The graph of y2 + 4x = 0 has symmetry with respect to the
a. x-axis only
b. y-axis only
c. origin only
d. all of a, b and c
528. If the eccentricity of a conic is greater than one, then it is a
a. an ellipse
b. a circle
c. a parabola
d. a hyperbola
529. The graph of Ax2 + Cy2 + Dx +Ey +F = 0 where A and C are not both zero is a parabola
if
a.
b.
c.
d.
AC = 0
AC > 0
AC < 0
AC ≠ 0
530. Which of the following curves is symmetric with respect to the x-axis?
a. y2 = 2x3
b. y = 2x3
c. xy = 2
d. y = 3x2
531. the graph of a limacon r = a + bcosθ has an inner loop if
a. a = b
b. 0 < a/b < 0
c. ab = 1
d. 0 < b/a < 1
532. Which of the following is the equation of a pair of parallel lines?
a. y2 – x2 = 0
b. x2 + y2 +7 = 0
c. y2 + 4y = 0
d. x2 – 6x + 9 = 0 533.
Which of the following is an equation of a pair of
semicubical parabola?
a.
b.
c.
d.
y = x3/2
y = x1/2
y = x4
y = 1/x
534.
The graph of 3x2 – y = y2 + 6x is
a. a parabola
b. an ellipse
c. a circle
d. a hyperbola
535.
The equation Ax2 + Cy2 + Dx + Ey +F = 0 is an ellipse if
a. both A and C are not zero, A = C and they have the same sign
b. neither A nor C is zero, A ≠ C and they have the same sign
c. both A and C are not zero, A = C and they have opposite signs
d. neither A nor C is zero, A ≠ C and they have opposite signs
536.
The distance between the foci of an ellipse 6x2 + 2y2 = 12
a. 4
b. 5
c. 6
d. 7
537.
The distance between the directrices of an ellipse in problem 40 is
a. 5
b. 6
c. 7
d. 8
538.
What is the polar equation of the line passing through (3, 0°) and perpendicular to the polar
axis?
a. r = 3cscθ
b. r = 3secθ
c. r = 3cosθ
d. r = 3sinθ
Find the equation of the radical axis of the following circles:
C1:
x2 + y2 – 5x +3y -2 = 0
539.
C2:
a.
b.
c.
d.
540.
Find
x2 + y2 + 4x – y – 7 = 0
9x + 4y – 5 = 0
9x – 4y + 5 = 0
9x – 4y – 5 = 0
9x + 4y + 5 = 0
the distance between the points A(-3,0) and B(-4,7).
a.
b.
c.
d.
541.
If the slope of the line determined by the points (x,5) and (1,8) is -3, find x.
a. 2
b. 1
c. 0
d. 3
542.
The focus of the parabola y2 = 4x is at
a. (4,0)
b. (0,4)
c. (1,0)
d. (0,1)
543.
The inclination of the line determined by the points (2,5) and (1,8) is
a. 106.41°
b. 107.42°
c. 108.43°
d. 109.44°
544.
The length of the latus rectum of 27x2 + 36y2 = 972 is
a. 8
b. 9
c. 10
d. 11
545.
The slope of the line through the points (-4,-5) and (2,7) is
a. 2
b. -2
c. 3
d. -3
The equivalent of x2 + y2 – y = 0 in polar form is
a. r = 2cosθ
b. r = 2sinθ
c. r2 = 2sinθ
d. r2 = 2cosθ
546.
547.
The area of the ellipse x2/64 + y2/16 = 1 is
a. 30π
b. 31π
c. 32π
d. 33π
548.
Find the equation of the ellipse which has the line 2x – 3y = 0 as one of its asymptotes. a.
2x2 – 3y2 = 6
b. 3y2 – 2y2 = 6
c. 4x2 – 9y2 = 36
d. 9y2 – 4x2 = 36
549.
The transverse axis of the hyperbola 36x2 – 25y2 = 900 is
a. 13
b. 12
c. 11
d. 10
550.
The parabola y = 3x2 – 6x + 5 has its vertex at
a. (0,5)
b. (1,2)
c. (-1,14)
d. (2,5)
551.
The line 4x – 6y + 14 = 0 is coincident with the line
a. 2x = 3y – 7
b. 2x = 3y + 7
c. 4x = 6y + 14
d. 4x = 14 – 6y
552.
Determine the axis of symmetry of the parabola (y + 5) 2 = 24x
a. y = 5
b. y = -5
c. x = 5
d. x = -5
Find the area of the triangle which the line 2x – 3y + 6 = 0 forms with the coordinate axes.
a. 2
b. 3
c. 4
d. 5
553.
554.
The directrix of the parabola is y = 5 and its focus is at (4,-3). What is the latus rectum?
a. 14
b. 15
c. 16
d. 17
555.
Find the equation of the circle containing the point (1,-4) and center at the origin. a.
16
b. x2 + y2 = 17
c. x2 + y2 = 18
d. x2 + y2 = 19
556.
Find the equation of the line containing the point (2,-3) and is parallel to the line 3x + y – 5 = 0.
a. 3x + y – 1 = 0
b. 3x + y – 4 = 0
c. 3x + y – 2 = 0
d. 3x + y – 3 = 0
557.
The distance from the point (2,1) to the line 4x – 3y + 5 = 0 is
a. -2
b. 2
c. -3
x2 + y2 =
d. 4
558.
If the slope of the line (k + 1)x + ky – 3 = 0 is -2, find k.
a. 2
b. 1
c. -3
d. -2
559.
Write the equation of the line with x-intercept -6 and y-intercept 3.
a. x + 2y – 6 = 0
b. x – 2y – 6 = 0
c. x – 2y + 6 = 0
d. x + 2y + 6 = 0
Write the equation of the tangent line to the circle x 2 + y2 = 80 at the point in the first quadrant
where x = 4.
560.
a.
b.
c.
d.
x – 2y – 20 = 0
x – 2y + 20 = 0
x + 2y – 20 = 0
x + 2y + 20 = 0
561.
If the distance between (8,7) and (3,y) is 13, what is the value of y?
a. -5 or 19
b. 5 or 19
c. 5 or -19
d. -5 or -19
562.
If the major axis of an ellipse is twice its minor axis, find its eccentricity.
a. 0.965
b. 0.866
c. 0.767
d. 0.668
563.
The center of the circle x2 + y2 – 18x +10y +25 = 0 is
a. (9,5)
b. (-9,5)
c. (-5,9)
d. (9,-5)
564.
Compute the area of the polygon with vertices at (6,1), (3,-10), (-3,-5) and (-2,0).
a. 60
b. 50
c. 40
d. 30
565.
A line with the equation y = mx + k passes through the points (-1/3,-6) and (2,1). Find m.
a. 2
b. 3
c. 4
d. 5
566.
Find the tangential distance from the point (8,5) to the circle (x – 2)2 + (y – 1)2 = 16.
a. 7
b. 8
c. 9
d. 6
Find the equation of the line through (-1,3) and is perpendicular to the line 5x – 2y + 3 =
0.
567.
a.
b.
c.
d.
568.
2x + 5y – 13 = 0
2x + 5y – 12 = 0
2x + 5y – 11 = 0
2x + 5y – 10 = 0
Find the distance between the two lines represented by the two linear equations 4x – 3y –
12 = 0 and 4x – 3y + 8 = 0.
8
6
5
4
The distance between the points (sinθ,cosθ) and (cosθ,-sinθ) is
a. 1
b. 2
a.
b.
c.
d.
569.
c.
d.
570.
Find the equation of the line parallel to 3x + 4y + 2 = 0 and -3 units from it.
a. 3x + 4y + 13 = 0
b. 3x + 4y – 13 = 0
c. 3x + 4y + 17 = 0
d. 3x + 4y – 17 = 0
571.
If the circle has its center (-3,1) and passes through (5,7), then its radius is
a. 7
b. 8
c. 9
d. 10
572.
Find the area of the triangle whose vertices lie at A, B and C whose coordinates are (4,1),
(6,2) and (2,-5), respectively.
a.
b.
c.
d.
4
5
6
7
573.
Express y3 = 4x2 in polar form
a. r = 4cot2θcscθ
b. r = 4cotθcsc2θ
c. r = 4cot2θcsc2θ
d. r = 4cotθcscθ
574.
If the slopes of the lines L1 and L2 are 3 and -1 respectively, find the angle between them
measured counterclockwise from L1 to L2.
a. 64.33°
b. 36.43°
c. 63.43°
d. 43.36°
575.
What is the length of the latus rectum of a hyperbola with foci at (-3,15) and (-3,-5) and a
transverse axis equal to 12?
a. 44/3
b. 54/3
c. 64/3
d. 74/3
576.
If the line through (-1,3) and (-3,-2) is perpendicular to the line through (-7,4) and (x,2), find x if
x is positive.
a. 3
b. 2
c. 4
d. 1
577.
Determine k so that the line y = kx – 3 will be parallel to the line 4x + 12y = 12.
a. 1/2
b. 1/3
c. -1/3
d. -3
578.
Find the equation of the parabola with focus at (0,8) and directrix y + 8 = 0.
a. x2 = -32y
b. x2 = 32y
c. y2 = -32x
d. y2 = 32x
579.
find the tangent of the angle from the line through (-2,-3) and (4,3) to the line through (1,6) and (3,-2)
a.
b.
c.
d.
580.
3
4
2
1
The second-degree equation
19x2 + 6xy 11y2 + 20x – 60y +80 = 0
represents a conic. To remove the xy-term, we rotate the coordinate axes through an angle of
a.
b.
c.
d.
16.40°
17.41°
18.43°
19.45°
581.
Find the value of k given that the slope of the line joining (3,1) and (5,k) is 2.
a. 2
b. 3
c. 4
d. 5
582.
If the focus of a parabola is at (-6,0) and its vertex is at (0,0), the equation of its directrix is
a. x + 6 = 0
b. x – 6 = 0
c. x + 3 = 0
d. x – 3 = 0
583.
For what value of k is the line 6y + (2k – 1 )x = 12 perpendicular to the line 3y – 2x = 6?
a. 5
b. 4
c. 3
d. 2
584.
The circumference of the circle x2 + y2 – 8x +2y + 8 = 0 is
a. 18.85
b. 17.85
c. 16.85
d. 15.85
585.
If the perpendicular distance from the line kx – 3y + 15 = 0 to the point (2,1) is -4, find k. a.
-4
b. -3
c. -2
d. -1
586.
The eccentricity of the hyperbola 16(y – 6)2 – 9(x – 7)2 = 144 is equal to
a. 4/3
b. 5/3
c. 7/3
d. 9/4
If the tangent of the angle from the line through (6,y) and (-4,2) to the line through (6,6) and
(3,0) is 8/9, find the value of y if y is positive.
a. 4
b. 5
c. 6
d. 7
587.
588.
Find the equation of the line which passes through the point (8,3) and forms with the
coordinate axes a triangle of area 54.
a. 4x + 3y – 41 = 0
b. 2x + 4y – 28 = 0
c. 5x + 2y – 46 = 0
d. 3x + 4y – 36 = 0
589.
If P0(x0,y0) is such that P1P0/P0P2 = 7/6 where P1(2,5) and P2(5 ,-1), find x0.
a.
b.
c.
d.
45/13
46/13
47/13
18/13
590.
Find the polar equation of the line perpendicular to θ = 20° and passing through the point
(6,20°).
a. r = 6sec(θ + 20°)
b. r = 6sec(θ – 20°)
c. r = -6sec(θ + 20°)
d. r = -6sec(θ – 20°)
591.
Determine b so that x2 + y2 + 2x – 3y – 5 = 0 and x2 + y2 + 4x + by + 2 = 0 are orthogonal.
a. 10/3
b. 11/3
c. 13/3
d. 14/3
592.
If the value of the invariant B2 – 4AC is negative, then the second-degree equation Ax2 +
Bxy +Cy2 + Dx + Ey + F = 0 represents either an ellipse or
a.
b.
c.
d.
593.
594.
595.
a pair of parallel lines
two intersecting lines
a point
a line
Find the distance between the points (4,40°) and (4,220°).
a. 7
b. 8
c. 10
d. 9
Identify the locus of the curve whose parametric equations are x = 3sinθ, y = 2cosθ. a. a circle
b. a parabola
c. an ellipse
d. a hyperbola
Find the equation of the line through the midpoint of AB where A(-3,1), B(2,-1) and is
perpendicular to AB.
a. 10x + 4y + 5 = 0
b. 10x + 4y – 5 = 0
c. 10x – 4y + 5 = 0
d. 10x – 4y – 5 = 0
596.
Find
+ 25
the length of the tangent line from the point P(4,-7) to the circle x2 + y2 – 10x – 4y
= 0.
a.
b.
c.
d.
597.
Find the equation of the circle with center at the midpoint of A(4,2), B(-1,-2) and having a
radius 3.
a. 4x2 + 4y2 + 12x + 27 = 0
b. 4x2 + 4y2 – 12x – 27 = 0
c. 4x2 + 4y2 + 12x – 27 = 0
d. 4x2 + 4y2 – 12x + 27 = 0
598.
Write the polar equation of the circle with center (-5,π) and radius 5.
a. r = 5cosθ
b. r = -5cosθ
c. r = 10cosθ
d. r = -10cosθ
599.
Give the Cartesian equation of the line whose parametric equations are x = 2t – 1, y = 3t + 5
where t is the parameter.
a. 3x – 2y + 13 = 0
b. 3x + 2y – 13 = 0
c. 3x – 2y – 13 = 0
d. 3x + 2y + 13 = 0
600.
Find the equation of the line through (6,-3) and parallel to the line through (2,8) and (5,1).
a. 3x + y + 15 = 0
b. 3x – y – 15 = 0
c. 3x – y – 15 = 0
d. 3x – y + 15 = 0
601.
The vertices of a triangle are A(4,6), B(2,-4) and C(-4,2). Find the length of the median of the
triangle from the vertex C to the side AB.
a.
b.
c.
d.
602.
Find the equation of the circle containing (1,-4) and center at the origin.
a. x2 + y2 = 14
b. x2 + y2 = 15
c. x2 + y2 = 16
d. x2 + y2 = 17
603.
If AB is perpendicular to CD and A(-1,0), B(2,5), C(3,-1), D(-3,a), find the value of a.
a. 13/4
b. 13/5
c. 13/6
d. 13/7
604.
Find the equation of the line through (4,0) and is parallel to the altitude from A to BC of the
triangle A(1,3), B(2,-6) and C(-3,0).
a. 5x + 6y + 20 = 0
b. 5x – 6y – 20 = 0
c. 5x + 6y – 20 = 0
d. 5x – 6y + 20 = 0
605.
Find the equation of the circle which has the line joining (4,7) and (2,-3) as diameter.
a. (x – 2)2 + (y – 3)2 = 26
b. (x – 2)2 + (y – 3)2 = 27
c. (x – 2)2 + (y – 3)2 = 28
d. (x – 2)2 + (y – 3)2 = 29
606.
Write the equation of the line with x-intercept -6 and y-intercept 3.
a. x – 2y – 6 = 0
b. x + 2y + 6 = 0
c. x – 2y + 6 = 0
d. x + 2y – 6 = 0
607.
Find the abscissa of the point P0 which divides P1P2 in the ratio P1P0/P0P2 = r1/r2 were P1(2,5),
P2(6,-3), r1 = 3, r2 = 4.
a. 25/7
b. 26/7
c. 27/7
d. 28/7
608.
Find the equation of the conic with eccentricity 7/4 and foci at (7,0) and (-7,0).
a. x2/33 + y2/16 = 1
b. x2/16 + y2/33 = 1
c. x2/33 – y2/16 = 1
d. x2/16 – y2/33 = 1
609.
Find the equation of the line passing trough (2,-3) and is parallel to the line 3x – y = 5. a. 3x + y –
2=0
b. 3x + y – 3 = 0
c. 3x + y – 4 = 0
d. 3x + y – 5 = 0
610.
If the slope of the line (k + 1)x + ky – 3 = 0 is arctan(-2), find the value of k.
a. 1
b. 2
c. 3
d. 4
611.
Find the equation of the line parallel to 5y – 5x + 12 = 0 and contains the point (0,-3). a. x – y +
3=0
b. x + y – 3 = 0
c. x – y – 3 = 0
d. x + y + 3 = 0
612.
Find k so that the circle x2 + y2 + 2kx + 4y – 5 = 0 will pass through the point (5,1).
a. -3/2
b. -5/2
c. -7/2
d. -9/2
613.
Find the equation of the line through the points (-7,-3) and (-1,9).
a. 2x – y + 11 = 0
b. 2x + y – 11 = 0
c. 2x + y + 11 = 0
d. 2x – y – 11 = 0
614.
The equation of the parabola with vertex (-1,2) and directrix at x = -3 is
a. (y – 2)2 = 8(x + 1)
b. (y + 2)2 = 8(x + 1)
c. (x + 1)2 = 8(y + 2)
d. (x – 1)2 = -8(y + 2)
615.
Find the length of the latus rectum of a parabola with focus at (-2,-6) and directrix x – 2 = 0.
a. 6
b. 4
c. 8
d. 10
616.
Write the equation of the line tangent to the circle x2 + y2 + 14x + 18 y – 39 = 0
at the point in the second quadrant where x = -2.
a.
b.
c.
d.
617.
5x + 12y + 26 = 0
5x – 12y – 26 = 0
5x + 12y – 26 = 0
5x – 12y + 26 = 0
The two points on the line 2x + 3y + 4 = 0 which are at a distance 2 from the line 3x + 4y
– 6 = 0 are
a.
b.
c.
d.
(7,-6) and (-11,6)
(-88,-8) and (-16,-16)
(64,-44) and (4,-4)
(-44,64) and (10,-10)
618.
Find the equation of the line which forms with the axes in the first quadrant a triangle of area 2
and whose intercepts differ by 3.
a. x + 4y – 4 = 0
b. x – 4y + 4 = 0
c. x + 4y + 4 = 0
d. x – 4y – 4 = 0
619.
What is the locus of a point which moves so that its distance from the line x = 8 is twice its
distance from the point (2,8)?
a. a circle
b. an ellipse
c. a parabola
d. a hyperbola
620.
Write the polar equation of a line which passes through the points (2,π/2) and (-1,0).
a. r(2cosθ + sinθ) – 2 = 0
b. r(2cosθ – sinθ) – 2 = 0
c. r(2cosθ + sinθ) + 2 = 0
d. r(2cosθ – sinθ) + 2 = 0
621.
The line segment with end points A(-1,-6) and B(3,0) is extended beyond point A to a point C so
that C is 4 times as far from B as from A. find the abscissa of point C.
a. -5/3
b. -7/3
c. -8/3
d. -4/3
622.
A semi-elliptic arch is 20-ft high at the center and as a span of 50-ft. find the height of the arch
at a point 10-ft from one end of the base.
a. 14 ft
b. 15 ft
c. 16 ft
d. 17 ft
623.
If the slope of a line 3x + y – 5 + k(x + 2y – 3) = 0 is 11/3, find k.
a. -4/5
b. -3/5
c. -2/5
d. -1/5
624.
The equation of the ellipse with vertices at (-3,-2) and (1,-2) and which passes through (2,-1) is
a.
b.
c.
d.
625.
Find the diameter of the ellipse 9x2 + 16y2 = 144 defined by the system of parallel chords of
slope 2.
a.
b.
c.
d.
626.
9x – 32y = 0
9x + 32y = 0
32x – 9y = 0
32x + 9y = 0
The locus of 4x2 + 4xy + y2 + 2x + y – 2 = 0 is a pair of parallel lines. What is the slope of each
line?
-1
-2
1
2
Find the area of a triangle with one vertex at the pole and the two others are (5,60°) and
a.
b.
c.
d.
627.
x2 + 3y2 + 2x + 12y + 9 = 0
3x2 + y2 + 2x + 12y – 9 = 0
x2 + 3y2 – 2x + 12y + 9 = 0
3x2 + y2 – 2x + 12y – 9 = 0
(4,-30°).
a.
b.
c.
d.
13
12
11
10
628.
Given A(3,7), B(-6,4), C(-2,8) and D(-7,0). Find the tangent of the angle measured
counterclockwise from AB to CD.
a. 17/23
b. 18/23
c. 19/23
d. 20/23
629.
Find the equation of the hyperbola with vertices at (4,0) and (-4,0) and asymptotes y = 2x and y
= -2x.
a. x2/64 – y2/16 = 1
b. x2/16 – y2/64 = 1
c. y2/64 – x2/16 = 1
d. y2/64 – x2/64 = 1
630.
The equation of the perpendicular bisector of the line segment joining the points (2,6) and
(-4,3) is
a.
b.
c.
d.
x + 2y – 8 = 0
4x + 2y – 5 = 0
x – 2y + 10 = 0
4x + 2y – 13 = 0
631.
Assume that power cables hang in a parabolic arc between two pole 100-ft apart. If the poles
are 40-ft high and if the lowest point on the suspended cable is 35-ft above the ground, find the
height of the cable at a point 20-ft from the pole.
a. 34.8 ft
b. 35.8 ft
c. 36.8 ft
d. 37.8 ft
632.
Transform the rectangular equation (x2 + y2)3 = 4x2y2 into polar coordinates.
a. r = 2sinθ
b. r = sin2θ
r = 2cosθ
r = cos2θ
What is the eccentricity of an equilateral hyperbola?
c.
d.
633.
a.
b.
c.
d.
634.
635.
1.5
2
Find the equation of the locus of a point which moves so that its distance from (4,0) is equal to
two thirds of its distance from the line x = 9.
a. 9x2 – 5y2 = 180
b. 5x2 – 9y2 = 180
c. 9x2 + 5y2 = 180
d. 5x2 + 9y2 = 180
Find the equation of the line through the point which divides A(-1,-1/2), B(6,3) in the ratio
AP/PB = 3/4 and through the point Q which is equidistant from C(1,-1), D(-3,1) and E(-1,3).
a. x – 8y – 6 = 0
b. x – 8y + 6 = 0
c. x + 8y + 6 = 0
d. x + 8y – 6 = 0
636.
Find the equation of the line tangent to the hyperbola 9x 2 – 2y2 = 18 at the point (-2,3).
a. 3x + y + 3 = 0
b. 3x – y + 3 = 0
c. 3x + y – 3 = 0
d. 3x – y – 3 = 0
637.
For the conic 2x2 – xy + x + y – 5 = 0, find the equation of the diameter defined by the cords of
slope ½.
a.
b.
c.
d.
638.
