1000 MMR
Author: Victor A. Tondo Jr., LPT
1000
MMR
Victor A. Tondo Jr., LPT
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1000 MMR
Author: Victor A. Tondo Jr., LPT
About the Author
Victor A. Tondo Jr. was a consistent Math
contest champion, a national finalist for MTAP and
PMO, a national awardee for the Intel Science Fair,
a number theorist, a coach for international
competitions, a review specialist, a licensed
teacher, a regional board top notcher, an
accomplished gamer, and a pyrographer.
This is a free reviewer. All rights reserved.
1000 MMR
Author: Victor A. Tondo Jr., LPT
Acknowledgment
To my mom, Analyn N. Torizuka, for your all-out support in my entire life, even after I spent more
than four years playing games on the computer. I love you so much! Without you, I am nothing.
To my sister, Vera-Lee A. Tondo, for always sensing my emotions and knowing everything that I
am going through. Your unwavering support kept me sane, especially after the lowest points in my life.
To my daughter, Aira Quineisha R. Tondo, for being the sweetest, most loving daughter I could
ever ask for. I have been loving you since the day you were born, and I will always be very proud of you.
To my teachers and professors, for molding me into the person I am today. You have taken good
care of me, even with my special needs. You did your best to understand me, brought out the best in me,
and inspired me to be in the same profession as you. I will always be grateful for your gift.
To Ma’am Romana Valdecasa, for turning a poor nobody into someone who speaks and breathes
Mathematics. You saw my potential and made me shine brighter than any star. You really are my second
mother – you figuratively gave birth to the successful Victor that I am now. I can never thank you enough.
To Ma’am Joaquina Birung, for always looking after me. You have been a very crucial factor to my
success during high school and even during college. You took me under your wings. You have been my
teacher, my coach, my mentor, my cooperating teacher, and of course, one of my favorite mothers.
To Sir Pedro Suyu, to Sir Manuel Belango, and to the beautiful Ma’am Apolinaria Andres, for
instilling in me the principles that a teacher must live by.
To my beloved Ma’am Sally Caldez, for never giving up on me. You are perhaps my second lola, and
I will always love you. May you rest in peace.
To Ma’am Loida Calonia, for teaching me your Christian ways. While others have given up when
they knew I am an atheist, you stood by my side patiently. When I was close to committing suicide, you
were there to guide me. You extended my life past the age of 30.
To my “daughters” Marone Khyme and Marby Alzate, for treating me not only as your tutor but
also as a friend, as your tatay, and as your mentor. I love you both, and I have always been proud of you.
To my students, tutees, reviewees, subscribers, and friends, for always believing in me.
At sa aking Inang Bayan, inaalay ko sa iyo itong unang lathalain ko. Ang tanging hangad ko lang ay
ang ika-uunlad mo.
Victor A. Tondo Jr., LPT
This is a free reviewer. All rights reserved.
1000 MMR
Author: Victor A. Tondo Jr., LPT
Tips for Reviewing
1. Expand your Mathematical vocabulary. You will find it difficult to learn concepts when your
Mathematical vocabulary is limited.
2. Do not memorize the answers. Instead, know how to use the solution, how it works, and when it should
be used.
3. Remember that there are many ways to deal with problems. Nobody wants a long solution when there
are several shortcuts out there.
4. Try answering the items first before reading the solutions or explanations. This will give you an idea on
your current standing and at the same time, what kinds of questions you should expect in the exam.
5. Familiarize yourself with your own scientific calculator. Know where the different buttons are, like nPr,
nCr, and n!. You should also know what these buttons do, and how to make them function.
Strategies in Answering the Mathematics Test
During the Licensure Exam
1. Keep track of the time. Before answering the questions, take note of the number of items and the time
allotted for the exam. Estimate the average time you should allot for each item.
2. Scan the items. Go over the test booklet quickly to get an idea of the types of questions or problems in
the test (such as word problems, analysis of graphs and tables, simple equations). This will help you
determine the appropriate pace to take in answering the test.
3. Pace yourself. Work quickly on topics you have mastered without being rash and careless. When you
encounter a difficult question, skip it and move on to less difficult ones. When you have finished
answering all the easy questions, go back to the skipped items.
4. Keep your pen going. On your second reading of the item, create a picture of the problem and draw that
picture on your scratch paper. Take down the relevant facts (numbers, units, dimensions), specially what
is asked for in the problem. Many items that seem difficult on first reading become easier to handle after
the facts have been taken down.
5. Visualize, draw diagrams. These help in answering word problems. Familiarize yourself with the types
of diagrams in every topic as these may prove useful in solving problems.
6. Eliminate choices. The right answer can be determined if all but one of the choices are eliminated.
Don’t easily give up on difficult items.
7. Work backwards. This method is useful if you find it difficult to start a solution to a problem. To work
backwards is similar to a trial-and-error on the choices. This can also be used to check if your answer is
correct.
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1000 MMR
Author: Victor A. Tondo Jr., LPT
1000
Questions
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1000 MMR
1. How many line segments can be made from 30
non-collinear points?
A. 900
B. 870
C. 450
D. 435
Author: Victor A. Tondo Jr., LPT
8. What are the missing terms in the series
5, 20, 80, ___,1280, ___, 20480?
A. 50; 210
B. 40; 160
C. 35; 135
D. 320; 5120
2. Calculate the mean absolute deviation of the
following numbers: 60, 80, 100, 75 and 95
A. 12.4
B. 14.2
C. 16.1
D. 18.9
9. At what rate per annum should P2400 be
invested so that it will earn an interest of P800 in
8 years?
A. 6 ½ %
B. 5 ½ %
C. 4.17 %
D. 6 %
3. Which of the following is the factorization of
the binomial x2 - 42?
A. (x + 4)(x + 2)
B. (x – 4)2
C. x(x + 2x + 2)
D. (x – 4)(x + 4)
4. What value of x will satisfy the equation:
0.4(5x – 1470) = x?
A. 490
B. 2,130
C. 1470
D. 588
5. Which of the following is ALWAYS true?
A. Vertical pairs of angles are supplementary.
B. Vertical pairs of angles are complementary.
C. Linear pairs of angles are congruent.
D. Linear pairs of angles are supplementary.
6. The average of 5 different counting numbers is
20. What is the highest possible value that one of
the numbers can have?
A. 20
B. 40
C. 30
D. 90
7. Three brothers inherited a cash amount of
P62,000 and they divided it among themselves in
the ratio of 5:4:1. How much more is the largest
share than the smallest share?
A. P75,000
B. P30,000
C. P24,800
D. P37,200
10. The area of a rectangle is (x2 + 2x - 8). If its
length is x + 4, what is its width?
A. x + 2
B. x - 2
C. x + 1
D. x + 6
11. What is the value of 12⅙ - 3 ⅜ - 5 ⅔ + 20 ¾?
A. 21
B. 22
C. 23
D. 21
12. The vertex angle of an isosceles triangle is
20°. What is the measure of one of the base
angles?
A. 150°
B. 60°
C. 75°
D. 80°
13. Ana and Beth do a job together in three
hours. Working alone, Ana does the job in 5
hours. How long will it take Beth to do the job
alone?
A. 3 and 1/3 hours B. 2 and 1/3 hours
C. 3 hours
D. 7 and 1/2 hours
14. How much greater is the sum of the first 100
counting numbers than the sum of the first 50
counting numbers?
A. 110
B. 3,775
C. 3,155
D. 1200
15. Which of the following has the largest value?
A. 85
B. 39
C. 65
D. 94
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1000 MMR
16. A water tank contains 18 liters when it is
20% full. How many liters does it contain when
50% full?
A. 60
B. 30
C. 58
D. 45
17. The edges of a rectangular solid have these
measures: 1.5 feet by 1½ feet by 3 inches. What
is its volume in cubic inches?
A. 324
B. 225
C. 972
D. 27
18. In a certain school, the ratio of boys to girls is
5 is to 7. If there are 180 boys and girls in the
school, how many boys are there?
A. 105
B. 90
C. 45
D. 75
19. Ruben’s grades in 6 subjects are 88, 90, 97,
90, 91 and 86. What is the grade that he should
aim for in the 7th subject if he has to have an
average of 91?
A. 97
B. 95
C. 92
D. 89
20. On a certain day, three computer technicians
took turns in manning a 24-hour internet shop.
The number of hours Cesar, Bert, and Danny
were on duty was in the ratio 3:4:5, respectively.
The shop owner pays them P50 per hour. How
much would Danny receive for that day?
A. P 230
B. P500
C. P160
D. P480
21. A retailer buys candies for P90.25. The pack
has 35 pieces of candies. If she sells each candy
for P3.25, how much profit does she make?
A. P11.50
B. P23.50
C. P37.50
D. P18.75
22. Determine the midpoint of the line segment
joining the points (7, -3) and (-1, 6).
A. (2, 3/2)
B. (2, -3/2)
C. (3, 3/2)
D. (1, 5/2)
Author: Victor A. Tondo Jr., LPT
23. One side of a 45° - 45° - 90° triangle
measures x cm. What is the length of its
hypotenuse?
A. x √3 cm
B. x cm
C. (x √3)/2 cm
D. x √2 cm
24. The legs of one right triangle are 9 and 12,
while those of another right triangle are 12 and
16. How much longer is the perimeter of the
larger triangle than the perimeter of the smaller
triangle?
A. 84
B. 7
C. 12
D. 14
25. An online shop sells a certain calculator for
P950 and charges P150 for shipping within
Manila, regardless of the number of calculators
ordered. Which of the following equations shows
the total cost (y) of an order as a function of the
number of calculators ordered (x)?
A. y = (950 + 150)x
B. y = 150x +950
C. x = 950y + 150
D. y = 950x + 150
26. Which of these has the longest perimeter?
A. A square 21 cm on a side
B. A rectangle 19 cm long and 24 cm wide
C. An equilateral triangle whose side is 28 cm
D. A right triangle whose two legs are 24 and 32
cm
27. How many square inches are in 2 square
yards?
A. 900
B. 144
C. 1296
D. 2,592
28. In a playground for Kindergarten kids, 18
children are riding tricycles or bicycles. If there
are 43 wheels in all, how many tricycles are
there?
A. 8
B. 9
C. 7
D. 11
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1000 MMR
29. Aira takes ¾ hour to dress and get ready for
school. It takes 4/5 hour to reach the school. If
her class starts promptly at 8:00 am; what is the
latest time she can jump out of bed in order not
to be late for school?
A. 6:42 am
B. 6:27 am
C. 6:57 am
D. 7:02 am
30. Which common fraction is equivalent to
0.215?
A. 43/200
B. 27/125
C. 21/50
D. 108/375
31. What are the next three terms in the
progression 1, 4, 16 …?
A. 64, 256, 1024
B. 67, 259, 1027
C. 48, 198, 1026
D. 65, 257, 1025
32. A man is 3 times as old as his son now. Four
years ago, the sum of their ages was 36. Find the
man’s age now.
A. 33
B. 11
C. 29
D. 36
33. What is the least common multiple of 12, 24
and 72?
A. 12
B. 72
C. 144
D. 36
34. The hypotenuse of a right triangle is 25 feet.
If one leg is 24 feet, what is the length of the
other leg?
A. 6 ft.
B. 5 ft.
C. 20 ft.
D. 7 ft.
35. If two variables X and Y are directly related,
which of these is NOT true?
A. When X is low, Y is also low.
B. As X increases, Y also increases.
C. When X increases, Y decreases.
D. A high Y is associated with a high X.
Author: Victor A. Tondo Jr., LPT
36. Find the domain of f(x) =
A. x
B. x = 1
C. x = -1
D. x
,x
.
-1
37. A car travels D km in H hours. Which of the
following expressions shows the distance
travelled by the car after M minutes?
A. MD/H
B. 60MD/H
C. MD/60H
D. 60HD/M
38. Find the surface area of a rectangular box
whose dimensions are 30 cm x 40 cm x 50 cm.
A. 4700 cm2
B. 7050 cm2
C. 9400 cm2
D. 11750 cm2
39. If x – y = 3, then (y-x)-3 = ___.
A. 9
B. -9
C. 1/27
D. -1/27
40. Factorize (x4 – 81) completely.
A. (x-3)4
B. (x – 3)2 (x + 3)2
C. (x+3) (x-3) (x2+9)
D. (x+3)3 (x-3)
41. √8 + √18 √2 = ____
A. 4√2
B. 5√2
C. √24
D. 2√6
42. By which property can we state the
following:
“If ax + b = c, then ax + b - b = c – b.”
A. transposition
B. transitive
C. additive inverse D. addition property
43. The midpoint of P and (-7, 4) is (-3, 1). What
are the coordinates of P?
A. (-5, 5/2)
B. (-11, 7)
C. (1, -2)
D. (-2, 3/2)
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1000 MMR
44. What is the slope of the line 3x – y = 11?
A. -1/3
B. 1/3
C. -3
D. 3
45. What is the minimum value of
f(x) = 3x2 + 6x + 7?
A. 1
B. -1
C. 4
D. -4
46. If xy = 23 and x2 + y2 = 75, find x + y.
A. 10.7845
B. 11
C. 11.2155
D. 11.7845
47. How much water must be evaporated from
90 ml of a 50% salt solution to increase its
concentration to 75%?
A. 40 ml
B. 38 ml
C. 35 ml
D. 30 ml
48. A and B form a vertical pair. If m A = 3x
and m B = 5x – 44, what is the value of x?
A. 50.5
B. 28
C. 22
D. 16.75
49. The angle of elevation from an observer to
the top of a building is 30o. If the building is 50
meters high, how far is the observer from the
building?
A. 25
B. 25√3
C. 50√3
D. 100
50. 1 and 3 are opposite angles in a
parallelogram. If m 1 = 40o, what is m 3?
A. 40o
B. 50o
C. 70o
D. 140o
51. Two parallel lines are cut by a transversal,
forming H and K. If the two angles are
exterior angles on the same side of the
transversal, what is the measure of H if the
measure of K is 50o?
A. 25o
B. 50o
o
C. 100
D. 130o
Author: Victor A. Tondo Jr., LPT
52. There are 33 red bags, 25 green bags, and 17
blue bags in a store. What percent of the bags is
red?
A. 33%
B. 44%
C. 66%
D. 67%
53. Given sin θ = 0.28, which of the following
could possibly be cos θ?
A. 0.72
B. -0.86
C. 0.96
D. 1.14
54. If the sum of the supplement and the
complement of an angle is 130 degrees, what is
the angle?
A. 65o
B. 70o
C. 50o
D. 25o
55. If today is a Saturday, what day is 125 days
from now?
A. Thursday
B. Friday
C. Sunday
D. Monday
56. Car A is traveling towards the east at a speed
of 35 kph, while car B is traveling towards the
west at 45 kph. If they left the same point at 1:00
PM, how far apart are they at 3:45 PM?
A. 240 km
B. 220 km
C. 200 km
D. 180 km
57. Mr. Santos left the house at 1:00 PM and
traveled east at an average speed of 40 kph. His
wife Mrs. Santos left the at 2:00 PM and traveled
west at an average speed of 30 kph. How far
apart are they at 4:00 PM?
A. 180 km
B. 140 km
C. 100 km
D. 60 km
58. Five consecutive even numbers have a sum of
120. What is the sum of the 2nd and 5th even
numbers?
A. 46
B. 48
C. 50
D. 52
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1000 MMR
59. If x = 3, which of the following is equal to 13?
A. 5x + 2
B. x2 + 2x + 1
3
C. x – 4x – 2
D. x2 + x + 2
60. If f(x) = x2 + 4x + 3, which of the following is
equal to 99?
A. f(11)
B. f(-12)
C. f(12)
D. f(-8)
61. Given f(x) = ln
A.
C. (2x+2) ln (x2+2x)
, what is f ‘(x)?
B.
D. 2x + 2
62. Which of the following could be the value of x
if x 3(mod 11)?
A. 33
B. 47
C. 52
D. 2
63. If = 6x2 + 8x – 7, which could be u?
A. 12x + 8
B. 3x3 + 4x2 – 7x + 11
3
2
C. 2x + 4x -7x +1 D. 12x2 + 8x - 10
64. What is the center of x2 + y2 – 8x + 6y = 0?
A. (-8.6)
B. (8, -6)
C. (-4, 3)
D. (4, -3)
65. Which of the following is a parabola that
opens to the right?
A. 6y = (x+9)2 - 8 B. -4y = (x-6)2 + 3
C. -5x + 3 = (y-2)2 D. 2x + 6 = (y+3)2
66. Factorize: 12x2 – 7x – 10.
A. (6x + 5) (2x – 2)
B. (6x – 2) (2x + 5)
C. (3x + 2) (4x – 5)
D. (3x – 2) (4x + 5)
67. For which value of k does 4x2 + kx + 49 have
only one root?
A. -28
B. -14
C. 7/2
D. -7/4
Author: Victor A. Tondo Jr., LPT
68. If A and B are the roots of x2 + 7x + 15, what
is AB?
A. 7√3 + 2
B. 2√3 + 7
C. 3√2 + 2√3
D. 15
69. 1 + 2 + 4 + 8 + … + 2048 = ____
A. 4095
B. 4096
C. 4097
D. 4098
70. 24 + 12 + 6 + 3 + 1.5 + … = ____
A. 48
B. 50
C. 54
D. 60
71. How many terms are there in the sequence
5, 13, 21, 29, …, 357?
A. 40
B. 44
C. 45
D. 70
72. How many ways can a group of 5 be selected
from 5 boys and 5 girls if the group must contain
3 boys and 2 girls?
A. 151,200
B. 1200
C. 252
D. 100
73. What is the probability of getting a sum of 9
when rolling 2 dice?
A. 1/9
B. 5/36
C. 1/6
D. 7/36
̅̅̅̅ where A is at (-3,4)
74. C is the midpoint of AB
and B is at (7,-10). Find the coordinates of C.
A. (5,-7)
B. (-5,7)
C. (2,-3)
D. (-2,3)
75. It is a line segment formed by connecting two
non-consecutive vertices of a polygon.
A. side
B. apothem
C. altitude
D. diagonal
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1000 MMR
76. Find the equation of the line perpendicular to
2x – 3y = 7, passing through (1,2).
A. 2x + 3y = 8
B. 3x + 2y = 7
C. 2x – 3y = -4
D. 3x – 2y = -1
77. Two parallel lines are cut by a transversal to
form X, Y, and Z. Given that X and Y are
alternate interior angles while Y and Z are
interior angles on the same side of the
transversal, find m Z if m X = 40o.
A. 40o
B. 50o
o
C. 130
D. 140o
78. The measure of each interior angle of a
regular polygon is 144o. How many vertices does
it have?
A. 36
B. 24
C. 12
D. 10
79. Solve: (x + 9) (x – 3) < 0
A. -9 < x < 3
B. x < -3 x > 9
C. x < -9 x > 3
D. x
; x -9, 3
80. The product of two consecutive even
counting numbers is 3248. Find the smaller
number.
A. 42
B. 46
C. 52
D. 56
81. Solve for x: 2log2 3 – log2 18 = x
A. ½
B. -1
C. -2
D. 1
82. Twinkle Bucks has four serving sizes for their
milk tea: Small, Medium, Large, and Extra Large.
What level of data are they using for their
serving sizes?
A. nominal
B. ordinal
C. interval
D. ratio
83. After receiving a 20% markup, a bag was sold
for P960. How much was it originally?
A. P1152
B. P4800
C. P800
D. P1200
Author: Victor A. Tondo Jr., LPT
84. Given ̅̅̅̅
BT bisects ABC and m ABT = 40o,
find m ABC.
A. 20o
B. 40o
C. 60o
D. 80o
85. A cone has a radius of 9 cm and a slant height
of 15 cm. Find its volume.
A. 243 π cm3
B. 324 π cm3
3
C. 405 π cm
D. 486 π cm3
86. If f(x) = x2 + 4x + 4 and g(x) = x-2, find
f(g(x)).
A. x2
B. x3 – 6x2 + 6x – 9
2
C. x + 8x + 16
D. x2 – 8x + 16
87. A 10 ft ladder leans against a wall, forming a
30o angle with it. How high on the wall does it
reach?
A. 5 ft
B. 5 √3 ft
C. 10 √3 ft
D. 10 √6 ft
88. How many ways can a committee of 5 be
selected from 9 people?
A. 126
B. 120
C. 3024
D. 15120
89. What is 60% of 80% of 500?
A. 480
B. 240
C. 120
D. 60
90. If 3x = 7 and 2y = 5, what is 6(x-y)?
A. -1
B. 1-√35
C. √7 - √5
D.
91. If two numbers have a product of 71 and the
sum of their squares is 147, what is their sum?
A. -17
B. 5
C. 12√3 + √5
D. 12 + √3
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1000 MMR
92. Find the median: 7, 9, 11, 10, 9, 13, 17, 14
A. 10 and 11
B. 9 and 10
C. 10.5
D. 9.5
93. How many 3-digit numbers can be formed
using the digits 0, 1, 2, 3, 4 and 5 if repetition is
not allowed?
A. 60
B. 80
C. 100
D. 120
94. How many ml of 20% acid must be added to
400 ml of 50% acid to make a 30% acid solution?
A. 1000 ml
B. 900 ml
C. 800 ml
D. 750 ml
95. How many ml each of 10% and 50% solution
should be mixed to make 500 ml of 18%
solution?
A. 400 ml of 10% and 100 ml of 50%
B. 350 ml of 10% and 150 ml of 50%
C. 300 ml of 10% and 200 ml of 50%
D. 200 ml of 10% and 300 ml of 50%
96. It takes 28 men a total of 24 days to build a
house. How long would it take 32 men to build a
similar house?
A. 28 days
B. 27 days
C. 21 days
D. 19 days
97. Evaluate: lim
A. undefined
C. 8
B. limit does not exist
D. +
98. A box contains 7 red, 8 blue, and 9 white
balls. When taking two balls in succession, what
is the probability that both balls are white?
A. 9/64
B. 9/69
C. 7/64
D. 7/69
Author: Victor A. Tondo Jr., LPT
99. Which of the following has two diagonals that
are perpendicular bisectors of each other?
A. kite
B. rectangle
C. rhombus
D. isosceles trapezoid
100. A pipe can fill a pool in 6 hours while
another pipe can drain empty the pool in 15
hours. How long will it take to fill the pool if both
pipes are open?
A. 9 hours
B. 9.125 hours
C. 9.45 hours
D. 10 hours
101. If log n – 1 = 2, find n.
A. 3
B. 1000
C. e3
D. 3e
102. log2 3 + 2 log2 7 – log2 5 = ______.
A. log2
B. log2
C. log2
D. log2
103. The surface areas of two spheres are 12 π
cm2 and 108 π cm2. What is the ratio of their
volumes?
A. 1:3√3
B. 1:9
C. 1:27
D. 2:3√3
104. The volume of a regular hexahedron is 64
in3. How long is each side?
A. 2 in
B. 4 in
C. 6 in
D. 8 in
105. Which of the following statements is
ALWAYS true?
A. The square of a prime number is odd.
B. The sum of two consecutive even numbers is
divisible by 4.
C. Any even number is composite.
D. The product of two consecutive even numbers
is divisible by 8.
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1000 MMR
106. Find the volume of a steel cylinder of radius
5 cm and height 12 cm.
A. 300 π cm3
B. 250 π cm3
C. 200 π cm3
D. 100 π cm3
107. A cube sits perfectly inside a sphere of
volume 108 √3 π cm3. Find the volume of the
cube.
A. 27 cm3
B. 54 cm3
C. 108 cm3
D. 216 cm3
108. Find the distance in cm of an 80 cm chord
from the center of a circle whose radius is 41 cm.
A. 41 - 2√10
B. 41 - 4√10
C. 9√2
D. 9
109. Which quadrilateral has two congruent
diagonals that bisect each other?
A. kite
B. isosceles trapezoid
C. rectangle
D. rhombus
110. What is the longest side of ∆MTC if m M =
40o and m C = 60o?
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅
A. MC
B. TC
C. MT
D. CT
111. Find the altitude to the hypotenuse of a
right triangle whose legs measure 10 cm and 24
cm.
A. 120 cm
B.
cm
C. 120√2 cm
D. 24√5 cm
112. Find the inverse of y = x2 + 10x.
A. y-1 = √
25 + 5
B. y-1 = √
25 – 5
C. y-1 = √ + 25 + 5
D. y-1 = √ + 25 – 5
113. Find the intersection of y = 2x + 3 and
y = 4x – 11.
A. (-4/3, 0)
B. (4/3, 0)
C. (7, 17)
D. (-7,-17)
Author: Victor A. Tondo Jr., LPT
114. Find the area of the triangle whose vertices
are (1,4), (2,3), and (3,0).
A. 0
B. 1
C. 5/3
D. 3/4
115. Find the tenth term: 3, 10, 17, 24, …
A. 66
B. 67
C. 68
D. 69
116. Find the remainder when
x4 – 5x3 + 6x2 + 2x + 1 is divided by (x – 2).
A. 17
B. 13
C. 9
D. 5
117. The sum of Fe’s age and Sita’s age is 60.
Twelve years ago, Fe was twice as old as Sita.
How old is Sita now?
A. 18
B. 24
C. 30
D. 36
118. If the length of a rectangle is increased by
20% while the width is decreased by 10%, what
will happen to its area?
A. decreased by 10%
B. increased by 10%
C. increased by 8%
D. decreased by 2%
119. The 19th term of an arithmetic sequence is
85 and the 12th term is 43. Find the common
difference.
A. 5
B. 6
C. 7
D. 8
120. If 2x = 3y and 4y = 5z, what is z in terms of
x?
A. z = x
B. z = x
C. z = x
D. z =
x
121. Victor had an average of 94 on his first four
Math tests. After taking the next test, his average
dropped to 93. Find his most recent grade.
A. 88
B. 89
C. 90
D. 91
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122. X is of Y and Y is of Z. What part of Z is X?
Author: Victor A. Tondo Jr., LPT
129. Evaluate
when x = ¾ and y = .
A. X = Z
B. X = Z
A. -38
C. X = Z
D. X = Z
123. Two buses leave the same station at 8:00
pm. One bus travels north at the rate of 30 kph
and the other travels east at 40 kph. How many
kilometers apart are the buses at 10 pm?
A. 140 km
B. 100 km
C. 70 km
D. 50 km
124. A bus drove for 6 hours at 75 kph and 4
hours at 80 kph. What was its average speed?
A. 76 kph
B. 77 kph
C. 77.5 kph
D. 78 kph
125. 18 students failed a quiz. They represent
30% of the class. How many students passed the
quiz?
A. 60
B. 42
C. 36
D. 24
126. Rationalize:
A.
√
C.
√
+1
C. 19
D. 38
130. Today, Vic is 11 years old while his father is
37. How many years from now will his father be
twice as old as he?
A. 15
B. 13
C. 11
D. 10
131. Carla and Diana are on a seesaw. Carla
weighs 50 kg and sits 168 cm to the left of the
fulcrum. If Diana weighs 60 kg, how far to the
right of the fulcrum must she sit to balance the
seesaw?
A. 140 cm
B. 170.8 cm
C. 201.6 cm
D. 210 cm
132. Twenty guests shake hands with each other.
If each guest is to shake hands with all the other
guests, how many handshakes will be made?
A. 400
B. 380
C. 200
D. 190
133. How many line segments can be made from
30 non-collinear points?
A. 900
B. 870
C. 450
D. 435
√
B. 2√5 – 4
D.
B. -19
√
127. RNHS has 130 quizzers. 67 of them are
Math, 60 are Science, and 20 are quizzers for
both Math and Science. How many quizzers are
neither Math nor Science?
A. 0
B. 13
C. 17
D. 23
128. Mr. Tondo has P100,000 to invest, from
which he wants to earn P5600 per year. Bank A
offers 5% per annum while Bank B offers 6%.
How much should he invest at Bank B?
A. P45,000
B. P50,000
C. P55,000
D. P60,000
134. The longest chord of a circle is 80 cm. How
long is its radius?
A. 20 cm
B. 30 cm
C. 20√2 cm
D. 40 cm
135. Find k such that 34k67 is divisible by 9.
A. 5
B. 6
C. 7
D. 8
136. Find the largest area of a rectangle whose
perimeter is 100 cm.
A. 2500 cm2
B. 2499 cm2
2
C. 625 cm
D. 624 cm2
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137. What time is 200 minutes past 10:30 PM?
A. 12:30 AM
B. 12:30 PM
C. 1:50 AM
D. 1:50 PM
Author: Victor A. Tondo Jr., LPT
146. Insert one term between 18 and 32 to make
a geometric sequence.
A. 20
B. 24
C. 25
D. 27
138. Find the product of two numbers whose
GCF is 24 and LCM is 120.
A. 2880
B. 1440
C. 720
D. 360
147. There are 100 pigs and chickens in a farm,
all of which are healthy. If there are 340 legs in
total, how many pigs are there?
A. 70
B. 65
C. 60
D. 55
139. The salary of 4 men for 5 days is P9,000.
How much is the salary of 5 men for 6 days?
A. P12,000
B. P12,600
C. P13,500
D. P14,400
140. The average grade of eleven students is 83.
If the average of six of these students is 88, what
is the average of the other 5 students?
A. 77
B. 78
C. 79
D. 80
141. If x is 80% of y, what percent of y is x?
A. 120%
B. 125%
C. 130%
D. 135%
142. Bus X left the terminal at 1 PM and traveled
at a speed of 60 kph. Bus Y left the same terminal
2 hours later and traveled 80 kph on the same
route. What time will Bus B catch up with Bus A?
A. 6 PM
B. 9 PM
C. 11 PM
D. 1 AM
143. What is the degree of the polynomial
-3 x2y3 + 21 x3y4 – 7 x5y6 – 15?
A. 4
B. 5
C. 11
D. 21
144. The average of x+5, 2x-4, and x+7 is 20.
Find x.
A. 18
B. 13
C. 9
D. 8
148. Adam can do a job alone in 8 hours, while
Bam can do the same job in 12 hours. One day,
they worked together for 1 hour before Bam left
Adam to finish the job. How long will it take
Adam to finish the remaining job?
A. 6 hrs 50 mins
B. 6 hrs 40 mins
C. 6 hrs 30 mins
D. 6 hrs 20 mins
149. Find x if 2748 = 9x.
A. 144
B. 81
150. Solve for x:
A. 1.142857
C. 1.5
C. 72
D. 60
49x = 343
B. 7
D. √7
151. What is the highest possible product of two
numbers if their sum is 45?
A. 506
B. 506.25
C. 506.5
D. 506.725
152. Which statistical test is used for comparing
observed frequencies to expected frequencies?
A. ANOVA
B. t-test
C. Pearson R
D. Chi Square
153. The product of two consecutive odd
counting numbers is 1443. What is their sum?
A. 76
B. 78
C. 80
D. 82
145. Mia is 16 years younger than Kia. 13 years
ago, Kia was thrice as old as Mia. What is Kia’s
present age?
A. 43
B. 40
C. 37
D. 34
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2 +1
4
154. Given ( ) = { 4
= 4,
7
4
( ).
find lim
A. 4
B. 9
C. 0
D. limit does not exist
155. If today is a Saturday, what day is 125 days
from now?
A. Friday
B. Sunday
C. Monday
D. Tuesday
156. If the sum of the supplement and the
complement of an angle is 124, what is the
angle?
A. 71
B. 72
C. 73
D. 74
157. Find + given x + y = 20 and xy = 81.
A.
B.
C.
D.
Author: Victor A. Tondo Jr., LPT
162. Which of the following angles in standard
position is coterminal with 40o?
A. 2200o
B. 1760o
C. 1520o
D. 1360o
163. Find the equation of the line passing
through (2,7) and (-3,-3).
A. y = 4x -1
B. y = 3x + 1
C. y = 3x + 6
D. y = 2x + 3
164. In which quadrant can we find θ if tan θ < 0
and sin θ > 0?
A. First Quadrant
B. Second Quadrant
C. Third Quadrant
D. Fourth Quadrant
165. Find the equation of the line passing
through the point of origin and (3,4).
A. y = x
B. y= x
C. y = x +
158. What is the remainder when
534,214,557,989,215 is divided by 4?
A. 0
B. 1
C. 2
D. 3
159. Dividing by 0.125 is the same as multiplying
by which number?
A. 5
B. 8
C. 10
D. 16
160. Find the surface area of a sphere whose
radius is 6 cm.
A. 72 π cm2
B. 108 π cm2
2
C. 144 π cm
D. 192 π cm2
161. Which of the following is the reference
angle of 216o?
A. 84o
B. 66o
C. 54o
D. 36o
D. y = x + 1
166. Find the range of f(x) = -2x2 + 4x.
A. y 2
B. y 2
C. y -2
D. y -2
167. If a3/2 – 1 = 7, what is a?
A. 4
B. 8
C. 9
D. 18
168. Which of the following is true?
A. A rectangle is a square.
B. A rhombus is a rectangle.
C. A trapezoid is a rhombus.
D. A square is a rhombus.
169. What is the measure of each exterior angle
of a pentagon?
A. 108o
B. 72o
C. 60o
D. 36o
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170. How many diagonals does a nonagon have?
A. 27
B. 36
C. 45
D. 54
171. What is the fractional equivalent of
0.123123123123…?
A.
B.
C.
D.
172. Mrs. Pasay saved P250 after buying a phone
with a 10% discount. How much did she pay for
the phone?
A. P2500
B. P2250
C. P2000
D. P1750
173. A book was sold for P270 after a 10%
discount was given. How much was the book
originally?
A. P330
B. P300
C. P297
D. P280
174. Find the area of an equilateral triangle
whose sides measure 12 cm each.
A. 36√3 cm2
B. 48√3 cm2
C. 60√3 cm2
D. 72√3 cm2
175. This is located at the intersection of the
angle bisectors of a triangle.
A. Incenter
B. Circumcenter
C. Centroid
D. Orthocenter
̅̅̅̅ is 9 cm long
176. ∆ABC is similar to ∆DEF. AB
̅̅̅̅
while DE is 12 cm long. If the area of ∆ABC is
27 cm2, what is the area of ∆DEF?
A. 36 cm2
B. 48 cm2
C. 60 cm2
D. 72 cm2
177. Find the remainder when x4 – 3x3 + 2x2 +
3x – 9 is divided by (x-3).
A. -18
B. -9
C. 9
D. 18
Author: Victor A. Tondo Jr., LPT
178. Which of the following has its incenter,
circumcenter, centroid, and orthocenter in just
one point?
A. Right Triangles
B. Equilateral Triangles
C. Isosceles Triangles
D. Scalene Triangles
179. Dexter is twice as heavy as Pablo. Ming is
4kg heavier than Pablo. The sum of their masses
is 164kg. How heavy is Dexter?
A. 40 kg
B. 44 kg
C. 80 kg
D. 88 kg
180. A circle is drawn inside a triangle such that
it is tangent to the sides of the triangle. Its center
will be the triangle’s ___________________.
A. Incenter
B. Circumcenter
C. Centroid
D. Orthocenter
181. Rayon can do a job in 3 hours, while Carlyn
can do the same job in 7 hours. How long will it
take them to finish the job by working together?
A. 2.1 hours
B. 2.5 hours
C. 5 hours
D. 10 hours
182. This line is perpendicular to one side of the
triangle passing through the opposite vertex.
A. Longitude
B. Median
C. Altitude
D. Bisector
183. How many ways can Lola Leonor arrange
her six meals on the Lazy Susan (the rotating
circular wooden server on top of the table)?
A. 720
B. 120
C. 36
D. 30
184. In parallelogram
MATH, m M = 7x – 12 and
m T = 5x + 32. Find m A.
A. 22
B. 38
C. 44
D. 142
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1000 MMR
185. Find the equation of the line perpendicular
to 2x + 5y = 7, passing through (1, 2).
A. 2x + 5y = 12
B. 2x – 5y = -8
C. 5x + 2y = 9
D. 5x – 2y = 1
186. How many ways can the letters of the word
BANANA be rearranged?
A. 720
B. 240
C. 120
D. 60
187. “The temperature in Baguio City is 20o
while the temperature in Tuguegarao City is
40o”. What level of data is temperature in
degrees Celsius?
A. Nominal
B. Ordinal
C. Interval
D. Ratio
188. What is formed by the intersection of two
planes?
A. a point
B. a line
C. a plane
D. space
189. What is formed when a plane intersects a
cone parallel to its circular base?
A. ellipse
B. hyperbola
C. circle
D. parabola
190. In which non-Euclidean model for geometry
can we have any given line ℓ and a point A which
is not on ℓ, wherein all lines through A will
intersect ℓ?
A. hyperbolic
B. elliptic
C. Saccheri
D. Pythagorean
191. Which numerical system is sexagesimal
(base-60)?
A. Mayan
B. Roman
C. Babylonian
D. Hindu-Arabic
Author: Victor A. Tondo Jr., LPT
193. Which of the following is false?
A. sin2 θ + cos2 θ = 1
B. sin θ (csc θ) = 1
C. sin θ ÷ cos θ = tan θ
D. sin θ (tan θ) = cos θ
194. If three-fourths of a number is 33 more than
its one-fifth, what is that number?
A. 240
B. 120
C.90
D. 60
195. Which of the following has the greatest
value:
A. 3 + 32 + (3 + 3)2
B. 33
2
2
C. [(3 + 3) ]
D. (3 + 3 + 3)2
196. Which of the following has an undefined
slope?
A. a vertical line
B. a horizontal line
C. a line parallel to the x-axis
D. a diagonal line
197. In solid geometry, what do you call a solid
bound by polygons?
A. multigon
B. tessellation
C. porygon
D. polyhedron
198. Tchr. Victor needs to randomly get 10 out of
his 50 students for drug testing. He proceeds by
making the students count off from 1 to 5. He
then randomly picks a number from 1 to 5.
Which sampling method did he use?
A. stratified
B. cluster
C. systematic
D. convenience
199. Which statistical test must be used in
testing the significance of group differences
between 2 or more groups?
A. Chi Square
B. t-test
C. ANOVA
D. Pearson R
192. Which numerical system makes use of dots
and horizontal lines, and shell shapes for zero?
A. Egyptian
B. Roman
C. Greek
D. Mayan
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1000 MMR
200. Which Mathematician is famous for the
Fibonacci sequence?
A. Ptolemy
B. Leonardo Pisano Bigollo
C. Pierre de Fermat
D. Luca Pacioli
Author: Victor A. Tondo Jr., LPT
207. Victor deposited an amount of P200,000 in
a bank that offers 5% interest compounded per
annum. How much will he have in his account
after 3 years?
A. P230,000
B. P231,525
C. P233,050
D. P234,575
201. Which Mathematician is famous for his last
theorem?
A. Pythagoras
B. Isaac Newton
C. Daniel Bernoulli D. Pierre de Fermat
208. Find the remainder when the polynomial
x4 – 3x3 + 2x2 – 5x + 8 is divided by (x – 3).
A. 5
B. 8
C. 11
D. 14
202. Which of the following is a square?
A. Polygon ABCD which has 4 congruent sides.
B. Polygon MATH which has 4 perpendicular
sides.
C. Quadrilateral HEAD which has one pair of
congruent perpendicular bisecting diagonals.
D. Quadrilateral FROG which has 4 right angles.
203. Which of the following is the set of points
whose sum of distance to two fixed points is
constant?
A. parabola
B. circle
C. ellipse
D. hyperbola
204. Which of the following is not a triangle
congruence postulate?
A. SAS
B. ASA
C. SAA
D. AAA
205. If A is at (-8,5) and B is at (4,-11), find C if C
is three-fourths the way from A to B.
A. (1, -7)
B. (-4, 1)
C. (1, 1)
D. (-4, -7)
206. CPCTC stands for “____________ parts of
congruent triangles are congruent”.
A. collinear
B. complementary
C. corresponding
D. conjugate
209. What is 60% of 120?
A. 50
B. 72
C. 180
D. 200
210. What percent of 80 is 55?
A. 145.45%
B. 135%
C. 68.75%
D. 44%
211. The hypotenuse of a right triangle measures
40 cm. Find its area if one angle measures 30o.
A. 100√3 cm2
B. 200√2 cm2
C. 200√3 cm2
D. 400√2 cm2
212. Nine cans of soda and four hamburgers cost
a total of P257. Five cans of soda and seven
hamburgers cost a total of P224. How much is a
can of soda?
A. P17
B. P19
C. P21
D. P23
213. The product of two consecutive even
numbers is 728. What is the smaller number?
A. 22
B. 24
C. 26
D. 28
214. What time is 219 minutes past 6:40 AM?
A. 8:59 AM
B. 9:19 AM
C. 9:49 AM
D. 10:19 AM
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1000 MMR
215. Find the vertex of y = 3x2 – 2x + 11.
A. ( , )
B. ( ,
)
C. (
, )
D. (
,
)
216. After getting a 20% discount, Mr. Lopez
paid P4,000 for a gadget. How much was its
original price?
A. P4,800
B. P5,000
C. P8,000
D. P20,000
Author: Victor A. Tondo Jr., LPT
223. Which of the following is not a function?
A. y = x2 + 2017x – 2017
B. y = |2017x| - 2017
C. y = √2017 + 2017
D. y2 = x + 2017
224. 12 + 17 + 22 + 27 + … + 117 = _____
A. 1409
B. 1414
C. 1419
D. 1424
217. When a number is increased by 3, its square
increases by 111. By what does its square
increase when the number is increased by 6?
A. 222
B. 240
C. 444
D. 480
225. Mr. G sold 80% of his apples and still had
213 apples left. How many apples did he have
originally?
A. 1704
B. 1065
C. 852
D. 293
218. How many prime numbers are there from 1
to 100?
A. 23
B. 24
C. 25
D. 26
226. When a number is increased by 4, its square
also increases by 168. What is this number?
A. 15
B. 19
C. 23
D. 27
219. Find the range of f(x) = 2x2 – 8x + 9.
A. y 0
B. y 1
C. y 9
D. y
227. Solve for k to make a perfect square
trinomial:
9x2 + kx + 25
A. 10
B. 15
C. 20
D. 30
220. Find the domain of y =
A. x ±7, ±10
B. x ±7
C. x ±10
D. x 1
228. Find the y-intercept of 2x + 3y = 4.
A.
B.
C.
D. 2
221. Solve for x: (x+3)2 = (x-4)2.
A. x = 0
B. x = ½
C. x = 1
D. no solution
222. The diagonal of a rectangular prism is 13
cm long. If it is 3 cm thick and 12 cm long, how
wide is it?
A. 3 cm
B. 4 cm
C.4√3 cm
D. 5 cm
229. Which of the following points is on the line
y = 2x + 5?
A. (1, 3)
B. (2, 9)
C. (0, 10)
D. (3, 10)
230. Find the intersection of y = -2x + 1 and
y = 3x + 16.
A. (-3, 7)
B. (-4, 9)
C. (3, -7)
D. (4, -9)
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1000 MMR
231. Find the slope of 3x + 5y = 7.
A.
B.
C.
Author: Victor A. Tondo Jr., LPT
239. Find the slope of the line tangent to
y = x3 – 6x2 + 2x + 7 at x = 4.
A. -8
B. -2
C. 2
D. 8
D.
240. Find the average rate of change of
y = x3 – 2x + 3 from x = 0 to x = 3.
A. 5
B. 6
C. 7
232. Which of the following is a polynomial?
A. √3 + 4 + 2
B. 2x + 3√
C.
+3
D. √3 x + 7
233. What is the degree of the polynomial
9x4 + 5x3 – 2x2 + 3x – 17?
A. 4
B. 5
C. 9
D. 10
234. log2 32√2 = __________.
A. 2.5
B. 3.5
C. 4.5
235. If y = √3
A. x = √
C. x = √
D. 5.5
+ 6 , what is x in terms of y?
B. x = √
–1
+1
D. x = √
+1
–1
236. Which of the following is a pair of parallel
lines?
A. y = 2 and x = 2
B. 12x + 13y = 14 and 13x + 14y = 15
C. y = 3x + 8 and 3y = x + 9
D. 4x + 5y = 6 and 8x + 10y = 21
237. Which of the following is a pair of
perpendicular lines?
A. x = 5 and y = 7
B. y = x and 2y = 4x + 5
C. x = 2y + 3 and 2x + 3y = 4
D. y = 5x + 6 and y = 0.2x – 8
238. Find the altitude to the hypotenuse of a
right triangle whose sides measure 5 cm, 12 cm,
and 13 cm.
A.
B.
C.
D. 26
D. 8
241. Find the radius of x2 + y2 + 2x – 4y = 44.
A. √39
B. 2√11
C. 7
D. 3√6
242. Gian has 8 more P5 coins than P1 coins. If
he has a total of P106, how many P5 coins does
he have?
A. 13
B. 15
C. 17
D. 19
243. After using half of her budget on bills, onethird on groceries, and P270 on a shirt, Mrs. D
still had P130 left. How much was her budget?
A. P2400
B. P2700
C. P3000
D. P3300
244. x varies directly as y and inversely as z. If
x = 24 when y = 32 and z = 4, what is x when
y = 21 and z = 7?
A. 3
B. 5
C. 7
D. 9
245. Find the mode of the following scores:
78
78
78
78
79
79
79
79
80
80
80
80
A. 79
B. 78, 79, and 80
C. 80
D. no mode
246. The average grade of 23 students in Section
A is 86, while the average grade of 27 students in
Section B is 91. What is the average grade of all
50 students in both sections?
A. 88.5
B. 88.6
C. 88.7
D. 88.8
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1000 MMR
247. Find the axis of symmetry of y = 3x2 – 5x.
A. x =
B. x =
C. x =
D. x =
248. Find the range of the following scores:
19
25
24
31
23
29
33
A. 12
B. 13
C. 14
D. 15
249. Mr. C travels for 2 hours at a speed of 38
kph and then north for 3 hours at a speed of 53
kph. What is his average speed?
A. 44 kph
B. 45.5 kph
C. 47 kph
D. 48.5 kph
250. Victor, Chris, and Aira volunteered to teach
at a nearby daycare. Chris worked for 2 hours
less than Aira. Victor worked twice as many
hours as Chris. Altogether, they worked for 58
hours. How many hours did Victor work?
A. 14
B. 16
C. 28
D. 32
Author: Victor A. Tondo Jr., LPT
254. Find the equation of the circle with center at
(2, 3), passing through (5, -1).
A. x2 + y2 + 4x + 6y = 0
B. x2 + y2 + 4x + 6y = 12
C. x2 + y2 – 4x – 6y = 0
D. x2 + y2 – 4x – 6y = 12
255. Find the equation of the vertical line passing
through (-3, 4).
A. x = -3
B. x = 4
C. y = -3
D. y = 4
256. Find the equation of the horizontal line
passing through (-3, 4).
A. x = -3
B. x = 4
C. y = -3
D. y = 4
257. Which of the following lines passes through
the point (3, -2)?
A. y = x + 5
B. y = 2x – 8
C. y = 5 – x
D. y = 5 – 2x
251. What conic figure does the equation
x2 + y2 + 4x = -4 form?
A. Real circle
B. Degenerate circle
C. Imaginary circle D. Ellipse
258. Given that I(2, -3) is the midpoint of V(-4, 5)
and C, find the coordinates of C.
A. (-1, 1)
B. (1, -1)
C. (8, -11)
D. (-10, 16)
252. What conic figure does the equation
x2 + y2 + 8x – 6y = -100 form?
A. Real circle
B. Degenerate circle
C. Imaginary circle D. Ellipse
259. The endpoints of the diameter of a circle are
A(9, -5) and B(-3, 11). What is the equation of
the circle?
A. (x – 3)2 + (y – 3)2 = 100
B. (x – 3)2 + (y – 3)2 = 400
C. (x + 3)2 + (y + 3)2 = 100
D. (x + 3)2 + (y + 3)2 = 400
253. Find the center of x2 + y2 + 6x – 10y = 2.
A. (6, -10)
B. (-6, 10)
C. (-3, 5)
D. (3, -5)
260. Find the distance between the line
3x + 4y – 5 = 0 and the point (8, -1).
A. 2
B. 3
C. 4
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D. 5
1000 MMR
261. B is one-fourth of the way from A(-13,9) to
C(7,-7). Find the coordinates of B.
A. (2, -3)
B. (-8, 5)
C. (2.5, -3.5)
D. (-7.5, 5.5)
261. Find the area of the triangle whose vertices
are X(-9, -3), Y(-2, 8), and Z(5, 1).
A. 61
B. 62
C. 63
D. 64
262. Which of the following is outside the circle
defined by the equation (x – 3)2 + y2 = 40?
A. (5, 6)
B. (7, 5)
C. (0, -5)
D. (-1, 4)
263. Which of the following is parallel to the line
defined by the equation y = 3x – 4?
A. y + 3x = 5
B. x + 3y = 6
C. y = x + 7
D. y = 3x + 8
Author: Victor A. Tondo Jr., LPT
267. Which of the following equations pertain to
a parabola that opens to the right?
A. 4(y + 3) = (x – 2)2
B. -3(y – 4) = (x + 5)2
C. (y – 6)2 = 5(x + 1)
D. (y + 8)2 = -2(x – 3)
268. Which of the following equations pertain to
a parabola that opens downward?
A. 4(y + 3) = (x – 2)2
B. -3(y – 4) = (x + 5)2
C. (y – 6)2 = 5(x + 1)
D. (y + 8)2 = -2(x – 3)
269. How long is the latus rectum of the parabola
defined by 12(y – 4) = (x + 3)2?
A. 12
B. 6
C. 4
D. 3
270. How far is the vertex from the directrix of
the parabola defined by 16y = x2?
A. 16
B. 8
C. 4
D. 2
264. Which of the following is perpendicular to
the line defined by the equation y = 3x – 4?
A. y + 3x = 9
B. -x + 3y = 10
C. y = x + 11
D. y = 3x – 12
271. Find the equation of the directrix of the
parabola defined by (y – 2)2 = -4(x + 3).
A. x = -2
B. x = 2
C. y = -4
D. y = -2
265. Which of the following is coincidental to the
line defined by the equation y = 2x + 13?
A. y + 2x = 13
B. 2x – y + 13 = 0
C. y = x + 13
D. 2y = 2x + 13
272. Find the coordinates of the focus of the
parabola defined by -12(y – 4) = (x + 5)2.
A. (-5, 7)
B. (-5, 1)
C. (-8, 4)
D. (-2, 4)
266. Which of the following equations pertain to
a parabola?
A. y2 + 5y = x
B. x2 + y2 + 3x – 4y = 0
C.
(
)
D.
(
)
+
(
)
=1
(
)
=1
273. In which quadrant would G(3,-4) fall?
A. First quadrant
B. Second quadrant
C. Third quadrant D. Fourth quadrant
274. Find the distance between the parallel lines
y = 3x + 9 and y = 3x – 12.
A. 7
B. 21
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C.
√
D.
√
1000 MMR
275. Find the intersection of the lines y = 2x + 5
and y = -4x + 23.
A. (3, 10)
B. (3, 11)
C. (4, 10)
D. (4, 11)
276. Which of the following pertains to a circle
that is concentric with (x – 3)2 + y2 = 24?
A. x2 + y2 – 6x + 3 = 0
B. x2 + y2 + 6y –15 = 0
C. x2 + y2 + 6x – 25 = 0
D. x2 + y2 + 6x + 6y – 24= 0
277. Which of the following is the equation of the
parabola that opens upward, whose latus rectum
is 12 units long, directrix is y = -3, and line of
symmetry is x = 7?
A. 12y2 = x – 7
B. 12y = (x – 7)2
C. 12(y + 3) = (x – 7)2
D. 12(y – 3) = (x – 7)2
278. Convert
A. 1.309o
C. 150o
rad to degrees.
B. 75o
D. 216o
279. Which of the following angles is coterminal
with 143o?
A. 217o
B. -37o
o
C. 323
D. 503o
280. Which of the following is NOT a
trigonometric identity?
A. sin2 θ + cos2 θ = 1
B. tan θ =
C. 1 + tan2 θ = sec2 θ
D. 1 – cot2 θ = csc2 θ
Author: Victor A. Tondo Jr., LPT
282. Which of the following is NOT a cofunction
identity?
A. cos θ = sin (90 – θ)
B. cot (90 – θ) = tan θ
C. sec θ = csc (90 – θ)
D. csc θ = sin (90 – θ)
283. The hypotenuse of a 30-60-90 triangle is 48
cm long. How long is its shortest side?
A. 24 cm
B. 24 √2 cm
C. 24 √3 cm
D. 16√3 cm
284. In a right triangle, the side opposite an
angle measuring 50o is 100 cm long. How long is
the side adjacent to the 50o angle?
A. 93.45 cm
B. 83.91 cm
C. 149.14 cm
D. 200 cm
285. A hundred cards are numbered 1 to 100.
What is the probability of drawing a card whose
number is divisible by seven?
A.
B.
C.
D.
286. What is the probability of rolling a sum of
10 when rolling two dice?
A.
B.
C.
D.
287. Factorize 3x2 + 5x – 2.
A. (3x – 1) (x + 2) B. (3x + 1) (x – 2)
C. (3x – 2) (x + 1) D. (3x + 2) (x – 1)
288. Six-sevenths of a number is 6 less than ninetenths of the same number. What is the number?
A. 130
B. 140
C. 200
D. 210
281. Which of the following is false?
A. tan θ =
B. csc θ =
C. cos θ =
D. sec θ =
This is a free reviewer. All rights reserved.
1000 MMR
289. A certain University has a dormitory. If 10
students stay in a room, 24 students will not
have a room. If 12 students stay in a room, there
will be 6 vacant beds. How many rooms are there
in the dormitory? How many students are
staying in the dormitory?
A. 116
B. 115
C. 114
D. 113
290. Which of the following is not between
and
Author: Victor A. Tondo Jr., LPT
295. Determine the relation that matches the
table of values.
x
1
2
3
4
5
y
13 11
9
7
5
A. y = 21 – x
B. y = 15 – 2x
C. y = 3x + 7
D. y = 2x + 11
296. Determine which polynomial expression
matches the algebra tile model.
?
A.
B.
C.
D.
291. Which value describes the position of C?
A. -0.75
B. -0.6
C. -1.25
D. -1.4
292. Choose the correct value of (x + y)(x – y)
when x = 3.5 and y = –8.7
A. -63.44
B. 63.44
C. -148.84
D. 10.4
A. 2x2 + x + 4
C. 3x2 – x + 4
B. 3x2 + x + 4
D. 3x2 + x – 5
297. Subtract: (–2x2 + 5x – 9) – (2x – 7)
A. 2x2 + 3x + 16
B. -2x2 + 3x + 2
C. -2x2 + 3x – 2
D. -2x2 + 7x + 2
298. Evaluate the polynomial 4x2 – 6x – 3 if x = 2.
A. -1
B. 1
C. 3
D. 5
299. Determine the measure of Y and Z.
293. Which two triangles are similar?
A. A and B
C. B and D
A. 45o, 55o
C. 48o, 48o
B. A and C
D. B and C
B. 40o, 50o
D. 50o, 50o
300. Which of the following is equal to x?
294. Determine which equation is equivalent to
4x – 1 = 11.
A. 4x = 10
B. 3x = 11
C. 4 – 1x = 11x
D. 4x = 12
A. 15 sin 72o
C. 72 sin 72o
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B. 15 sin 18o
D. 72 sin 18o
1000 MMR
301. Find the points of intersection of the graphs
of y = x2 and y = 3x – 2.
A. (1, 1) and (1, 4)
B. (1, 1) and (2, 4)
C. (1, -1) and (2, 4)
D. (-2, 4) and (1, 1)
Author: Victor A. Tondo Jr., LPT
C.
D.
302. An approximate value for
.
.
is:
.
A. 2
B. 20
C. 200
D. 2000
308. Which expression is equivalent to (9-2)8?
A. -8132
B. -818
C.
D.
303. The graph below represents the motion of a
car. The graph shows us that the car is:
309. What is 5 × 10–4 written in standard
notation?
A. 0.00005
B. 0.0005
C. 5,000
D. 50,000
A. accelerating
B. standing still
C. travelling north-east
D. travelling at a constant speed
304. The units digit of the number 543444 is:
A. 3
B. 9
C. 7
D. 1
305. 4n+1 (4n+2) equals
A. 42n + 3
B. 82n + 3
2n
+
3
C. 16
D. 4n + 3
310. What is the value of 54 × 5-6?
A. -25
B.
C.
D. 25
311. . Which comparison is true?
A. 4 < 180.5 < 4.5
B. 4.5 < 180.5 < 5
0.5
C. 8.5 < 18 < 9.5
D. 17 < 180.5 < 19
312. A weather station recorded the amount of
rain that fell during an 8-hour time frame using a
rain gauge. The findings are recorded in the
graph below.
306. The greatest number of Fridays that can
occur in a 75 day period is:
A. 10
B. 11
C. 12
D. 13
307. Which model is not a function?
A.
B.
Between which hours was the rate at which the
rain fell greater than the rate at which the rain
fell between hours 0 and 1?
A. between hours 3 and 4
B. between hours 4 and 5
C. between hours 5 and 6
D. between hours 7 and 8
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1000 MMR
312. Each day of the month, Carl earns an
allowance, in cents, equal to the square of that
date of the month. Which is a number of cents
Carl could earn in a single day?
A. 21
B. 31
C. 64
D. 111
Author: Victor A. Tondo Jr., LPT
316. Praetor jogged on a path that was 2 miles
long, took a break, and then jogged back along
the same path to where he started. He jogged at
different speeds for different distances along the
path as shown in the graph.
313. Which set of ordered pairs models a
function?
A. {(2, 9), (7, 5), (3, 14), (2, 6)}
B. {(5, 10), (5, 15), (5, 20), (5, 25)}
C. {(3, 10), (4, 15), (5, 20), (3, 25)}
D. {(–10, 20), (–20, 30), (–30, 40), (–40, 10)}
314. Rayon has a piece of rectangular paper that
is 12 inches wide by 16 inches long. He drew a
straight line along the diagonal of the paper.
What is the length of the line Rayon drew?
A. √28 inches
B. √192 inches
C. 20 inches
D. 28 inches
315. Which equation has infinitely many
solutions?
A. 2x + 4 = 7x + 9
B. 3(2x + 5) = 6x + 15
C. 4x + 13 = 5x + (20 – x)
D. x + 3 = 5x – 21
Between which times did Praetor jog the fastest?
A. 0 minutes and 10 minutes
B. 10 minutes and 25 minutes
C. 25 minutes and 30 minutes
D. 30 minutes and 60 minutes
317. Which expression has a value of -2?
A. |2| + |-4|
B. |-2| – |4|
C. |4| – |-2|
D. |-4| + |2|
318. Reion is tossing a six-sided number cube
labeled 1, 2, 3, 4, 5, and 6. What is the probability
of tossing 6 twice in a row?
A.
B.
C.
D.
319. Which represents the value of x in
6 – 4x 26?
A. x -8
B. x -8
C. x -5
D. x -5
320. The table below shows the resting heart
rates in beats per minutes of six students. The
rate, 40 beats per minute, seems to be an outlier.
Which measure of central tendency changes the
least by dropping 40 from the data?
Heart
Rate
78
71
A. mean
C. mode
This is a free reviewer. All rights reserved.
79
80
B. median
D. range
40
71
1000 MMR
321. The sum of a number, n, and 5 is subtracted
from 8. Which expression represents this
statement?
A. 8 – (n + 5)
B. (n + 5) + 8
C. (n + 5) – 8
D. 8 + (n + 5)
Author: Victor A. Tondo Jr., LPT
328. Which of the following is closest to the
value of the expression below?
(20.0143642359)2 x 8π
A. 1,000
B. 10,000
C. 100,000
D. 1,000,000
322. How is 0.5600 written in scientific notation?
A. 5.6 × 10
B. 5.6 × 10-1
C. 5.6 × 10-2
D. 5.6 × 10-3
329. Which of the following expressions has a
value of 0?
A. (2 – 3) – (2 – 3)
B. (2 – 3) – |2 – 3|
C. (2 – 3) + (-3 + 2)
D. |2 – 3| – (2 – 3)
323. What is the value of x in 3(x – 4) = –21?
A. x = –11
B. x = –3
C. x = 3
D. x = 11
324. In the spinner, what is the probability of the
arrow NOT landing on the space with the ∆?
330. What is the factorization of 10x2 – x – 21?
A. (5x – 7) (2x + 3)
B. (5x + 7) (2x – 3)
C. (5x + 3) (2x – 7)
D. (5x – 3) (2x – 7)
331. Evaluate: (√343)2
A. 7√7
B. 49
C. 49√7
D. 343
332. Find the length of the latus rectum of the
A.
B.
C.
D.
325. Which values of x and y make the system of
equations below true?
2x - y = -1
3x - y = -3
A. x = -4; y = -7
C. x = 2; y = 5
B. x =-2; y = -3
D. x = 4; y = 15
326. The lengths of two sides of a triangle are 8
inches and 13 inches. Which of the following
represents x, the possible length in inches of the
remaining side of the triangle?
A. 5 < x < 21
B. 5 x 21
C. x < 5 or x > 21
D. x 5 or x 21
327. What is the value of the expression
below?
80 ÷ ( 6 + (3 – 5) x 2)
A. -8
B. 8
C. 10
ellipse defined by
A.
B.
(
)
+
(
)
C.
= 1.
D.
333. Let A be a set such that A = {v, w, x, y, z}.
How many subsets does set A have?
A. 5
B. 10
C. 25
D. 32
334. Solve for x:
2 (5x – 11) + 7 = 3 (x – 7) – 15
A. x = 3
B. x = 1
C. x = -1
D. x = -3
335. What is the 4th term in the expansion of
(2x + 3y)7?
A. 15120 x4y3
B. 7560 x4y3
C. 3780 x3y4
D. 1890 x3y4
D. 40
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1000 MMR
336. The shell shape
, as used in the Mayan
numeral system, is the symbol for which
number?
A. 100
B. 10
C. 1
D. 0
337. Which of the following is irrational?
̅̅̅̅̅
A. 0.125
B. 43.29%
C. √200
D. √343
338. Aira is six years older than Zayne. Six years
ago, she was twice as old as he. How old is Aira
now?
A. 21
B. 18
C. 15
D. 12
339. The two parallel sides of a trapezoidal lot
measure 100m and 70m. If these sides are 80m
apart, what is the area of the lot?
A. 13600 m2
B. 6800 m2
C. 3400 m2
D. 2400 m2
340. If y = x and y = 2x + 2, find the value of x.
A. x = -2
B. x = -1
C. x = 0
D. x = 1
341. If the difference between the squares of two
consecutive counting numbers is 49, what is the
larger number?
A. 99
B. 49
C. 25
D. 7
342. Rayon needed to find the perimeter of an
equilateral triangle whose sides measure x + 4
cm each. Jake realized that he could multiply
3 (x + 4) = 3x + 12 to find the total perimeter in
terms of x. Which property did he use to
multiply?
A. Associative Property of Addition
B. Distributive Property of Multiplication over
Addition
C. Commutative Property of Multiplication
D. Inverse Property of Addition
Author: Victor A. Tondo Jr., LPT
343. A ride in a Feak Taxi costs P25.00 for the
first km and P10.00 for each additional km.
Which of the following could be used to calculate
the total cost, y, of a ride that was x km?
A. y = 25x + 10
B. y = 10x + 25
C. y = 25(x 1) + 10
D. y = 10(x 1) + 25
344. Which of the following points is in the
fourth quadrant?
A. (3, 4)
B. (-3, 4)
C. (3, -4)
D. (-3, -4)
345. The distance from the sun to the earth is
approximately 9.3 × 107 miles. What is this
distance expressed in standard notation?
A. 9,300,000,000
B. 930,000,000
C. 93,000,000
D. 651
346. The square of a number added to 25 equals
10 times the number. What is the number?
A. -10
B. -5
C. 5
D. 10
347. The sum of the square of a number and 12
times the number is 27. What is the smaller
possible value of this number?
A. -9
B. -3
C. 3
D. 9
348. Let x = 1. Find the corresponding y given
that 2x 3y = 5.
A. y = -1
B. y = 1
C. y = 3
D. y = -3
349. The sum of two consecutive even integers is
126. What is the smaller integer?
A. 63
B. 62
B. 61
D. 60
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1000 MMR
350. Factorize: a2 – a – 90
Author: Victor A. Tondo Jr., LPT
358. The cost of renting a bike at the local bike
shop can be represented by the equation
y = 2x + 2, where y is the total cost and x is the
number of hours the bike is rented. Which of the
following ordered pairs would be possible
number of hours rented and the corresponding
total cost?
A. (0, 2)
B. (2, 6)
C. (6, 2)
D. ( 2, 6)
A. (a – 10) (a + 9)
B. (a + 10) (a + 9)
C. (a + 10) (a – 9)
D. (a – 10) (a – 9)
351. Evaluate 10P5.
A. 2
B. 100,000
C. 1,024
D. 30,240
352. Jay bought twenty-five P4.57 stamps. How
much did he spend?
A. P 104.25
B. P 114.25
C. P 119.75
D. P124.25
353. Given f(x) = x3 + kx2 – 7, find k if f(2) = 41.
A. 5
B. 10
C. 15
D. 20
354. If y = x and y = 2x + 2, find x + y.
A. -8
B. -4
C. 0
D. 4
355. Mulan and Lilo are competing to see who
can sell the most candy bars for a fundraiser.
Mulan sold 4 candy bars on the first day and 2
each day after that. Lilo sold 7 on the first day
and 1 each day after that. On what day will they
have the same number of candy bars sold?
A. 7th
B. 6th
C. 4th
D. 3rd
356. Which of the following is not a polynomial?
A. -3x2 + x – 9
C.
√
B. √2x + π
D. 7
359. The distance from the earth to the moon is
approximately 240,000 miles.
What is this distance expressed in scientific
notation?
A. 24 × 104
B. 2.4 × 104
5
C. 2.4 × 10
D. 2.4 × 10 5
360.
=
A.
B.
C.
361. The sum of two angles is 180°. The measure
of one angle is 34° greater than the measure of
the other angle. What is the measure of the
smaller angle?
A. 74°
B. 73°
C. 72°
D. 71°
362. Solve the following system of linear
equations:
3x + y = -9
3x 2y = 12
A. ( 2, 3)
C. (3, 2)
B. (2, 3)
D. ( 3, 2)
+9
357. Factorize: 3p2 – 2p – 5
A. (3p – 5) (p + 1)
B. (3p + 5) (p – 1)
C. (3p + 1) (p – 5)
D. (3p – 1) (p + 5)
D.
363. Find the x-intercept of 3x + 2y = 24
A. x = 8
B. x = -8
C. y = 12
D. y = -12
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1000 MMR
364. A circle is drawn such that ̅̅̅̅ is a diameter
and its midpoint is O. Given that C is a point on
the circle, what is the measure of ACB?
A. 180o
B. 90o
o
C. 60
D. not enough info
Author: Victor A. Tondo Jr., LPT
370. Ten factorial is equal to _____.
A. 100
B. e10
C. 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
D. 10e
365. Samantha owns a rectangular field that has
an area of 3,280 square meters. The length of the
field is 2 more than twice the width. What is the
width of the field?
A. 40 m
B. 41 m
C. 82 m
D. 84 m
371. How many 3-digit numbers can be made
using the digits 5, 6, 7, 8, 9, and 0 if repetition is
not allowed?
A. 80
B. 100
C. 120
D. 140
366. Rayon used the following mathematical
statement to show he could change an
expression and still get the same answer on both
sides:
10 × (6 × 5) = (10 × 6) × 5
Which mathematical property did Rayon use?
A. Identity Property of Multiplication
B. Commutative Property of Multiplication
C. Distributive Property of Multiplication over
Addition
D. Associative Property of Multiplication
372. A researcher is curious about the IQ of
students at the Utrecht University. The entire
group of students is an example of a:
A. parameter
B. statistic
C. population
D. sample
373. Jordan filled a bottle with grains until it was
1/4 full and weighed 8 kg. He added more grains
into the bottle until it was 7/8 full. It now
weighed 18 kg. What is the mass of the empty
bottle?
A. 16
B. 8
C. 4
D. 2
374. If 37 – 4x < 17, then
A. x < 5
B. x > 5
C. x < -5
D. x > -5
367. Factorize x3 – 27y3.
A. (x – 3y) (x2 – 3x + 9y2)
B. (x – 3y) (x2 + 3x + 9y2)
C. (x + 3y) (x2 + 3x – 9 y2)
D. (x + 3y) (x2 – 3x + 9 y2)
368. What is the intersection of the lines
x + 3y = 5 and -2x + 4y = 0?
A. (2, 4)
B. (-2, 1)
C. (2, 1)
D. (-2, 4)
375. How many solutions are there for the
following system of linear equations?
-3x + 5y = 6
6x 10y = 0
A. only one solution
B. two solutions
C. infinitely many solutions
D. no solution
369. Given f(x) = 7x3 – 3x2 + 2x – 9, f(2) =
A. 56
B. 48
C. 44
D. 39
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1000 MMR
376. Find a and b so that the system below has
the unique solution (-2, 3).
ax + by = 17
2ax by = 11
A. a = 3, b = -1
C. a = 1, b = 3
B. a = 1, b = 5
D. a = 1, b = 5
377. What is the probability choosing only one
vowel when three letters are randomly selected
from the word NUMBERS?
A.
B.
C.
D.
378. If A > B, which is always true?
A.
B. A2 > B2
C. A < B + 2
D. A – B > 0
379. Statistical techniques that summarize and
organize the data are classified as what?
A. Population statistics
B. Sample statistics
C. Descriptive statistics
D. Inferential statistics
380. Five-point Likert scales (strongly disagree,
disagree, neutral, agree, strongly agree) are
frequently used to measure motivations and
attitudes. A Likert scale is a:
A. Discrete variable.
B. Ordinal variable.
C. Categorical variable.
D. All of the above
381. What is the radius of the circle defined by
(x + 2)2 + (y – 3)2 = 16?
A. 256
B. 16
C. 8
D. 4
Author: Victor A. Tondo Jr., LPT
382. A teacher asks students to identity their
favorite reality television show. What type of
measurement scale do the different television
shows make up?
A. Nominal
B. Ordinal
C. Interval
D. Ratio
383. What is the center of the circle defined by
x2 + y2 – 8x + 6y – 10 = 0?
A. (-8, 6)
B. (8, -6)
C. (-4, 3)
D. (4, -3)
384. Find the equation of the line passing (1, 4)
with slope equal to 5.
A. y = 5x + 3
B. y = 5x + 1
C. y = 5x – 1
D. y = 5x – 3
385. The seminar rooms in the library are
identified by the letters A to H. A researcher
records the number of classes held in each room
during the first semester. What kind of graph
would be appropriate to present the frequency
distributions of these data?
A. Histogram
B. Scatterplot
C. Bar chart
D. Box plot
386. Find the slope of the line passing the points
A(2, 3) and B(-7, -15).
A. 1
B. -1
C. ½
D. 2
387. Factorize: 3n2 – 8n + 4
A. (3n – 2) (n – 2)
B. (3n – 2) (n + 2)
C. (3n + 2) (n + 2)
D. (3n + 2) (n – 2)
388. In now many ways can the letters
AAABBCDEEE be arranged in a straight line?
A. 50,400
B. 25,200
C. 12,600
D. 6,300
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1000 MMR
389. Find the smaller angle formed by the x-axis
and the line y = 5x.
A. 78.69o
B. 63.48o
C. 54.15o
D. 41.32o
Author: Victor A. Tondo Jr., LPT
395. Find the number of subsets having 4
elements of the set {1,2,3,4,5,6,7,8,9,10,11}.
A. 165
B. 330
C. 660
D. 1320
390. Convert 48o to radians.
A. π rad
B. π rad
396. There are ten true - false questions in an
exam. How many responses are possible?
A. 1024
B. 256
C. 20
D. 10
C.
π rad
D.
π rad
391. The median is always:
A. The most frequently occurring score in set of
data
B. The middle score when results are ranked in
order of magnitude
C. The same as the average
D. The difference between the maximum and
minimum scores.
392. A teacher gave a statistics test to a class of
Geography students and computed the measures
of central tendency for the test scores. Which of
the following statements cannot be an accurate
description of the scores?
A. The majority of students had scores above the
mean.
B. The majority of students had scores above the
median.
C. The majority of students had scores above the
mode.
D. All of the above options (A, B and C) are false
statements.
393. Find the area of a semicircle whose radius
measures 28 cm.
A. 784 π cm2
B. 392 π cm2
2
C. 28 π cm
D. 14 π cm2
394. Find the length of each side of an equilateral
triangle whose perimeter is 90 cm.
A. 45 cm
B. 30 cm
C. 22.5 cm
D. 10 cm
397. In a 500m speed skating race, time results
would be considered an example of which level
of measurement?
A. Nominal
B. Ordinal
C. Interval
D. Ratio
398. In how many ways can 4 girls and 5 boys be
arranged in a row so that all the four girls are
together?
A. 4,320
B. 8,640
C. 17,280
D. 34,560
399. A box contains 8 batteries, 5 of which are
good and the other 3 are defective. Two batteries
are selected at random and inserted into a toy. If
the toy only functions with two good batteries,
what is the probability that the toy will function?
A.
B.
C.
D.
400. IQ tests are standardized so that the mean
score is 100 for the entire group of people who
take the test. However, if you select a group of 50
who took the test, you probably would not get
100. What statistical concept explains the
difference between the two means?
A. Statistical error
B. Inferential error
C. Residual error
D. Sampling error
401. Which Mathematician pioneered the study
of conic sections?
A. Euclid
B. Apollonius
C. Archimedes
D. Hipparchus
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1000 MMR
402. A researcher studies the factors that
determine the number of children future couples
decide to have. The variable ‘number of children’
is a:
A. Discrete variable
B. Continuous variable
C. Categorical variable
D. Ordinal variable
403. Surface area and volume, center of gravity,
and hydrostatics are some of the studies of
which Mathematician?
A. Apollonius
B. Euclid
C. Archimedes
D. Hipparchus
404. The book Philosophiæ Naturalis Principia
Mathematica, more fondly known simply as
Principia, is the work of which Mathematician?
A. Euclid
B. Newton
C. Einstein
D. Archimedes
405. A researcher is interested in the travel time
of Rayon’s University students to college. A
group of 50 students is interviewed. Their mean
travel time is 16.7 minutes. For this study, the
mean of 16.7 minutes is an example of a
A. parameter
B. statistic
C. population
D. sample
406. Who is considered by many Mathematicians
as “The Last Universalist”?
A. Jules Henri Poincare
B. Hendrik Lorentz
C. Georg Cantor
D. Gottfried Wilhelm Leibniz
Author: Victor A. Tondo Jr., LPT
408. Which of the following sets of scores has the
greatest variability or range?
A. 2, 5, 8, 11
B. 13, 13, 13, 13
C. 20, 25, 26 ,27
D. 42, 43, 44, 45
409. This Mathematician was the first to describe
a pinwheel calculator in 1685 and invented the
wheel named in his honor, which was used in the
arithmometer, the first mass-produced
mechanical calculator. He also refined the binary
number system, which is the foundation of all
digital computers. Which Mathematician is this,
who is also crucial to the development of
computers?
A. Gottfried Wilhelm Leibniz
B. Charles Babbage
C. Ada Lovelace
D. Alexander Graham Bell
410. Solve for x, given 9x – 10 = 11x + 30
A. x = 40
B. x = 20
C. x = -20
D. x = -40
411. Using Calculus, this Mathematician
explained why tides occur, why the shapes of
planetary orbits are conic sections, and how to
get the shape of a rotating body of fluid, among
many other things. Which Mathematician is this?
A. Kepler
B. Euclid
C. Apollonius
D. Newton
412. Which of the following terms does NOT
describe the number 9?
A. rational number B. integer
C. real number
D. prime number
407. In the theory he developed, there are
infinite sets of different sizes (called
413. Which expression below is equal to 5?
cardinalities). Which Mathematician formalized
A. (1 + 2)2
B. 9 – 22
many ideas related to infinity and infinite sets
C. 11 10 × 5
D. 45 ÷ 3 × 3
during the late 19th and early 20th centuries?
A. Jules Henri Poincare
B. Hendrik Lorentz
C. Georg Cantor
D. Gottfried Wilhelm Leibniz
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1000 MMR
414. A bus picks up a group of tourists at a hotel.
The sightseeing bus travels 2 blocks north, 2
blocks east, 1 block south, 2 blocks east, and 1
block south. Where is the bus in relation to the
hotel?
A. 2 blocks north
B. 1 block west
C. 3 blocks south
D. 4 blocks east
415. When five is added to three more than a
certain number, the result is 29. What is the
number?
A. 24
B. 21
C. 8
D. 4
416. The math club is electing new officers.
There are 3 candidates for president, 4
candidates for vice-president, 4 candidates for
secretary, and 2 candidates for treasurer. How
many different combinations of officers are
possible?
A. 13
B. 96
C. 480
D. 17,160
417. Twelve points lie on a circle. How many
cyclic quadrilaterals can be drawn by using these
points? [Note: Cyclic quadrilaterals are
quadrilaterals whose vertices are on a circle.]
A. 48
B. 495
C. 11,880
D. 1,663,200
418. What is the variance for the following set of
scores? 143 143 143 143 143 143
A. 0
B. 2
C. 4
D. 25
Author: Victor A. Tondo Jr., LPT
421. Of the following Z-score values, which one
represents the location closest to the mean?
A. Z = +0.5
B. Z = +1.0
C. Z = -1.5
D. Z = -0.3
422. The sum of five consecutive integers is 215.
What is the largest of these integers?
A. 43
B. 44
C. 45
D. 46
423. You go to the cafeteria for lunch and have a
choice of 4 entrees, 5 sides, 5 drinks, and 4
desserts. Assuming you have one of each
category, how many different lunches could be
made?
A. 18
B. 81
C. 40
D. 400
424. What can be said about the following
statements?
i. Any quadrilateral with four congruent sides is
a square.
ii. Any square has four congruent sides.
A. Only the first statement is true.
B. Only the second statement is true.
C. Both statements are true.
D. Both statements are fall.
425. Out of 6 boys and 4 girls, a committee of 5
has to be formed. In how many ways can this be
done if we take 2 girls and 3 boys?
A. 120
B. 186
C. 240
D. 256
426. In the figure, which of the following will
yield the value of the hypotenuse x?
419. When 18 is subtracted from six times a
certain number, the result is 42. What is the
number?
A. 10
B. 4
C. -4
D. -10
420. Find the equation of the line passing (2, 3)
and (-7, -15)
A. y = 2x + 1
B. y = 2x – 1
C. y = x + 2
D. y = x – 2
A. x =
B. x =
C. x =
D. x = 10 tan 35o
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1000 MMR
427. The shortest side of a 30-60-90 triangle is
20.19 cm long. How long is the hypotenuse?
A. 40.38 cm
B. 30.29 cm
C. 34.97 cm
D. 17.48 cm
428. The hypotenuse of a 30-60-90 triangle is
34.96 cm long. How long is the shortest side?
A. 40.38 cm
B. 30.29 cm
C. 34.97 cm
D. 17.48 cm
Author: Victor A. Tondo Jr., LPT
434. Which triangle has the centroid, incenter,
circumcenter, orthocenter, and nine-point-center
at the same location?
A. isosceles right triangle
B. 30-60-90 triangle
C. equilateral triangle
D. hyperbolic triangle
435. Given cos θ =
A.
B.
and θ
QIV, find tan θ.
C.
429. The second angle of a triangle is three times
as large as the first. The measure of the third
angle is 40 degrees greater than that of the first
angle. How large is the first angle?
A. 28o
B. 30o
C. 35o
D. 38o
436. Find the measure of T:
430. Normally distributed data are normally
referred to as:
A. Bell-shaped
B. Asymmetrical
C. Skewed
D. Peaked
A. 59o
431. A population has a mean of μ=35 and a
standard deviation of σ=5. After 3 points are
added to every score of the population, what are
the new values for the mean and standard
deviation?
A. μ=35 and σ=5
B. μ=35 and σ=8
C. μ=38 and σ=5
D. μ=38 and σ=8
432. Given sin θ =
A.
B.
, find cos θ.
C.
D.
433. In a triangle, what is located 2/3 of the
distance from each vertex to the midpoint of the
opposite side?
A. centroid
B. incenter
C. circumcenter
D. orthocenter
B. 69o
C. 79o
D.
D. 89o
437. If the scores on a test have a mean of 26 and
a standard deviation of 4, what is the z-score for
a score of 18?
A. -1.41
B. 11
C. -2
D. 2
438. If a researcher sets a level of significance at
0.05 (i.e. 5%), what does this mean?
A. Five times out of 100, a significant result will
be found that is due to chance alone and not to
true relationship.
B. Ninety-five times out of 100, a significant
result will be found that is due to chance alone
and not to true relationship.
C. Five times out of 100, a significant result will
be found that is not due to chance, but to true
relationship.
D. None of the above
439. When does a researcher risk a Type I error?
A. Anytime the decision is ‘fail to reject’.
B. Anytime H0 is rejected.
C. Anytime Ha is rejected.
D. All of the above options
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1000 MMR
440. Solve for x:
A. 4
B. 5
Author: Victor A. Tondo Jr., LPT
446. How much water must be evaporated from
2000 mL of 30% acid solution to make a 50%
acid solution?
A. 800 mL
B. 850 mL
C. 900 mL
D. 950 mL
C. 6
D. 7
441. Which of the following is equidistant from
the vertices of the triangle?
A. circumcenter
B. orthocenter
C. incenter
D. centroid
442. Which of the following is equidistant from
the sides of the triangle?
A. circumcenter
B. centroid
C. orthocenter
D. incenter
443. Although rarely used in proving, what is the
extra line or line segment drawn in a figure to
help in a proof?
A. base line
B. auxiliary line
C. converse line
D. Euler’s line
444. What is the measure of V in the following
figure?
A. 60o
B. 65o
C. 70o
445. What is the intersection of all three
altitudes of a triangle?
A. incenter
B. centroid
C. orthocenter
D. circumcenter
D. 75o
447. Which of the following is the intersection of
angle bisectors of a triangle?
A. circumcenter
B. incenter
C. centroid
D. orthocenter
448. In terms of a conditional statement, what is
the statement formed by exchanging and
negating the antecedent and the consequent?
A. inverse
B. converse
C. adverse
D. contrapositive
449. What is formed when the hypothesis and
the conclusion of the conditional statement are
interchanged?
A. converse
B. inverse
C. adverse
D. contrapositive
450. What is formed when both the hypothesis
and the conclusion of the conditional statement
are negated?
A. converse
B. inverse
C. adverse
D. contrapositive
451. Which of the following is the converse of the
following statement?
“If two angles are congruent,
then they have the same measure.”
A. If two angles are not congruent, then they do
not have the same measure.
B. If two angles have the same measure, then
they are congruent.
C. If two angles do not have the same measure,
then they are not congruent.
D. If two angles are not congruent, then they
have the same measure.”
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1000 MMR
452. In which geometry are there no parallel
lines?
A. elliptic geometry
B. hyperbolic geometry
C. spatial geometry
D. solid geometry
453. What do we call the ratio of two numbers
(larger number: smaller number) whose ratio to
each other is equal to the ratio of their sum to
the larger number? [Note: This is applied in
Fibonacci sequences]
A. pi
B. golden ratio
C. 1.618
D. Euler’s ratio
454. Which of the following pertains to the law of
cosines?
A. c2 = a2 + b2 – 2 ab cos C
B. c2 = a2 + b2 + 2 ab cos C
C. c2 = a2 + b2 – ab cos C
D. c2 = a2 + b2 + ab cos C
455. Solve for x:
Author: Victor A. Tondo Jr., LPT
458. Which lines are not in the same plane and
do not intersect but are not parallel?
A. asymptotes
B. tangent lines
C. skew lines
D. directrices
459. Two adjacent angles whose distinct sides lie
on the same line are called what?
A. linear pair
B. vertical pair
C. alternate
D. corresponding
460. The point of concurrency of a triangle’s
three altitudes is called _____.
A. circumcenter
B. incenter
C. orthocenter
D. centroid
461. What do we call three positive integers with
the property that the sum of the squares of two
of the integers equals the square of the third?
A. Euclid’s triple
B. Pythagorean triple
C. Newton’s triple
D. Cartesian triple
462. Which of the following expressions will give
the value of x?
A. x = 8
C. x = 10
B. x = 9
D. x = 11
A. 50 tan 37o
C. 50 sin 37o
456. What do we call an angle formed by two
chords of the circle with a common endpoint
(the vertex of the angle)?
A. inscribed angle
B. tangential angle
C. circumscribed angle
D. interior angle
457. Find the inverse of y =
A. y-1 =
B. y-1 =
C. y-1 =
D. y-1 =
B. 50 cos 37o
D. 50 cot 37o
463. Solve for x:
.
A. 6
B. 7
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C. 7.5
D. 8
1000 MMR
464. A bus travels 600 km in 7 hrs and another
300 km in 5 hrs. What is its average speed?
A. 72.86 kph
B. 75 kph
C. 77.86 kph
D. 80 kph
465. A sniper on a cliff observes that the angle of
depression to his target is 30o. If the cliff is 10
meters high, how far must the bullet travel to hit
the sniper’s target?
A. 20 meters
B. 10√3 meters
C. 10√2 meters
D. 10 meters
466. Rowena received a total of 25 bills. These
bills are either P20 or P50 bills. If Rowena
received an amount of P800, how many P20 bills
did she receive?
A. 10
B. 13
C. 15
D. 17
467. How many ways can the word PILIPINAS be
rearranged?
A. 302
B. 3,024
C. 30,240
D. 302,400
468. A coin is tossed 60 times. Head appeared 27
times. Find the experimental probability of
getting heads.
A.
B.
C.
D.
469. A parabola is defined by the equation
5x = -3y2 – 4y + 2. Which of the following is true
about the parabola?
A. It opens to the left.
B. It opens to the right.
C. It opens upward.
D. It opens downward.
Author: Victor A. Tondo Jr., LPT
470. Find the equation of the circle whose center
is at (7, -24) given that it passes the point of
origin.
A. (x + 7)2 – (y – 24)2 = 961
B. (x – 7)2 + (y + 24)2 = 961
C. (x + 7)2 + (y – 24)2 = 625
D. (x – 7)2 + (y + 24)2 = 625
471. Ana left their house and jogged at a speed of
60 meters per minute. Bea followed her two
minutes later and jogged at a speed of 70 meters
per minute. How many minutes after Bea left
would she catch up with Ana?
A. 14
B. 13.5
C. 13
D. 12
472. How many mL of 40% acid must be added
to 1000 mL of 10% acid solution to make a 20%
acid solution?
A. 250
B. 500
C. 600
D. 750
473. The hypotenuse of a 30-60-90 triangle is
432 cm long. How long is the leg opposite the 30o
angle?
A. 216 cm
B. 216 √2 cm
C. 216 √3 cm
D. 432 cm
474. The shortest leg of a 30-60-90 triangle is
123 cm long. How long is the leg adjacent to the
30o angle?
A. 123 cm
B. 123√2 cm
C. 123√3 cm
D. 246 cm
475. How many odd 4digit numbers can be
formed using the digits 7, 6, 5, 4, 3, 2, and 1 if
repetition is not allowed?
A. 720
B. 480
C. 240
D. 120
476. There are 70 dogs and geese in a farm. If
there are a total of 200 legs, how many dogs are
there?
A. 25
B. 30
C. 35
D. 40
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1000 MMR
477. Find the maximum area of a rectangle if the
perimeter is set at 350 cm.
A. 8656.25 cm2
B. 7656.25 cm2
C. 6656.25 cm2
D. 5656.25 cm2
478. A rectangle is 60 cm long and 45 cm wide.
How long is its diagonal?
A. 75 cm
B. 85 cm
C. 95 cm
D. 105 cm
479. Find the length of the diagonal of a cube
given each side measures 17 cm.
A. √290 cm
B. 17√2 cm
C. 17√3 cm
D. 34 cm
480. Find the measure of each interior angle of a
regular 20-sided polygon.
A. 162o
B. 150o
C. 144o
D. 126o
481. How many diagonals does a regular 14sided polygon have?
A. 77
B. 91
C. 96
D. 101
482. How many ways can 14 people be seated in
a Ferris wheel given that each cart can only
contain one person?
A. 15!
B. 14!
C. 13!
D. 12!
483. There are 24 mangoes in a basket, of which
7 are rotten. What is the probability that when
randomly getting two mangoes at the same time,
both are rotten?
A.
B.
C.
D.
484. In a gathering of gamers and admins, there
are 24 gamers of which 6 are females, and 3
admins of which one is female. If a female is
randomly called, what is the probability that she
is an admin?
A.
B.
C.
D.
Author: Victor A. Tondo Jr., LPT
485. What is the remainder when 3x6+ 4x5 – 5x4
+ 6x3 + 7x2 – 8x + 3 is divided by (x – 1)?
A. 8
B. 9
C. 10
D. 11
486. Seven people have an average weight of 49
kg. A child was added to the group and the
average became 45 kg. How heavy is the child?
A. 15 kg
B. 16 kg
C. 17 kg
D. 18 kg
487. Six numbers have an average of 71. If 85 is
added to the group, what is the new average?
A. 72
B. 73
C. 74
D. 75
488. In an arithmetic sequence, the 7th term is 25
and the 10th term is 67. What is the common
difference?
A. 42
B. 21
C. 14
D. 7
489. Triangle ABC has sides measuring 20 cm, 20
cm, and 29 cm. What kind of triangle is ABC?
A. acute
B. right
C. obtuse
D. reflex
490. Find the surface area of a sphere given that
the sphere sits perfectly inside a cube whose
sides measure 20 cm each.
A. 400 cm2
B. 800 cm2
2
C. 1200 cm
D. 2400 cm2
491. Given f(x) = (25 x20 – 24x10)(x2 – 9x + 3),
find f ’(x).
A. f ‘(x) = 550 x21 – 4725 x20 + 1500 x19 – 288 x11
+ 2376 x10 – 720 x9
B. f ‘(x) = 450 x21 – 4725 x20 + 150 x19 + 288 x11
+ 2376 x10 – 720 x9
C. f ‘(x) = 550 x21 – 4725 x20 + 150 x19 – 288 x11
+ 2376 x10 – 720 x9
D. f ‘(x) = 450 x21 – 4725 x20 + 1500 x19 – 288 x11
+ 2376 x10 – 720 x9
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1000 MMR
492. Simplify: eln 2019 x
A. 2019x
B. 2019 x
C.
x
D.
Author: Victor A. Tondo Jr., LPT
499. Mocha can finish a job in 24 hours, while
her sister Tiramisu can do the same job in only
20 hours. How long will it take them to finish the
job by working together?
A.
hrs
B.
hrs
493. If the roots of a quadratic equation are
and , which of the following could be the
quadratic equation?
A. 63x2 + 22x – 21 = 0
B. 63x2 – 22x – 21 = 0
C. 63x2 + 22x + 21 = 0
D. 63x2 – 22x + 21 = 0
C. 11 hrs
494. If three more than twice a number is
seventeen less than seven times the number,
what is the number?
A. 2
B. 3
C. 4
D. 5
495. A team is to be made from a group of seven
teachers and six scientists. If the team is to be
composed two teachers and two scientists, how
many different ways can they form a team?
A. 325
B. 315
C. 300
D. 285
496. Find the measure of the smaller angle
formed by the hands of the clock at 11:20.
A. 130o
B. 135o
C. 140o
D. 145o
D.
hrs
500. What conic figure does the equation
x2 + y2 + 10x – 16y = -100 form?
A. Real circle
B. Degenerate circle
C. Imaginary circle D. Ellipse
End of first 500 items.
Please, take a break.
Remember:
Nasa Diyos ang awa,
Nasa tao ang gawa.
497. How many even 3-digit even numbers can
be formed using the digits 7, 6, 5, 4, 3, 2, 1, and 0
if repetition is not allowed?
A. 150
B. 160
C. 170
D. 180
498. There are 50 students in a class. Twenty of
them have a laptop. Thirty-two of them have a
smartphone. Seven of them have both a laptop
and a smartphone. How many of them have
neither a laptop nor a smartphone?
A. 4
B. 5
C. 6
D. 7
This is a free reviewer. All rights reserved.
1000 MMR
501. Victor, Praetor, and Rowena volunteered to
teach at a nearby daycare. Praetor worked for
twice as long as Rowena did. Victor worked
twice as many hours as Praetor. Altogether, they
worked for 56 hours. For how many hours did
Victor work?
A. 14
B. 16
C. 28
D. 32
502. Which of the following is the factorization of
the binomial x4 – 20194?
A. (x + 2019)2 (x – 2019)2
B. (x – 2019)4
C. (x2 + 20192) (x + 2019) (x – 2019)
D. (x – 4)(x + 4)
503. The average of 5 different counting
numbers is 143. What is the highest possible
value that one of the numbers can have?
A. 706
B. 705
C. 704
D. 703
504. What value of x will satisfy the equation:
0.25(20x – 2020) = x?
A. 125.25
B. 125.75
C. 126.25
D. 126.5
505. Three brothers inherited a cash amount of
P5,670,000 and they divided it among
themselves in the ratio of 2:3:4. How much more
is the largest share than the smallest share?
A. P1,260,000
B. P1,270,000
C. P630,000
D. P635,000
506. Rayon and Wena can do a job together in
four hours. Working alone, Rayon does the job in
six hours. How long will it take Wena to do the
job alone?
A. 12 hours
B. 11 hours
C. 10 hours
D. 9 hours
Author: Victor A. Tondo Jr., LPT
507. What is the sum of the first 2019 counting
numbers?
A. 2,021,019
B. 2,027,190
C. 2,039,190
D. 2,043,190
508. In a certain school, the ratio of boys to girls
is 4 is to 9. If there are 260 boys and girls in the
school, how many boys are there?
A. 100
B. 90
C. 85
D. 80
509. Solve for x: 8x + 9y = 17
9x + 10y = 18
A. x = 9
C. x = -9
B. x = 8
D. x = -8
510. Mr. Park is 20 years older than his son now.
Five years from now, the Mr. Park’s age will be 5
less than twice his son’s age. Find the son’s
present age.
A. 18
B. 20
C. 38
D. 40
510. What is the minimum value of
f(x) = 3x2 + 6x + 11?
A. 1
B. 2
C. 4
D. 8
511. What is the sum of the roots of
5x2 + 10x – 17?
A.
B.
C. 2
D. -2
512. If xy = 17 and x2 + y2 = 135, find x + y.
A. 13
B. 12.8749
C. 12.5249
D. 12.5
513. How many mL of 50% acid solution must be
added to 200mL of 10% acid solution to make a
40% acid solution?
A. 400 mL
B. 500 mL
C. 600 mL
D. 700 mL
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1000 MMR
514. Nine consecutive even numbers have a sum
of 180. What is the smallest of these even
numbers?
A. 18
B. 16
C. 14
D. 12
Author: Victor A. Tondo Jr., LPT
523. If two numbers have a product of 267 and
the sum of their squares is 250, what is their
sum?
A. -30√3
B. 28
C. 7√3 + 2√5
D. 30
515. If x = 13, which of the following is equal to
200?
A. 15x + 2
B. x2 + 2x + 5
3
C. x – 4x – 2
D. x2 + x + 2
524. How many ml of 10% acid must be added to
500 ml of 40% acid to make a 30% acid solution?
A. 1000 ml
B. 750 ml
C. 500 ml
D. 250 ml
516. For which value of k does 9x2 – kx + 25
have only one root?
A. 3.75
B. 7.5
C. 15
D. 30
525. 3log2 3 + 2 log2 5 – log2 7 = ______.
A. log2
B. log2
517. If A and B are the roots of x2 + 19x + 20,
what is AB?
A. 20
B. 2√5 + 5
C. 3√2 + 2√3
D. -19
518. 0.25 + 0.5 + 1 + 2 + 4 + … + 1024 = ____
A. 2046.75
B. 2047.75
C. 2048.75
D. 2049.75
519. How many terms are there in the sequence
2, 9, 16, 23, …, 345?
A. 40
B. 45
C. 50
D. 70
520. If 2x + 3 = 25, then what is (2x + 3)2 – 25?
A. 650
B. 625
C. 600
D. 575
521. What is 50% of 200% of 2019?
A. 1009.5
B. 2019
C. 4038
D. 8076
522. If 3x = 5 and 2y = 11, what is 12(x – y)?
A. -46
B. -45
C. 45
D. 46
C. log2
D. log2
526. Find the intersection of y = 3x + 4 and
y = 5x – 8.
A. (8/5, 0)
B. (-4/3, 0)
C. (1, 7)
D. (6, 22)
527. Find the remainder when
x5 – 3x3 + 5x2 – 7x + 9 is divided by (x – 1).
A. 11
B. 9
C. 7
D. 5
528. Multiply: (3x – 7) (5x + 9)
A. 15x2 – 8x – 63
B. 15x2 – 8x + 63
2
C. 15x + 8x – 63
D. 15x2 + 8x + 63
529. Multiply: (2x2 – 5x + 3) (3x + 4)
A. 6x3 – 7x2 – 11x + 12
B. 6x3 + 7x2 + 11x + 12
C. 6x3 – 7x2 + 11x + 12
D. 6x3 + 7x2 – 11x + 12
530. x varies directly as y and inversely as z. If
x = 40 when y = 8 and z = 2, what is x when
y = 24 and z = 4?
A. 120
B. 90
C. 60
D. 30
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1000 MMR
531. Pedro has 7 more P10 coins than P5 coins. If
he has a total of P475, how many P5 coins does
he have?
A. 23
B. 25
C. 27
D. 29
532. When a number is increased by 5, its square
also increases by 255. What is this number?
A. 22
B. 23
C. 24
D. 25
533. Find k such that 16x2 + kx + 81 is a perfect
square trinomial.
A. 144
B. 72
C. 36
D. 18
Author: Victor A. Tondo Jr., LPT
540. Rowena can do a job in 15 hours, while
Victor can do the same job in 25 hours. How long
will it take them to finish the job by working
together?
A. 9.375 hours
B. 9.25 hours
C. 9.125 hours
D. 8.875 hours
541. Armel is twice as heavy as Kuku. Gabbi is
17kg heavier than Kuku. The sum of their masses
is 217kg. How heavy is Kuku in kg?
A. 40
B. 45
C. 50
D. 55
542. Find + given x + y = 30 and xy = 219.
534. Solve for x: (x + 4)2 = (x – 6)2.
A. x = 0
B. x = ½
C. x = 1
D. no solution
535. When a number is increased by 3, its square
increases by 135. By what does its square
increase when the number is increased by 5?
A. 225
B. 235
C. 445
D. 485
536. Ten cans of soda and six hamburgers cost a
total of P440. Five cans of soda and seven
hamburgers cost a total of P360. How much is a
hamburger?
A. P31
B. P33
C. P35
D. P37
537. The product of two consecutive odd
numbers is 1763. What is the smaller number?
A. 31
B. 37
C. 41
D. 47
538. Find the remainder when the polynomial
x4 – 5x3 + 7x2 – 11x + 19 is divided by (x – 2).
A. 5
B. 3
C. 1
D. -1
539. If three-fourths of a number is 44 more than
its one-fifth, what is that number?
A. 120
B. 100
C.80
D. 60
A.
B.
C.
D.
543. The product of two consecutive odd
counting numbers is 1023. What is their sum?
A. 60
B. 64
C. 68
D. 72
544. There are 250 pigs and chickens in a farm,
all of which are healthy. If there are 720 legs in
total, how many pigs are there?
A. 110
B. 100
C. 90
D. 80
545. Solve for x given 13x + 17 = 15x – 21.
A. x = 17
B. x = 18
C. x = 19
D. x = 20
546. Factorize: x3 + 3x2 – 4x – 12
A. (x + 2) (x + 3) (x – 2)
B. (x + 4) (x + 3) (x – 1)
C. (x + 2) (x – 3) (x + 2)
D. (x + 4) (x – 3) (x + 1)
547. Factorize (x4 – 16) completely.
A. (x – 2)4
B. (x – 2)2 (x + 2)2
C. (x + 2)2 (x – 2)2
D. (x + 2) (x – 2) (x2 + 4)
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1000 MMR
548. √125 + √45
A. 6√5
C. 2√5
√245 = ____
B. 4√5
D. √5
549. The product of two consecutive even
counting numbers is 3968. Find the smaller
number.
A. 42
B. 46
C. 54
D. 62
550. A pipe can fill a pool in 5 hours while
another pipe can drain empty the pool in 10
hours. How long will it take to fill the pool if both
pipes are open?
A. 8 hours
B. 9.25 hours
C. 9.5 hours
D. 10 hours
551. Today, Rayon is 13 years old while his
father is thrice his age. How many years from
now will his father be twice as old as he?
A. 15
B. 13
C. 11
D. 10
552. Roy and Wena are on a seesaw. Roy weighs
48 kg and sits 160 cm to the left of the fulcrum. If
Wena weighs 60 kg, how far to the right of the
fulcrum must she sit to balance the seesaw?
A. 115 cm
B. 123 cm
C. 128 cm
D. 135 cm
553. Insert one term between 81 and 169 to
make a geometric sequence.
A. 113
B. 117
C. 121
D. 125
554. Find the common difference of an
arithmetic sequence whose 21st term is 1987 and
29th term is 2019.
A. 2.5
B. 3
C. 3.5
D. 4
555. Find the general term An of the sequence
7, 16, 25, 34, 43, 52, 61, …
A. An = 9n – 2
B. An = 8n – 1
C. An = 9n + 2
D. An = 8n + 1
Author: Victor A. Tondo Jr., LPT
556. Find the remainder when x4 – 4x3 + 3x2 +
5x – 6 is divided by (x – 2).
A. 3
B. 2
C. 1
D. 0
557. 2 + 9 + 16 + 23 + … + 135 = ____
A. 1353
B. 1360
C. 1370
D. 1377
558. Factorize: (2x + y) (2x + y + 3) + 2.
A. (2x + y + 1) (2x + y + 2)
B. (2x + y + 1) (2x + y + 6)
C. (2x + y + 2) (2x + y + 3)
D. (2x + y + 3) (2x + y + 4)
559. The area of a rectangle is (x2 – x – 30). If its
length is x + 5, what is its width?
A. x + 4
B. x – 3
C. x + 2
D. x – 6
560. Factorize 81x2 – 225y2 completely.
A. (9x + 15y)(9x – 15y)
B. 9(9x2 – 25y2)
C. 9(3x – 5y)2
D. 9(3x + 5y)(3x – 5y)
561. What are the missing terms in the series
3, 6, 12, 24, ___, 96, ____?
A. 48; 192
B. 36, 120
C. 46, 196
D. 46, 192
562. In a parking lot, there are 65 tricycles and
motorcycles. If there are 150 wheels in all, how
many tricycles are there?
A. 18
B. 19
C. 20
D. 21
563. Which fraction is equivalent to 0.425?
A. 21/50
B. 27/50
C. 27/40
D. 17/40
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1000 MMR
564. What are the zeroes of 6x2 – x – 35?
A. and
B. and
C. and
D. and
565. Seven more than four times a number is 79.
What is five less than three times the number?
A. 59
B. 49
C. 39
D. 29
Author: Victor A. Tondo Jr., LPT
571. Factorize 27x3 + 125.
A. (3x + 5)(9x2 + 15x + 25)
B. (3x + 5)(9x2 – 15x + 25)
C. (3x – 5)(9x2 + 15x + 25)
D. (3x – 5)(9x2 – 15x + 25)
572. What is the 5th term of (2x + 3)7?
A. 22680x3
B. 11340x3
C. 21600x3
D. 10800x3
566. Mr. Park is five times as old as his son. Eight
years from now, he will only be thrice as old as
his son. How old is Mr. Park now?
A. 30
B. 35
C. 40
D. 45
573. In the equation 2x + 5y = 50, what is x
when y is 8?
A. 2
B. 3
C. 4
D. 5
567. Mr. Yu is four times as old as his daughter.
Ten years from now, he will only be thrice as old
as her. How old is his daughter now?
A. 10
B. 15
C. 20
D. 25
574. X varies directly as Y and inversely as Z. If X
is 24 when Y is 30 and Z is 5, find X when Y is 36
and Z is 4.
A. 36
B. 32
C. 24
D. 16
568. The speed of the current of a river is 9 kph.
Rayon rows his boat upstream for 8 hours, and
then travels back to his original position by
rowing downstream for 4 hours. What is Rayon’s
speed on calm water?
A. 9 kph
B. 18 kph
C. 27 kph
D. 36 kph
575. What is the 4th term of (3x – 2)5?
A. 360x2
B. 240x2
2
C. -240x
D. -360x2
569. Thrice Rayon’s age 16 years ago is equal to
his age 16 years from now. How old is Rayon
now?
A. 16
B. 24
C. 32
D. 40
570. Divide 8x5 – 4x3 + 2x2 – 3x + 4 by x + 2.
A. 8x4 – 16x3 + 28x2 – 54x + 105 r. -206
B. 8x4 + 16x3 + 28x2 – 54x + 105 r. -206
C. 8x4 – 16x3 + 28x2 – 54x – 105 r. 206
D. 8x4 + 16x3 + 28x2 – 54x – 105 r. 206
576. Factorize 64x6 – y6.
A. (2x – y) (4x2 + 2xy + y2) (2x + y) (4x2 – 2xy + y2)
B. (2x – y) (4x2 + 2xy + y2) (2x + y) (4x2 + 2xy + y2)
C. (2x + y) (4x2 – 2xy – y2) (2x + y) (4x2 – 2xy + y2)
D. (2x + y) (4x2 – 2xy – y2) (2x + y) (4x2 – 2xy + y2)
577. Factorize x2 + 6x – y2 + 2y +8.
A. (x + y + 2) (x + y + 4)
B. (x + y + 2) (x – y + 4)
C. (x + y + 2) (x + y – 4)
D. (x + y + 2) (x – y – 4)
578. What value of x will satisfy the equation:
2.5(4x – 504) = x?
A. 150
B. 145
C. 140
D. 135
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1000 MMR
579. Factorize (2x + y) (2x + y + 6) + 5
A. (2x + y + 5) (2x + y + 1)
B. (2x + y – 5) (2x + y – 1)
C. (2x + y + 5) (2x – y + 1)
D. (2x – y + 5) (2x – y + 1)
580. Simplify: (2x3y4z2)5
A. 10x15y20z10
B. 25x15y20z10
C. 32x15y20z10
D. 100x15y20z10
5x2
581. Find k such that
+ 30x + k has two
unequal real roots.
A. k > 6
B. k < 6
C. k > 45
D. k < 45
582. Which of the following is not a polynomial?
A. π r2 + 2 π r
B. m – n
C. √2 x + √3 y – √5 z
D. ab √
cd
583. Which of the following is irrational?
A. 0.193193193193…
B.
C. √1023
D. √2401
584. If x + y = 17 and x2 + y2 = 135, find xy.
A. 78
B. 77
C. 76
D. 75
585. If H and K are the roots of 5x2 – 8x + 9, find
H + K – HK.
A. -2/5
B. -1/5
C. 0
D. 1/5
586. If H and K are the roots of 3x2 + 4x + 5, find
H2 + K2.
A. -5/9
B. -1
C. -14/9
D. -2
587. Factorize 9x2 + 24xy + 16y2 – 81z2.
A. (3x + 4y + 9z) (3x + 4y – 9z)
B. (3x – 4y + 9z) (3x + 4y – 9z)
C. (3x + 4y + 9z) (3x – 4y – 9z)
D. (3x – 4y + 9z) (3x – 4y – 9z)
Author: Victor A. Tondo Jr., LPT
588. Solve for x: 2x + 3y = -8, and 5x – 7y = 67
A. 5
B. 6
C. -5
D. -6
589. A farmer has 15 goats, 23 pigs, and a few
chickens in his farm. If he counted a total of 200
legs in his farm (excluding his, of course), how
many chickens does he have?
A. 96
B. 72
C. 48
D. 24
590. Rationalize
√
√
A.
√
√
√
B.
√
√
√
C.
√
√
√
D.
√
√
√
591. If 3x + 2 = 4y and 5y – 3 = z, express z in
terms of x.
A. 15x + 9
B. 15x + 7
C.
D.
592. If 72x + 3 = B, what is B ÷ 49x?
A. 343
B. 21
C. 7
D. 3
593. Multiply: (7x + 4) (2x – 1) (3x + 5)
A. 42x3 + 73x2 + 7x – 20
B. 42x3 – 73x2 + 7x – 20
C. 42x3 + 73x2 – 7x – 20
D. 42x3 – 73x2 – 7x – 20
594. Rayon invested P1,600,000.00 in a bank
that offers 2.5% interest per annum. How much
will his account hold after 4 years?
A. P160,000
B. P1,160,000
C. P1,460,000
D. P1,760,000
595. Solve for x: 45x+3 = 82x+6
A. 2
B. 3
C. 4
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D. 5
1000 MMR
596. Given
=
, which of the following
expressions is equal to x?
A.
B.
C.
D.
597. Add: (2x2 + 4x + 5) + (5 – 6x – x2)
A. x2 + 2x + 10
B. 3x2 + 2x + 10
2
C. x – 2x + 10
D. 3x2 + 2x + 10
598. Solve for x: 272x–8 = 9x+10
A. 9
B. 10
C. 11
Author: Victor A. Tondo Jr., LPT
605. Which of these has the longest perimeter?
A. A regular pentagon with sides 21 cm long
B. A rectangle 29 cm long and 24 cm wide
C. An equilateral triangle with sides 36 cm long
D. A square whose area is 1024 cm2
606. The hypotenuse of a right triangle is 82 feet.
If one leg is 80 feet, what is the length of the
other leg in feet?
A. 15
B. 16
C. 17
D. 18
D. 12
599. If 25x – 49y = 2019, what is 49y – 25x?
A. 2018
B. -2019
C.
D. cannot be determined
600. If x + y = 90 and x2 + y2 = 2020, find xy.
A. 3025
B. 3030
C. 3035
D. 3040
601. Which of the following is ALWAYS true?
A. Vertical pairs of angles are supplementary.
B. Vertical pairs of angles are congruent.
C. Linear pairs of angles are congruent.
D. Linear pairs of angles are complementary.
602. How many line segments can be made from
28 non-collinear points?
A. 378
B. 394
C. 412
D. 422
603. The vertex angle of an isosceles triangle is
70°. What is the measure of one of the base
angles?
A. 45°
B. 50°
C. 55°
D. 60°
604. A shoebox measures 20 inches by 15 inches
by 9 inches. What is its volume in cubic inches?
A. 900
B. 1350
C. 1650
D. 2700
607. Find the surface area of a rectangular box
whose dimensions are 25 cm x 35 cm x 45 cm.
A. 7000 cm2
B. 7045 cm2
2
C. 7150 cm
D. 7745 cm2
608. A and B form a linear pair. If m A = 3x
and m B = 5x + 40, what is the value of x?
A. 17.5
B. 18
C. 19
D. 19.75
609. 1 and 3 are opposite angles in a
parallelogram. If m 1 = 50o, what is m 3?
A. 40o
B. 50o
C. 140o
D. 130o
610. A and C are consecutive angles in a
parallelogram. If m A = 60o, what is m C?
A. 40o
B. 50o
C. 150o
D. 120o
611. Two parallel lines are cut by a transversal,
forming R and V. If the two angles are
corresponding angles, what is the measure of R
if the measure of V is 70o?
A. 35o
B. 70o
o
C. 160
D. 110o
612. If the sum of the supplement and the
complement of an angle is 100 degrees, what is
the angle?
A. 65o
B. 70o
C. 75o
D. 85o
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1000 MMR
613. It is the perpendicular bisector of a regular
polygon’s side, passing through the center.
A. side
B. apothem
C. asymptote
D. diagonal
Author: Victor A. Tondo Jr., LPT
620. In the figure, A is the center of the circle.
What is the measure of BDC?
A. 160o
B. 80o
C. 40o
D. 20o
614. In the figure,
// . If m AFE = 130o,
find the measure of CGH.
621. Find the measure of E using the figure of
parallelogram WENA below.
A. 50o
B. 130o
C. 65o
D. 25o
615. The measure of each interior angle of a
regular polygon is 170o. How many vertices does
it have?
A. 36
B. 24
C. 12
D. 10
616. How many diagonals does a regular
dodecahedron have?
A. 45
B. 54
C. 60
D. 66
617. Find the area of a square whose perimeter
is 480 cm.
A. 14400 cm2
B. 28800 cm2
C. 57600 cm2
D. 270400 cm2
618. A circle has a diameter of 34 cm. What is its
area?
A. 17 π cm2
B. 34 π cm2
C. 289 π cm2
D. 1156 π cm2
619. The perimeter of a rectangle is 84. If its
length is 29 cm, find its area.
A. 377 cm2
B. 1218 cm2
C. 2436 cm2
D. 4872 cm2
A. 15
B. 65
C. 115
D. 85
622. In parallelogram
MATH, m A = 7x – 32 and
m T = 5x + 32. Find m A.
A. 15
B. 63
C. 73
D. 117
623. The diagonal of a rectangular prism is 17
cm long. If it is 9 cm thick and 12 cm long, how
wide is it?
A. 15 cm
B. 10 cm
C.8√3 cm
D. 8 cm
624. Statement 1: A square is a rhombus.
Statement 2: A square is a rectangle.
A. Only the first statement is true.
B. Only the second statement is true.
C. Both statements are true.
D. Both statements are false.
625. Which of the following has its incenter,
circumcenter, centroid, and orthocenter in just
one point?
A. Equilateral Triangle
B. Right Triangle
C. Scalene Triangle
D. Obtuse Triangle
This is a free reviewer. All rights reserved.
1000 MMR
626. In triangles, this line segment is drawn from
midpoint of one side to its opposite vertex.
A. Median
B. Altitude
C. Bisector
D. Longitude
627. The radius of a cone measures 12 cm and its
height is 15 cm. Find its volume.
A. 240 π cm3
B. 360 π cm3
C. 480 π cm3
D. 720 π cm3
628. This is located at the intersection of the
perpendicular bisectors of a triangle.
A. Centroid
B. Circumcenter
C. Incenter
D. Orthocenter
Author: Victor A. Tondo Jr., LPT
634. Find the volume of a rectangular pyramid
given that its base has length 21 cm and width 10
cm, and its height is 15 cm.
A. 700 cm3
B. 1050 cm3
3
C. 1575 cm
D. 1400 cm3
635. The diameter of a cylinder is 24 cm. If its
height is 40 cm, find its surface area.
A. 312 π cm2
B. 624 π cm2
2
C. 936 π cm
D. 1248 π cm2
636. In the figure, AC is a diagonal of rhombus
ABCD. If m B = 118o, what is m CAD?
A. 124o
B. 118o
C. 62o
D. 31o
629. The radius of a cylinder measures 9 cm and
its height is 15 cm. Find its volume.
A. 1215 π cm3
B. 810 π cm3
C. 540 π cm3
D. 360 π cm3
630. What is the measure of each exterior angle
of a regular decagon?
A. 108o
B. 72o
C. 60o
D. 36o
631. Find the area of an isosceles trapezoid given
bases measuring 20 cm and 30 cm, with each of
the congruent slants measuring 13 cm.
A. 250 cm2
B. 275 cm2
2
C. 300 cm
D. 325 cm2
632. Which of the following side lengths belong
to an acute triangle?
A. 5 cm, 7 cm, 9 cm
B. 6 cm, 8 cm, 10 cm
C. 6 cm, 9 cm, 12 cm
D. 7 cm, 9 cm, 11 cm
633. A prism has a right triangle as its base. The
two legs of the right triangular base are 6 cm and
8 cm long. If the prism is 2 cm thick, find its
lateral surface area.
A. 24 cm2
B. 48 cm2
2
C. 56 cm
D. 72 cm2
637. A rhombus has diagonals measuring 20 cm
and 30 cm. What is its area?
A. 75 cm2
B. 150 cm2
C. 300 cm2
D. 600 cm2
638. A rhombus as diagonals measuring 30 cm
and 16 cm. What is its perimeter?
A. 136 cm
B. 92 cm
C. 68 cm
D. 46 cm
639. Find the approximate distance around a
semicircular park with radius 100m.
A. ~1028 m
B. ~771 m
C. ~514 m
D. ~257 m
640. Find the equation of the line perpendicular
to 7x + 4y = 12, passing through (-2, 9).
A. 4x – 7y = -71
B. 4x + 7y = 71
C. 4x – 7y = 71
D. 4x + 7y = -71
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1000 MMR
641. Find the equation of the line perpendicular
to 9x –7y = -6, passing through (4, -3).
A. 7x + 9y = 1
B. 7x + 9y = -1
C. 7x – 9y = 1
D. 7x – 9y = -1
642. A rectangle is drawn with dimensions 28 cm
by 42 cm. A larger rectangle is drawn by adding
5 cm margins from each side of the original
rectangle. What is the area of the larger
rectangle?
A. 1456 cm2
B. 1551 cm2
2
C. 1556 cm
D. 1976 cm2
Author: Victor A. Tondo Jr., LPT
646. In the figure below, m CAB = 2x + 4, while
m COB = 5x – 7. Find x.
A.
B. -2
C.
D. 15
647. In the figure below,
//
. If m AGE =
7x + 11 and m CHF = 5x + 1, what is m AGE?
643. Find the length of the diagonals of a
rectangular prism 6 cm thick, 15 cm long, and 10
cm wide.
A. 17√3 cm
B. 18√2 cm
C. 19 cm
D. 20√5 cm
A. 119o
B. 109o
C. 71o
D. 61o
644. Find the volume of the following pool.
648. Find the sum of all interior angles of a
regular 15-sided polygon.
A. 5400o
B. 2700o
o
C. 2340
D. 1170o
A. 700 m3
C. 1400 m3
B. 1050 m3
D. 2800 m3
645. What is the slope of the line defined by the
equation 3x + 5y = 11?
A.
B.
C. 3
D. 5
649. If each square in the figure has an area of 36
square cm, find the perimeter of the figure.
A. 864 cm
C. 144 cm
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B. 792 cm
D. 132 cm
1000 MMR
650. Find the area of the shaded part of the
circle.
A. 9720 π cm2
C. 540 π cm2
657. A square has a diagonal measuring 28 cm.
Find its area.
A. 784 cm2
B. 392 cm2
C. 196 cm2
D. 98 cm2
B. 972 π cm2
D. 270 π cm2
651. In the figure, ̂ = 120o and m E = 28o.
Find ̂ .
A. 46o
B. 54o
C. 64o
Author: Victor A. Tondo Jr., LPT
656. Find the equation of the line passing
through (2, -7) with a slope of -3.
A. y = -3x – 1
B. y = -3x + 1
C. y = -3x – 2
C. y = -3x + 2
D. 72o
652. Find the volume of a cone whose radius is
30 cm and height is 170 cm.
A. 289,000 π
B. 153,000 π
C. 51,000 π
D. 17,000 π
658. Symbol for line.
A.
B.
C.
D.
659. Symbol for ray.
A.
B.
C.
D.
660. Find the volume of the swimming pool
drawn below.
A. 2000 cu. ft.
B. 2100 cu. ft.
C. 2250 cu. ft.
D. 2500 cu. ft.
653. Find the center of a circle given one of its
longest chords has endpoints at M(-11,8) and
N(23,-24).
A. (12, -16)
B. (6, -8)
C. (-17,16)
D. (17, -16)
654. Find the radius of a circle if its diameter has
endpoints at M(3,5) and N(-12,13).
A. 8
B. 8.5
C. 9
D. 9.5
655. Find the equation of the line passing
through (7, 5) and (5, 1).
A. y = 2x + 9
B. y = 2x – 9
C. y = ½ x + 9
D. y = ½ x – 9
661. Which geometric figure extends indefinitely
in two opposite directions?
A. angle
B. ray
C. line
D. line segment
This is a free reviewer. All rights reserved.
1000 MMR
662. In the figure, ACB is formed by the
tangents AC and BC. If m ACB = 34o, what is the
measure of the minor intercepted arc?
Author: Victor A. Tondo Jr., LPT
670. Find the equation of a circle if its diameter
has endpoints at M(3,5) and N(-12,13).
A. (x + 4.5)2 + (y + 9)2 = 72.25
B. (x – 4.5)2 + (y + 9)2 = 72.25
C. (x + 4.5)2 + (y – 9)2 = 72.25
D. (x – 4.5)2 + (y – 9)2 = 72.25
A. 112o
671. A bicycle’s wheel is 24 inches in diameter. If
it revolves 625 times, how far does the bicycle
travel?
A. 7,500 π
B. 15,000 π
C. 30,000 π
D. 60,000 π
B. 146o
C. 68o
D. 34o
663. X(7,9) is the midpoint of A(1,2) and B. Find
the coordinates of B.
A (-13, -16)
B. (13, -16)
C. (-13, 16)
D. (13, 16)
664. Point O is two-thirds the way from A(-4,5)
to B(8,-10). Find the coordinates of point O.
A. (0, 0)
B. (4, -5)
C. (3, -6)
D. (3, -4)
665. H and K form a vertical pair. If m H =
4x + 20 and m K = 6x – 10, find x.
A. 13
B. 15
C. 17
D. 19
672. A right triangle is inscribed in a circle. If the
triangle’s legs are 16 cm and 30 cm, what is the
area of the circle?
A. 289 π cm2
B. 480 cm2
C. 480 π cm2
D. 1156 π cm2
673. In the figure, a rectangle is
inscribed in a circle. If the rectangle
is 20 cm by 48 cm, what is the area of
the shaded figure?
A. 676π – 960 cm2
B. 1352π – 960 cm2
2
C. 2304π – 960 cm
D. insufficient data
No item #666, as requested by many subscribers.
667. F and G form a linear pair. If m F =
4x + 20 and m G = 6x – 10, find x.
A. 13
B. 15
C. 17
D. 19
668. Point C is two-fifths the way from M(-3,4) to
N(7,-11). Find the coordinates of point C.
A. (0, 0)
B. (2, -1)
C. (1, -2)
D. (1, -1)
674. What is the area of the red trapezium in the
given figure?
A. 23 sq. units
C. 25 sq. units
669. Find the slope of the line 5x – 6y = 7.
A.
B.
C.
D.
This is a free reviewer. All rights reserved.
B. 24 sq. units
D. 26 sq. units
1000 MMR
675. Which of the following are not congruent?
A. Corresponding angles formed by two parallel
lines cut by a transversal.
B. Alternate exterior angles formed by two
parallel lines cut by a transversal.
C. Pairs of vertical angles.
D. Interior angles on the same side of the
transversal cutting two parallel lines.
Author: Victor A. Tondo Jr., LPT
682. Which of the following is false?
A. Radii of the same circle are always congruent.
B. Any two right angles are congruent.
C. An inscribed right angle always intercepts a
semicircle.
D. There are infinitely many lines that may be
drawn parallel to a given line passing through a
given point.
676. A rectangle is 24 cm long. If its diagonal is
25 cm, what is its perimeter in cm?
A. 49
B. 62
C. 64
D. 98
683. The volume of a cube is 343 cm3. What is its
surface area?
A. 24 cm2
B. 24√3 cm2
C. 48 √13 cm2
D. 294 cm2
677. Square FACE is drawn such that one of its
sides AC is the diagonal of a rectangle, ABCD. If
AB = 20 and BC = 43, find the area of FACE.
A. 2209 cm2
C. 2249 cm2
B. 2229 cm2
D. 2269 cm2
684. Find the volume of a rectangular plank of
wood that is half-inch thick, six inches wide, and
6 feet long.
A. 18 cubic inches B. 36 cubic inches
C. 108 cubic inches D. 216 cubic inches
678. If the vertex angle of an isosceles triangle is
50o, what is the measure of each base angle?
A. 130o
B. 80o
C. 65o
D. 25o
685. In the following figure, O is the center of the
circle. If m CAB = 36o, what is m DBA?
679. Which of the following is a square?
A. VCTR whose sides measure 20 cm each
B. LUVS whose angles are 90o each
C. RWNA whose diagonals are 50 cm each
D. AYIE whose congruent diagonals
perpendicularly bisect each other
A. 18o
680. The following are measures of sides of four
triangles. Which of them are taken from an
obtuse triangle?
A. 20, 20, 29
B. 20, 21, 29
C. 20, 22, 28
D. 20, 23, 30
681. Find the length of the intercepted arc of a
central angle measuring 60o given that the radius
of the circle is 12 cm.
A. 4 π cm
B. 6 π cm
C. 8 π cm
D. 12 π cm
B. 36o
C. 54o
D. 72o
686. What is the measure of each interior angle
of a regular nonagon?
A. 135o
B. 140o
o
C. 144
D. 145o
687. Find the surface area of a ball whose radius
is 24 cm.
A. 576 π cm2
B. 1152 π cm2
2
C. 2304 π cm
D. 4608 π cm2
This is a free reviewer. All rights reserved.
1000 MMR
688. Give the converse of the conditional
statement “If two angles are vertical angles, then
they are congruent.”
A. “If they are are congruent, then two angles are
vertical angles.”
B. “If two angles not vertical angles, then they are
not congruent.”
C. “If they are not congruent, then two angles are
not vertical angles.”
D. “If two angles are linear angles, then they are
supplementary.”
689. Give the inverse of the conditional
statement “If two angles are vertical angles, then
they are congruent.”
A. “If they are are congruent, then two angles are
vertical angles.”
B. “If two angles not vertical angles, then they are
not congruent.”
C. “If they are not congruent, then two angles are
not vertical angles.”
D. “If two angles are linear angles, then they are
supplementary.”
Author: Victor A. Tondo Jr., LPT
693. How many diagonals does a regular
decagon have?
A. 30
B. 35
C. 40
D. 45
694. Each side of a regular hexagon is 256 cm
long. Find the distance around it.
A. 1536 cm
B. 1280 cm
C. 1024 cm
D. 718 cm
695. What can be said about these statements?
i. Any rectangle has two congruent diagonals.
ii. Any rhombus has two perpendicular
diagonals.
A. Only the first statement is true.
B. Only the second statement is true.
C. Both statements are true.
D. Both statements are false.
696. In the figure, A is the midpoint of
and
. By which triangle congruence postulate can
we prove that △CAB △DAE?
690. In which geometry do two distinct lines
intersect in two points?
A. Euclidean Geometry
B. Hyperbolic Geometry
C. Elliptic Geometry
D. Plane Geometry
691. In which geometry can the sum of the
measures of the angles of a triangle be less than
180 degrees?
A. Euclidean Geometry
B. Hyperbolic Geometry
C. Elliptic Geometry
D. Plane Geometry
A. SSS Congruence Postulate
B. SAS Congruence Postulate
C. ASA Congruence Postulate
D. ASS Congruence Postulate
697. Find the circumference of a coin whose
radius is 13 cm.
A. 13 π cm
B. 26 π cm
C. 169 π cm
D. 676 π cm
692. In which geometry are the summit angles of
a Saccheri quadrilateral obtuse angles?
A. Euclidean Geometry
B. Hyperbolic Geometry
C. Elliptic Geometry
D. Plane Geometry
This is a free reviewer. All rights reserved.
1000 MMR
698. Find the area of a square whose perimeter
is 280 cm.
A. 1120 cm2
B. 4900 cm2
C. 19600cm2
D. 78400 cm2
Author: Victor A. Tondo Jr., LPT
707. The longest side in a 30-60-90 triangle is
480 cm. How long is the shortest side?
A. 120 cm
B. 160 cm
C. 240 cm
D. 360 cm
699. Find the altitude to the hypotenuse of a
right triangle whose legs are 60 cm and 80 cm.
A. 40 cm
B. 48 cm
C. 96 cm
D. 100 cm
708. If the hypotenuse of a 45-45-90 triangle is
50 cm long, how long is a leg?
A. 25 cm
B. 25√2 cm
C. 25√3 cm
D. 50√2 cm
700. This is formed by two rays with a common
end point.
A. polygon
B. angle
C. line
D. line segment
709. Which of the ff. is coterminal with 143o?
A. 2403o
B. 2303o
o
C. 2103
D. 2003o
701. Convert
A. 210o
C. 630o
710. What is the measure of the smaller angle
formed by the hands of the clock at 5:55?
A. 180o
B. 174.75o
o
C. 163.625
D. 152.5o
rad to degrees.
B. 420o
D. 840o
702. What is the reference angle of 212o?
A. 32o
B. 68o
C. 106o
D. 12o
703. A negative sine and a positive cosine are
properties of angles in which quadrant?
A. QI
B. QII
C. QIII
D. QIV
704. Which of the following equal to sec 50o?
A. sin 40o
B. cos 40o
o
C. csc 40
D. cot 40o
705. Which of the ff. is coterminal with 2019o?
A. 219o
B. 209o
C. 199o
D. 189o
711. What is the measure of the smaller angle
formed by the hands of the clock at 3:50?
A. 185o
B. 175o
o
C. 170
D. 165o
712. Which of the following is false?
A. tan θ =
B. csc θ =
C. sec θ =
D. sin θ =
714. If sin x = 0.936, which of the following could
be cos x?
A. 0.064
B. 0.8
C. 0.016
D. -0.352
706. The angle of elevation to the top of a
building is 30o. If the observer is 50√3 meters
away from the building, how tall is the building?
A. 25 m
B. 25√3 m
C. 50 m
D. 100 m
This is a free reviewer. All rights reserved.
1000 MMR
715. A man stands 20 meters away from a
building. If the angle of elevation to the top of the
building is 60o, how tall is the building?
A. 20√2 m
B. 20√3 m
C. 40 m
D. 20√5 m
Author: Victor A. Tondo Jr., LPT
724. A sniper is on top of a 20-meter cliff. He
spots a target at an angle of depression of 30o.
How far must his bullet travel to hit the target?
A. 20 meters
B. 20 √2 meters
C. 20 √3 meters
D. 40 meters
716. If csc x =
725. A meter stick leans to a wall and reaches a
point 50 cm away from the wall. What is the
measure of the angle formed by the meter stick
and the wall?
A. 30o
B. 45o
C. 60o
D. 75o
A.
, what is sin x?
B.
C.
D.
717. If sec x = , which of the following could be
sin x?
A.
B.
C.
D.
718. Which of the following is false?
A. cos A (tan A) = sin A
B. csc A (cos A) = cot A
C. sin A (sec A) = tan A
D. tan A (csc A) = cos A
719. Which of the ff. is coterminal with 63o?
A. 7503o
B. 7303o
o
C. 7103
D. 6903o
726. A tight rope connects the top of a 500 cm
pole to the ground. If the angle formed by the
ground and the tight rope is 45o, how long is the
tight rope?
A. 500 cm
B. 500√2 cm
C. 500√3 cm
D. 500√5 cm
727. Which of the
following can be used
to solve for x in the
figure?
A. sin
B. cos
720. Convert 160o to radians.
A. π
B. π
C. π
C. arcsin
D. π
D. arccos
721. What is the reference angle of 156o?
A. 14o
B. 24o
C. 56o
D. 66o
728. Evaluate: 2 sin 30 – 4 cos 60 + 3 tan 45.
A. -10.2378
B. -12.1426
C. 2
D. 1
722. In which quadrants is the sine function
negative?
A. QI and QII
B. QII and QIII
C. QIII and QIV
D. QIV and QI
729. Which of the following is true?
A. sin x = sin (-x)
B. cos x = cos (90 – x)
C. –sin x = sin (-x) D. tan x = (sin x)(cos x)
723. What is the reference angle if 2019o?
A. 19o
B. 39o
C. 51o
D. 59o
This is a free reviewer. All rights reserved.
1000 MMR
730. Which of the following is true?
A. sin x = csc (90 – x)
B. cos x = csc (90 – x)
C. tan x = cot (90 – x)
D. sec x = cos (90 – x)
Author: Victor A. Tondo Jr., LPT
740. Give the range of f(x) = 2020.
A. y | y
B. y 2020
C. y = 2020
D. y = { }
741. Give the range of f(x) =
731. Evaluate: lim
A. 2019
C. 0
A. y | y
C. y 2
2019
B. -2019
D. 1
5x2)
732. Evaluate: lim (20 –
A. 1
B. 15
C. -15
733. Evaluate: lim (
A. 3
B. 18
3
734. Evaluate: lim
A. LDNE
B. 2
735. Evaluate: lim
A. +
B. -
D. 10
– 5x2)
C. -18
D. 72
C. 4
D. 0
C. LDNE
B. -
C. LDNE
B. x
D. x
743. Give the range of f(x) = x125.
A. y | y
B. y 0
C. y 2019
D. y = { }
744. Give the range of f(x) =
A. y | y
B. y
C. y
D. y
-2
D.
746. Give the range of f(x) = 2019x – 2019.
A. y | y
B. y 1
C. y = 2019
D. y = { }
-2
738. Give the domain of f(x) = √2018
A. x | x
B. x
C. x > 0
742. Give the range of f(x) = 20x – 20.
A. y | y
B. y 1
C. y = 20
D. y = { }
745. Give the range of f(x) = x2019.
A. y | y
B. y 0
C. y 2019
D. y = { }
737. Give the domain of f(x) =
A. x | x
C. x 2
-2
D. 2
736. Evaluate: lim
A. +
B. y
D. y
747. Give the range of f(x) =
2019.
A. y | y
B. y
C. y
D. y
D. x
748. Give the domain of f(x) =
A. x | x
739. Give the domain of f(x) = x2 – 20x + 19.
C. x 2
A. x | x
B. x 1, 9
C. x > -81
D. x -81
This is a free reviewer. All rights reserved.
B. x
D. x
-3
1000 MMR
749. What is 0! equal to?
A. 0
B. 1
C. undefined
D. +
Author: Victor A. Tondo Jr., LPT
758. Find the remainder when 1998! is divided
by 2000.
A. 1024
B. 256
C. 64
D. 0
750. Give the range of f(x) = 2019x + 2020.
A. y | y
B. y 2020
C. y 2019
D. y 2020
759. In Mathematics, what does the delta ∆
symbol mean or refer to?
A. the change in
B. the summation of
C. therefore
D. belonging to
751. Find the first derivative: f(x) = (2x + 3)5.
A. f ’(x)= 5 (2x + 3)4
B. f ’(x)= 10 (2x + 3)4
C. f ’(x)= 15 (2x + 3)4
D. f ’(x)= 20 (2x + 3)4
752. Find f ‘(x) given f(x) = (3x2 – 2x + 1 )7.
A. f ’(x)= (42x – 14) (3x2 – 2x + 1)6
B. f ’(x)= (42x – 28) (3x2 – 2x + 1)6
C. f ’(x)= (42x + 14) (3x2 – 2x + 1)6
D. f ’(x)= (42x + 28) (3x2 – 2x + 1)6
753. Find the maximum area that can be
enclosed by a rectangle given its perimeter
should only be 202 meters.
A. 2550 m2
B. 2550.25 m2
C. 2550.5 m2
D. 2550.75 m2
754. Find the lowest possible value of the
function f(x) = x2 – 8x + 7.
A. -10
B. -9
C. -8
D. -7
755. Find the instantaneous rate of change for
f(x) = 5x3 – 2x2 + 7x – 9 at x = 2.
A. 37
B. 48
C. 59
D. 70
756. Find the instantaneous rate of change for
f(x) = 4x3 + 5x2 – 7x – 11 at x = 1.
A. -15
B. 0
C. 15
D. 30
760. In Mathematics, what does the sigma
symbol mean or refer to?
A. the change in
B. the summation of
C. therefore
D. belonging to
761. In Mathematics, what does the epsilon
symbol mean or refer to?
A. such that
B. element of
C. therefore
D. ergo
762. In set theory, what does the symbol
or refer to?
A. not O
B. not zero
C. does not
D. empty
mean
763. Which of the following is ALWAYS an odd
number?
A. The difference of two odd numbers.
B. The sum of two odd numbers.
C. The product of two odd numbers.
D. The product of two even numbers.
764. Given 2x + 2x = 2y, express y in terms of x.
A. y = 2x
B. y = x2
C. y = x + 2
D. y = x + 1
765. In set theory, what does the symbol mean
or refer to?
A. no elements
B. common elements
C. all elements
D. half of the elements
757. Find the slope of the line tangent to
y = 3x5 – 4x3 + 5x2 – 10x + 9 at x = 2.
A. 50.5
B. 101
C. 151.5
D. 202
This is a free reviewer. All rights reserved.
1000 MMR
766. In Mathematics, what does the symbol
mean?
A. since
B. prove
C. therefore
D. previously
767. Which of the following is the Geometric
symbol for congruent?
A.
B.
C.
D. //
768. Which of the following is the obelus sign?
A. !
B. ÷
C. ⊙
D. θ
769. Given X = {1, 4, 16, 64} and Y = {1, 2, 3, 4},
find X Y.
A. {1, 4}
B. {16, 64}
C. {2, 3}
D. {1, 2, 3, 4, 16, 64}
770. Given X = {1, 2, 4, 8} and Y = {1, 2, 3, 4}, find
X – Y.
A. {1, 4, 8}
B. {8}
C. {3, 8}
D. {1, 2, 3, 4, 8}
771. Given X = {1, 2, 4, 8} and Y = {1, 2, 3, 4}, find
Y – X.
A. {1, 4, 8}
B. {3}
C. {3, 8}
D. {1, 2, 3, 4, 8}
772. Given X = {6, 5, 4, 3} and Y = {1, 2, 3, 4}, find
X Y.
A. {3, 4}
B. {6, 5}
C. {1, 2}
D. {1, 2, 3, 4, 5, 6}
773. Which of the
following
represents the
shaded region in
the Venn diagram?
A. A B
C. A – B
B. A B
D. B – A
Author: Victor A. Tondo Jr., LPT
774. Which of the
following
represents the
shaded region in
the Venn diagram?
A. A’ B’
B. A’ B’
C. (A B)’
D. (A B)’
775. In a class of 50, there are 20 students who
excel in Math, 24 who excel in Science, and 4 who
excel in both Math and Science. How many of
them excel in neither?
A. 2
B. 4
C. 7
D. 10
776. There are 50 students in a class. 39 of them
have a cellphone, 17 of them have a laptop, and 5
of them have neither. How many have both a
cellphone and a laptop?
A. 6
B. 7
C. 9
D. 11
777. Rayon has 10 shirts, 5 pairs of pants, and 4
pairs of shoes. How many ways can he pick what
to wear for today?
A. 19
B. 20
C. 190
D. 200
778. Using the digits 1, 2, 3, 4, 5, and 6 without
repetition, how many 3-digit numbers can be
made?
A. 480
B. 240
C. 120
D. 60
779. Using the digits 1, 2, 3, 4, 5, 6, and 7 without
repetition, how many 4-digit numbers can be
made?
A. 1680
B. 840
C. 420
D. 210
780. Using the digits 0, 1, 2, 3, 4, 5, and 6 without
repetition, how many 3-digit numbers can be
made?
A. 360
B. 180
C. 90
D. 45
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1000 MMR
781. Using the digits 0, 1, 2, 3, 4, 5, 6, and 7
without repetition, how many 4-digit numbers
can be made?
A. 1680
B. 1470
C. 840
D. 735
782. Using the digits 0, 1, 2, 3, 4, 5, 6, and 7
without repetition, how many 3-digit numbers
can be made?
A. 672
B. 588
C. 336
D. 294
783. Using the digits 0, 1, 2, 3, 4, 5, and 6 without
repetition, how many odd 3-digit numbers can
be made?
A. 180
B. 150
C. 90
D. 75
784. Using the digits 0, 1, 2, 3, 4, 5, 6, and 7
without repetition, how many odd 4-digit
numbers can be made?
A. 1470
B. 1440
C. 735
D. 720
785. Using the digits 0, 1, 2, 3, 4, 5, 6, and 7
without repetition, how many odd 3-digit
numbers can be made?
A. 336
B. 288
C. 168
D. 144
786. Using the digits 0, 1, 2, 3, 4, 5, 6, and 7
without repetition, how many even 3-digit
numbers can be made?
A. 162
B. 156
C. 150
D. 144
787. Using the digits 0, 1, 2, 3, 4, 5, 6, and 7
without repetition, how many even 4-digit
numbers can be made?
A. 765
B. 750
C. 735
D. 720
788. Using the digits 0, 1, 2, 3, 4, 5, and 6 without
repetition, how many even 3-digit numbers can
be made?
A. 120
B. 105
C. 90
D. 75
Author: Victor A. Tondo Jr., LPT
789. Evaluate: log6 8 + 3 log6 3.
A. 24
B. 14
C. 9.5
D. 3
790. How many ways can the word ABABA be
rearranged?
A. 120
B. 60
C. 20
D. 10
791. There are 10 different beads and a locking
mechanism to be used in making a bracelet. How
many different bracelet patterns can be made?
A. 3,628,800
B. 362,880
C. 1,814,400
D. 181,440
792. There are six different Math books, four
different Science books, and two different
English books to be arranged on a shelf. How
many ways can this be done?
A. 12C3
B. 12!
C. 48
D. 3!
793. There are six different Math books, four
different Science books, and two different
English books to be arranged on a shelf. If books
of the same subject must be together, how many
ways can this be done?
A. 479,001,600
B. 207,360
C. 34560
D. 144
794. There are six people to be seated on a bench
for a picture. A certain couple, Vic and Rowena,
are to be seated next to each other. How many
ways can this be done?
A. 120
B. 240
C. 360
D. 720
795. There are six people to be seated on a bench
for a picture. A certain trio, Vic, Rayon, and Aira,
do not want to be separated. How many ways
can this be done?
A. 60
B. 72
C. 144
D. 180
This is a free reviewer. All rights reserved.
1000 MMR
796. TNB Gaming has five “carry” players, four
“offlane” players, and six “support” players. If a
game calls for two carry, one offlane, and two
support players, how many possible lineups do
they have?
A. 1200
B. 600
C. 300
D. 150
797. If today is a Sunday, what day is 125 days
from now?
A. Friday
B. Saturday
C. Sunday
D. Monday
798. If today is a Monday, what day is 200 days
from now?
A. Friday
B. Saturday
C. Sunday
D. Monday
Author: Victor A. Tondo Jr., LPT
804. There are ten red, nine blue, and six black
pens in a bag. If a pen is randomly drawn from
the bag, what is the probability that it is red?
A.
B.
C.
D.
805. There are ten red, nine blue, and six black
pens in a bag. If a pen is randomly drawn from
the bag, what is the probability that it is black?
A. 0.06
B. 0.6
C. 0.24
D. 0.7
806. What does a probability of 0 pertain to?
A. an impossible event
B. a very unlikely event
C. a very likely event
D. a sure event
799. If three-fourths of a number is 33 more than
one-fifth of itself, then what is the number?
A. 80
B. 60
C. 40
D. 20
807. Simplify: √10 + 2√21.
A. √2 + √5
B. √3 + √7
C. √3 √2
D. √5 √2
800. What time is 2019 hours after 3:00PM?
A. 6:00 AM
B. 12:00 NN
C. 6:00 PM
D. 12:00 MN
808. Simplify: √11
A. √6 √5
C. √3 2√2
801. In a certain fastfood chain, soft drinks are
served in Small, Medium, and Large cups.
What level of data is this?
A. Nominal
B. Ordinal
C. Interval
D. Ratio
802. The prime factorization of a number is
given as 23 x 52 x 133. How many factors does it
have?
A. 96
B. 48
C. 36
D. 18
809. ∫(3 + 4)5 dx
A. (3x + 4)6+ c
B.
(3x + 4)6+ c
C. 5(3x + 4)4+ c
D.
(3x + 4)6+ c
810. ∫(6 + 6
6) dx
3
2
A. 2x + 3x + 6x + c
C. 2x3 + 3x2 – 6x + c
B. 2x3 – 3x2 + 6x + c
D. 2x3 – 3x2 – 6x + c
811. ∫
803. The prime factorization of a number is
given as 22 x 32 x 7 x 11. How many factors does
it have?
A. 96
B. 48
C. 36
D. 18
2√30.
B. √6 + √5
D. 1
dx
A. + c
B. + c
C.
D.
+c
This is a free reviewer. All rights reserved.
+c
1000 MMR
812. ∫ dx
A.
C.
+c
+c
B.
+c
D.
+c
Author: Victor A. Tondo Jr., LPT
818. (x2 + 3x – 2)4 = _________
A. (x2 + 3x – 2)3
B. 4(x2 + 3x – 2)3
C. (8x + 12) (x2 + 3x – 2)3
D. (2x + 3) (x2 + 3x – 2)3
813. Mr. dela Cruz is going to present the growth
of a certain plant over a span of 10 weeks. Which
graph would be best suited for this?
A. bar graph
B. line graph
C. pie graph
D. pictograph
814. Rayon is going to present the sales of six
different teams in the company from January to
June. Which graph would be best suited for this?
A. bar graph
B. line graph
C. pie graph
D. pictograph
815. Ms. Rowena is tasked to send 8 randomly
selected students from her class of 40 to the
clinic for drug test. She decides to let the
students count off from 1 to 5, after which she
randomly picks a number from 1 to 5. All
students whose number is that which she picked
will be sent to the clinic. What sampling method
did Ms. Rowena use?
A. cluster
B. systematic
C. stratified
D. quota
816. What is the 7th decile of the following data?
14
15
16
20
24
25
26
26
27
29
31
33
34
34
35
A. 31
B. 31.4
C. 31.5
D. 32
817. Find the remainder when f(x) is divided by
(x – 3) given f(x) = x3 – 3x2 + 5x – 13.
A. 1
B. 2
C. 3
D. 4
819.
A.
√
2 + 3 = _________
B. √
√
C. √
D.
√
820. (2x + 3)(4x – 5) = _________
A. 8x2 + 2x + 15
B. 8x2 + 2x – 15
2
C. 8x – 2x – 15
D. 8x2 – 2x + 15
821.
(
) = _________
A. (
)
B. (
)
C. (
)
D. (
)
822. There are 17 red bags, 19 green bags, and
14 blue bags in a store. What percent of the bags
is blue?
A. 38%
B. 34%
C. 28%
D. 27%
823. Which of the following is true?
A. (sin x) (cot x) = sec x
B. (cos x) (csc x) = tan x
C. (tan x) (csc x) = sec x
D. (cot x) (sec x) = cos x
824. Which of the following could be the value of
x if x 4 (mod 10)?
A. 46
B. 49
C. 52
D. 54
825. Which of the following could be the value of
x if x 7 (mod 9)?
A. 146
B. 149
C. 151
D. 154
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1000 MMR
826. Given ̅̅̅̅
BF bisects ABC and m ABF = 34o,
find m ABC.
A. 17o
B. 34o
C. 51o
D. 68o
827. If f(x) = x2 – 2x + 3 and g(x) = x + 1, find
fog(x).
A. x2 – 1
B. x2 – 2x + 4
2
C. x – 3x + 5
D. x2 + 2
828. How many 3-digit numbers can be formed
using the digits 0, 1, 2, 3, 4, 5, and 6 if repetition
is not allowed?
A. 160
B. 180
C. 200
D. 210
829. What is the longest side of ∆VAC if m V =
45o and m C = 65o?
A. ̅̅̅̅
VC
B. ̅̅̅̅
AC
C. ̅̅̅̅
VA
D. ̅̅̅̅
CA
830. Which quadrilateral has two congruent
diagonals that are perpendicular to each other?
A. kite
B. isosceles trapezoid
C. rectangle
D. rhombus
831. If the length of a rectangle is increased by
30% while the width is decreased by 30%, what
will happen to its area?
A. It stays the same.
B. It is increased by 9%.
C. It is decreased by 9%.
D. It is decreased by 6%.
832. Rayon had an average of 93 on his first five
Math tests. After taking the next test, his average
increased to 94. Find his most recent grade.
A. 96
B. 97
C. 98
D. 99
833. A bus drove for 7 hours at 73 kph and 3
hours at 88 kph. What was its average speed?
A. 75.5 kph
B. 76.5 kph
C. 77.5 kph
D. 78.5 kph
Author: Victor A. Tondo Jr., LPT
834. Fifteen guests shake hands with each other.
If each guest is to shake hands with all the other
guests, how many handshakes will be made?
A. 105
B. 150
C. 210
D. 225
835. The salary of 5 men for 6 days is P9,000.
How much is the salary of 7 men for 8 days?
A. P15,000
B. P15,600
C. P16,800
D. P17,400
836. The average grade of eleven students is 84.
If the average of seven of these students is 82,
what is the average of the other four students?
A. 88
B. 87.5
C. 86.5
D. 86
837. A radius of a circle is 25 cm long. How long
is its longest chord?
A. 20√2 cm
B. 40 cm
C. 50 cm
D. It depends,
838. Find the largest area of a rectangular piece
of land that can be enclosed with 800 meters of
fencing material.
A. 40,000 m2
B. 56,250 m2
2
C. 62,500 m
D. 80,000 m2
839. Which statistical test is used for testing for
relationship between two variables?
A. ANOVA
B. t-test
C. Pearson R
D. Chi Square
2 +4
5
840. Given ( ) = { 5
= 5,
9
5
( ).
find lim
A. 5
B. 14
C. 16
D. limit does not exist
841. Find the equation of the line passing
through (3, 7) and (-3, -5).
A. y = 2x + 1
B. y = 3x + 1
C. y = 2x – 1
D. y = 3x – 1
This is a free reviewer. All rights reserved.
1000 MMR
842. In which non-Euclidean model for geometry
can we have any given line ℓ and a point A which
is not on ℓ, wherein all lines through A will
intersect ℓ?
A. hyperbolic
B. elliptic
C. Saccheri
D. Pythagorean
843. If A is at (-9, 11) and B is at (6,-9), find C if C
is three-fifths the way from A to B.
A. (1, -1)
B. (0, 1)
C. (1, 1)
D. (0, -1)
D. ( ,
)
845. Mr. G sold 70% of his chickens and still had
129 chickens left. How many chickens did he
have originally?
A. 344
B. 430
C. 598
D. 860
846. Find the average rate of change of
y = x3 – 6x2 + 5x – 20 from x = 1 to x = 6.
A. 5
B. 6
C. 7
D. 8
847. What is the remainder when
534,214,557,989,215 is divided by 4?
A. 1
B. 2
C. 3
850. An item is sold for P7,280 after being
marked down by 30%. What was its original
price?
A. P11,400
B. P10,900
C. P10,400
D. P9,900
851. Which of the following lines passes through
the point (-4, 5)?
A. y = x + 1
C. y = 1 – x
844. Find the vertex of y = 5x2 – 4x – 7.
A. ( , )
B. ( , )
C. ( , )
Author: Victor A. Tondo Jr., LPT
A. 65
B. 61
C. 57
D. 53
D. 0
848. What do you call an equation in which only
integer solutions are allowed?
A. Diophantine equations
B. Euclidean equations
C. Fibonacci equations
D. Newtonian equations
B. y = 2x – 3
D. y = 3 – 2x
852. Which of the following is outside the circle
defined by the equation (x + 2)2 + y2 = 50?
A. (-9, 1)
B. (-7, 5)
C. (-5, 7)
D. (2, 4)
853. Which of the following equations pertain to