7x + 2y – 3 = 0
7x + 2y + 3 = 0
7x – 2y + 3 = 0
7x – 2y – 3 = 0
The equation of the hyperbola with foci at (0,9) and (0,-9) and conjugate axis 10 units is a.
x2/56 – y2/25 = 1
b. x2/25 – y2/56 = 1
c. y2/56 – x2/25 = 1
d.
y2/25 – x2/56 = 1
639.
An arch is in the form of a semi-ellipse with major axis as the span. If the span is 24.4 m and the
maximum eight is 9.2 m, find the height of the arch at a point 4.6 m from the semi-minor axis.
a. 6.9 m
b. 5.9 m
c. 8.9 m
d. 7.9 m
640.
If the area of the quadrilateral with vertices at (-5,-1), (x,2), (10,-4) and (-2,7) is 78.5, find x if x is
positive.
a. 5
b. 6
c. 7
d. 8
641.
Find the value of k so that the radius of the circle x2 + y2 – kx + 6y – 3 = 0 is equal to 4.
a. 3
b. 4
c. 5
d. 6
642.
A parabolic segment is 32 dm high and its base is 16 dm. What is the focal distance?
a. 0.5 dm
b. 0.4 dm
c. 0.6 dm
d. 0.3 dm
643.
Write the equation of the hyperbola conjugate to the hyperbola 4x 2 – 3y2 + 32x + 18y +
25 = 0.
a.
b.
c.
d.
644.
4x2 – 3y2 + 32x + 18y – 49 = 0
4x2 – 3y2 + 32x + 18y – 36 = 0
4x2 – 3y2 + 32x + 18y – 16 = 0
4x2 – 3y2 + 32x + 18y – 64 = 0
Find the abscissa of the point P on the line segment AP for A(-8,4) and B(-13,6) if AP:PB
= 3:2.
a.
b.
c.
d.
-10
-11
-9
-12
645.
Find the point on the parabola x2 = 16y at which there is a tangent with a slope ½.
a. (8,4)
b. (-8,4)
c. (4,1)
d. (-4,1)
646.
What is the equation of the line tangent to the hyperbola
if the slope
of the line is 2?
2x + y + 23 = 0
2x + y – 23 = 0
2x – y + 23 = 0
2x – y – 23 = 0
Find the eccentricity of an ellipse whose latus rectum is 2/3 of the major axis.
a. 0.58
b. 0.68
c. 0.78
d. 0.88
a.
b.
c.
d.
647.
648.
The vertices of a triangle are (2,4), (x,-6) and (-3,5). If x is negative and the area of the triangle is
28.5, find x.
a. -5
b. -6
c. -4
d. -7
649.
A parabolic arch has a span of 20 m and a maximum height of 15 m. how high is the arch
4 m from the center of the span?
a.
b.
c.
d.
650.
10.6 m
11.6 m
12.6 m
13.6 m
Determine the value of k so the following circles are orthogonal:
C1:
x2 + y2 + 2x – 3y – 5 = 0
C2:
x2 + y2 + 4x + ky + 2 = 0
a.
b.
c.
d.
11/2
13/3
14/3
10/3
651.
An ellipse has its foci at (0,c) and (0,-c) and its eccentricity is ½. Find the length of the latus
rectum.
a. 2c
b. 3c
c. 4c
d. 5c
652.
The earth’s orbit is an ellipse with eccentricity 1/60. If the semi-major axis of the orbit is 93M
miles and the sun is at one of the foci, what is the shortest distance between the earth and the
sun?
a. 89.43M mi
b. 90.44M mi
c. 91.45M mi
d. 92.46M mi
653.
If the length of the latus rectum of an ellipse is ¾ of the length of its minor axis, then its
eccentricity is
a. 0.46
b. 0.56
c. 0.66
d. 0.76
654.
If the point P(9,2) divides the line segment from A(6,8) to B(x,y) such that AP:AB =
3:10, find y.
a.
b.
c.
d.
655.
-11
-10
-9
-12
Find the rectangular equation for the curve whose parametric equations are x = 2cosθ, y =
cos2θ.
a. x2 = 2(y + 1)
b. x2 = 2(y – 1)
c. y2 = 2(x + 1)
d. y2 = 2(x – 1)
656.
A parabolic arch spans 200-ft wide. How high must the arch be above the stream to give a
minimum clearance of 40-ft over a tunnel in the center which is 120-ft wide?
a. 60.5 ft
b. 61.5 ft
c. 62.5 ft
d. 63.5 ft
657.
In the parabola x2 = 4y, an equilateral triangle is inscribed with one vertex at the origin. Find the
length of each side of the triangle.
a.
b.
c.
d.
658.
13.86
12.85
11.84
10.83
The foci of a hyperbola are (4,3) and (4,-9) and the length of the conjugate axis is
.
Find its eccentricity.
a.
b.
c.
d.
659.
1.3
1.5
1.7
1.9
Find the length of the common chord of the curves whose equations are x 2 + y2 = 48 and x2 + 8y
= 0.
a.
b.
c.
d.
660.
The point (8,5) bisects a chord of the circle whose equation is x 2 + y2 – 4x + 8y = 110. Find the
equation of the cord.
a.
b.
c.
3x + 2y = 0
3x – 2y = 14
2x + 3y = 31
d.
661.
Find the length of the latus rectum of the parabola with focus at (-2,-6) and directrix x – 2
= 0.
a.
b.
c.
d.
662.
2x – 3y = 1
8
7
6
4
Find the distance between (1,2,-5) and (-1,-1,4).
a.
b.
c.
d.
663.
What is the distance from the origin to the point (4,-3,2)? a.
b.
c.
d.
664.
Find the direction numbers of the line through (4,-1,-3) and (0,1,4).
a. 4,-2,-7
b. -4,2,-7
c. -4,-2,7
d. -4,2,-7
665.
The direction numbers of two lines are 2,-1,4 and -3,y,2 respectively. Find y if the lines are
perpendicular to each other.
a. -1
b. 3
c. -2
d. 2
666.
Transform p = 6θ to spherical coordinates.
a. r2 – z2 = 36θ2
b. r2 – z2 = 6θ
c. r2 + z2 = 36θ2
d. r2 + z2 = 6θ
667.
the surface described by the equation 4x2 + y2 + 26z = 100 is an
a. elliptic hyperboloid
b. elliptic paraboloid
c. ellipsoid
d. elliptic cone
668.
Find the Cartesian coordinates of the point having the cylindrical coordinates (3,π/2,5). a.
(5,0,3)
b. (3,0,5)
c. (0,5,3)
d. (0,3,5)
669.
Find the cylindrical coordinates of the point having the rectangular coordinates (4,4,-2).
a. (
b. (
c. (
d. (
670.
The distance of the point (-4,5,2) from the x-axis is
a.
b.
c.
d.
671.
The equivalent of (3,4,5) in the cylindrical coordinate system is
a. (5,31.53°,5)
b. (5,51.33°,5)
c. (5,53.13°,5)
d. (5,35.31°,5)
672.
If one end of a line is (-2,4,8) and its midpoint is (1,-2,5), find the x-coordinate of the other end.
a. 4
b. 3
c. 5
d. 6
673.
Find the value of k such that the plane x + ky – 2z – 9 = 0 shall pass through the point
(5,-4,-6).
a.
b.
c.
d.
2
1
3
4
674.
The locus of 9x2 – 4z2 – 36y = 0 is a/an
a. elliptic cone
b. hyperbolic paraboloid
c. parabolic cylinder
d. ellipsoid
675.
The trace of x2 + 4z2 – 8y = 0 on the xy-plane is
a. a hyperbola
b. an ellipse
c. a parabola
d. a point
676.
The locus of y2 + z2 – 4x = 0 has symmetry with respect to
a. xz-plane only
b. yz- and xy-planes
c. z-axis
d. xz- and xy-planes
If the plane curve b2x2 + a2y2 = a2b2 is revolved about the x-axis, the surface generated is a/an
677.
a.
b.
c.
d.
678.
ellipsoid of revolution
hyperbolic paraboloid
paraboloid of revolution
parabolic cylinder
The rectangular coordinates for the point whose cylindrical coordinates are (6,120°,-2) are
a. (3,3
,-2)
b. (2,3
,-3)
c. (-3,3
,-2)
d. (-2,3
,-3)
679.
Which of the following has a locus that is a hyperbolic paraboloid?
a. x2 + y2 – 2z = 0
b. x2 + 5z2 – 6y = 0
c. z2 – 2y2 + 4x = 0
d. 4x2 + y2 – 4z = 0
680.
Find the z-coordinate of the midpoint of the segment whose end points are (4,5,6) and (3,1,2).
a.
b.
c.
d.
681.
The traces of the surface
a.
b.
c.
d.
682.
3
4
5
6
on the coordinate planes are
circles
ellipses
parabolas
hyperbolas
Transform the equation θ = tanφ to cylindrical coordinates.
a. r = zθ
b. z = rθ
c. θ = rz
d. r = zφ
683.
Which of the following is a quadric cone?
a. x2 – y2 – 4z2 = 0
b. x2 – y2 – 4z = 0
c. x2 + y2 – 4z2 = 0
d. x2 + y2 – 4z = 0
684.
Transform z 2r = 1 to spherical coordinates.
a. pcosφ – 2sinφ = 1
b. p(sinφ – 2cosφ) = 1
c. cosφ – 2psinφ = 1
d. p(cosφ – 2sinφ) = 1
685.
If z = 0 in the equation 2y2 + 3z2 – x2 = 0, then the trace of the surface on the xy-plane is a a. pair
of parallel lines
b. pair of intersecting lines
c. line
d. point
686.
Find the cylindrical coordinates for the point (6,3,2).
a. (
b. (
c. (
d. (
687.
A line makes an angle of 45 degrees with the x-axis and an angle of 60 degrees with the y-axis.
What angle does it make with the z-axis?
a. 30°
b. 45°
c. 60°
d. 55°
688.
Two directions cosines of a line are 1/3 and -2/3. What is the third?
a. 2/3
b. 4/3
c. 5/3
d. 7/3
689.
A line makes equal angles with the coordinate axes. Find the angle.
a. 44.64°
b. 54.74°
c. 64.84°
d. 74.94°
690.
Find the distance of the point (6,2,3) from the x-axis.
a.
b.
c.
d.
691.
What is the locus of any equation of the form x2 + y2 = f(z)?
a. hyperboloid of revolution
b. ellipsoid of revolution
c. paraboloid of revolution
d. cylinder of revolution
692.
The radius of the sphere x2 + y2 + z2 – 6x + 4z – 3 = 0 is
a. 2
b. 3
c. 5
d. 4
693.
The direction numbers of two lines are 2,-1,k and -3,2,2 respectively. Find k if the lines are
perpendicular.
a. 4
b. 2
c. 5
d. 3
694.
Find the equation of the locus of a point which moves so that it is 4 units in front of the xzplane.
a. y +4 = 0
b. z – 4 = 0
c. x + 4 = 0
d. y – 4 = 0
695.
The equation x2 + z2 = 5y is a paraboloid of revolution that is symmetric with respect to a.
x-axis
b. y-axis
c. z-axis
d. origin
696.
The equation of the plane through the point (-1,2,4) and parallel to the plane 2x – 3y – 5z
+ 6 = 0.
a.
b.
c.
d.
697.
2x – 3y – 5z + 27 = 0
2x – 3y – 5z + 26 = 0
2x – 3y – 5z + 28 = 0
2x – 3y – 5z + 29 = 0
Find the distance of the point (6,2,3) from the z-axis.
a.
b.
c.
d. 7
698.
A line drawn from the origin to the point (-6,2,3). Find the angle which the line makes with the
z-axis.
a. 147°
b. 149°
c. 151°
d. 150°
699.
Find
b.
c.
the length of the line segment whose end points are (3,5,-4) and (-1,1,2). a.
d.
700.
Find the locus of a point whose distance from the point (-3,2,1) is 4.
a. x2 + y2 + z2 + 6x – 4y – 2z + 3 = 0
b. x2 + y2 + z2 + 6x – 4y – 2z – 4 = 0
c. x2 + y2 + z2 + 6x – 4y – 2z + 1 = 0
d. x2 + y2 + z2 + 6x – 4y – 2z – 2 = 0
701.
Find the center of the sphere x2 + y2 + z2 – 6x + 4y – 8z = 7.
a. C(3,2)
b. C(-3,2)
c. C(3,-2)
d. C(-3,-2)
Find the rectangular coordinates for the point (4,210°,30°).
702.
a. (
b. (
c. (
d. (
703.
The vertices of a triangle are A(2,-3,1), B(-6,5,3) and C(8,7,-7). Find the length of the median
drawn from A to BC.
a.
b.
c.
d.
704.
Find the angle between the line L1 with direction numbers 3,4,1 and the line L2 with direction
numbers 5,3,-6.
a. 55.41°
b. 60.51°
c. 65.61°
d. 70.71°
705.
Find spherical coordinates for the point (-2,2,-1).
a. (3,315°,109.5°)
b. (3,240°,107.5°)
c. (3,300°,110°)
d. (3,215°,100°)
706.
Find
the distance from the plane 2x + 7y + 4z – 3 = 0 to the point (2,3,3). a.
b.
c.
d.
707.
Transform psinφsinθtanθ = 5 to rectangular coordinates.
a. x2 = 5y
b. y2 = 5x2
c. y2 = 5x
d. y = 5x2
708.
Two direction angles of a line are 45 degrees and 60 degrees. Find the third direction angle.
a. 30°
b. 35°
c. 40°
d. 45°
709.
Find m so that the plane 5x – 6y – 7z = 0 and the plane 3x + 2y – mz + 1 = 0 are parallel. a.
-5/3
b. -7/3
c. -4/3
d. -2/3
710.
Transform y2 = 4ax to cylindrical coordinates.
a. rcosθtanθ = 4a
b. rcosθcotθ = 4a
c. rsinθtanθ = 4a
d. rsinθcotθ = 4a
711.
The triangle with vertices (3,5,-4),(-1,1,2) and (-5,-5,-2) is
a. equilateral
b. isosceles
c. right
d. equiangular
712.
The sphere x2 + y2 + z2 – 2x + 6y +2z – 14 = 0 has a radius
a. 2
b. 4
c. 5
d. 3
713.
Find the x-coordinate of a point which is 10 units from the origin and has direction cosines cosβ
= 1/3 and cosγ = -2/3.
a. 19/3
b. 20/3
c. 17/3
d. 22/3
714.
Give the equivalent spherical coordinates of (3,4,6).
a.
(
b.
c.
d. (
715.
If the line L1 has direction numbers x,-2x3 and line L2 has direction numbers -2,x,4 and if L1 is
perpendicular to L2, find x.
a. 5
b. 4
c. 3
d. 2
716.
Find the cosine of the angle between the line directed from (3,2,5) to (8,6,2) and the line
directed from (-4,5,3) to (-3,4,3).
a. 1/12
b. 1/10
c. 1/11
d. 1/13
717.
Find the angle between the planes 3x – y + z – 5 = 0 and x + 2y + 2z + 2 = 0.
a. 69.42°
b. 70.43°
c. 71.44°
d. 72.45°
718.
Find the coordinates of the point P(x,y,z) which divides the line segment P1P2 where P1(2,5,-3)
and P2(-4,0,1) in the ratio 2:3.
a. (2/5,-3,-7/5)
b. (-2/5,3,7/5)
c. (-2/5,3,-7/5)
d. (-2/5,-3,-7/5)
719.
Find the Cartesian coordinates of the point having the spherical coordinates (4,
.
a. (
b. (
c. (
d. (
720.
Find the equations of the line through (2,-1,3) and parallel to the x-axis.
a. y + 1 = 0, z – 3 = 0
b. y – 1 = 0, z + 3 = 0
c. y – 1 = 0, z – 3 = 0
d. y + 1 = 0, z + 3 = 0
721.
Give the polar coordinates for the point (1,-2,2).
a. (3,48.2°,131.8°,70.5°)
b. (3,70.5°,131.8°,48.2°)
c. (3,48.2°,70.5°,131.8°)
d. (3,131.8°,70.5°,48.2°)
722.
Transform the equation cosγ = p(cos2α – cos2β) to rectangular coordinates.
a. y = x2 – z2
b. x = y2 – z2
c. z = x2 – y2
d. z = x2 + y2
723.
A point P(x,y,z) moves so that its distance from the z-axis is 4 times its distance from the x-axis.
Find the equation of the locus.
a.
b.
c.
d.
15y2 + 16z2 – x2 = 0
15y2 – 16z2 + x2 = 0
15y2 – 16z2 – x2 = 0
15y2 + 16z2 + x2 = 0
724.
Write the equation in rectangular coordinates of p = 5acosφ.
a. x2 – y2 + z2 = 5az
b. x2 + y2 – z2 = 5az
c. x2 – y2 – z2 = 5az
d. x2 + y2 + z2 = 5az
725.
The rectangular coordinates for the point (2,90°,30°,60°) is
a. (0,
b. (0,
c. (1,
d. (1,
727.
Find the equations of the line through (1,-1,6) with direction numbers 2,-1,1.
a. x = 2z + 11, y = z – 5
b. x = 2z – 11, y = z + 5
c. x = 2z – 11, y = 5 – z
d. x = 2z + 11, y = 5 – z
If the angle between two lines with direction numbers 1,4,-8 and x,3x-6 respectively is
arccos(62/63),find x.
a. 4
b. 5
c. 2
d. 3
728.
Find the polar coordinates of the point (0,-2,-2)
726.
a. (2
b. (2
c. (
d. (
729.
Find the point where the line through the points (3,-1,0) and (1,3,4) pierces the xz-plane. a.
(1,0,1)
b. (1.5,0,1)
c. (2,0,1)
d. (2.5,0,1)
730.
Find the equation of the plane such that the foot of the perpendicular from the origin to the
plane is (-6,3,6).
a. 2x + y + 2z – 27 = 0
b. 2x – y – 2z + 27 = 0
c. 2x – y + 2z + 27 = 0
d. 2x + y – 2z – 27 = 0
731.
Find angle A of the triangle whose vertices are A(4,6,1), B(6,4,0) and C(-2,3,3).
a. 112.39°
b. 111.38°
c. 110.37°
d. 109.36°
732.
Find the equation of the plane that passes through (3,-2,1), (2,4,-2) and (-1,3,2).
a. 21x + 13y + 19z – 56 = 0
b. 21x + 13y – 19z – 56 = 0
c. 21x + 13y + 19z + 56 = 0
d. 21x – 13y – 19z – 56 = 0
733.
Find the acute angle between the lines x + y + z + 1 = 0, x – y + z + 1 = 0 and x – y – z –
1 = 0, x + y = 0.
a.
b.
c.
d.
71.20°
72.21°
73.22°
74.23°
734.
Find the equation of the plane through the point (-1,2,3) and perpendicular to the line for
which cosα = 2/3, cosβ = -1/3, cosγ = 2/3.
a. 2x – y + 2z – 2 = 0
b. 2x – y – 2z + 2 = 0
c. 2x + y – 2z – 2 = 0
d. 2x + y + 2z – 2 = 0
735.
Find the area of the triangle with vertices (1,3,3), (0,1,0) and (4,-1,0).
a.
b.
c.
d.
736.
If the acute angle between the planes 2x – y + z – 7 = 0 and x + y + kz – 11 = 0 is 60°, find k.
a. 4
b. 3
c. 1
d. 2
737.
Transform the cylindrical coordinates (8,120°,6) to spherical coordinates.
a. (10,120°,53.13°)
b. (11,120°,53.13°)
c. (12,120°,53.31°)
d. (10,120°,51.33°)
738.
Find the locus of the point equidistant from the plane y = 7 and the point (0,5,0).
a. x2 – z2 + 4y – 24 = 0
b. x2 – z2 – 4y + 24 = 0
c. x2 + z2 + 4y – 24 = 0
d. x2 + z2 – 4y + 24 = 0
739.
Find the direction numbers of the line 2x – y + 3z + 4 = 0, 3x + 2y – z + 7 = 0.
a. 5,-11,7
b. -5,11,7
c. -5,7,11
d. 5,-7,11
Find the equation of the plane perpendicular to the line joining (2,5,-3) and (4,-1,0) and which
passes through the point (1,4,-7).
a. 2x – 6y – 3z + 43 = 0
b. 2x + 6y – 3z + 43 = 0
c. 2x – 6y + 3z + 43 = 0
d. 2x + 6y + 3z + 43 = 0
740.
741.
Find the equation of the line which passes through (-1,-3,6) and which is parallel to the plane 4x
– 9y + 7z + 2 = 0.
a. 4x – 9y + 7z – 65 = 0
b. 4x + 9y + 7z – 65 = 0
c. 4x – 9y – 7z + 65 = 0
d. 4x + 9y – 7z + 65 = 0
742.
Find the value of m so that the line passing through (-m,-1,2) and (0,2,4) be perpendicular to
the line through (1,m,1) and (m+1,0,2).
a. 1 or 5
b. 1 or 4
c. 1 or 3
d. 1 or 2
743.
Find the acute angle between the line
and the line
. a.
b.
c.
d.
744.
If the angle between the planes 2x – 3y + 6z = 18 and 2x – y + kz = 12 is arccos(19/21), find k.
a. 4
b. 3
c. 2
d. 1
745.
A plane contains the point P1(4,-4,2) and is perpendicular to the line segment from P1 to
P2(0,6,6). Find the equation of the plane.
a. 2x + 5y + 2z – 24 = 0
b. 2x + 5y – 2z + 24 = 0
c. 2x – 5y + 2z + 24 = 0
d. 2x – 5y – 2z – 24 = 0
746.
A line whose parametric equations are
perpendicular to the plane 2x + ky + 12z = 3. Find the value of k.
is
a.
b.
c.
d.
-3
-4
-5
-6
747.
Write the equations of the line through (-2, 2, -3) and (2, -2, 3).
a. x – y = 0, 3y + 2z = 0
b. x + y = 0, 3y + 2z = 0
c. x – y = 0, 3y – 2z = 0
d. x + y = 0, 3y – 2z = 0
748.
Find the equation of the paraboloid with vertex at (0, 0, 0), axis along the y-axis and passing
through (1, 1, 1) and (3/2, 7/12, 1/2).
a. x2 + 5z2 = 6y
b. x2 + 6z2 = 5y
c. 5x2 + z2 = 6y
d. 6z2 + z2 = 5y
749.
Find the equation of the plane determined by the points (6,-4,1), (0,1,-3) and (2,2,-7). a. x + 2y –
z+1=0
b. x – 2y + z – 1 = 0
c. x + 2y + z + 1 = 0
d. x – 2y – z – 1 = 0
750.
What is the locus of the moving point, the difference of whose distance from (0,0,3) and (0,0,3) is 4?
a.
b.
c.
d.
751.
Find the piercing point in the xy-plane of the line x + y – z – 3 = 0, x + 2y + z – 4 = 0. a.
b. (1,0,2)
c. (2,0,1)
d. (2,1,0)
752.
Find the acute angle between the line
(1,2,0)
and the plane 2x – 2y + z – 3 =
0.
a.
b.
c.
d.
25.3°
26.4°
27.5°
28.6°
753.
Find the equation of the plane through (1,-2,3) and perpendicular to the line of intersections of
the plane 3x + 2y – 2z = 12 and x + 2y + 2z = 0.
a. 2x – 2y – z – 9 = 0
b. 2x – 2y + z – 9 = 0
c. 2x + 2y – z + 9 = 0
d. 2x + 2y + z + 9 = 0
754.
A plane contains the points (3,1,7) and (-3,-2,3) and as an x-intercept equal to three times its zintercepts. Find the equation of the plane.
a. x + 6y – 3z + 18 = 0
b. x – 6y – 3z + 18 = 0
c. x – 6y + 3z – 18 = 0
d. x + 6y – 3z – 18 = 0
755.
Find the acute angle between the lines through the points (-2,3,1) and (4,6,7) and the plane x +
4y + z – 10 = 0.
a. 35.64°
b. 36.74°
c. 37.84°
d. 38.94°
756.
Find the equation of the plane which contains the line x – 2y + z = 1, 2x = y – z and is
perpendicular to the plane 3x + 2y – 3z = 0.
a. 9x – 6y + 5z – 1 = 0
b. 9x + 6y – 5z + 1 = 0
c. 9x + 6y – 5z – 1 = 0
d. 9x + 6y + 5z + 1 = 0
757.
Find the equation of the plane which is perpendicular to the xy-plane and which passes through
(2,-1,0) and (3,0,5).
a. x + y + 3 = 0
b. x – y – 3 = 0
c. x + y – 3 = 0
d. x – y + 3 = 0
758.
Find the acute angle between the lines
and 2x + 2y + z – 4 = 0, x – 3y +
2z = 0.
a.
b.
c.
d.
759.
Find the equations of the line through (2,-3,4) and perpendicular to the plane 3x – y + 2z
= 4.
a.
b.
c.
d.
760.
46°24’
47°25’
48°26’
49°27’
x = 3y – 7, z = 2y – 2
x = 3y + 7, z = 2y + 2
x = -3y – 7, z = -2y – 2
x = -3y + 7, z = -2y + 2
Find the point of intersection of the plane 3x + 2y + z = 1 and the line
(1,0,1)
b. (1,1,0)
c. (-1,1,0)
d. (1,-1,0)
761.
Transform 3x2 – 3y2 = 8z to spherical coordinates.
a. 2psin2φcos2θ = 8pcosφ
b. 2psin2φcos2θ = 8pcosφ
c. 2p2sin2φcos2θ = 8pcosφ
d. 2p2sin2φcos2θ = 8pcosφ
. a.
762.
Find the equation of the sphere whose center is (2,1,-1) and which is tangent to the plane x – 2y
+ z + 7 = 0.
a. x2 + y2 – 4z – 2y + 2z = 0
b. x2 + y2 – 4z + 2y + 2z = 0
c. x2 + y2 + 4z – 2y – 2z = 0
d. x2 + y2 + 4z + 2y – 2z = 0
763.
If the line
a.
b.
c.
d.
is parallel to the plane 6x + ky – 5z – 8 = 0, find the value of k.
2
3
-2
-3
764.
Find the equation of the plane that is perpendicular to the yz-plane and having 5 and -2 as its yand z-intercepts respectively.
a. 2y + 5z – 10 = 0
b. 2y – 5z – 10 = 0
c. 2y + 5z + 10 = 0
d. 2y – 5z + 10 = 0
765.
Find the angle between the line with direction numbers 1,-1,-1 and the plane 3x – 4y + 2z
– 5 = 0.
a.
b.
c.
d.
32.42°
34.22°
42.32°
43.22°
766.
Find the equation of the locus of a point whose distance from the xy-plane is equal to its
distance from (-1,2,-3).
a. x2 + y2 – 2x + 4y – 6z – 14 = 0
b. x2 + y2 – 2x – 4y + 6z + 14 = 0
c. x2 + y2 + 2x + 4y – 6z – 14 = 0
d. x2 + y2 + 2x – 4y + 6z + 14 = 0
767.
Given the points A(k,1,-1), B(2k,0,2) and C(2+2k,k,1). Find k so that the line segment AB shall be
perpendicular to the line segment BC.
a. 3
b. 1
c. 2
d. 4
768.
The angle between two lines with direction numbers 4,3,5 and x,-1,2 respectively is 45 degrees.
Find x.
a. 4
b. 5
c. 2
d. 3
769.
At the minimum point, the slope of the tangent line to a curve is
a. positive
b. negative
c. zero
d. infinity
770.
A curve y = f(x) is concave downward if the value of y’’ is
a. negative
b. positive
c. unity
d. zero
771.
The point where the concavity of a curve changes is called the
a. maximum point
b. minimum point
c. inflection point
d. tangent point
772.
If the 1st derivative of a function is a constant, then its graph is
a. a point
b. a line
c. a parabola
d. a circle
773.
At the minimum point of y = f(x), the value of d2y/dx2 is
a. zero
b. undefined
c. positive
d. negative
774.
If at x = a, f’’(a) is positive, then f’(x) increases as x
a. increases
b. decreases
a.
b.
c.
d.
c. becomes infinite
d. becomes zero
775.
776.
If the first derivative of a function is a constant, then the function is sinusoidal
exponential linear quadratic
A function f(x) is said to be an even function if its graph is symmetric with respect to a.
the x-axis
b. the y-axis
c. the origin
d. both axes
777.
Which of the following is an odd function?
a. f(x) = xcosx
b. f(x) = xsinx
c. f(x) = ecosx
d. f(x) = sin2x
778.
The notation f’(x) was invented by
a. Leibniz
b. Newton
c. Wallis
d. Lagrange
779.
At the inflection point of y = f(x) where x = a,
a. f”(a) < 0
b. f”(a) = 0
c. f”(a) > 0
d. f”(a) = ∞
780.
If a function f(x) is concave downward on the interval (1,10), then f(8) and f(3)
a. may be true
b. cannot be true
c. must be true
d. is never true
a.
b.
c.
d.
781.
If a tangent to a curve y = f(x) is horizontal at x = a, then f’(a) is
a. positive
b. negative
c. zero
d. infinity
782.
For a function y = f(x), if f”(x) = -f(x), then the function is logarithmic exponential
transcendental sinusoidal
Which of the following notations is an open interval?
a. (-3,4)
b. [-3,4]
c. [-3,∞)
d. (-∞,4)
783.
784.
The graph of y = x5 – x will cross the x-axis
a. twice
b. 3 times
c. 4 times
d. 5 times
785.
The derivative of an increasing function f(x) must be
a. strictly positive
b. always positive
c. nonnegative
d. negative
786.
If the function f(x) increases at x = a, then which of the following is definitely true?
a. f'(a) = 0 or f’(a) > 0
b. f’(a) = 0 or f’(a) < 0
c. f’(a) ≠ 0 or f’(a) > 0
d. f’(a) ≠ 0 or f’(a) < 0
787.
At the maximum point, the value of the 2nd derivative of a function is
a. positive
a.
b.
c.
d.
b. negative
c. zero
d. infinite
788.
At the inflection point, the value of y” is
a. zero
b. positive
c. negative
d. unity
Which of the following functions will have an inflection point? y = x 4 y = x3 y = x2 y = x
The function y = f(x) has a maximum value of x = 2 if f’(2) = 0 and f”(2) is
a. equal to zero
b. less than zero
c. greater than zero
d. unity
789.
790.
791.
At the maximum point, the tangent line is
a. slanting upward
b. oblique
c. horizontal
d. vertical
792.
Which of the following is true?
a. ∞ – ∞ = 0
b. ∞ + ∞ = ∞
c. ∞/∞ = ∞
d. both a and b
793.
Which of the following functions is neither even nor odd?
a. h(x) = x2
b. g(x) = x3
c. f(x) = x2 + x
d. t(x) = x3 + x
794.
Find the rate of change of the volume of a cube with respect to its side when the side is 6 cm.
a. 108 cm3/cm
b. 107 cm3/cm
a.
b.
c.
d.
c. 106 cm3/cm
d. 105 cm3/cm
795.
If f(x) = e –x+1, then f’(1) is equal to
a. 0
b. 1
c. -1
d. ∞
796.
If f(x) = Aekx, f(0) = 5 and f(3) = 10, find k.
a. 0.1184
b. 0.1285
c. 0.1386
d. 0.1487
797.
The function
a.
b.
c.
d.
is discontinuous at x =
1 or -3
1 or -2
-1 or 2
-1 or 3
798.
Find the slope of the line tangent to y = 4/x at x = 2.
a. 1
b. -1
c. 2
d. -2
799.
If y = cos24x, find dy/dx.
a. 2cos4x
b. 2sin4x
c. -4sin8x
d. -8sin4x
800.
Evaluate the limit of ln(1 – x)/x as x approaches zero.
a. 0
b. -1
c. 1
d. ∞
a.
b.
c.
d.
801.
Evaluate
a.
b.
c.
d.
802.
.
∞
0
½
2
The rate of change of the area of a circle with respect to its radius when the diameter is
6cm is
4π cm2/cm
5π cm2/cm
6π cm2/cm
7π cm2/cm
At what point of the curve y = x3 + 3x are the values of y’ and y” equal?
a. (0,0)
b. (-1,-4)
c. (2,14)
d. (1,4)
a.
b.
c.
d.
803.
804.
If f(x) = ln x and g(x) = log x and if g(x) = kf(x), find k.
a. 0.4433
b. 0.3434
c. 0.3344
d. 0.4343
805.
If N(x) = sin x – sin θ and D(x) = x – θ, find the limit of N(x)/D(x) as x approaches θ.
a. sinθ
b. cosθ
c. zero
d. no limit
806.
Given z2 + x2 + y2 = 0, find
a.
b.
c.
d.
807.
x/z
–x/z
z/x
–z/x
What is the 50th derivative of y = cosx
a. sinx
b. –sinx
a.
b.
c.
d.
c. cosx
d. –cosx
808.
Which of the following has no horizontal asymptote?
a.
b.
c.
d.
809.
If f(x) =
a.
b.
c.
d.
810.
811.
if f(x) = x – 2 and g(x) = x2 – 1.
∞
0
½
¼
Evaluate
a.
b.
c.
d.
812.
.
infinity
unity
zero
undefined
Evaluate
a.
b.
c.
d.
, find
0
∞
1
e
If z = xy2 + yx3, find zxyx.
a. 6yx
b. 6x
a.
b.
c.
d.
.
c. 3xy
d. 3x2
813.
If y = x2, find ∆y – dy when x = 2 and dx = 0.01.
a. 0.0001
b. 0.001
c. 0.0002
d. 0.002
814.
If f(x) = x3 + 2x, find f”(2).
a. 10
b. 11
c. 12
d. 13
815.
The motion of a particle along the x-axis is given by the equation x = 2t3 – 3t2. Find the velocity of the particle
when t = 2.
a.
b.
c.
d.
10
9
11
12
816.
Find x for which the line tangent to the parabola y = 4x – x2 is horizontal.
a. 4
b. -4
c. 2
d. -2
817.
The slope of the tangent to y = 2 – x2 at the point (1,1) is
a. -2
b. -1
c. 0
d. -4
818.
If y = sin2x, the derivative dy/dx is equal to
a. cos2x
b. sin2x
c. 2cosx
d. 2sinx
819.
If y = x3 – 2x2 + 3x – 1, then d2y/dx2 is equal to
a. 6x
b. 6x + 4
c. 6x – 4
a.
b.
c.
d.
d. 3x – 4
820.
If y = x2 – 2x and x changes from 2 to 2.01, find ∆y.
a. 0.0102
b. 0.0210
c. 0.0120
d. 0.0201
821.
The radius R of a circle is increasing at the rate of 1cm per sec. how fast is the area changing when R = 4cm?
a. 8π cm2/s
b. 10π cm2/s
c. 6π cm2/s
d. 12π cm2/s
822.
Find the slope of y = 1 – x3 at the point where y = 9.
a. -11
b. -12
c. -10
d. -13
823.
If an error of 1 percent is made in measuring the edge of a cube, what is the percentage error in the computed
volume?
a. 3%
b. 2%
c. 4%
d. 5%
824.
Find the derivative of y with respect to x of y = xlnx – x.
a. 1
b. x
c. lnx
d. lnx – 1
825.
For what value of x will the curve y = x3 – 3x2 + 4 be concave upward?
a. 1
b. 2
c. 3
d. 4
826.
How fast does the diagonal of a cube increase if each edge of the cube increases at a constant rate of 5cm/s?
a.
b.
c.
d.
a.
b.
c.
d.
827.
6.7 cm/s
7.7 cm/s
8.7 cm/s
9.7 cm/s
If f(x) = tanx – x and g(x) = x3, evaluate the limit of f(x)/g(x) as x approaches zero.
a.
b.
c.
d.
0
∞
3
1/3
828.
Find the 3rd derivative of y = xlnx.
a. -1/x
b. -1/x2
c. -1/x3
d. -1
829.
Evaluate
a.
b.
c.
d.
.
∞
1
e
1/e
830.
If xy3 + x3y = 2, find dy/dx at the point (1,1).
a. 1
b. -1
c. 2
d. -2
831.
The tangent line to the curve y = x3 at the point (1,1) will intersect the x-axis at x =
a. 2/3
b. 4/3
c. 1/3
d. 5/3
832.
If y = ex + xe + xx, find y’ at x = 1.
a. e +1
b. e – 1
c. 2e + 1
d. 2e – 1
a.
b.
c.
d.
833.
Evaluate
a.
b.
c.
d.
.
0
∞
½
1
834.
Find the value of x for which y = x3 – 3x2 has a minimum value.
a. 1
b. 2
c. 0
d. -2
835.
Find
the angle of intersection between the curve y = x2 and x = y2. a.
b.
c.
d.
836.
If z = xy2, and x changes from 1 to 1.0, and y changes from 2 to 1.98, find the approximate change in z.
a.
b.
c.
d.
837.
-0.0202
-0.0303
-0.0404
-0.0505
A ball is thrown vertically upward from a roof 112-ft above the ground. The height s of the ball above the roof is
given by the equation s = 96t -16t2
where s is measured in ft and the time t in sec. calculate its velocity wen it strikes the ground.
a.
b.
c.
d.
-130 fps
-128 fps
-126 fps
-124 fps
838.
If y = ln(tanhx), find dy/dx.
a. 2sech2x
b. 2sech2x
c. 2csch2x
d. 2coth2x
839.
Find the approximate surface area of a sphere of radius 5.02 cm.
a.
b.
c.
d.
a.
b.
c.
d.
317 sq. cm
315 sq. cm
313 sq. cm
311 sq. cm
840.
Find the value of x for which y = x5 – 5x3 – 20x – 2 will have a maximum point.
a. -1
b. -2
c. 1
d. 2
841.
A man is walking at a rate of 1.5 m/s toward a street light which is 5 m above the level ground. At what rate is
the tip of his shadow moving if the man is 2 m tall?
a. -1.5 m/s
b. -2.5 m/s
c. -3.5 m/s
d. -5 m/s
842.
If y = ln(x2ex), find y”.
a. -1/x2
b. -2/x2
c. -1/x
d. -2/x
843.
Find the radius of curvature of y = x3 at the point (1,1).
a. 3.25
b. 4.26
c. 5.27
d. 6.25
844.
A particle moves along the circumference of a circle of radius 10-ft in such a manner that its distance measured
along the circumference from a fixed point at the end of t sec is given by the equation s = t 2. Find the angular
velocity at the end of 3 seconds.
a. 0.40 rad/s
b. 0.50 rad/s
c. 0.60 rad/s
d. 0.70 rad/s
845.
Find the point on the curve y = x3 – 3xfor which the tangent line is parallel to the x-axis. a.
b. (2,2)
c. (1,2)
d. (0,0)
846.
If y = 1/2tan2x + ln(cosx), find y’.
a.
b.
c.
d.
(-1,2)
a.
b.
c.
d.
tan3x
tanx – sinx
tanxsec2x
0
847.
If S = 4πR2, find ∆S – dS when R = 2 and ∆R = 0.01.
a. 0.0021
b. 0.0102
c. 0.0210
d. 0.0012
848.
Find two numbers whose sum is 8 if the product of one number and the cube of the other is a maximum.
a. 3 and 5
b. 4 and 4
c. 2 and 6
d. 1 and 7
849.
Find the approximate height of the curve y = x3 – 2x2 + 7 at the point where x = 2.98.
a. 14.8
b. 15.7
c. 16.6
d. 17.5
850.
If y =
, find x for which dy/dx = 0.
a.
b.
c.
d.
851.
Te volume of a cube is increasing at the rate of 6 cm 3/min. How fast is the surface increasing when the length of
each edge is 12 cm?
a.
b.
c.
d.
a.
b.
c.
d.
3 cm2/min
4 cm2/min
2 cm2/min
5 cm2/min
852.
If u =
, find the approximate change in u as x changes from 10 to 10.02 and y changes from 4 to
4.01.
a.
b.
c.
d.
-0.00170
-0.00701
-0.00107
-0.00017
y = 3x – 14
853.
Find the equation of the line tangent to y = x2 – 3x – 5 and parallel to the line y = 3x – 2. a.
b. y = 3x – 13
c. y = 3x – 12
d. y = 3x – 11
854.
A garden is in the form of an ellipse with semi-major axis 4 and semi-minor axis 3. If the axes are increased by
0.18 unit each, find the approximate increase in the area.
a. 3.92
b. 3.94
c. 3.96
d. 3.98
Find the relative error in the computed area of an equilateral triangle due to an error of 3 percent in measuring
the edge of the triangle.
a. 0.05
b. 0.06
c. 0.07
d. 0.08
855.
856.
A body is thrown vertically upward from the ground. After 2 seconds, its velocity is 10 ft/sec. Find its initial
velocity.
a. 54 fps
b. 64 fps
c. 74 fps
d. 84 fps
857.
In problem 345, find the rate at which the length of the shadow of the man is shortening. a.
b. -1.5 cm/s
c. -2 cm/s
d. -2.5 cm/s
858.
A rectangular field is fenced off, an existing wall being used as one side. If the area of the field is 7,200 sq. ft,
find the least amount of fencing needed.
a. 250 ft
b. 240 ft
c. 230 ft
a.
b.
c.
d.
-1 cm/s
d. 220 ft
859.
The side of an equilateral triangle is increasing at the rate of 0.50 cm/s. Find the rate at which its altitude is
increasing.
a. 0.334 cm/s
b. 0.443 cm/s
c. 0.433 cm/s
d. 0.343 cm/s
860.
Find C co that the line y = 4x + 3 is tangent to the curve y = x2 + C.
a. 3
b. 4
c. 5
d. 6
861.
862.
At what acute angle does the curve y = 1 – 1/2x2 cut the x-axis?
a. 34.54°
b. 44.64°
c. 54.74°
d. 64.84°
The angle θ, made by a swinging pendulum with the vertical direction, is given at time t by the equation θ =
asin(bt + c), where a, b and c are constants. Find the angular acceleration at time t.
a. –a2θ
b. –b2θ
c. –aθ
d. –bθ
863.
If y =
a.
b.
c.
d.
find y’ at x = 5.
1/13
1/14
1/15
1/16
864.
Find the equation of the line with slope -1/2 and tangent to the ellipse x2 + y2 = 8.
a. x + 2y – 4 = 0
b. x – 2y + 4 = 0
c. x + 2y + 4 = 0
d. x – 2y – 4 = 0
865.
Find the second derivative (y”) of 4x2 + 9y2 = 36 by implicit differentiation.
a. -16y3/9
b. -16/9y3
c. -9y3/16
a.
b.
c.
d.
d. -9/16y3
866.
Approximate the root of 3x + x – 2 = 0 by Newton’s Method of Approximation.
a. 0.420
b. 0.419
c. 0.421
d. 0.418
867.
The volume of a sphere is increasing at the rate of 6 cm 3/hr. at what rate is its surface area increasing when the
radius is 40 cm?
a.
b.
c.
d.
868.
0.30 cm2/hr
0.40 cm2/hr
0.50 cm2/hr
0.60 cm2/hr
If f(x) = ex – e-x – 2x and g(x) = x – sinx, evaluate the limit of f(x)/g(x) as x approaches zero.
a.
b.
c.
d.
∞
0
1
2
869.
Find the point of inflection of y = 4 + 3x – x3.
a. (1,6)
b. (0,4)
c. (-2,4)
d. (2,2)
870.
Find the volume of the largest right circular cone that can be cut from a sphere of radius R.
a. 1.421 R3
b. 1.124 R3
c. 1.241 R3
d. 1.412 R3
871.
If s = x2 + 2y2 + 3z2 and x +y +z = 5, find the minimum value of s.
a. 148/11
b. 149/11
c. 150/11
d. 151/11
a.
b.
c.
d.
872.
The cost of fuel per hour in operating a luxury liner is proportional to the square of its speed and is Php.
12,000.00 per hour for a speed of 10-kph. Other costs amount to Php. 48,000.00 per hour independent of the
speed. Calculate the speed at which the cost per kilometer is a minimum.
a. 35 kph
b. 30 kph
c. 25 kph
d. 20 kph
873.
Find the slope of the tangent to the curve
a.
b.
c.
d.
at the point (1,1).
-1/5
-2/5
-3/5
-4/5
874.
If y = 1/2x(sin(lnx) – cos(lnx)), find dy/dx.
a. sin(lnx)
b. cos(lnx)
c. –sin(lnx)
d. –cos(lnx)
875.
If x = et and y = 2e-t, find d2y/dx2.
a. 4e-t
b. 4e-2t
c. 4e-3t
d. 4e-4t
876.
Two corridors 6 m and 4 m wide respectively, intersect at right angles. Find the length of the longest ladder that
will go horizontally around the corner.
a. 13 m
b. 14 m
c. 15 m
d. 16 m
877.
An angle φ of a right triangle is given by the equation φ = arcsin(y/x). If x is increasing at the rate of 1 in/sec and
y is decreasing at 0.10 in/sec, how fast is φ changing?
a. -0.06892 rad/sec
b. -0.08926 rad/sec
c. -0.09268 rad/sec
d. -0.06928 rad/sec
878.
Find the maximum capacity of a conical vessel whose slant height is 9 cm.
a. 293.84 cm3
a.
b.
c.
d.
b. 283.94 cm3
c. 284.93 cm3
d. 294.83 cm3
879.