a parabola?
A. y2 – 5y = x2 + 4
B. x2 + 5x – 4y = 13
C.
(
)
D.
(
)
+
(
)
=1
(
)
=1
854. Which of the following equations pertain to
a parabola that opens to the left?
A. -2(y + 7) = (x – 3)2
B. 5(y – 9) = (x – 5)2
C. (y – 4)2 = 2(x + 7)
D. (y + 3)2 = -3(x + 4)
855. The lengths of two sides of a triangle are 7
inches and 15 inches. Which of the following
represents x, the possible length in inches of the
remaining side of the triangle?
A. 8 < x < 22
B. 8 x 22
C. x < 8 or x > 22
D. x 8 or x 22
849. The sum of two numbers is 73 and their
difference is 41. What is the larger number?
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1000 MMR
856. Rayon has 46 coins in P1 and P5
denominations for a total worth of P186. How
many P5 coins does he have?
A. 25
B. 30
C. 35
D. 40
Author: Victor A. Tondo Jr., LPT
865. Which of the following Mathematicians had
a method “Sieve” for identifying prime numbers?
A. Pythagoras
B. Euclid
C. Eratosthenes
D. Archimedes
857. If x = 2 and y = 3, what is x2 – y2 + 3xy?
A. 31
B. 13
C. -5
D. -13
866. Which Mathematician developed a formula
for finding the area of a triangle using its side
lengths?
A. Hipparchus
B. Pythagoras
C. Heron
D. Euclid
858. In an arithmetic sequence, the first term is
200 and the common difference is -3. What is the
64th term?
A. 20
B. 17
C. 14
D. 11
859. In an arithmetic sequence, the 23rd term is
157 and the common difference is -4. What is the
34th term?
A. 113
B. 117
C. 121
D. 125
860. Find the inverse of y = x2 – 6x + 5.
A. √ + 4 + 3
B. √ + 4 – 3
C. √
4+3
D. √
4–3
861. ln e4 = _____
A. 24.1295
C. 6.032375
862.
A. 5
eln 5
= _____
B. 6.25
863. What is ln e?
A. 0
B. 1
868. This Mathematician invented natural
logarithms.
A. Isaac Newton
B. John Napier
C. Marin Mersenne D. Rene Descartes
869. Prime numbers that are 1 less than a power
of 2 are called ______.
A. Pythagorean Primes
B. Mersenne Primes
C. Fermat Primes
D. Pascal Primes
B. 12.06475
D. 4
C. 12.5
D. 25
C. e
D. 10
864. Who are the first to use papyrus?
A. Sumerians
B. Egyptians
C. Chinese
D. Greeks
867. Who refined and perfected the decimal
place value number system?
A. Greeks
B. Romans
C. Chinese
D. Indians
870. Which Mathematician developed the
triangle of binomial coefficients?
A. Gottfried Leibniz
B. Isaac Newton
C. Blaise Pascal
D. Johann Bernoulli
871. Two Mathematicians are known for the
theory of hyperbolic geometry although they
worked independently from each other. Who are
these Mathematicians?
A. Euclid and Pythagoras
B. Euclid and Descartes
C. Lobachevsky and Bolyai
D. Babbage and Bolyai
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1000 MMR
872. Which Mathematician developed the use of
AND, OR, and NOT operators in Algebra?
A. George Boole
B. Janos Bolyai
C. Charles Babbage
D. August Mobius
873. Which Mathematician is known for his last
theorem?
A. Andrew Wiles
B. Pierre de Fermat
C. John Wallis
D. Rene Descartes
874. Who have the first fully-developed base-10
number system in use?
A. Romans
B. Egyptians
C. Indians
D. Sumerians
875. Notched tally bones proved their use of
Mathematics around 35000 BCE.
A. Africans
B. Asians
C. Europeans
D. Americans
876. Which sampling method is best used when
the population has subgroups?
A. systematic sampling
B. stratified sampling
C. quota sampling
D. cluster sampling
877. Mr. Lazada drove for 2 hours at a speed of
48 kph, 3 hours at 58 kph, and then 5 hours at 64
kph. What was his average speed?
A. 57 kph
B. 58 kph
C. 59 kph
D. 60 kph
878. Find the domain of y = 3x.
A. x | x
B. x 0
C. x 3
D. x 0
Author: Victor A. Tondo Jr., LPT
879. Convert rad to degrees.
A. 70o
B. 105o
C. 210o
D. 420o
880. Find the volume of a cylinder whose radius
is 50 cm and height is 24 cm.
A. 20,000 π cm3
B. 30,000 π cm3
C. 45,000 π cm3
D. 60,000 π cm3
881. Find the altitude to the hypotenuse of a
right triangle whose legs measure 16 cm and 30
cm.
A.
cm
B. 240 cm
C. 34√2 cm
D. 24√3 cm
882. How many line segments can be made from
25 non-collinear points?
A. 900
B. 600
C. 450
D. 300
883. Dividing by 0.25 is the same as multiplying
by which number?
A. 2
B. 4
C. 5
D. 16
884. Find the surface area of a sphere whose
radius is 24 cm.
A. 192 π cm2
B. 768 π cm2
C. 1152 π cm2
D. 2304 π cm2
885. Which of the following is the reference
angle of 295o?
A. 25o
B. 45o
C. 65o
D. 75o
886. Which numerical system is sexagesimal
(base-60)?
A. Roman
B. Babylonian
C. Mayan
D. Hindu-Arabic
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1000 MMR
887. In solid geometry, what do you call a solid
bound by polygons?
A. tessellation
B. multigon
C. polyhedron
D. soligon
Author: Victor A. Tondo Jr., LPT
894. The amount of money you have falls under
what level of data?
A. nominal
B. ordinal
C. interval
D. ratio
888. Two triangles have a pair of congruent
angles. Which of the following is true about these
triangles?
A. They are isosceles triangles.
B. They are equilateral triangles.
C. They are congruent triangles.
D. They are similar triangles.
895. To study tooth decay a researcher takes a
sample at random but with the stipulation that
all age groups are represented proportionally.
What sampling method did the researcher use?
A. systematic
B. cluster
C. stratified
D. convenience
889. Rayon deposited an amount of P500,000 in
a bank that offers 4% interest compounded per
annum. How much will he have in his account
after 3 years?
A. P720,000
B. P600,000
C. P560,000
D. P562,432
890. Find the range of the following scores:
37
40
44
45
41
39
48
A. 7
B. 9
C. 10
D. 11
891. What is the 9th decile of the following data?
13
15
17
19
21
23
24
26
28
29
30
33
35
36
40
A. 34
B. 35.5
C. 37.6
D. 38
896. By what property do we say that when
A = B and B = C, then A = C?
A. reflexive
B. symmetrical
C. transitive
D. closure
897. In number theory, what do we call a
polynomial equation with integer coefficients
that also allows the variables and solutions to be
integers only?
A. Integral equations
B. Differential equations
C. Diophantine equations
D. Bolyai-Lobachevsky equations
898. How long is the latus rectum of the parabola
defined by (y + 3)2 = 20x – 196?
A. 3
B. 12
C. 20
D. 19
892. -20192 is NOT equal to ___.
A. (-2019)2
C. (-20192)
B. –(2019 x 2019)
D. –(2019)2
899. In the set of real numbers, what do we call
the set that includes only the counting numbers
and zero?
A. rational numbers B. integers
C. whole numbers D. irrational numbers
893. The product of two numbers is 1425. If each
of the two numbers is doubled, the product of
these larger numbers is _____.
A. 2,850
B. 5,700
C. 8,550
D. 11.400
900. Find the sum of the first 60 counting even
numbers.
A. 3660
B. 3600
C. 1830
D. 900
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1000 MMR
901. If M and N are complementary angles,
which of the following is true?
A. cos M = sec N
B. sin M = cos N
C. tan M = csc N
D. sec M = -cos N
902. The top ten students of a graduating class
got the following scores in their final
examination in Calculus:
89
84
83
89
89
85
88
93
83
98
What is their mean score?
A. 87.4
B. 87.5
C. 88.1
D. 89.3
903. The top ten students of a graduating class
got the following scores in their final
examination in Calculus:
89
84
83
89
89
85
88
93
83
98
What is the modal score?
A. 83
B. 85
C. 89
D. 98
904. The sum of five consecutive integers is 980.
What is the value of the greatest integer?
A. 192
B. 194
C. 196
D. 198
905. x varies directly as y and inversely as z. If
x = 20 when y = 10 and z = 3, what is x when
y = 15 and z = 9?
A. 10
B. 15
C. 25
D. 30
906. Factorize 15x2 – 13x – 20.
A. (5x – 4)(3x – 5)
B. (5x – 4)(3x + 5)
C. (5x + 4)(3x – 5)
D. (5x + 4)(3x + 5)
907. Which of the following is the axis of
symmetry of the parabola defined by the
equation y = x2 – 6x + 5?
A. x = 5
B. x = -5
C. x = 3
D. x = -3
Author: Victor A. Tondo Jr., LPT
908. Find the radius of x2 + y2 + 12x – 14y = 36.
A. 6
B. 7
C. 11
D. 13
909. Find the instantaneous rate of change for
f(x) = 4x3 + 7x2 – 27x – 49 at x = 1.
A. -2
B. -1
C. 1
D. 2
910. log3 24 – 3 log3 2 = _____
A. 1
B. 0
C. -1
D. e
911. After using one-sixth of her budget on bills,
two-fifths on groceries, and P1700 on books and
magazines, Mrs. Lazada still had P7400 left. How
much was her budget?
A. P20,000
B. P21,000
C. P22,500
D. P30,000
912. An assistant’s response times were
recorded on the table below. What is her average
response time in minutes?
Response Time Frequency
1 minute
3
2 minutes
7
3 minutes
11
4 minutes
9
5 minutes
5
6 minutes
3
7 minutes
2
A. 3.625
B. 3.575
C. 3.525
D. 3.5
913. The average grade of 22 students in Section
Abaca is 95, while the average grade of 28
students in Section Acacia is 89. What is the
average grade of all 50 students in both sections?
A. 92.36
B. 91.64
C. 90.72
D. 89.8
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1000 MMR
914. Find the slope of 2x – 6y = 13.
A. 3
B.
C. -3
D.
915. Find the intersection of y = -3x – 2 and
y = 2x + 23.
A. (5, 13)
B. (-5, -13)
C. (5, -13)
D. (-5, 13)
916. A book was sold for P630 after a 10%
discount was given. How much was the book
originally?
A. P800
B. P750
C. P700
D. P690
917. If the sum of the supplement and the
complement of an angle is 146, what is the
angle?
A. 61
B. 62
C. 63
D. 64
918. A bag contains some marbles. When the
marbles are grouped by 2, 3, 4, 5, or 6, there is
always one marble left. Which of the following
could be the number of marbles in the bag?
A. 31
B. 41
C. 51
D. 61
919. Amitaf, Bernard, and Chloe share a total of
P2,460 in the ratio 3:5:4 respectively. How much
is Amitaf’s share?
A. P600
B. P615
C. P630
D. P660
920. Fifty-four kilometers per hour is equal to
how many meters per second?
A. 15
B. 30
C. 54
D. 60
921. Simplify
A. 1
.
B.
C.
Author: Victor A. Tondo Jr., LPT
922. If x = 10, which of the following is equal to
219?
A. 23x – 1
B. 2x2 + 5x + 5
C. 2x2 + 2x – 1
D. x2 + x + 2
923. In the equation 3x – 4y = 25, what is x when
y is 5?
A. 18
B. 15
C. 12
D. 9
924. What is the measure of each interior angle
of a regular 30-sided polygon?
A. 158o
B. 162o
o
C. 165
D. 168o
925. Simplify:
A. csc x – tan x
C. csc x – cot x
926. Simplify:
A.
B. csc x + tan x
D. csc x + cot x
x
B.
C.
D.
927. Which of the following is a diagonal matrix?
1 6 9
0 2 4
A. [ 2 2 6]
B. [1 0 5]
3 4 3
2 3 0
4 0 0
3 2 3
C. [0 5 0 ]
D. [2 3 6]
0 0
6
1 4 3
928. Which of the following is a scalar matrix?
0 2 3
5
7 9
A. [ 2
]
B.
[
4 0 7]
8 6
3
8 0
3 4 3
7 0 0
7 0 0
C. [0
D. [0 7 0]
5 0]
0 0 6
0 0 7
D. 2
This is a free reviewer. All rights reserved.
1000 MMR
929. Which of the following is an identity matrix?
1 0 0
0
5 3
A. [0 1 0]
B. [ 4 0 9]
0 0 1
7 10 0
6 1
1
0 1 1
C. [1 12 1 ]
D. [1 0 1]
1 1 18
1 1 0
Author: Victor A. Tondo Jr., LPT
936. Which of the following graphs have a
positive leading coefficient and an even degree?
A.
930. Which of the following is the transposition
1 4
3
of A = *
+?
5 10 15
5 10 15
4 5 15
A. AT = *
+
B. AT = *
+
1 4
3
10 1 3
1 4
3
1 5
C. AT = [4 10]
B. AT = [5 10 15]
1 1
1
3 15
B.
931. When dealing with matrices A, B, and C,
which of the following is always true?
A. AB = BA
B. If AB = 0, then A = 0 or B = 0
C. If AB = AC, then B = C
D. None of the above.
C.
932. If the numbers x-1, x+2 and 2x+4 are
consecutive terms of a geometric sequence, what
is x?
A. 1
B. 2
C. 4
D. 8
D.
933. What are the zeros of 12x2 – x – 35?
A. and
B. and
C. and
D.
and
934. What is the sum of the interior angles of a
regular dodecagon?
A. 1620o
B. 1800o
C. 1980o
D. 2160o
935. 45x – 20 = _____.
A. 45x – 20
B. 5(45x – 20)
5x
–
20
C. 20(4
)
D. (5x – 20)(45x – 20)
This is a free reviewer. All rights reserved.
1000 MMR
937. Which of the following graphs have a
negative leading coefficient and an odd degree?
Author: Victor A. Tondo Jr., LPT
939. What can be said about the equation f(x) of
the following graph?
A.
A. The leading coefficient is positive.
B. The constant is positive.
C. The degree is 4.
D. The function is logarithmic.
B.
+1
4
940. If ( ) {2 + 1 = 4 , find lim
5
3
4
A. Limit does not exist.
B. 9
C. 13
D. 17
( ).
2 +1
3
7
=
3
941. If ( ) {
, find lim
( ).
3
3
A. 3
B. 6
C. 7
D. Limit does not exist.
C.
942. Give the domain of f(x) = √8 + 24.
A. x | x
B. x -3
C. x > 24
D. x -3
D.
938. Given f(x) = (x + 2)(x – 1)(2x – 3)(x – 5),
where can we find f(3)?
A. On the x-axis.
B. Above the x-axis.
C. Below the x-axis.
D. On the intersection of the x- and y-axis.
943. What is the 3rd term in the expansion of
(A + B)5?
A. 5A4B
B. 10A3B2
2
3
C. 10A B
C. 5A3B2
944. Find the remainder when
2x4 – 5x3 + 3x2 – 5x + 7 is divided by (x – 1).
A. 1
B. 2
C. 4
D. 7
945. Give the range of f(x) =
A. y | y
B. y
C. y
D. y
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0
1000 MMR
946. In the arithmetic sequence -2, 1, 4, 7, …,
which term is 100?
A. 34th
B. 35th
C. 36th
D. 37th
947. Find the length of the intercepted arc of a
central angle measuring 45o given the radius is
80 cm.
A. 20 π cm
B. 30 π cm
C. 40 π cm
D. 50 π cm
948. If f(x) = 5x2 – 4x + 9, find the average rate
of change from x = 1 to x = 4.
A. 20
B. 21
C. 22
D. 23
Author: Victor A. Tondo Jr., LPT
954. The freezing temperature of water is 0o
Celsius. What level of data is temperature in
Celsius?
A. nominal
B. ordinal
C. interval
D. ratio
955. Factorize: x3 + 5x2 – 9x – 45
A. (x + 1)(x + 9)(x – 5)
B. (x + 1)(x – 9)(x + 5)
C. (x + 3)(x + 3)(x – 5)
D. (x + 3)(x – 3)(x + 5)
956. Which of these is equal to A-1 + B-1?
A.
B. +
(
949. If f(x) = 4x2 + x – 5, find the instantaneous
rate of change at x = 5.
A. 39
B. 40
C. 41
D. 42
950. How is 67500 written in scientific notation?
A. 67.5 x 103
B. 6.75 x 103
C. 67.5 x 104
D. 6.75 x 104
951. Find the mean of the following data:
24
24
25
26
28
29
A. 24
B. 25.5
C. 26
D. 29
952. Rayon’s Social Security System (SSS) ID
number is 02-2525255-2. What level of data is
the SSS ID?
A. nominal
B. ordinal
C. interval
D. ratio
953. A tree is 4 meters tall, while the flagpole is 8
meters tall. What level of data is used?
A. nominal
B. ordinal
C. interval
D. ratio
)
C.
D.
957. Find the equation of the line with slope 3,
passing through (2, 5).
A. y = 3x – 2
B. y = 3x – 1
C. y = 3x + 1
D. y = 3x + 2
958. Find the distance of the point (7, 8) from
the line 2x – 3y + 4 = 0
A.
√
C. 4√73
B.
√
D. 5√13
959. Find the distance between the parallel lines
y = 2x – 1 and y = 2x + 4.
A. 5
B. 2√5
C. √5
D. ½
960. There are 80 cows and ducks in a farm, all
of which are healthy. If there are 190 legs in
total, how many cows are there?
A. 5
B. 10
C. 15
D. 20
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1000 MMR
961. Which of the following angles in standard
position is coterminal with 150o?
A. 2570o
B. 5490o
C. 7830o
D. 9870o
962. Which of the following is false?
A. sin x =
B. tan x =
C. cos x =
D. csc x =
963. V and R form a vertical pair. If m V = 7x
and m R = 3x + 60, find m V.
A. 105o
B. 84o
C. 20o
D. 15o
964. V and R form a linear pair. If m V = 13x
and m R = 5x + 72, find m R.
A. 112o
B. 102o
C. 78o
D. 68o
965. Evaluate: tan 45o – sin 60o + cos 30o
A. -1
B. 0
C. 1
D. 2
966. Evaluate: lim
A. LDNE
B. undefined C. 0
D. 27
967. Which of the following circles is concentric
with x2 + y2 – 7x + 9y = 25?
A. (x – 3)2 + (y + 5)2 = 25
B. (x – 7)2 + (y + 9)2 = 50
C. (x + 3.5)2 + (y – 4.5)2 = 100
D. (x – 3.5)2 + (y + 4.5)2 = 125
968. Find k such that (x + 2) is a factor of the
polynomial 5x3 + 11x2 – kx + 12.
A. 11
B. 7
C. -3
D. -8
969. If P = 4S and A = S2, express A in terms of P.
A. A =
C. A = 16P2
B. A =
D. A = 4P2
Author: Victor A. Tondo Jr., LPT
970. The distance D of a
projectile from the
ground t seconds
after launch is given
D = 8t – t2. How
many seconds
after launch does
the projectile hit
the ground?
A. 4
B. 6
C. 7
as
D. 8
971. The distance D of a projectile from the
ground t seconds after launch is given as
D = 14t – t2. How many seconds after launch
does it attain its peak?
A. 4
B. 6
C. 7
D. 8
972. The distance D of a projectile (in meters)
from the ground t seconds after launch is given
as D = 16t – 2t2. What is its maximum height?
A. 16 m
B. 24 m
C. 32 m
D. 40 m
973. Find the slope of the line tangent to the
graph of f(x) = x3 – 3x2 + 5x – 1 at x = 1.
A. -1
B. 0
C. 1
D. 2
974. Which of the following is a diagonal matrix?
9 0 0
0 1
4
A. [ 0 3 0]
B. [ 1 0
7]
0 0 6
2 3 0
7 6 9
4 2 9
C. [4
D. [ 1
3 8]
8 7]
5 4 5
2
4 6
975. Which of the following is a scalar matrix?
5 1
2
5 0
0
A. [ 4
9 10 ] B. [ 0
5 0]
8
5
3
0
0
5
7 1 1
0
2
5
C. [1
D. [ 9 0
5 1]
1]
1 1 6
1
8 0
This is a free reviewer. All rights reserved.
1000 MMR
976. Which of the following is an identity matrix?
4 1 1
0 6
3
A. [1 8 1 ]
B. [ 4 0 2 ]
1 1 12
7 9 0
1 0 0
0 1 1
B. [0 1 0]
D. [1 0 1]
0 0 1
1 1 0
977. Find the quotient when the polynomial
7x5 – 3x3 + 2x2 – 9x + 3 is divided by x – 1.
A. 7x4 + 7x3 + 4x2 + 6x – 3
B. 7x4 + 7x3 + 4x2 – 6x – 3
C. 7x4 + 7x3 – 4x2 – 6x – 3
D. 7x4 – 7x3 – 4x2 – 6x – 3
Author: Victor A. Tondo Jr., LPT
983. A whole number is 3 more than another
number. The sum of their squares is 4145. Find
the larger number.
A. 47
B. 46
C. 45
D. 44
984. The numbers x, y, z, and w have an average
of 30. If x, y and z have an average 35, what is w?
A. 5
B. 10
C. 15
D. 20
985. A is a constant. Find A such that the
equation 2x + 1 = 2A + 3(x + A) has a solution
at x = 2.
A. -0.4
B. -0.2
C. 0
D. 0.2
978. Find k such that 9x2 + 36x + k = 0 has only
one unique root.
A. 4
B. 16
C. 18
D. 36
986. It takes 4 men 9 days to build 2 houses. How
many days will it take 6 men to build 5 houses?
A. 12
B. 13
C. 14
D. 15
979. There are 210 candies in a jar. The ratio of
red candies to blue is 2:3, and the ratio of blue to
yellow is 4:5. How many yellow candies are
there?
A. 35
B. 45
C. 60
D. 90
987. If 860 = 32x, what is x?
A. 18
B. 24
C. 20
980. A certain bacteria doubles its population
after 3 minutes. If the bacteria in a petri dish is
512,000 at 8:45 AM, at what time was its
population count 125?
A. 6:30 AM
B. 7:15 AM
C. 8:09 AM
D. 9:21 AM
981. A square picture was framed and given 3 cm
margins. If the total area of the margin is 324
cm2, what is the area of the picture?
A. 441 cm2
B. 576 cm2
2
C. 729 cm
D. 900 cm2
982. Given f(x) = x2 + 10x + 25 and
g(x) = 3x + 1, find f(g(x)).
A. 3x2 + 30x + 26
B. 9x2 + 36x + 36
2
C. 9x + 30x + 26
D. 3x2 + 36x + 36
D. 36
988. In a certain university, a student’s grade is
computed as the sum of 25% of his prelims
grade, 30% of his midterms grade, and 45% of
his finals grade. He knows that his prelims grade
is 72 and his midterms grade is 70. What is his
grade for the finals if he got a grade of 75 in
Calculus?
A. 75
B. 77
C. 79
D. 80
989. Mr. Lazada bought 75 pieces of Elunium for
8,000 per piece and sold them for a total of
720,000. What is his mark-up rate?
A. 20%
B. 25%
C. 32.5%
D. 40%
990. Triangular numbers are numbers that can
be shown by triangular arrangements of dots.
The triangular numbers are 1, 3, 6, 10, 15, 21, …
What is the 20th triangular number?
A. 200
B. 205
C. 210
D. 220
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1000 MMR
991. What is a pattern of shapes that covers a
surface completely without overlaps or gaps?
A. translation
B. tessellation
C. constellation
D. transposition
992. Find the equation of the line perpendicular
to 5x + 3y = 12, passing through (3, -7).
A. 3x – 5y = -44
B. 3x + 5y = 44
C. 3x – 5y = 44
D. 3x + 5y = -44
993. Point V is two-fifths the way from A(9, -7)
to B(-6, 13). Find the coordinates of point V.
A. (2, 2)
B. (2, 0)
C. (3, 1)
D. (3, 2)
Author: Victor A. Tondo Jr., LPT
999. A certain Math challenge gives the
competitors a score of 4 for each correct answer,
and a deduction of 1 point for each wrong
answer. If a contestant answered all 100 items
and got a score of 200, how many items did the
contestant answer correctly?
A. 45
B. 50
C. 60
D. 35
1000. What is 111011012 in decimal?
A. 257
B. 237
C. 217
D. 197
994. Find the angle of inclination of the line
passing through V(5, -5) and T(2, 4)
A. 88.145o
B. 92.429o
C. 97.593o
D. 108.435o
995. The first term of an arithmetic sequence is 3
and the 20th term is 98. What is the 25th term?
A. 108
B. 115
C. 118
D. 123
996. An educational psychologist classifies
students as high, medium and low intelligence.
What kind of scale is being used?
A. nominal scale
B. ordinal scale
C. interval scale
D. ratio scale
997. The GCF of two numbers is 8 and their LCM
is 80. What is their product?
A. 80
B. 160
D. 320
D. 640
998. If c1.5 – 4 = 7, what is c3?
A. 784
B. 121
C. 49
End of 1000MMR.
Congratulations!!
D. 16
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1000 MMR
Author: Victor A. Tondo Jr., LPT
Solutions
and
Explanations
This is a free reviewer. All rights reserved.
1000 MMR
1. How many line segments can be made from 30
non-collinear points?
A. 900
B. 870
C. 450
D. 435
Solution:
( )
30C2 = 435; or
= 435
................................................
2. Calculate the mean absolute deviation of the
following numbers: 60, 80, 100, 75 and 95
A. 12.4
B. 14.2
C. 16.1
D. 18.9
Solution:
Mean = (60 + 80 + 100 + 75 + 95)/5 = 82
Mean absolute deviation daw, ibig sabihin, mean
or average ng absolute value ng x- ̅ .
MAD = (|60-82| + |80 – 82| + | 100 – 82| +
|75 – 82| + |95 – 82|) / 5 = 62/5 = 12.4
................................................
3. Which of the following is the factorization of
the binomial x2 - 42?
A. (x + 4)(x + 2)
B. (x – 4)2
C. x(x + 2x + 2)
D. (x – 4)(x + 4)
Explanation:
The factors of the difference of two squares are
the sum and difference of their roots.
................................................
4. What value of x will satisfy the equation:
0.4(5x – 1470) = x?
A. 490
B. 2,130
C. 1470
D. 588
Solution:
0.4(5x - 1470) = x
2x – 588 = x
2x – x = 588
x = 588
Author: Victor A. Tondo Jr., LPT
5. Which of the following is ALWAYS true?
A. Vertical pairs of angles are supplementary.
B. Vertical pairs of angles are complementary.
C. Linear pairs of angles are congruent.
D. Linear pairs of angles are supplementary.
Explanation:
Linear pairs are supplementary, while vertical
pairs are congruent.
................................................
6. The average of 5 different counting numbers is
20. What is the highest possible value that one of
the numbers can have?
A. 20
B. 40
C. 30
D. 90
Solution:
The 5 different counting numbers will assume
the values of 1, 2, 3, 4, and N. Since the average is
20, the sum is 5(20) or 100.
1+2+3+4+N = 100
10 + N = 100
N = 90
................................................
7. Three brothers inherited a cash amount of
P62,000 and they divided it among themselves in
the ratio of 5:4:1. How much more is the largest
share than the smallest share?
A. P75,000
B. P30,000
C. P24,800
D. P37,200
Solution:
Let the three numbers 5x, 4x, and x so that the
ratio will still be 5:4:1.
5x + 4x + x = 62000
10x = 62000;
x = 6200
Difference: 5x – x = 4x; 4x = 4(6200) = 24,800
................................................
8. What are the missing terms in the series
5, 20, 80, ___,1280, ___, 20480?
A. 50; 210
B. 40; 160
C. 35; 135
D. 320; 5120
This is a free reviewer. All rights reserved.
1000 MMR
Solution:
Since the common ratio is 4, then next terms
should be 80(4) and 1280(4), or 320 and 5120.
................................................
9. At what rate per annum should P2400 be
invested so that it will earn an interest of P800 in
8 years?
A. 6 ½ %
B. 5 ½ %
C. 4.17 %
D. 6 %
Solution:
i=PRT
800 = 2400 x R x 8
800 = 19200 R
0.0416666 = R
................................................
10. The area of a rectangle is (x2 + 2x - 8). If its
length is x + 4, what is its width?
A. x + 2
B. x - 2
C. x + 1
D. x + 6
Explanation:
Just factorize.
................................................
11. What is the value of 12⅙ - 3 ⅜ - 5 ⅔ + 20 ¾?
A. 21
B. 22
C. 23
D. 21
Solution:
–3
Solution:
(180-20)/2 = 160/2 = 80
................................................
13. Ana and Beth do a job together in three
hours. Working alone, Ana does the job in 5
hours. How long will it take Beth to do the job
alone?
A. 3 and 1/3 hours B. 2 and 1/3 hours
C. 3 hours
D. 7 and 1/2 hours
Solution:
Time to finish a job by working together =
=3
where A = 5
=3
5B = 15 + 3B
2B = 15
B = 7.5
................................................
14. How much greater is the sum of the first 100
counting numbers than the sum of the first 50
counting numbers?
A. 110
B. 3,775
C. 3,155
D. 1200
Solution:
LCD = 24
12
Author: Victor A. Tondo Jr., LPT
12. The vertex angle of an isosceles triangle is
20°. What is the measure of one of the base
angles?
A. 150°
B. 60°
C. 75°
D. 80°
–5
+ 20
–3
Sum of the first N counting numbers =
Sum of the first 100 counting numbers:
(1002 + 100)/2 = 5050
= 12
+ 20
–5
= 32
–8
Sum of the first 50 counting numbers:
(502 + 50)/2 = 1275
= 24
or 24
5050 – 1275= 3775
................................................
=23
................................................
15. Which of the following has the largest value?
A. 85
B. 39
C. 65
D. 94
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1000 MMR
Explanation: (just use your calculator)
85 = 32,768
39 =19,683
5
6 = 7,776
94 = 6,561
................................................
16. A water tank contains 18 liters when it is
20% full. How many liters does it contain when
50% full?
A. 60
B. 30
C. 58
D. 45
Solution:
18:20 = n : 50
18(50) = 20n
900 = 20n
n = 45
................................................
17. The edges of a rectangular solid have these
measures: 1.5 feet by 1½ feet by 3 inches. What
is its volume in cubic inches?
A. 324
B. 225
C. 972
D. 27
Solution:
Convert the side measures from feet to inches
before proceeding with multiplication
1.5 ft = 1.5(12) or 18 in
Vol = 18 (18) (3) = 972
................................................
18. In a certain school, the ratio of boys to girls is
5 is to 7. If there are 180 boys and girls in the
school, how many boys are there?
A. 105
B. 90
C. 45
D. 75
Solution:
Author: Victor A. Tondo Jr., LPT
19. Ruben’s grades in 6 subjects are 88, 90, 97,
90, 91 and 86. What is the grade that he should
aim for in the 7th subject if he has to have an
average of 91?
A. 97
B. 95
C. 92
D. 89
Solution:
91(7) – (88+90+97+90+91+86) = N
637 – 542 = 95
................................................
20. On a certain day, three computer technicians
took turns in manning a 24-hour internet shop.
The number of hours Cesar, Bert, and Danny
were on duty was in the ratio 3:4:5, respectively.
The shop owner pays them P50 per hour. How
much would Danny receive for that day?
A. P 230
B. P500
C. P160
D. P480
Solution:
Let their respective times be 3x, 4x, and 5x for a
total of 24 hours.
3x + 4x + 5x = 24
12x = 24
x=2
.: Danny works for 10 hours at P50/hr, or P500
for that day.
................................................
21. A retailer buys candies for P90.25. The pack
has 35 pieces of candies. If she sells each candy
for P3.25, how much profit does she make?
A. P11.50
B. P23.50
C. P37.50
D. P18.75
Solution:
Let 5x = boys
7x = girls
Profit = 35(3.25) – 90.25 = 113.75 – 90.25
Profit = 23.50
................................................
5x + 7x = 180
12x = 180
x = 15
5x = 5(15) = 75
................................................
22. Determine the midpoint of the line segment
joining the points (7, -3) and (-1, 6).
A. (2, 3/2)
B. (2, -3/2)
C. (3, 3/2)
D. (1, 5/2)
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1000 MMR
Solution:
x = (7+ -1)/2 = 3
y = (6 + -3)/2 = 3/2
................................................
23. One side of a 45° - 45° - 90° triangle
measures x cm. What is the length of its
hypotenuse?
A. x √3 cm
B. x cm
C. (x √3)/2 cm
D. x √2 cm
Explanation:
In a 45-45-90 triangle, the hypotenuse is √2
times of the leg.
................................................
24. The legs of one right triangle are 9 and 12,
while those of another right triangle are 12 and
16. How much longer is the perimeter of the
larger triangle than the perimeter of the smaller
triangle?
A. 84
B. 7
C. 12
D. 14
Solution:
Solve for the hypotenuse of the two triangles.
The first one will have 15, while the other will
have 20.
Get their respective perimeters. The first triangle
has a perimeter of 9+12+15 or 36. The other
triangle’s perimeter is 12+16+20 or 48.
48 – 36 = 12
................................................
25. An online shop sells a certain calculator for
P950 and charges P150 for shipping within
Manila, regardless of the number of calculators
ordered. Which of the following equations shows
the total cost (y) of an order as a function of the
number of calculators ordered (x)?
A. y = (950 + 150)x
B. y = 150x +950
C. x = 950y + 150
D. y = 950x + 150
Explanation:
The cost of each calculator is P950, so x
calculators cost P950x. Add the constant
shipping cost which is P150 and that’s D.
Author: Victor A. Tondo Jr., LPT
26. Which of these has the longest perimeter?
A. A square 21 cm on a side
B. A rectangle 19 cm long and 24 cm wide
C. An equilateral triangle whose side is 28 cm
D. A right triangle whose two legs are 24 and 32
cm
Solution:
A. P = 4S;
4(21) = 84
B. P = 2(L+W)
2(24+19) = 86
C. P = 3S
3(28) = 84
D. P = L1 + L2 + H
24 + 32 + 40 = 96
................................................
27. How many square inches are in 2 square
yards?
A. 900
B. 144
C. 1296
D. 2,592
Solution:
1 yard = 3 feet = 3(12) or 36 inches
1 square yard = 362 or 1296 square inches
.: 2 square yards = 2(1296) = 2592 sq in
................................................
28. In a playground for Kindergarten kids, 18
children are riding tricycles or bicycles. If there
are 43 wheels in all, how many tricycles are
there?
A. 8
B. 9
C. 7
D. 11
Solution:
T + B = 18
3T + 2B = 43
 2T + 2B = 36
 3T + 2B = 43
T=7
................................................
29. Aira takes ¾ hour to dress and get ready for
school. It takes 4/5 hour to reach the school. If
her class starts promptly at 8:00 am; what is the
latest time she can jump out of bed in order not
to be late for school?
A. 6:42 am
B. 6:27 am
C. 6:57 am
D. 7:02 am
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Solution:
Author: Victor A. Tondo Jr., LPT
Then, solve algebraically.
¾ hr = 45 mins, while 4/5 hr = 48 mins
45+48 = 93 mins, 93 mins = 1 hr 33 mins
8:00
7:60
- 1:33 
1:33
6:27
.: 1hr 33 mins before 8:00 AM is 6:27 AM
................................................
(3x – 4) + (x – 4) = 36
4x – 8 = 36
4x = 44;
30. Which common fraction is equivalent to
0.215?
A. 43/200
B. 27/125
C. 21/50
D. 108/375
x = 11;
3x = 3(11) = 33
................................................
33. What is the least common multiple of 12, 24
and 72?
A. 12
B. 72
C. 144
D. 36
Explanation:
0.215 is read as 215 thousandths. In fraction
form, that’s
. In simplest form, .
Use continuous division.
12
24
72
6
2
4
12
2
1
2
6
2
1
1
3
Alternative Method:
6 x 2 x 2 x 1 x 1 x 3 = 72
................................................
Explanation:
Just use your calculator.
................................................
31. What are the next three terms in the
progression 1, 4, 16 …?
A. 64, 256, 1024
B. 67, 259, 1027
C. 48, 198, 1026
D. 65, 257, 1025
Explanation:
Use the Pythagorean Theorem.
................................................
Explanation:
Each term is 4 times its precedent.
................................................
32. A man is 3 times as old as his son now. Four
years ago, the sum of their ages was 36. Find the
man’s age now.
A. 33
B. 11
C. 29
D. 36
Solution:
35. If two variables X and Y are directly related,
which of these is NOT true?
A. When X is low, Y is also low.
B. As X increases, Y also increases.
C. When X increases, Y decreases.
D. A high Y is associated with a high X.
Explanation:
C refers to an inverse or indirect relation.
................................................
First, create a table.
Man
Son
34. The hypotenuse of a right triangle is 25 feet.
If one leg is 24 feet, what is the length of the
other leg?
A. 6 ft.
B. 5 ft.
C. 20 ft.
D. 7 ft.
Age Now
Age 4 Yrs Ago
3x
x
3x-4
x-4
36. Find the domain of f(x) =
A. x
B. x = 1
C. x = -1
D. x
,x
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.
-1
1000 MMR
Explanation:
The given function is a rational algebraic
expression (RAE). When facing RAE, just look at
the denominator and see if it can be equated to 0
to make the RAE undefined.
The RAE will have an undefined value at
x = -1. Otherwise, it will always be equal to a real
number.
................................................
37. A car travels D km in H hours. Which of the
following expressions shows the distance
travelled by the car after M minutes?
A. MD/H
B. 60MD/H
C. MD/60H
D. 60HD/M
Solution:
Distance = Speed x Time (the unit of time should
be consistent)
The car is traveling at a speed of D/H km per hr.
The time is M minutes or M/60 hrs (for
consistency).
Distance = (D/H) (M/60) = MD/60H
................................................
38. Find the surface area of a rectangular box
whose dimensions are 30 cm x 40 cm x 50 cm.
A. 4700 cm2
B. 7050 cm2
2
C. 9400 cm
D. 11750 cm2
Solution:
SA = 2 (LW + WH + LH)
SA = 2 (50x40 + 40x30 + 30x50)
SA = 2 (2000 + 1200 + 1500) = 9400
................................................
39. If x – y = 3, then (y-x)-3 = ___.
A. 9
B. -9
C. 1/27
D. -1/27
Solution:
Since x – y = 3, then y – x = -3.
(-3)-3 = 1/(-3)3 = 1/-27 or -1/27
................................................
Author: Victor A. Tondo Jr., LPT
40. Factorize (x4 – 81) completely.
A. (x-3)4
B. (x – 3)2 (x + 3)2
C. (x+3) (x-3) (x2+9)
D. (x+3)3 (x-3)
Solution:
(x4 – 81) = (x2 – 9) (x2 + 9)
(x4 – 81) = (x+3) (x-3) (x2 + 9)
................................................
41. √8 + √18 √2 = ____
A. 4√2
B. 5√2
C. √24
D. 2√6
Solution:
√8 + √18 √2 = 2√2 + 3√2 √2 = 4√2
................................................
42. By which property can we state the
following:
“If ax + b = c, then ax + b - b = c – b.”
A. transposition
B. transitive
C. additive inverse D. addition property
Explanation:
We added –b to both sides of the equation, thus
we used APE (addition property of equality).
................................................
43. The midpoint of P and (-7, 4) is (-3, 1). What
are the coordinates of P?
A. (-5, 5/2)
B. (-11, 7)
C. (1, -2)
D. (-2, 3/2)
Solution:
Let P be at (x,y). By Midpoint formula:
(-7 + x)/2 = -3
(4 +y)/2 = 1
-7 + x = -6
4+y=2
x = -6 + 7
y=2–4
x=1
y = -2
................................................
44. What is the slope of the line 3x – y = 11?
A. -1/3
B. 1/3
C. -3
D. 3
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1000 MMR
Solution:
Isolate y on one side of the equation to rewrite
the equation in the form y = mx + b.
3x – y = 11
-y = -3x + 11
y = 3x – 11
................................................
45. What is the minimum value of
f(x) = 3x2 + 6x + 7?
A. 1
B. -1
C. 4
D. -4
Solution:
Min Value = c – b2/4a
That’s 7 – 36/12 or 7-3=4
................................................
46. If xy = 23 and x2 + y2 = 75, find x + y.
A. 10.7845
B. 11
C. 11.2155
D. 11.7845
Solution:
x2 + 2xy +y2 = x2 + y2 + 2xy
x2 + 2xy +y2 = 75 + 2(23)
x2 + 2xy +y2 = 121
x + y = 11
................................................
47. How much water must be evaporated from
90 ml of a 50% salt solution to increase its
concentration to 75%?
A. 40 ml
B. 38 ml
C. 35 ml
D. 30 ml
Solution:
V1 C1 + V2 C2 = VR CR
Since we are evaporating water, we will be
adding a NEGATIVE volume of water (or simply
put, we are subtracting water, diba?)
90(50) + (-X)(0) = (90-X)(75)
4500 + 0 = 6750 – 75X
75X = 6750 – 4500
75X = 2250;
X = 30
................................................
Author: Victor A. Tondo Jr., LPT
48. A and B form a vertical pair. If m A = 3x
and m B = 5x – 44, what is the value of x?
A. 50.5
B. 28
C. 22
D. 16.75
Solution:
Since the two angles form a vertical pair, then
they are congruent.
3x = 5x – 44
44 = 5x – 3x
44 = 2x;
22 = x
................................................
49. The angle of elevation from an observer to
the top of a building is 30o. If the building is 50
meters high, how far is the observer from the
building?
A. 25
B. 25√3
C. 50√3
D. 100
Solution:
Use a 30-60-90 triangle. The side opposite of the
30o angle will represent the building.
................................................
50. 1 and 3 are opposite angles in a
parallelogram. If m 1 = 40o, what is m 3?
A. 40o
B. 50o
C. 70o
D. 140o
Explanation:
Opposite angles of a parallelogram are
congruent.
................................................
51. Two parallel lines are cut by a transversal,
forming H and K. If the two angles are
exterior angles on the same side of the
transversal, what is the measure of H if the
measure of K is 50o?
A. 25o
B. 50o
o
C. 100
D. 130o
Explanation:
Exterior angles on the same side of the
transversal are supplementary.
*Mnemonic: SST (same side of transversal)
means supplementary. ALTERNATE (either
interior or exterior) means congruent. Also,
CORRESPONDING angles are congruent.
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1000 MMR
52. There are 33 red bags, 25 green bags, and 17
blue bags in a store. What percent of the bags is
red?
A. 33%
B. 44%
C. 66%
D. 67%
Solution:
33/(33+25+17) = 33/75 or 11/25
11/25 in percent is 44%
................................................
53. Given sin θ = 0.28, which of the following
could possibly be cos θ?
A. 0.72
B. -0.86
C. 0.96
D. 1.14
Solution:
sin2 θ + cos2 θ = 1
(0.28)2 + cos2 θ = 1
cos2 θ = 1 – 0.0784
cos2 θ = 0.9216
cos θ = √0.9216 = ±0.96
................................................
54. If the sum of the supplement and the
complement of an angle is 130 degrees, what is
the angle?
A. 65o
B. 70o
C. 50o
D. 25o
Solution:
(90-x) + (180-x) = 130
270 – 2x = 130
270 – 130 = 2x
140 = 2x
70 = x
................................................
55. If today is a Saturday, what day is 125 days
from now?
A. Thursday
B. Friday
C. Sunday
D. Monday
Solution:
Every 7 days, it would be a Saturday again.
Author: Victor A. Tondo Jr., LPT
The nearest multiple of 7 to 125 is 126. That
means 126 days after today is Saturday again,
and 125 days after today should be Friday.
................................................
56. Car A is traveling towards the east at a speed
of 35 kph, while car B is traveling towards the
west at 45 kph. If they left the same point at 1:00
PM, how far apart are they at 3:45 PM?
A. 240 km
B. 220 km
C. 200 km
D. 180 km
Solution:
Time spent driving: 1:00 to 3:45 = 2.75 hrs
(45 mins in decimals is 45/60 since there are 60
mins in 1 hr)
Car A distance from mid: 2.75 (35) = 96.25
Car B distance from mid: 2.75 (45) = 123.75
Total distance: 123.75 + 96.25 = 220 km
Alternative Solution:
Since the two cars are traveling in two opposite
directions, add their speeds and multiply by
elapsed time.
2.75 (45+35) = 2.75 (80) = 220 km
................................................
57. Mr. Santos left the house at 1:00 PM and
traveled east at an average speed of 40 kph. His
wife Mrs. Santos left the at 2:00 PM and traveled
west at an average speed of 30 kph. How far
apart are they at 4:00 PM?
A. 180 km
B. 140 km
C. 100 km
D. 60 km
Solution:
Mr. Santos’s data:
Speed: 40 kph
Elapsed time: 1PM to 4PM = 3 hrs
Distance: 40kph (3hrs) = 120 km
Mrs. Santos’s data:
Speed: 30 kph
Elapsed time: 2PM to 4PM = 2 hrs
Distance: 30 kph (2hrs) = 60 km
Total Distance: 60 + 120 = 180 km
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1000 MMR
58. Five consecutive even numbers have a sum of
120. What is the sum of the 2nd and 5th even
numbers?
A. 46
B. 48
C. 50
D. 52
Author: Victor A. Tondo Jr., LPT
61. Given f(x) = ln
, what is f ‘(x)?
Solution:
Solution:
You can rewrite ln
as x2 + 2x since ln is
the natural logarithm (the logarithm whose base
is the natural number, e).
Let x = lowest even number
x + (x+2) + (x+4) + (x+6) + (x+8) = 120
5x + 20 = 120
5x = 100
x = 20
.: numbers are 20, 22, 24, 26, 28
22 + 28 = 50
Alternative Solution:
The middle (3rd) even number is 120/5 or 24.
That means the 2nd even number is 24 – 2 or 22,
and the 5th is 24 + 2(2) or 28.
................................................
59. If x = 3, which of the following is equal to 13?
A. 5x + 2
B. x2 + 2x + 1
C. x3 – 4x – 2
D. x2 + x + 2
Explanation:
Just substitute x with 3.
................................................
A.
C. (2x+2) ln (x2+2x)
B.
D. 2x + 2
Remember: ln eu = u, wherein u is the exponent
to which e is being raised.
The derivative of x2 + 2x is, of course, 2x + 2.
................................................
62. Which of the following could be the value of x
if x 3(mod 11)?
A. 33
B. 47
C. 52
D. 2
Solution:
Just divide the numbers by 11 and see which one
gives a remainder of 3.
................................................
63. If = 6x2 + 8x – 7, which could be u?
A. 12x + 8
B. 3x3 + 4x2 – 7x + 11
C. 2x3 + 4x2 -7x +1 D. 12x2 + 8x - 10
Explanation:
60. If f(x) = x2 + 4x + 3, which of the following is
equal to 99?
A. f(11)
B. f(-12)
C. f(12)
D. f(-8)
Anti-derivatives. If you already forgot how to do
that, simply check which choice has a derivative
of 6x2 + 8x – 7.
................................................
Solution:
64. What is the center of x2 + y2 – 8x + 6y = 0?
A. (-8.6)
B. (8, -6)
C. (-4, 3)
D. (4, -3)
x2 + 4x + 3 = 99
x2 + 4x + 3 + 1 = 99 + 1
x2 + 4x + 4 = 100
√( + 2) = 100
x + 2 = ± 10
x = -2 ± 10
Solution:
The center, C(h,k) is given as h = -D/2 and
k = -E/2 wherein D and E are from the equation
x2 + y2 + Dx + Ey + F = 0.
That’s -2+10 or 8, and -2-10 or -12
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1000 MMR
65. Which of the following is a parabola that
opens to the right?
A. 6y = (x+9)2 - 8 B. -4y = (x-6)2 + 3
C. -5x + 3 = (y-2)2 D. 2x + 6 = (y+3)2
Explanation:
When x is the squared variable, the parabola
opens upward when the coefficient of y is
positive (example: A).
When x is the squared variable, the parabola
opens downward when the coefficient of y is
negative (example: B).
When y is the squared variable, the parabola
opens to the left when the coefficient of x is
negative (example: C).
When y is the squared variable, the parabola
opens to the right when the coefficient of x is
positive (example: D).
................................................
66. Factorize: 12x2 – 7x – 10.
A. (6x + 5) (2x – 2)
B. (6x – 2) (2x + 5)
C. (3x + 2) (4x – 5)
D. (3x – 2) (4x + 5)
................................................
67. For which value of k does 4x2 + kx + 49 have
only one root?
A. -28
B. -14
C. 7/2
D. -7/4
Explanation:
You may use Completing Square Trinomials. The
middle term is twice the product of the square
roots of the first and third terms. In the problem,
the middle term is twice the product of √4 and
√49. That’s 2(2x)(7) or 28x. Don’t forget that the
middle term could be positive or negative.
You may also use the discriminant to answer
this:
b2 – 4ac = 0 when there’s only one root,
b2 – 4ac > 0 when there are two real roots
b2 – 4ac < 0 when there are no real roots
Author: Victor A. Tondo Jr., LPT
Explanation:
Since A and B are the roots, then AB pertains to
the product of the roots which is given as c/a.
................................................
69. 1 + 2 + 4 + 8 + … + 2048 = ____
A. 4095
B. 4096
C. 4097
D. 4098
Solution:
You may use the Geometric Series formula which
is ∑
= (
), where r is the common
ratio, n is the number of terms, and a1 is the first
term.
Alternative Solution:
In this problem, however, you cannot easily use
the GS formula since you don’t know n, the
number of terms.
I will personally use the shortcut for the sum of a
geometric sequence wherein the ratio is 2 or ½.
The shortcut is SUM = 2(largest) – smallest. In
this problem, that’s 2(2048)-1 = 4095.
You may also apply this in the next item, #70.
70. 24 + 12 + 6 + 3 + 1.5 + … = ____
A. 48
B. 50
C. 54
D. 60
Solution:
You may use the Infinite Geometric Series
formula which is ∑
= ( ), where r is the
common ratio and a1 is the first term.
Alternative Solution:
In this problem, I would still be using the
shortcut since the ratio is ½. Since this is an
infinite geometric sequence, then the last term
won’t have any significant value. Thus, the sum is
simply twice the first term. That’s 2(24) = 48.
68. If A and B are the roots of x2 + 7x + 15, what
is AB?
A. 7√3 + 2
B. 2√3 + 7
C. 3√2 + 2√3
D. 15
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1000 MMR
71. How many terms are there in the sequence
5, 13, 21, 29, …, 357?
A. 40
B. 44
C. 45
D. 70
Author: Victor A. Tondo Jr., LPT
74. C is the midpoint of ̅̅̅̅
AB where A is at (-3,4)
and B is at (7,-10). Find the coordinates of C.
A. (5,-7)
B. (-5,7)
C. (2,-3)
D. (-2,3)
Solution:
Solution:
Midpoint Formula: (
An = A1 + (n – 1)d
357 = 5 + (n – 1)(8)
357-5 = 8(n – 1)
352 =8(n – 1)
44 = n – 1
45 = n
,
)
Midpoint: (
,
)
................................................
75. It is a line segment formed by connecting two
non-consecutive vertices of a polygon.
A. side
B. apothem
C. altitude
D. diagonal
Alternative Solution:
(this is the “y=mx+b” solution I taught my grade
3 student for Singapore. Yes, Grade 3.)
Before anything else, since this might be “new”
to you, your d is our m, your An is our y, your n is
our x, and b is your A1 – d.
357 = 8x +(5 – 8)
357 = 8x – 3
360 = 8x
45 = x
................................................
72. How many ways can a group of 5 be selected
from 5 boys and 5 girls if the group must contain
3 boys and 2 girls?
A. 151,200
B. 1200
C. 252
D. 100
Explanation:
A side is formed by connecting two consecutive
vertices of a polygon.
The apothem is only for regular polygons. It is
the perpendicular bisector of one of its sides,
passing through the center.
A diagonal is a line segment formed by
connecting two non-consecutive vertices of a
polygon.
................................................
76. Find the equation of the line perpendicular to
2x – 3y = 7, passing through (1,2).
A. 2x + 3y = 8
B. 3x + 2y = 7
C. 2x – 3y = -4
D. 3x – 2y = -1
Solution:
Solution:
A group, committee, or team (any set with no
hierarchy of members) calls for Combinations.
To pick 3 boys from a total of 5 boys, use 5C3
and that’s 10. To pick 2 girls from a total of 5
girls, use 5C2 and that’s 10. Lastly, 10x10 = 100.
................................................
Simply interchange the numerical coefficients of
x and y in the original equation, then change the
operation between them. 2x – 3y becomes
3x + 2y.
For the constant, simply substitute the
x and y values of the point ((1 ,2) in this
problem) and solve for the constant.
3(1) + 2(2)=7.
Thus, 3x+2y=7.
73. What is the probability of getting a sum of 9
when rolling 2 dice?
A. 1/9
B. 5/36
C. 1/6
D. 7/36
The only pairs with a sum of 9 are (3,6), (4,5),
(5,4), and (6,3). There are only 4 pairs out of 36.
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1000 MMR
77. Two parallel lines are cut by a transversal to
form X, Y, and Z. Given that X and Y are
alternate interior angles while Y and Z are
interior angles on the same side of the
transversal, find m Z if m X = 40o.
A. 40o
B. 50o
o
C. 130
D. 140o
Explanation:
Alternate, corresponding, and vertical pairs
automatically suggest that the two angles are
congruent. Linear pairs and angles on the same
side of transversal (SST) are supplementary.
m X = 40, .: m Y = 40 since alternate interior
angles
m Z = 180-40 = 140 since Y and Z are
interior angles on the same side of the
transversal.
................................................
78. The measure of each interior angle of a
regular polygon is 144o. How many vertices does
it have?
A. 36
B. 24
C. 12
D. 10
Solution:
MIA =
=
(
)
Alternative Solution:
Personally, I always go for the exterior angle first
to get the number of sides or vertices. Since the
exterior and interior are supplementary, then
each exterior measures 180-144 or 36. The
formula for number of sides or vertices given the
measure of each exterior is 360÷MEA, so that’s
360÷36 or 10 vertices.
By the way, you may derive this solution by
(
)
manipulating the formula for MIA:
. That
becomes 180 –
. That means
= 180 – MIA,
or
= n.
................................................
79. Solve: (x + 9) (x – 3) < 0
A. -9 < x < 3
B. x < -3 x > 9
C. x < -9 x > 3
D. x
; x -9, 3
Author: Victor A. Tondo Jr., LPT
Solution:
Usually, people would
straight go for the Test
Point Table method which
we use in Calculus.
However, since this is the
licensure exam, I’d prefer
that you use a simpler and
quicker approach to this
problem.
First, identify the zeros of
the inequality by equating
each factor to 0. Our zeros
are -9 and 3.
Next, identify the opening
of the parabola. Since the
leading coefficient would
be positive, then the
parabola opens upwards.
Since the parabola opens upwards, then the
parts less than 0 should be between the zeros of
the inequality. That means x should be between
-9 and 3.
................................................
80. The product of two consecutive even
counting numbers is 3248. Find the smaller
number.
A. 42
B. 46
C. 52
D. 56
Solution:
x(x+2) = 3248
x2 + 2x = 3248
x2 + 2x + 1 = 3249
√ + 2 + 1 = √3249
x + 1 = 57
x = 56
WAIS Solution:
Get your scientific calculator, extract √3248 and
then scrape the decimals or round down.
#2EZ4U
................................................
81. Solve for x: 2log2 3 – log2 18 = x
A. ½
B. -1
C. -2
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D. 1
1000 MMR
Solution:
Rewrite the logarithm as a single logarithm by
applying the rules of logarithms.
2log2 3 becomes log2 , or log2 ½
log2 ½ = -1
................................................
82. Twinkle Bucks has four serving sizes for their
milk tea: Small, Medium, Large, and Extra Large.
What level of data are they using for their
serving sizes?
A. nominal
B. ordinal
C. interval
D. ratio
................................................
83. After receiving a 20% markup, a bag was sold
for P960. How much was it originally?
A. P1152
B. P4800
C. P800
D. P1200
Solution:
Selling Price = Original Price (1 + Markup Rate)
960 = OP (1 + 0.20)
960/1.2 = OP
800 = OP
................................................
̅̅̅̅ bisects ABC and m ABT = 40o,
84. Given BT
find m ABC.
A. 20o
B. 40o
C. 60o
D. 80o
Explanation:
ABT is formed after the bisection of ABC. That
means ABT is half of ABC, or ABC is twice of
ABT.
................................................
85. A cone has a radius of 9 cm and a slant height
of 15 cm. Find its volume.
A. 243 π cm3
B. 324 π cm3
3
C. 405 π cm
D. 486 π cm3
Explanation:
Be careful with cones. Tendency kasi sa LET that
they will give the slant height while looking for
volume and the height while looking for the
Author: Victor A. Tondo Jr., LPT
surface area. Just remember that the slant height
is always longer than the height. The slant height
is the hypotenuse, while the height is one of the
legs with the radius as the other. Just use the
Pythagorean formula to solve for whichever is
missing.
The height is 12 cm (after using Pythagorean
formula).
Vol = π r2 h = π (92) (12) = 324 π cm2
................................................
86. If f(x) = x2 + 4x + 4 and g(x) = x-2, find
f(g(x)).
A. x2
B. x3 – 6x2 + 6x – 9
C. x2 + 8x + 16
D. x2 – 8x + 16
Solution:
f(g(x)) = f(x-2)
= (x-2)2 + 4(x-2) + 4
= (x2 – 4x + 4)+ (4x – 8) + 4 = x2
................................................
87. A 10 ft ladder leans against a wall, forming a
30o angle with it. How high on the wall does it
reach?
A. 5 ft
B. 5 √3 ft
C. 10 √3 ft
D. 10 √6 ft
Solution:
Draw the problem first. The ladder and the wall
form a 30o angle with each other and the wall is
of course perpendicular to the ground.
That means the ladder forms a 60o angle with the
ground. The ladder is the hypotenuse, while its
reach on the wall is adjacent to the 30o angle or
simply put, the longer side. The smaller side
measures half of 10 or 5 ft, therefore the longer
side must be 5√3 ft.
................................................
88. How many ways can a committee of 5 be
selected from 9 people?
A. 126
B. 120
C. 3024
D. 15120
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1000 MMR
Explanation:
Just use your scientific calculator: 9C5.
................................................
89. What is 60% of 80% of 500?
A. 480
B. 240
C. 120
D. 60
Solution:
(0.6)(0.8)(500) = 240
................................................
90. If 3x = 7 and 2y = 5, what is 6(x-y)?
A. -1
B. 1-√35
C. √7 - √5
D.
Solution:
6(x-y) = 6x – 6y = 2(3x) – 3(2y)
2(3x) – 3(2y) = 2(7) – 3(5) = 14-15 = -1
................................................
91. If two numbers have a product of 71 and the
sum of their squares is 147, what is their sum?
A. -17
B. 5
C. 12√3 + √5
D. 12 + √3
Solution:
Let A and B be our two numbers.
AB = 71;
A2 + B2 = 147
.: A2 + B2 + 2AB = 147 + 2(71) = 289
(A + B)2 = 289;
A + B = ±17
................................................
Author: Victor A. Tondo Jr., LPT
93. How many 3-digit numbers can be formed
using the digits 0, 1, 2, 3, 4 and 5 if repetition is
not allowed?
A. 60
B. 80
C. 100
D. 120
Solution:
Use FCP (Fundamental Counting Principle):
__ x __ x __
For the first digit, we cannot use 0. That means
we only have 5 choices for the first digit.
For the second digit, we can now use 0. Since we
have already used one digit for the first, that
means we have 5 choices for the second digit.
For the last digit, since we have already used two
digits, we only have 4 choices.
5 x 5 x 4 = 100
................................................
94. How many ml of 20% acid must be added to
400 ml of 50% acid to make a 30% acid solution?
A. 1000 ml
B. 900 ml
C. 800 ml
D. 750 ml
Solution:
C1 V1 + C2 V2 = CR VR
20 (V) + 50 (400) = 30 (V + 400)
20V + 20,000 = 30V + 12,000
20,000 – 12,000 = 30V – 20 V
8,000 = 10V;
800 = V
................................................
92. Find the median: 7, 9, 11, 10, 9, 13, 17, 14
A. 10 and 11
B. 9 and 10
C. 10.5
D. 9.5
95. How many ml each of 10% and 50% solution
should be mixed to make 500 ml of 18%
solution?
A. 400 ml of 10% and 100 ml of 50%
B. 350 ml of 10% and 150 ml of 50%
C. 300 ml of 10% and 200 ml of 50%
D. 200 ml of 10% and 300 ml of 50%
Solution:
Solution:
Rearrange the numbers from least to greatest:
7, 9, 9, 10, 11, 13, 14, 17
 there are 8 nos.
The median is the
th or 4.5th number. That
means we have to get half the sum of our 4th and
5th numbers. (10+11)/2 = 10.5
Since our resultant volume is 500, then our two
volumes will be x and (500-x).
C1 V1 + C2 V2 = CR VR
10(x) + 50(500-x) = 18(500)
10x + 25,000 – 50x = 9,000
25,000 – 9,000 = 50x – 10x
16,000 = 40x;
400 = x
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1000 MMR
96. It takes 28 men a total of 24 days to build a
house. How long would it take 32 men to build a
similar house?
A. 28 days
B. 27 days
C. 21 days
D. 19 days
Solution:
This is an indirect or inverse proportion.
Let x = number of days it would take the 32 men
to build the house
28(24) = 32 x
672 = 32 x
21 = x
................................................
97. Evaluate: lim
A. undefined
C. 8
B. limit does not exist
D. +
Author: Victor A. Tondo Jr., LPT
A rectangle’s diagonals are congruent and they
bisect each other. However, they are not
perpendicular.
An isosceles trapezoid has congruent diagonals,
however, they do not bisect each other, nor are
they perpendicular.
A rhombus has diagonals that are perpendicular
and that bisect each other. However, they are not
congruent.
PS: A square has diagonals that are congruent,
perpendicular, and that bisect each other.
................................................
100. A pipe can fill a pool in 6 hours while
another pipe can drain empty the pool in 15
hours. How long will it take to fill the pool if both
pipes are open?
A. 9 hours
B. 9.125 hours
C. 9.45 hours
D. 10 hours
Solution:
Explanation:
You may simplify the function first before
substituting x with 4.
................................................
98. A box contains 7 red, 8 blue, and 9 white
balls. When taking two balls in succession, what
is the probability that both balls are white?
A. 9/64
B. 9/69
C. 7/64
D. 7/69
This is similar to our “Working Together”
problem, except instead of adding their times, we
will subtract (since the draining pipe is doing the
opposite of helping).
AB/(A-B) = 15(6)/(15-6) = 90/9 = 10 hrs
................................................
101. If log n – 1 = 2, find n.
A. 3
B. 1000
C. e3
D. 3e
Solution:
Solution:
First white ball: 9/24
Second white ball: 8/23
9/24 x 8/23 = 9/69
................................................
99. Which of the following has two diagonals that
are perpendicular bisectors of each other?
A. kite
B. rectangle
C. rhombus
D. isosceles trapezoid
Explanation:
A kite’s diagonals are perpendicular but only one
diagonal will bisect the other.
log n – 1 = 2
log n = 2 + 1 = 3
(note that the base of the log is 10)
log n = 3 translates to 103 = n
Therefore n = 1000
................................................
102. log2 3 + 2 log2 7 – log2 5 = ______.
A. log2
B. log2
C. log2
D. log2
Explanation:
Just apply the laws of logarithms.
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1000 MMR
103. The surface areas of two spheres are 12 π
cm2 and 108 π cm2. What is the ratio of their
volumes?
A. 1:3√3
B. 1:9
C. 1:27
D. 2:3√3
Author: Victor A. Tondo Jr., LPT
106. Find the volume of a steel cylinder of radius
5 cm and height 12 cm.
A. 300 π cm3
B. 250 π cm3
C. 200 π cm3
D. 100 π cm3
Solution:
Vol = π r2 h = 52 (12) π = 300 π cm3
................................................
Ratio of surface areas: 12:108 or 1:9
Ratio of radii: √1: √9 or 1:3
Ratio of volumes: 13:33 or 1:27
................................................
104. The volume of a regular hexahedron is 64
in3. How long is each side?
A. 2 in
B. 4 in
C. 6 in
D. 8 in
Explanation:
A regular hexahedron is simply a cube.
................................................
Solution:
107. A cube sits perfectly inside a sphere of
volume 108 √3 π cm3. Find the volume of the
cube.
A. 27 cm3
B. 54 cm3
3
C. 108 cm
D. 216 cm3
Solution:
Volume of sphere = 108√3 π cm3
π r3 = 108√3 π
r3 = (108)√3
105. Which of the following statements is
ALWAYS true?
A. The square of a prime number is odd.
B. The sum of two consecutive even numbers is
divisible by 4.
C. Any even number is composite.
D. The product of two consecutive even numbers
is divisible by 8.
r3 = 81√3; r = 3√3;
d = 6√3
Diagonal of cube = s√3 = 6√3
.: s = 6;
volume = s3 = 63 = 216
Explanation:
N=
A. Counterexample: The prime number, 2. The
square of 2 is 4 which is even.
B. Always false. One example is 2 and 4. Their
sum, 6, is not divisible by 4.
C. Counterexample: The prime even number, 2.
D. Proof by Algebra: Let the first even number be
2x. The second even number will be 2x + 2.
Their product will be 4x2 + 4x.
If x is an odd number, x = 2y + 1 where y is a
counting number. 4x2 + 4x = 4(4y2 + 4y + 1) +
4(2y + 1) = 16y2 + 16y + 4 + 8y + 4 = 16y2 +
24y + 8, which is divisible by 8.
If x is even, x = 2y where y is any counting
number. 4(4y2) + 4(2y) = 16y2 + 8y, which is
also divisible by 8.
Either ways, the statement holds true.
................................................
Alternative Solution:
Ratio of volume of cube to sphere (cube is inside
sphere) = 2 : √3 π
N : 108 √3 π = 2 : √3 π
(
√ )( )
√
= 216
108. Find the distance in cm of an 80 cm chord
from the center of a circle whose radius is 41 cm.
A. 41 - 2√10
B. 41 - 4√10
C. 9√2
D. 9
Solution:
The chord is perpendicularly bisected by a
segment connected to the center of the circle,
whose length is the distance we are looking for.
If the radius is drawn connected to one endpoint
of the chord, we can form a right triangle whose
hypotenuse is the radius and one leg is half of the
chord. Using the Pythagorean theorem, the
distance is √41
40 or simply, 9.
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1000 MMR
109. Which quadrilateral has two congruent
diagonals that bisect each other?
A. kite
B. isosceles trapezoid
C. rectangle
D. rhombus
................................................
110. What is the longest side of ∆MTC if m M =
40o and m C = 60o?
̅̅̅̅
A. ̅̅̅̅
MC
B. ̅̅̅̅
TC
C. MT
D. ̅̅̅̅
CT
Explanation:
m T = 180-(40+60) = 80
The longest side is opposite the largest angle, T.
................................................
111. Find the altitude to the hypotenuse of a
right triangle whose legs measure 10 cm and 24
cm.
A. 120 cm
B.
cm
C. 120√2 cm
D. 24√5 cm
Solution:
Find the hypotenuse first. That would be 26 cm.
Altitude to the Hyp = (L1 L2)/Hyp = 24(10)/26
Altitude to the Hyp = 120/13
................................................
112. Find the inverse of y = x2 + 10x.
A. y-1 = √
25 + 5
B. y-1 = √
25 – 5
C. y-1 = √ + 25 + 5
D. y-1 = √ + 25 – 5
Solution:
y = x2 + 10x
y + 25 = x2 + 10x + 25
y + 25 = (x+5)2
√ + 25 = x + 5
√ + 25 – 5 = x
√ + 25 – 5 = y-1
................................................
Author: Victor A. Tondo Jr., LPT
113. Find the intersection of y = 2x + 3 and
y = 4x – 11.
A. (-4/3, 0)
B. (4/3, 0)
C. (7, 17)
D. (-7,-17)
Solution:
y = 2x + 3
- y = 4x – 11
0 = -2x + 14
2x = 14;
x=7
................................................
114. Find the area of the triangle whose vertices
are (1,4), (2,3), and (3,0).
A. 0
B. 1
C. 5/3
D. 3/4
Solution:
1 2 3 1
|
| |
4 3 0 4
= ( 2) =
1
Note: If the result is negative, that means your
points are simply arranged clockwise. Just get
the absolute value of the answer.
................................................
115. Find the tenth term: 3, 10, 17, 24, …
A. 66
B. 67
C. 68
D. 69
Solution:
A10 = 3 + (10-1) (7) = 3 + 63 = 66
................................................
116. Find the remainder when
x4 – 5x3 + 6x2 + 2x + 1 is divided by (x – 2).
A. 17
B. 13
C. 9
D. 5
Solution:
24 – 5(23) + 6(22) + 2(2) + 1
= 16 – 40 + 24 + 4 + 1 = 5
................................................
117. The sum of Fe’s age and Sita’s age is 60.
Twelve years ago, Fe was twice as old as Sita.
How old is Sita now?
A. 18
B. 24
C. 30
D. 36
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1000 MMR
Solution:
Fe
Sita
Age Now
x
60-x
Age 12 yrs ago
x – 12
(60 – x) – 12 or 48 – x
x – 12 = (2)(48 – x)
x – 12 = 96 – 2x
x + 2x = 96 + 12
x + 2x = 108
x = 36
Author: Victor A. Tondo Jr., LPT
By transitive property of equality,
8x = 15z, or x = z
................................................
121. Victor had an average of 94 on his first four
Math tests. After taking the next test, his average
dropped to 93. Find his most recent grade.
A. 88
B. 89
C. 90
D. 91
Solution:
60 – x = 60 – 36 = 24
................................................
118. If the length of a rectangle is increased by
20% while the width is decreased by 10%, what
will happen to its area?
A. decreased by 10%
B. increased by 10%
C. increased by 8%
D. decreased by 2%
New Score =(New Number)(New Average) –
(Old Number)(Old Average)
New Score = 5(93) – 4(94) = 465 – 376 = 89
................................................
122. X is of Y and Y is of Z. What part of Z is X?
A. X = Z
B. X = Z
C. X = Z
D. X = Z
Solution:
Solution:
(L x 1.2) (W x 0.9) = (1.08 x LW)
................................................
X= Y
119. The 19th term of an arithmetic sequence is
85 and the 12th term is 43. Find the common
difference.
A. 5
B. 6
C. 7
D. 8
Solution:
d=
=
=
=6
X = ( Z) = Z or Z
................................................
123. Two buses leave the same station at 8:00
pm. One bus travels north at the rate of 30 kph
and the other travels east at 40 kph. How many
kilometers apart are the buses at 10 pm?
A. 140 km
B. 100 km
C. 70 km
D. 50 km
Solution:
120. If 2x = 3y and 4y = 5z, what is z in terms of
x?
A. z = x
B. z = x
C. z = x
D. z =
x
Solution:
Make two equations wherein y will have the
same numerical coefficients.
2x = 3y

8x = 12y
4y = 5z

12y = 15z
From 8 to 10 PM is 2 hours. After two hours, one
bus will have travelled 60 km while the other, 80
km. Since the two buses are traveling on
perpendicular directions, we can use the
Pythagorean Theorem to find their distance.
D = √60 + 80 = 100 km
................................................
124. A bus drove for 6 hours at 75 kph and 4
hours at 80 kph. What was its average speed?
A. 76 kph
B. 77 kph
C. 77.5 kph
D. 78 kph
This is a free reviewer. All rights reserved.
1000 MMR
Solution:
Get the total distance and the total time first.
6 hrs x 75 kph = 450 km
4 hrs x 80 kph = 320 km
Total distance = 770 km, total time = 10 hrs
Average spd =
=
= 77 kph
................................................
125. 18 students failed a quiz. They represent
30% of the class. How many students passed the
quiz?
A. 60
B. 42
C. 36
D. 24
Solution:
18:30% = N:70%
18(70)/30 = N
42 = N
................................................
126. Rationalize:
A.
√
C.
√
+1
B. 2√5 – 4
√
To rationalize this, multiply both numerator and
denominator by the conjugate of the
denominator. By doing this, we are sure to have
a rational denominator.
√
x
√
=
√
Let x = investment in Bank B
.: 100,000 – x = investment in Bank A
0.05(100,000 – x) + 0.06x = 5,600
(5,000 – 0.05x) + 0.06x = 5,600
0.01x = 600;
x = 60,000
................................................
129. Evaluate
A. -38
=
when x = ¾ and y = .
B. -19
C. 19
D. 38
Solution:
=
;
=
=
÷ = x = 19
................................................
Solution:
√
Solution:
+ =
√
D.
√
Author: Victor A. Tondo Jr., LPT
128. Mr. Tondo has P100,000 to invest, from
which he wants to earn P5600 per year. Bank A
offers 5% per annum while Bank B offers 6%.
How much should he invest at Bank B?
A. P45,000
B. P50,000
C. P55,000
D. P60,000
√
................................................
127. RNHS has 130 quizzers. 67 of them are
Math, 60 are Science, and 20 are quizzers for
both Math and Science. How many quizzers are
neither Math nor Science?
A. 0
B. 13
C. 17
D. 23
Solution:
(A B)’ = U - (A B)
(A B)’ = 130 – [A + B – (A B)]
(A B)’ = 130 – (67 + 60 – 20) = 130 – 107 = 23
130. Today, Vic is 11 years old while his father is
37. How many years from now will his father be
twice as old as he?
A. 15
B. 13
C. 11
D. 10
Solution:
Let x = number of years from now
2(11+x) = 37+x
22 + 2x = 37 + x
2x – x = 37 – 22;
15 = x
................................................
131. Carla and Diana are on a seesaw. Carla
weighs 50 kg and sits 168 cm to the left of the
fulcrum. If Diana weighs 60 kg, how far to the
right of the fulcrum must she sit to balance the
seesaw?
A. 140 cm
B. 170.8 cm
C. 201.6 cm
D. 210 cm
This is a free reviewer. All rights reserved.
1000 MMR
Solution:
Seesaw problems call for inverse or indirect
proportion.
50(168) = 60N
8400 = 60N
140 = N
................................................
132. Twenty guests shake hands with each other.
If each guest is to shake hands with all the other
guests, how many handshakes will be made?
A. 400
B. 380
C. 200
D. 190
Solution:
20C2 = 190
................................................
133. How many line segments can be made from
30 non-collinear points?
A. 900
B. 870
C. 450
D. 435
Solution:
30C2 = 435
................................................
134. The longest chord of a circle is 80 cm. How
long is its radius?
A. 20 cm
B. 30 cm
C. 20√2 cm
D. 40 cm
Explanation:
The longest chord is the diameter, and the radius
is half the diameter.
................................................
135. Find k such that 34k67 is divisible by 9.
A. 5
B. 6
C. 7
D. 8
Author: Victor A. Tondo Jr., LPT
136. Find the largest area of a rectangle whose
perimeter is 100 cm.
A. 2500 cm2
B. 2499 cm2
C. 625 cm2
D. 624 cm2
Solution:
Instead of jumping to differential calculus
(minima and maxima) to solve this, simply make
it a square. That’s the shortcut for this kind of
question.
................................................
137. What time is 200 minutes past 10:30 PM?
A. 12:30 AM
B. 12:30 PM
C. 1:50 AM
D. 1:50 PM
Solution:
200 minutes = 3 hrs 20 mins
10:30 PM
+ 3:20
13:50 PM
 1:50 AM
................................................
138. Find the product of two numbers whose
GCF is 24 and LCM is 120.
A. 2880
B. 1440
C. 720
D. 360
Explanation:
The product of two numbers is equal to the
product of their LCM and GCF.
24 x 120 = 2880
................................................
139. The salary of 4 men for 5 days is P9,000.
How much is the salary of 5 men for 6 days?
A. P12,000
B. P12,600
C. P13,500
D. P14,400
Solution:
Solution:
Remember that for a number to be divisible by 9,
the sum of its digits must be equal to 9.
3+4+k+6+7 = 20+k
2+0+k =
2 + k = 9;
k=7
................................................
First, find the cost of each “man-day”.
4 men x 5 days = 20 man-days
P9,000 ÷ 20 man-days = P450 per man-day
You may now solve the problem.
5 men x 6 days = 30 man-days
30 man-days x P450 per man-day = P13,500
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1000 MMR
140. The average grade of eleven students is 83.
If the average of six of these students is 88, what
is the average of the other 5 students?
A. 77
B. 78
C. 79
D. 80
Solution:
Solution:
(
Sum of grades of 11 students: 11 x 83 = 913
Sum of grades of 6 students: 6 x 88 = 528
Sum of grades of other five: 913 – 528 = 385
Average of grades of other five: 385 ÷ 5 = 77
................................................
141. If x is 80% of y, what percent of y is x?
A. 120%
B. 125%
C. 130%
D. 135%
Solution:
x = 0.8 y
=y
 1÷0.8 = 1.25
.
1.25x = y
................................................
142. Bus X left the terminal at 1 PM and traveled
at a speed of 60 kph. Bus Y left the same terminal
2 hours later and traveled 80 kph on the same
route. What time will Bus B catch up with Bus A?
A. 6 PM
B. 9 PM
C. 11 PM
D. 1 AM
)
(
)
(
)
= 20
(x + 5) + (2x – 4) + (x + 7) = 60
4x + 8 = 60
4x = 52
x = 13
................................................
145. Mia is 16 years younger than Kia. 13 years
ago, Kia was thrice as old as Mia. What is Kia’s
present age?
A. 43
B. 40
C. 37
D. 34
Solution:
Kia
Mia
Age today
x
x – 16
Age 13 years ago
x – 13
x – 16 – 13 or x – 29
(x – 13) = 3(x – 29)
x – 13 = 3x – 87
-13 + 87= 3x - x
74 = 2x
37 = x
................................................
146. Insert one term between 18 and 32 to make
a geometric sequence.
A. 20
B. 24
C. 25
D. 27
Solution:
Let x = running time for Bus X
60x = 80(x-2)
60x = 80x – 160
160 = 20x;
8=x
Author: Victor A. Tondo Jr., LPT
144. The average of x+5, 2x-4, and x+7 is 20.
Find x.
A. 18
B. 13
C. 9
D. 8
 Bus Y left 2 hrs later
8 hours after Bus X left the terminal is 9AM.
................................................
143. What is the degree of the polynomial
-3 x2y3 + 21 x3y4 – 7 x5y6 – 15?
A. 4
B. 5
C. 11
D. 21
Explanation:
Solution:
Shortcut for inserting one term is √AB. This is
also the formula for the geometric mean.
√18(32) = √576 = 24
................................................
147. There are 100 pigs and chickens in a farm,
all of which are healthy. If there are 340 legs in
total, how many pigs are there?
A. 70
B. 65
C. 60
D. 55
The degree of a polynomial is the highest sum of
exponents in a term.
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1000 MMR
Solution:
Author: Victor A. Tondo Jr., LPT
150. Solve for x:
49x = 343
A. 1.142857
B. 7
C. 1.5
D. √7
Let P = number of pigs;
C = number of chickens
P + C = 100
4P + 2C = 340
 since pigs have four legs
and chickens have two
2(P + C = 100)

2P + 2C = 200
4P + 2C = 340
- 4P + 2C = 340
-2P
= - 140
P = 70
................................................
148. Adam can do a job alone in 8 hours, while
Bam can do the same job in 12 hours. One day,
they worked together for 1 hour before Bam left
Adam to finish the job. How long will it take
Adam to finish the remaining job?
A. 6 hrs 50 mins
B. 6 hrs 40 mins
C. 6 hrs 30 mins
D. 6 hrs 20 mins
Solution:
(
)
First, express both numbers as powers of the
same base.
49x = 343

(72)x = 73
Next, apply the laws of exponents.
(72)x = 73
72x = 73
2x = 3;
x = 3/2 or 1.5
................................................
151. What is the highest possible product of two
numbers if their sum is 45?
A. 506
B. 506.25
C. 506.5
D. 506.725
Solution:
(
,
Solution:
(
)
)
=
= or 6
That’s 6 hrs. and 20 mins.
hrs
Mnemonic:
For questions like this (about working together
and then someone leaves), use PuTS U. When
someone leaves you, “PuTS U” !!
PuTS U stands for P(u)roduct, Time, Sum, Umalis
................................................
149. Find x if 2748 = 9x.
A. 144
B. 81
C. 72
D. 60
Solution:
Express both sides as a power of 3.
(33)48 = (32)x
3144 = 32x
144 = 2x
72 = x
................................................
Instead of jumping straight to minima and
maxima under differential calculus, simply make
your numbers equal to maximize their product.
45/2 = 22.5
Both numbers will be 22.5, so their product is
22.5 x 22.5 = 506.25
................................................
152. Which statistical test is used for comparing
observed frequencies to expected frequencies?
A. ANOVA
B. t-test
C. Pearson R
D. Chi Square
Explanation:
Observed vs Expected: Chi Square
Relationship: Pearson R (R for relationship)
Group differences: ANOVA (variance =
differences)
Comparing sets of normal distributions: T-test
................................................
153. The product of two consecutive odd
counting numbers is 1443. What is their sum?
A. 76
B. 78
C. 80
D. 82
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1000 MMR
Solution:
Author: Victor A. Tondo Jr., LPT
157. Find + given x + y = 20 and xy = 81.
Let x = first number;
x+2 = next number
x(x+2) = 1443
x2 + 2x = 1443
x2 + 2x + 1 = 1443 + 1
√ + 2x + 1 = √1444
x+1 = 38;
x = 37
x+2 = 39
................................................
A.
2 +1
4
154. Given ( ) = { 4
= 4,
7
4
( ).
find lim
A. 4
B. 9
C. 0
D. limit does not exist
Solution:
Limit from the left: 2(4) + 1 = 9
Limit from the right: 42 – 7 = 9
Since both limits are equal, then the limit is 9.
................................................
155. If today is a Saturday, what day is 125 days
from now?
A. Friday
B. Sunday
C. Monday
D. Tuesday
B.
C.
D.
Solution:
+ =
+
=
=
................................................
158. What is the remainder when
534,214,557,989,215 is divided by 4?
A. 0
B. 1
C. 2
D. 3
Explanation:
The divisibility rule for 4 tells us that our
concern would only be the last 2 digits.
15 ÷ 4 = 3 r. 3
................................................
159. Dividing by 0.125 is the same as multiplying
by which number?
A. 5
B. 8
C. 10
D. 16
Explanation:
Solution:
Just use 1 as your test number.
1÷ 0.125 = 8
................................................
This is an application of modulo.
125 6 (mod 7)
or
125÷7 = 17 r. 6
6 days after Saturday is Friday
................................................
160. Find the surface area of a sphere whose
radius is 6 cm.
A. 72 π cm2
B. 108 π cm2
C. 144 π cm2
D. 192 π cm2
156. If the sum of the supplement and the
complement of an angle is 124, what is the
angle?
A. 71
B. 72
C. 73
D. 74
Solution:
Solution:
(180 – x) + (90 – x) = 124
270 – 2x = 124
270 – 124 = 2x
146 = 2x;
73 = x
................................................
Surface Area = 4 π r2 = 4 (62) π = 144 π
161. Which of the following is the reference
angle of 216o?
A. 84o
B. 66o
C. 54o
D. 36o
Explanation:
The reference angle for angles from the different
quadrants are as follows:
QI: the angle θ itself
QII. 180 – θ
QIII. θ – 180
QIV. 360 – θ
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1000 MMR
162. Which of the following angles in standard
position is coterminal with 40o?
A. 2200o
B. 1760o
C. 1520o
D. 1360o
Explanation:
Coterminal angles are congruent, modulo 360.
That means they will leave the same remainder
when divided by 360.
In textbooks, θ is coterminal with any angle
expressed as 360N + θ wherein N is an integer.
To easily tackle this question, simply subtract 40
from each of the choices, then see if any of those
is divisible by 360 (or leaves a remainder of 0
when divided by 360) using your calculator.
2200 – 40 = 2160
2160 ÷ 360 = 6
.: 2200o is coterminal with 40o
................................................
163. Find the equation of the line passing
through (2,7) and (-3,-3).
A. y = 4x -1
B. y = 3x + 1
C. y = 3x + 6
D. y = 2x + 3
Solution:
Two-point form of linear equations:
y – y1 =
(x x )
y–7=
(x 2)
y – 7 = 2(x – 2)
y – 7 = 2x – 4
y = 2x + 3
................................................
164. In which quadrant can we find θ if tan θ < 0
and sin θ > 0?
A. First Quadrant
B. Second Quadrant
C. Third Quadrant
D. Fourth Quadrant
Explanation:
Use the CAST
mnemonic.
Author: Victor A. Tondo Jr., LPT
165. Find the equation of the line passing
through the point of origin and (3,4).
A. y = x
B. y= x
C. y = x +
D. y = x + 1
Solution:
y – y1 =
y–0=
(x
(x
x );
(0,0) and (3,4)
0)
y= x
................................................
166. Find the range of f(x) = -2x2 + 4x.
A. y 2
B. y 2
C. y -2
D. y -2
Explanation:
Since this is a quadratic function, you need to
know two things to determine its range: its
opening and k of its vertex (h,k).
The parabola opens downward since a = -2.
k = c – (b2/4a) = -16/(-8) = 2
Since the parabola opens downward, the graph
starts from - , going to k which is 2.
Thus, y 2.
................................................
167. If a3/2 – 1 = 7, what is a?
A. 4
B. 8
C. 9
D. 18
Solution:
a3/2 – 1 = 7
a3/2 = 8
(a3/2 = 23)2/3
a = 22 = 4
................................................
168. Which of the following is true?
A. A rectangle is a square.
B. A rhombus is a rectangle.
C. A trapezoid is a rhombus.
D. A square is a rhombus.
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1000 MMR
169. What is the measure of each exterior angle
of a pentagon?
A. 108o
B. 72o
C. 60o
D. 36o
Solution:
MEA = 360/N = 360/5 = 72o
................................................
170. How many diagonals does a nonagon have?
A. 27
B. 36
C. 45
D. 54
Solution:
Diagonals = N(N-3)/2 = 9(6)/2 = 27
................................................
171. What is the fractional equivalent of
0.123123123123…?
A.
B.
C.
D.
Author: Victor A. Tondo Jr., LPT
Solution:
P250 = discount
10% = discount rate
Original Price (OP) = ???
Selling Price = ???
DC = OP x DC Rate
250 = OP x (0.1)
250/0.1 = OP
2500 = OP
Selling Price = OP – DC
Selling Price = 2500 – 250 = P2250
................................................
173. A book was sold for P270 after a 10%
discount was given. How much was the book
originally?
A. P330
B. P300
C. P297
D. P280
Solution:
Algebraic Solution:
Let x = 0.123123123123…
1000x = 123.123123123123…
1000x – x = 123
999x = 123
x = 123/999
x = 41/333
Alternative Solution:
Write a fraction whose numerator is the
repeated digits (123) and whose denominator
has the same number of digits but is made of 9s
(123 is 3-digit, so use 999).
Thus, 123/999 or 41/333.
................................................
172. Mrs. Pasay saved P250 after buying a phone
with a 10% discount. How much did she pay for
the phone?
A. P2500
B. P2250
C. P2000
D. P1750
SP = OP (1-DC Rate)
270 = OP (0.9)
270/0.9 = OP
300 = OP
................................................
174. Find the area of an equilateral triangle
whose sides measure 12 cm each.
A. 36√3 cm2
B. 48√3 cm2
C. 60√3 cm2
D. 72√3 cm2
Solution:
√
√
AreaEqTri =
=
= 36√3
................................................
175. This is located at the intersection of the
angle bisectors of a triangle.
A. Incenter
B. Circumcenter
C. Centroid
D. Orthocenter
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1000 MMR
Explanation:
Author: Victor A. Tondo Jr., LPT
Solution:
Incenter: intersection of angle bisectors
Circumcenter: intersection of perpendicular
bisectors
Centroid: intersection of medians
Orthocenter: intersection of altitudes
................................................
D = 2P
176. ∆ABC is similar to ∆DEF. ̅̅̅̅
AB is 9 cm long
̅̅̅̅ is 12 cm long. If the area of ∆ABC is
while DE
27 cm2, what is the area of ∆DEF?
A. 36 cm2
B. 48 cm2
2
C. 60 cm
D. 72 cm2
D = 2(40) = 80
................................................
Solution:
Ratio of sides = 9:12 or 3:4
Ratio of areas = 32:42 or 9:16
9:16 = 27:N
16(27) = 9N;
48 = N
................................................
177. Find the remainder when x4 – 3x3 + 2x2 +
3x – 9 is divided by (x-3).
A. -18
B. -9
C. 9
D. 18
Solution:
Use the remainder theorem.
x-a = x-3