If the semi-axes of the ellipse 4x2 + 9y2 = 36 are each increased by 0.15 cm, find the approximate increase in its
area.
a.
b.
c.
d.
2.36 cm2
2.46 cm2
2.56 cm2
2.66 cm2
880.
If y = 4/(2x – 1)3, find y” at x = 1.
a. 190
b. 191
c. 192
d. 193
881.
The side of an equilateral triangle increases at the rate of 2 cm/hr. At what rate is the area of the triangle
changing at the instant when the side is 4 cm?
a.
b. 4
c. 5
d. 6
882.
Find the value of x and y which satisfy 2x + 3y = 8 and whose product is a minimum. a. 1 and 2
b. 3 and 2/3
c. 3/2 and 5/3
d. 2 and 4/3
883.
If ln(ln y) + ln y = ln x, find dy/dx.
a.
a.
b.
c.
d.
b.
c.
d.
884.
If x = 2sinθ, y = 1 – 4cosθ, then dy/dx is equal to
a. 2cotθ
b. 2tanθ
c. 2cscθ
d. 2secθ
885.
The upper and lower edges of a picture frame hanging on a wall are 8 feet and 2 feet above an observer’s eye
level respectively. How far from the wall must the observer stand in order that the angle subtended by the
picture is a maximum?
a. 3.5 ft
b. 4 ft
c. 4.5 ft
d. 5 ft
If x increases at the rate of 30 cm/s, at what rate is the expression (x + 1) 2 increasing when x becomes 6 cm?
886.
a.
b.
c.
d.
400 cm2/s
410 cm2/s
420 cm2/s
430 cm2/s
887.
Find the radius of a right circular cylinder of maximum volume that can be inscribed in a right circular cone of
radius R.
a. R/3
b. R/2
c. 3R/4
d. 2R/3
888.
Find the area of the triangle bounded by the coordinate axes and the tangent to the parabola y = x 2 at the point
(2,4).
a. 2
b. 3
c. 4
d. 5
a.
b.
c.
d.
889.
What is the maximum value of y = 3sinx + 4cosx ?
a. 8
b. 7
c. 6
d. 5
890.
Find the maximum point of the curve y = 4 + 3x – x3.
a. (-2,6)
b. (0,4)
c. (1,6)
d. (-3,22)
891.
Water flows into a cylindrical tank at the rate of 20 m 3/s. How fast is the water surface rising in the tank if the
radius of the tank if the radius of the tank is 2 m?
a.
b.
c.
d.
5/π
6/π
3/π
4/π
892.
If (0,4) and (1,6) are critical points of y = a + bx + cx 3, find the value of c.
a. 1
b. 2
c. -1
d. -2
893.
Intensity of light is proportional to the cosine of the angle of incidence and inversely proportional to the square
of the distance from the source of light. A lamp is directly over the center of a circular table of radius 3 feet.
How high above should the lamp be placed so that there will be maximum illumination around the edge of the
table? a. 2.18 ft
b. 2.16 ft
c. 2.14 ft
d. 2.12 ft
894.
Find the value of x so that the determinant given below will have a minimum value.
a.
b.
c.
d.
a.
b.
c.
d.
5
6
7
8
895.
Find the area of the largest triangle that can be formed by the tangent to the curve y = e-x and the coordinate
axes.
a.
b.
c.
d.
1/e
2/e
3/e
4/e
896.
A bus company planning a tour knows from experience that at Php. 20.00 per person, all 30 seats in the bus will
be taken but for each increase of Php. 1.00, two seats will become vacant. The expenses of the tour are Php.
100.00 plus Php. 11.00 per person. What price should the company charge to maximize the profit?
a. Php. 23.00
b. Php. 24.00
c. Php. 25.00
d. Php. 26.00
897.
An isosceles triangle has legs 26 cm long. The base decreases at the rate of 12 cm/s. Find the rate of change of
the angle at the apex when the base is 48 cm.
a. -1.4 cm/s
b. -1.3 cm/s
c. -1.2 cm/s
d. -1.1 cm/s
898.
Find the weight of the heaviest cylinder that can be cut out from a sphere which weighs
12 kg.
a.
b.
c.
d.
899.
If
find dy/dx.
a.
b.
c.
d.
900.
4.93 kg
5.93 kg
6.93 kg
7.93 kg
eaxcosbx
eaxsinbx
–eaxcosbx
-eaxsinbx
A weight is attached to one end of a 29-m rope passing over a small pulley 17 m above the ground. A man
keeping his hand 5 m above the ground holds the other end of the rope and walks away at a rate of 3 m/s. How
fast is the weight rising at the instant when the man is 9 m from the point directly below the pulley?
a. 1.2 m
b. 1.4 m
c. 1.6 m
a.
b.
c.
d.
d. 1.8 m
901.
A right triangle as a hypotenuse of length 13 and one leg of length 5. Find the area of the largest rectangle that
can be inscribed in the triangle if it has one side along the hypotenuse of the triangle.
a. 15
b. 16
c. 17
d. 18
902.
Evaluate
a.
b.
c.
d.
903.
904.
905.
∞
1
e-2
e2
The sum of two numbers is K. Find the minimum value of the sum of their cubes.
a. K3
b. K3/2
c. K3/3
d. K3/4
A chord of a circle 4 m in diameter is increasing at the rate of 0.60 m/min. Find the rate of change of the smaller
arc subtended by the chord when the chord is 3 m long.
a. 0.81 m/min
b. 0.71 m/min
c. 0.91 m/min
d. 0.61 m/min
A manufacturer estimates that he can sell 1,000 units of a certain product per week if he sets the price per unit
at Php. 3.00 and that his sale will rise by 100 units with each Php.
0.10 decrease in price. Find his maximum revenue.
a.
b.
c.
d.
906.
.
Php. 3,000
Php. 4,000
Php. 5,000
Php. 6,000
The volume of a pyramid is increasing at the rate of 30 cm3/s and the area of the base is increasing at the rate
of 5 cm2/s. How fast is the altitude increasing at the instant when the area of the base is 100 cm 2 and the
altitude is 8 cm?
a. 0.50 cm/s
b. 0.40 cm/s
c. 0.60 cm/s
d. 0.70 cm/s
a.
b.
c.
d.
907.
A closed right circular cylinder has a surface area of 100 cm2. What sould be its radius in order to provide the
largest possible volume?
a.
b.
c.
d.
908.
909.
910.
3.320 cm
2.330 cm
3.203 cm
2.303 cm
A ship 5 km from a straight shore and travelling at the rate of 36 kph is moving parallel to the shore. How fast is
the ship coming closer to a fort on the shore when it is 13 km from the fort?
a. 34.24 km
b. 33.23 km
c. 32.21 km
d. 31.20 km
The sum of the base and the altitude of a trapezoid is 36 cm. Find the altitude if its area is to be maximum.
a. 18 cm
b. 20 cm
c. 19 cm
d. 17 cm
Find the equation of the line parallel to the line x + 2y = 6 and tangent to the ellipse x 2 +
4y2 = 8 in the first quadrant?
a.
b.
c.
d.
x + 2y + 4 = 0
x – 2y + 4 = 0
x + 2y – 4 = 0
x – 2y – 4 = 0
911.
A sector with perimeter of 24 cm is to be cut from a circle. What should be the radius of the circle if the area of
the sector is to be a maximum?
a. 6 cm
b. 7 cm
c. 5 cm
d. 4 cm
912.
Find the equation of the line tangent to the curve y = x 3 – 6x2 at its point of inflection. a. 3x + y + 2 = 0
b. 3x – y + 2 = 0
c. 3x + y – 2 = 0
d. 3x – y – 2 = 0
913.
Find the radius of a right circular cylinder of greatest lateral surface area that can be inscribed in a sphere of
radius 4.
a. 2.53
b. 2.63
c. 2.73
a.
b.
c.
d.
d. 2.83
914.
Evaluate
a.
b.
c.
d.
.
zero
one
infinity
none
915.
Two posts 30 m apart are 10 m and 15 m high respectively. A transmission wire passing through the tops of the
post is used to brace the posts at a point on level ground between them. How far from the 10-m post must that
point be located in order to use the least amount of wire?
a. 10 m
b. 11 m
c. 12 m
d. 13 m
916.
Three sides of a trapezoid are each 8 cm long. How long is the fourth side when the area of the trapezoid has
the largest value?
a. 14 cm
b. 15 cm
c. 16 cm
d. 17 cm
917.
A spherical iron ball 8 inches in diameter is coated with a layer of ice of uniform thickness. If the ice melts at a
rate of 10 cu in per min, how fast is the outer surface of the ice decreasing when the ice is 2 inches thick?
a. -3.39 in2/min
b. -3.33 in2/min
c. -3.36 in2/min
d. -3.31 in2/min
918.
A circular filter paper of radius 15 cm is folded into a conical filter, the radius of whose base is x. Find the value
of x for which the conical filter will have the greatest volume. a. 11.25 cm
b. 12.25 cm
c. 13.25 cm
d. 14.25 cm
919.
Water flows out of a hemispherical tank at the constant rate of 18 cu cm per min. If the radius of the tank is 8
cm, how fast is the water level falling when the water is 4 cm deep?
a. -0.1491 cm/min
b. -0.1941 cm/min
c. -0.1194 cm/min
d. -0.1149 cm/min
a.
b.
c.
d.
920.
Find the area of the largest isosceles triangle that can be inscribed in a circle of radius 2. a.
b. 6.3
c. 4.1
d. 3.8
921.
Sand is poured at the rate of 10 ft3/min so as to form a conical pile whose altitude is always equal to the radius
of its base. At what rate is the area of the base increasing when its radius is 5 ft?
a. 3 ft3/min
b. 4 ft3/min
c. 5 ft3/min
d. 6 ft3/min
922.
Find the altitude of the largest right circular cone that can be cut from a sphere of radius R.
a. 7R/3
b. 5R/3
c. 4R/3
d. 8R/3
923.
A light is placed 3 ft above the ground and 32 ft from a building. A man 6 ft tall walks from the light toward the
building at the rate of 6 ft/sec. Find the rate at which the length of his shadow is decreasing when he is 8 ft.
a. -1 fps
b. -1.5 fps
c. -2 fps
d. -2.5 fps
924.
An open box is made by cutting squares of side x inches from four corners of a sheet of cardboard that is 24
inches by 32 inches and then folding up the sides. What should x be to maximize the volume of the box?
a. 16.3 in
b. 15.2 in
c. 13.8 in
d. 14.1 in
925.
Let f be a function defined by f(x) = Ax2 + Bx + C with the following properties: f(0) = 2, f’(2) = 10 and f”(10) = 4.
Find the value of B.
a.
b.
c.
d.
926.
5.2
1
2
3
4
A rectangle has its base on the x-axis and its two upper corners on the curve y = 2(1 – x2). What is the maximum
perimeter of the rectangle?
a.
b.
c.
d.
a.
b.
c.
d.
927.
4
5
6
7
Find the maximum vertical distance between y = cosx and y =
sinx over the interval
[0,2].
a.
b.
c.
d.
928.
1.5
2
2.5
3
A baseball diamond is a square 90 ft on the side. A runner travels from home plate to first base at the rate of 20
ft/sec. how fast is the runner’s distance from the second base changing when the runner is halfway to the first
base?
a.
b.
c.
d.
929.
If the line to the curve y = x – lnx at x = a, passes through the origin, find a.
a. 2.72
b. 2.83
c. 2.91
d. 2.69
930.
Find the radius of curvature of the ellipse 4x2 + 5y2 = 20 at (0,2).
a. -1.5
b. -2.5
c. -3.5
d. -4.5
931.
If sin(x/y) = y/x, find dy/dx.
a.
b.
c.
d.
a.
b.
c.
d.
x/y
–x/y
y/x
–y/x
932.
Water is running into a right circular cone with vertical angle equal to 60 degrees (at the bottom) at the rate of
2 cubic feet per second, and at the same time water is leaking out at a rate which is 4.8 times the square root of
its depth. How high will the water rise? a. 0.1637 ft
b. 0.1367 ft
c. 0.1673 ft
d. 0.1736 ft
933.
If
,
a.
b.
c.
d.
934.
and
evaluate
25
26
27
28
Find the area bounded by x = y + 2, x = 1 – y2, y = 1 and y = -1 with or without integration.
a.
b.
c.
d.
11/3
8/3
7/3
5/3
935.
Find the upper area bounded by the curves r = cscθ and r = 4sinθ.
a. 9.10
b. 10.11
c. 11.12
d. 12.13
936.
If f(x) = x1/2 and g(x) = (2x + 1)5/2, evaluate
a.
b.
c.
d.
937.
37/324
36/324
35/324
43/324
Find the perimeter of the cardioid r = 1 – cosθ.
a. 7
b. 9
c. 6
a.
b.
c.
d.
from x = ½ to x = 4.
d. 8
938.
Find the centroid of the volume of a cone formed by revolving about the y-axis the part of the line
intercepted between the coordinate axes.
(0,1)
(0,2)
(0,3)
(0,4)
A barrel has the shape of an ellipsoid of revolution with equal pieces but off ends. If the barrel is 10 units long
with circular ends of radius 2 units and the midsection of radius 4 units, find the volume of the barrel with or
without integration.
a. 100π
b. 110π
c. 120π
d. 130π
a.
b.
c.
d.
939.
940.
Each of the vertical ends of a trough is a parabolic segment with base 4 m and altitude 1
m. Find the force against one of the trough if it is full of water.
a.
b.
c.
d.
11.43 kN
12.44 kN
11.45 kN
10.46 kN
941.
If the trough in problem 444 is 5 m long, how long will it take a 0.50-hp pump to empty the trough by pumping
all of the water to the top of the trough?
a. 2.1 min
b. 1.2 min
c. 1.4 min
d. 2.4 min
942.
Find the centroid of a semicircular arc of radius r by placing its diameter along the y-axis. a.
b. (3r/π,0)
c. (2r/π,0)
d. (4r/π,0)
943.
Find the moment of inertia of the semicircular arc in problem 446 with respect to its diameter.
a.
a.
b.
c.
d.
r5
(r/π,0)
944.
b.
r4
c.
r3
d.
r2
If
, find the value of m.
a.
b.
c.
d.
2
3
4
5
945.
A dog is tied to a 4-m circular tank with a cord 3 m long. The point where the cord is attached to the tank is at
the same level as the dog’s collar. Compute the total area in which the dog can move.
a. 18.64 m2
b. 16.84 m2
c. 14.85 m2
d. 16.48 m2
946.
If
find f(x).
a.
b.
c.
d.
x3/3
x4/4
x3
x4
947.
An equilateral triangle of side 8 ft is immersed in water with its plane vertical. If one side is horizontal, and the
vertex opposite that side is in the surface of the water, find the force of pressure on the face of the triangle.
a. 8,500 lb
b. 8,000 lb
c. 7,500 lb
d. 7,000 lb
948.
The area bounded by y = x2 and y = 2 – x2 is revolved about the x-axis. Find the volume of the solid generated
with or without integration.
a.
b.
c.
d.
a.
b.
c.
d.
14π/3
16π/3
17π/3
19π/3
949.
Find the perimeter of the curve x2/3 + y2/3 = 4.
950.
46
47
48
49
Evaluate the integral of cos4xdx from x = -π/2 to x = π/2.
a. 3π/8
b. 4π/5
c. 5π/6
d. 9π/4
a.
b.
c.
d.
951.
Find the surface area generated by revolving the length of the arc of r = 1 + cosθ from 0 to π about the polar
axis.
a. 23.13
b. 22.15
c. 21.12
d. 20.11
952.
Evaluate
a.
b.
c.
d.
953.
5/3
7/3
2/3
4/3
Find the area of the region that is inside the curve r = 8cosθ but is outside the curve r =
4cosθ with or without integration.
a.
b.
c.
d.
954.
.
10π
11π
12π
13π
A conoid is a solid having a circular base such that every plane section perpendicular to the diameter of the
base is an isosceles triangle. Find the volume of the conoid having a radius of 2 m and the altitude of the
triangle is 4 m.
a. 6 m3
b. 7 m3
c. 8 m3
d. 9 m3
a.
b.
c.
d.
955.
956.
A rectangular plate 5 ft long and 4 ft wide is submerged in a liquid at an angle of 60 degrees with the vertical. If
the liquid weighs w lb per cu ft, find the force of pressure on the plate if the longer edge is parallel to the
surface of the liquid and is 2 ft below the surface.
a. 45w lb
b. 50w lb
c. 55w lb
d. 60w lb
Find the moment of inertia of the volume of a right circular cylinder with base radius r and altitude h relative to
its base.
a.
b.
c.
d.
957.
Evaluate
a.
b.
c.
d.
.
1/23
1/24
1/25
1/26
958.
Find the area bounded by yx2 = 1, x = 1 and the x-axis.
a. ½
b. 1
c. 3/2
d. 2
959.
If a 10-lb weight could be lifted from the surface of the earth to a height of 4000 miles above the surface of the
earth, how much work would have to be done? Assume the force of gravitation to vary inversely as the square
of the distance from the center of the earth and take the radius of the earth to be 4000 miles.
a. 20,000 mi-lb
b. 21,000 mi-lb
c. 22,000 mi-lb
d. 23,000 mi-lb
a.
b.
c.
d.
960.
Find the value of
.
0.4049
0.4409
0.4094
0.4904
The cross section of a certain solid made by any plane perpendicular to the x-axis is an equilateral triangle with
the ends of one of its sides on the parabolas y = x2 + 5 and y =
2x2 + 1. Find the volume of this solid between the points of intersection of the parabolas. a.
12.76
a.
b.
c.
d.
961.
b. 13.77
c. 14.78
d. 15.79
962.
A hole of radius 3 units is bored through the center of a sphere of radius 5 units. Find the volume of the part of
the sphere with or without integration.
a. 278.2
b. 268.1
c. 258.4
d. 248.3
963.
Find the x-coordinate (or ) of the centroid of the area in the first quadrant bounded by the curves y = 2 – x2
and y = x2.
a.
b.
c.
d.
3/8
1/4
2/3
4/9
964.
Find the length of the arc of the curve r = 2(1 + cosθ) from θ = 0 to θ = π.
a. 6
b. 7
c. 8
d. 9
965.
A solid has a circular base of radius 3 units. Find the volume of the solid if every plane section perpendicular to
a fixed diameter of the base is an isosceles triangle with its altitude equal to its base. Solve with or without
integration.
a. 42
b. 52
a.
b.
c.
d.
c. 62
d. 72
966.
A pit is to be dug in the form of an inverted right circular cone, 4 m deep, and 6 m in diameter at the surface of
the ground. Find the number of kilojoules of work to be done if the material weighs w kN/cm 3.
a. 15πw kJ
b. 14πw kJ
c. 12πw kJ
d. 13πw kJ
967.
A trough 6 m long as its vertical cross section in the form of an isosceles trapezoid. The upper and lower bases
are 6 m and 4 m respectively and its altitude is 2 m. if the trough is full of liquid with specific weight 9.81 kN per
cu m, find the forces against the slant side of the trough.
a. 111.41 kN
b. 121.51 kN
c. 131.61 kN
d. 141.71 kN
968.
Evaluate
a.
b.
c.
d.
.
4.9348
4.3894
4.4938
4.8439
969.
The stretch of a spring is proportional to the force applied. If a force of 5 pounds produces a stretch of onetenth the original length, how much work will be done in stretching the spring to double its original length? (Let
L = original length)
a. 20L
b. 22L
c. 24L
d. 25L
970.
Find the volume of the ring-shaped solid generated by revolving about the x-axis the portion of the plane
bounded by the line y = 5 and the parabola y = 9 – x2.
a. 342.24
b. 442.34
c. 542.44
d. 642.54
a.
b.
c.
d.
971.
972.
A uniform chain that weighs 4 N/m has a leaky 15-L bucket attached to it. The bucket contains a liquid that
weighs 9 N/L. If the bucket is full when 8 m of the chain is out and half full when no chain is out, how much
work was done in winding the chain on a windlass. Assume that the liquid leaks out at a uniform rate.
a. 893 J
b. 938 J
c. 398 J
d. 839 J
Find the volume of the torus generated by revolving a circle of radius r about a line on the same plane of the
circle and whose distance is 2r from the center of the circle. Solve with or without integration.
a. 4πr2
b. 4πr3
c. 4π2r2
d. 4π2r3
973.
A plate in the shape of a right triangle is submerged vertically in the water and the base 3 m long is in the
surface of the water. Find the altitude of the triangle if the force due to the water pressure against one face of
the plate is 50w kN where w is the specific weight of the water.
a. 8 m
b. 9 m
c. 10 m
d. 11 m
974.
Find the volume of the torus generated by revolving about the x-axis the area bounded by x2 + (y – 4)2 = 4.
a. 315.83
b. 314.73
c. 313.63
d. 312.53
975.
The cross section of a deep well containing mineral water is a circle of radius 1.2 m. the cost of pumping the
water to an outlet at the top of the well is 2 pesos per joule of work. The mineral water weighs 9810 newton
per cubic meter. If the surface of the water is one meter below the top of the well and the water is sold 50,000
pesos per cubic meter, find the depth to which the water is to be pumped out to realize maximum profit.
a. 2.35 m
b. 2.45 m
c. 2.55 m
d. 2.65 m
a.
b.
c.
d.
976.
A wedge is cut from a circular tree whose diameter is 2 m by a horizontal cutting plane up to the vertical axis
and another cutting plane which is inclined by 45 degrees from the previous plane. Find the volume of the
wedge with or without integration.
a. 3/5 m3
b. 2/3 m3
c. 3/4 m3
d. 2/5 m3
977.
Find the moment of inertia of a circle 5 cm in diameter about an axis through its centroid. a.
b. 31.58 cm4
c. 32.48 cm4
d. 33.38 cm4
978.
Find the moment of inertia of the circle in problem 481 relative to the line tangent to the circle.
a. 76.47 cm4
b. 77.57 cm4
c. 78.67 cm4
d. 79.77 cm4
979.
Find the perimeter of the astroid whose parametric equations are x = acos 3t, y = asin3t.
a. 5a
b. 6a
c. 7a
d. 8a
980.
The axes of two right circular cylinders of equal radii 9 cm each intersect at right angles. Find the volume of the
common part of the cylinders.
a. 3666 cm3
b. 3777 cm3
c. 3888 cm3
d. 3999 cm3
981.
A hemispherical tank is full of oil weighing 7.85 kN/m 3. The oil is to be pumped to the top of the tank. Find the
work done if the radius of the tank if 0.60 m.
a.
b.
c.
d.
a.
b.
c.
d.
0.799 kJ
0.688 kJ
0.577 kJ
0.466 kJ
30.68 cm4
982.