a=3
M=P+4
D + P + M = 164
2P + P + (P+4) = 164
4P + 4 = 164
4P = 160;
P = 40,
180. A circle is drawn inside a triangle such that
it is tangent to the sides of the triangle. Its center
will be the triangle’s ___________________.
A. Incenter
B. Circumcenter
C. Centroid
D. Orthocenter
Explanation:
If the circle is inside the triangle, its center is the
INcenter. If the circle circumscribes the triangle,
its center is the CIRCUMcenter.
................................................
181. Rayon can do a job in 3 hours, while Carlyn
can do the same job in 7 hours. How long will it
take them to finish the job by working together?
A. 2.1 hours
B. 2.5 hours
C. 5 hours
D. 10 hours
Remainder = f(3)
Solution:
f(3) = 34 – 3(33) + 2(32) + 3(3) – 9
f(3) = 81 – 81 + 18 + 9 – 9 = 18
................................................
= = 2.1
................................................
178. Which of the following has its incenter,
circumcenter, centroid, and orthocenter in just
one point?
A. Right Triangles
B. Equilateral Triangles
C. Isosceles Triangles
D. Scalene Triangles
................................................
179. Dexter is twice as heavy as Pablo. Ming is
4kg heavier than Pablo. The sum of their masses
is 164kg. How heavy is Dexter?
A. 40 kg
B. 44 kg
C. 80 kg
D. 88 kg
182. This line is perpendicular to one side of the
triangle passing through the opposite vertex.
A. Longitude
B. Median
C. Altitude
D. Bisector
Explanation:
Altitude: perpendicular to one side, passing
through the opposite vertex.
Median: line segment from midpoint of one side
to opposite vertex
Bisector: line segment that bisects an angle of a
triangle
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1000 MMR
183. How many ways can Lola Leonor arrange
her six meals on the Lazy Susan (the rotating
circular wooden server on top of the table)?
A. 720
B. 120
C. 36
D. 30
Explanation:
This problem is about Circular Permutations or
arrangements on a circle. The formula is (N-1)!.
................................................
184. In parallelogram
MATH, m M = 7x – 12 and
m T = 5x + 32. Find m A.
A. 22
B. 38
C. 44
D. 142
Author: Victor A. Tondo Jr., LPT
186. How many ways can the letters of the word
BANANA be rearranged?
A. 720
B. 240
C. 120
D. 60
Solution:
This is a permutation with repeated elements:
!
P = ! ! !… where n is the total number of
elements (or letters) and a!, b!, c!, … are the
number of times the different elements (or
letters) were repeated.
BANANA has 6 letters: 1 B, 3 A, and 2 N
!
P = ! ! ! = ( )( ) = 60
................................................
187. “The temperature in Baguio City is 20o
while the temperature in Tuguegarao City is
40o”. What level of data is temperature in
degrees Celsius?
A. Nominal
B. Ordinal
C. Interval
D. Ratio
Solution:
M and T are opposite angles,
therefore m M = m T.
7x – 12 = 5x + 32
7x – 5x = 32 + 12
2x = 44
x = 22
Explanation:
.: m M = 7(22)-12 = 154-12 = 142
M and A are consecutive angles,
therefore m M + m A = 180.
142 + m A = 180
m A = 180 – 142
m A = 38
................................................
185. Find the equation of the line perpendicular
to 2x + 5y = 7, passing through (1, 2).
A. 2x + 5y = 12
B. 2x – 5y = -8
C. 5x + 2y = 9
D. 5x – 2y = 1
Solution:
Just like what we did in item #76, simply
interchange the numerical coefficients of x and y
in the original equation, then change the
operation between them. For the constant,
simply substitute the x and y values of the point
and solve.
Since you cannot infer that Tuguegarao City is
TWICE AS HOT as Baguio City, then the data is
not ratio.
Remember: Temperature in degrees Celsius or
Fahrenheit is interval, but temperature in Kelvin
is ratio.
................................................
188. What is formed by the intersection of two
planes?
A. a point
B. a line
C. a plane
D. space
................................................
189. What is formed when a plane intersects a
cone parallel to its circular base?
A. ellipse
B. hyperbola
C. circle
D. parabola
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1000 MMR
Explanation:
The conic sections are formed by intersecting a
cone with a plane.
Parallel to its base: circle
Perpendicular to its base: parabola
Slanted relative to the base: ellipse
................................................
190. In which non-Euclidean model for geometry
can we have any given line ℓ and a point A which
is not on ℓ, wherein all lines through A will
intersect ℓ?
A. hyperbolic
B. elliptic
C. Saccheri
D. Pythagorean
Explanation:
In Euclidean geometry, only one line will pass
through A. In elliptic geometry, all lines will pass
through A and intersect ℓ.
................................................
191. Which numerical system is sexagesimal
(base-60)?
A. Mayan
B. Roman
C. Babylonian
D. Hindu-Arabic
Explanation:
Mayans: base-20 (vigesimal)
Babylonians: base-60 (sexagesimal)
Romans and Hindu-Arabic: base-10 (decimal)
................................................
192. Which numerical system makes use of dots
and horizontal lines, and shell shapes for zero?
A. Egyptian
B. Roman
C. Greek
D. Mayan
................................................
193. Which of the following is false?
A. sin2 θ + cos2 θ = 1
B. sin θ (csc θ) = 1
C. sin θ ÷ cos θ = tan θ
D. sin θ (tan θ) = cos θ
Author: Victor A. Tondo Jr., LPT
194. If three-fourths of a number is 33 more than
its one-fifth, what is that number?
A. 240
B. 120
C.90
D. 60
Solution:
x=
x + 33
20 ( x =
x + 33) 
15x = 4x + 660
11x = 660
x = 60
................................................
195. Which of the following has the greatest
value:
A. 3 + 32 + (3 + 3)2
B. 33
2
2
C. [(3 + 3) ]
D. (3 + 3 + 3)2
Solution:
A. 3 + 32 + (3 + 3)2 = 3 + 9 + 36 = 48
B. 33 = 27
C. [(3 + 3)2]2 = 362 = 1296
D. (3 + 3 + 3)2 = 92 = 81
................................................
196. Which of the following has an undefined
slope?
A. a vertical line
B. a horizontal line
C. a line parallel to the x-axis
D. a diagonal line
Horizontal line (parallel to x-axis): m = 0
Vertical line (parallel to y-axis): m is undefined
Slanted downwards to the right: m is negative
Slanted upwards to the right: m is positive
................................................
197. In solid geometry, what do you call a solid
bound by polygons?
A. multigon
B. tessellation
C. porygon
D. polyhedron
Explanation:
= tan ; or sin θ = tan θ (cos θ)
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1000 MMR
198. Tchr. Victor needs to randomly get 10 out of
his 50 students for drug testing. He proceeds by
making the students count off from 1 to 5. He
then randomly picks a number from 1 to 5.
Which sampling method did he use?
A. stratified
B. cluster
C. systematic
D. convenience
Explanation:
Stratified: /strata/ population has hierarchy or
sub-classifications
Cluster: for homogenous population in a large
area
Systematic: counting off
................................................
199. Which statistical test must be used in
testing the significance of group differences
between 2 or more groups?
A. Chi Square
B. t-test
C. ANOVA
D. Pearson R
................................................
200. Which Mathematician is famous for the
Fibonacci sequence?
A. Ptolemy
B. Leonardo Pisano Bigollo
C. Pierre de Fermat
D. Luca Pacioli
................................................
201. Which Mathematician is famous for his last
theorem?
A. Pythagoras
B. Isaac Newton
C. Daniel Bernoulli D. Pierre de Fermat
................................................
202. Which of the following is a square?
A. Polygon ABCD which has 4 congruent sides.
B. Polygon MATH which has 4 perpendicular
sides.
C. Quadrilateral HEAD which has one pair of
congruent perpendicular bisecting diagonals.
D. Quadrilateral FROG which has 4 right angles.
Author: Victor A. Tondo Jr., LPT
Explanation:
A. ABCD is a rhombus
B. MATH is a rectangle
D. FROG is a rectangle
................................................
203. Which of the following is the set of points
whose sum of distance to two fixed points is
constant?
A. parabola
B. circle
C. ellipse
D. hyperbola
Explanation:
Parabola: set of points equidistant to a fixed
point (focus) and a fixed line (directrix)
Circle: set of points equidistant to a fixed point
(center)
Ellipse: set of points equidistant to two fixed
points (foci)
Hyperbola: set of points whose difference of
distances to two fixed points is constant
................................................
204. Which of the following is not a triangle
congruence postulate?
A. SAS
B. ASA
C. SAA
D. AAA
Explanation:
AAA is a triangle similarity postulate.
................................................
205. If A is at (-8,5) and B is at (4,-11), find C if C
is three-fourths the way from A to B.
A. (1, -7)
B. (-4, 1)
C. (1, 1)
D. (-4, -7)
Solution:
Since C is three-fourths the way from A to B, then
its coordinates are:
x = -8 + [4 – (-8)]
x = -8 + 9
x=1
y = 5 + (-11 – 5)
y = 5 + (-12)
y=7
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1000 MMR
206. CPCTC stands for “____________ parts of
congruent triangles are congruent”.
A. collinear
B. complementary
C. corresponding
D. conjugate
................................................
207. Victor deposited an amount of P200,000 in
a bank that offers 5% interest compounded per
annum. How much will he have in his account
after 3 years?
A. P230,000
B. P231,525
C. P233,050
D. P234,575
Solution:
Since the interest is compounded annually,
Acct = Principal x (1 + rate)time
Acct = 200,000 x (1.053)
Acct = 200,000 x 1.157625 = 231,525
................................................
208. Find the remainder when the polynomial
x4 – 3x3 + 2x2 – 5x + 8 is divided by (x – 3).
A. 5
B. 8
C. 11
D. 14
Use the Remainder Theorem.
Our divisor is (x – 3), so a = 3.
f(3) = 34 – 3 (33) + 2 (32) – 5(3) + 8
f(3) = 81 – 81 + 18 – 15 + 8 = 11
................................................
C. 180
D. 200
Solution:
60% of 120 translates to (0.60) x 120, or 72.
................................................
210. What percent of 80 is 55?
A. 145.45%
B. 135%
C. 68.75%
D. 44%
Solution:
The formulas are:
Rate: R = x 100%
Part: P = B x R (convert rate to decimal first)
Base: B =
(convert rate to decimal first)
We are looking for the Rate in this problem, so
R = x 100% = 0.6875 x 100% = 68.75%
................................................
211. The hypotenuse of a right triangle measures
40 cm. Find its area if one angle measures 30o.
A. 100√3 cm2
B. 200√2 cm2
C. 200√3 cm2
D. 400√2 cm2
Solution:
Since the hypotenuse is 40 cm, then the leg
opposite 30o is 20 cm (half the hypotenuse), and
the leg opposite 60o is 20√3 cm (√3 times the
short leg). The area of a right triangle is given by
=
.
√
Solution:
209. What is 60% of 120?
A. 50
B. 72
Author: Victor A. Tondo Jr., LPT
Rate comes with the word “percent” or the
percent symbol (%).
=
= 200√3 cm2
................................................
212. Nine cans of soda and four hamburgers cost
a total of P257. Five cans of soda and seven
hamburgers cost a total of P224. How much is a
can of soda?
A. P17
B. P19
C. P21
D. P23
Solution:
Let
C = price of a can of soda,
B = price of a hamburger
9C + 4B = 257
x7  63C + 28B = 1799
5C + 7B = 224
x4  20C + 28B = 896
63C + 28B = 1799
20C + 28B = 896
43C = 903
43
43
C = 21
Identify the Part (in textbooks, they use the word
“Percentage”), the Base, and the Rate when you
face questions like this. Part comes with the
word “is”, Base comes with the word “of”, while
This is a free reviewer. All rights reserved.
1000 MMR
213. The product of two consecutive even
numbers is 728. What is the smaller number?
A. 22
B. 24
C. 26
D. 28
Solution:
Let x = smaller number;
x + 2 = larger number
x (x+2) = 728
x2 + 2x = 728;
2
x + 2x + 1 = 729
√x + 2x + 1 = √729
x + 1 = 27
x = 26
................................................
214. What time is 219 minutes past 6:40 AM?
A. 8:59 AM
B. 9:19 AM
C. 9:49 AM
D. 10:19 AM
Solution:
219 minutes = 3 hrs 39 mins
6:40
+
3:39
9:79 or 10:19
................................................
215. Find the vertex of y = 3x2 – 2x + 11.
A. ( , )
B. ( ,
)
C. (
, )
D. (
,
)
Solution:
The vertex is at (h,k) where
h=
h=
;
(
( )
k = 11 –
(
Sell Price = Orig Price x (1 - Disct Rate)
4000 = OP x (1 – 0.2)
4000 = OP x (0.8)
4000 ÷ 0.8 = OP
5000 = OP
................................................
217. When a number is increased by 3, its square
increases by 111. By what does its square
increase when the number is increased by 6?
A. 222
B. 240
C. 444
D. 480
Solution:
Let’s find the original number first.
(x+3)2 – x2 = 111
x2 + 6x + 9 – x2 = 111
6x + 9 = 111
6x
= 102
x
= 17
172 = 289; (17+6)2 = 232 = 529
529 – 289 = 240
................................................
218. How many prime numbers are there from 1
to 100?
A. 23
B. 24
C. 25
D. 26
................................................
219. Find the range of f(x) = 2x2 – 8x + 9.
A. y 0
B. y 1
C. y 9
D. y
Solution:
k=c)
Author: Victor A. Tondo Jr., LPT
Solution:
)
( )
h=
k=
................................................
216. After getting a 20% discount, Mr. Lopez
paid P4,000 for a gadget. How much was its
original price?
A. P4,800
B. P5,000
C. P8,000
D. P20,000
This is a quadratic function so the graph is a
parabola opening upwards (since A = 2). Solve
for k to find its minimum value.
k = c – b2/4a = 9 – 64/8
k=9–8=1
Therefore, y 1.
220. Find the domain of y =
A. x ±7, ±10
B. x ±7
C. x ±10
D. x 1
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1000 MMR
Explanation:
The denominator should not be equal to 0.
x2 – 49 0
x2 49
x ±7
................................................
221. Solve for x: (x+3)2 = (x-4)2.
A. x = 0
B. x = ½
C. x = 1
D. no solution
Solution:
(x+3)2 = (x-4)2
x2 + 6x + 9 = x2 – 8x + 16
6x + 8x = 16 – 9
14x = 7
x=½
................................................
222. The diagonal of a rectangular prism is 13
cm long. If it is 3 cm thick and 12 cm long, how
wide is it?
A. 3 cm
B. 4 cm
C.4√3 cm
D. 5 cm
Solution:
Diagonal2 = Length2 + Width2 + Height2
132 = 122 + W2 + 32
169 = 144 + W2 + 9
169 – 144 – 9 = W2
16 = W2;
4=W
................................................
223. Which of the following is not a function?
A. y = x2 + 2017x – 2017
B. y = |2017x| - 2017
C. y = √2017 + 2017
D. y2 = x + 2017
Explanation:
When y is raised to an even number, it
automatically becomes not a function.
Author: Victor A. Tondo Jr., LPT
224. 12 + 17 + 22 + 27 + … + 117 = _____
A. 1409
B. 1414
C. 1419
D. 1424
Solution:
n=
n=
+1
+1 = 22
Sum =
(n)
Sum =
(22) = 1419
................................................
225. Mr. G sold 80% of his apples and still had
213 apples left. How many apples did he have
originally?
A. 1704
B. 1065
C. 852
D. 293
Solution:
213 = 20% x Original number of apples
213 ÷ 0.2 = Original number of apples
1065 = Original number of apples
................................................
226. When a number is increased by 4, its square
also increases by 168. What is this number?
A. 15
B. 19
C. 23
D. 27
Solution:
(x+4)2 – x2 = 168
x2 + 8x + 16 – x2 = 168
8x + 16 = 168
8x = 152
x = 19
................................................
227. Solve for k to make a perfect square
trinomial:
9x2 + kx + 25
A. 10
B. 15
C. 20
D. 30
Solution:
9x2 + kx + 25 = (3x)2 + 2(3x)(5) + 52
= (3x)2 + 30x + 52
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1000 MMR
228. Find the y-intercept of 2x + 3y = 4.
A.
B.
C.
D. 2
Solution:
y-intercept is taken when x = 0.
2(0) + 3y = 4
3y = 4
y=
................................................
229. Which of the following points is on the line
y = 2x + 5?
A. (1, 3)
B. (2, 9)
C. (0, 10)
D. (3, 10)
Explanation:
Just substitute the x and y-values of each point
and see which one makes a true equation.
A. (1, 3)  3 = 2(1) + 5
False
B. (2, 9)  9 = 2(2) + 5
True
C. (0, 10)  10 = 2(0) + 5
False
D. (3, 10)  10 = 2(3) + 5
False
................................................
230. Find the intersection of y = -2x + 1 and
y = 3x + 16.
A. (-3, 7)
B. (-4, 9)
C. (3, -7)
D. (4, -9)
Author: Victor A. Tondo Jr., LPT
Solution:
Convert it to its slope-intercept form.
3x + 5y = 7 
5y = -3x + 7
y= x+
.: m =
................................................
232. Which of the following is a polynomial?
A. √3 + 4 + 2
B. 2x + 3√
C.
+3
D. √3 x + 7
Explanation:
A polynomial accepts only WHOLE numbers as
exponents of the variable/s. Only D has whole
numbers as exponents of x.
................................................
233. What is the degree of the polynomial
9x4 + 5x3 – 2x2 + 3x – 17?
A. 4
B. 5
C. 9
D. 10
Explanation:
Degree refers to the highest exponent or sum of
exponents of the variables in any term of a
polynomial.
................................................
234. log2 32√2 = __________.
A. 2.5
B. 3.5
C. 4.5
Solution:
y = -2x + 1
y = 3x + 16
0 = -5x – 15
5x = -15
x = -3;
y = -2(-3) + 1
y = 6+1 = 7
................................................
(-)
Solution:
32 = 25
and √2 = 21/2
5 + ½ = 5.5
................................................
235. If y = √3
231. Find the slope of 3x + 5y = 7.
A.
B.
A. x = √
C.
C. x = √
D.
D. 5.5
+ 6 , what is x in terms of y?
B. x = √
–1
+1
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D. x = √
+1
–1
1000 MMR
Solution:
Square the equation
y = √3 + 6
y2 = 3x2 + 6x
Divide by 3
= x2 + 2x
Complete the square
+ 1=
x2
+ 2x + 1
=
x2
+ 2x + 1
Extract the root
√
=x+1
Finally,
√
- 1= x
................................................
236. Which of the following is a pair of parallel
lines?
A. y = 2 and x = 2
B. 12x + 13y = 14 and 13x + 14y = 15
C. y = 3x + 8 and 3y = x + 9
D. 4x + 5y = 6 and 8x + 10y = 21
Explanation:
Parallel lines have the same slope. Both lines
have a slope of -4/5 in D.
................................................
237. Which of the following is a pair of
perpendicular lines?
A. x = 5 and y = 7
B. y = x and 2y = 4x + 5
C. x = 2y + 3 and 2x + 3y = 4
D. y = 5x + 6 and y = 0.2x – 8
Explanation:
Perpendicular lines have negative reciprocal
slopes. For choice A, x=5 is horizontal and y=7 is
vertical, therefore they are perpendicular.
................................................
238. Find the altitude to the hypotenuse of a
right triangle whose sides measure 5 cm, 12 cm,
and 13 cm.
A.
B.
C.
D. 26
Solution:
Author: Victor A. Tondo Jr., LPT
239. Find the slope of the line tangent to
y = x3 – 6x2 + 2x + 7 at x = 4.
A. -8
B. -2
C. 2
D. 8
Solution:
Get the first derivative of y.
y' = 3x2 – 12x + 2
Substitute x = 4 for every x in y’.
3(4)2 – 12(4) + 2 = 48 – 48 + 2 = 2
................................................
240. Find the average rate of change of
y = x3 – 2x + 3 from x = 0 to x = 3.
A. 5
B. 6
C. 7
D. 8
Solution:
Average Rate of Change =
(
)
(
)
=
=7
................................................
241. Find the radius of x2 + y2 + 2x – 4y = 44.
A. √39
B. 2√11
C. 7
D. 3√6
Solution:
Complete the squares on the left side of the
equation to return it to its center-radius form.
x2 + y2 + 2x – 4y = 44
x2 + 2x + y2– 4y = 44
(x2 + 2x + 1)+ (y2– 4y +4)= 44 + 1 + 4
(x2 + 2x + 1)+ (y2– 4y +4)= 49
................................................
242. Gian has 8 more P5 coins than P1 coins. If
he has a total of P106, how many P5 coins does
he have?
A. 13
B. 15
C. 17
D. 19
Solution:
Let x = number of P1 coins
.: x+8 = number of P5 coins
1(x) + 5 (x+8) = 106
x + 5x + 40 = 106
6x + 40 = 106
6x = 66
x = 11
.: He has 11 P1 coins and (11+8) or 19 P5 coins.
Altitude to the Hyp = (L1 L2)/Hyp = 5(12)/13
Altitude to the Hyp = 60/13
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1000 MMR
243. After using half of her budget on bills, onethird on groceries, and P270 on a shirt, Mrs. D
still had P130 left. How much was her budget?
A. P2400
B. P2700
C. P3000
D. P3300
Author: Victor A. Tondo Jr., LPT
246. The average grade of 23 students in Section
A is 86, while the average grade of 27 students in
Section B is 91. What is the average grade of all
50 students in both sections?
A. 88.5
B. 88.6
C. 88.7
D. 88.8
Solution:
Solution:
Let x = budget
x + x + 270 + 130 = x
Average =
=
= 88.7
................................................
(
x + 400 = x
400 = x – x
400 = x
2400 = x
................................................
244. x varies directly as y and inversely as z. If
x = 24 when y = 32 and z = 4, what is x when
y = 21 and z = 7?
A. 3
B. 5
C. 7
D. 9
Solution:
x = ky/z
24 = k(32)/4
24 = 8k
3=k
x = 3y/z
x = 3 (21) / 7
x = 63/7
x=9
................................................
245. Find the mode of the following scores:
78
78
78
78
79
79
79
79
80
80
80
80
A. 79
B. 78, 79, and 80
C. 80
D. no mode
)
(
)
247. Find the axis of symmetry of y = 3x2 – 5x.
A. x =
B. x =
C. x =
D. x =
Explanation:
The axis of symmetry is located at x = .
................................................
248. Find the range of the following scores:
19
25
24
31
23
29
33
A. 12
B. 13
C. 14
D. 15
Explanation:
Range = Highest Score – Lowest Score
................................................
249. Mr. C travels for 2 hours at a speed of 38
kph and then north for 3 hours at a speed of 53
kph. What is his average speed?
A. 44 kph
B. 45.5 kph
C. 47 kph
D. 48.5 kph
Solution:
Average Speed =
Average Speed =
=
= 47 kph
Explanation:
Mode, by definition, is the score with the highest
frequency. Since each score has a frequency of 4,
then there is no mode.
................................................
250. Victor, Chris, and Aira volunteered to teach
at a nearby daycare. Chris worked for 2 hours
less than Aira. Victor worked twice as many
hours as Chris. Altogether, they worked for 58
hours. How many hours did Victor work?
A. 14
B. 16
C. 28
D. 32
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1000 MMR
Solution:
Aira = n hours
.: Chris = (n – 2) hours
.: Victor = 2(n-2) hours
n + (n-2) + 2(n-2) = 58
n + n - 2 + 2n - 4 = 58
4n – 6 = 58
4n = 64
n = 16
.: Victor worked for 2(16-2) or 28 hours.
................................................
251. What conic figure does the equation
x2 + y2 + 4x = -4 form?
A. Real circle
B. Degenerate circle
C. Imaginary circle D. Ellipse
Solution:
Use CTS (completing trinomial squares) to
convert the equation to its center-radius form.
x2 + y2 + 4x = -4
2
(x + 4x + 4) + y2 = -4 + 4
(x + 2)2 + y2 = 0
Author: Victor A. Tondo Jr., LPT
Solution:
The center of x2 + y2 + Dx + Ey + F = 0
is at ( , ).
................................................
254. Find the equation of the circle with center at
(2, 3), passing through (5, -1).
A. x2 + y2 + 4x + 6y = 0
B. x2 + y2 + 4x + 6y = 12
C. x2 + y2 – 4x – 6y = 0
D. x2 + y2 – 4x – 6y = 12
Solution:
Find the radius first by using the center-radius
form.
(x – 2)2 + (y – 3)2 = r2
(5 – 2)2 + (-1 – 3)2 = r2
9 + 16 = r2
25 = r2
5=r
Since r2 = 0, then the radius is also 0, making it a
degenerate circle.
................................................
Substitute this to your center-radius form and
simplify.
(x – 2)2 + (y – 3)2 = 52
x2 – 4x + 4 + y2 – 6y + 9 = 25
x2 + y2 – 4x – 6y + 13 = 25
x2 + y2 – 4x – 6y = 12
................................................
252. What conic figure does the equation
x2 + y2 + 8x – 6y = -100 form?
A. Real circle
B. Degenerate circle
C. Imaginary circle D. Ellipse
255. Find the equation of the vertical line passing
through (-3, 4).
A. x = -3
B. x = 4
C. y = -3
D. y = 4
Explanation:
Solution:
Use CTS (completing trinomial squares) to
convert the equation to its center-radius form.
x2 + y2 + 8x – 6y = -100
(x2 + 8x + 16) + (y2 – 6y + 9) = -100 + 16 + 9
(x + 4)2 + (y – 3)2 = -75
Since r2 = -75, then the radius is imaginary,
making it an imaginary circle.
................................................
x2
y2
Equation of vertical lines: x = abscissa of your
point
................................................
256. Find the equation of the horizontal line
passing through (-3, 4).
A. x = -3
B. x = 4
C. y = -3
D. y = 4
Explanation:
Equation of horizontal lines: y = ordinate of your
point
253. Find the center of + + 6x – 10y = 2.
A. (6, -10)
B. (-6, 10)
C. (-3, 5)
D. (3, -5)
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1000 MMR
257. Which of the following lines passes through
the point (3, -2)?
A. y = x + 5
B. y = 2x – 8
C. y = 5 – x
D. y = 5 – 2x
Author: Victor A. Tondo Jr., LPT
of the circle. The second option would be more
preferable as it already gives us the equation of
the circle.
Solution:
Using the point A(9, -5), we get
(9 – 3)2 + (-5 – 3)2 = r2
36 + 64 = r2
100 = r2
Simply substitute x and y from your point. If the
equation holds true, then the given line passes
through your point.
y=x+5

(-2) 3 + 5
y = 2x – 8

(-2) = 2(3) – 8
y=5–x

(-2) 5 - 3
y = 5 – 2x

(-2) 5 – 2(3)
................................................
258. Given that I(2, -3) is the midpoint of V(-4, 5)
and C, find the coordinates of C.
A. (-1, 1)
B. (1, -1)
C. (8, -11)
D. (-10, 16)
(
.: (x – 3)2 + (y – 3)2 = 100
................................................
260. Find the distance between the line
3x + 4y – 5 = 0 and the point (8, -1).
A. 2
B. 3
C. 4
, ym =
D. 5
Solution:
Use the formula for distance of point (x1, y1)
from line Ax + By + C = 0: D = √
D=
Solution:
Use the midpoint formula: xm =
(x – 3)2 + (y – 3)2 = r2
( )
(
)
√
D= =3
................................................
)
2=
-3 =
4 = (-4) + x2
-6 = 5 + y2
8 = x2
-11 = y2
................................................
259. The endpoints of the diameter of a circle are
A(9, -5) and B(-3, 11). What is the equation of
the circle?
A. (x – 3)2 + (y – 3)2 = 100
B. (x – 3)2 + (y – 3)2 = 400
C. (x + 3)2 + (y + 3)2 = 100
D. (x + 3)2 + (y + 3)2 = 400
Solution:
Find the midpoint of the diameter to get the
center, C(h,k).
( )
h=
k=
h = 3, k = 3
Next, find the length of the radius. You can do
this by either A) getting half of the length of the
diameter AB, or B) using the center-radius form
261. B is one-fourth of the way from A(-13,9) to
C(7,-7). Find the coordinates of B.
A. (2, -3)
B. (-8, 5)
C. (2.5, -3.5)
D. (-7.5, 5.5)
Solution:
In this question, it is important to note the
“from” point and the “to” point.
Abscissa of point B = abscissa of “from” point +
¼ (abscissa of “to” point minus abscissa of
“from” point)
Abscissa of point B = -13 + ¼ (7 – (-13))
= -13 + ¼ (20) = -8
Ordinate of point B = ordinate of “from” point +
¼ (ordinate of “to” point minus ordinate of
“from” point)
Ordinate of point B = 9 + ¼ (-7 – 9)
= 9 + ¼ (-16) = 5
................................................
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1000 MMR
261. Find the area of the triangle whose vertices
are X(-9, -3), Y(-2, 8), and Z(5, 1).
A. 61
B. 62
C. 63
D. 64
Solution:
The biggest mistake you could make in solving
this problem is by getting the equation of one the
line formed by one pair of vertices, getting the
length of the said side, and getting the distance of
the third point from the line you first obtained,
then multiplying the obtained distance by the
length of the line segment and then dividing by
two. That’s a very long solution.
Simply use the formula A∆ = ½ *
+
9
2 5
A∆ = ½ *
+
3 8 1
A∆ = ½ [(-72 + -2 + -15) – (6 + 40 + -9)]
A∆ = ½ |(-89 – 37)|
A∆ = ½ (126) = 63
................................................
262. Which of the following is outside the circle
defined by the equation (x – 3)2 + y2 = 40?
A. (5, 6)
B. (7, 5)
C. (0, -5)
D. (-1, 4)
Solution:
A point is outside the circle when its distance
from the center is greater than the radius.
A: (5, 6)  (5 – 3)2 + 62 = 40
This point is ON the circle.
B: (7, 5)  (7 – 3)2 + 52 > 40
This point is OUTSIDE the circle.
C: (0, -5)  (0 – 3)2 + (-5)2 < 40
This point is IN the circle.
D: (-1, 4)  (-1 – 3)2 + 42 < 40
This point is IN the circle.
................................................
263. Which of the following is parallel to the line
defined by the equation y = 3x – 4?
A. y + 3x = 5
B. x + 3y = 6
C. y = x + 7
D. y = 3x + 8
Author: Victor A. Tondo Jr., LPT
Solution:
Two lines are parallel when their slopes are
equal. Check the slopes of the lines by converting
them to their slope-intercept form (y = mx + b).
A) y + 3x = 5 
y = -3x + 5 m = -3
B) x + 3y = 6 
y= x+2 m=
C) y = x + 7
m=
D) y = 3x + 8 
m=3
................................................
264. Which of the following is perpendicular to
the line defined by the equation y = 3x – 4?
A. y + 3x = 9
B. -x + 3y = 10
C. y = x + 11
D. y = 3x – 12
Solution:
Two lines are perpendicular when their slopes
are negative reciprocals of each other, or when
m1 x m2 = -1. The slope of the given line is 3, so
the slope of the other line is . Check the slopes
of the lines by using their slope-intercept form
(y = mx + b).
A) y + 3x = 9 
B) -x + 3y = 10
y = -3x + 9
y= x+
m = -3
m=
C) y = x + 11
m=
D) y = 3x – 12 
m=3
................................................
265. Which of the following is coincidental to the
line defined by the equation y = 2x + 13?
A. y + 2x = 13
B. 2x – y + 13 = 0
C. y = x + 13
D. 2y = 2x + 13
Solution:
Two lines are coincidental when their equations
are equivalent to each other, or when their
slopes and y-intercepts are equal.
A) y + 2x = 13

y = -2x + 13
B) 2x – y + 13 = 0 
y = 2x + 13
C) y = x + 13

y = x + 13
D) 2y = 2x + 13
This is a free reviewer. All rights reserved.

y=x+
1000 MMR
266. Which of the following equations pertain to
a parabola?
A. y2 + 5y = x
B. x2 + y2 + 3x – 4y = 0
C.
(
)
D.
(
)
+
(
)
=1
(
)
=1
Author: Victor A. Tondo Jr., LPT
Explanation:
In the form 4A(y – k) = (x – h)2, the length of the
latus rectum is 4A.
................................................
270. How far is the vertex from the directrix of
the parabola defined by 16y = x2?
A. 16
B. 8
C. 4
D. 2
Explanation:
A parabola’s equation includes two variables,
one of which is squared.
................................................
267. Which of the following equations pertain to
a parabola that opens to the right?
A. 4(y + 3) = (x – 2)2
B. -3(y – 4) = (x + 5)2
C. (y – 6)2 = 5(x + 1)
D. (y + 8)2 = -2(x – 3)
Explanation:
The parabola opens to the right when the
squared variable is y and the coefficient of x is
positive. It opens to the left when the squared
variable is y and the coefficient of x is negative.
................................................
268. Which of the following equations pertain to
a parabola that opens downward?
A. 4(y + 3) = (x – 2)2
B. -3(y – 4) = (x + 5)2
C. (y – 6)2 = 5(x + 1)
D. (y + 8)2 = -2(x – 3)
Explanation:
The parabola opens upward when the squared
variable is x and the coefficient of y is positive. It
opens to the downward when the squared
variable is x and the coefficient of y is negative.
................................................
269. How long is the latus rectum of the parabola
defined by 12(y – 4) = (x + 3)2?
A. 12
B. 6
C. 4
D. 3
Explanation:
The distance of the vertex from the directrix is A
(from 4A, which is the length of the latus
rectum). The distance of the vertex from the
focus is also A. The vertex is between the focus
and the directrix, which means from focus to
directrix is 2A. The distance from focus to any
endpoint of the latus rectum is also 2A.
................................................
271. Find the equation of the directrix of the
parabola defined by (y – 2)2 = -4(x + 3).
A. x = -2
B. x = 2
C. y = -4
D. y = -2
Explanation:
To find the directrix,
first get the vertex
and the opening of
the parabola. The
vertex is at (-3, 2)
and the parabola
opens to the left,
which means the
directrix is vertical.
The directrix is A units from the vertex. Since
4A = -4, then A = -1. The directrix is 1 unit to the
right of the vertex (-3, 2), meaning it passes
through (-2, 2). The vertical line passing through
(-2, 2) is x = -2. See attached image.
................................................
272. Find the coordinates of the focus of the
parabola defined by -12(y – 4) = (x + 5)2.
A. (-5, 7)
B. (-5, 1)
C. (-8, 4)
D. (-2, 4)
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1000 MMR
Explanation:
The parabola opens downward, with vertex at
(-5, 4). From the equation, 4A = -12, or A = -3.
This means the focus is 3 units downward from
the vertex. Therefore, the vertex is at (-5, 1).
................................................
Solution:
273. In which quadrant would G(3,-4) fall?
A. First quadrant
B. Second quadrant
C. Third quadrant D. Fourth quadrant
Explanation:
Quadrant
Abscissa
Ordinate
First
Positive
Positive
Second
Negative
Positive
Third
Negative
Negative
Fourth
Positive
Negative
................................................
274. Find the distance between the parallel lines
y = 3x + 9 and y = 3x – 12.
A. 7
B. 21
C.
√
D.
√
Solution:
Use the distance between two parallel lines
D=√
D=
(
√
)
=
√
=
√
................................................
275. Find the intersection of the lines y = 2x + 5
and y = -4x + 23.
A. (3, 10)
B. (3, 11)
C. (4, 10)
D. (4, 11)
Solution:
Here we have a system of linear equations in two
variables, namely x and y. Solve for x and y by
using either substitution or elimination.
y = 2x + 5
y = -4x + 23
2x + 5 = -4x + 23
6x = 18
x=3
y = 2x + 5 = 2(3) + 5 = 11
Author: Victor A. Tondo Jr., LPT
276. Which of the following pertains to a circle
that is concentric with (x – 3)2 + y2 = 24?
A. x2 + y2 – 6x + 3 = 0
B. x2 + y2 + 6y –15 = 0
C. x2 + y2 + 6x – 25 = 0
D. x2 + y2 + 6x + 6y – 24= 0
Concentric circles have the same center. That
means the center we are looking for is (3, 0).
A. x2 + y2 – 6x + 3 = 0
Center: (3, 0)
B. x2 + y2 + 6y –15 = 0
Center: (0, -3)
2
2
C. x + y + 6x – 25 = 0
Center: (-3, 0)
D. x2 + y2 + 6x + 6y – 24= 0Center: (-3, -3)
................................................
277. Which of the following is the equation of the
parabola that opens upward, whose latus rectum
is 12 units long, directrix is y = -3, and line of
symmetry is x = 7?
A. 12y2 = x – 7
B. 12y = (x – 7)2
C. 12(y + 3) = (x – 7)2
D. 12(y – 3) = (x – 7)2
Solution:
Since the latus rectum is 12 units long, then
4A = 12, or A = 3. The parabola opens upward,
which means the vertex must be 3 units above
the directrix y = -3. Therefore, the ordinate of
the vertex (k) is equal to (-3) + 3 or 0. The line
of symmetry is x = 7, which means that the
abscissa of the vertex (h) is also 7. The parabola
opens upward, therefore the equation should be
4A(y – k) = (x – h)2.
................................................
278. Convert
A. 1.309o
C. 150o
rad to degrees.
B. 75o
D. 216o
Solution:
π rad = 180o
Therefore
rad = (180o) = 75o
................................................
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1000 MMR
279. Which of the following angles is coterminal
with 143o?
A. 217o
B. -37o
C. 323o
D. 503o
Explanation:
Coterminal angles share the same initial side and
terminal sides. Finding coterminal angles is done
by adding or subtracting multiples of 360 or 2π
rad to each angle, depending on whether the
given angle is in degrees or radians.
503o = 143o + (1)(360o).
................................................
280. Which of the following is NOT a
trigonometric identity?
A. sin2 θ + cos2 θ = 1
B. tan θ =
C. 1 + tan2 θ = sec2 θ
D. 1 – cot2 θ = csc2 θ
Explanation:
The correct identity goes 1 + cot2 θ = csc2 θ.
................................................
281. Which of the following is false?
A. tan θ =
B. csc θ =
C. cos θ =
D. sec θ =
Explanation:
Just use the SohCahToa – ChoShaCao mnemonic.
The reciprocal of sine is cosecant (csc), the
reciprocal of cosine is secant (sec), and the
reciprocal of tangent is cotangent (cot).
................................................
282. Which of the following is NOT a cofunction
identity?
A. cos θ = sin (90 – θ)
B. cot (90 – θ) = tan θ
C. sec θ = csc (90 – θ)
D. csc θ = sin (90 – θ)
Author: Victor A. Tondo Jr., LPT
Explanation:
The value of a trigonometric function of an angle
equals the value of the cofunction of the
complement of the angle. Cofunctions are sine
and COsine, tangent and COtangent, and secant
and COsecant.
cos θ = sin (90 – θ)
sec θ = csc (90 – θ)
tan θ = cot (90 – θ)
sin θ = cos (90 – θ)
csc θ = sec (90 – θ)
cot θ = tan (90 – θ)
................................................
283. The hypotenuse of a 30-60-90 triangle is 48
cm long. How long is its shortest side?
A. 24 cm
B. 24 √2 cm
C. 24 √3 cm
D. 16√3 cm
Explanation:
For any 30-60-90 triangle, the ratio of the sides
are as follows:
Short leg (opposite of the 30o angle) = n
Long leg (opposite of the 60o angle) = n√3
Hypotenuse = 2n
Since the hypotenuse 2n is equal to 48, then
n equals 24.
................................................
284. In a right triangle, the side opposite an
angle measuring 50o is 100 cm long. How long is
the side adjacent to the 50o angle?
A. 93.45 cm
B. 83.91 cm
C. 149.14 cm
D. 200 cm
Solution:
The given are an angle and the sides opposite
and adjacent to it. Use the tangent function.
tan 50o =
tan 50o =
x=
= 83.91 cm
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1000 MMR
285. A hundred cards are numbered 1 to 100.
What is the probability of drawing a card whose
number is divisible by seven?
A.
B.
C.
D.
Solution:
There are 14 multiples of 7 from 1 to 100 since
100 ÷ 7 = 14 r 2.
Therefore the probability is
or .
................................................
286. What is the probability of rolling a sum of
10 when rolling two dice?
A.
B.
C.
D.
Solution:
The preferred outcomes are (4,6), (5,5), and
(6,4). There are 36 possible outcomes. Therefore
the probability is 3/36 or 1/12.
................................................
287. Factorize 3x2 + 5x – 2.
A. (3x – 1) (x + 2) B. (3x + 1) (x – 2)
C. (3x – 2) (x + 1) D. (3x + 2) (x – 1)
................................................
288. Six-sevenths of a number is 6 less than ninetenths of the same number. What is the number?
A. 130
B. 140
C. 200
D. 210
Solution:
Let n = the number of students
and x = the number of rooms
n = 10x + 24
=
10(6x) = 7(9x – 60)
60x = 63x – 420
420 = 63x – 60x
420 = 3x
140 = x
n = 12x + 6
Next, n = 10x + 24 = 10(9) + 24 = 114
Therefore, there are 114 students.
................................................
290. Which of the following is not between
and
?
A.
B.
C.
D.
Solution:
= -0.4 and
B.
6
and
10x + 24 = 12x + 6
24 – 6 = 12x – 10x
18 = 2x
9=x
A.
Solution:
=
Author: Victor A. Tondo Jr., LPT
289. A certain University has a dormitory. If 10
students stay in a room, 24 students will not
have a room. If 12 students stay in a room, there
will be 6 vacant beds. How many rooms are there
in the dormitory? How many students are
staying in the dormitory?
A. 116
B. 115
C. 114
D. 113
= -0.75.
= -0.8
= -0.65
C.
= -0.5
D.
= -0.6
Only A is beyond -0.4 to -0.75.
................................................
291. Which value describes the position of C?
................................................
A. -0.75
B. -0.6
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C. -1.25
D. -1.4
1000 MMR
292. Choose the correct value of (x + y)(x – y)
when x = 3.5 and y = –8.7
A. -63.44
B. 63.44
C. -148.84
D. 10.4
Author: Victor A. Tondo Jr., LPT
297. Subtract: (–2x2 + 5x – 9) – (2x – 7)
A. 2x2 + 3x + 16
B. -2x2 + 3x + 2
2
C. -2x + 3x – 2
D. -2x2 + 7x + 2
................................................
Solution:
298. Evaluate the polynomial 4x2 – 6x – 3 if x = 2.
A. -1
B. 1
C. 3
D. 5
(x + y)(x – y) = (3.5 + (-8.7))(3.5 – (-8.7))
= (-5.2)(12.2)
= -63.44
................................................
293. Which two triangles are similar?
Solution:
4x2 – 6x – 3 when x = 2
= 4(2)2 – 6(2) – 3
=1
................................................
299. Determine the measure of Y and Z.
A. A and B
C. B and D
B. A and C
D. B and C
Explanation:
B and D are isosceles right triangles, or 45-45-90
triangles. Since they have congruent
corresponding angles, then they are similar.
................................................
294. Determine which equation is equivalent to
4x – 1 = 11.
A. 4x = 10
B. 3x = 11
C. 4 – 1x = 11x
D. 4x = 12
................................................
295. Determine the relation that matches the
table of values.
x
1
2
3
4
5
y
13 11
9
7
5
A. y = 21 – x
B. y = 15 – 2x
C. y = 3x + 7
D. y = 2x + 11
................................................
296. Determine which polynomial expression
matches the algebra tile model.
A. 2x2 + x + 4
C. 3x2 – x + 4
B. 3x2 + x + 4
D. 3x2 + x – 5
A. 45o, 55o
C. 48o, 48o
B. 40o, 50o
D. 50o, 50o
Since Y and Z intercept the same arc as X,
then they are congruent to X.
................................................
300. Which of the following is equal to x?
A. 15 sin 72o
C. 72 sin 72o
B. 15 sin 18o
D. 72 sin 18o
Solution:
x is the adjacent side to the 72o angle and 15 is
the hypotenuse. The trigonometric function for
adjacent and hypotenuse is cosine. However,
none of the choices made use of cos 72o.
Therefore we must use the other angle which
measures 18o. Its opposite side is x and the
hypotenuse is still 15. Therefore,
sin 18o =
15 sin 18o = x
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1000 MMR
301. Find the points of intersection of the graphs
of y = x2 and y = 3x – 2.
A. (1, 1) and (1, 4)
B. (1, 1) and (2, 4)
C. (1, -1) and (2, 4)
D. (-2, 4) and (1, 1)
Solution:
By transitive property, we can say x2 = 3x – 2.
x2 – 3x + 2 = 0
(x – 2)(x – 1) = 0
x = 1, 2
Author: Victor A. Tondo Jr., LPT
Solution:
The last digit of the powers of 3 (the last digit of
543) are 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, …
The last digit repeats after four powers.
The last digit of 31, 35, 39, … is 3.
The last digit of 32, 36, 310, … is 9.
The last digit of 33, 37, 311,… is 7.
The last digit of 34, 38, 312, … is 1.
Therefore the units digit of 543444 is 1.
................................................
When x = 1, then y = 12 = 1.
When x = 2, then y = 22 = 4.
Therefore, the points of intersection of the
graphs are (1,1) and (2,4).
................................................
302. An approximate value for
.
.
is:
.
A. 2
B. 20
C. 200
D. 2000
305. 4n+1 (4n+2) equals
A. 42n + 3
B. 82n + 3
2n
+
3
C. 16
D. 4n + 3
Explanation:
Apply the laws of exponents.
................................................
306. The greatest number of Fridays that can
occur in a 75 day period is:
A. 10
B. 11
C. 12
D. 13
Explanation:
Ignore the decimals of the large numbers and use
only the first few decimals of 0.0403289925.
Using your calculator, 302476 x 0.04 ÷ 5962 2.
................................................
303. The graph below represents the motion of a
car. The graph shows us that the car is:
A. accelerating
B. standing still
C. travelling north-east
D. travelling at a constant speed
................................................
304. The units digit of the number 543444 is:
A. 3
B. 9
C. 7
D. 1
Explanation:
Let the first day be Friday. The last Friday will be
on the 71st day. That’s a total of 11 Fridays.
................................................
307. Which model is not a function?
A.
B.
C.
D.
Explanation:
In a function, no x-value can ever have two or
more corresponding y-values.
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1000 MMR
308. Which expression is equivalent to (9-2)8?
A. -8132
B. -818
C.
D.
Solution:
(9-2)8 = ( ) = ( ( )) =
................................................
309. What is 5 × 10–4 written in standard
notation?
A. 0.00005
B. 0.0005
C. 5,000
D. 50,000
................................................
310. What is the value of
A. -25
B.
54
×
C.
5-6?
D. 25
Solution:
54 × 5-6 = 5-2 =
................................................
311. . Which comparison is true?
A. 4 < 180.5 < 4.5
B. 4.5 < 180.5 < 5
C. 8.5 < 180.5 < 9.5
D. 17 < 180.5 < 19
Explanation:
180.5 = √18 4.24
................................................
312. A weather station recorded the amount of
rain that fell during an 8-hour time frame using a
rain gauge. The findings are recorded in the
graph below.
Author: Victor A. Tondo Jr., LPT
A. between hours 3 and 4
B. between hours 4 and 5
C. between hours 5 and 6
D. between hours 7 and 8
................................................
312. Each day of the month, Carl earns an
allowance, in cents, equal to the square of that
date of the month. Which is a number of cents
Carl could earn in a single day?
A. 21
B. 31
C. 64
D. 111
Explanation:
In the choices, only 64 is a perfect square.
................................................
313. Which set of ordered pairs models a
function?
A. {(2, 9), (7, 5), (3, 14), (2, 6)}
B. {(5, 10), (5, 15), (5, 20), (5, 25)}
C. {(3, 10), (4, 15), (5, 20), (3, 25)}
D. {(–10, 20), (–20, 30), (–30, 40), (–40, 10)}
Explanation:
Only D has no repeated x-value.
................................................
314. Rayon has a piece of rectangular paper that
is 12 inches wide by 16 inches long. He drew a
straight line along the diagonal of the paper.
What is the length of the line Rayon drew?
A. √28 inches
B. √192 inches
C. 20 inches
D. 28 inches
Explanation:
Use the Pythagorean theorem.
................................................
315. Which equation has infinitely many
solutions?
A. 2x + 4 = 7x + 9
B. 3(2x + 5) = 6x + 15
C. 4x + 13 = 5x + (20 – x)
D. x + 3 = 5x – 21
Between which hours was the rate at which the
rain fell greater than the rate at which the rain
fell between hours 0 and 1?
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1000 MMR
Explanation:
A. 2x + 4 = 7x + 9
4 – 9 = 5x – 2x
-5 = 3x
-5/3 = x
 One unique solution
B. 3(2x + 5) = 6x + 15
6x + 15 = 6x + 15
0=0
 True equation. This
means there are infinitely
many solutions.
C. 4x + 13 = 5x + (20 – x)
4x + 13 = 4x – 20
4x – 4x = -20 – 13
0 = -33
 False equation. There is
no solution.
D. x + 3 = 5x – 21
3 + 21 = 5x – x
24 = 4x
6=x
 One unique solution
................................................
316. Praetor jogged on a path that was 2 miles
long, took a break, and then jogged back along
the same path to where he started. He jogged at
different speeds for different distances along the
path as shown in the graph.
Author: Victor A. Tondo Jr., LPT
317. Which expression has a value of -2?
A. |2| + |-4|
B. |-2| – |4|
C. |4| – |-2|
D. |-4| + |2|
Explanation:
A. |2| + |-4| = 2 + 4 = 6
B. |-2| – |4| = 2 – 4 = -2
C. |4| – |-2| = 4 – 2 = 2
D. |-4| + |2| = 4 + 2 = 6
................................................
318. Reion is tossing a six-sided number cube
labeled 1, 2, 3, 4, 5, and 6. What is the probability
of tossing 6 twice in a row?
A.
B.
C.
D.
................................................
319. Which represents the value of x in
6 – 4x 26?
A. x -8
B. x -8
C. x -5
D. x -5
................................................
320. The table below shows the resting heart
rates in beats per minutes of six students. The
rate, 40 beats per minute, seems to be an outlier.
Which measure of central tendency changes the
least by dropping 40 from the data?
Heart
Rate
78
71
A. mean
C. mode
79
80
40
71
B. median
D. range
Explanation:
Between which times did Praetor jog the fastest?
A. 0 minutes and 10 minutes
B. 10 minutes and 25 minutes
C. 25 minutes and 30 minutes
D. 30 minutes and 60 minutes
Explanation:
The steepest part of the graph is between 0 and
10 minutes.
The original mean is 69.833. When 40 is
removed from the data, it becomes 75.8. The
mean decreases by 5.967.
The original median is 74.5. When 40 is removed
from the data, it becomes 78. The median
increases by -3.5.
The original mode is 71. When 40 is removed
from the data, it stays the same.
The original range is 40. When 40 is removed
from the data, it becomes 9. The range decreases
by 31.
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1000 MMR
321. The sum of a number, n, and 5 is subtracted
from 8. Which expression represents this
statement?
A. 8 – (n + 5)
B. (n + 5) + 8
C. (n + 5) – 8
D. 8 + (n + 5)
................................................
Author: Victor A. Tondo Jr., LPT
2x - y = -1
2(-2) - y = -1
-4 - y = -1
-4 + 1 = y
-3 = y
................................................
322. How is 0.5600 written in scientific notation?
A. 5.6 × 10
B. 5.6 × 10-1
C. 5.6 × 10-2
D. 5.6 × 10-3
................................................
323. What is the value of x in 3(x – 4) = –21?
A. x = –11
B. x = –3
C. x = 3
D. x = 11
326. The lengths of two sides of a triangle are 8
inches and 13 inches. Which of the following
represents x, the possible length in inches of the
remaining side of the triangle?
A. 5 < x < 21
B. 5 x 21
C. x < 5 or x > 21
D. x 5 or x 21
Solution:
Explanation:
3(x – 4) = –21
x – 4 = -7
x = -7 + 4 = -3
................................................
The third side of any triangle must be between
the difference and the sum of the two other
sides.
................................................
324. In the spinner, what is the probability of the
arrow NOT landing on the space with the ∆?
327. What is the value of the expression
below?
80 ÷ ( 6 + (3 – 5) x 2)
A. -8
B. 8
C. 10
D. 40
................................................
A.
B.
C.
D.
................................................
325. Which values of x and y make the system of
equations below true?
2x - y = -1
3x - y = -3
A. x = -4; y = -7
C. x = 2; y = 5
Solution:
(-)
2x - y = -1
3x - y = -3
-x = 2
x = -2
B. x =-2; y = -3
D. x = 4; y = 15
328. Which of the following is closest to the
value of the expression below?
(20.0143642359)2 x 8π
A. 1,000
B. 10,000
C. 100,000
D. 1,000,000
Explanation:
Round off and solve.
202 x 8 x 3.14 = 10,048
................................................
329. Which of the following expressions has a
value of 0?
A. (2 – 3) – (2 – 3)
B. (2 – 3) – |2 – 3|
C. (2 – 3) + (-3 + 2)
D. |2 – 3| – (2 – 3)
................................................
330. What is the factorization of 10x2 – x – 21?
A. (5x – 7) (2x + 3)
B. (5x + 7) (2x – 3)
C. (5x + 3) (2x – 7)
D. (5x – 3) (2x – 7)
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1000 MMR
331. Evaluate: (√343)2
A. 7√7
B. 49
C. 49√7
D. 343
................................................
332. Find the length of the latus rectum of the
ellipse defined by
A.
B.
(
)
+
(
)
C.
= 1.
D.
Solution:
Length of latus rectum of ellipse =
Since the lower denominator is 16, then b = 4.
Since the higher denominator is 25, then a = 5.
2
2(4 ) 32
=
=
5
5
................................................
333. Let A be a set such that A = {v, w, x, y, z}.
How many subsets does set A have?
A. 5
B. 10
C. 25
D. 32
Explanation:
Let n = number of elements
Number of subsets = 2n
................................................
Author: Victor A. Tondo Jr., LPT
Solution:
The mth term of the expansion of (a + b)n is given
as nC(m-1) an-m+1bm-1.
7C4 (2x)4(3y)3 = 35 (16x4) (27y3)
= 15120 x4y3
................................................
336. The shell shape
, as used in the Mayan
numeral system, is the symbol for which
number?
A. 100
B. 10
C. 1
D. 0
Explanation:
The Mayan numerals consisted of only three
symbols: zero, represented as a shell shape; one,
a dot; and five, a bar.
................................................
337. Which of the following is irrational?
̅̅̅̅̅
A. 0.125
B. 43.29%
C. √200
D. √343
Explanation:
̅̅̅̅̅ in fraction is
A. 0.125
B. 43.29% in fraction is
C. √200 cannot be rewritten as a fraction with
whole numbers as numerator and denominator
D. √343 is 7.
................................................
334. Solve for x:
2 (5x – 11) + 7 = 3 (x – 7) – 15
A. x = 3
B. x = 1
C. x = -1
D. x = -3
Solution:
2 (5x – 11) + 7 = 3 (x – 7) – 15
10x – 22 + 7 = 3x – 21 – 15
10x –15 = 3x – 36
7x = -21
x = -3
................................................
335. What is the 4th term in the expansion of
(2x + 3y)7?
A. 15120 x4y3
B. 7560 x4y3
3
4
C. 3780 x y
D. 1890 x3y4
338. Aira is six years older than Zayne. Six years
ago, she was twice as old as he. How old is Aira
now?
A. 21
B. 18
C. 15
D. 12
Solution:
Let A = Aira’s present age
A – 6 = Zayne’s present age
.: A – 6 = Aira’s age 6 years ago
A – 12 = Zayne’s age 6 years ago
A – 6 = 2 (A – 12)
A – 6 = 2A – 24
-6 + 24 = 2A – A
18 = A
This is a free reviewer. All rights reserved.
1000 MMR
339. The two parallel sides of a trapezoidal lot
measure 100m and 70m. If these sides are 80m
apart, what is the area of the lot?
A. 13600 m2
B. 6800 m2
2
C. 3400 m
D. 2400 m2
Solution:
Area of Trapezoid =
(
)
(
(
)
)
=
(80)= 6800
................................................
340. If y = x and y = 2x + 2, find the value of x.
A. x = -2
B. x = -1
C. x = 0
D. x = 1
Solution:
Since y = x and y = 2x + 2, then by transitive
property of equality,
x = 2x + 2
-2 = 2x - x
-2 = x
................................................
341. If the difference between the squares of two
consecutive counting numbers is 49, what is the
larger number?
A. 99
B. 49
C. 25
D. 7
Solution:
Let the consecutive counting numbers
be x and x + 1.
(x + 1)2 – x2 = 49
2
(x + 2x + 1) – x2 = 49
2x + 1 = 49
2x = 48
x =24
.: The larger number is 24 + 1 = 25.
................................................
Author: Victor A. Tondo Jr., LPT
342. Rayon needed to find the perimeter of an
equilateral triangle whose sides measure x + 4
cm each. Jake realized that he could multiply
3 (x + 4) = 3x + 12 to find the total perimeter in
terms of x. Which property did he use to
multiply?
A. Associative Property of Addition
B. Distributive Property of Multiplication over
Addition
C. Commutative Property of Multiplication
D. Inverse Property of Addition
................................................
343. A ride in a Feak Taxi costs P25.00 for the
first km and P10.00 for each additional km.
Which of the following could be used to calculate
the total cost, y, of a ride that was x km?
A. y = 25x + 10
B. y = 10x + 25
C. y = 25(x 1) + 10
D. y = 10(x 1) + 25
................................................
344. Which of the following points is in the
fourth quadrant?
A. (3, 4)
B. (-3, 4)
C. (3, -4)
D. (-3, -4)
Explanation:
A point in the fourth quadrant has a positive
abscissa (x-value) and a negative ordinate
(y-value).
................................................
345. The distance from the sun to the earth is
approximately 9.3 × 107 miles. What is this
distance expressed in standard notation?
A. 9,300,000,000
B. 930,000,000
C. 93,000,000
D. 651
................................................
346. The square of a number added to 25 equals
10 times the number. What is the number?
A. -10
B. -5
C. 5
D. 10
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1000 MMR
Solution:
Author: Victor A. Tondo Jr., LPT
350. Factorize: a2 – a – 90
x2 + 25 = 10x
x2 – 10x + 25 = 0
10 + 25 =√0
√
x–5=0
x=5
................................................
A. (a – 10) (a + 9)
B. (a + 10) (a + 9)
C. (a + 10) (a – 9)
D. (a – 10) (a – 9)
................................................
347. The sum of the square of a number and 12
times the number is 27. What is the smaller
possible value of this number?
A. -9
B. -3
C. 3
D. 9
Let x = the number
x2 + 12x = -27
2
x + 12x + 27 = 0
(x + 9) (x + 3) = 0
x = -9 or -3
................................................
348. Let x = 1. Find the corresponding y given
that 2x 3y = 5.
A. y = -1
B. y = 1
C. y = 3
D. y = -3
Solution:
2x 3y = 5
2(1) – 3y = 5
2 – 3y = 5
2 – 5 = 3y
-3 = 3y
-1 = y
................................................
349. The sum of two consecutive even integers is
126. What is the smaller integer?
A. 63
B. 62
B. 61
D. 60
Let x = smaller even integer
.: x + 2 = next even integer
x + (x + 2) = 126
2x + 2 = 126
2x = 124
x = 62
Solution:
10P5 = (
Solution:
Solution:
351. Evaluate 10P5.
A. 2
B. 100,000
C. 1,024
D. 30,240
!
)!
=
!
!
= 30,240
Or simply use your calculator to evaluate 10P5
by using the nPr button.
................................................
352. Jay bought twenty-five P4.57 stamps. How
much did he spend?
A. P 104.25
B. P 114.25
C. P 119.75
D. P124.25
Solution:
25 x 4.57 = 114.25
If you’re using a scientific calculator and it shows
you , simply press then S D button.
................................................
353. Given f(x) = x3 + kx2 – 7, find k if f(2) = 41.
A. 5
B. 10
C. 15
D. 20
Solution:
Just substitute x with 2.
f(2) = 23 + k(22) – 7
41 = 8 + 4k – 7
41 = 4k + 1
40 = 4k
10 = k
................................................
354. If y = x and y = 2x + 2, find x + y.
A. -8
B. -4
C. 0
This is a free reviewer. All rights reserved.
D. 4
1000 MMR
Solution:
Since y = x and y = 2x + 2, then by transitive
property of equality,
x = 2x + 2
-2 = 2x - x
-2 = x
.: y = -2 as well
.: x + y = -2 + (-2) = -4
................................................
355. Mulan and Lilo are competing to see who
can sell the most candy bars for a fundraiser.
Mulan sold 4 candy bars on the first day and 2
each day after that. Lilo sold 7 on the first day
and 1 each day after that. On what day will they
have the same number of candy bars sold?
A. 7th
B. 6th
C. 4th
D. 3rd
Solution:
Let x = number of days to pass after the 1st day
4 + 2x = 7 + x
2x – x = 7 – 4
x=3
They will have the same number of candy bars
sold 3 days after the first day, or on the 4th day.
................................................
356. Which of the following is not a polynomial?
A.
-3x2
C.
√
+ x–9
B. √2x + π
D. 7
+9
Author: Victor A. Tondo Jr., LPT
358. The cost of renting a bike at the local bike
shop can be represented by the equation
y = 2x + 2, where y is the total cost and x is the
number of hours the bike is rented. Which of the
following ordered pairs would be possible
number of hours rented and the corresponding
total cost?
A. (0, 2)
B. (2, 6)
C. (6, 2)
D. ( 2, 6)
Explanation:
Simply substitute the x- and y-values of the point
into the equation y = 2x + 2. Whichever point
holds a true equation is the correct answer.
................................................
359. The distance from the earth to the moon is
approximately 240,000 miles.
What is this distance expressed in scientific
notation?
A. 24 × 104
B. 2.4 × 104
C. 2.4 × 105
D. 2.4 × 10 5
................................................
360.
=
A.
B.
C.
D.
Solution:
(
)(
)
(
)(
)
=
................................................
Explanation:
In a polynomial, x cannot be raised to exponents
that are fractions or irrational numbers.
................................................
357. Factorize: 3p2 – 2p – 5
A. (3p – 5) (p + 1)
B. (3p + 5) (p – 1)
C. (3p + 1) (p – 5)
D. (3p – 1) (p + 5)
................................................
361. The sum of two angles is 180°. The measure
of one angle is 34° greater than the measure of
the other angle. What is the measure of the
smaller angle?
A. 74°
B. 73°
C. 72°
D. 71°
Solution:
Let x = measure of the smaller angle
.: x + 34 = measure of the larger angle
x + (x + 34) = 180
2x + 34 = 180
2x = 146
x = 73
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1000 MMR
362. Solve the following system of linear
equations:
3x + y = -9
3x 2y = 12
A. ( 2, 3)
C. (3, 2)
B. (2, 3)
D. ( 3, 2)
Solution:
Solution:
(+)
Author: Victor A. Tondo Jr., LPT
365. Samantha owns a rectangular field that has
an area of 3,280 square meters. The length of the
field is 2 more than twice the width. What is the
width of the field?
A. 40 m
B. 41 m
C. 82 m
D. 84 m
Let w = width of the field
.: 2w + 2 = length of the field
3x + y = -9
3x 2y = 12
–y = 3
y = -3
Using the first equation 3x + y = 9,
3x + (-3) = -9
3x = -6
x = -2
................................................
363. Find the x-intercept of 3x + 2y = 24
A. x = 8
B. x = -8
C. y = 12
D. y = -12
Solution:
To find the x-intercept, let y = 0.
3x + 2y = 24  3x + 2(0) = 24
3x = 24
x=8
................................................
364. A circle is drawn such that ̅̅̅̅ is a diameter
and its midpoint is O. Given that C is a point on
the circle, what is the measure of ACB?
A. 180o
B. 90o
C. 60o
D. not enough info
Explanation:
Any inscribed angle that opens
to a diameter or a semicircle is
a right angle, or measures 90o.
w (2w + 2) = 3280
2w2 + 2w = 3280  divide equation by 2
w2 + w = 1640  complete the square
2
w + w + 0.25 = 1640 + 0.25
w2 + w + 0.25 = 1640.25
√ + + 0.25 = √1640.25
w + 0.5 = 40.5
w = 40
................................................
366. Rayon used the following mathematical
statement to show he could change an
expression and still get the same answer on both
sides:
10 × (6 × 5) = (10 × 6) × 5
Which mathematical property did Rayon use?
A. Identity Property of Multiplication
B. Commutative Property of Multiplication
C. Distributive Property of Multiplication over
Addition
D. Associative Property of Multiplication
................................................
367. Factorize x3 – 27y3.
A. (x – 3y) (x2 – 3x + 9y2)
B. (x – 3y) (x2 + 3x + 9y2)
C. (x + 3y) (x2 + 3x – 9 y2)
D. (x + 3y) (x2 – 3x + 9 y2)
................................................
368. What is the intersection of the lines
x + 3y = 5 and -2x + 4y = 0?
A. (2, 4)
B. (-2, 1)
C. (2, 1)
D. (-2, 4)
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1000 MMR
Solution:
2(x + 3y = 5) 
2x + 6y = 10
-2x + 4y = 0  (+) -2x + 4y = 0
10y = 10
y=1
x + 3y = 5
x + 3(1) = 5
x+3=5
x=2
................................................

369. Given f(x) = 7x3 – 3x2 + 2x – 9, f(2) =
A. 56
B. 48
C. 44
D. 39
Solution:
Simply substitute x with 2.
Author: Victor A. Tondo Jr., LPT
372. A researcher is curious about the IQ of
students at the Utrecht University. The entire
group of students is an example of a:
A. parameter
B. statistic
C. population
D. sample
Explanation:
Entirety is population, part of it is sample.
................................................
373. Jordan filled a bottle with grains until it was
1/4 full and weighed 8 kg. He added more grains
into the bottle until it was 7/8 full. It now
weighed 18 kg. What is the mass of the empty
bottle?
A. 16
B. 8
C. 4
D. 2
f(x) = 7x3 – 3x2 + 2x – 9
f(2) = 7(23) – 3(22) + 2(2) – 9
f(2) = 56 – 12 + 4 – 9
f(2) = 39
................................................
Solution:
370. Ten factorial is equal to _____.
A. 100
B. e10
C. 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
D. 10e
................................................
Eqn 2: B + G = 18
Let B = weight of empty bottle,
G = weight of capacity of bottle
Eqn 1: B + G = 8
Subtract Eqn 1 from Eqn 2:
B + G = 18
(–)
B+ G=8
___________________________________________________
371. How many 3-digit numbers can be made
using the digits 5, 6, 7, 8, 9, and 0 if repetition is
not allowed?
A. 80
B. 100
C. 120
D. 140
G = 10
5G = 80
G = 16
Solving for B,
Solution:
There are only five possible choices for the
hundreds digit since we cannot start with 0.
There are five choices for the tens since out of
the six, we have used one for the hundreds, and
we can use 0 for the tens digit.
We have used two for the hundreds and tens, so
we only have four choices for the units digit.
5 x 5 x 4 = 100
B + ¼ (16) = 8
B + 4 = 8;
B=4
................................................
374. If 37 – 4x < 17, then
A. x < 5
B. x > 5
C. x < -5
D. x > -5
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1000 MMR
Solution:
Author: Victor A. Tondo Jr., LPT
Solution:
Substitute x and y with -2 and 3 respectively.
37 – 4x < 17
–4x < 17 – 37
–4x < –20
-¼ (–4x) < -¼ (–20) *
x>5
ax + by = 17
2ax by = 11
Solve for a and b by elimination.
*Remember to change your inequality sign after
multiplying or dividing the inequality by a
negative number.
................................................
375. How many solutions are there for the
following system of linear equations?
-3x + 5y = 6
6x 10y = 0
A. only one solution
B. two solutions
C. infinitely many solutions
D. no solution
Explanation:
Try to solve for x and y first.
-2 (-3x + 5y = 6)  6x
6x 10y = 0  6x
(–)
6x
6x
10y = -12
10y = 0
0 = -12
10y = -12
10y = 0
-2(-2a + 3b = 17)
-4a – 3b = -11
 4a – 6b = -34
 -4a – 3b = -11
4a – 6b = -34
(+) -4a – 3b = -11
-9b = -45
b=5
Using -2a + 3b = 17 and substituting b = 5,
-2a + 3(5) = 17
-2a + 15 = 17
-2a = 17 – 15
-2a = 2
a = -1
................................................
377. What is the probability choosing only one
vowel when three letters are randomly selected
from the word NUMBERS?
A.
B.
C.
D.
Solution:
which is false.
When you get a false equation, it means there is
no solution for the given system of linear
equations.
Also, when there’s no solution for a system of
linear equations, the lines defined by the two
equations are parallel – they will never intersect.
................................................
376. Find a and b so that the system below has
the unique solution (-2, 3).
ax + by = 17
2ax by = 11
A. a = 3, b = -1
C. a = 1, b = 3
 -2a + 3b = 17
 -4a – 3b = -11
NUMBERS has two vowels and five consonants.
The number of ways we can get only one vowel
out of three is taken as 2C1 x 5C2, or 20.
The number of ways we can get three letters
from seven is 7C3 or 35.
Therefore, the answer is or .
................................................
378. If A > B, which is always true?
A.
B. A2 > B2
C. A < B + 2
D. A – B > 0
B. a = 1, b = 5
D. a = 1, b = 5
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1000 MMR
Explanation:
Use counter-examples to disprove the choices.
A.
When A = 2 and B = 1,
is false.
B. A2 > B2
When A = 2 and B = -3, then 22 > (-3)2 is false.
C. A < B + 2
When A = 1 and B = 0, then 1 < 0 + 2 is false.
D. A – B > 0
This will always be true for any A > B.
................................................
379. Statistical techniques that summarize and
organize the data are classified as what?
A. Population statistics
B. Sample statistics
C. Descriptive statistics
D. Inferential statistics
................................................
380. Five-point Likert scales (strongly disagree,
disagree, neutral, agree, strongly agree) are
frequently used to measure motivations and
attitudes. A Likert scale is a:
A. Discrete variable.
B. Ordinal variable.
C. Categorical variable.
D. All of the above
................................................
Author: Victor A. Tondo Jr., LPT
382. A teacher asks students to identity their
favorite reality television show. What type of
measurement scale do the different television
shows make up?
A. Nominal
B. Ordinal
C. Interval
D. Ratio
................................................
383. What is the center of the circle defined by
x2 + y2 – 8x + 6y – 10 = 0?
A. (-8, 6)
B. (8, -6)
C. (-4, 3)
D. (4, -3)
Explanation:
Since the equation of the circle is already in the
form x2 + y2 + Dx + Ey + F = 0, then the center
( )
(h, k) is at ( , ) or (
, ) or (4, -3).
................................................
384. Find the equation of the line passing (1, 4)
with slope equal to 5.
A. y = 5x + 3
B. y = 5x + 1
C. y = 5x – 1
D. y = 5x – 3
Solution:
Use the form y = mx + b and substitute the xand y- values of the point.
x = 1, y = 4, m = 5
y = mx + b
4 = 5(1) + b
4–5=b
-1 = b
381. What is the radius of the circle defined by
(x + 2)2 + (y – 3)2 = 16?
A. 256
B. 16
C. 8
D. 4
y = mx + b
y = 5x + (-1) or y = 5x – 1
................................................
Explanation:
385. The seminar rooms in the library are
identified by the letters A to H. A researcher
records the number of classes held in each room
during the first semester. What kind of graph
would be appropriate to present the frequency
distributions of these data?
A. Histogram
B. Scatterplot
C. Bar chart
D. Box plot
The equation of the circle is already in the
center-radius form.
Since r2 = 16, then r = 4.
................................................
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1000 MMR
386. Find the slope of the line passing the points
A(2, 3) and B(-7, -15).
A. 1
B. -1
C. ½
D. 2
Author: Victor A. Tondo Jr., LPT
Solution:
Solution:
48o x
=
=
................................................
Use the formula m =
m=
=
m=2
................................................
387. Factorize: 3n2 – 8n + 4
A. (3n – 2) (n – 2)
B. (3n – 2) (n + 2)
C. (3n + 2) (n + 2)
D. (3n + 2) (n – 2)
................................................
388. In now many ways can the letters
AAABBCDEEE be arranged in a straight line?
A. 50,400
B. 25,200
C. 12,600
D. 6,300
Solution:
!
10!
= 50,400
! ! ! ! 3! 2! 3!
................................................
=
389. Find the smaller angle formed by the x-axis
and the line y = 5x.
A. 78.69o
B. 63.48o
C. 54.15o
D. 41.32o
Solution:
tan
=5
= tan-1 5
= 78.69o
................................................
390. Convert 48o to radians.
A. π rad
B. π rad
C.
π rad
D.
π rad
To convert from degrees to radians, simply
multiply the given degree measure by
.
391. The median is always:
A. The most frequently occurring score in set of
data
B. The middle score when results are ranked in
order of magnitude
C. The same as the average
D. The difference between the maximum and
minimum scores.
................................................
392. A teacher gave a statistics test to a class of
Geography students and computed the measures
of central tendency for the test scores. Which of
the following statements cannot be an accurate
description of the scores?
A. The majority of students had scores above the
mean.
B. The majority of students had scores above the
median.
C. The majority of students had scores above the
mode.
D. All of the above options (A, B and C) are false
statements.
Explanation:
A. If majority of the students had scores above
the mean, then this is an example of a negativelyskewed data. This can happen.
B. 50% of the class is always above the median,
and the other 50% of the class is below the
median.
C. The mode can pop up anywhere. That means
the students may have a low modal score.
................................................
393. Find the area of a semicircle whose radius
measures 28 cm.
A. 784 π cm2
B. 392 π cm2
2
C. 28 π cm
D. 14 π cm2
This is a free reviewer. All rights reserved.
1000 MMR
Solution:
A = ½ π r2
A = ½ π (28)2
A = 392 π cm2
................................................
394. Find the length of each side of an equilateral
triangle whose perimeter is 90 cm.
A. 45 cm
B. 30 cm
C. 22.5 cm
D. 10 cm
Solution:
An equilateral triangle has three congruent sides.
Its perimeter is given as P = 3s.
3s = 90
s = 30
................................................
395. Find the number of subsets having 4
elements of the set {1,2,3,4,5,6,7,8,9,10,11}.
A. 165
B. 330
C. 660
D. 1320
Author: Victor A. Tondo Jr., LPT
398. In how many ways can 4 girls and 5 boys be
arranged in a row so that all the four girls are
together?
A. 4,320
B. 8,640
C. 17,280
D. 34,560
Solution:
Let 4 girls be one unit. So now, there are 6 units
in all: 5 boys and the solo unit made by the four
girls. They can be arranged in 6! ways.
In each of these arrangements 4 girls can be
arranged in 4! ways.
Total number of arrangements in which girls
are always together
= 6! × 4!
= 720 × 24
= 17,280
................................................
Solution:
399. A box contains 8 batteries, 5 of which are
good and the other 3 are defective. Two batteries
are selected at random and inserted into a toy. If
the toy only functions with two good batteries,
what is the probability that the toy will function?
A.
B.
C.
D.
11C4 = 330
................................................
Solution:
396. There are ten true - false questions in an
exam. How many responses are possible?
A. 1024
B. 256
C. 20
D. 10
Solution:
There are two possible responses per item, and
there are ten items. 210 = 1024.
................................................
397. In a 500m speed skating race, time results
would be considered an example of which level
of measurement?
A. Nominal
B. Ordinal
C. Interval
D. Ratio
................................................
=
................................................
400. IQ tests are standardized so that the mean
score is 100 for the entire group of people who
take the test. However, if you select a group of 50
who took the test, you probably would not get
100. What statistical concept explains the
difference between the two means?
A. Statistical error
B. Inferential error
C. Residual error
D. Sampling error
................................................
401. Which Mathematician pioneered the study
of conic sections?
A. Euclid
B. Apollonius
C. Archimedes
D. Hipparchus
This is a free reviewer. All rights reserved.
1000 MMR
Explanation:
Apollonius of Perga was a Greek geometer and
astronomer known for his theories on the topic
of conic sections. His definitions of the terms
ellipse, parabola, and hyperbola are the ones in
use today.
................................................
402. A researcher studies the factors that
determine the number of children future couples
decide to have. The variable ‘number of children’
is a:
A. Discrete variable
B. Continuous variable
C. Categorical variable
D. Ordinal variable
................................................
403. Surface area and volume, center of gravity,
and hydrostatics are some of the studies of
which Mathematician?
A. Apollonius
B. Euclid
C. Archimedes
D. Hipparchus
................................................
404. The book Philosophiæ Naturalis Principia
Mathematica, more fondly known simply as
Principia, is the work of which Mathematician?
A. Euclid
B. Newton
C. Einstein
D. Archimedes
................................................
405. A researcher is interested in the travel time
of Rayon’s University students to college. A
group of 50 students is interviewed. Their mean
travel time is 16.7 minutes. For this study, the
mean of 16.7 minutes is an example of a
A. parameter
B. statistic
C. population
D. sample
Explanation:
The 50 students in the above case is only part of
the population of students of Rayon’s University,
thus, they are only a sample. Data taken from a
sample is called statistic.
Author: Victor A. Tondo Jr., LPT
406. Who is considered by many Mathematicians
as “The Last Universalist”?
A. Jules Henri Poincare
B. Hendrik Lorentz
C. Georg Cantor
D. Gottfried Wilhelm Leibniz
Explanation:
Both Poincare and Leibniz are polymaths, but
Jules Henri Poincare is fondly referred to by
fellow Mathematicians as the Last Universalist.
................................................
407. In the theory he developed, there are
infinite sets of different sizes (called
cardinalities). Which Mathematician formalized
many ideas related to infinity and infinite sets
during the late 19th and early 20th centuries?
A. Jules Henri Poincare
B. Hendrik Lorentz
C. Georg Cantor
D. Gottfried Wilhelm Leibniz
................................................
408. Which of the following sets of scores has the
greatest variability or range?
A. 2, 5, 8, 11
B. 13, 13, 13, 13
C. 20, 25, 26 ,27
D. 42, 43, 44, 45
Explanation:
Range = highest score – lowest score
................................................
409. This Mathematician was the first to describe
a pinwheel calculator in 1685 and invented the
wheel named in his honor, which was used in the
arithmometer, the first mass-produced
mechanical calculator. He also refined the binary
number system, which is the foundation of all
digital computers. Which Mathematician is this,
who is also crucial to the development of
computers?
A. Gottfried Wilhelm Leibniz
B. Charles Babbage
C. Ada Lovelace
D. Alexander Graham Bell
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1000 MMR
410. Solve for x, given 9x – 10 = 11x + 30
A. x = 40
B. x = 20
C. x = -20
D. x = -40
Author: Victor A. Tondo Jr., LPT
Solution:
Solution:
9x – 10 = 11x + 30
-10 – 30 = 11x – 9x
-40 = 2x
................................................
411. Using Calculus, this Mathematician
explained why tides occur, why the shapes of
planetary orbits are conic sections, and how to
get the shape of a rotating body of fluid, among
many other things. Which Mathematician is this?
A. Kepler
B. Euclid
C. Apollonius
D. Newton
Explanation:
Newton explained the above-mentioned topics in
his work Principia.
................................................
412. Which of the following terms does NOT
describe the number 9?
A. rational number B. integer
C. real number
D. prime number
Explanation:
Its factors are 1, 3, and 9. Therefore, 9 is a
composite number.
................................................
413. Which expression below is equal to 5?
A. (1 + 2)2
B. 9 – 22
C. 11 10 × 5
D. 45 ÷ 3 × 3
................................................
414. A bus picks up a group of tourists at a hotel.
The sightseeing bus travels 2 blocks north, 2
blocks east, 1 block south, 2 blocks east, and 1
block south. Where is the bus in relation to the
hotel?
A. 2 blocks north
B. 1 block west
C. 3 blocks south
D. 4 blocks east
415. When five is added to three more than a
certain number, the result is 29. What is the
number?
A. 24
B. 21
C. 8
D. 4
Solution:
(n + 3) + 5 = 29
n + 8 = 29
n = 21
................................................
416. The math club is electing new officers.
There are 3 candidates for president, 4
candidates for vice-president, 4 candidates for
secretary, and 2 candidates for treasurer. How
many different combinations of officers are
possible?
A. 13
B. 96
C. 480
D. 17,160
Solution:
Apply the fundamental counting principle.
3 x 4 x 4 x 2 = 96
................................................
417. Twelve points lie on a circle. How many
cyclic quadrilaterals can be drawn by using these
points? [Note: Cyclic quadrilaterals are
quadrilaterals whose vertices are on a circle.]
A. 48
B. 495
C. 11,880
D. 1,663,200
Solution:
12C4 = 495
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1000 MMR
418. What is the variance for the following set of
scores? 143 143 143 143 143 143
A. 0
B. 2
C. 4
D. 25
Author: Victor A. Tondo Jr., LPT
422. The sum of five consecutive integers is 215.
What is the largest of these integers?
A. 43
B. 44
C. 45
D. 46
Explanation:
Solution:
The scores are all the same, so the variance is 0.
................................................
Let x = smallest number
.: The numbers will be expressed as x, x + 1,
x + 2, x + 3, and x + 4.
419. When 18 is subtracted from six times a
certain number, the result is 42. What is the
number?
A. 10
B. 4
C. -4
D. -10
Solution:
6x – 18 = -42
6x = -42 + 18
6x = -24
x = -4
................................................
420. Find the equation of the line passing (2, 3)
and (-7, -15)
A. y = 2x + 1
B. y = 2x – 1
C. y = x + 2
D. y = x – 2
Solution:
Use the two-point form.
y – y1 =
(x x )
y–3=
(x 2)
y – 3 = 2(x – 2)
y – 3 = 2x – 4
y = 2x – 1
................................................
421. Of the following Z-score values, which one
represents the location closest to the mean?
A. Z = +0.5
B. Z = +1.0
C. Z = -1.5
D. Z = -0.3
Explanation:
A lower absolute value of Z-score means the item
is closer to the mean. The Z-score with the
lowest absolute value is D, Z = -0.3.
................................................
x + (x + 1) + (x + 2) + (x + 3) + (x + 4) = 215
5x + 10 = 215
5x = 205
x = 41
.: The numbers are 41, 42, 43, 44, and 45.
................................................
423. You go to the cafeteria for lunch and have a
choice of 4 entrees, 5 sides, 5 drinks, and 4
desserts. Assuming you have one of each
category, how many different lunches could be
made?
A. 18
B. 81
C. 40
D. 400
Solution:
Apply the fundamental counting principle.
4 x 5 x 5 x 4 = 400
................................................
424. What can be said about the following
statements?
i. Any quadrilateral with four congruent sides is
a square.
ii. Any square has four congruent sides.
A. Only the first statement is true.
B. Only the second statement is true.
C. Both statements are true.
D. Both statements are fall.
Explanation:
i. Any quadrilateral with four congruent sides is
a RHOMBUS.
ii. Any square has four congruent sides. TRUE.
................................................
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1000 MMR
425. Out of 6 boys and 4 girls, a committee of 5
has to be formed. In how many ways can this be
done if we take 2 girls and 3 boys?
A. 120
B. 186
C. 240
D. 256
Solution:
6C3 x 4C2 = 20 x 6 = 120
................................................
426. In the figure, which of the following will
yield the value of the hypotenuse x?
A. x =
B. x =
C. x =
D. x = 10 tan 35o
Solution:
The side measuring 10 units is opposite the
given angle, while x is the hypotenuse. The
trigonometric ratio that involves the opposite
side and the hypotenuse is the sine function.
From the mnemonic SohCahToa, we can recall
that sin =
.
sin 35o =
x sin 35o = 10
x=
................................................
427. The shortest side of a 30-60-90 triangle is
20.19 cm long. How long is the hypotenuse?
A. 40.38 cm
B. 30.29 cm
C. 34.97 cm
D. 17.48 cm
Explanation:
In a 30-60-90 triangle, the hypotenuse is always
twice the shortest side.
................................................
428. The hypotenuse of a 30-60-90 triangle is
34.96 cm long. How long is the shortest side?
A. 40.38 cm
B. 30.29 cm
C. 34.97 cm
D. 17.48 cm
Author: Victor A. Tondo Jr., LPT
429. The second angle of a triangle is three times
as large as the first. The measure of the third
angle is 40 degrees greater than that of the first
angle. How large is the first angle?
A. 28o
B. 30o
C. 35o
D. 38o
Solution:
Let x = measure of the first angle
.: 3x = measure of the second angle
x + 40 = measure of the third angle
x + 3x + x + 40 = 180
5x + 40 = 180
5x = 180 – 40 = 140
x = 28
................................................
430. Normally distributed data are normally
referred to as:
A. Bell-shaped
B. Asymmetrical
C. Skewed
D. Peaked
................................................
431. A population has a mean of μ=35 and a
standard deviation of σ=5. After 3 points are
added to every score of the population, what are
the new values for the mean and standard
deviation?
A. μ=35 and σ=5
B. μ=35 and σ=8
C. μ=38 and σ=5
D. μ=38 and σ=8
Explanation:
When a constant is added or subtracted to every
score of a population, the mean increases or
decreases by the same amount. The standard
deviation, however, stays the same.
................................................
432. Given sin θ =
A.
B.
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, find cos θ.
C.
D.
1000 MMR
Solution:
Since sin θ =
Author: Victor A. Tondo Jr., LPT
435. Given cos θ = and θ QIV, find tan θ.
, then θ = sin-1
Therefore, cos θ = cos (sin-1
A.
.
)=
.
B.
C.
D.
Solution:
Alternative Solution:
Since cos θ =
Draw a right triangle containing the angle θ.
Mark the side opposite θ as 7 and the
hypotenuse as 25 since sin θ =
.
opposite = √13
and cos θ =
Since tan θ =
5 = √144 =
, then
12.
, adjacent = 5, and
opposite = 12, by using the CAST mnemonic,
then tan θ =
since θ QIV.
................................................
436. Find the measure of T:
Solve for the missing leg by using the
Pythagorean theorem. You will get the measure
of the adjacent side as √25
7 , or √625 49
which is equal to 24.
A. 59o
B. 69o
C. 79o
D. 89o
................................................
Now that we know the measure of the adjacent
side, we can substitute cos θ =
= .
437. If the scores on a test have a mean of 26 and
a standard deviation of 4, what is the z-score for
a score of 18?
A. -1.41
B. 11
C. -2
D. 2
................................................
Solution:
433. In a triangle, what is located 2/3 of the
distance from each vertex to the midpoint of the
opposite side?
A. centroid
B. incenter
C. circumcenter
D. orthocenter
................................................
Z-score =
434. Which triangle has the centroid, incenter,
circumcenter, orthocenter, and nine-point-center
at the same location?
A. isosceles right triangle
B. 30-60-90 triangle
C. equilateral triangle
D. hyperbolic triangle
Z-score =
= = 2
................................................
438. If a researcher sets a level of significance at
0.05 (i.e. 5%), what does this mean?
A. Five times out of 100, a significant result will
be found that is due to chance alone and not to
true relationship.
B. Ninety-five times out of 100, a significant
result will be found that is due to chance alone
and not to true relationship.
C. Five times out of 100, a significant result will
be found that is not due to chance, but to true
relationship.
D. None of the above
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1000 MMR
Explanation:
The level of significance is also known as margin
for error.
................................................
439. When does a researcher risk a Type I error?
A. Anytime the decision is ‘fail to reject’.
B. Anytime H0 is rejected.
C. Anytime Ha is rejected.
D. All of the above options
Explanation:
Type I errors happen when we reject a true null
hypothesis.
Type II errors happen when we fail to reject a
false null hypothesis
................................................
B. 5
444. What is the measure of V in the following
figure?
A. 60o
B. 65o
C. 70o
D. 75o
Solution:
440. Solve for x:
A. 4
Author: Victor A. Tondo Jr., LPT
443. Although rarely used in proving, what is the
extra line or line segment drawn in a figure to
help in a proof?
A. base line
B. auxiliary line
C. converse line
D. Euler’s line
................................................
90 + 86 + 114 + x = 360
290 + x = 360
x = 70
................................................
C. 6
D. 7
Solution:
445. What is the intersection of all three
altitudes of a triangle?
A. incenter
B. centroid
C. orthocenter
D. circumcenter
................................................
90 + 90 + 98 + 17x – 3 = 360
275 + 17x = 360
17x = 85
x=5
................................................
446. How much water must be evaporated from
2000 mL of 30% acid solution to make a 50%
acid solution?
A. 800 mL
B. 850 mL
C. 900 mL
D. 950 mL
441. Which of the following is equidistant from
the vertices of the triangle?
A. circumcenter
B. orthocenter
C. incenter
D. centroid
................................................
Solution:
442. Which of the following is equidistant from
the sides of the triangle?
A. circumcenter
B. centroid
C. orthocenter
D. incenter
Let x = amount of water to be evaporated
C1V1 + C2V2 = CrVr
30%(2000) + 0% (-x) = 50%(2000 – x)
600 + 0 = 1000 – 0.5 x
0.5 x = 1000 – 600
0.5 x = 400
x = 800
Note: Water is 0% acid, and evaporation means
removal of volume thus –x.
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1000 MMR
447. Which of the following is the intersection of
angle bisectors of a triangle?
A. circumcenter
B. incenter
C. centroid
D. orthocenter
................................................
448. In terms of a conditional statement, what is
the statement formed by exchanging and
negating the antecedent and the consequent?
A. inverse
B. converse
C. adverse
D. contrapositive
Explanation:
Statement
If p, then q.
Converse
If q, then p.
Inverse
If not p, then not q.
Contrapositive If not q, then not p.
................................................
449. What is formed when the hypothesis and
the conclusion of the conditional statement are
interchanged?
A. converse
B. inverse
C. adverse
D. contrapositive
................................................
450. What is formed when both the hypothesis
and the conclusion of the conditional statement
are negated?
A. converse
B. inverse
C. adverse
D. contrapositive
................................................
451. Which of the following is the converse of the
following statement?
“If two angles are congruent,
then they have the same measure.”
A. If two angles are not congruent, then they do
not have the same measure.
B. If two angles have the same measure, then
they are congruent.
C. If two angles do not have the same measure,
then they are not congruent.
D. If two angles are not congruent, then they
have the same measure.”
Author: Victor A. Tondo Jr., LPT
452. In which geometry are there no parallel
lines?
A. elliptic geometry
B. hyperbolic geometry
C. spatial geometry
D. solid geometry
................................................
453. What do we call the ratio of two numbers
(larger number: smaller number) whose ratio to
each other is equal to the ratio of their sum to
the larger number? [Note: This is applied in
Fibonacci sequences]
A. pi
B. golden ratio
C. 1.618
D. Euler’s ratio
................................................
454. Which of the following pertains to the law of
cosines?
A. c2 = a2 + b2 – 2 ab cos C
B. c2 = a2 + b2 + 2 ab cos C
C. c2 = a2 + b2 – ab cos C
D. c2 = a2 + b2 + ab cos C
................................................
455. Solve for x:
A. x = 8
B. x = 9
C. x = 10
D. x = 11
................................................
456. What do we call an angle formed by two
chords of the circle with a common endpoint
(the vertex of the angle)?
A. inscribed angle
B. tangential angle
C. circumscribed angle
D. interior angle
................................................
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1000 MMR
457. Find the inverse of y =
A. y-1 =
B. y-1 =
C. y-1 =
D. y-1 =
.
Author: Victor A. Tondo Jr., LPT
462. Which of the following expressions will give
the value of x?
Solution:
First, solve for x in terms of y.
y=
2020 y = 2018 x + 2019
2020 y – 2019 = 2018 x
=x
Then switch y and x.
=y
Lastly, replace y with
A. 50 tan 37o
C. 50 sin 37o
B. 50 cos 37o
D. 50 cot 37o
Solution:
tan 37o =
50 tan 37o = x
................................................
463. Solve for x:
y-1.
= y-1.
................................................
458. Which lines are not in the same plane and
do not intersect but are not parallel?
A. asymptotes
B. tangent lines
C. skew lines
D. directrices
................................................
459. Two adjacent angles whose distinct sides lie
on the same line are called what?
A. linear pair
B. vertical pair
C. alternate
D. corresponding
................................................
A. 6
B. 7
C. 7.5
D. 8
Solution:
94 + 86 + 94 + 11x – 2 = 360
272 + 11x = 360
11x = 360 – 72 = 88
x=8
................................................
460. The point of concurrency of a triangle’s
three altitudes is called _____.
A. circumcenter
B. incenter
C. orthocenter
D. centroid
................................................
464. A bus travels 600 km in 7 hrs and another
300 km in 5 hrs. What is its average speed?
A. 72.86 kph
B. 75 kph
C. 77.86 kph
D. 80 kph
461. What do we call three positive integers with
the property that the sum of the squares of two
of the integers equals the square of the third?
A. Euclid’s triple
B. Pythagorean triple
C. Newton’s triple
D. Cartesian triple
Average speed = total distance / total time
Average speed = (600 + 300) / (7 + 5)
Average speed = 900/12 = 75 kph
................................................
Solution:
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1000 MMR
465. A sniper on a cliff observes that the angle of
depression to his target is 30o. If the cliff is 10
meters high, how far must the bullet travel to hit
the sniper’s target?
A. 20 meters
B. 10√3 meters
C. 10√2 meters
D. 10 meters
Solution:
Author: Victor A. Tondo Jr., LPT
Solution:
This problem is on permutation with repeated
elements. PILIPINAS has nine letters, of which
two are P, three are I, and the other letters are
singular.
9!
= 30,240
2! 3!
................................................
468. A coin is tossed 60 times. Head appeared 27
times. Find the experimental probability of
getting heads.
A.
B.
C.
D.
Let trajectory = x
sin 30o =
Explanation:
0.5 =
x = . = 20
................................................
466. Rowena received a total of 25 bills. These
bills are either P20 or P50 bills. If Rowena
received an amount of P800, how many P20 bills
did she receive?
A. 10
B. 13
C. 15
D. 17
Solution:
The experimental probability of an event is the
ratio of the number of times the event occurs to
the total number of trials.
................................................
469. A parabola is defined by the equation
5x = -3y2 – 4y + 2. Which of the following is true
about the parabola?
A. It opens to the left.
B. It opens to the right.
C. It opens upward.
D. It opens downward.
Explanation:
Let x = number of P20 bills
25 – x = number of P50 bills
20(x) + 50(25 – x) = 800
20x + 1250 – 50x = 800
1250 – 800 = 50x – 20x
450 = 30x
15 = x
................................................
467. How many ways can the word PILIPINAS be
rearranged?
A. 302
B. 3,024
C. 30,240
D. 302,400
The squared variable is y and the coefficient of y2
is negative. Therefore it opens to the left.
................................................
470. Find the equation of the circle whose center
is at (7, -24) given that it passes the point of
origin.
A. (x + 7)2 – (y – 24)2 = 961
B. (x – 7)2 + (y + 24)2 = 961
C. (x + 7)2 + (y – 24)2 = 625
D. (x – 7)2 + (y + 24)2 = 625
Explanation:
Use the center-radius form. The radius is the
distance from the center (7, -24) and the point of
origin (0, 0).
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1000 MMR
471. Ana left their house and jogged at a speed of
60 meters per minute. Bea followed her two
minutes later and jogged at a speed of 70 meters
per minute. How many minutes after Bea left
would she catch up with Ana?
A. 14
B. 13.5
C. 13
D. 12
Author: Victor A. Tondo Jr., LPT
474. The shortest leg of a 30-60-90 triangle is
123 cm long. How long is the leg adjacent to the
30o angle?
A. 123 cm
B. 123√2 cm
C. 123√3 cm
D. 246 cm
Explanation:
Solution:
Let x = number of minutes after Bea left to catch
up with Ana
60 (x + 2) = 70 (x)
60x + 120 = 70x
120 = 70x – 60x
120 = 10x
12 = x
................................................
The leg adjacent to the 30o angle in a 30-60-90
triangle is always √3 times the length of the
shortest leg.
................................................
475. How many odd 4digit numbers can be
formed using the digits 7, 6, 5, 4, 3, 2, and 1 if
repetition is not allowed?
A. 720
B. 480
C. 240
D. 120
472. How many mL of 40% acid must be added
to 1000 mL of 10% acid solution to make a 20%
acid solution?
A. 250
B. 500
C. 600
D. 750
Solution:
Solution:
476. There are 70 dogs and geese in a farm. If
there are a total of 200 legs, how many dogs are
there?
A. 25
B. 30
C. 35
D. 40
Let x = amount to be added in mL
40 (x) + 10 (1000) = 20 (x + 1000)
40x + 10,000 = 20x + 20,000
40x – 20x = 20,000 – 10,000
20x = 10,000
x = 500
................................................
473. The hypotenuse of a 30-60-90 triangle is
432 cm long. How long is the leg opposite the 30o
angle?
A. 216 cm
B. 216 √2 cm
C. 216 √3 cm
D. 432 cm
Explanation:
The shorter leg in a 30-60-90 triangle is always
half the length of the hypotenuse.
................................................
6 x 5 x 4 x 4 = 480
................................................
Solution:
Leg D = number of dogs
G = number of geese
D + G = 70
4D + 2G = 200
2D + 2G = 140
4D + 2G = 200
-2D
= -60
D = 30
................................................


477. Find the maximum area of a rectangle if the
perimeter is set at 350 cm.
A. 8656.25 cm2
B. 7656.25 cm2
2
C. 6656.25 cm
D. 5656.25 cm2
Shortcut:
Make it a square.
P = 4S = 350
S = 87.5
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.: A = S2 = 87.52
A = 7656.25
1000 MMR
478. A rectangle is 60 cm long and 45 cm wide.
How long is its diagonal?
A. 75 cm
B. 85 cm
C. 95 cm
D. 105 cm
Solution:
The diagonal D is given as √
+
.
= √45 + 60 = √2025 + 3600
D = √5625 = 75
................................................
479. Find the length of the diagonal of a cube
given each side measures 17 cm.
A. √290 cm
B. 17√2 cm
C. 17√3 cm
D. 34 cm
Solution:
In a cube, the diagonal D is √3 times the measure
of each side.
................................................
480. Find the measure of each interior angle of a
regular 20-sided polygon.
A. 162o
B. 150o
C. 144o
D. 126o
Explanation:
MIA = 180 where n = number of sides of
regular polygon
................................................
481. How many diagonals does a regular 14sided polygon have?
A. 77
B. 91
C. 96
D. 101
Solution:
(
)
(
)
D=
=
= 77
................................................
482. How many ways can 14 people be seated in
a Ferris wheel given that each cart can only
contain one person?
A. 15!
B. 14!
C. 13!
D. 12!
Author: Victor A. Tondo Jr., LPT
Explanation:
This problem is on circular permutations so use
(n – 1)!.
................................................
483. There are 24 mangoes in a basket, of which
7 are rotten. What is the probability that when
randomly getting two mangoes at the same time,
both are rotten?
A.
B.
C.
D.
Solution:
=
................................................
484. In a gathering of gamers and admins, there
are 24 gamers of which 6 are females, and 3
admins of which one is female. If a female is
randomly called, what is the probability that she
is an admin?
A.
B.
C.
D.
Explanation:
There are only seven females, of which one is an
admin.
................................................
485. What is the remainder when 3x6+ 4x5 – 5x4
+ 6x3 + 7x2 – 8x + 3 is divided by (x – 1)?
A. 8
B. 9
C. 10
D. 11
Solution:
Use the Remainder Theorem:
3(1)6+ 4(1)5 – 5(1)4 + 6(1)3 + 7(1)2 – 8(1) + 3
= 3 + 4 – 5 + 6 + 7 – 8 + 3 = 10
................................................
486. Seven people have an average weight of 49
kg. A child was added to the group and the
average became 45 kg. How heavy is the child?
A. 15 kg
B. 16 kg
C. 17 kg
D. 18 kg
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1000 MMR
Solution:
Author: Victor A. Tondo Jr., LPT
Solution:
Let N = child’s weight
( )
45 =
The diameter of the sphere is of the same length
as each side of the cube. That means the
diameter of the sphere is 20 cm, and
subsequently, the radius is 10 cm.
SA = 4 r2
SA = 4 (102) = 400 cm2
................................................
45 =
360 = 343 + N
17 = N
................................................
487. Six numbers have an average of 71. If 85 is
added to the group, what is the new average?
A. 72
B. 73
C. 74
D. 75
Solution:
Sum of the first six numbers: 6(71) = 426
New average =
=
= 73
................................................
488. In an arithmetic sequence, the 7th term is 25
and the 10th term is 67. What is the common
difference?
A. 42
B. 21
C. 14
D. 7
491. Given f(x) = (25 x20 – 24x10)(x2 – 9x + 3),
find f ’(x).
A. f ‘(x) = 550 x21 – 4725 x20 + 1500 x19 – 288 x11
+ 2376 x10 – 720 x9
B. f ‘(x) = 450 x21 – 4725 x20 + 150 x19 + 288 x11
+ 2376 x10 – 720 x9
C. f ‘(x) = 550 x21 – 4725 x20 + 150 x19 – 288 x11
+ 2376 x10 – 720 x9
D. f ‘(x) = 450 x21 – 4725 x20 + 1500 x19 – 288 x11
+ 2376 x10 – 720 x9
Solution:
Use the product rule.
Solution:
Let u = (25 x20 – 24x10)
and v = (x2 – 9x + 3)
D=
=
= = 14
................................................
.: du = 500 x19 – 240 x9
dv = 2x – 9
489. Triangle ABC has sides measuring 20 cm, 20
cm, and 29 cm. What kind of triangle is ABC?
A. acute
B. right
C. obtuse
D. reflex
Explanation:
For any triangle with sides X, Y, and Z, given
that X Y Z, if X2 + Y2 > Z2, then the triangle is
obtuse.
................................................
490. Find the surface area of a sphere given that
the sphere sits perfectly inside a cube whose
sides measure 20 cm each.
A. 400 cm2
B. 800 cm2
C. 1200 cm2
D. 2400 cm2
f ‘(x) = u dv + v du
= (25 x20 – 24x10) (2x – 9) + (x2 – 9x + 3)
(500 x19 – 240 x9)
= 50 x21 – 48 x11 – 225 x20 + 216 x10 +
500 x21 – 4500 x20 + 1500 x19 – 240 x11 +
2160 x10 – 720 x9
= 550 x21 – 4725 x20 + 1500 x19 – 288 x11
+ 2376 x10 – 720 x9
Alternative Solution:
f(x) = (25 x20 – 24x10)(x2 – 9x + 3)
= 25x22 – 225x21 + 75x20 – 24x12 + 216 x11
– 72 x10
f ‘(x) = 550 x21 – 4725 x20 + 1500 x19 – 288 x11
+ 2376 x10 – 720 x9
................................................
This is a free reviewer. All rights reserved.
1000 MMR
492. Simplify: eln 2019 x
A. 2019x
B. 2019 x
C.
x
D.
Author: Victor A. Tondo Jr., LPT
Solution:
Explanation:
496. Find the measure of the smaller angle
formed by the hands of the clock at 11:20.
A. 130o
B. 135o
C. 140o
D. 145o
7C2 x 6C2 = 21 x 15 = 315
................................................
Remember that eln u = u.
In the question, u = 2019 x.
Therefore, eln 2019 x = 2019 x
................................................
493. If the roots of a quadratic equation are
and , which of the following could be the
quadratic equation?
A. 63x2 + 22x – 21 = 0
B. 63x2 – 22x – 21 = 0
C. 63x2 + 22x + 21 = 0
D. 63x2 – 22x + 21 = 0
Solution:
x=
9x = -7
9x + 7 = 0
Solution:
A = 30h -
m = 30(11) - (20)
= 330 – 110
= 220
Since the angle taken from our formula is a reflex
angle and we are looking for the smaller angle,
then 360 – 220 = 140.
................................................
497. How many even 3-digit even numbers can
be formed using the digits 7, 6, 5, 4, 3, 2, 1, and 0
if repetition is not allowed?
A. 150
B. 160
C. 170
D. 180
x=
7x = 3
7x – 3 = 0
(9x + 7) (7x – 3) = 0
63x2 + 49x – 27x – 21 = 0
63x2 + 22x – 21 = 0
................................................
494. If three more than twice a number is
seventeen less than seven times the number,
what is the number?
A. 2
B. 3
C. 4
D. 5
Solution:
2x + 3 = 7x – 17
3 + 17 = 7x – 2x
20 = 5x
4=x
................................................
495. A team is to be made from a group of seven
teachers and six scientists. If the team is to be
composed two teachers and two scientists, how
many different ways can they form a team?
A. 325
B. 315
C. 300
D. 285
Solution:
Number of 3-digit numbers:
7 x 7 x 6 = 294
Number of 3-digit odd numbers:
6 x 6 x 4 = 144
.: Number of 3-digit even numbers:
294 – 144 = 150
................................................
498. There are 50 students in a class. Twenty of
them have a laptop. Thirty-two of them have a
smartphone. Seven of them have both a laptop
and a smartphone. How many of them have
neither a laptop nor a smartphone?
A. 4
B. 5
C. 6
D. 7
Solution:
50 – (20 + 32 – 7) = 50 – 45 = 5
This is a free reviewer. All rights reserved.
1000 MMR
499. Mocha can finish a job in 24 hours, while
her sister Tiramisu can do the same job in only
20 hours. How long will it take them to finish the
job by working together?
A.
hrs
B.
hrs
C. 11 hrs
D.
hrs
Solution:
( )
=
=
=
hrs
................................................
500. What conic figure does the equation
x2 + y2 + 10x – 16y = -100 form?
A. Real circle
B. Degenerate circle
C. Imaginary circle D. Ellipse
Solution:
Use CTS (completing trinomial squares) to
convert the equation to its center-radius form.
x2 + y2 + 10x – 16y = -100
x2 + 10x
+ y2 – 16y = -100
(x2 + 10x + 25) + (y2 – 16y + 64) = -100 + 25 +
64
(x + 5)2 + (y – 8)2 = -11
Since r2 = -11, then the radius is imaginary,
making it an imaginary circle.
End of first 500 items.
Author: Victor A. Tondo Jr., LPT
What is the difference between
a function and a relation?
“Sa function, loyal si x.
In a relation, kiber
kung ilang beses nagamit si x.”
What are gametes?
Gametes are sex cells.
Ako nga, ilang beses na akong
na-gamete eh.
There are only 10 kinds of people:
Those who know binary
And those who don’t.
Bakit daw nagalit ang mga Pinoy
nung namatay si Rizal?
Kasi wala siyang pang-buyback.
Mabilis ako kumuha ng square root
ng malalaking numbers.
Ayaw mo maniwala?
Akin na cellphone number mo,
kukunin ko square root.
Mr. Lazada bought 200 cars.
Please, take a break.
OMG ang yaman ni Mr. Lazada!
Remember:
Nabutas ang isang water tank.
Normies: OMG! Sayang ang tubig!
Math major: *computes the rate
at which the tank is being emptied*
Nasa Diyos ang awa,
Nasa tao ang gawa.
This is a free reviewer. All rights reserved.
1000 MMR
501. Victor, Praetor, and Rowena volunteered to
teach at a nearby daycare. Praetor worked for
twice as long as Rowena did. Victor worked
twice as many hours as Praetor. Altogether, they
worked for 56 hours. For how many hours did
Victor work?
A. 14
B. 16
C. 28
D. 32
Solution:
Rowena = n hours
.: Praetor = 2n hours
.: Victor = 2(2n) hours
n + 2n + 4n = 56
7n = 56
n=8
.: Victor worked for 2(2(8)) or 32 hours.
................................................
502. Which of the following is the factorization of
the binomial x4 – 20194?
A. (x + 2019)2 (x – 2019)2
B. (x – 2019)4
C. (x2 + 20192) (x + 2019) (x – 2019)
D. (x – 4)(x + 4)
Solution:
x4 – 20194 as a difference of squares
= (x2)2 – (20192)2
= (x2 + 20192) (x2 – 20192)
= (x2 + 20192) (x + 2019) (x – 2019)
................................................
503. The average of 5 different counting
numbers is 143. What is the highest possible
value that one of the numbers can have?
A. 706
B. 705
C. 704
D. 703
Author: Victor A. Tondo Jr., LPT
504. What value of x will satisfy the equation:
0.25(20x – 2020) = x?
A. 125.25
B. 125.75
C. 126.25
D. 126.5
Solution:
0.25(20x – 2020) = x
5x – 505 = x
5x – x = 505
4x = 505
x = 156.25
................................................
505. Three brothers inherited a cash amount of
P5,670,000 and they divided it among
themselves in the ratio of 2:3:4. How much more
is the largest share than the smallest share?
A. P1,260,000
B. P1,270,000
C. P630,000
D. P635,000
Solution:
Let the three numbers 2x, 3x, and 4x so that the
ratio will still be 2:3:4.
2x + 3x + 4x = 5,670,000
9x = 5,670,000
x = 630,000
Difference: 4x – 2x = 2x
2x = 2(630,000) = 1,260,000
................................................
506. Rayon and Wena can do a job together in
four hours. Working alone, Rayon does the job in
six hours. How long will it take Wena to do the
job alone?
A. 12 hours
B. 11 hours
C. 10 hours
D. 9 hours
Solution:
Solution:
The 5 different counting numbers will assume
the values of 1, 2, 3, 4, and N. Since the average is
143, the sum is 5(143) or 715.
1+2+3+4+N = 715
10 + N = 715
N = 705
Formula for “working together”: AB/(A+B)
AB/(A+B) = 4; but A = 6
6B/(6+B) = 4
6B = 24 + 4B
2B = 24
B = 12
This is a free reviewer. All rights reserved.
1000 MMR
507. What is the sum of the first 2019 counting
numbers?
A. 2,021,019
B. 2,027,190
C. 2,039,190
D. 2,043,190
Solution:
Sum of the first N counting numbers =
Sum of the first 2019 counting numbers:
=
= 2,039,190
................................................
508. In a certain school, the ratio of boys to girls
is 4 is to 9. If there are 260 boys and girls in the
school, how many boys are there?
A. 100
B. 90
C. 85
D. 80
Solution:
Let 4x = boys, 9x = girls
4x + 9x = 13x = 260
.: x = 20
Number of boys = 4x = 4(20) = 80
................................................
Mr. Park
Son
Age Now
Age 5 years
from now
x + 20
x
x + 20 + 5
x+5
x + 20 + 5 = 2(x + 5) – 5
x + 25 = 2x + 5
25 – 5 = 2x – x
20 = x
................................................
510. What is the minimum value of
f(x) = 3x2 + 6x + 11?
A. 1
B. 2
C. 4
D. 8
Solution:
Minimum Value = c – b2/4a
11 – 36/12 or 11 – 3 = 8
................................................
511. What is the sum of the roots of
5x2 + 10x – 17?
A.
B.
C. 2
D. -2
Solution:
509. Solve for x: 8x + 9y = 17
9x + 10y = 18
A. x = 9
C. x = -9
Author: Victor A. Tondo Jr., LPT
Solution:
Sum of roots = =
= -2
................................................
B. x = 8
D. x = -8
Solution:
10 (8x + 9y = 17) 
9 (9x + 10y = 18) 
80x + 90y = 170
(-) 81x + 90y = 162
-x
=8
x = -8
................................................
510. Mr. Park is 20 years older than his son now.
Five years from now, the Mr. Park’s age will be 5
less than twice his son’s age. Find the son’s
present age.
A. 18
B. 20
C. 38
D. 40
512. If xy = 17 and x2 + y2 = 135, find x + y.
A. 13
B. 12.8749
C. 12.5249
D. 12.5
Solution:
x2 + 2xy +y2 = x2 + y2 + 2xy
x2 + 2xy +y2 = 133 + 2(17)
x2 + 2xy +y2 = 169
x + y = 13
................................................
513. How many mL of 50% acid solution must be
added to 200mL of 10% acid solution to make a
40% acid solution?
A. 400 mL
B. 500 mL
C. 600 mL
D. 700 mL
This is a free reviewer. All rights reserved.
1000 MMR
Solution:
C1 V1 + C2 V2
= CR VR
50(x) + 10(200) = 40(x + 200)
50x + 2000 = 40x + 8000
50x – 40x = 8000 – 2000
10x = 6000
x = 600
................................................
514. Nine consecutive even numbers have a sum
of 180. What is the smallest of these even
numbers?
A. 18
B. 16
C. 14
D. 12
Solution:
Let x = lowest even number
x + (x+2) + (x+4) + (x+6) + (x+8) + (x+10)
+ (x+12) + (x+14) + (x+16) = 180
9x + 72 = 180
9x = 108
x = 12
................................................
515. If x = 13, which of the following is equal to
200?
A. 15x + 2
B. x2 + 2x + 5
3
C. x – 4x – 2
D. x2 + x + 2
Explanation:
Just substitute x with 13.
A. 15(13) + 2 = 197
B. (13)2 + 2(13) + 5 = 200
C. (13)3 – 4(13) – 2 = 2143
D. (13)2 + (13) + 2 = 184
................................................
516. For which value of k does 9x2 – kx + 25
have only one root?
A. 3.75
B. 7.5
C. 15
D. 30
Explanation:
Author: Victor A. Tondo Jr., LPT
517. If A and B are the roots of x2 + 19x + 20,
what is AB?
A. 20
B. 2√5 + 5
C. 3√2 + 2√3
D. -19
Explanation:
Since A and B are the roots, then AB pertains to
the product of the roots which is given as c/a.
................................................
518. 0.25 + 0.5 + 1 + 2 + 4 + … + 1024 = ____
A. 2046.75
B. 2047.75
C. 2048.75
D. 2049.75
Solution:
You may use the Geometric Series formula which
is ∑
= (
), where r is the common
ratio, n is the number of terms, and a1 is the first
term.
Alternative Solution:
Use the shortcut for the sum of a geometric
sequence wherein the ratio is 2 or ½, which is
SUM = 2(largest) – smallest.
In this problem, that’s 2(1024) – 0.25 = 2047.75.
................................................
519. How many terms are there in the sequence
2, 9, 16, 23, …, 345?
A. 40
B. 45
C. 50
D. 70
Solution:
An = A1 + (n – 1)d
345 = 2 + (n – 1)(7)
345 – 2 = 7(n – 1)
343 = 7n – 7
350 = 7n
50 = n
................................................
520. If 2x + 3 = 25, then what is (2x + 3)2 – 25?
A. 650
B. 625
C. 600
D. 575
b2 – 4ac = 0 when there’s only one root.
k2 – 4(9)(25) = 0
k2 = 900
k = 30
This is a free reviewer. All rights reserved.
1000 MMR
Solution:
Author: Victor A. Tondo Jr., LPT
Solution:
(2x + 3)2 – 25 = 252 – 25 = 625 – 25 = 600
................................................
C1 V1 + C2 V2 = CR VR
10 (V) + 40 (500) = 30 (V + 500)
10V + 20,000 = 30V + 15,000
20,000 – 15,000 = 30V – 10V
5,000 = 20V
250 = V
................................................
521. What is 50% of 200% of 2019?
A. 1009.5
B. 2019
C. 4038
D. 8076
Solution:
(0.5)(2)(2019) = 2019
................................................
522. If 3x = 5 and 2y = 11, what is 12(x – y)?
A. -46
B. -45
C. 45
D. 46
525. 3log2 3 + 2 log2 5 – log2 7 = ______.
A. log2
B. log2
C. log2
D. log2
Explanation:
Solution:
Apply the laws of logarithms.
12(x – y) = 12x – 12y
= 4(3x) – 6(2y)
= 4(5) – 6(11)
= 20 – 66
= -46
................................................
3log2 3 + 2 log2 5 – log2 7 = log2
= log2
................................................
523. If two numbers have a product of 267 and
the sum of their squares is 250, what is their
sum?
A. -30√3
B. 28
C. 7√3 + 2√5
D. 30
Solution:
Let A and B be our two numbers.
AB = 267;
A2 + B2 = 250
.: A2 + B2 + 2AB = 250 + 2(267)
= 784
(A + B)2 = 784;
A + B = ±28
................................................
524. How many ml of 10% acid must be added to
500 ml of 40% acid to make a 30% acid solution?
A. 1000 ml
B. 750 ml
C. 500 ml
D. 250 ml
(
)(
)
526. Find the intersection of y = 3x + 4 and
y = 5x – 8.
A. (8/5, 0)
B. (-4/3, 0)
C. (1, 7)
D. (6, 22)
Solution:
y = 3x + 4
- y = 5x – 8
0 = -2x + 12
2x = 12
x=6
y = 3x + 4
= 3(6) + 4
= 22
................................................
527. Find the remainder when
x5 – 3x3 + 5x2 – 7x + 9 is divided by (x – 1).
A. 11
B. 9
C. 7
D. 5
Solution:
Use the Remainder Theorem.
14 – 3(13) + 5(12) – 7(1) + 9 = r
1–3+5–7+9=r
5=r
This is a free reviewer. All rights reserved.
1000 MMR
528. Multiply: (3x – 7) (5x + 9)
A. 15x2 – 8x – 63
B. 15x2 – 8x + 63
2
C. 15x + 8x – 63
D. 15x2 + 8x + 63
Solution:
3x
-7
times
5x
+9
27x
-63
15x2
-35x
15x2
-8x
-63
................................................
529. Multiply: (2x2 – 5x + 3) (3x + 4)
A. 6x3 – 7x2 – 11x + 12
B. 6x3 + 7x2 + 11x + 12
C. 6x3 – 7x2 + 11x + 12
D. 6x3 + 7x2 – 11x + 12
Solution:
2x2
530. x varies directly as y and inversely as z. If
x = 40 when y = 8 and z = 2, what is x when
y = 24 and z = 4?
A. 120
B. 90
C. 60
D. 30
Solution:
x = ky/z
Let x = number of P5 coins
.: x+7 = number of P10 coins
5(x) + 10(x+7) = 475
5x + 10x + 70 = 475
15x + 70 = 475
15x = 405
x = 27
.: He has twenty-seven P5 coins.
................................................
532. When a number is increased by 5, its square
also increases by 255. What is this number?
A. 22
B. 23
C. 24
D. 25
Solution:
(x+)2 – x2 = 255
+ 10x + 25 – x2 = 255
10x + 25 = 255
10x = 230
x = 23
................................................
x2
-5x
+3
3x
+4
2
8x
-20x
+12
6x3
-15x2
+9x
6x3
-7x2
-11x
+12
................................................
times
Author: Victor A. Tondo Jr., LPT
Solution:
40 = k(8)/2
40 = 4k
10 = k
x = 10y/z
x = 10(24)/4
x = 240/4
x = 60
................................................
531. Pedro has 7 more P10 coins than P5 coins. If
he has a total of P475, how many P5 coins does
he have?
A. 23
B. 25
C. 27
D. 29
533. Find k such that 16x2 + kx + 81 is a perfect
square trinomial.
A. 144
B. 72
C. 36
D. 18
Solution:
16x2 + kx + 81 = (4x)2 + 2(4x)(9) + 92
= (4x)2 + 72x + 92
................................................
534. Solve for x: (x + 4)2 = (x – 6)2.
A. x = 0
B. x = ½
C. x = 1
D. no solution
Solution:
(x + 4)2 = (x – 6)2
x2 + 8x + 16 = x2 – 12x + 36
8x + 12x = 36 - 16
20x = 20
x=1
................................................
This is a free reviewer. All rights reserved.
1000 MMR
535. When a number is increased by 3, its square
increases by 135. By what does its square
increase when the number is increased by 5?
A. 225
B. 235
C. 445
D. 485
Solution:
Let’s find the original number first.
(x+3)2 – x2 = 135
2
x + 6x + 9 – x2 = 135
6x + 9 = 135
6x = 126
x = 21
212
(21+5)2
262
= 441;
=
= 676
676 – 441 = 235
................................................
536. Ten cans of soda and six hamburgers cost a
total of P440. Five cans of soda and seven
hamburgers cost a total of P360. How much is a
hamburger?
A. P31
B. P33
C. P35
D. P37
Solution:
Let
C = price of a can of soda
B = price of a hamburger
10C + 6B = 440
2(5C + 7B) = 360

10C + 6B = 440
 (-) 10C + 14B = 720
-8B = -280
B = 35
................................................
537. The product of two consecutive odd
numbers is 1763. What is the smaller number?
A. 31
B. 37
C. 41
D. 47
Solution:
Let x = smaller number;
x + 2 = larger number
x (x+2) = 1763
x2 + 2x = 1763
2
x + 2x + 1 = 1764
√x + 2x + 1 = √1764
x+1 = 42
x = 41
Author: Victor A. Tondo Jr., LPT
538. Find the remainder when the polynomial
x4 – 5x3 + 7x2 – 11x + 19 is divided by (x – 2).
A. 5
B. 3
C. 1
D. -1
Solution:
Use the Remainder Theorem.
Our divisor is (x – 2), so a = 2.
f(2) = 24 – 5(23) + 7(22) – 11(2) + 19
f(2) = 16 – 40 + 28 – 22 + 19 = 1
................................................
539. If three-fourths of a number is 44 more than
its one-fifth, what is that number?
A. 120
B. 100
C.80
D. 60
Solution:
x=
x + 44
20 ( x =
x + 44) 
15x = 4x + 880
11x = 880
x = 80
................................................
540. Rowena can do a job in 15 hours, while
Victor can do the same job in 25 hours. How long
will it take them to finish the job by working
together?
A. 9.375 hours
B. 9.25 hours
C. 9.125 hours
D. 8.875 hours
Solution:
( )
=
=
= 9.375
................................................
541. Armel is twice as heavy as Kuku. Gabbi is
17kg heavier than Kuku. The sum of their masses
is 217kg. How heavy is Kuku in kg?
A. 40
B. 45
C. 50
D. 55
Solution:
A = 2K
G = K + 17
A + G + K = 217
2K + (K + 17) + K = 217
4K + 17 = 217
4K = 200
K = 50
This is a free reviewer. All rights reserved.
1000 MMR
542. Find + given x + y = 30 and xy = 219.
Author: Victor A. Tondo Jr., LPT
Solution:
A.
13x + 17 = 15x – 21
17 + 21 = 15x – 13x
38 = 2x
19 = x
................................................
B.
C.
D.
Solution:
+ =
+
=
=
=
................................................
543. The product of two consecutive odd
counting numbers is 1023. What is their sum?
A. 60
B. 64
C. 68
D. 72
Solution:
Let x = first number;
√
546. Factorize: x3 + 3x2 – 4x – 12
A. (x + 2) (x + 3) (x – 2)
B. (x + 4) (x + 3) (x – 1)
C. (x + 2) (x – 3) (x + 2)
D. (x + 4) (x – 3) (x + 1)
Solution:
x+2 = next number
x(x+2) = 1023
x2 + 2x = 1023
x2 + 2x + 1 = 1023 + 1
+ 2x + 1 = √1024
x+1 = 32
Therefore x = 31 and x+2 = 33
The numbers are 31 and 33, so their sum is 64.
................................................
544. There are 250 pigs and chickens in a farm,
all of which are healthy. If there are 720 legs in
total, how many pigs are there?
A. 110
B. 100
C. 90
D. 80
Solution:
Let P = number of pigs
C = number of chickens
P + C = 250
4P + 2C = 720
 since pigs have four legs
and chickens have two
2(P + C = 250)

2P + 2C = 500
4P + 2C = 720
- 4P + 2C = 720
-2P
= -220
P = 110
................................................
545. Solve for x given 13x + 17 = 15x – 21.
A. x = 17
B. x = 18
C. x = 19
D. x = 20
Factorize by grouping.
x3 + 3x2 – 4x – 12
= (x3 + 3x2) – (4x + 12)
= x2 (x + 3) – 4 (x + 3)
= (x2 – 4) (x + 3)
= (x + 2) (x – 2) (x + 3)
................................................
547. Factorize (x4 – 16) completely.
A. (x – 2)4
B. (x – 2)2 (x + 2)2
C. (x + 2)2 (x – 2)2
D. (x + 2) (x – 2) (x2 + 4)
Solution:
(x4 – 16) = (x2 – 4) (x2 + 4)
= (x + 2) (x – 2) (x2 + 4)
................................................
548. √125 + √45
A. 6√5
C. 2√5
√245 = ____
B. 4√5
D. √5
Solution:
√125 + √45 √245 = 5√5 + 3√5 7√5 = √5
................................................
549. The product of two consecutive even
counting numbers is 3968. Find the smaller
number.
A. 42
B. 46
C. 54
D. 62
This is a free reviewer. All rights reserved.
1000 MMR
Solution:
Author: Victor A. Tondo Jr., LPT
Solution:
x(x+2) = 3968
x2 + 2x = 3968
2
x + 2x + 1 = 3968
√ + 2 + 1 = √3969
x + 1 = 63
x = 62
................................................
Seesaw problems call for inverse or indirect
proportion.
48(160) = 60N
7680 = 60N
128 = N
................................................
550. A pipe can fill a pool in 5 hours while
another pipe can drain empty the pool in 10
hours. How long will it take to fill the pool if both
pipes are open?
A. 8 hours
B. 9.25 hours
C. 9.5 hours
D. 10 hours
Solution:
This is similar to our “Working Together”
problem, except instead of adding their times, we
will subtract (since the draining pipe is doing the
opposite of helping).
AB/(A-B) = 10(5)/(10-5) = 50/5 = 10 hrs
................................................
551. Today, Rayon is 13 years old while his
father is thrice his age. How many years from
now will his father be twice as old as he?
A. 15
B. 13
C. 11
D. 10
Solution:
Let x = number of years from now
2(13+x) = 39+x
26 + 2x = 39 + x
2x – x = 39 – 26
x = 13
................................................
552. Roy and Wena are on a seesaw. Roy weighs
48 kg and sits 160 cm to the left of the fulcrum. If
Wena weighs 60 kg, how far to the right of the
fulcrum must she sit to balance the seesaw?
A. 115 cm
B. 123 cm
C. 128 cm
D. 135 cm
553. Insert one term between 81 and 169 to
make a geometric sequence.
A. 113
B. 117
C. 121
D. 125
Solution:
Shortcut for inserting one term is √AB. This is
also the formula for the geometric mean.
√81(169) = 9(13) = 117
................................................
554. Find the common difference of an
arithmetic sequence whose 21st term is 1987 and
29th term is 2019.
A. 2.5
B. 3
C. 3.5
D. 4
Solution:
d=
=
= =4
................................................
555. Find the general term An of the sequence
7, 16, 25, 34, 43, 52, 61, …
A. An = 9n – 2
B. An = 8n – 1
C. An = 9n + 2
D. An = 8n + 1
Solution:
An = dn – (d – A1)
= 9n – (9 – 7)
= 9n – 2
................................................
556. Find the remainder when x4 – 4x3 + 3x2 +
5x – 6 is divided by (x – 2).
A. 3
B. 2
C. 1
D. 0
This is a free reviewer. All rights reserved.
1000 MMR
Solution:
Use the remainder theorem.
x – a = x – 2  That means a = 2
Remainder = f(a) = f(2)
= 24 – 4(23) + 3(22) + 5(2) – 6
= 16 – 32 + 12 + 10 – 6
................................................
557. 2 + 9 + 16 + 23 + … + 135 = ____
A. 1353
B. 1360
C. 1370
D. 1377
Solution:
n=
n=
Sum =
+1
+ 1 = 20
Author: Victor A. Tondo Jr., LPT
560. Factorize 81x2 – 225y2 completely.
A. (9x + 15y)(9x – 15y)
B. 9(9x2 – 25y2)
C. 9(3x – 5y)2
D. 9(3x + 5y)(3x – 5y)
Solution:
81x2 – 225y2 = 9(9x2 – 25y2)
= 9(3x + 5y)(3x – 5y)
................................................
561. What are the missing terms in the series
3, 6, 12, 24, ___, 96, ____?
A. 48; 192
B. 36, 120
C. 46, 196
D. 46, 192
Solution:
(n)
Sum =
(20) = 1370
................................................
558. Factorize: (2x + y) (2x + y + 3) + 2.
A. (2x + y + 1) (2x + y + 2)
B. (2x + y + 1) (2x + y + 6)
C. (2x + y + 2) (2x + y + 3)
D. (2x + y + 3) (2x + y + 4)
562. In a parking lot, there are 65 tricycles and
motorcycles. If there are 150 wheels in all, how
many tricycles are there?
A. 18
B. 19
C. 20
D. 21
Solution:
T + M = 65
3T + 2M = 150
Solution:
Let 2x + y = A
(2x + y) (2x + y + 3) + 2
Since the common ratio is 2, then next terms
should be 24(2) and 96(2), or 48 and 192.
................................................
= A (A + 3) + 2
= A2 + 3A + 2
= (A + 2)(A + 1)
Substitute A with 2x + y and you’ll get
(2x + y + 2) (2x + y + 1).
................................................
 2T + 2M = 130
 3T + 2M = 150
T = 20
................................................
563. Which fraction is equivalent to 0.425?
A. 21/50
B. 27/50
C. 27/40
D. 17/40
Explanation:
559. The area of a rectangle is (x2 – x – 30). If its
length is x + 5, what is its width?
A. x + 4
B. x – 3
C. x + 2
D. x – 6
0.425 is read as 425 thousandths. In fraction
form, that’s
or in its simplest form, .
Solution:
Just input 0.425 and press your scientific
calculator’s S D button.
x2 – x – 30 = (x + 5) W
(x + 5) (x – 6) = (x + 5) W
x–6=W
Alternative Solution:
This is a free reviewer. All rights reserved.
1000 MMR
564. What are the zeroes of 6x2 – x – 35?
A. and
B. and
Author: Victor A. Tondo Jr., LPT
Solution:
Age 10 yrs
from now
4x + 10
x + 10
Person
Age Now
Solution:
Mr. Yu
Daughter
4x
x
6x2 – x – 35 = 0
(3x + 7) (2x – 5) = 0
3x + 7 = 0
2x – 5 = 0
x=
x=
................................................
4x + 10 = 3(x + 10)
4x + 10 = 3x + 30
4x – 3x = 30 – 10
x = 20
................................................
565. Seven more than four times a number is 79.
What is five less than three times the number?
A. 59
B. 49
C. 39
D. 29
568. The speed of the current of a river is 9 kph.
Rayon rows his boat upstream for 8 hours, and
then travels back to his original position by
rowing downstream for 4 hours. What is Rayon’s
speed on calm water?
A. 9 kph
B. 18 kph
C. 27 kph
D. 36 kph
C. and
D. and
Solution:
4x + 7 = 79
4x = 72
x = 18
3x – 5 = 3(18) – 5 = 49
................................................
566. Mr. Park is five times as old as his son. Eight
years from now, he will only be thrice as old as
his son. How old is Mr. Park now?
A. 30
B. 35
C. 40
D. 45
Solution:
Person
Age Now
Mr. Park
Son
5x
x
Age 8 yrs
from now
5x + 8
x+8
5x + 8 = 3(x + 8)
5x + 8 = 3x + 24
5x – 3x = 24 – 8
2x = 16
x=8
5x = 40
................................................
567. Mr. Yu is four times as old as his daughter.
Ten years from now, he will only be thrice as old
as her. How old is his daughter now?
A. 10
B. 15
C. 20
D. 25
Solution:
8(x – 9) = 4(x + 9)
8x – 72 = 4x + 36
8x – 4x = 36 + 72
4x = 108
x = 27
................................................
569. Thrice Rayon’s age 16 years ago is equal to
his age 16 years from now. How old is Rayon
now?
A. 16
B. 24
C. 32
D. 40
Solution:
3(x – 16) = x + 16
3x – 48 = x + 16
3x – x = 16 + 48
2x = 64
x = 32
................................................
570. Divide 8x5 – 4x3 + 2x2 – 3x + 4 by x + 2.
A. 8x4 – 16x3 + 28x2 – 54x + 105 r. -206
B. 8x4 + 16x3 + 28x2 – 54x + 105 r. -206
C. 8x4 – 16x3 + 28x2 – 54x – 105 r. 206
D. 8x4 + 16x3 + 28x2 – 54x – 105 r. 206
This is a free reviewer. All rights reserved.
1000 MMR
Solution:
8x
x+
2
-16x3
+28x
4
2
8x
-4x3
-54x
+2x2
+10
5
-3x
Author: Victor A. Tondo Jr., LPT
573. In the equation 2x + 5y = 50, what is x
when y is 8?
A. 2
B. 3
C. 4
D. 5
+4
5
8x
+16x
5
4
-16x4
-16x4
-4x3
-32x3
28x3
28x3
Solution:
2x + 5(8) = 50
2x + 40 = 50
2x = 50 – 40
2x = 10
x=5
................................................
+2x2
+56x
2
-54x2
-54x2
-3x
108x
105x
105x
Alternative: Use Synthetic division.
-2
8
0
-4
2
+4
+21
0
-206
-3
4
-16
32
-56 108
210
8
-16
28
-54 105
206
................................................
571. Factorize 27x3 + 125.
A. (3x + 5)(9x2 + 15x + 25)
B. (3x + 5)(9x2 – 15x + 25)
C. (3x – 5)(9x2 + 15x + 25)
D. (3x – 5)(9x2 – 15x + 25)
Solution:
X = kY/Z
X = 4Y/Z
24 = 30k/5
X = 4(36)/9
24 = 6k
X = 144/9
4=k
X= 16
................................................
575. What is the 4th term of (3x – 2)5?
A. 360x2
B. 240x2
2
C. -240x
D. -360x2
Solution:
5C3 (3x)2 (-2)3 = 5 (9x2) (-8)
= -360x2
................................................
Explanation:
576. Factorize 64x6 – y6.
A3 + B3 = (A + B)(A2 – AB + B2)
................................................
572. What is the 5th term of (2x + 3)7?
A. 22680x3
B. 11340x3
C. 21600x3
D. 10800x3
Solution:
574. X varies directly as Y and inversely as Z. If X
is 24 when Y is 30 and Z is 5, find X when Y is 36
and Z is 4.
A. 36
B. 32
C. 24
D. 16
A. (2x – y) (4x2 + 2xy + y2) (2x + y) (4x2 – 2xy + y2)
B. (2x – y) (4x2 + 2xy + y2) (2x + y) (4x2 + 2xy + y2)
C. (2x + y) (4x2 – 2xy – y2) (2x + y) (4x2 – 2xy + y2)
D. (2x + y) (4x2 – 2xy – y2) (2x + y) (4x2 – 2xy + y2)
Solution:
64x6 – y6 = (8x3 – y3)(8x3 + y3)
= (2x – y) (4x2 + 2xy + y2) (2x + y) (4x2 – 2xy + y2)
(7C4)(2x)3(3)4 = 35 (8x3) (81)
= 22680x3
This is a free reviewer. All rights reserved.
1000 MMR
577. Factorize x2 + 6x – y2 + 2y +8.
A. (x + y + 2) (x + y + 4)
B. (x + y + 2) (x – y + 4)
C. (x + y + 2) (x + y – 4)
D. (x + y + 2) (x – y – 4)
Solution:
x2 + 6x – y2 + 2y +8
= x2 + 6x – (y2 – 2y) +8
= x2 + 6x +9 – (y2 – 2y + 1)
= (x + 3)2 – (y – 1)2
= (x + 3 + y – 1) (x + 3 – y + 1)
= (x + y + 2) (x – y + 4)
................................................
578. What value of x will satisfy the equation:
2.5(4x – 504) = x?
A. 150
B. 145
C. 140
D. 135
Solution:
2.5(4x – 504) = x
10x – 1260 = x
10x – x = 1260
9x = 1260
x = 140
................................................
579. Factorize (2x + y) (2x + y + 6) + 5
A. (2x + y + 5) (2x + y + 1)
B. (2x + y – 5) (2x + y – 1)
C. (2x + y + 5) (2x – y + 1)
D. (2x – y + 5) (2x – y + 1)
Author: Victor A. Tondo Jr., LPT
Solution:
25 = 32
(x3y4z2)5 = x15y20z10
.: (2x3y4z2)5 = 32 x15y20z10
................................................
581. Find k such that 5x2 + 30x + k has two
unequal real roots.
A. k > 6
B. k < 6
C. k > 45
D. k < 45
Solution:
A=5
B = 30
C=k
B2 – 4AC > 0
– 4(5)k > 0
900 – 20k > 0
-20k > -900
k < 45
................................................
302
582. Which of the following is not a polynomial?
A. π r2 + 2 π r
B. m – n
C. √2 x + √3 y – √5 z
D. ab √
cd
Explanation:
Variables cannot bear an irrational exponent.
................................................
583. Which of the following is irrational?
A. 0.193193193193…
B.
C. √1023
D. √2401
Solution: Let 2x + y = A.
Explanation:
(2x + y) (2x + y + 6) + 5
= A (A + 6) + 5
= A2 + 6A + 5
= (A + 5) (A + 1) Substitute A with 2x + y.
= (2x + y + 5) (2x + y + 1)
................................................
A. 0.193193… is
580. Simplify: (2x3y4z2)5
A. 10x15y20z10
B. 25x15y20z10
C. 32x15y20z10
D. 100x15y20z10
584. If x + y = 17 and x2 + y2 = 135, find xy.
A. 78
B. 77
C. 76
D. 75
B.
in fraction form
is already a fraction
C. √1023 cannot be expressed as a fraction
D. √2401 is equal to 49
................................................
This is a free reviewer. All rights reserved.
1000 MMR
Solution:
(x + y)2 = x2 + 2xy + y2
172 = x2 + y2 + 2xy
289 = 135 + 2xy
289 – 135 = 2xy
154 = 2xy
77 = xy
................................................
585. If H and K are the roots of 5x2 – 8x + 9, find
H + K – HK.
A. -2/5
B. -1/5
C. 0
D. 1/5
Solution:
H + K = sum of roots = 8/5
HK = product of roots = 9/5
H + K – HK = 8/5 – 9/5 = -1/5
................................................
586. If H and K are the roots of 3x2 + 4x + 5, find
H2 + K2.
A. -5/9
B. -1
C. -14/9
D. -2
Solution:
H + K = -4/3
HK = 5/3
(H + K)2 = 16/9
H2 + K2 + 2HK = 16/9
H2 + K2 + 2(5/3) = 16/9
H2 + K2 = 16/9 – 10/3
H2 + K2 = -14/9
................................................
9x2
16y2
587. Factorize
+ 24xy +
–
A. (3x + 4y + 9z) (3x + 4y – 9z)
B. (3x – 4y + 9z) (3x + 4y – 9z)
C. (3x + 4y + 9z) (3x – 4y – 9z)
D. (3x – 4y + 9z) (3x – 4y – 9z)
81z2.
Solution:
Author: Victor A. Tondo Jr., LPT
588. Solve for x: 2x + 3y = -8, and 5x – 7y = 67
A. 5
B. 6
C. -5
D. -6
Solution:
7(2x + 3y = -8)
3(5x – 7y = 67)
14x + 21y = -56
15x – 21y = 201
29x
= 145
x=5
................................................