Find the area of one loop of the curve r2 = 8cos2θ.
a. 3
b. 4
c. 5
d. 6
983.
Find
of the centroid of the solid generated by revolving about the y-axis, the first quadrant area bounded by
y2 = 12x, x = 3 and y = 0.
a.
b.
c.
d.
984.
2.3
2.5
2.7
2.9
The angle between 90 degrees and 180 degrees has
A. negative cotangent and cosecant
B. negative sine and tangent
C. negative secant and tangent
D. negative sine and cosine
985.
It is defined as the angle subtended by a circular arc whose length is equal to the radius of the circle.
A. mil
B. radian
C. degree
D. grade
986.
In what quadrant does an angle terminate if its cosine and tangent are both negative?
A. first
B. second
C. third
D. fourth
987.
Which of the following angles in standard position is a quadrantal angle ?
A. 540 degrees
B. 480 degrees
C. -135 degrees
D. -390 degrees
988.
It is an angular unit that is equal to 1/6400 of four right angles.
a.
b.
c.
d.
A. mil
B. grade
C. radian
D. rpm
Relative to a right triangle ABC where C = 90 degrees, which of the following is not true ?
C. cos A = sec B
A. sin A = cos B
B. tan A = cot B
D. csc A = sec B
990. If the value of sin A is a negative fraction, then angle A terminates in
989.
A. quadrants II and III C. quadrants III and IV
B. quadrants I and III D. quadrants II and IV
991.
The secant is the cofunction of
A. sine
992.
B. cosine
C. cotangent
D. cosecant
Which of the following is an undirected distance ?
A. The distance of a point from the x-axis.
B. The distance of a point from the y-axis.
C. The distance of a point from the origin.
D. The distance of a point from a line.
Which of the following systems of angle measurements uses the degree as the unit of measure?
C. sexagesimal system
A. circular system
B. mil system
D. grade system
994. In what quadrant will angle A terminate if sec A is positive and csc A is negative.
993.
A. I
995.
B. II
C. III
D. IV
Which of the following relations is not true ?
A. sinx = (tanx/secx) C. cotx = cscx cosx
B. (cotx/cscx) = (sinx/tanx)
D. (secx/tanx) = (cosx/cotx)
996.
Within what limits between between 0 degrees and 360 degrees must the angle θ lie if cos θ = -2/5 ?
A. between 0 degrees and 180 degrees
B. between 90 degrees and 180 degrees
C. between 90 degrees and 270 degrees
D. between 90 degrees and 360 degrees
a.
b.
c.
d.
997.
The coreference angle of any angle A is the positive acute angle determined by the terminal side of A and the yaxis. What is the coreference angle of 290 degrees ?
A. 70 degrees
B. 50 degrees
C. 30 degrees
D. 20 degrees
998.
A measure of 3200 mils is equal to
A. 90 deg
B. 45 deg
C. 180 deg
D. 120 deg
999.
The value of vers θ is equal to
A. 1 - cosθ
B. 1 - sinθ
C. 1 + cosθ
D. 1 +sinθ
1000. To find the interior angles of a triangle whose sides are given, use the law of
A. sine
1001. The point P(x,y) where x
A. I or IV
C. tangent
B. cosine
D. secant
0 and y > 0 is located in quadrant
B. II or III
C. I or II
D. III or IV
1002. Which of the following relations is true for any angle θ ?
A. sin(-θ) = sin θ
B. sec(-θ) = sec θ 1003.
Coversine A is equal to
C. tan(-θ) = tan θ
D. csc(-θ) = csc θ
A. 1 - cosA
C. 1 + cosA
B. 1 - sin A
D. 1 + sin A
1004. The terminal side of -1,500 degrees will lie in quadrant
A. one B. two C. three D. four
1005. Which of the following is false as the angle A increases from 0 degrees to 90 degrees ?
A. sin A increases from zero to one
a.
b.
c.
d.
B. tan A increases from zero to infinity
C. cos A decreases from one to zero
D. sec A decreases from one to infinity
1006. Which of the following functions is positive if angle A terminates in the second quadrant ?
A. csc A
B. tan A C. sec A D. cos A
1007. An angle in standard position and whose terminal side falls along one of the coordinate axes is called a
C. quadrantal angle
A. reference angle
B. vertical angle
D. central angle
1008. Which of the following pairs of angles in standard positions are coterminal angles ?
A. 710 degrees and -10 degrees
B. 120 degrees and 60 degrees
C. -240 degrees and 30 degrees
D. 325 degrees and -40 degrees
1009. The gradient of the line in the figure is
A. tan θ
B. -1/tan θ
C. -tan θ
D. cot θ
1010. Which of the following is true in quadrants III and IV ?
A. negative cosecant C. negative cotangent
B. positive sine D. positive tangent
a.
b.
c.
d.
1011. Which of the following is not a first quadrant angle ?
A. 450 degrees C. -330 degrees
B. 60 degrees D. -120 degrees
1012. If tan θ > 0 and cosθ < 0, then θ is a
A. first quadrant angle C. third quadrant angle
B. second quadrant angle
D. fourth quadrant angle
1013. If an angle is in the standard position and its measure is 215 degrees, the its reference angle is
A. 25 degrees B. 30 degrees C. 35 degrees D. 40 degrees
1014. In the second quadrant, which of the following is true ?
A. The tangent and secant are positive
B. The sine and cosecant are positive
C. The cotangent and cosecant are positive
D. The sine and tangent are positive
1015. In what quadrant can we locate the point (x, -4) if x is positive ?
A. I
B. II
C. III
D. IV
1016. In what quadrants do the secant and cosecant of an angle have the same algebraic sign?
A. II and IV
B. I and II
C. I and III
D. III and IV
1017. If cos 3A + sin A = 0, find the value of A.
A. 30 degrees B. 45 degrees C. 60 degrees D. 90 degrees
1018. If tan A = 2 and tan B = 1/2, find A + B.
A. 90 degrees B. 30 degrees C. 45 degrees D. 60 degrees
1019. If sin x = 5/13 , find sin 2x.
A. 120/169
B. 25/169
C. 10/13
D. 12/13
1020. If cot θ = square root of 3 and cos θ < 0, find csc θ.
A. 2
B. -2
1021. If sin A = -5/13 and A in quadrant III, find cot A.
a.
b.
c.
d.
C. 1/2
D. -1/2
A. 12/5
B. -12/5
C. 5/12
D. -5/12
C. 17/19
D. 8/17
C. 4 pi
D. 6 pi
1022. Find the value of sin(Arecos 15/17).
A. 8/9
B. 8/2
1023. The cosecant of 960 degrees is equal to
A. -2( square root of 3 / 3)
B. 2( square root of 3 / 3)
C. 1/2
D. -1/2
1024. If sin 3A = cos 6B, then
A. A - 2B = 90 degrees
B. A + 2B = 90 degrees
C. A + B = 180 degrees
D. A + 2B = 30 degrees
1025. What is the period of y = 3 sin(x/2) ?
A. 2 pi
B. 3 pi
1026. If the product of cot 2θand cot 68 degrees is equal to unity, find θ.
A. 13 degrees
B. 12 degrees
C. 11 degrees
D. 10 degrees
1027. Sec A - cos A is identically equal to
A. sin A cot A
B. cos A tan A
C. sin A tan A
D. cos A cot A
1028. Simplify ( sin θ/ 1 - cos θ) - ( 1 + cos θ/ sin θ)
A. sin²θ
B. cos²θ
C. 1
D. 0
1029. If tan x = 1/2 and tan y = 1/3, find tan (x + y).
A. 1
B. 2/3
1030. If cos θ= 3/5 and θ in quadrant IV, find cos2θ
a.
b.
c.
d.
C. 2
D. 1/2
A. 7/25
D. -24/25
C. 24/25
B. -7/25
1031. simplify (sinθ + cosθtanθ)/(cosθ)
A. tanθ
B. 2cotθ
D. 2tanθ
C. sinθ
1032. If Arcsin(2x) = 30 degrees, find x.
A. 0.20
D. 0.35
C. 0.3
B. 0.25
1033. If sin 40 degrees + sin 20 degrees = sin θ, find the value of θ.
A. 20 degrees
B. 60 degrees
C. 80 degrees
D.120 degrees
1034. The angle that is equal to one half of its supplement is
A. 60 degrees
B. 90 degrees
C. 80 degrees
1035. Find the equivalent value of y in the equation y = (1 + cos 2θ) / (cot θ)
A. sin2θ
B. cos2θ
C. sinθ D. cosθ
1036. If tan A = -3 and tan B = 2/3, find tan(A - B).
A. -11/9
B. -10/9 C. -13/9 D. -12/9
1037. If cos 65 degrees + cos 55 degrees = cos θ, find the θ in radians.
A. 1.832
B. 1.658
C. 0.7853
D. 0.0873
1038. If tan (A / 4) = cot A, find A.
A. 52 degrees B. 72 degrees C. 42 degrees D. 62 degrees
1039. Simplify cos^4 x- sin^4 x
A. cos 4x
B. sin 4x
C. sin 2x
D. cos 2x
1040. If tan 4x = cot 6y, then
A. 2x - 3y = 45 degrees C. 4x - 6y = 90 degrees
B. 2x + 3y = 45 degrees
D. 6y - 4x = 90 degrees
1041. Simplify
Arctan(1/3) + Arctan(1/5)
A. Arctan (7/4) C. Arctan (8/15) B. Arctan (4/7) D. Arctan (1/15)
1042. If sin A =3.5x and cos A = 5.5x, find angle A.
a.
b.
c.
d.
D. 45 degrees
A. 32.47 degrees
B. 33.47 degrees
C. 34.47 degrees
D. 35.47 degrees
1043. If the tangent of an angle x is 3/4, find the value of the cosine of 2x.
A. 0.60 B. 0.28 C. 0.8 D. 0.38
1044. Find the angle which a 9-m ladder will make with the ground if it is leaned against a window still 6m high.
A. 21.8 degrees
C. 41.8 degrees
B. 31.8 degrees
D. 51.8 degrees
1045. The expression (1 -sinx) / (cosx) is equal to
A.tanx
B.1
C.(1 - cosx)/sinx
D.(cosx) / (1 + sinx)
1046. A tree 30 m long casts a shadow 36 m long. Find the angle of elevation of the sun.
A. 39.41 degrees
B. 39.51 degrees
C. 39.81 degrees
D. 39.61 degrees
1047. Which of the following is true ?
C. tan(90 degrees + θ) = -tanθ B. tan(180 degrees - θ) = -tan θ D.
A. tan(180 degrees + θ) = - tanθ
tan(270 degrees - θ) = - tanθ
1048. Express 3i + 5 + (square root of -16) in the standard form.
A. 5 - 7i
B. 5 + 7i
C. -5 + 7i
D. -5 - 7i
1049. Write (square root of 2) cis 135 degrees in rectangular form.
A. 1 -i B. -1 + i
C. -1 - i D. 1 + i
1050. Give the conjugate of 2 + (square root of -25) in the standard form.
A. 2 - 5i
B. 2 + 5i C. -2 + 5i
D. -2 -5i
1051. For the trigometric function y = a sin(bx +c), the absolute value of the ratio c/b is called
A. amplitude B. period
C. argument
D. phase shift
1052. If sin2x sin4x = cos2x cos4x, find the value of x.
A. 13° B. 14° C. 15° D. 16°
1053. If sin θ = 3.5x and cos θ = 5.5x, find x.
A. 0.1532
a.
b.
c.
d.
B. 0.1534
C. 0.1536
D. 0.1538
1054. Find θ if 2tan θ = ( 1 - tan² θ) cot 56° .
A. 18° B. 16° C. 19° D. 17°
1055. Solve for x if Arctan ( 1 – x ) + Arctan ( 1 + x ) = Arctan ( 1/8 ).
A. 2
B. 4
C. 6
D. 3
1056. If A + B = 180°, then which of the following is true ?
sin A = sin B
cos A = cos B
tan A = tan B
B. (2) only
A. (1) only
1057. Simplify
C. (3) only
D. all of them
(sin ½x – cos ½x) ²
A. 1 + sin x
B. 1 – sin x
C. 1 + cos x
D. 1 – cos x
1058. Find the value of Arctan 2cos(Arcsin √3/ 2) .
A. 30°
B. 45°
C. 60°
D. 90°
1059. If sin A = -7/25 where 180° < A < 270°, find tan(A/2).
A. -1/5
B. -5
C. -1/7
D. -7
1060. If sin²x + y = m and cos²x + y = n, find y.
A. (m + n + 1)/2
B. (m + n – 1)/2
C. (m+n)/2 – 1
D. (m+n)/2 +1
1061. Given cos θ = √3/2, find the value of 1 - tan² θ.
A. -2
B. -1/3
C. ½
1062. What is the value of A between 270° and 360° if 2sin² A – sin A = 1 ?
D. 2/3
A. 290° B. 275° C. 300° D. 330°
1063. Evaluate ( sin 0° + sin 1° + sin 2° + … + sin 90°) / ( cos 0° + cos 1° + cos 2° + … + cos 90°)
a.
b.
c.
d.
A. 0
B. 1
C. 2
D. 3
1064. If the supplement of an angle θ is 5/2 of its complement. Find the value of θ.
B. 25°
A. 30°
C. 20°
D. 15°
1065. Express -4 - 4√3 i in trigonometric form.
A. 8 cis 120°
C. 8 cis 150°
B. 8 cis 240°
D. 8 cis 300°
1066. If cos A = -15/17 and A is in quadrant III, find cos ½ A.
A. 0.29054
B. 0.24125
C. 0.24254
D.0.24354
1067. If sin A = 3/5 and cos B = 5/13, find sin (A + B).
A. 0.388
B. 0.865
1068. Simplify (sin 2x) / ( 1 + cos 2x)
A. cot x
B. tan x C. tan 2x
C. 0.650
D. 0.969
D. 1
1069. A pole which leans to the sun by 10° 15’ from the vertical casts a shadow of 9.43 m on the level ground when the
angle of elevation of the sun is 54°50’. The length of the pole is
A. 15.3 m
B. 16.3 m
C. 17.3 m
D. 18.3 m
1070. Triangle ABC has sides a, b and c. If a = 75 m, b = 100 m and the angle opposite side a is 32°, find the angle
opposite side c.
A. 93° B. 80° C. 103° D. 100°
1071. If the cosine of angle x is 3/5, then the value of the sine of x/2 is
A. 0.500
B. 0.361
C. 0.215
D. 0.447
1072. If 82° + 0.35x = Arctan( cot 0.45x ), find x.
A. 11° B. 10° C. 12° D. 13°
1073. If sec A = -5/4, A in quadrant II, find tan 2A.
A.24/7
B.25/7
C.-25/7
1074. Evaluate cos( Arcsin 3/5 + Arctan 8/15 )
A. 34/85
B. 35/85
C. 36/85
1075. If sin x = ¼ , find the value of 4sin(x/2)cos(x/2).
a.
b.
c.
d.
D.37/85
D.-24/7
A. 1/8 B. 1/3 C. ½
D. 1/6
1076. If Arcsin( x – 2 ) = π/6, find x.
A. 5/4 B. 5/3 C. 5/2 D. 5/6
1077. The trigonometric expression ( 1 - tan²x ) / ( 1 + tan²x ) is equal to
A. sin1/2x
B. sin2x C. cos1/2x
D. cos2x
1078. If x + y = 90°, then ( sinx tan y ) / ( sin y tan x ) is equal to
A. tanx B. 1/tanx
C. –tanx
D. -1/tanx
1079. Twelve round holes are bored through a square piece of steel plate. Their centers are equally spaced on the
circumference of a circle 18 cm in diameter. Find the distance between the centers of two consecutive holes.
A. 4.33 cm
B. 4.44 cm
C. 4.55 cm
D. 4.66 cm
1080. Two sides and the included angle of a triangle are measured to be 11 cm, 20 cm and 112° respectively. Find the
length of the third side.
A. 26.19 cm
B. 24.14 cm
C. 23.16 cm
D. 22.15 cm
1081. The rationalized value of ( 4 - 4√3 i ) / ( -2√3 + 2i ) is
A. √3 + i
B. -√3 + i
C. -√3 – i
D. √3 – i
1082. If Arctan(2x) + Arctan(x) = π/4, find x.
A. 0.261
B. 0.271
C. 0.281
D. 0.291
1083. A ladder leans against the wall of a building with its lower end 4 m from the building. How long is the ladder if it
makes an angle of 70° with the ground?
A. 12.3 m
B. 13.5 m
C. 11.7 m
D. 10.8 m
1084. Find the product of (4cis120°)(2cis30°) in rectangular form.
A. -4(√3 + i)
B. -4(√3 – i)
C. 4(√3 + i)
D. 4(√3 – i)
1085. Solve for x if x = (tanθ + cotθ) ² sinθ - tan²θ
A. 4
B. 3
C. 2
1086. If ysinx = a and ycosx = b, find y in terms of a and b.
a.
b.
c.
d.
D. 1
A. a + b
B. a² + b²
C. √a² + b²
D. √a + b
1087. If tan(Arctanx + Arctan ¼) = 7/11, find x.
A. 1/3
B. ¼
C. 1/5
1088. if tanθ = √3, θ in quadrant III, find the value of (1 + cosθ) / (1 – cosθ).
D. ½
A. ½
B. ¼
C. 1/3 D. 1/5
1089. From the top of a lighthouse 37 m above sea level, the angle of depression of a boat is 15°.How far is the boat
from the lighthouse?
A. 138.1 m
B. 137.2 m
C. 136.3 m
D. 135.4 m
1090. The angles B and C of a triangle ABC are 50°30’ and 122°09’ respectively and BC = 9, find the length of AB.
A. 57.36
B. 58.46
C. 59.56
D. 60.66
1091. If the product of csc(x/2) and cos(x/3 + 60°) is equal to 1, find the value of x.
A. 46°
B. 36°
C. 26°
D. 16°
1092. If Arctanx + Arctan(1/3) = 45°, find x.
A. ½
B. ¼
C. 1/5
D. 1/6
1093. If cscθ = 2 and cosθ < 0, then ( secθ + tanθ ) / ( secθ – tanθ ) =
A. 2
B. 3
C. 4
D. 5
1094. Evaluate [6( cos80° + isin80° ) / 3( cos35° + isin35° )]
A. √2 ( 1 + i )
B. √2 ( 1 – i )
C. 2 ( 1 + i )
D. 2 ( 1- i )
1095. If sin(x + 10°) = cos3x, then x =
A. 23°
B. 22°
C. 21°
D. 20°
1096. If cos(x + y) = 0.17 and cosx = 0.50, find sin y.
A. 0.2355
B. 0.3455
C. 0.4344
D. 0.4233
1097. If sin A + sin B = 1 and sin A – sin B = 1, find A.
A. 60°
a.
b.
c.
d.
B. 70°
C. 80°
D. 90°
1098. At a certain instant, a lighthouse is 4 miles north of a ship which is traveling directly east. If after 10 minutes, the
bearing of the lighthouse is found to be North 21 degree 15 minutes West, find the speed of the ship in miles per
hour.
A. 11.3 mph
B. 10.3 mph
C. 9.3 mph
D. 8.3 mph
C. cot A
D. sin A
C. 8i
D. -8i
C. -16
D. 16
1099. Simplify ( sec A + csc A ) / ( 1 + tan A )
B. sec A
A. csc A
1100. Evaluate [2(cos60° + isin60°)]³
A. 8
B. -8
1101. Evaluate (1 + i)^8
A. -16i
B. 16i
1102. Two buildings with flat roofs are 15 m apart. From the edge of the roof of the lower building, the angle of
elevation of the edge of the roof of the taller building is 32°. How high is the taller building if the lower building is
18 m high?
A. 26.4 m
B. 27.4 m
C. 28.4 m
D. 29.4 m
1103. If two sides of a triangle are each equal to 8 units and the included angle is 70°, find the third side.
A. 6.15 B. 7.16 C. 8.17 D. 9.18
1104. Express sin(2Arccosx) in terms of x.
A. 2x√1 + x²
B. 3x√1 + x²
C. 2x√1 - x²
D. 3x√1 - x²
1105. Transform Arctanx + Arctany = pi/4 into an algebraic equation
A. x + xy + y = 1
B. x + xy –y = 1 C. x – xy + = 1
D. x – xy-y =1
1106. A tower 28.65 m high is situated on the bank of a river. The angle of depression of an object on the opposite bank
of the river is 25°20’. Find the width of the river.
A. 62.50 m
B. 60.52 m
C. 65.20 m
D. 63.25 m
1107. Two cars start at the same time from the same station and move along straight roads that form an angle of 30°,
one car at the rate of 30 kph and the other at the rate of 40 kph. How far apart are the cars at the end of half an
hour ?
A. 10.17 km
B. 10.27 km
C. 10.37 km
1108. Given: sec2θ = √10 and 2θ in quadrant IV
Find : cos4θ
a.
b.
c.
d.
D. 10.47 km
A. -0.60
B. -0.70 C. -0.80
D. -0.90
1109. The bearing of B from A is N20°E, the bearing of C from B is S30°E and the bearing of A from C is S40°W. If AB = 10,
find the area of triangle ABC.
A. 14.95
C. 12.93
B. 13.94
D. 11.92
1110. Two ships start from the same point, one going south and the other North 28° East. If the speed of the first ship is
12 kph and the second ship is 16 kph, find the distance between them after 45 minutes.
D. 20.4 km
A. 17.3 km
B. 18.5 km
C. 19.2 km
1111. If tanθ = ½ and θ is in the 1st quadrant, find tan 4θ.
A. -24/7
B. -20/7
C. -23/7
D.-22/7
1112. Find the height of a tree if the angle of elevation of its top changes from 20° to 40° as the observer advances 23m
toward its base.
A. 138.5 m
B. 148.5 m
C. 158.5 m
D. 159.5 m
C. 19°
D. 18°
1113 If 77° + (2x/5) = Arccos(sin x/4) , find x.
A. 21°
B. 20°
1114. Evaluate tan (Arccos(12/13) – Arcsin(4/5))
A. -33/56
B. -33/55
C. -33/54
D. -33/53
1115. Three times the sine of an angle is equal to twice the square of the cosine of the same angle. Find the angle.
A. 20° B. 25° C. 30° D. 35°
1116. Stations A and B are 1000 m apart on a straight road running from eat to west. From A, the bearing of a tower at
C is 32° west of north and from B, the bearing of C is 26° north of east. Find the shortest distance of the tower at C from
the road.