589. A farmer has 15 goats, 23 pigs, and a few
chickens in his farm. If he counted a total of 200
legs in his farm (excluding his, of course), how
many chickens does he have?
A. 96
B. 72
C. 48
D. 24
Solution:
15(4) + 23(4) + n(2) = 200
60 + 92 + 2n = 200
2n = 200 – 152
2n = 48
n = 24
................................................
√
590. Rationalize
√
A.
√
√
√
B.
√
√
√
C.
√
√
√
D.
√
√
√
Solution:
√
√
√
√
=
√
√
√
√
√
√
=
................................................
591. If 3x + 2 = 4y and 5y – 3 = z, express z in
terms of x.
A. 15x + 9
B. 15x + 7
C.
D.
9x2 + 24xy + 16y2 – 81z2
= (3x + 4y)2 – (9z)2
= (3x + 4y + 9z) (3x + 4y – 9z)
................................................
This is a free reviewer. All rights reserved.
1000 MMR
Solution:
Author: Victor A. Tondo Jr., LPT
Solution:
3x + 2 = 4y
=y
5y – 3 = z
5(
)–3=z
A = P (1 + tr)
A = 1,600,000 [1 + 4(0.025)]
A = 1,600,000 (1 + 0.1)
A = 1,600,000 (1.1)
A = 1,760,000
................................................
=z
=z
................................................
595. Solve for x: 45x+3 = 82x+6
A. 2
B. 3
C. 4
592. If 72x + 3 = B, what is B ÷ 49x?
A. 343
B. 21
C. 7
Solution:
D. 3
Solution:
49x
Express each side as a power of a common
number before solving for x.
45x+3 = 82x+6
= (23)2x+6
210x+6 = 26x+18
72x + 3 = B
2
(7 )x (73) = B
49x (343) = B
(343) 49x = B
(22)5x+3
49x
49x
B÷
= (343)
÷
= 343
................................................
593. Multiply: (7x + 4) (2x – 1) (3x + 5)
A. 42x3 + 73x2 + 7x – 20
B. 42x3 – 73x2 + 7x – 20
C. 42x3 + 73x2 – 7x – 20
D. 42x3 – 73x2 – 7x – 20
Solution:
7x
+4
times
2x
-1
-7x -4
14x2
8x
14x2
+x
-4
times
3x
+5
2
70x
+5x -20
42x3 +3x2 -12x
42x3 +73x2 -7x -20
................................................
594. Rayon invested P1,600,000.00 in a bank
that offers 2.5% interest per annum. How much
will his account hold after 4 years?
A. P160,000
B. P1,160,000
C. P1,460,000
D. P1,760,000
D. 5
 Apply laws of exponents.
10x + 6 = 6x + 18
10x – 6x = 18 – 6
4x = 12
x=3
................................................
596. Given
=
, which of the following
expressions is equal to x?
A.
B.
C.
D.
Solution:
=
2cxy = 3abz
x=
 Cross-multiply.
 Isolate x.
................................................
597. Add: (2x2 + 4x + 5) + (5 – 6x – x2)
A. x2 + 2x + 10
B. 3x2 + 2x + 10
2
C. x – 2x + 10
D. 3x2 + 2x + 10
................................................
598. Solve for x: 272x–8 = 9x+10
A. 9
B. 10
C. 11
This is a free reviewer. All rights reserved.
D. 12
1000 MMR
Solution:
Express each side as a power of a common
number before solving for x.
272x–8 = 9x+10
(33)2x–8 = (32)x+10
26x–24 = 22x+20
 Apply laws of exponents.
6x – 24 = 2x + 20
6x – 2x = 20 + 24
4x = 44
x = 11
................................................
599. If 25x – 49y = 2019, what is 49y – 25x?
A. 2018
B. -2019
C.
D. cannot be determined
Explanation:
49y – 25y = (-1)(25x – 49y)
= (-1)(2019)
= -2019
................................................
600. If x + y = 90 and x2 + y2 = 2020, find xy.
A. 3025
B. 3030
C. 3035
D. 3040
Solution:
(x + y)2 = x2 + 2xy + y2
902 = x2 + y2 + 2xy
8100 = 2020 + 2xy
8100 – 2020 = 2xy
6080 = 2xy
3040 = xy
................................................
601. Which of the following is ALWAYS true?
A. Vertical pairs of angles are supplementary.
B. Vertical pairs of angles are congruent.
C. Linear pairs of angles are congruent.
D. Linear pairs of angles are complementary.
Explanation:
Linear pairs are supplementary, while vertical
pairs are congruent.
Author: Victor A. Tondo Jr., LPT
602. How many line segments can be made from
28 non-collinear points?
A. 378
B. 394
C. 412
D. 422
Solution:
28C2 = 378
................................................
603. The vertex angle of an isosceles triangle is
70°. What is the measure of one of the base
angles?
A. 45°
B. 50°
C. 55°
D. 60°
Solution:
(180 – 70)/2 = 110/2 = 55
................................................
604. A shoebox measures 20 inches by 15 inches
by 9 inches. What is its volume in cubic inches?
A. 900
B. 1350
C. 1650
D. 2700
Solution:
V=LxWxH
= 20 x 15 x 9 = 2700
................................................
605. Which of these has the longest perimeter?
A. A regular pentagon with sides 21 cm long
B. A rectangle 29 cm long and 24 cm wide
C. An equilateral triangle with sides 36 cm long
D. A square whose area is 1024 cm2
Solution:
A. P = 5 (21) = 105
B. P = 2 (29 + 24) = 106
C. P = 3 (36) = 108
D. S = 32, so P = 4 (32) = 128
................................................
606. The hypotenuse of a right triangle is 82 feet.
If one leg is 80 feet, what is the length of the
other leg in feet?
A. 15
B. 16
C. 17
D. 18
This is a free reviewer. All rights reserved.
1000 MMR
Explanation:
Author: Victor A. Tondo Jr., LPT
Explanation:
Use the Pythagorean Theorem: A2 + B2 = C2
Consecutive angles of a parallelogram are
supplementary.
................................................
A2 + 802 = 822
A2 = 6724 – 6400
A2 = 324
A = 18
................................................
607. Find the surface area of a rectangular box
whose dimensions are 25 cm x 35 cm x 45 cm.
A. 7000 cm2
B. 7045 cm2
2
C. 7150 cm
D. 7745 cm2
Solution:
SA = 2 (LW + WH + LH)
SA = 2 (45x35 + 35x25 + 25x45)
SA = 2 (1575 + 875 + 1125) = 7150
................................................
608. A and B form a linear pair. If m A = 3x
and m B = 5x + 40, what is the value of x?
A. 17.5
B. 18
C. 19
D. 19.75
Solution:
Since the two angles form a linear pair, then they
are supplementary.
3x + 5x + 40 = 180
8x + 40 = 180
8x = 140
x = 17.5
................................................
609. 1 and 3 are opposite angles in a
parallelogram. If m 1 = 50o, what is m 3?
A. 40o
B. 50o
C. 140o
D. 130o
611. Two parallel lines are cut by a transversal,
forming R and V. If the two angles are
corresponding angles, what is the measure of R
if the measure of V is 70o?
A. 35o
B. 70o
o
C. 160
D. 110o
Explanation:
Corresponding angles formed by two parallel
lines cut by a transversal are congruent.
................................................
612. If the sum of the supplement and the
complement of an angle is 100 degrees, what is
the angle?
A. 65o
B. 70o
C. 75o
D. 85o
Solution:
(90 – x) + (180 – x) = 100
270 – 2x = 100
270 – 100 = 2x
170 = 2x
85 = x
................................................
613. It is the perpendicular bisector of a regular
polygon’s side, passing through the center.
A. side
B. apothem
C. asymptote
D. diagonal
Explanation:
Explanation:
Opposite angles of a parallelogram are
congruent.
610. A and C are consecutive angles in a
parallelogram. If m A = 60o, what is m C?
A. 40o
B. 50o
C. 150o
D. 120o
The apothem is only for regular polygons. It is
the perpendicular bisector of one of its sides,
passing through the center.
A diagonal is a line segment formed by
connecting two non-consecutive vertices of a
polygon.
A side is formed by connecting two consecutive
vertices of a polygon.
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1000 MMR
614. In the figure,
// . If m AFE = 130o,
find the measure of CGH.
Author: Victor A. Tondo Jr., LPT
618. A circle has a diameter of 34 cm. What is its
area?
A. 17 π cm2
B. 34 π cm2
C. 289 π cm2
D. 1156 π cm2
Solution:
A. 50o
B. 130o
C. 65o
D. 25o
Explanation:
D = 2r
34 = 2r
17 = r
A = π r2
A = π (172)
A = 289 π
................................................
AFE and CGH are exterior angles on the same
side of the transversal. Therefore, they are
supplementary.
................................................
619. The perimeter of a rectangle is 84. If its
length is 29 cm, find its area.
A. 377 cm2
B. 1218 cm2
C. 2436 cm2
D. 4872 cm2
615. The measure of each interior angle of a
regular polygon is 170o. How many vertices does
it have?
A. 36
B. 24
C. 12
D. 10
Solution:
Solution:
N=
=
= 36
................................................
P = 2(L + W)
84 = 2(L + W)
42 = (L + W)
A=LxW
42 = 29 + W
A = 29 x 13
13 = W
A = 377
................................................
620. In the figure, A is the center of the circle.
What is the measure of BDC?
A. 160o
B. 80o
C. 40o
D. 20o
616. How many diagonals does a regular
dodecahedron have?
A. 45
B. 54
C. 60
D. 66
Solution:
(
)
(
)
D=
=
= 54
................................................
Explanation:
617. Find the area of a square whose perimeter
is 480 cm.
A. 14400 cm2
B. 28800 cm2
C. 57600 cm2
D. 270400 cm2
BAC is a central angle measuring 40o. That
means ̂ = 40o. Since BDC is an inscribed
angle intercepting ̂ , then its measure is half of
̂ , or 20o.
Solution:
P = 4S
480 = 4S
120 = S
A = S2
A = 1202
A = 14400
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1000 MMR
621. Find the measure of E using the figure of
parallelogram WENA below.
A. 15
B. 65
C. 115
D. 85
Solution:
7x – 40 = 4x + 5
7x – 4x = 40 + 5
3x = 45
x = 15
So m W = 4x + 5 = 65, which means
m E = 180 – 65 = 115
................................................
Author: Victor A. Tondo Jr., LPT
Solution:
Diagonal2 = Length2 + Width2 + Height2
172 = 92 + W2 + 122
289 = 81 + W2 + 144
289 – 144 – 81 = W2
64 = W2
8=W
................................................
624. Statement 1: A square is a rhombus.
Statement 2: A square is a rectangle.
A. Only the first statement is true.
B. Only the second statement is true.
C. Both statements are true.
D. Both statements are false.
................................................
625. Which of the following has its incenter,
circumcenter, centroid, and orthocenter in just
one point?
A. Equilateral Triangle
B. Right Triangle
C. Scalene Triangle
D. Obtuse Triangle
................................................
622. In parallelogram
MATH, m A = 7x – 32 and
m T = 5x + 32. Find m A.
A. 15
B. 63
C. 73
D. 117
Solution:
A and T are consecutive angles,
therefore m A + m T = 180
7x – 32 + 5x + 32 = 180
7x + 5x = 180
12x = 180
x = 15
626. In triangles, this line segment is drawn from
midpoint of one side to its opposite vertex.
A. Median
B. Altitude
C. Bisector
D. Longitude
Explanation:
m A = 7x – 32
= 7(15) – 32
= 105 – 32 = 73
................................................
623. The diagonal of a rectangular prism is 17
cm long. If it is 9 cm thick and 12 cm long, how
wide is it?
A. 15 cm
B. 10 cm
C.8√3 cm
D. 8 cm
Altitude: perpendicular to one side, passing
through the opposite vertex.
Median: line segment from midpoint of one side
to opposite vertex
Bisector: line segment that bisects an angle of a
triangle
................................................
627. The radius of a cone measures 12 cm and its
height is 15 cm. Find its volume.
A. 240 π cm3
B. 360 π cm3
C. 480 π cm3
D. 720 π cm3
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1000 MMR
Solution:
Author: Victor A. Tondo Jr., LPT
Solution:
V = πr2h
Find the height of the trapezoid using the
Pythagorean Theorem first.
= π (144) (15)
= 720 π
................................................
628. This is located at the intersection of the
perpendicular bisectors of a triangle.
A. Centroid
B. Circumcenter
C. Incenter
D. Orthocenter
Explanation:
Incenter: intersection of angle bisectors
Circumcenter: intersection of perpendicular
bisectors
Centroid: intersection of medians
Orthocenter: intersection of altitudes
................................................
629. The radius of a cylinder measures 9 cm and
its height is 15 cm. Find its volume.
A. 1215 π cm3
B. 810 π cm3
3
C. 540 π cm
D. 360 π cm3
Solution:
V = πr2h
= π (81) (15)
= 1215 π
................................................
630. What is the measure of each exterior angle
of a regular decagon?
A. 108o
B. 72o
C. 60o
D. 36o
Solution:
MEA = 360/N
= 360/10 = 36o
................................................
631. Find the area of an isosceles trapezoid given
bases measuring 20 cm and 30 cm, with each of
the congruent slants measuring 13 cm.
A. 250 cm2
B. 275 cm2
C. 300 cm2
D. 325 cm2
52 + h2 = 132
25 + h2 = 169
h2 = 144
h = 12
Area =
( )
Area =
(12)
Area = 25(12) = 300
................................................
632. Which of the following side lengths belong
to an acute triangle?
A. 5 cm, 7 cm, 9 cm
B. 6 cm, 8 cm, 10 cm
C. 6 cm, 9 cm, 12 cm
D. 7 cm, 9 cm, 11 cm
Solution:
A. 52 + 72 < 92
Obtuse
2
2
2
B. 6 + 8 = 10
Right
C. 62 + 92 < 122
Obtuse
2
2
2
D. 7 + 9 > 11
Acute
................................................
633. A prism has a right triangle as its base. The
two legs of the right triangular base are 6 cm and
8 cm long. If the prism is 2 cm thick, find its
lateral surface area.
A. 24 cm2
B. 48 cm2
2
C. 56 cm
D. 72 cm2
Solution:
The hypotenuse of the base is 10 cm.
Lateral surface area = 2(6 + 8 + 10) = 48 cm2
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1000 MMR
634. Find the volume of a rectangular pyramid
given that its base has length 21 cm and width 10
cm, and its height is 15 cm.
A. 700 cm3
B. 1050 cm3
3
C. 1575 cm
D. 1400 cm3
Solution:
V= LxWxH
V = (21)(10)(15)
V = 1050
................................................
635. The diameter of a cylinder is 24 cm. If its
height is 40 cm, find its surface area.
A. 312 π cm2
B. 624 π cm2
C. 936 π cm2
D. 1248 π cm2
Author: Victor A. Tondo Jr., LPT
637. A rhombus has diagonals measuring 20 cm
and 30 cm. What is its area?
A. 75 cm2
B. 150 cm2
C. 300 cm2
D. 600 cm2
Solution:
A = ½ D1 D2
A = ½ (600) = 300
................................................
638. A rhombus as diagonals measuring 30 cm
and 16 cm. What is its perimeter?
A. 136 cm
B. 92 cm
C. 68 cm
D. 46 cm
Solution:
Solution:
SA = 2πr2 + 2πrh
SA = 2π (144) + 2π(12)(40)
SA = 288π + 960π
SA = 1248π
................................................
636. In the figure, AC is a diagonal of rhombus
ABCD. If m B = 118o, what is m CAD?
A. 124o
B. 118o
C. 62o
D. 31o
By Pythagorean Theorem, the hypotenuse of
each right triangle is 17 cm. That means each
side of the rhombus is 17 cm, and its perimeter is
4 x 17 or 68 cm.
................................................
639. Find the approximate distance around a
semicircular park with radius 100m.
A. ~1028 m
B. ~771 m
C. ~514 m
D. ~257 m
Solution:
Solution:
Since ABCD is a rhombus, then AC bisects BAD,
and BAD and B are supplementary.
Diameter = 2r =
200
Semicircle = r = 100π ~ 314
Perimeter =
~514
................................................
m B + m BAD = 180
118 + m BAD = 180
m BAD = 62
640. Find the equation of the line perpendicular
to 7x + 4y = 12, passing through (-2, 9).
A. 4x – 7y = -71
B. 4x + 7y = 71
C. 4x – 7y = 71
D. 4x + 7y = -71
m CAD = ½ (62) = 31
................................................
Solution:
The perpendicular line for Ax + By = C is given
as Bx – Ay = B(xp) – A(yp).
4x – 7y = 4(-2) – 7(-9)
4x – 7y = -71
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1000 MMR
641. Find the equation of the line perpendicular
to 9x –7y = -6, passing through (4, -3).
A. 7x + 9y = 1
B. 7x + 9y = -1
C. 7x – 9y = 1
D. 7x – 9y = -1
Author: Victor A. Tondo Jr., LPT
644. Find the volume of the following pool.
Solution:
The perpendicular line for Ax + By = C is given
as Bx – Ay = B(xp) – A(yp).
-7x – 9y = -7(4) – 9(-3)
-7x – 9y = -7(4) – 9(-3)
-7x – 9y = -1
But since the new A is negative, then we will
change all signs of the new equation, giving us
7x + 9y = 1.
................................................
642. A rectangle is drawn with dimensions 28 cm
by 42 cm. A larger rectangle is drawn by adding
5 cm margins from each side of the original
rectangle. What is the area of the larger
rectangle?
A. 1456 cm2
B. 1551 cm2
C. 1556 cm2
D. 1976 cm2
Explanation:
The dimensions of the larger rectangle would be
(28 + 5 + 5) by (42 + 5 + 5), or 38 by 52. That
means its area is 38 x 52 or 1976.
................................................
643. Find the length of the diagonals of a
rectangular prism 6 cm thick, 15 cm long, and 10
cm wide.
A. 17√3 cm
B. 18√2 cm
C. 19 cm
D. 20√5 cm
A. 700 m3
C. 1400 m3
B. 1050 m3
D. 2800 m3
Solution:
The pool is a trapezoid prism. Its area is given as
A=
A=
(8)(10)
A = 1400 m3
................................................
645. What is the slope of the line defined by the
equation 3x + 5y = 11?
A.
B.
C. 3
D. 5
Solution:
3x + 5y = 11
 Ax + By = C
5y = -3x + 11
y= x+
 y = mx + b
................................................
646. In the figure below, m CAB = 2x + 4, while
m COB = 5x – 7. Find x.
Solution:
D=√ +
+
D = √15 + 10 + 6
D = √225 + 100 + 36
D = √361
D = 19
A.
B. -2
Solution:
2(2x + 4) = 5x – 7
4x + 8 = 5x – 7
8 + 7 = 5x – 4x
15 = x
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C.
D. 15
1000 MMR
647. In the figure below,
//
. If m AGE =
7x + 11 and m CHF = 5x + 1, what is m AGE?
Author: Victor A. Tondo Jr., LPT
Solution:
Each square has a side of 6 cm. The perimeter of
the figure has 22 segments measuring 6 cm each.
Therefore the perimeter is 22 x 6 or 132 cm.
................................................
650. Find the area of the shaded part of the
circle.
A. 119o
B. 109o
C. 71o
D. 61o
Solution:
Since AGE and CHF are exterior angles on the
same side of the transversal, then they are
supplementary.
m AGE + m CHF = 180
7x + 11 + 5x + 1 = 180
12x + 12 = 180
12x = 168
x = 14
A=
648. Find the sum of all interior angles of a
regular 15-sided polygon.
A. 5400o
B. 2700o
o
C. 2340
D. 1170o
Solution:
180 (n – 2) = SIA
180 (15 – 2) = SIA
2340 = SIA
................................................
649. If each square in the figure has an area of 36
square cm, find the perimeter of the figure.
B. 792 cm
D. 132 cm
B. 972 π cm2
D. 270 π cm2
Solution:
m AGE = 7x + 11
m AGE = 7(14) + 11
m AGE = 109
................................................
A. 864 cm
C. 144 cm
A. 9720 π cm2
C. 540 π cm2
π r2
A = (900 π)
A = 270 π
................................................
651. In the figure, ̂ = 120o and m E = 28o.
Find ̂ .
A. 46o
B. 54o
C. 64o
D. 72o
Solution:
m E=
28 =
̂
̂
̂
56 = 120 ̂
̂ = 120 – 56
̂ = 64
................................................
652. Find the volume of a cone whose radius is
30 cm and height is 170 cm.
A. 289,000 π
B. 153,000 π
C. 51,000 π
D. 17,000 π
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1000 MMR
Solution:
Author: Victor A. Tondo Jr., LPT
655. Find the equation of the line passing
through (7, 5) and (5, 1).
A. y = 2x + 9
B. y = 2x – 9
C. y = ½ x + 9
D. y = ½ x – 9
V = π r2 h
V = π (30)2 170
V = (153,000 π)
V = 51,000 π
................................................
Solution:
653. Find the center of a circle given one of its
longest chords has endpoints at M(-11,8) and
N(23,-24).
A. (12, -16)
B. (6, -8)
C. (-17,16)
D. (17, -16)
y–5=
Solution:
The longest chord of a circle is the diameter and
its midpoint is the center.
Midpoint Formula: (
,
)
Midpoint: (
,
(
)
)
Midpoint: (6, 8)
................................................
654. Find the radius of a circle if its diameter has
endpoints at M(3,5) and N(-12,13).
A. 8
B. 8.5
C. 9
D. 9.5
Solution:
The diameter’s midpoint is the center.
Center: (
,
)
Center: ( 4.5,9)
Radius is the distance between the center and
any of its endpoints.
R = √( 4.5 3) + (9 5)
R = √( 7.5) + 4
R = √56.25 + 16
R = √72.25
R = 8.5
Better Solution:
Use the two-point form.
y – y1 =
(x – x1)
(x – 7)
y – 5 = (x – 7)
y – 5 = 2 (x – 7)
y – 5 = 2x – 14
y = 2x – 9
................................................
656. Find the equation of the line passing
through (2, -7) with a slope of -3.
A. y = -3x – 1
B. y = -3x + 1
C. y = -3x – 2
C. y = -3x + 2
Solution:
y = mx + b
-7 = -3(2) + b
-7 = -6 + b
-1 = b
where x = 2, y = -7, m = -3
.: y = -3x – 1
................................................
657. A square has a diagonal measuring 28 cm.
Find its area.
A. 784 cm2
B. 392 cm2
2
C. 196 cm
D. 98 cm2
Solution:
A=
A=
A = 392
................................................
658. Symbol for line.
A.
B.
C.
D.
R=½D
where D = length of diameter
D = √( 12 3) + (13 5)
D = √225 + 64
D = 17
Therefore, R = ½ (17) or 8.5
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1000 MMR
659. Symbol for ray.
A.
B.
C.
D.
................................................
660. Find the volume of the swimming pool
drawn below.
A. 2000 cu. ft.
B. 2100 cu. ft.
C. 2250 cu. ft.
D. 2500 cu. ft.
Author: Victor A. Tondo Jr., LPT
662. In the figure, ACB is formed by the
tangents AC and BC. If m ACB = 34o, what is the
measure of the minor intercepted arc?
A. 112o
B. 146o
C. 68o
D. 34o
Explanation:
When two tangents to a circle form an angle, the
intercepted arcs are 180 ± θ, where θ is the
measure of the angle formed by the tangents.
................................................
663. X(7,9) is the midpoint of A(1,2) and B. Find
the coordinates of B.
A (-13, -16)
B. (13, -16)
C. (-13, 16)
D. (13, 16)
Solution:
Solution:
7=
9=
14 = x + 1
18 = y + 2
13 = x
16 = y
................................................
Let’s rebuild the pool as a figure that is left by
removing a triangular prism (gray figure) from a
rectangular prism (whole figure).
The rectangular prism is 25 ft by 8 ft by 15 ft. Its
volume is 3000 cubic feet.
The triangular prism’s base is 6 ft by 20 ft. Its
base area is 60 square feet. Since its length is 15
feet, then its volume is 900 cubic feet.
Remove 900 from 3000 and you’ll get 2100 cubic
feet.
................................................
661. Which geometric figure extends indefinitely
in two opposite directions?
A. angle
B. ray
C. line
D. line segment
664. Point O is two-thirds the way from A(-4,5)
to B(8,-10). Find the coordinates of point O.
A. (0, 0)
B. (4, -5)
C. (3, -6)
D. (3, -4)
Solution:
x = -4 + (8 – (-4))
x = -4 + (12)
x = -4 + 8
x=4
y = 5+ (-10 – 5)
y = 5+ (-15)
y = 5+ (-15)
y = 5 + (-10)
y = -5
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1000 MMR
665. H and K form a vertical pair. If m H =
4x + 20 and m K = 6x – 10, find x.
A. 13
B. 15
C. 17
D. 19
Solution:
Vertical pairs of angles are congruent.
4x + 20 = 6x – 10
20 + 10 = 6x – 4x
30 = 2x
15 = x
................................................
Author: Victor A. Tondo Jr., LPT
Solution:
5x – 6y = 7
-6y = -5x + 7
y= x–
................................................
670. Find the equation of a circle if its diameter
has endpoints at M(3,5) and N(-12,13).
A. (x + 4.5)2 + (y + 9)2 = 72.25
B. (x – 4.5)2 + (y + 9)2 = 72.25
C. (x + 4.5)2 + (y – 9)2 = 72.25
D. (x – 4.5)2 + (y – 9)2 = 72.25
No item #666, as requested by many subscribers.
................................................
Solution:
667. F and G form a linear pair. If m F =
4x + 20 and m G = 6x – 10, find x.
A. 13
B. 15
C. 17
D. 19
The diameter’s midpoint is the center.
Center: (
,
)
Center: ( 4.5,9)
Solution:
Linear pairs of angles are supplementary.
4x + 20 + 6x – 10 = 180
10x + 10 = 180
10x = 170
x = 17
................................................
668. Point C is two-fifths the way from M(-3,4) to
N(7,-11). Find the coordinates of point C.
A. (0, 0)
B. (2, -1)
C. (1, -2)
D. (1, -1)
Solution:
x = -3+ (7 – (-3))
y = 4 + (-11 – 4)
x = -3+ (10)
x = -3 + 4
x=1
y = 4 + (-15)
y = 4 + (-6)
y = -2
................................................
669. Find the slope of the line 5x – 6y = 7.
A.
B.
C.
D.
Radius is the distance between the center and
any of its endpoints.
R = √( 4.5 3) + (9 5)
R = √( 7.5) + 4
R = √56.25 + 16
R = √72.25
R = 8.5
................................................
671. A bicycle’s wheel is 24 inches in diameter. If
it revolves 625 times, how far does the bicycle
travel?
A. 7,500 π
B. 15,000 π
C. 30,000 π
D. 60,000 π
Solution:
Circumference = π D
Circumference = 24 π
That means the bicycle travels 24 π inches per
revolution.
625 x 24 = 15,000 π
................................................
672. A right triangle is inscribed in a circle. If the
triangle’s legs are 16 cm and 30 cm, what is the
area of the circle?
A. 289 π cm2
B. 480 cm2
C. 480 π cm2
D. 1156 π cm2
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1000 MMR
Solution:
The hypotenuse of the right
triangle is the diameter of the
circle since it is inscribed in the
circle. By Pythagorean Theorem,
the diameter is 34 cm, meaning the radius is 17
cm. The area is given as π r2, meaning the area is
289 π cm2.
................................................
673. In the figure, a rectangle is
inscribed in a circle. If the rectangle
is 20 cm by 48 cm, what is the area of
the shaded figure?
A. 676π – 960 cm2
B. 1352π – 960 cm2
C. 2304π – 960 cm2
D. insufficient data
Solution:
The rectangle’s diagonal is the diameter of the
circle. By Pythagorean Theorem, the diameter is
52 cm, meaning the radius is 26 cm. Therefore,
the area of the entire circle is 676 π cm2.
The area of the rectangle is 20 x 48 or 960 cm2.
That means the area of the shaded figure is equal
to 676π – 960 cm2.
................................................
674. What is the area of the red trapezium in the
given figure?
A. 23 sq. units
C. 25 sq. units
Solution:
B. 24 sq. units
D. 26 sq. units
Author: Victor A. Tondo Jr., LPT
The white triangle in the lower right area is 3 x 3
units, giving it an area of 4.5 sq. units.
48 – (6 + 7.5 + 5 + 4.5) = 25
................................................
675. Which of the following are not congruent?
A. Corresponding angles formed by two parallel
lines cut by a transversal.
B. Alternate exterior angles formed by two
parallel lines cut by a transversal.
C. Pairs of vertical angles.
D. Interior angles on the same side of the
transversal cutting two parallel lines.
Explanation:
When two parallel lines are cut by a transversal,
corresponding and alternate interior/exterior
angles are congruent. Vertical pairs of angles are
also congruent. However, interior/exterior
angles on the same side of the transversal are
supplementary.
................................................
676. A rectangle is 24 cm long. If its diagonal is
25 cm, what is its perimeter in cm?
A. 49
B. 62
C. 64
D. 98
Solution:
Solve for the width first using the Pythagorean
Theorem.
W2 + 242 = 252
W2 + 576 = 625
W2 = 625 – 576
W2 = 49
W=7
P = 2(L + W)
P = 2(24 + 7)
P = 2(31) = 62
................................................
677. Square FACE is drawn such that one of its
The entire figure is 8 x 6, or 48 sq. units.
sides AC is the diagonal of a rectangle, ABCD. If
The white triangle in the upper left area is 3 x 4
AB = 20 and BC = 43, find the area of FACE.
units, giving it an area of 6 sq. units.
The white triangle in the upper right area is 5 x 3
A. 2209 cm2
B. 2229 cm2
units, giving it an area of 7.5 sq. units.
C. 2249 cm2
D. 2269 cm2
The white triangle in the lower left area is 5 x 2
units, giving it an area of 5 sq. units.
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1000 MMR
Solution:
(AC)2 = 202 + 432
(AC)2 = 400 + 1849
(AC)2 = 2249
................................................
678. If the vertex angle of an isosceles triangle is
50o, what is the measure of each base angle?
A. 130o
B. 80o
C. 65o
D. 25o
Solution:
50 + x + x = 180
50 + 2x = 180
2x = 130
x = 65
................................................
679. Which of the following is a square?
A. VCTR whose sides measure 20 cm each
B. LUVS whose angles are 90o each
C. RWNA whose diagonals are 50 cm each
D. AYIE whose congruent diagonals
perpendicularly bisect each other
Explanation:
Author: Victor A. Tondo Jr., LPT
681. Find the length of the intercepted arc of a
central angle measuring 60o given that the radius
of the circle is 12 cm.
A. 4 π cm
B. 6 π cm
C. 8 π cm
D. 12 π cm
Solution:
(2 π r) = (24 π) = 4 π
................................................
682. Which of the following is false?
A. Radii of the same circle are always congruent.
B. Any two right angles are congruent.
C. An inscribed right angle always intercepts a
semicircle.
D. There are infinitely many lines that may be
drawn parallel to a given line passing through a
given point.
Explanation:
There can only be one line that may be drawn
parallel to a given line passing through a given
point.
................................................
A. VCTR is a rhombus
B. LUVS is a rectangle
C. RWNA is a rectangle
D. Only squares have congruent diagonals that
are perpendicular bisectors of each other.
................................................
683. The volume of a cube is 343 cm3. What is its
surface area?
A. 24 cm2
B. 24√3 cm2
C. 48 √13 cm2
D. 294 cm2
680. The following are measures of sides of four
triangles. Which of them are taken from an
obtuse triangle?
A. 20, 20, 29
B. 20, 21, 29
C. 20, 22, 28
D. 20, 23, 30
S3 = 343
S=7
Explanation:
If A < B < C, then A2 + B2 < C2 if ABC is an
obtuse triangle.
A. 202 + 202 < 292
B. 202 + 212 = 292
C. 202 + 222 > 282
D. 202 + 232 > 302
 Obtuse
 Right
 Acute
 Acute
Solution:
Surface Area = 6s2
= 6 (72)
= 294
................................................
684. Find the volume of a rectangular plank of
wood that is half-inch thick, six inches wide, and
6 feet long.
A. 18 cubic inches B. 36 cubic inches
C. 108 cubic inches D. 216 cubic inches
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1000 MMR
Solution:
Author: Victor A. Tondo Jr., LPT
Solution:
V=LxWxH
V = 72 x 6 x ½
 6 feet is 72 inches
V = 216
................................................
MIA = 180 –
685. In the following figure, O is the center of the
circle. If m CAB = 36o, what is m DBA?
A. 18o
B. 36o
C. 54o
D. 72o
Solution:
Most people would solve for m DBA by doing
this long solution:
Measure of arc BC = 2 x 36 = 72
Since chord AC passes through center O, then AC
is a diameter, and arc CBA is a semicircle. That
means arc AB is 180 – 72 = 108.
Since chord BD passes through center O, then BD
is a diameter too, making arc BAD a semicircle.
That means arc AD = 180 – 108 = 72.
Finally, since DBA intercepts arc AD, then
m DBA = ½ x 72 = 36.
That’s a long solution. The best way to deal with
this is to remember that OA and OB are both
radii of the same circle, making them congruent.
That means △OAB is an isosceles triangle, with a
base angle at OAB (same angle as CAB). Since
the base angles of an isosceles triangle are
congruent, then OBA (same angle as DBA) is
the same measure as OAB, or 36o.
................................................
MIA = 180 –
MIA = 180 – 40
MIA = 140
................................................
687. Find the surface area of a ball whose radius
is 24 cm.
A. 576 π cm2
B. 1152 π cm2
2
C. 2304 π cm
D. 4608 π cm2
Solution:
SA = 4 π r2
SA = 4 π (242)
SA = 2304 π
................................................
688. Give the converse of the conditional
statement “If two angles are vertical angles, then
they are congruent.”
A. “If they are are congruent, then two angles are
vertical angles.”
B. “If two angles not vertical angles, then they are
not congruent.”
C. “If they are not congruent, then two angles are
not vertical angles.”
D. “If two angles are linear angles, then they are
supplementary.”
Explanation:
The converse of a conditional is formed by
interchanging the if-statement (hypothesis) and
the then-statement (conclusion).
................................................
686. What is the measure of each interior angle
of a regular nonagon?
A. 135o
B. 140o
o
C. 144
D. 145o
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1000 MMR
689. Give the inverse of the conditional
statement “If two angles are vertical angles, then
they are congruent.”
A. “If they are are congruent, then two angles are
vertical angles.”
B. “If two angles not vertical angles, then they are
not congruent.”
C. “If they are not congruent, then two angles are
not vertical angles.”
D. “If two angles are linear angles, then they are
supplementary.”
Explanation:
The inverse of a conditional is formed by
negating both the if-statement (hypothesis) and
the then-statement (conclusion).
................................................
690. In which geometry do two distinct lines
intersect in two points?
A. Euclidean Geometry
B. Hyperbolic Geometry
C. Elliptic Geometry
D. Plane Geometry
Explanation:
Author: Victor A. Tondo Jr., LPT
692. In which geometry are the summit angles of
a Saccheri quadrilateral obtuse angles?
A. Euclidean Geometry
B. Hyperbolic Geometry
C. Elliptic Geometry
D. Plane Geometry
Explanation:
Euclidean: right angles
Hyperbolic: acute angles
Elliptic: obtuse angles
................................................
693. How many diagonals does a regular
decagon have?
A. 30
B. 35
C. 40
D. 45
Solution:
D = n(n – 3)/2
................................................
694. Each side of a regular hexagon is 256 cm
long. Find the distance around it.
A. 1536 cm
B. 1280 cm
C. 1024 cm
D. 718 cm
In Elliptic Geometry, two distinct lines intersect
in two points. In both Euclidean and Hyperbolic
Geometry, two distinct lines will intersect at only
one point.
................................................
Solution:
P=nS
P = 6 (256)
P = 1536
................................................
691. In which geometry can the sum of the
measures of the angles of a triangle be less than
180 degrees?
A. Euclidean Geometry
B. Hyperbolic Geometry
C. Elliptic Geometry
D. Plane Geometry
695. What can be said about these statements?
i. Any rectangle has two congruent diagonals.
ii. Any rhombus has two perpendicular
Explanation:
diagonals.
A. Only the first statement is true.
B. Only the second statement is true.
C. Both statements are true.
D. Both statements are false.
................................................
Euclidean: sum is exactly 180 degrees
Hyperbolic: sum is less than 180 degrees
Elliptic: sum is more than 180 degrees
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1000 MMR
696. In the figure, A is the midpoint of
and
. By which triangle congruence postulate can
we prove that △CAB △DAE?
Author: Victor A. Tondo Jr., LPT
Solution:
P = 4S
280 = 4S
A = S2
70 = S
A = 702 = 4900
................................................
699. Find the altitude to the hypotenuse of a
right triangle whose legs are 60 cm and 80 cm.
A. 40 cm
B. 48 cm
C. 96 cm
D. 100 cm
Solution:
A. SSS Congruence Postulate
B. SAS Congruence Postulate
C. ASA Congruence Postulate
D. ASS Congruence Postulate
Solve for the hypotenuse first using Pythagorean
Theorem.
602 + 802 = H2
3600 + 6400 = H2
10000 = H2
100 = H
Explanation:
AB = AE and AD = AC since A is the midpoint of
the segments
and
. Their included angles,
DAE and CAB, are also congruent because
vertical angles are congruent.
When two sides and an included angle of a
triangle are congruent to two sides and an
included angle of another triangle, then the two
triangles are congruent by SAS Congruence
Postulate.
................................................
697. Find the circumference of a coin whose
radius is 13 cm.
A. 13 π cm
B. 26 π cm
C. 169 π cm
D. 676 π cm
Solution:
C=2πr
C = 2 π (13) = 26 π
................................................
698. Find the area of a square whose perimeter
is 280 cm.
A. 1120 cm2
B. 4900 cm2
2
C. 19600cm
D. 78400 cm2
Then use the following formula:
(
)
Alt to Hyp =
(
)
Alt to Hyp =
= 48
................................................
700. This is formed by two rays with a common
end point.
A. polygon
B. angle
C. line
D. line segment
................................................
701. Convert
A. 210o
C. 630o
rad to degrees.
B. 420o
D. 840o
Solution:
rad x
= 210o
................................................
702. What is the reference angle of 212o?
A. 32o
B. 68o
C. 106o
D. 12o
Solution:
Since 212o is in QIII, then 212o – 180o = 32o
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1000 MMR
703. A negative sine and a positive cosine are
properties of angles in which quadrant?
A. QI
B. QII
C. QIII
D. QIV
Author: Victor A. Tondo Jr., LPT
Explanation:
Use the CAST mnemonic. See item #164.
................................................
704. Which of the following equal to sec 50o?
A. sin 40o
B. cos 40o
o
C. csc 40
D. cot 40o
Explanation:
Apply the cofunctions. The COfunction for secant
is COsecant. The COmplement of 50o is 40o.
................................................
705. Which of the ff. is coterminal with 2019o?
A. 219o
B. 209o
o
C. 199
D. 189o
Solution:
2019/360 = 5.608333…
 Grab 5 only
2019 – 5(360) = 219
................................................
706. The angle of elevation to the top of a
building is 30o. If the observer is 50√3 meters
away from the building, how tall is the building?
A. 25 m
B. 25√3 m
C. 50 m
D. 100 m
Solution:
tan 30 =
√
=
50√3 = x√3
50 = x
................................................
707. The longest side in a 30-60-90 triangle is
480 cm. How long is the shortest side?
A. 120 cm
B. 160 cm
C. 240 cm
D. 360 cm
Solution:
2x = 480
x = 240
................................................
708. If the hypotenuse of a 45-45-90 triangle is
50 cm long, how long is a leg?
A. 25 cm
B. 25√2 cm
C. 25√3 cm
D. 50√2 cm
Solution:
x√2 = 50
x = 25√2
................................................
709. Which of the ff. is coterminal with 143o?
A. 2403o
B. 2303o
C. 2103o
D. 2003o
Solution:
√
50√3 (√3) = 3x
150 = 3x
50 = x
A. 2403 – 143 = 2260 (not divisible by 360)
B. 2303 – 143 = 2160 (divisible by 360)
C. 2103 – 143 = 2060 (not divisible by 360)
D. 2003 – 143 = 1960 (not divisible by 360)
Alternative Solution:
Remember:
Use the 30-60-90 map. (You have to know this by
heart.)
Two angle measures are coterminal if and only if
their difference a multiple of 360.
................................................
√
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1000 MMR
710. What is the measure of the smaller angle
formed by the hands of the clock at 5:55?
A. 180o
B. 174.75o
C. 163.625o
D. 152.5o
Solution:
A = |30H – 5.5M|
A = |30(5) – 5.5(55)|
A = |150 – 302.5|
A = 152.5
................................................
711. What is the measure of the smaller angle
formed by the hands of the clock at 3:50?
A. 185o
B. 175o
C. 170o
D. 165o
Author: Victor A. Tondo Jr., LPT
Explanation:
Refer to this mnemonic hexagon. Any
trigonometric function is equal to (katabi)
divided by (susunod sa katabi).
That means tan = sin ÷ cos or tan = sec ÷ csc,
sin = cos ÷ cot or sin = tan ÷ sec, etc.
................................................
714. If sin x = 0.936, which of the following could
be cos x?
A. 0.064
B. 0.8
C. 0.016
D. -0.352
Solution:
A = |30H – 5.5M|
A = |30(3) – 5.5(50)|
A = |90 – 275|
A = 185
Solution:
However, 185o is a reflex angle. The smaller
angle is 360 – 185 or 175o.
sin2 x + cos2 x = 1
(0.936)2 + cos2 x = 1
cos2 x = 1 – 0.876096
cos2 x = 0.123904
cos x = ±0.352
 Trig. Identity
Alternative Solution:
................................................
712. Which of the following is false?
A. tan θ =
B. csc θ =
C. sec θ =
D. sin θ =
Input cos (sin-1 (0.936)) in your calculator.
................................................
715. A man stands 20 meters away from a
building. If the angle of elevation to the top of the
building is 60o, how tall is the building?
A. 20√2 m
B. 20√3 m
C. 40 m
D. 20√5 m
Explanation:
Plot your entries in the 30-60-90 triangle and
solve. See item #706.
................................................
716. If csc x =
A.
, what is sin x?
B.
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C.
D.
1000 MMR
Explanation:
sin x is the reciprocal of csc x.
................................................
717. If sec x = , which of the following could be
sin x?
A.
B.
C.
D.
D. π
Solution:
160o x
= π
................................................
721. What is the reference angle of 156o?
A. 14o
B. 24o
C. 56o
D. 66o
Solution:
Since sec x = , then cos x = .
sin2 x + cos2 x = 1
sin2 x + ( )2 = 1
sin2 x +
Author: Victor A. Tondo Jr., LPT
720. Convert 160o to radians.
A. π
B. π
C. π
Explanation:
Since 156o is in QII, then the reference angle is
180 – 156 or 24o.
................................................
=1
sin2 x = 1 –
sin2 x =
sin x =
................................................
722. In which quadrants is the sine function
negative?
A. QI and QII
B. QII and QIII
C. QIII and QIV
D. QIV and QI
Explanation:
718. Which of the following is false?
A. cos A (tan A) = sin A
B. csc A (cos A) = cot A
C. sin A (sec A) = tan A
D. tan A (csc A) = cos A
Use the CAST mnemonic. See item #164.
................................................
Explanation:
Explanation:
Refer to the mnemonic hexagon in #712. Kahit
aling trig function ay product ng dalawang trig
functions na katabi niya.
................................................
The coterminal angle for 2019o is 219o (see item
#705). Since 219o is in QIII, then its reference
angle is 219 – 180 or 39o.
................................................
719. Which of the ff. is coterminal with 63o?
A. 7503o
B. 7303o
o
C. 7103
D. 6903o
724. A sniper is on top of a 20-meter cliff. He
spots a target at an angle of depression of 30o.
How far must his bullet travel to hit the target?
A. 20 meters
B. 20 √2 meters
C. 20 √3 meters
D. 40 meters
Solution:
A. 7503 – 63 = 7440 (not divisible by 360)
B. 7303 – 63 = 7240 (not divisible by 360)
C. 7103 – 63 = 7040 (not divisible by 360)
D. 6903 – 63 = 6840 (divisible by 360)
................................................
723. What is the reference angle if 2019o?
A. 19o
B. 39o
C. 51o
D. 59o
Solution:
x = 20 m
.: 2x = 40 m
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1000 MMR
725. A meter stick leans to a wall and reaches a
point 50 cm away from the wall. What is the
measure of the angle formed by the meter stick
and the wall?
A. 30o
B. 45o
C. 60o
D. 75o
Author: Victor A. Tondo Jr., LPT
728. Evaluate: 2 sin 30 – 4 cos 60 + 3 tan 45.
A. -10.2378
B. -12.1426
C. 2
D. 1
Solution:
2 sin 30 – 4 cos 60 + 3 tan 45
= 2 (0.5) – 4 (0.5) + 3 (1)
=1–2+3
=2
................................................
Solution:
=
sin x = 0.5
x = arcsin 0.5
x = 30o
................................................
729. Which of the following is true?
A. sin x = sin (-x)
B. cos x = cos (90 – x)
C. –sin x = sin (-x) D. tan x = (sin x)(cos x)
726. A tight rope connects the top of a 500 cm
pole to the ground. If the angle formed by the
ground and the tight rope is 45o, how long is the
tight rope?
A. 500 cm
B. 500√2 cm
C. 500√3 cm
D. 500√5 cm
Explanation:
Solution:
730. Which of the following is true?
A. sin x = csc (90 – x)
B. cos x = csc (90 – x)
C. tan x = cot (90 – x)
D. sec x = cos (90 – x)
=
sin 45 =
x sin 45 = 500
x=
A/C: –sin x = sin (-x)
B: cos x = sin (90 – x)
D. tan x = (sin x) ÷ (cos x)
................................................
Explanation:
x = 500√2
................................................
727. Which of the
following can be used
to solve for x in the
figure?
A. sin
B. cos
Definition of cofunctions. See #704.
................................................
731. Evaluate: lim
A. 2019
C. 0
2019
B. -2019
D. 1
Explanation:
The limit of a constant function is the constant
itself.
................................................
C. arcsin
D. arccos
Explanation:
732. Evaluate: lim (20 – 5x2)
A. 1
B. 15
C. -15
is the ratio of the adjacent side to the
hypotenuse. That’s the cosine value for x. Since
the angle x is missing, then we will use arccos .
Solution:
Substitute x with 1.
20(1) – 5(12) = 20 – 5 = 15
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D. 10
1000 MMR
733. Evaluate: lim (
A. 3
B. 18
3
–
5x2)
C. -18
D. 72
Solution:
Substitute x with 3.
33 – 5(32) = 27 – 45 = -18
................................................
734. Evaluate: lim
A. LDNE
B. 2
C. 4
D. 0
lim
B. x
D. x
-2
Explanation:
The given function is a rational function. It will
be undefined when the denominator is 0.
Therefore, 3x – 6 0, or x 2.
................................................
C. x > 0
(
)(
735. Evaluate: lim
A. +
B. -
C. LDNE
D. 2
Explanation:
Retain only the highest term of the numerator
and the denominator.
lim
 lim
= lim
D. x
2
736. Evaluate: lim
B. -
Explanation:
The given function is a radical function, more
specifically a square root function. If the
radicand 2018x – 2019 is negative, f(x) becomes
imaginary. That means 2018x – 2019 must be
greater than or equal to 0.
2018x – 2019 0
2018x 2019
x
................................................
739. Give the domain of f(x) = x2 – 20x + 19.
A. x | x
B. x 1, 9
C. x > -81
D. x -81
Since x approaches + , and the function is 2x2,
then the limit is + .
................................................
A. +
2019.
)
= lim
= lim ( + 2)
=4
................................................
C. LDNE
D.
Explanation:
Retain only the highest term of the numerator
and the denominator.
lim
A. x | x
C. x 2
738. Give the domain of f(x) = √2018
A. x | x
B. x
Solution:
lim
Author: Victor A. Tondo Jr., LPT
737. Give the domain of f(x) =
Explanation:
The given function has no denominator and no
radicand. Therefore f(x) exists for all real values
of x.
................................................
740. Give the range of f(x) = 2020.
A. y | y
B. y 2020
C. y = 2020
D. y = { }
Explanation:
 lim
lim
................................................
The given function is a constant function. That
means y = 2020 for all values of x.
................................................
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1000 MMR
741. Give the range of f(x) =
Author: Victor A. Tondo Jr., LPT
744. Give the range of f(x) =
A. y | y
C. y 2
A. y | y
B. y
C. y
D. y
B. y
D. y
-2
Solution:
-2
Solution:
First, express y in terms of x.
y=
3xy – 6y = 2x + 3
3xy – 2x = 6y + 3
x(3y – 2) = 6y + 3
x=
First, express y in terms of x.
y=
6xy + 7y = 5x - 4
6xy – 5x = -7y – 4
x (6y – 5) = -7y - 4
x=
The denominator 3y – 2 cannot be equal to 0.
3y – 2 0
3y 2
y
Shortcut:
The denominator 6y – 5 cannot be equal to 0.
6y – 5 0
6y 5
y
Shortcut:
Since the numerator and denominator are of the
same degree, just take the leading coefficients of
the numerator (2) and denominator (3).
Proof:
Since the numerator and denominator are of the
same degree, just take the leading coefficients of
the numerator (5) and denominator (6).
Proof:
 Cross-multiply.
6x + 9 6x – 12
9 + 12 6x – 6x
21 0
................................................
742. Give the range of f(x) = 20x – 20.
A. y | y
B. y 1
C. y = 20
D. y = { }
Explanation:
All linear functions have a range of y | y .
................................................
743. Give the range of f(x) = x125.
A. y | y
B. y 0
C. y 2019
D. y = { }
 Cross-multiply.
5(6x + 7) 6(5x – 4)
30x + 35 30x – 24
30x – 30x -24 – 35
0 -59
................................................
745. Give the range of f(x) = x2019.
A. y | y
B. y 0
C. y 2019
D. y = { }
Explanation:
(See item #743.)
................................................
Explanation:
The function has an odd degree (125). All odddegree functions have a range of y | y .
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1000 MMR
746. Give the range of f(x) = 2019x – 2019.
A. y | y
B. y 1
C. y = 2019
D. y = { }
Explanation:
All linear functions have a range of y | y .
................................................
B. y
C. y
D. y
Explanation:
(See item #741.)
................................................
748. Give the domain of f(x) =
A. x | x
C. x 2
B. x
D. x
Solution:
Let u = 2x + 3 so we can rewrite f(x) = u5.
Since du = 2, and f ’(x)= 5u4 du, then by
substitution,
f ’(x)= 5u4 du
f ’(x)= 5(2x + 3)4 (2)
f ’(x)= 10 (2x + 3)4
................................................
747. Give the range of f(x) =
A. y | y
Author: Victor A. Tondo Jr., LPT
751. Find the first derivative: f(x) = (2x + 3)5.
A. f ’(x)= 5 (2x + 3)4
B. f ’(x)= 10 (2x + 3)4
C. f ’(x)= 15 (2x + 3)4
D. f ’(x)= 20 (2x + 3)4
-3
752. Find f ‘(x) given f(x) = (3x2 – 2x + 1 )7.
A. f ’(x)= (42x – 14) (3x2 – 2x + 1)6
B. f ’(x)= (42x – 28) (3x2 – 2x + 1)6
C. f ’(x)= (42x + 14) (3x2 – 2x + 1)6
D. f ’(x)= (42x + 28) (3x2 – 2x + 1)6
Solution:
Explanation:
The given function is a rational function. It will
be undefined when the denominator is 0.
Therefore, 3x + 9 0, or x -3.
................................................
749. What is 0! equal to?
A. 0
B. 1
C. undefined
D. +
................................................
750. Give the range of f(x) = 2019x + 2020.
A. y | y
B. y 2020
C. y 2019
D. y 2020
Explanation:
The function is an exponential function
increased by a constant (2020). The exponential
part 2019x is asymptotic to 0 (it will come super
close to 0 but will never be equal to 0) and can
only be positive. Add this to the constant 2020
and you’ll get y 2020.
Let u = 3x2 – 2x + 1 so we can rewrite f(x) = u7.
Since du = 6x – 2, and f ’(x)= 7u6 du, then by
substitution,
f ’(x)= 7u6 du
f ’(x)= 7(3x2 – 2x + 1)6 (6x – 2)
f ’(x)= (42x – 14) (3x2 – 2x + 1)6
................................................
753. Find the maximum area that can be
enclosed by a rectangle given its perimeter
should only be 202 meters.
A. 2550 m2
B. 2550.25 m2
2
C. 2550.5 m
D. 2550.75 m2
Solution:
Max area for such a question is taken by making
the rectangle a square.
P = 4S
4S = 202
S = 50.5
A = S2
S2 = 50.52
50.52 = 2550.25
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1000 MMR
754. Find the lowest possible value of the
function f(x) = x2 – 8x + 7.
A. -10
B. -9
C. -8
D. -7
Author: Victor A. Tondo Jr., LPT
757. Find the slope of the line tangent to
y = 3x5 – 4x3 + 5x2 – 10x + 9 at x = 2.
A. 50.5
B. 101
C. 151.5
D. 202
Solution:
Solution:
Get the critical value of x first by getting the first
derivative and equating it to 0.
Find f ’(2).
f ’(x) = 15x4 – 12x2 + 10x – 10
f ’(2) = 15(2)4 – 12(2)2 + 10(2) – 10
f ’(2) = 240 – 48 + 20 – 10
f ’(2) = 202
................................................
f ‘(x) = 2x – 8
2x – 8 = 0
2x = 8
x=4
 Critical Point
Get the minimum by get f(crit. value).
f(4) = 42 – 8(4) + 7
f(4) = 16 – 32 + 7
f(4) = -9
758. Find the remainder when 1998! is divided
by 2000.
A. 1024
B. 256
C. 64
D. 0
Explanation:
Shortcut:
The minimum (for A>0) or maximum (for A<0)
of f(x) = Ax2 + Bx + C is given by C –
.
C – = 7 – = 7 – 16 = -9
................................................
755. Find the instantaneous rate of change for
f(x) = 5x3 – 2x2 + 7x – 9 at x = 2.
A. 37
B. 48
C. 59
D. 70
Solution:
Find f ’(2).
f ’(x) = 15x2 – 4x + 7
f ’(2) = 15(22) – 4(2) + 7
f ’(2) = 59
................................................
756. Find the instantaneous rate of change for
f(x) = 4x3 + 5x2 – 7x – 11 at x = 1.
A. -15
B. 0
C. 15
D. 30
Solution:
Find f ’(1).
f ’(x) = 12x2 + 10x – 7
f ’(1) = 12(12) + 10(1) – 7
f ’(1) = 15
1998! = 1 x 2 x 3 x … x 1000 x …
Therefore 1998 is divisible by 2000.
Also Note:
(n – 2)! is divisible by n if and only if n is a
composite number greater than 4.
................................................
759. In Mathematics, what does the delta ∆
symbol mean or refer to?
A. the change in
B. the summation of
C. therefore
D. belonging to
................................................
760. In Mathematics, what does the sigma
symbol mean or refer to?
A. the change in
B. the summation of
C. therefore
D. belonging to
................................................
761. In Mathematics, what does the epsilon
symbol mean or refer to?
A. such that
B. element of
C. therefore
D. ergo
................................................
762. In set theory, what does the symbol
or refer to?
A. not O
B. not zero
C. does not
D. empty
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mean
1000 MMR
763. Which of the following is ALWAYS an odd
number?
A. The difference of two odd numbers.
B. The sum of two odd numbers.
C. The product of two odd numbers.
D. The product of two even numbers.
................................................
764. Given 2x + 2x = 2y, express y in terms of x.
A. y = 2x
B. y = x2
C. y = x + 2
D. y = x + 1
Explanation:
2x + 2x = 2y
2(2x) = 2y
2x+1 = 2y
................................................
765. In set theory, what does the symbol mean
or refer to?
A. no elements
B. common elements
C. all elements
D. half of the elements
................................................
766. In Mathematics, what does the symbol
mean?
A. since
B. prove
C. therefore
D. previously
................................................
767. Which of the following is the Geometric
symbol for congruent?
A.
B.
C.
D. //
................................................
768. Which of the following is the obelus sign?
A. !
B. ÷
C. ⊙
D. θ
................................................
769. Given X = {1, 4, 16, 64} and Y = {1, 2, 3, 4},
find X Y.
A. {1, 4}
B. {16, 64}
C. {2, 3}
D. {1, 2, 3, 4, 16, 64}
Explanation:
X – Y means tanggalin mo sa X lahat ng elements
na meron din sa Y.
................................................
771. Given X = {1, 2, 4, 8} and Y = {1, 2, 3, 4}, find
Y – X.
A. {1, 4, 8}
B. {3}
C. {3, 8}
D. {1, 2, 3, 4, 8}
................................................
772. Given X = {6, 5, 4, 3} and Y = {1, 2, 3, 4}, find
X Y.
A. {3, 4}
B. {6, 5}
C. {1, 2}
D. {1, 2, 3, 4, 5, 6}
Explanation:
X Y means the union of all the elements of X
and Y.
773. Which of the
following
represents the
shaded region in
the Venn diagram?
A. A B
C. A – B
B. A B
D. B – A
774. Which of the
following
represents the
shaded region in
the Venn diagram?
A. A’ B’
C. (A B)’
B. A’ B’
D. (A B)’
Explanation:
Explanation:
X
Author: Victor A. Tondo Jr., LPT
770. Given X = {1, 2, 4, 8} and Y = {1, 2, 3, 4}, find
X – Y.
A. {1, 4, 8}
B. {8}
C. {3, 8}
D. {1, 2, 3, 4, 8}
Y refers to the common elements of X and Y.
The area that is not shaded is A B. The shaded
region is its complement, (A B)’.
This is a free reviewer. All rights reserved.
1000 MMR
775. In a class of 50, there are 20 students who
excel in Math, 24 who excel in Science, and 4 who
excel in both Math and Science. How many of
them excel in neither?
A. 2
B. 4
C. 7
D. 10
Author: Victor A. Tondo Jr., LPT
Solution:
7 x 6 x 5 x 4 = 840
................................................
50 – (20 – 4) – (24 – 4) – 4 = 10
780. Using the digits 0, 1, 2, 3, 4, 5, and 6 without
repetition, how many 3-digit numbers can be
made?
A. 360
B. 180
C. 90
D. 45
Solution B:
Solution:
50 – (20 + 24 – 4) = 10
................................................
6 x 6 x 5 = 180
................................................
776. There are 50 students in a class. 39 of them
have a cellphone, 17 of them have a laptop, and 5
of them have neither. How many have both a
cellphone and a laptop?
A. 6
B. 7
C. 9
D. 11
781. Using the digits 0, 1, 2, 3, 4, 5, 6, and 7
without repetition, how many 4-digit numbers
can be made?
A. 1680
B. 1470
C. 840
D. 735
Solution A:
Solution:
(39 + 17) – (50 – 5) = 56 – 45 = 11
................................................
777. Rayon has 10 shirts, 5 pairs of pants, and 4
pairs of shoes. How many ways can he pick what
to wear for today?
A. 19
B. 20
C. 190
D. 200
Solution:
10 x 5 x 4 = 200
................................................
778. Using the digits 1, 2, 3, 4, 5, and 6 without
repetition, how many 3-digit numbers can be
made?
A. 480
B. 240
C. 120
D. 60
Solution:
6 x 5 x 4 = 120
................................................
779. Using the digits 1, 2, 3, 4, 5, 6, and 7 without
repetition, how many 4-digit numbers can be
made?
A. 1680
B. 840
C. 420
D. 210
Solution:
7 x 7 x 6 x 5 = 1470
................................................
782. Using the digits 0, 1, 2, 3, 4, 5, 6, and 7
without repetition, how many 3-digit numbers
can be made?
A. 672
B. 588
C. 336
D. 294
Solution:
7 x 7 x 6 = 294
................................................
783. Using the digits 0, 1, 2, 3, 4, 5, and 6 without
repetition, how many odd 3-digit numbers can
be made?
A. 180
B. 150
C. 90
D. 75
Solution:
5 x 5 x 3 = 75
................................................
784. Using the digits 0, 1, 2, 3, 4, 5, 6, and 7
without repetition, how many odd 4-digit
numbers can be made?
A. 1470
B. 1440
C. 735
D. 720
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1000 MMR
Solution:
6 x 6 x 5 x 4 = 720
................................................
785. Using the digits 0, 1, 2, 3, 4, 5, 6, and 7
without repetition, how many odd 3-digit
numbers can be made?
A. 336
B. 288
C. 168
D. 144
Solution:
6 x 6 x 4 = 144
................................................
786. Using the digits 0, 1, 2, 3, 4, 5, 6, and 7
without repetition, how many even 3-digit
numbers can be made?
A. 162
B. 156
C. 150
D. 144
D. 3
Solution:
log6 8 + 3 log6 3
= log6 8 + log6 33
= log6 8 + log6 27
= log6 (8x27)
= log6 216
=3
................................................
790. How many ways can the word ABABA be
rearranged?
A. 120
B. 60
C. 20
D. 10
Solution:
!
= 10
(See item #186.)
................................................
! !
Solution:
When asked about even n-digit numbers tapos
may 0 sa usable digits, do this:
3-digit #s:
7 x 7 x 6 = 294
odd 3-digit #s:
6 x 6 x 4 = 144
even 3-digit #s:
150
................................................
787. Using the digits 0, 1, 2, 3, 4, 5, 6, and 7
without repetition, how many even 4-digit
numbers can be made?
A. 765
B. 750
C. 735
D. 720
Solution:
4-digit #s:
7 x 7 x 6 x 5 = 1470
odd 4-digit #s:
6 x 6 x 5 x 4 = 720
even 4-digit #s:
750
................................................
788. Using the digits 0, 1, 2, 3, 4, 5, and 6 without
repetition, how many even 3-digit numbers can
be made?
A. 120
B. 105
C. 90
D. 75
Solution:
3-digit #s:
odd 3-digit #s:
even 3-digit #s:
Author: Victor A. Tondo Jr., LPT
789. Evaluate: log6 8 + 3 log6 3.
A. 24
B. 14
C. 9.5
791. There are 10 different beads and a locking
mechanism to be used in making a bracelet. How
many different bracelet patterns can be made?
A. 3,628,800
B. 362,880
C. 1,814,400
D. 181,440
Solution:
n = 11 here since the locking mechanism is
considered as another unit
(
)!
!
The formula is
. Thus, there are or
1,814,400 different patterns.
................................................
792. There are six different Math books, four
different Science books, and two different
English books to be arranged on a shelf. How
many ways can this be done?
A. 12C3
B. 12!
C. 48
D. 3!
Explanation:
There are 12 different books to be arranged in a
linear fashion. There are 12! ways to do this.
6 x 6 x 5 = 180
5 x 5 x 3 = 75
105
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1000 MMR
793. There are six different Math books, four
different Science books, and two different
English books to be arranged on a shelf. If books
of the same subject must be together, how many
ways can this be done?
A. 479,001,600
B. 207,360
C. 34560
D. 144
Explanation:
There are 3 x 2 x 1 or 3! ways to arrange the
subjects Math, Science, and English. There are 6!
ways to arrange the Math books, 4! ways for
Science, and 2! ways for English.
3! x 6! x 4! x 2! = 6 x 720 x 24 x 2 = 207360
................................................
794. There are six people to be seated on a bench
for a picture. A certain couple, Vic and Rowena,
are to be seated next to each other. How many
ways can this be done?
A. 120
B. 240
C. 360
D. 720
Explanation:
Let Vic and Rowena take one unit, and then add
the other four people to make five units. Since
Vic and Rowena can be arranged in 2! ways and
the five units can be arranged in 5! ways, then
the total number of ways this can be done is the
product of 2! and 5!, or 240.
................................................
795. There are six people to be seated on a bench
for a picture. A certain trio, Vic, Rayon, and Aira,
do not want to be separated. How many ways
can this be done?
A. 60
B. 72
C. 144
D. 180
Explanation:
Author: Victor A. Tondo Jr., LPT
796. TNB Gaming has five “carry” players, four
“offlane” players, and six “support” players. If a
game calls for two carry, one offlane, and two
support players, how many possible lineups do
they have?
A. 1200
B. 600
C. 300
D. 150
Solution:
5C2 x 4C1 x 6C2 = 10 x 4 x 15 = 600
................................................
797. If today is a Sunday, what day is 125 days
from now?
A. Friday
B. Saturday
C. Sunday
D. Monday
Explanation:
Exactly 18 weeks or 126 days from now would
be Sunday again, therefore, 125 days from now
would be Saturday.
................................................
798. If today is a Monday, what day is 200 days
from now?
A. Friday
B. Saturday
C. Sunday
D. Monday
Explanation:
Exactly 28 weeks or 196 days from now would
be Monday again, therefore, 200 days from now
would be Friday.
................................................
799. If three-fourths of a number is 33 more than
one-fifth of itself, then what is the number?
A. 80
B. 60
C. 40
D. 20
Solution:
Let the trio take one unit, and then add the other
three people to make four units. The trio can be
arranged in 3! ways and the four units can be
arranged in 4! ways. 3! x 4! = 144.
................................................
= + 33  Multiply equation by 20.
15x = 4x + 660
11x = 660
x = 60
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1000 MMR
800. What time is 2019 hours after 3:00PM?
A. 6:00 AM
B. 12:00 NN
C. 6:00 PM
D. 12:00 MN
Explanation:
2019 ÷ 24 = 84 r. 3
3 hours after 3 PM is 6 PM
................................................
801. In a certain fastfood chain, soft drinks are
served in Small, Medium, and Large cups.
What level of data is this?
A. Nominal
B. Ordinal
C. Interval
D. Ratio
Explanation:
The data is presented according to the order of
sizes, thus, ordinal.
................................................
802. The prime factorization of a number is
given as 23 x 52 x 133. How many factors does it
have?
A. 96
B. 48
C. 36
D. 18
Solution:
Add 1 to the exponent of each prime factor, then
multiply to get the number of factors.
(3 + 1)(2 + 1)(3 + 1) = 4 x 3 x 4 = 48
................................................
803. The prime factorization of a number is
given as 22 x 32 x 7 x 11. How many factors does
it have?
A. 96
B. 48
C. 36
D. 18
Solution:
(2 + 1)(2 + 1)(1 + 1)(1 + 1) = 36
................................................
804. There are ten red, nine blue, and six black
pens in a bag. If a pen is randomly drawn from
the bag, what is the probability that it is red?
A.
B.
C.
D.
Author: Victor A. Tondo Jr., LPT
805. There are ten red, nine blue, and six black
pens in a bag. If a pen is randomly drawn from
the bag, what is the probability that it is black?
A. 0.06
B. 0.6
C. 0.24
D. 0.7
................................................
806. What does a probability of 0 pertain to?
A. an impossible event
B. a very unlikely event
C. a very likely event
D. a sure event
................................................
807. Simplify: √10 + 2√21.
A. √2 + √5
B. √3 + √7
C. √3 √2
D. √5 √2
Explanation:
(√a + √b)2 = a + b + 2√
Therefore,
√a + b + 2√ab = √a + √b
In the problem, a + b = 10 and ab = 21. The only
pair that has a sum of 10 and a product of 21 is 3
and 7.
................................................
808. Simplify: √11
A. √6 √5
C. √3 2√2
2√30.
B. √6 + √5
D. 1
Explanation:
( √a
√b)2 = a + b – 2√
Therefore,
√a + b 2√ab = √a
√b
In the problem, a + b = 11 and ab = 30. The only
pair that has a sum of 11 and a product of 30 is 6
and 5.
................................................
809. ∫(3 + 4)5 dx
A. (3x + 4)6+ c
B.
(3x + 4)6+ c
C. 5(3x + 4)4+ c
D.
(3x + 4)6+ c
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1000 MMR
Explanation:
∫(3 + 4)5 dx = (
)( )
(3x + 4)5+1 + c
∫(3 + 4)5 dx = (3x + 4)6 + c
................................................
810. ∫(6 + 6
6) dx
A. 2x3 + 3x2 + 6x + c
C. 2x3 + 3x2 – 6x + c
Explanation:
B. 2x3 – 3x2 + 6x + c
D. 2x3 – 3x2 – 6x + c
Solution:
∫(6
+6
6) dx
=
+
– 6x + c
= 2x3 + 3x2 – 6x + c
................................................
811. ∫
dx
A. + c
B. + c
C.
D.
+c
dx = ∫
dx
=
+c
=
+c
+c
= +c
................................................
812. ∫
A.
C.
dx
+c
+c
+c
D.
+c
814. Rayon is going to present the sales of six
different teams in the company from January to
June. Which graph would be best suited for this?
A. bar graph
B. line graph
C. pie graph
D. pictograph
................................................
816. What is the 7th decile of the following data?
14
15
16
20
24
25
26
26
27
29
31
33
34
34
35
A. 31
B. 31.4
C. 31.5
Solution:
Solution:
∫
B.
Line graphs are best suited for presenting data
on development over time.
................................................
815. Ms. Rowena is tasked to send 8 randomly
selected students from her class of 40 to the
clinic for drug test. She decides to let the
students count off from 1 to 5, after which she
randomly picks a number from 1 to 5. All
students whose number is that which she picked
will be sent to the clinic. What sampling method
did Ms. Rowena use?
A. cluster
B. systematic
C. stratified
D. quota
................................................
Solution:
∫
Author: Victor A. Tondo Jr., LPT
813. Mr. dela Cruz is going to present the growth
of a certain plant over a span of 10 weeks. Which
graph would be best suited for this?
A. bar graph
B. line graph
C. pie graph
D. pictograph
dx = ∫
dx
=
+c
=
+c
=
+c
(15 + 1)th data or 11.2th data
11th data is 31, while 12th data is 33
11.2th data = 31 + 0.2(33-31)
= 31 + 0.4
= 31.4
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D. 32
1000 MMR
817. Find the remainder when f(x) is divided by
(x – 3) given f(x) = x3 – 3x2 + 5x – 13.
A. 1
B. 2
C. 3
D. 4
Solution:
Since the divisor is x – 3, then the remainder is
given by f(3).
f(3) = 33 – 3(32) + 5(3) – 13
f(3) = 27 – 27 + 15 – 13
f(3) = 2
................................................
818. (x2 + 3x – 2)4 = _________
A. (x2 + 3x – 2)3
B. 4(x2 + 3x – 2)3
C. (8x + 12) (x2 + 3x – 2)3
D. (2x + 3) (x2 + 3x – 2)3
B. (
)
C. (
)
D. (
)
Solution:
u = 2x + 1
v = 3x – 4
du = 2
dv = 3
=
(
)( ) (
=
(
(
) (
=(
.: du = 2x + 3
=
du
= 4 (x2 + 3x – 2)3 (2x + 3)
= (8x + 12) (x2 + 3x – 2)3
................................................
A.
)
(
)( )
)
)
)
)
................................................
Let u = x2 + 3x – 2
819.
A. (
( ) =
Solution:
u4
Author: Victor A. Tondo Jr., LPT
821. (
) = _________
4u3
√
2 + 3 = _________
Solution:
B. √
√
C. √
D.
14/(17 + 19 + 14) = 14/50
14/50 = 28%
................................................
√
Solution:
Let u = x3 – 2x + 3
.: du = 3x2 – 2
u1/2 = ½ u-1/2 du
= (√
822. There are 17 red bags, 19 green bags, and
14 blue bags in a store. What percent of the bags
is blue?
A. 38%
B. 34%
C. 28%
D. 27%
)(3x2 – 2)
823. Which of the following is true?
A. (sin x) (cot x) = sec x
B. (cos x) (csc x) = tan x
C. (tan x) (csc x) = sec x
D. (cot x) (sec x) = cos x
= √
................................................
820. (2x + 3)(4x – 5) = _________
A. 8x2 + 2x + 15
B. 8x2 + 2x – 15
C. 8x2 – 2x – 15
D. 8x2 – 2x + 15
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1000 MMR
Explanation:
Refer to the trigonometric hexagon. Any
trigonometric function is the product of the two
functions beside it.
Author: Victor A. Tondo Jr., LPT
827. If f(x) = x2 – 2x + 3 and g(x) = x + 1, find
fog(x).
A. x2 – 1
B. x2 – 2x + 4
C. x2 – 3x + 5
D. x2 + 2
Solution:
f(g(x)) = f(x + 1)
= (x + 1)2 – 2 (x + 1) + 3
= (x2 + 2x + 1) – (2x + 2) + 3
= x2 + 2
................................................
That means:
(sin x) (cot x) = cos x;
(cos x) (csc x) = cot x;
(tan x) (csc x) = sec x; and
(cot x) (sec x) = csc x.
................................................
828. How many 3-digit numbers can be formed
using the digits 0, 1, 2, 3, 4, 5, and 6 if repetition
is not allowed?
A. 160
B. 180
C. 200
D. 210
Solution:
We are looking for a number x such that x leaves
a remainder of 4 when divided by 10. Simply put,
the number’s units digit is 4.
................................................
Use FCP (Fundamental Counting Principle):
__ x __ x __
For the first digit, we cannot use 0. That means
we only have 6 choices for the first digit.
For the second digit, we can now use 0. Since we
have already used one digit for the first, that
means we have 6 choices for the second digit.
For the last digit, since we have already used two
digits, we only have 5 choices.
6 x 6 x 5 = 180
................................................
825. Which of the following could be the value of
x if x 7 (mod 9)?
A. 146
B. 149
C. 151
D. 154
829. What is the longest side of ∆VAC if m V =
45o and m C = 65o?
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅
A. VC
B. AC
C. VA
D. CA
Explanation:
Explanation:
We are looking for a number x such that x leaves
a remainder of 7 when divided by 9.
................................................
m A = 180 – (45 + 65) = 70
The longest side is opposite the largest angle, A.
................................................
̅̅̅̅ bisects ABC and m ABF = 34o,
826. Given BF
find m ABC.
A. 17o
B. 34o
C. 51o
D. 68o
830. Which quadrilateral has two congruent
diagonals that are perpendicular to each other?
A. kite
B. isosceles trapezoid
C. rectangle
D. rhombus
824. Which of the following could be the value of
x if x 4 (mod 10)?
A. 46
B. 49
C. 52
D. 54
Explanation:
Explanation:
ABF is formed after the bisection of ABC. That
means ABF is half of ABC, or ABC is twice of
ABF.
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1000 MMR
831. If the length of a rectangle is increased by
30% while the width is decreased by 30%, what
will happen to its area?
A. It stays the same.
B. It is increased by 9%.
C. It is decreased by 9%.
D. It is decreased by 6%.
Author: Victor A. Tondo Jr., LPT
835. The salary of 5 men for 6 days is P9,000.
How much is the salary of 7 men for 8 days?
A. P15,000
B. P15,600
C. P16,800
D. P17,400
Solution:
(L x 1.3) (W x 0.7) = (0.91 x LW)
................................................
5 men x 6 days = 30 units worth a total of P9000.
That means each unit is worth P300.
7 men x 8 days = 56 units.
56 units at P300 per unit costs P16800.
................................................
832. Rayon had an average of 93 on his first five
Math tests. After taking the next test, his average
increased to 94. Find his most recent grade.
A. 96
B. 97
C. 98
D. 99
836. The average grade of eleven students is 84.
If the average of seven of these students is 82,
what is the average of the other four students?
A. 88
B. 87.5
C. 86.5
D. 86
Solution:
Solution:
New Score = (New Number)(New Average) –
(Old Number)(Old Average)
New Score = 6(94) – 5(93) = 564 – 465 = 99
................................................
Sum of grades of 11 students:
11 x 84 = 924
Sum of grades of 7 students:
7 x 82 = 574
Sum of grades of other four:
924 – 574 = 350
Average of grades of other four: 350 ÷ 4 = 87.5
................................................
Solution:
833. A bus drove for 7 hours at 73 kph and 3
hours at 88 kph. What was its average speed?
A. 75.5 kph
B. 76.5 kph
C. 77.5 kph
D. 78.5 kph
Solution:
Get the total distance and the total time first.
7 hrs x 73 kph = 511 km
3 hrs x 88 kph = 264 km
Total distance = 775 km, total time = 10 hrs
Average spd =
=
= 77.5 kph
................................................
834. Fifteen guests shake hands with each other.
If each guest is to shake hands with all the other
guests, how many handshakes will be made?
A. 105
B. 150
C. 210
D. 225
Solution:
15C2 = 105
837. A radius of a circle is 25 cm long. How long
is its longest chord?
A. 20√2 cm
B. 40 cm
C. 50 cm
D. It depends,
Explanation:
The longest chord is the diameter, and the radius
is half the diameter.
................................................
838. Find the largest area of a rectangular piece
of land that can be enclosed with 800 meters of
fencing material.
A. 40,000 m2
B. 56,250 m2
C. 62,500 m2
D. 80,000 m2
Solution:
Instead of jumping to differential calculus
(minima and maxima) to solve this, simply make
the rectangle a square. That’s the shortcut for
this kind of question.
This is a free reviewer. All rights reserved.
1000 MMR
839. Which statistical test is used for testing for
relationship between two variables?
A. ANOVA
B. t-test
C. Pearson R
D. Chi Square
Explanation:
Observed vs Expected: Chi Square
Relationship: Pearson R (R for relationship)
Group differences: ANOVA (variance =
differences)
Comparing sets of normal distributions: T-test
................................................
Author: Victor A. Tondo Jr., LPT
842. In which non-Euclidean model for geometry
can we have any given line ℓ and a point A which
is not on ℓ, wherein all lines through A will
intersect ℓ?
A. hyperbolic
B. elliptic
C. Saccheri
D. Pythagorean
Explanation:
In Euclidean geometry, only one line will pass
through A. In elliptic geometry, all lines will pass
through A and intersect ℓ.
................................................
2 +4
5
5
840. Given ( ) = {
= 5,
9
5
(
).
find lim
A. 5
B. 14
C. 16
D. limit does not exist
843. If A is at (-9, 11) and B is at (6,-9), find C if C
is three-fifths the way from A to B.
A. (1, -1)
B. (0, 1)
C. (1, 1)
D. (0, -1)
Solution:
Since C is three-fifths the way from A to B, then
its coordinates are:
x = -9 + [6 – (-9)] = -9 + 9 = 0
Limit from the left: 2(5) + 4 = 14
Limit from the right: 52 – 9 = 16
Since the limits are not equal, then the limit does
not exist.
................................................
841. Find the equation of the line passing
through (3, 7) and (-3, -5).
A. y = 2x + 1
B. y = 3x + 1
C. y = 2x – 1
D. y = 3x – 1
Solution:
y = 11 + [(-9) – 11] = 11 + (-12) = -1
................................................
844. Find the vertex of y = 5x2 – 4x – 7.
A. ( , )
B. ( , )
C. ( , )
D. ( ,
)
Solution:
Solution:
Two-point form of linear equations:
y – y1 =
(x x )
y–7=
(x 3)
y – 7 = 2(x – 3)
y – 7 = 2x – 6
y = 2x + 1
................................................
The vertex is at (h,k) where
h=
h=
;
(
k=c)
( )
;
k = (-7)
h=
;
k = (-7)
h=
;
k = (-7)
h= ;
k = (-7)
h= ;
k=
This is a free reviewer. All rights reserved.
(
)
( )
1000 MMR
845. Mr. G sold 70% of his chickens and still had
129 chickens left. How many chickens did he
have originally?
A. 344
B. 430
C. 598
D. 860
Solution:
129 = 30% x Original number of chickens
129 ÷ 0.3 = Original number of chickens = 430
................................................
846. Find the average rate of change of
y = x3 – 6x2 + 5x – 20 from x = 1 to x = 6.
A. 5
B. 6
C. 7
D. 8
Solution:
Average Rate of Change =
Author: Victor A. Tondo Jr., LPT
Solution:
Let x = larger number
y = smaller number
x + y = 73
(+) x – y = 41
2x = 114
x = 57
................................................
850. An item is sold for P7,280 after being
marked down by 30%. What was its original
price?
A. P11,400
B. P10,900
C. P10,400
D. P9,900
Solution:
(
)
(
(
)
)
Average Rate of Change =
=6
................................................
847. What is the remainder when
534,214,557,989,215 is divided by 4?
A. 1
B. 2
C. 3
D. 0
(100% - 30%) OP = 7280
70% OP = 7280
0.7 OP = 7280
OP = 7280 ÷ 0.7
OP = 10,400
................................................
851. Which of the following lines passes through
the point (-4, 5)?
Explanation:
A. y = x + 1
C. y = 1 – x
Simply divide the last two digits, 15, by 4 and get
the remainder.
................................................
Solution:
848. What do you call an equation in which only
integer solutions are allowed?
A. Diophantine equations
B. Euclidean equations
C. Fibonacci equations
D. Newtonian equations
................................................
849. The sum of two numbers is 73 and their
difference is 41. What is the larger number?
A. 65
B. 61
C. 57
D. 53
B. y = 2x – 3
D. y = 3 – 2x
Simply substitute x and y from your point. If the
equation holds true, then the given line passes
through your point.
y=x+1