A. 243.92 m
B. 253.92 m
C. 263.92 m
D. 273.92 m
1117.If tan35° = y, then (tan145° - tan125°) / (1 + tan145°tan125°) =
A.(1 + y²) / 2y
B.(1 - y²) / 2y
C.(y²-1) / 2y
D. (2y-1)/2y
1118. A tree stands vertically on a hillside which makes an angle of 22° with the horizontal. From a point 60 ft down the
hill directly from the base of the tree, the angle of elevation of the top of the tree is 55°. How high is the tree ?
A. 56.97 ft
B. 57.96 ft
1119. If cos 2A = √m , find cos 8A.
a.
b.
c.
d.
C. 59.76 ft
D. 57.69 ft
A. 8m² + 8m + 1
B. 8m² + 8m – 1 C. 8m² -8m + 1 D. 8m² - 8m -1
1120. The angle of triangle ABC are in the ratio 5:10:21 and the side opposite the smallest angle is 5. Find the side
opposite the largest angle.
A. 13.41
B. 14.31
C. 13.14
D. 11.43
1121. On the top of a cliff, the farthest distance that can be seen on the surface of the earth is 60 miles. How high is the
cliff if the radius of the earth is taken to be 4000 miles ?
A. 0.41 mi
B. 0.43 mi
C. 0.45 mi
D. 0.47
1122. Two towers are of equal height. At a point P on level ground between them, the angle of elevation of the top of
the nearer tower is 60° and at a point M 24 meters directly away from point P, the angle of elevation of the top of
the nearer tower is 45°. How high is each tower ?
A. 20.8 m
B. 19.8 m
C. 18.8 m
D. 17.8 m
1123. A quadrilateral ABCD has its side AB perpendicular to side BC at B and its side AD perpendicular to side CD at D. If
angle BAD equals 60°, AB = 10 m and AD = 12 m, find the distance (diagonal) from A to C.
A. 11.96 m
B. 12.86 m
C. 13.76 m
D. 14.66 m
1124. The sides of triangle ABC are AB = 5, BC = 12 and AC = 10. Find the length of the line segment drawn from vertex A
and bisecting BC.
A. 5.15 B. 5.25 C. 5.35 D.5.45
1125. Express 1/2 (1 - √3 i ) in trigonometric form.
A. cis 120°
B. cis 240°
C. cis 300°
D. cis 315°
1126. If versinθ = x and 1 – sinθ = ½ , find x if θ < 90°.
A. 0.124
B. 0.134
C. 0.154
D. 0.164
1127. Two points A and B, 150 m apart lie on the same side of a tower on a hill and in a horizontal line passing directly
under the tower. The angles of elevation of the top and bottom of the tower viewed from B are 42° and 34°
respectively and at A, the angle of elevation of the bottom is 10°. Find the height of the tower.
A. 7.3 m
B. 8.3 m
C. 9.3 m
D. 10.3 m
1128. A point P is at a distance of 4, 5 and 6 from the vertices of an equilateral triangle of side of x. Find x.
A. 8.5 B. 9.5 C. 7.5 D. 10.5
a.
b.
c.
d.
1129. A quadrilateral ABCD has its sides AB and BC perpendicular to each other at B. Side AD makes an angle of 45° with
the vertical while side CD makes an angle of 70° with the horizontal. If AB = 15 and BC = 10, find the length of side
CD.
A. 31.5 B. 51.5 C. 61.5 D. 41.5
1130. A clock has a minute hand 16 cm long and an hour hand 11 cm long. Find the distance between the outer tips of
the hands at 2:30 o’clock.
A. 19.6 cm
B. 20.6 cm
C. 21.6 cm
D. 22.6 cm
1131. If rcosxsiny = a, rcosxcosy = b and rsinx = c, find r.
A. √a² - b² - c² B. √a² + b² -c²
C. √a² - b² + c² D. √a²+b²+c²
1132. From the top of a tower 18 m high, the angles of depression of two objects situated in the horizontal line with the
base of the tower and on the same side, are 30 and 45 degrees. Find the distance between the two objects.
A. 13.18 m
B. 13.28 m
C. 13.38 m
D. 13.48 m
1133. The sum of the sines of two angles A and B is 3/2 while the sum of the cosines of the angles is √3 /2 . Find A.
A. 60°
B. 30°
C. 90°
D. 45°
C. ¾
D. 3/6
1134. Evaluate tan( Arcsec √5 – Arccot 2 )
A. 3/7
B. 3/5
1135. What is the greatest distance on the surface of the earth that can be seen from the top of Mayon volcano which
is 2.4 kilometers high if the radius of the earth is 6370 km ?
A. 159.7 km
B. 174.8 km
C. 179.7 km
D. 189.7 km
1136. A pole stands on a plane which makes an angle of 15° with the horizontal. A wire from the top of the pole is
anchored on a point 8 m from the foot of the pole. If the angle between the wire and the plane is 30 degrees, find the
length of the wire.
A. 10.93 m
B. 11.93 m
C. 12.93 m
D. 13.93 m
1137. If sin x + sin y = ½ and cos x – cos y = 1, find x.
A. 15°
B. 20°
C. 25°
D. 30°
1138. A tower standing on level ground is due north of point A and due east of point B. At A and B, the angles of
elevation of the top of the tower are 60° and 45° respectively. If AB = 20 , find the height of the tower.
a.
b.
c.
d.
A. 18.32 m
B. 17.32 m
C. 16.32 m
D. 15.32 m
C. 90°
D. 45°
1139. If cot(80° - x/2) cot(2x/3) = 1, find x.
A. 30°
B. 60°
1140. If Arctan z = x/2, find cos x in terms of z.
A. (1 + z²) / (1 - z²)
B. (1 - z²) / (1 + z²)
C. (z² + 1) / (z² - 1)
D. (z² - 1) / (z² + 1)
1141. A flagstaff stands on the top of a house 15 m high. From a point on the plane on which theehouse stands., the
angles of elevation of the top and bottom of the flagstaff are found to be 60° and 45° respectively. Find the height
of the flagstaff.
A. 10.98 m
B. 11.87 m
C. 12.76 m
D. 13.25 m
1142. Two observers 100m apart and facing each other on a horizontal plane, observer at the same time the angles of
elevation of a balloon in their vertical to be 58° and 44°. Find the height of the balloon .
A. 60.23 m
B. 59.34 m
C. 61.31 m
D. 58.75 m
1143. From a point outside an equilateral triangle, the distances of the vertices are 10 m, 18 m and 10 m respectively.
Find the side of the triangle.
A. 19.94 m
B. 20.94 m
C. 21.94 m
D. 22.94 m
1144. A spherical triangle which contains at least one side equal to a right angle is called
A. a right triangle
B. a polar triangle
C. an isosceles triangle
D. a quadrantal triangle
1145. If A, B and C are the angles of a spherical triangle, then which of the following is true ?
A. 180° < A + B + C < 360°
B. 180° < A + B + C < 540°
C. 0° < A + B + C < 360°
D. 0° < A + B + C < 180°
1146. The angular distance of the horizon from the zenith is equal to how many degrees ?
A. 45° B. 60° C. 90° D. 180°
1147. The point on the celestial sphere directly above the observer is called the
A. zenith
B. nadir C. pole D. azimuth
1148. The small circle parallel to the equator is called the
a.
b.
c.
d.
A. equinox
B.parallel of latitude C.meridian
D.horizon
1149. If a, b and c are the sides of a spherical triangle, then
A. a + b + c < 180°
B. a + b + c < 360°
C. a + b + c < 540°
D. a+b+c< 90°
1150. The point on the celestial sphere diametrically opposite the zenith is called the
A. south pole B. nadir
C. azimuth
D. north pole
1151. It is the angular distance of a heavenly body from the celestial equator.
A. declination B. altitude
C. latitude
D. colatitude
1152. At sunset or at sunrise, the astronomical triangle is
A. an isosceles triangle
B. a quadrantal triangle
1153. The azimuth angle is always less than
C. a right triangle
D. an oblique triangle
A. 90° B. 180° C. 360° D. 540°
1154. A great circle which passes through the celestial poles and a heavenly body B is called the ________ of B.
A. vertical circle
B. hour circle C. longitude
D. horizon
1155. The angular distance of a point on the celestial sphere from the horizon is called its
A. longitude
B. altitude
C. latitude
D. polar distance
1156. It is the angle at the zenith from the upper branch of the observer's meridian toward the east to the vertical circle
of the heavenly body.
A. quadrantal angle C. hour angle
B. polar angle D. azimuth
1157. The zenith distance of a star is the complement of its
A. declination C. altitude
B. polar distance
D. latitude
a.
b.
c.
d.
1158. Which of the following given sets of parts of a spherical triangle is possible in order to define the triangle ?
A. A = 55°, B = 65°, C = 60°
B. a = 110°, b = 135°, c = 130°
C. A = 160°, B = 65°, C = 90°
D. a = 120°, b = 150°, c = 60°
1159. The complement of the declination of a star is called the
A. polar distance
C. longitude
B. zenith distance
D. altitude
1160. A 90-degree arc on the terrestrial sphere is equal to how many nautical miles ?
A. 3400
B. 4400 C. 5400 D. 6400
1161. How far in statute miles is a place at latitude 40° N from the equator ?
A. 2764.8
B. 2846.7
C. 2684.7
D. 2486.7
1162. Find the distance in nautical miles between A ( 40°30'N, 60°E ) and B (80°20'S, 60°E)
A. 6250
B. 7250 C. 8250 D. 9250
1163. Express 82°26' in nautical miles.
A. 4946
B. 4694 C. 4964 D. 4496
1164. If a place is 12°S of the equator, find its distance in nautical miles from the north pole.
A. 5130
B. 6120 C. 7110 D. 8100
1165. Find the difference in longitude between the following
places: M(34°54'33" N, 56°12'51" W) P(30°20'46" N, 87°18'20" W)
A. 31°05'29" B. 31°06'28"
C. 31°07'27"
D. 31°08'26"
1166. Find the difference in latitude between the places given in problem 22.
A. 4°32'46"
B. 4°33'47"
C. 4°31'48"
D. 4°30'49"
1167. If an observer is 840 nautical miles south of the equator, find his latitude.
A. 12° S
B. 13° S C. 14° S
D. 15° S
1168. How far apart are two points on the equator one in longitude 40° East and the other in longitude 150° West ?
a.
b.
c.
d.
A. 190°
B. 180°
C. 170°
D. 160°
C. 47°45'58"
D. 47°58'45"
1169. Express 3^h 11^m 55^s in angle units.
A. 45°47'58"
B. 58°47'45"
1170. Express 260°34' in time units.
C. 17^h 26^m 21^s
D. 17^h 21^m 26^s
1171. The plane of a small circle on a sphere of radius 25 cm is 7 cm from the center of the sphere. Find the radius of
the small circle.
A. 17^h 22^m 16^s
B. 17^h 16^m 22^s
A. 22 cm
B. 23 cm
C. 24 cm
D. 25 cm
1172. Find the area of a spherical triangle ABC on the surface of a sphere of raidus 10 where A = 119°37', B = 38°43' and
C = 34°23'.
A. 23.18
B. 22.19
C. 21.16
D. 24.13
1173. An hour-angle of one hour is equal to
A. 14°
B. 13°
D. 16°
C. 15°
1174.The plane of a small circle on a sphere of radius 10 cm is 5 cm from the center of the sphere. Find the area of the
small circle.
A. 55π B. 65π C. 75π D. 85π
1175. If the radius of the earth is 3960 miles, find the radius of a parallel of latitude 50° north.
A. 2445.44 mi B. 2554.44 mi C. 2455.44 mi D. 2545.44 mi
1176. Use Napier's rule to find a formula for finding angle B of a right spherical triangle when angle A and side c are
given.
C. cot B = cos c tan A
A. tan B = cos c tan A
B. cot B = sin c tan A
D. tan B = sin c cot A
1177. Given a right triangle with angles A = 63°15' and B = 135°34'. Find side b.
A. 134.1°
B. 143.1°
C. 131.4°
D. 141.3°
1178. The two sides of a right spherical triangle are 86°40' and 32°41'. Find the angle opposite the first given side.
A. 88°12'
B. 87°11'
C. 86°10'
D. 85°09'
1179. If the angles of a spherical triangle are A = 74°21' , B = 83°41' and C = 58°39', find side c.
A. 55°54'
a.
b.
c.
d.
B. 54°55'
C. 45°55'
D. 55°45'
1180. The sides of an oblique spherical triangle ABC are given as follows: a = 51°31' , b = 36°47'and c = 80°12'. Find A.
A. 32.35°
B. 33.45°
C. 34.55°
D. 35.56°
1181. Find the distance of Manila(14°36' N, 121°05' E) from Hongkong(22°18' N, 114°10' E) in kilometers.
A. 1123.42 km B.1124.32 km C.1231.24 km D.1321.42km
1182. If a boat sails N 30° W until the departure is 20 miles, what distance does it sail?
A. 55 mi
B. 50 mi
C. 45 mi
D. 40 mi
1183. A ship in latitude 50° N sails 80 nautical miles due East. Find the resulting change in longitude.
A. 2.05° E
B. 2.07° E
C. 2.09° E
D. 2.03° E
1184. Find the longitude of an observer if his local apparent time is 10:36:41 AM and the local Greenwich time is
4:23:12 AM.
A. 93°22'15" E
C. 91°22'15" E
B. 92°22'15" E
D. 90°22'15" E
1185. A ship in latitude 32° N sails due East intil it has made good a difference in longitude of 2°35' . Find the departure.
A. 128.42 nm B. 129.43 nm
C. 130.44 nm
D. 131.45 nm
1186. Given a spherical triangle ABC with a = 68°27' , b = 87°32' and C = 97°53'. Find c.
A. 96.41°
B. 95.14°
C. 94.61°
D. 93.65°
1187. Find the area of a spherical triangle on the surface of a sphere of radius 10 where a =
140°30', b = 70°15' and C = 116°45'
A. 301.7
B. 370.2
C. 300.7
D. 307.1
1188. If the difference in longitude between two places A and B on the earth is 50° and their latitudes are each 30° N.
Find the distance AB in nautical miles.
A. 2589
B. 2598 C. 2985 D. 2895
1189. A ship leaves A(45°15' N, 140°38' W) and arrives at a place B(48°45' N, 137°12' W). Find the distance AB in
nautical miles using middle latitude sailing.
B. 140.47
C. 140.45
D. 140.43
1190. An arc of one degree on the surface of the earth is approximately equal to how many statute miles ?
A. 140.49
A. 67.1 B. 68.1 C. 69.1 D. 70.1
a.
b.
c.
d.
1191. How many miles away is Manila(14°36' N, 121°05' E) from San Francisco(37°48' N, 122°24' W) ?
A. 7051
B. 8051 C. 9051 D. 10051
1192. A ship sails on a course between south and east making a difference in latitude of 13 nautical miles and a
departure of 20 nautical miles. Find the course of the ship.
C. 56°58'34" E
A. 54°56'43" E
B. 55°53'84" E
D. 58°54'36" E
1193. Leaving point A(49°57' N, 15°16' W) , a ship sails between south and west till the departure is 38 nautical miles
and the latitude is 49°38' N. Find the distance traveled.
A. 42.49 n miles
B. 43.48 n miles
C. 44.47 n miles
D. 45.46 n miles
1194. Find the initial course of a flight from Manila(14°36' N, 121°05' E) to Tokyo(35°40' N, 139°46' E).
A. 35°06'
B. 36°05'
C. 30°56'
D. 30°65'
1195. Given a quadrantal triangle with B = 117°54', a = 95°42' and c = 90°. Find angle A.
A. 95.64°
B. 96.46°
C. 97.54°
D. 94.56°
1196. The initial course of a ship sailing from a place at latitude 40°40' N and longitude 73°58' W is due east. After it has
sailed 600 nautical miles on a great-circle track, find its latitude.
A. 37°54' N
B. 38°54' N
C. 39°54' N
D. 36°54' N
1197. If an airplane is to fly from Manila ( 14°36' N, 121°05' E) to Hongkong(22°18' N, 114°10' E) at an average speed of
200 nautical miles per hour, how long should the trip take ?
A. exactly 3 hours
C. almost 3 hours
D. about 3 hours
B. less than 3 hours
1198. Find the local apparent time of sunrise at Paris ( lat 48°50' N) when the declination of the sun is 14°38'.
C. 4:50:41 AM
A. 4:40:31 AM
B. 4:45:30 AM
D. 4:55:40 AM
1199. Find the local apparent time when an observer at latitude 37°52' N finds that the sun's altitude in the eastern sky
is 50°10' and the sun's declination is 12°30' N.
A. 9:46:51 AM C. 9:56:45 AM
B. 9:45:56 AM D. 9:41:56 AM
1200. An airplane leaves Guam ( 13°24' N, 144°38' E) with an initial course of 36°40' for a great-circle track. Locate the
point on the great-circle track which is nearest to the north pole.
A. (54°09' N, 80°12' W)
a.
b.
c.
d.
C. (45°10' N, 81°02' W)
B. (59°04' N, 82°10' W) D. (49°05' N, 80°21' W)
1201. The declination of a star is 22°; its hour angle is 15°10' and the place of observation is Berlin ( 52°32' N, 13°25' E).
Find the altitude of the star.
A. 56.32°
B. 57.32°
C. 58.32°
D. 59.32°
1202. At 8:56 AM, the altitude and declination of the sun are found to be 36°18' and 14°35' respectively. If the
observation is done in the northern hemisphere, find the latitude of the place of observation.
A. 52°56' N
B. 53°57' N
C. 54°58' N
D. 55°59' N
1203. An airplane flew from Manila (14°36' N, 121°05' E) at an average speed of 300 mph on a course S 32° E. At what
point will it cross the equator ?
A. 130°02' E
B. 140°03' E
C. 150°04' E
D. 160°05' E
1204. A ship sails from A( 38° N, 120° W) on a course 300° for a distance of 140 nautical miles to point B. Find the
position of B by middle latitude sailing method.
A. (107° N, 125°44' W) C. (108° N, 126°55' W)
B. (106° N, 126°54' W) D. (109° N, 127°45' W)
1205. Find the azimuth of a star at 5:30 PM at a place whose latitude is 41° if the star's declination is 24°.
A. 284.18°
B. 274.18°
C. 264.18°
D. 254.18°
1026.Which of the following statements is false ?
A. The diagonals of a rhombus are perpendicular to each other.
B. The diagonals of a rectangle are equal.
C. The diagonals of a rhombus are equal.
D. The diagonals of a parallelogram bisect each other.
1207. The angle inscribed in a semicircle is
A. an obtuse angle
B. an acute angle
C. a straight angle
D. a right angle
1208. Which of the following points is equidistant from the vertices of a triangle ?
A. incenter
B. centroid
C. orthocenter
D.circumcenter
1209. The point of intersection of the altitudes of a triangle is called the
a.
b.
c.
d.
A. incenter
B. centroid
C. orthocenter
D.circumcenter
1210. The point of concurrency of the angle bisectors of a triangle is called the
A. incenter
B. centroid
C. orthocenter
D.circumcenter
1211. The point inside a triangle that is equidistant from its sides is called the
A. incenter
C. orthocenter
B. centroid
D.circumcenter
1212. The point of intersection of the medians of a triangle is called
A. incenter
B. centroid
C. orthocenter
D.circumcenter
1213.The line segment which joins the midpoints of two sides of a triangle is parallel to the third side and is what part of
the third side ?
A. one half
B. one third
C. one fourth
D. two thirds
1214. The sum of the interior angles of a convex polygon of n sides is equal to how many right angles ?
A. 2(n-1)
B. 2(n-2)
C. 2(n-3)
D.2(n-4)
1215. A convex polygon is a polygon each interior angle of which is less than
A. 45° B. 60° C. 180° D. 90°
1216. Which of the following points is two thirds of the distance from each vertex of a triangle tothe midpoint of the
corresponding opposite side ?
A. incenter
B. centroid
C. orthocenter
D.circumcenter
1217. It is a quadrilateral two and only two of whose sides are parallel
A. rectangle
B. rhombus
C. trapezoid
D. parallelogram
1218. It is a quadrilateral whose four sides are equal and with no angle equal to a right angle.
A. rectangle
B. rhombus
C. trapezoid
D. parallelogram
1219. The area of a circle is 6 times its circumference. What is its radius ?
a.
b.
c.
d.
A. 12 B. 11
C. 10
D. 13
1220. In a circle of radius 6, a sector has an area of 15 pi. What is the length of the arc of the sector ?
A. 3 pi B. 4 pi C. 5 pi D. 6 pi
1221. If the length of a side of a square is increased by 100%, its perimeter is increased by
A. 100 %
B. 150 %
C. 200 %
D. 250 %
1222. The side of a regular hexagon measures 10 cm. The radius of the circumscribing circle is
A. 8 cm
B. 10 cm
C. 12 cm
D. 14 cm
1223. The median of a trapezoid is 8 and one base is 5. How long is the other base ?
A. 13 B. 12
C. 11
D. 10
1224. What is the value of θ in the figure ?
A. 20° C. 30°
B. 10° D. 15°
1225. The area of the triangle inscribed in a circle is 40 sq. cm. and the radius of the circumscribed circle is 7 cm. If two
sides of the triangle are 8 cm and 10 cm, find the third side.
A. 10 cm
B. 12 cm
C. 13 cm
D. 14 cm
1226. The altitude of an equilateral triangle is 4. Find the length of each side.
A. 3.62
B. 4.62 C. 5.62 D. 6.62
1227. Find the side of a square whose area is equal to that of a rectangle with sides 32 and 18 cm.
A. 21 cm
B. 22 cm
C. 23 cm
D. 24 cm
1228. A railroad curve is to be laid out on a circle. What radius should be used if the tract is to change direction by 25° in
a distance of 36 m ?
A. 82.51 m B. 81.52 m
a.
b.
c.
d.
C. 85.21 m
D. 81.25 m
1229. Find the area of a rhombus whose diagonals are 32 cm and 40 cm.
A. 540 cm²
B. 340 cm²
C. 640 cm²
D. 440 cm²
1230. The altitude of a triangle is half the base. Find the base if the area is 64.
A. 15 B. 16 C. 17 D. 18
1231. Find the area of a triangle whose sides are 9, 12 and 15.
A. 54 B. 53 C. 52 D. 51
1232. An isosceles trapezoid has two base angles of 45° and its bases are 6 and 10. Find its area.
A. 12 B. 14 C. 16 D. 18
1233. Find the altitude of a trapezoid of area 180 cm² if the bases are 16 and 14 cm.
A. 11 cm
B. 12 cm
C. 13 cm
D. 14 cm
1234. Find the area of a sector of a circle of radius 10 cm and whose central angle is 15°.
A. 193.32 cm² B. 194.33 cm² C. 195.34 cm² D. 196.35 cm²
1235. Find the length of an arc of a circle of radius 20 which subtends a central angle of 30°.
A. 10 pi/3
B. 11 pi/3
C. 13 pi/3
D. 14 pi/3
1236. Find the length of a chord which is 2 units from the center of a circle of radius 6 units.
A. 6√2 B. 7√2 C. 8√2 D. 9√2
1237. How many sides has a convex polygon if the sum of the measure of its angles is 1080°?
A. 8 B. 7
C. 6
D. 5
1238. What is the measure of each interior angle of a regular pentagon ?
A. 106°
B. 109° C. 107° D. 108°
1239. What is the radius of a circle if its circumference is equal to its area ?
A. 4 B. 3
C. 2
D. 1
1240. What is the radius of a circle if the length of a 72° arc is 4π ?
A. 11π B. 10π C. 9π D. 8π
1241. Find the area of a parallelogram of sides 15 and 16 if one of its angles is 60°.
A. 206.82
B. 207.85
C. 208.81
D. 205.83
1242. Each side of a rhombus is 7 and one angle is 42° . What is its area ?
A. 30.69
B. 31.59
C. 32.79
D. 33.89
1243. In triangle ABC, if a = 10 and b = 12 and angle C = 150° , find the area of the triangle.
A. 30 B. 31 C. 32 D. 33
a.
b.
c.
d.
1244. The diagonals of a rhombus are 6 cm and 8 cm long. Find the perimeter of the rhombus.
A. 20 cm
B. 24 cm
C. 22 cm
D. 26 cm
1245. The angles between the diagonals of a rectangle is 30° and each diagonal is 12 cm long.
Find the area of the rectangle.
A. 26 cm²
B. 36 cm²
C. 46 cm²
D. 56 cm²
1246. The sides of triangle ABC are a = 14, b = 12 and c = 10. Find the length of the median from vertex A to side a.
A. 8.34
B. 8.44 C. 8.54 D.8.64
1247. The minute hand of a large clock is 2 m long. Find the distance traveled by the tip of the minute hand in 5
minutes.
A. pi/4B. pi/2 C. pi/6 D. pi/3
1248. Find the area of a parallelogram whose sides are 128 and 217 if an included angle is 136°.
A, 16942.38
B. 17492.83
C. 19294.83
D. 18249.38
1249. The area of a sector of a circle, having a central angle of 60° is 24 pi. Find the radius of thecircle.
A. 11 B. 12 C. 13 D. 14
1250. Two circles, each of radius 6 units, have their centers 8 units apart. Find the length of their common chord.
A. 2√5 B. 3√5 C. 4√5 D. 5√5
1251. What is the apothem of a regular polygon having an area 225 sq. cm. and a perimeter 60 cm?
A. 7.5 cm
B. 6.5 cm
C. 8.5 cm
D. 4.5 cm
1252. Find the area of a regular hexagon of side 3 cm.
A. 22.28 cm² B. 23.38 cm² C. 24.48 cm² D. 25.58 cm²
1253. A triangle has sides 3, 6 and 9. Find the shortest side of a similar triangle whose longest side is 15.
A. 6 B. 10 C. 8
D. 5
1254. The perimeter of an octagon is 32 and its longest side is 6. What is the longest side of a similar octagon whose
perimeter is 24 ?
A. 3.5 B. 4
C. 4.5 D. 5
a.
b.
c.
d.
1255. In the figure AB = AC. The value of θ is
A. 31 C. 33
B. 32 D. 34
1256. A hexagon is circumscribed about a circle of radius 5. If the perimeter of the hexagon is 38, what is the area of the
hexagon ?
A. 75 B. 65
C. 85
D. 95
1257. The circumference of two circles are 6π and 10π. What is the ratio of their areas ?
A. 9/25
B. 8/25 C. 7/25 D.6/25
1258. If a regular polygon has 54 diagonals, then it has how many sides ?
A. 10 B. 11 C. 14 D. 12
1259. If AB is parallel to CD where CD is the diameter of the circle as shown in the figure, find angle θ.
A. 20°
B. 10°
C. 25°
D. 15°
1260. What is the diameter of a circle that is circumscribed about an equilateral triangle of side 7.4 cm.
A. 8.64 cm
B. 8.54 cm
C. 9.54 cm
D. 9.64 cm
a.
b.
c.
d.
1261. If the perimeter of a rhombus is 40 and one of its diagonals is 12, find the other diagonal.
A. 16 B. 15 C. 18 D. 17
1262. Given a circle as shown. The length of arc AB is
A. 1.527
B. 1.725
C. 1.257
D. 1.275
1263. Find the area of the annulus bounded by the inscribed and circumscribed circles of an equilateral triangle with a
side of length 6.
A. 11π B. 8π C. 10π D. 9π
1264. Find the area of a regular octagon inscribed in a circle whose radius is 10 cm.
A. 822.8 cm² B. 282.8 cm² C. 828.2 cm² D. 228.8 cm²
1265. In the figure shown, find the shaded area.
A. 4π
B. 5π
C. 6π
a.
b.
c.
d.
D. 7π
1266. If the perimeter of a regular hexagon is 24, what is the apothem ?
A. 3√3 B. 4√3 C. 2√3 D. 5√3
1267. The ratio of the sum of the exterior angles to the sum of the interior angles of a polygon is 1:3. Identify the
polygon.
A. hexagon
B. heptagon
C. octagon
D. nonagon
1268. A circular sector has a radius of 6 cm and whose central angle is 60°. If it is bent to form a right circular cone, the
radius of the cone is
A. 1 cm
B. 2 cm C. 3 cm D. 4 cm
1269. A square is inscribed in a 90° sector of a circle as shown. Find the area of the shaded region.
A. 1.214
B. 1.412
C. 1.124
D. 1.142
1270. A regular octagon is inscribed in a circle of radius 6. Find the perimeter of the octagon.
A. 34.54
B. 35.64
C. 36.74
D. 37.84
1271. If four angles of a pentagon have measures 100°, 96°, 87° and 97°, find the measure of the fifth angle.
A. 150°
B. 160° C. 140° D. 130°
a.
b.
c.
d.
1272. If the sum two exterior angles of a triangle is 230°, find the measure of the third exterior angle.
A. 130°
B. 120° C. 110° D. 100°
1273. Given are two concentric circles with line segment AB = 10 cm which is always tangent tothe small circle. Find the
area of the shaded region (see figure).
A. 50 pi cm²
B. 45 pi cm²
C. 25 pi cm²D. 30 pi cm²
1274. A circle whose radius is 10 cm is inscribed in a regular hexagon. The area of the hexagon is
A. 346.4 cm² B. 634.4 cm² C. 364.4 cm² D. 436.6 cm²
1275. The area of a parabolic segment having a base width of 10 cm and a height of 27 cm is
A. 270 cm²
B. 150 cm²
C. 210 cm²
D. 180 cm²
1276. A side of a regular hexagon is 6. What is the circumference of its circumscribed circle?
A. 12 pi
B. 11 pi C. 13 pi D. 10 pi
1277. Two chords PQ and RS of a circle meet when extended through Q and S at a point T. If QP= 7, TQ = 9 , TS = 6, find
SR.
A. 16 B. 17
C. 18 D. 19
1278. What is the angle at the center of a circle if the subtending chord is equal to two thirds of the radius.
A. 39.95°
B. 38.94°
C. 37.93°
D. 36.92°
1279. The area of a rhombus is 250 and one of the angles is 37°25'. What is the length of each side?
A. 20.18
B. 20.28
C. 20.38
D. 20.48
a.
b.
c.
d.
1280. If in triangle ABC, A = 76°30', B = 81°40' and c = 368, find the diameter of the circumscribed circle.
A. 989.5
B. 959.8
C. 395.8
D. 958.9
1281. Given a parallelogram ABCD such that AB = 7, AC = 10 and angle BAC = 36°07'. Find the length of BC.
A. 4.992
B. 5.992
C. 6.992
D. 7.992
1282. What is the diameter of the circle that is circumscribed about an isosceles triangle whose vertical angle is 18° and
the sum of the two equal sides is 18 units ?
A. 7.11
B. 8.11 C.9.11 D.10.11
1283. The diagonals of a quadrilateral are 34 and 56 intersecting at an angle of 67°. Find its area.
A. 837.62
B. 863.72
C. 826.37
D. 876.32
1284. Find the radius of a circle in which is inscribed a regular nonagon whose perimeter is 417.6 cm.
A. 68.37 cm B. 67.83 cm C. 63.87 cm
D. 68.73 cm
1285. If each interior angle of a regular polygon has a measure of 160°, how many sides has the polygon ?
A. 16 B. 17 C. 19 D. 18
1286. The sides of a right triangle are a, b and c where c is the hypotenuse. Find the radius of the circle that is inscribed
in the triangle.
A. 1/2 (a+b+c) C. 1/2(a-b+c)
B. 1/2(a+b-c) D. 1/2(a-b-c)
1287. Two sides of a parallelogram are 20 and 30 and the included angle is 36°. Find the length of the longer diagonal.
A. 74.65
B. 64.75
C. 57.46
D. 47.65
1288. The sides of a triangle are 17, 21 and 28. Find the length of the line segment bisecting the longest side and drawn
from the opposite angle.
A. 11 B. 12 C. 13 D. 14
1289. Two tangent circles of radii 6 and 2 have a common external tangent as shown in the figure. Find the length of
this external tangent.
A. 4√3
B. 5√3
C. 6√3
D.
7√3
1290. PQ and RS are secants of a circle which when extended beyond Q and S at a point T outside the circle. Given that
arc PR = 105° and arc QS = 67°, find the angle QTS.
A. 18° B. 19° C. 20° D. 21°
a.
b.
c.
d.
1291. A bridge across a river is in the form of an arc of a circle. If the span is 40 ft and the midpoint of the arc is to be 8
ft higher than the ends, what is the radius of the circle?
A. 27 ft
C. 29 ft
B. 28 ft
D. 30 ft
1292. Find the angle formed by the secant and tangent to a circle if one intercepted arc is 30° more than the other and
the secant passes through the center of the circle.
A. 15° B. 16° C. 17° D. 18°
1293. Find the radius of a circle whose area is equal to the area of the annulus formed by two consecutive circles with
radii 5 and 13.
A. 10 B. 11
C. 12
D. 13
1294. A circle in inscribed in an equilateral triangle. If the circumference of the circle is 3, find the perimeter of the
equilateral triangle.
A. 9.246
B. 6.294
C. 2.946
D. 4.962
1295. Given a square ABCD as shown where E is the midpoint of side AD and G is the midpoint of side BC. If arc DF has
its center at E and arc FB has its center at G, find the shaded area.
A. 6
a.
b.
c.
d.
B. 8
C. 10
D. 12
1296. Two concentric circles each contains an inscribed square. The larger square is also circumscribed about the
smaller circle. If the circumference of the larger circle is 12 pi, what is the circumference of the smaller circle ?
A. 6√2 pi
B. 5√2 pi
C. 4√2 pi
D. 3√2 pi
1297. A regular cross is inscribed in a circle as shown. Find the area ( shaded) between the regular cross and the circle.
A. 43.44
B. 44.55
C. 45.66
D. 46.77
1298. Point P is a point on the minor arc AB of a circle with center at 0 as shown. If the angle APB is x degree and angle
A0E is y degrees, find an equation connecting x and y.
A. 2x - y = 360°
a.
b.
c.
d.
B. 2x + y = 360°
C. x - 2y = 360° D. x+2y=360°
1299. The quadrilateral ABCD is inscribed in a circle and its diagonal AC is drawn so that angle DAC = 34°, angle CAB =
38° and angle DBA = 65°. Find arc AB.
A. 96° B. 86° C. 76° D. 66°
1300. PORS is a quadrilateral that is inscribed in a circle. If angle SQR = 23° and the angle between the side SP and the
tangent line through the point P is 64°, find angle PSR.
A. 86° B. 87° C. 88° D. 89°
1301. The lines TA and TB are tangent to a circle at points A and B respectively. IF angle T = 42°and P is a point on the
major arc AB, find angle APB.
A. 69° B. 68° C. 67° D. 66°
1302. A secant and a tangent to a circle intersect an angle of 38° degrees. If the measures of the arc intercepted
between the secant and tangent are in the ratio 2:1, find the measure of the third arc.
A. 129°
B. 130° C. 131° D. 132°
1303. Compute the difference between the perimeters of a regular pentagon and a regular hexagon if the area of each
is 12.
A. 0.31
B. 0.21 C.0.41 D.0.51
1304. The area of the sector of a circle having a central angle of 60° is 24π. Find the perimeter ofthe sector.
A. 34.4
B. 35.5 C. 36.6 D.37.7
1305. Find the common area of two intersecting circles of radii 12 and 18 respectively if their common chord is 14 long.
A. 34.19
a.
b.
c.
d.
B. 35.29
C. 36.39
D. 37.49
1306. In a parallelogram ABCD, the diagonal AC makes with the angle 27°10' and 32°43' respectively. If side AB is 2.8 m
long, what is the area of the parallelogram ?
A. 4.7 m²
B. 5.7 m²
C. 6.7 m²
D. 8.7m²
1307. The sum of the sides of a triangle is 100. The angle at A double that of B and the angle at Bis double that of C. Find
side c.
A. 41.5
B. 42.5 C. 43.5 D. 44.5
1308. A diagonal of a parallelogram is 56.38 ft long and makes an angle of 26°13' and 16°24' respectively with the sides.
Find the area of the parallelogram.
A. 595 ft²
B. 585 ft²
C. 575 ft²
D. 565 ft²
1309. Find the area of a regular five-pointed star that is inscribed in a circle of radius 10.
A. 121.62
B. 112.26
C. 122.16
D. 126.21
1310. What is the difference in the areas between an inscribed and a circumscribed regular octagon if the radius of the
circle is 6?
A. 15.27
B. 16.37
C. 17.47
D. 18.57
1311. If BC = 2(AB), what fraction of the circle is shaded?
A. 1/4 B. 1/3 C. 1/2 D. 1/5
a.
b.
c.
d.
1312. In the figure, the small circle is tangent to 4 circular arcs. Find the area of the shaded region if the radius of the
larger circle is 10.
A. 34.94
B. 35.49
C. 31.94
D. 32.49
1313. A regular five-pointed star is inscribed in a circle of radius b cm. Find the area between thecircle and the star.
A. 4.04 b²
B. 3.03 b²
C. 1.01 b²
D. 2.02 b²
1314. From a point outside of an equilateral triangle, the distances of the vertices are 12, 20 and 12 respectively. Find
the length of each side of the triangle.
a.
b.
c.
d.
A. 23.95
B. 22.85
C. 21.78
D. 20.68
1315. Using the vertices of a square, four arcs are drawn as shown in the figure. If each edge is 10 units long, find the
shaded portion (common area).
A. 21.5
B. 31. 5 C. 41.5 D. 51.5
1316. Assuming that the earth is a perfect sphere of radius 6370 kilometers, a person at a point T on top of a tower 60
meters high looks at a point P on the surface of the earth. What is the approximate distance of P from T ?
A. 24.3 km
B. 25.4 km
C. 26.5 km
D. 27.6 km
1317. Each of four circles ( see figure ) is tangent to the other three. If the radius of each of the smaller circles is 3, what
is the radius of the largest circle ?
A. 6.46
a.
b.
c.
d.
B. 6.64 C. 4.64 D.4.46
1318. In the figure, if arc AB = 50°, arc BC = 80° and arc AD = 90°, find θ.
A. 85° B. 65° C. 95° D. 75°
1319. In the figure, if PB = 6, PC = 10, PA = 5 and θ = 30°, find the area of the quadrilateral
ABCD.
A. 21.5
a.
b.
c.
d.
B. 22.5 C. 23.5 D. 24.5
1320. The solid formed by revolving a circle about an external axis in its plane is called.
A. annulus
B. conoid
C. torus
D. prismatoid
1321. The intersection of two faces of a pyramid is called the
A. lateral edge
B. slant height C. altitude
D. hypotenuse
1322. It is a polyhedron of which one face is a polygon and the other faces are triangles which have a common vertex.
A. prism
B. pyramid
C. cone D. prismatoid
1323. The altitude of any of the lateral faces of a regular pyramid is called the
A lateral edge
B.altitude
C.median
D.slant height
1324. It is a polyhedron whose faces are all squares.
A. tetrahedron C. octahedron
B. hexahedron D.dodecahedron
1325. The dihedral angle is the angle between two intersecting
A.lines
B. arcs
C.planes
D.chords
1326. Which of the regular polyhedrons has faces that are regular pentagons ?
A. tetrahedron
C. octahedron
B. dodecahedron
D. icosahedron
1327. If the base of a solid is a circle and every section perpendicular to the base is an isosceles triangle, the solid is
called
A. conicoid
B. prismoid
C. conoid
D. astroid
1328. The radius of a sphere that is inscribed in a regular hexahedron of edge e is equal to
a.
b.
c.
d.
A. e/2 B. e/3 C. e/4 D. e/5
1329. It is a polyhedron of which two faces are equal polygons in parallel plane and the other faces are parallelogram.
A.tetrahedron
B. prism
C.pyramid
D.prismoid
1330. A spherical wooden ball 15 cm in diameter sinks to a depth of 12 cm in a certain liquid.
The area exposed above the liquid is
A. 54 pi cm²
B. 15 pi cm²
C. 45 pi cm²
D. 35 pi cm²
1331. What is the total area of a cube whose edge is 5 cm?
A. 150 cm²
B. 145 cm²
C. 140 cm²
D. 135 cm²
1332. Find the volume of the frustum of a right circular cone whose altitude is 6 and whose base radii are 2 and 3.
A. 35π B. 36π C. 37π D. 38π
1333. The angle of a lune is 60° and the radius of the sphere is 15 cm. Find the volume of the spherical wedge whose
base is the given lune.
A. 750π cm³
B. 700π cm³
C. 650π cm³
D. 600π cm³
1334. A sphere of radius R is inscribed in a cube of edge e. What is the ratio of the volume of the sphere to the volume
of the cube?
A. 0.6523
B. 0.5236
C. 0.3652
D. 0.2635
1335. The slant height of a right circular cone is 13 and the altitude is 12. Find the radius of the base.
A. 8
B. 7
C. 6
D. 5
1336. A hemispherical bowl of radius 10 cm is filled with water to a depth of 5 cm. Find the volume of the water.
A. 615π/3 cm³ B. 620π/3 cm³ C. 625π/3 cm³ D. 630π/3 cm³
1337. The area of a lune is 4π m² and the radius of the sphere is 3 m. Find the angle of the lune.
A. 40° B. 45° C. 50° D. 55°
1338. The volume of a sphere whose diameter is 20 cm is
A. 4198.87 cm³ B. 4179.88 cm³ C. 4188.79 cm³ D.4187.89 cm³
a.
b.
c.
d.
1339 Find the length of the diagonal of a rectangular box whose edges are 6, 8 and 10.
A. 7√2
B. 8√2
D. 10√2
C. 9√2
1340. Find the area of the base of a prism whose volume is 516.6 cu. ft and whose height is 16.4 in.
A.
372 ft² B. 374 ft²
C. 376 ft²
D. 378 ft²
1341. Find the slant height of a regular pyramid each of whose faces is enclosed by an equilateraltriangle with side 8.
A.
6.73
B. 6.93 C. 6.83 D. 6.63
1342. What is the volume of a pyramid whose altitude is 27 and whose base is a square 8 on a side ?
A.
756
B. 657 C. 576 D. 675
1343. A concrete pedestal is in the form of a frustum of a regular square pyramid whose altitude is 1.2 cm and base
edges 0.40 m and 0.70 m respectively. Find the volume of the pedestal.
A.
0.372 m³
B. 0.327 m³
C. 0.273 m³
D. 0.723 m³
1344. The base radii of the frustum of a cone are 6 cm and 10 cm respectively. Find the altitude of the frustum if its
volume is 1176π cu. cm ?
16 cm B. 17 cm
C. 18 cm
which its volume is equal to its surface area?
A.
A. 7
D. 19 cm 1345. What is the diameter of a sphere for
C. 5
B. 6
D. 4
1346. Find the volume of a spherical wedge whose angle is 36° on a sphere of radius 6 cm.
A. 28.8π cm³ B. 27.7π cm³ C. 26.6π cm³ D. 25.5π cm³
1347. Find the volume of a right circular cone whose base radius is 8 cm and whose altitude is 15 cm.
A. 320 pi cm³ B. 330 pi cm³
C. 340 pi cm³
D. 350 pi cm³
1348. The volume of a sphere of radius 1.2 m is
A. 8.372 m³
B. 2.783 m³
C. 3.872 m³
D. 7.238 m³
1349. The volume of a square pyramid is 384 cm³ and its altitude is 8 cm. How long is an edge of the base?
A. 11 cm
B. 12 cm
C. 13 cm
D. 14 cm
1350. Find the altitude of a right prism flow which the area of the lateral surface is 338 and the perimeter of the base is
13.
a.
b.
c.
d.
A. 25 B. 26
C. 27
D. 28
1351. A conical tank is 10.5 m deep and its circular top has a radius of 5 cm. How many liters of water will it hold?
A. 260500π
B. 261500π
C. 262500π
D. 263500π
1352. Find the diameter of a sphere whose surface area is 324π.
A. 16
B. 17
C. 18
D. 19
1353. Find the area of a zone of a sphere whose radius is 6 if the altitude of the zone is 2.
A. 21 pi
B. 22 pi C. 23 pi D. 24 pi
1354. The volume of a 10-cm high conical paper weight is 180 cm³. The radius of the base is
A. 4.15 cm
B. 4.17 cm
C. 4.19 cm
D. 4.21 cm
1355. The volume of the frustum of a cone which is 25 cm high and whose base radii are 7.5 cm and 5 cm long
respectively is
A. 3108.87 cm³ B. 3107.88 cm³ C. 3170.88 cm³ D.3180.78 cm³
1356. Find the volume of a cube whose total area is 384 cm².
A. 212 cm³
B. 312 cm³
C. 412 cm³
D. 512 cm³
1357. Find the total area of a tetrahedron 3 units on an edge.
A. 8√3
B. 9√3
C. 10√3
D. 11√3
1358. The volume of a pyramid is 256 cm³ and its altitude is 24 cm. Find its base area.
A. 52 cm²
B. 42 cm²
C. 32 cm²
D. 22 cm²
1359. The radii of the bases of the frustum of a right circular cone are 6 and 9 respectively and the altitude is 4. Find the
lateral area.
A. 75π B. 65π C. 95π D. 85π
1360. The altitude of a parallelepiped is 20 and the base is a rhombus with diagonals 10 and 16. Find the volume of the
parallelepiped.
A. 1500
B. 1600 C. 1700 D. 1800
1361. A sphere of radius r just fits into a cylindrical box. Find the empty space inside the box.
A. 2π r³/3
a.
b.
c.
d.
B. 8π r³/9
C. 4π r³/9
D. 20π r³/27
1362. Find the volume of a pyramid having a pentagonal base with sides each equal to 12 cm andan apothem of 3 cm.
The altitude of the pyramid is 36 cm.
A. 2660 cm³
B. 2770 cm³
C. 2880 cm³
D. 2990 cm³
1363. Find the volume of the frustum of a regular triangular pyramid whose altitude is 3 and whose base edges are 4
and 8 respectively.
A. 25√3
B. 26√3 C. 27√3 D. 28√3
1364. The lateral area of a right circular cone with a radius of 20 cm and a height of 30 cm is
A. 2265.43 cm² B. 2236.45 cm² C. 2245.63 cm² D.2253.46 cm²
1365. The base of a prism is a rectangle with sides 3 and 5. If its lateral area is 64, find its altitude.
A. 3
B. 4
C. 5
D. 6
1366. Find the number of degrees on a dihedral angle of a regular tetrahedron.
A. 68.33°
B. 69.43°
C. 70.53°
D. 71.63°
1367. Find the volume of a spherical cone in a sphere of radius 17 cm if the radius of the zone is 8 cm.
A.1126π/3 cm³B.1136π/3 cm³C.1146π/3cm³
D.1156π/3 cm³
1368. Find the volume of a regular square pyramid whose slant height is 10 and whose base edgeis 12.
A. 384 B. 374 C. 364 D. 354
1369. The base of a prism is a rhombus whose sides are each 10 cm and whose shorter diagonal is 12 cm. If the altitude
is 12 cm, find its volume.
A. 1132 cm³
B. 1142 cm³
C. 1152 cm³
D. 1162 cm³
1370. Find the volume of a triangular prism whose altitude is 20 cm and whose sides are 6 cm, 8 cm and 12 cm.
A. 426.61 cm³ B. 421.66 cm³ C. 461.26 cm³ D. 416.62 cm³
1371. Find the volume of the solid as shown.
A. 2330
a.
b.
c.
d.
B. 2440 C. 2660 D. 2770
1372. Find the volume of a spherical segment if the radii of the bases are 3 and 4 respectively and its altitude is 2.
A. 83.27
B. 87.32
C. 83.72
D. 82.73
1373. A stone is dropped into a circular tub 40 inches in diameter, causing the water therein to rise 20 inches. What is
the volume of the stone ?
B. 7000π in³
C. 8000π in³ D. 9000π in³
1374. The base of a right parallelepiped is a rhombus whose sides are each 10 cm long and one of whose angles is 60
degrees. If the altitude of the parallelepiped is 4 cm, find its volume .
A. 6000π in³
A. 100√3 cm³ B. 200√3 cm³ C. 300√3 cm³
D. 400√3 cm³
1375. Find the volume of the largest cube that can be out from a circular log whose radius is 30.
A. 76367.53
B. 75567.33
C. 73675.36
D. 77653.36
1376. Find the lateral area of a pyramid whose altitude is 27 cm and whose base is a square 8 cm on a side.
A. 437.62 cm² B. 436.72 cm² C. 432.76 cm² D. 427.63 cm²
1377. The diagonal of a cube is 2√3. Find its volume.
A. 9
B. 7
C. 8
D. 6
1378. Find the lateral area of the frustum of a regular square pyramid whose base edges are 6 and12 and whose
altitude is 4.
A. 150 B. 160 C. 170 D. 180
1379. If the radius of a sphere is 8 and if a plane passes through the sphere at a distance of 5 from its center. what is the
area of the circle of intersection ?
A. 38 pi
B. 39 piC. 40 pi D. 41 pi
1380. Find the lateral area of a right circular cone that can be inscribed in a cube whose volume is 64.
A. 28.1 B. 26.1 C. 24.1 D.22.1
a.
b.
c.
d.
1381. Find the lateral edge of a regular square pyramid whose slant height is 8 and whose base edge is 6.
A. 6.54 B. 7.54 C. 8.54 D.9.54
1382. The base edges of a triangular pyramid are 12, 14 and 16. If its altitude is 22, what is the volume of the pyramid ?
A. 594.64
B. 564.94
C. 544.69
D. 596.44
1383. The volume of the frustum of a right circular cone is 78 pi. The upper base radius is 2 and the lower base radius is
5. What is the altitude of the frustum ?
A. 5
B. 6
C. 7
D. 8
1384. The volume of a right circular cone having a slant height of 13 and altitude 12 is
A. 100π
B. 150π C. 200π D. 250π
1385. Find the lateral area of a regular triangular pyramid whose base edge is 4 and its lateral edge is 6.
A. 21√2
B. 22√2 C. 23√2 D. 24√2
1386. Find the height of a pyramid whose volume is 35 and whose base is a triangle with sides 4,7 and 5.
A. 11.72
B. 10.72
C. 8.72 D. 9.72
1387. The radii of the bases of the frustum of a right circular cone are 6 and 9 respectively and itsaltitude is 4. Find its
lateral area.
A. 75π B. 85π C. 95π D. 65π
1388. Find the volume of a sphere whose surface area is 64π.
A. 256π/3
B. 254π/3
C. 252π/3
D. 250π/3
1389. What is the area of a sphere if a zone on it having an area of 18 and has an altitude of 2?
A. 79 pi
B. 80 pi C. 81 pi
D. 82 pi
1390. A spherical bowl of radius 8 inches contains water to a depth of 3 inches. Find the volume of the water in the
bowl.
A. 199.72 in³
B. 197.92 in³
C. 179.29 in³
D. 192.27 in³
1391. The volume of a pyramid is 256 cm³ and its altitude is 24 cm. Find the area of its base.
B. 34 cm²
C. 31 cm²
D. 33 cm²
1392. Find the lateral area of a right circular cone if its slant height is 22 and the circumference of its base is 8.
A. 32 cm²
a.
b.
c.
d.
A. 55 B. 66
C. 77
D. 88
1393. What is the diameter of a sphere for which its volume is equal to its surface area ?
A. 5
B. 6
C. 7
D. 8
1394. The lateral area of a regular pyramid is 2048 and the perimeter of the base is 128. Find the slant height.
A. 42 B. 22
C. 32
D. 52
1395. The area of the base of a right circular cone is 144π . If its altitude is 14, find its slant height.
A. 18.44
B. 17.33
C. 16.22
D. 15.11
1396. Find the approximate change in the volume of a cube if each edge x of the cube is increased by one percent.
A. 0.02 x³
B. 0.03 x³
C. 0.04 x³
D. 0.05 x³
1397. The area of a diagonal section of a cube is 4√2 cm². Find the edge of the cube.
A. 3 cm
B. 2 cm C. 4 cm D. 1 cm
1398. Find the volume of the largest circular cylinder that can be inscribed in a cube whose volume is 64 cu. cm.
A. 13π cm³
B. 14π cm³
C. 15π cm³
D. 16π cm³
1399. The altitude of a square pyramid is 10 and a side of the base is 15. Find the area of a cross section at a distance of
6 from the vertex.
A. 81 B. 82
C. 83
D. 84
1400. The diameter of one solid ball is 3 times the diameter of another ball of the same material. If the weight of the
smaller ball is 250 pounds, what is the weight of the larger ball ?
A. 6957 lb
B. 6750 lb
C. 6507 lb
D. 6570 lb
1401. Find the volume of a regular tetrahedron whose edges are each equal to 6.
A. 16√2
B. 17√2 C. 18√2D. 19√2
1402. The lateral area of a regular pyramid is 514.5 and the slant height is 42. Find the perimeter of the base.
A. 24.5 B. 26.5 C. 22.5 D.28.5
a.
b.
c.
d.
1403. A wedge is cut from a circular tree whose diameter is 2 m by a horizontal plane up to the vertical axis and another
cutting plane which is inclined at 45 degrees from the previous plane.
The volume of the wedge is
A. 1/4 B. 1/2 C. 2/3 D. 3/4
1404. The zone of a spherical cone has a altitude of 2 cm and a radius of 4 cm. Find the volume of the spherical cone.
A. 115π/3 cm³ B. 110π/3 cm³ C. 105π/3 cm³ D. 100π/3 cm³
1405. The base of a prism is the triangle ABC with A = 35 degrees, B = 68 degrees and c = 25. If the altitude of the prism
is 10, find the volume of the prism.
A. 1607.5
B. 1705.6
C. 1507.6
D. 1076.5
1406. The capacities of two hemispherical tanks are in the ratio 64:125. If 4.8 kg of paint is required to paint the outer
surface of the smaller tank, then how many kg of paint would be needed to paint the outer surface of the larger
tank ?
A. 6.5 kg
B. 7.5 kg
C. 8.5 kg
D. 9.5 kg
1407. Find the volume of a sphere that is circumscribed about a cube of edge 4.
A. 30√3 π
B. 32√3 π
C. 34√3 π
D. 36√3 π
1408. A sphere is inscribed in a right circular cone. The slant height of the cone is equal to the diameter of its base. If
the altitude of the cone is 15, find the surface area of the sphere.
A. 125π
B. 120π C. 110π D. 100π
1409. The base of a tetrahedron is a triangle whose sides are 10, 24 and 26. If the altitude of the tetrahedron is 20, find
the area of a cross-section whose distance from the base is 15.
A. 9.5 B. 8.5 C. 7.5 D. 6.5
1410. If the length of each edge of a cube is increased by 3 cm, its volume is increased by 387 cucm. Find the length of
each edge of the original cube.
A. 5 cm
B. 6 cm C. 7 cm D. 8 cm
1411.The lateral area of a regular pyramid is 2048 and the perimeter of the base is 128. If its base is a regular octagon,
find the altitude of the pyramid.
A. 24.5
B. 25.5
C. 26.5
D. 27.5
1412. Find the area illuminated by a candle h meters from the surface of a ball r meters in radius.
a.
b.
c.
d.
A. (2πrh²) / (r+h)
B. (2πrh²) / (r-h)
C. (2πr²h) / (r+h)
D. (2πr²h) / (r-h)
1413. Find the volume of the frustum of a pyramid whose bases are regular hexagons with base edges 5 cm and 10 cm
respectively and the altitude is 15 cm.
A. 2273.31 cm³ C. 2327.13 cm³
B. 2171.33 cm³ D. 2713.32 cm³
1414. What is the volume of the cube if the number of cubic units in its volume is twice the number of square units in
its total surface area ?
A. 1827
B. 1287 C. 1872 D. 1728
1415. Find the lateral area of a regular hexagonal pyramid whose lateral edges are each 13 cm and whose base has
sides 10 cm each.
A. 350 B. 360 C. 370 D. 380
1416. The ratio of the volume of two spheres is 8:27. What is the ratio of their surface areas?
A. 2/9 B. 4/9 C. 5/9 D. 7/9
1417. Each edge of the upper base of the frustum of a regular quadrangular pyramid is 2 less than an edge of the lower
base. Find the edge of the lower base if the slant height is 10 and the total area is 160.
A. 3
B. 4
C. 5
D. 6
1418. Find the volume of a solid formed by revolving an equilateral triangle with side e about an altitude.
A. (√3π e³) / 24
C. (√2π e³) / 24
B. (√3π e³) / 12
D. (√2π e³) / 12
1419. If the diameter of a sphere is increased by 40 percent, by what percent is the volume increased ?
A. 144.7%
B. 147.4%
C. 177.4%
D. 174.4%
1420. The radii of two spheres are in the ratio 3:4 and the sum of their surfaces is 2500. Find the radius of the smaller
sphere.
A. 14 B. 15
C. 16
D. 17
1421. If the ratio of the lateral area of the frustum of a cone to its volume is 15:28, find the altitude of the frustum if its
base radii are 3 and 6 respectively.
A. 6
a.
b.
c.
d.
B. 5
C. 4
D. 3
1422. The lateral area of a right circular cone is 3 times the area of its base. Find the angle at which the slant height of
the cone is inclined with the base.
A. 71.35°
B. 72.15°
C. 70.53°
D. 73.25°
1423. The volume of a rectangular parallelepiped is 162. The three dimensions are in the ratio 1:2:3. Find the total area.
A. 198 B. 197 C. 196 D. 195
1424. The base edge of a square pyramid is 3 m and its altitude is 10 m. Find the area of a section parallel to the base
and 6 m from it.
A. 1.22 m²
B. 1.33 m²
C. 1.44 m²
D. 1.55 m²
1425. The area of the base of a pyramid is 25 and its altitude is 10. What is the distance from the base of a section
parallel to the base whose area is 9 ?
A. 4
B. 3
C. 5
D. 2
1426. The edge of a regular tetrahedron is 5. Find the edge of a cube which has the same volume as the tetrahedron.
A. 2.35 B. 2.45 C. 2.55 D. 2.65
1427. The segment of a paraboloid of revolution( see figure ) is a solid in which every section parallel to the base is a
circle the radius R of which is the mean proportional between the distance H from the vertex and the radius r of
the base. Find the volume of the segment of altitude h.
A. 1/2 πr²h
a.
b.
c.
d.
B. 1/3 πr²h
C. 1/2 πrh²
D. 1/3 πrh²
1428. A right circular cone whose slant height is 18 cm and the circumference of whose base is 6 cm is cut by a plane
parallel to the base such that the cone is cut off, has a slant height of 4 cm. Find the lateral area of the frustum
formed.
A. 48.3 B. 49.1 C. 50.2 D. 51.3
1429. A tank has the form of a cylinder of revolution whose diameter is 60 cm and whose height is 244 cm. The tank is
in horizontal position and is filled with water to a depth of 46 cm. Find the approximate number of liters of water
in the tank.
A. 566 B. 567 C. 568 D. 569
1430. A solid gas a circular base of radius 20. Find the volume of the solid if every section perpendicular to a certain
diameter is an equilateral triangle.
B. 14871.52
C. 17845.12
D. 15781.25
1431. In a cone of altitude h and elliptic base A, every section parallel to the base has an area Ay = Ay² / h² where y is
the distance from the vertex to the section ( see figure ). Find the volume of the elliptic cone.
A. 18475.21
A. πabh / 3
a.
b.
c.
d.
B. πabh / 2
C. πabh / 4
D. πabh / 5
1432. Find the total area of a regular hexagonal pyramid whose slant height is 5 ft and whose base is 4 ft.
A. 105.71 ft²
B. 107.15 ft²
C. 101.57 ft²
D. 110.75 ft²
1433. For the solid shown, every section perpendicular to the edge AB is a circle. If arc ACB is asemicircle of diameter
18, find the volume of the solid ( see figure ).
A. 342 pi
B. 423 pi
C. 432 pi
D. 243pi
1434. A solid consists of a hemisphere surmounted by a right circular cone. Find the vertical angle of the cone if the
volume of the conical and spherical portions are equal.
A. 51.13°
B. 52.13°
C. 53.13°
D. 54.13°
1435. The slant height of the frustum of a right circular cone makes an angle of 60° with the larger base. If the slant
height is 30 cm and the radius of the smaller base is 5 cm, find the volume of the frustum.
A. 15283.7 cm³ B. 14283.7 cm³ C. 13283.7 cm³ D.12283.7 cm³
1436. The lateral area of the frustum of a regular pyramid is 336 sq cm. If the lower base is a square having a side of 8
cm; the upper base is a square of side x cm and its slant height is 12 cm, find the value of x.
a.
b.
c.
d.
B. 4
A. 6
C. 7
D. 5
1437. If the area of the base of a regular hexagonal prism is 3√3 / 2 sq cm and the total area is a 45√3 sq cm, find the
volume of the prism.
A. 20.5 cm³
C. 21.5 cm³
B. 31.5 cm³
D. 30.5 cm³
1438. If a cylinder has a lateral area of 88 pi and a volume of 176 pi, what is its total area ?
A. 120 pi
B. 125 pi
C. 130 pi
D. 135 pi
1439. A rectangular prism has a width of 2 cm, a height of 4 cm and a length of 3√3 cm. If its volume is equal to the
volume of a cube with diagonal d, find the value of d.
A. 8 cm
B. 7 cm C. 6 cmD. 5 cm
1440. The axes of two right circular cylinders of equal radii 3 m long, intersect at right angles. Find the volume of their
common part.
A. 122 m³
B. 133 m³
D. 155 m³
C. 144 m³
1441. Which of the following statements is false ?
A. Any two integrals of a given function differ by a constant.
n
B. The integral of sec xdx where n is an odd integer requires integration by parts.
C. If f(x) is an even function, then the integral of f(x)dx from x = -a to x = a is equal to zero.
D. The key connection between the derivative and integral is known as the fundamental theorem of calculus.
1442. Which of the following differentials must be integrated by parts ?
A. (lnx/x)dx
B. sin²(3x)dx
C. x²cos(x³)dx D. (lnx)²dx
1443. The process of finding the function f(x) whose differential f'(x)dx is given, is called integration or
A. involution 1444.
B. evolution
C.antidifferentiation D. exponentiation
x
Evaluate ∫xe dx
x
x
x
B. xe – 1 + c
A. e (x-1) + c
C. e – x + c
x
D. xe – x + c
k
1445. For some constant k, the antiderivative of x is equal to
A. (x
k+1
)/(k+1)
B. [(x
k+1
)/(k+1)]+c
C. [(x
2k
) / 2k] +c
D. A or B
1446. The mathematician who first give a modern definition of the definite integral is
a.
b.
c.
d.
B. Leibniz
A. Riemann
C. Newton
D. Gauss
4
1447. To integrate ∫(xdx) / (1+x ) by the u-substitution method, let u =
A. 1 + x²
B. x²
C. 1 + x
4
D. x
4
1448. Which of the following is correct ?
3
4
C. ∫sin xdx = [(sin x) / 4] + c
A. ∫cos2xdx = -sin2x + c
x
2
x
D. ∫e cosxdx = e sinx + c
B. ∫sin2xdx = sin x + c
1449. Who proved that the area under a parabolic arch is 2bh/3 where b is the width of the base of the arch and h is the
height ?
A. Wallis
1
B. Newton
C. Riemann
D. Archimedes
C. -20/3
D. -28/3
C. lncoshu + c
D. lncothu + c
2
1450. Evaluate ∫ -1(x – 4) dx
A. -25/3
B. -22/3
1451. The antiderivative of tanhudu is
A. lnsinhu + c
B. lnsechu + c
2 2
1452. using the theorem of Pappus, find the volume of the torus generated by revolving the area of the circle x + y =
2
a about the line x = b where b > a.
2
A. 2π a b
B. 2π ab
2
2 2
C. 2π ab
2 2
D. 2π a b
3
1
1453. If f(x) = x – 1and g(x) = x – 1, evaluate ∫ 0 [f(x) / g(x) ] dx.
A. 11/6 B. 13/6 C. 10/6 D. 14/6
x
1454. Find the area bounded by the curve y = e , the lines x = -1, x = 1 and the x-axis.
A. 2.15 B. 2.25 C. 2.35 D. 2.45
2
2
1455. If the area bounded by y = x and y = 2 – x is revolved about the x-axis and a vertical rectangular element is
taken, the element of volume generated is a
A. disk B. washer
a.
b.
c.
d.
C. shell D. torus
5
5
5
5
1456. If ∫ -2 f(x)dx = 18, ∫ -2 g(x)dx = 5, and ∫ -2 h(x)dx = -11, evaluate∫ -2 [f(x)+g(x)h(x)]dx.
A. 32 B. 33
C. 34
D. 35
1457. Find the length of the curve y = coshx from x = -1 to x = 1.
A. 2.15 B. 2.25 C. 2.35 D. 2.45
2
1458. Evaluate ∫ -1 (2x-(2/x)+(x/2) dx
A. 2.3637
B. 2.3763
C. 2.3367
D. 2.6733
2
1459. If the area bounded by the parabola y = x and the line y = x is revolved about the x-axis, the volume of the solid
formed may be found by using which of the following methods ?
A. washer method only C. shell or washer method
B. washer or disk method
D. shell or disk method
n x^2
1460. The differential x e dx is integrable if n is
A. an even integer
B. an odd integer
C. any positive integer
D. any whole number
2
2
1461. If a vertical element of area is used in finding the area bounded by the parabolas y = x – 7 and y = 1 – x , then
the elemental area dA =
2
2
2
2
A. (2x – 8)dx B. (8 – 2x )dx C. (2x – 6)dx D. (6 – 2x )dx
x
1462. If ∫ 0 sin2ycos2ydy = ¼, then x is equal to
A. pi/2 B. pi/6 C. pi/3 D. pi/4
2
2
1463. If the area bounded by the ellipse 9x + 4y = 36 is revolved about the line 2x + y = 8 and a horizontal rectangular
element is taken, the element of volume generated is a
A. washer or circular ring
C. circular disk
B. cylindrical shell
D. none of A, B or C
a.
b.
c.
d.
1464. Which of the following cannot be evaluated by the power rule formula ?
2
A. ∫ (dx) / x (1 + (2/x))
3
C. ∫ (ln(x+1)dx) / (x+1)
B. ∫ (√1 + sinx)dx / (secx)
1465. Evaluate ∫
2
2
D. ∫ (x √x + 4 ) dx
2π 1
0∫ 0 rdrdθ
A. 3π/4
B. π/4
C. π/6
D. 2π/3
2
1466. Find the area bounded by y = x , the x-axis and the lines x = 1, x = 3.
B. 25/3
A. 26/3
C. 23/3
D. 20/3
2
1467.Evaluate the integral of xsin(x )dx from x = 0 to x = √π
A. -1
B. 0
C. -1/2
D. 1/3
3
1468. Find the volume of the solid generated by revolving about the x-axis, the area bounded by y = x , the x-axis and
the line x = 1.
A.π/3
B. π/5
1469. The integral of e
4lnx
C. π/7
D. π/9
C. ¼
D. 1/5
dx from x = 0 to x = 1 is
A. ½
B. 1/3
2
1470. Evaluate ∫ sec xtanx dx
2
2
A. ½ tan x + c C. ½ sec x + c
3
B. 1/3 sec x + c
2 x
D. A or C
1471. To integrate ∫ x e dx by parts, it is wise to choose u =
a.
b.
c.
d.
A. x
B. x
2
x
x
C. e
D. xe
C. 2.197
D. 2.791
2
1472. Evaluate ∫sin xdx
3
A. 1/2(x-sinxcosx) + c C. 1/3sin x + c
B. ½ x - ¼ sin2x + c
D. A or B
1 x x
1473. Evaluate ∫ 0 2 3 dx
A. 2.971
a.
b.
c.
d.
B. 2.719
a.
b.
c.
d.