5 -4 + 1
y = 2x – 3

5 2(-4) – 3
y=1–x

5 = 1 – (-4)
y = 3 – 2x

5 3 – 2(-4)
................................................
852. Which of the following is outside the circle
defined by the equation (x + 2)2 + y2 = 50?
A. (-9, 1)
B. (-7, 5)
C. (-5, 7)
D. (2, 4)
This is a free reviewer. All rights reserved.
1000 MMR
Solution:
Author: Victor A. Tondo Jr., LPT
Explanation:
A point is outside the circle when its distance
from the center is greater than the radius.
A: (-9, 1)  (-9 + 2)2 + 12 = 50
This point is ON the circle.
The third side of any triangle must be between
the difference and the sum of the two other
sides.
................................................
B: (-7, 5)  (-7 + 2)2 + 52 = 50
This point is ON the circle.
C: (-5, 7)  (-5 + 2)2 + 72 > 50
This point is OUTSIDE the circle.
D: (2, 4)  (2 + 2)2 + 42 < 50
This point is IN the circle.
................................................
853. Which of the following equations pertain to
a parabola?
A. y2 – 5y = x2 + 4
B. x2 + 5x – 4y = 13
C.
(
)
D.
(
)
+
(
)
=1
(
)
=1
Explanation:
A parabola’s equation includes two variables,
one of which is squared.
................................................
854. Which of the following equations pertain to
a parabola that opens to the left?
A. -2(y + 7) = (x – 3)2
B. 5(y – 9) = (x – 5)2
C. (y – 4)2 = 2(x + 7)
D. (y + 3)2 = -3(x + 4)
Explanation:
The parabola opens to the left when the squared
variable is y and the coefficient of x is negative.
................................................
855. The lengths of two sides of a triangle are 7
inches and 15 inches. Which of the following
represents x, the possible length in inches of the
remaining side of the triangle?
A. 8 < x < 22
B. 8 x 22
C. x < 8 or x > 22
D. x 8 or x 22
856. Rayon has 46 coins in P1 and P5
denominations for a total worth of P186. How
many P5 coins does he have?
A. 25
B. 30
C. 35
D. 40
Solution:
Let x = number of P5 coins
.: 46 – x = number of P1 coins
5(x) + 1(46 – x) = 186
5x + 46 – x = 186
5x – x = 186 – 46
4x = 140
x = 35
................................................
857. If x = 2 and y = 3, what is x2 – y2 + 3xy?
A. 31
B. 13
C. -5
D. -13
Solution:
x2 – y2 + 3xy = 22 – 32 + 3(2)(3)
x2 – y2 + 3xy = 4 – 9 + 18 = 13
................................................
858. In an arithmetic sequence, the first term is
200 and the common difference is -3. What is the
64th term?
A. 20
B. 17
C. 14
D. 11
Solution:
An = A1 + (n – 1)d
A64 = A1 + (64 – 1)(-3)
A64 = 200 + 63(-3)
A64 = 200 – 189
A64 = 11
................................................
859. In an arithmetic sequence, the 23rd term is
157 and the common difference is -4. What is the
34th term?
A. 113
B. 117
C. 121
D. 125
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1000 MMR
Solution:
An = Am + (n – m)d
A34 = A23 + (34 – 23)(-4)
A34 = 157 + 11(-4)
A34 = 157 + (-44)
A34 = 113
................................................
860. Find the inverse of y = x2 – 6x + 5.
A. √ + 4 + 3
B. √ + 4 – 3
C. √
4+3
D. √
4–3
Solution:
x2
y = – 6x + 5
y – 5 = x2 – 6x
y – 5 + 9 = x2 – 6x + 9
y + 4 = x2 – 6x + 9
y + 4 = (x – 3)2
3)
√ + 4 = √(
√ +4=x–3
√ +4+3=x
√ + 4 + 3 = y-1
................................................
e4
861. ln = _____
A. 24.1295
C. 6.032375
B. 12.06475
D. 4
Explanation:
ln ea = a
................................................
862. eln 5 = _____
A. 5
B. 6.25
C. 12.5
D. 25
Explanation:
eln a = a
................................................
863. What is ln e?
A. 0
B. 1
C. e
D. 10
................................................
Author: Victor A. Tondo Jr., LPT
865. Which of the following Mathematicians had
a method “Sieve” for identifying prime numbers?
A. Pythagoras
B. Euclid
C. Eratosthenes
D. Archimedes
................................................
866. Which Mathematician developed a formula
for finding the area of a triangle using its side
lengths?
A. Hipparchus
B. Pythagoras
C. Heron
D. Euclid
................................................
867. Who refined and perfected the decimal
place value number system?
A. Greeks
B. Romans
C. Chinese
D. Indians
................................................
868. This Mathematician invented natural
logarithms.
A. Isaac Newton
B. John Napier
C. Marin Mersenne D. Rene Descartes
................................................
869. Prime numbers that are 1 less than a power
of 2 are called ______.
A. Pythagorean Primes
B. Mersenne Primes
C. Fermat Primes
D. Pascal Primes
................................................
870. Which Mathematician developed the
triangle of binomial coefficients?
A. Gottfried Leibniz
B. Isaac Newton
C. Blaise Pascal
D. Johann Bernoulli
................................................
871. Two Mathematicians are known for the
theory of hyperbolic geometry although they
worked independently from each other. Who are
these Mathematicians?
A. Euclid and Pythagoras
B. Euclid and Descartes
C. Lobachevsky and Bolyai
D. Babbage and Bolyai
864. Who are the first to use papyrus?
A. Sumerians
B. Egyptians
C. Chinese
D. Greeks
This is a free reviewer. All rights reserved.
1000 MMR
872. Which Mathematician developed the use of
AND, OR, and NOT operators in Algebra?
A. George Boole
B. Janos Bolyai
C. Charles Babbage
D. August Mobius
................................................
Author: Victor A. Tondo Jr., LPT
Solution:
873. Which Mathematician is known for his last
theorem?
A. Andrew Wiles
B. Pierre de Fermat
C. John Wallis
D. Rene Descartes
................................................
Average speed =
Average speed = 59 kph
................................................
874. Who have the first fully-developed base-10
number system in use?
A. Romans
B. Egyptians
C. Indians
D. Sumerians
................................................
Explanation:
2 x 48 = 96
3 x 58 = 174
5 x 64 = 320
Total distance = 590
Average speed =
878. Find the domain of y = 3x.
A. x | x
B. x 0
C. x 3
D. x 0
The domain of any exponential function – a
function where a polynomial in terms of x is used
as the exponent of a constant – is x | x .
................................................
875. Notched tally bones proved their use of
Mathematics around 35000 BCE.
A. Africans
B. Asians
C. Europeans
D. Americans
................................................
879. Convert
A. 70o
C. 210o
876. Which sampling method is best used when
the population has subgroups?
A. systematic sampling
B. stratified sampling
C. quota sampling
D. cluster sampling
rad x
= 105o
880. Find the volume of a cylinder whose radius
is 50 cm and height is 24 cm.
A. 20,000 π cm3
B. 30,000 π cm3
C. 45,000 π cm3
D. 60,000 π cm3
Explanation:
Solution:
The root word of the word stratified is stratum,
meaning subgroups.
................................................
V = π r2 h
= 502 (24) π
= 60,000 π cm3
................................................
877. Mr. Lazada drove for 2 hours at a speed of
48 kph, 3 hours at 58 kph, and then 5 hours at 64
kph. What was his average speed?
A. 57 kph
B. 58 kph
C. 59 kph
D. 60 kph
rad to degrees.
B. 105o
D. 420o
Solution:
881. Find the altitude to the hypotenuse of a
right triangle whose legs measure 16 cm and 30
cm.
A.
cm
B. 240 cm
C. 34√2 cm
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D. 24√3 cm
1000 MMR
Solution:
Author: Victor A. Tondo Jr., LPT
Explanation:
Find the hypotenuse first.
295o in in QIV. Therefore, its reference angle is
360 – 295 = 65o.
................................................
162 + 302 = h2
256 + 900 = h2
1156 = h2
34 = h
886. Which numerical system is sexagesimal
(base-60)?
A. Roman
B. Babylonian
C. Mayan
D. Hindu-Arabic
Altitude to the Hyp =
Altitude to the Hyp =
(
)
Altitude to the Hyp =
................................................
882. How many line segments can be made from
25 non-collinear points?
A. 900
B. 600
C. 450
D. 300
Solution:
25C2 = 300
................................................
883. Dividing by 0.25 is the same as multiplying
by which number?
A. 2
B. 4
C. 5
D. 16
Explanation:
Just use 1 as your test number.
1÷ 0.25 = 4
................................................
884. Find the surface area of a sphere whose
radius is 24 cm.
A. 192 π cm2
B. 768 π cm2
C. 1152 π cm2
D. 2304 π cm2
Solution:
Surface Area = 4 π r2
Surface Area = 4 (242) π = 2304 π
885. Which of the following is the reference
angle of 295o?
A. 25o
B. 45o
C. 65o
D. 75o
Explanation:
Romans and Hindu-Arabic: base-10 (decimal)
Mayans: base-20 (vigesimal)
Babylonians: base-60 (sexagesimal)
................................................
887. In solid geometry, what do you call a solid
bound by polygons?
A. tessellation
B. multigon
C. polyhedron
D. soligon
................................................
888. Two triangles have a pair of congruent
angles. Which of the following is true about these
triangles?
A. They are isosceles triangles.
B. They are equilateral triangles.
C. They are congruent triangles.
D. They are similar triangles.
................................................
889. Rayon deposited an amount of P500,000 in
a bank that offers 4% interest compounded per
annum. How much will he have in his account
after 3 years?
A. P720,000
B. P600,000
C. P560,000
D. P562,432
Solution:
Since the interest is compounded annually,
Acct = Principal x (1 + rate)time
Acct = 500,000 x (1.043)
Acct = 500,000 x 1.124864 = 562,432
................................................
This is a free reviewer. All rights reserved.
1000 MMR
890. Find the range of the following scores:
37
40
44
45
41
39
48
A. 7
B. 9
C. 10
D. 11
Explanation:
Range = Highest Score – Lowest Score
................................................
891. What is the 9th decile of the following data?
13
15
17
19
21
23
24
26
28
29
30
33
35
36
40
A. 34
B. 35.5
C. 37.6
D. 38
Solution:
(15 + 1)th data or 14.4th data
14th data is 36, while 15th data is 40
14.4th data = 36 + 0.4(40 – 36)
= 36 + 1.6
= 37.6
................................................
892. -20192 is NOT equal to ___.
A. (-2019)2
C. (-20192)
B. –(2019 x 2019)
D. –(2019)2
Explanation:
(-2019)2 = 4,076,361
-20192 = -4,076,361
................................................
893. The product of two numbers is 1425. If each
of the two numbers is doubled, the product of
these larger numbers is _____.
A. 2,850
B. 5,700
C. 8,550
D. 11.400
Solution:
Let xy = 1425
(2x)(2y) = 4xy
(2x)(2y) = 4(1425)
(2x)(2y) = 5700
Author: Victor A. Tondo Jr., LPT
894. The amount of money you have falls under
what level of data?
A. nominal
B. ordinal
C. interval
D. ratio
................................................
895. To study tooth decay a researcher takes a
sample at random but with the stipulation that
all age groups are represented proportionally.
What sampling method did the researcher use?
A. systematic
B. cluster
C. stratified
D. convenience
Explanation:
The age groups are the different strata or
subgroups of the population presented
proportionally.
................................................
896. By what property do we say that when
A = B and B = C, then A = C?
A. reflexive
B. symmetrical
C. transitive
D. closure
................................................
897. In number theory, what do we call a
polynomial equation with integer coefficients
that also allows the variables and solutions to be
integers only?
A. Integral equations
B. Differential equations
C. Diophantine equations
D. Bolyai-Lobachevsky equations
................................................
898. How long is the latus rectum of the parabola
defined by (y + 3)2 = 20x – 196?
A. 3
B. 12
C. 20
D. 19
Explanation:
The length of the latus rectum is equal to the
coefficient of the non-squared variable.
................................................
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1000 MMR
899. In the set of real numbers, what do we call
the set that includes only the counting numbers
and zero?
A. rational numbers B. integers
C. whole numbers D. irrational numbers
................................................
900. Find the sum of the first 60 counting even
numbers.
A. 3660
B. 3600
C. 1830
D. 900
Solution:
Sum of the first N counting numbers is given as
N2 + N.
................................................
901. If M and N are complementary angles,
which of the following is true?
A. cos M = sec N
B. sin M = cos N
C. tan M = csc N
D. sec M = -cos N
Explanation:
Sine and Cosine are trigonometric cofunctions.
................................................
902. The top ten students of a graduating class
got the following scores in their final
examination in Calculus:
89
84
83
89
89
85
88
93
83
98
What is their mean score?
A. 87.4
B. 87.5
C. 88.1
D. 89.3
Solution:
Sum of scores = 881
881 ÷ 10 = 88.1
................................................
903. The top ten students of a graduating class
got the following scores in their final
examination in Calculus:
89
84
83
89
89
85
88
93
83
98
What is the modal score?
A. 83
B. 85
C. 89
D. 98
Author: Victor A. Tondo Jr., LPT
Explanation:
The score 89 has the highest frequency.
................................................
904. The sum of five consecutive integers is 980.
What is the value of the greatest integer?
A. 192
B. 194
C. 196
D. 198
Solution:
x + (x + 1) + (x + 2) + (x + 3) + (x + 4) = 980
5x + 10 = 980
5x
= 970
x = 194
x + 4 = 194 + 4 = 198
................................................
905. x varies directly as y and inversely as z. If
x = 20 when y = 10 and z = 3, what is x when
y = 15 and z = 9?
A. 10
B. 15
C. 25
D. 30
Solution:
x = ky/z
20 = k(10)/3
60 = 10k
6=k
x = 6y/z
x = 6(15)/9
x = 90/9
x = 10
................................................
906. Factorize 15x2 – 13x – 20.
A. (5x – 4)(3x – 5)
B. (5x – 4)(3x + 5)
C. (5x + 4)(3x – 5)
D. (5x + 4)(3x + 5)
................................................
907. Which of the following is the axis of
symmetry of the parabola defined by the
equation y = x2 – 6x + 5?
A. x = 5
B. x = -5
C. x = 3
D. x = -3
Explanation:
The parabola opens upward. Therefore, its axis
of symmetry is x = -B/2A.
This is a free reviewer. All rights reserved.
1000 MMR
908. Find the radius of x2 + y2 + 12x – 14y = 36.
A. 6
B. 7
C. 11
D. 13
Author: Victor A. Tondo Jr., LPT
Solution:
Let x = budget
x + x + 1700 + 7400 = x
Solution:
x + 9100 = x
Complete the squares on the left side of the
equation to return it to its center-radius form.
x2 + y2 + 12x – 14y = 36
x2 + 12x
+ y2 – 14y
= 36
2
(x + 12x + 36)+ (y2 – 14y +49)= 36 + 36 + 49
(x2 + 12x + 36)+ (y2 – 14y +49)= 121
................................................
909. Find the instantaneous rate of change for
f(x) = 4x3 + 7x2 – 27x – 49 at x = 1.
A. -2
B. -1
C. 1
D. 2
Solution:
Find f ’(1).
f ’(x) = 12x2 + 14x – 27
f ’(1) = 12(12) + 14(1) – 27
f ’(1) = -1
................................................
9100 = x –
x
9100 = x
21000 = x
................................................
912. An assistant’s response times were
recorded on the table below. What is her average
response time in minutes?
Response Time Frequency
1 minute
3
2 minutes
7
3 minutes
11
4 minutes
9
5 minutes
5
6 minutes
3
7 minutes
2
A. 3.625
B. 3.575
C. 3.525
D. 3.5
Solution:
910. log3 24 – 3 log3 2 = _____
A. 1
B. 0
C. -1
D. e
First, insert a column for fx, which is the product
of the frequency and response time in each row.
Response Time Frequency fx
1 minute
3
3
2 minutes
7
14
3 minutes
11
33
4 minutes
9
36
5 minutes
5
25
6 minutes
3
18
7 minutes
2
14
Solution:
log3 24 – 3 log3 2 = log3
= log3 3
=1
................................................
911. After using one-sixth of her budget on bills,
two-fifths on groceries, and P1700 on books and
magazines, Mrs. Lazada still had P7400 left. How
much was her budget?
A. P20,000
B. P21,000
C. P22,500
D. P30,000
∑
= 143
n = 40
143 ÷ 40 = 3.575
................................................
913. The average grade of 22 students in Section
Abaca is 95, while the average grade of 28
students in Section Acacia is 89. What is the
average grade of all 50 students in both sections?
A. 92.36
B. 91.64
C. 90.72
D. 89.8
This is a free reviewer. All rights reserved.
1000 MMR
Solution:
Author: Victor A. Tondo Jr., LPT
Solution:
(
)
(
)
Average =
=
= 91.64
................................................
914. Find the slope of 2x – 6y = 13.
A. 3
B.
C. -3
D.
Solution:
(180 – x) + (90 – x) = 146
270 – 2x = 146
270 – 146 = 2x
124 = 2x
62 = x
................................................
Convert it to its slope-intercept form.
2x – 6y = 13 
-6y = -2x + 13
y= x+
918. A bag contains some marbles. When the
marbles are grouped by 2, 3, 4, 5, or 6, there is
always one marble left. Which of the following
could be the number of marbles in the bag?
A. 31
B. 41
C. 51
D. 61
.: m = or
................................................
Explanation:
915. Find the intersection of y = -3x – 2 and
y = 2x + 23.
A. (5, 13)
B. (-5, -13)
C. (5, -13)
D. (-5, 13)
Solution:
y = -3x – 2
(-)
y = 2x + 23
0 = -5x – 25
5x = -25
x = -5;
y = 2(-5) + 23
y = 13
................................................
916. A book was sold for P630 after a 10%
discount was given. How much was the book
originally?
A. P800
B. P750
C. P700
D. P690
N – 1 should be divisible by 2, 3, 4, 5, and 6.
................................................
919. Amitaf, Bernard, and Chloe share a total of
P2,460 in the ratio 3:5:4 respectively. How much
is Amitaf’s share?
A. P600
B. P615
C. P630
D. P660
Solution:
3x + 5x + 4x = 2460
12x = 2460
x = 205
Amitaf’s share is 3(205) or P615.
................................................
920. Fifty-four kilometers per hour is equal to
how many meters per second?
A. 15
B. 30
C. 54
D. 60
Solution:
Solution:
SP = OP (1 – DC Rate)
630 = OP (0.9)
630/0.9 = OP
700 = OP
................................................
=
=
................................................
917. If the sum of the supplement and the
complement of an angle is 146, what is the
angle?
A. 61
B. 62
C. 63
D. 64
921. Simplify
A. 1
.
B.
This is a free reviewer. All rights reserved.
C.
D. 2
1000 MMR
Solution:
(
Author: Victor A. Tondo Jr., LPT
Solution:
)
=
=
=1
................................................
922. If x = 10, which of the following is equal to
219?
A. 23x – 1
B. 2x2 + 5x + 5
C. 2x2 + 2x – 1
D. x2 + x + 2
Explanation:
= csc x – cot x
................................................
926. Simplify:
A.
x
B.
C.
D.
Solution:
Just substitute x with 10.
A. 23(10) – 1 = 229
B. 2(10)2 + 5(10) + 5 = 255
C. 2(10)2 + 2(10) – 1 = 219
D. (10)2 + (10) + 2 = 112
................................................
923. In the equation 3x – 4y = 25, what is x when
y is 5?
A. 18
B. 15
C. 12
D. 9
Solution:
3x – 4(5) = 25
3x – 20 = 25
3x = 25 + 20
3x = 45
x = 15
................................................
924. What is the measure of each interior angle
of a regular 30-sided polygon?
A. 158o
B. 162o
C. 165o
D. 168o
Solution:
MIA = 180 –
MIA = 180 –
MIA = 180 – 12
MIA = 168
................................................
925. Simplify:
A. csc x – tan x
C. csc x – cot x
=
x
=
x
=
x
=
x
................................................
927. Which of the following is a diagonal matrix?
1 6 9
0 2 4
A. [ 2 2 6]
B. [1 0 5]
3 4 3
2 3 0
4 0 0
3 2 3
C. [0 5 0 ]
D. [2 3 6]
0 0
6
1 4 3
Explanation:
A diagonal matrix is a square matrix where all its
non-diagonal elements are 0.
................................................
928. Which of the following is a scalar matrix?
0 2 3
5
7 9
A. [ 2
B. [4 0 7]
8 6]
3
8 0
3 4 3
7 0 0
7 0 0
C. [0
D. [0 7 0]
5 0]
0 0 6
0 0 7
Explanation:
A scalar matrix is a diagonal matrix where all its
leading diagonal elements are of equal value.
................................................
B. csc x + tan x
D. csc x + cot x
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1000 MMR
929. Which of the following is an identity matrix?
1 0 0
0
5 3
A. [0 1 0]
B. [ 4 0 9]
0 0 1
7 10 0
6 1
1
0 1 1
C. [1 12 1 ]
D. [1 0 1]
1 1 18
1 1 0
Explanation:
An identity matrix is a scalar matrix where the
elements on its leading diagonal are 1 and the
rest are of value 0.
................................................
930. Which of the following is the transposition
1 4
3
of A = *
+?
5 10 15
5 10 15
4 5 15
A. AT = *
+
B. AT = *
+
1 4
3
10 1 3
1 4
3
1 5
T
T
C. A = [4 10]
B. A = [5 10 15]
1 1
1
3 15
Explanation:
Transpose A by interchanging the rows and
columns.
................................................
931. When dealing with matrices A, B, and C,
which of the following is always true?
A. AB = BA
B. If AB = 0, then A = 0 or B = 0
C. If AB = AC, then B = C
D. None of the above.
................................................
932. If the numbers x-1, x+2 and 2x+4 are
consecutive terms of a geometric sequence, what
is x?
A. 1
B. 2
C. 4
D. 8
Author: Victor A. Tondo Jr., LPT
Solution:
=
=2
x + 2 = 2(x – 1)
x + 2 = 2x – 2
2 + 2 = 2x – x
4=x
................................................
933. What are the zeros of 12x2 – x – 35?
A. and
B. and
C. and
D.
and
Solution:
12x2 – x – 35 = 0
(4x – 7)(3x + 5) = 0
4x – 7 = 0
3x + 5 = 0
4x = 7
3x = -5
x=
x=
................................................
934. What is the sum of the interior angles of a
regular dodecagon?
A. 1620o
B. 1800o
C. 1980o
D. 2160o
Solution:
SIA = 180(n – 2)
SIA = 1800
................................................
935. 45x – 20 = _____.
A. 45x – 20
B. 5(45x – 20)
C. 20(45x – 20)
D. (5x – 20)(45x – 20)
Solution:
au = au du
Let u = 5x – 20
.: du =5
45x – 20 = 45x – 20 (5)
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1000 MMR
936. Which of the following graphs have a
positive leading coefficient and an even degree?
Author: Victor A. Tondo Jr., LPT
937. Which of the following graphs have a
negative leading coefficient and an odd degree?
A.
A.
B.
B.
C.
C.
D.
D.
Explanation:
Explanation:
A positive leading coefficient means the right tail
must point upward. An even degree means that
the left tail points to the same direction (up) as
the right tail.
................................................
A negative leading coefficient means the right
tail must point downward. An odd degree means
that the left tail points to the opposite direction
as the right tail.
................................................
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1000 MMR
938. Given f(x) = (x + 2)(x – 1)(2x – 3)(x – 5),
where can we find f(3)?
A. On the x-axis.
B. Above the x-axis.
C. Below the x-axis.
D. On the intersection of the x- and y-axis.
Author: Victor A. Tondo Jr., LPT
2 +1
3
= 3 , find lim
941. If ( ) { 7
( ).
3
3
A. 3
B. 6
C. 7
D. Limit does not exist.
Explanation:
lim
( ) = 2(3) + 1 = 7
f(3) = 5(2)(3)(-2)
f(3) = -60
Since f(3) is negative, then it is found below the
x-axis.
................................................
lim
( ) = 32 – 3 = 6
939. What can be said about the equation f(x) of
the following graph?
942. Give the domain of f(x) = √8 + 24.
A. x | x
B. x -3
C. x > 24
D. x -3
Solution:
Since lim
( ) lim
( ), then
therefore, limit does not exist.
................................................
Explanation:
The given function is a radical function, more
specifically a square root function. If the
radicand 8x + 24 is negative, f(x) becomes
imaginary. That means 8x + 24 must be greater
than or equal to 0.
A. The leading coefficient is positive.
B. The constant is positive.
C. The degree is 4.
D. The function is logarithmic.
Explanation:
Since the right tail is pointing downward, the
leading coefficient is negative.
The left tail points upward (opposite of right
tail); therefore, the degree is odd.
Since the graph touches the y-axis above the
origin, then the constant is positive.
................................................
+1
4
940. If ( ) {2 + 1 = 4 , find lim
5
3
4
A. Limit does not exist.
B. 9
C. 13
D. 17
( ).
8x + 24 0
8x -24
x -3
................................................
943. What is the 3rd term in the expansion of
(A + B)5?
A. 5A4B
B. 10A3B2
2
3
C. 10A B
C. 5A3B2
Solution:
First, use the Pascal triangle to find the
expansion of (A + B)5:
1
Solution:
lim
( ) = 42 + 1 = 17
lim
( ) = 5(4) – 3 = 17
Since lim
( ) = lim
lim
( ) = 17.
1
1
1
5
1
4
1
3
10
2
6
1
3
10
1
4
1
5
1
A5 + 5A4B + 10A3B2 + 10A2B3 + 5AB4 + B5
( ), then
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1
1000 MMR
944. Find the remainder when
2x4 – 5x3 + 3x2 – 5x + 7 is divided by (x – 1).
A. 1
B. 2
C. 4
D. 7
Author: Victor A. Tondo Jr., LPT
946. In the arithmetic sequence -2, 1, 4, 7, …,
which term is 100?
A. 34th
B. 35th
C. 36th
D. 37th
Solution:
Solution:
Use the Remainder Theorem.
2(14) – 5(13) + 3(12) – 5(1) + 7 = r
2–5+3–5+7=r
2=r
................................................
An = A1 + (n – 1)d
100 = -2 + (n – 1)(3)
100 = -2 + 3n – 3
100 = 3n – 5
105 = 3n
35 = n
................................................
945. Give the range of f(x) =
A. y | y
B. y
C. y
D. y
947. Find the length of the intercepted arc of a
central angle measuring 45o given the radius is
80 cm.
A. 20 π cm
B. 30 π cm
C. 40 π cm
D. 50 π cm
0
Solution:
First, express y in terms of x.
y=
(5x – 8)y = 3x + 4
5xy – 8y = 3x + 4
5xy – 3x = 8y + 4
x (5y – 3) = 8y + 4
x=
Solution:
Arc length =
(2πr)
Arc length =
(160 π)
Arc length = 20 π
................................................
The denominator 5y – 3 cannot be equal to 0.
5y – 3 0
5y 3
y
948. If f(x) = 5x2 – 4x + 9, find the average rate
of change from x = 1 to x = 4.
A. 20
B. 21
C. 22
D. 23
Shortcut:
f(1) = 5 – 4 + 9 = 10
f(4) = 80 – 16 + 9 = 73
Since the numerator and denominator are of the
same degree, just take the leading coefficients of
the numerator (3) and denominator (5).
Proof:
 Cross-multiply.
5(3x + 4)
15x + 20
15x – 15x
0
3(5x – 8)
15x – 24
-24 – 20
-44
Solution:
( )
( )
ARoC =
=
= 21
................................................
949. If f(x) = 4x2 + x – 5, find the instantaneous
rate of change at x = 5.
A. 39
B. 40
C. 41
D. 42
Solution:
f ‘(x) = 8x + 1
f ‘(5) = 8(5) + 1
f ‘(5) = 41
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1000 MMR
950. How is 67500 written in scientific notation?
A. 67.5 x 103
B. 6.75 x 103
4
C. 67.5 x 10
D. 6.75 x 104
................................................
951. Find the mean of the following data:
24
24
25
26
28
29
A. 24
B. 25.5
C. 26
D. 29
Solution:
Mean =
= 26
................................................
952. Rayon’s Social Security System (SSS) ID
number is 02-2525255-2. What level of data is
the SSS ID?
A. nominal
B. ordinal
C. interval
D. ratio
Explanation:
ID numbers fall under the nominal level of data.
................................................
953. A tree is 4 meters tall, while the flagpole is 8
meters tall. What level of data is used?
A. nominal
B. ordinal
C. interval
D. ratio
Explanation:
We can infer that the flagpole is literally twice as
tall as the tree. Height or tallness falls under the
ratio level of data.
................................................
954. The freezing temperature of water is 0o
Celsius. What level of data is temperature in
Celsius?
A. nominal
B. ordinal
C. interval
D. ratio
Explanation:
0o Celsius does not mean the absence of
temperature or heat. Temperature in Celsius
(and also in Fahrenheit) falls under the interval
level of data.
Author: Victor A. Tondo Jr., LPT
955. Factorize: x3 + 5x2 – 9x – 45
A. (x + 1)(x + 9)(x – 5)
B. (x + 1)(x – 9)(x + 5)
C. (x + 3)(x + 3)(x – 5)
D. (x + 3)(x – 3)(x + 5)
Solution:
x3 + 5x2 – 9x – 45
= x2 (x + 5) – 9 (x + 5)
= (x2 – 9)(x + 5)
= (x + 3)(x – 3)(x + 5)
................................................
956. Which of these is equal to A-1 + B-1?
A. ( )
B. +
C.
D.
................................................
957. Find the equation of the line with slope 3,
passing through (2, 5).
A. y = 3x – 2
B. y = 3x – 1
C. y = 3x + 1
D. y = 3x + 2
Solution:
y = mx + b
y = 3x + b
 slope is 3
5 = 3(2) + b
 passing (2, 5)
-1 = b
.: y = 3x – 1
 m = 3, b = -1
................................................
958. Find the distance of the point (7, 8) from
the line 2x – 3y + 4 = 0
A.
√
B.
C. 4√73
√
D. 5√13
Solution:
Use the formula for distance of point (x1, y1)
from line Ax + By + C = 0:
D= √
( )
D=
D=
√
√
( )
(
=
)
√
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1000 MMR
959. Find the distance between the parallel lines
y = 2x – 1 and y = 2x + 4.
A. 5
B. 2√5
C. √5
D. ½
Solution:
Use the distance between two parallel lines
D=√
D=
(
)
√
=
√
= √5
................................................
960. There are 80 cows and ducks in a farm, all
of which are healthy. If there are 190 legs in
total, how many cows are there?
A. 5
B. 10
C. 15
D. 20
Let C = number of cows;
D = number of ducks
2(C + D = 80)
4C + 2D = 190
 since cows have four legs
and ducks have two

2C + 2D = 160
- 4C + 2D = 190
-2C
= -30
C = 15
................................................
961. Which of the following angles in standard
position is coterminal with 150o?
A. 2570o
B. 5490o
o
C. 7830
D. 9870o
Solution:
= 6.722…
 Not a whole number
= 14.833…  Not a whole number
= 21.333…  Not a whole number
= 27
C. cos x =
D. csc x =
Explanation:
You may use the mnemonics SohCahToa and
ChoShaCao to remember that sin x and csc x are
reciprocals of each other. So are cos x and sec x,
and tan x and cot x.
................................................
963. V and R form a vertical pair. If m V = 7x
and m R = 3x + 60, find m V.
A. 105o
B. 84o
C. 20o
D. 15o
Solution:
Solution:
C + D = 80
4C + 2D = 190
Author: Victor A. Tondo Jr., LPT
962. Which of the following is false?
A. sin x =
B. tan x =
 Whole number
Therefore, 9870o is coterminal with 150o.
Since the angles form a vertical pair, then their
measures are equal.
m V=m R
7x = 3x + 60
7x – 3x = 60
4x = 60
x = 15
m V = 7(15) = 105
................................................
964. V and R form a linear pair. If m V = 13x
and m R = 5x + 72, find m R.
A. 112o
B. 102o
C. 78o
D. 68o
Solution:
Since the angles form a linear pair, then the sum
of their measures is 180.
m V + m R = 180
13x + 5x + 72 = 180
18x + 72 = 180
18x = 180 – 72
18x = 108
x=6
.:m R = 5(6) + 72 = 102
................................................
965. Evaluate: tan 45o – sin 60o + cos 30o
A. -1
B. 0
C. 1
D. 2
Explanation:
Just use your calculator.
This is a free reviewer. All rights reserved.
1000 MMR
966. Evaluate: lim
A. LDNE
B. undefined C. 0
Author: Victor A. Tondo Jr., LPT
Solution:
D. 27
Solution:
P = 4S
=S
A = S2
A = ( )2
A=
................................................
lim
(
)(
)
= lim
= lim ( + 3 + 9)
= 32 + 3(3) + 9
= 27
................................................
967. Which of the following circles is concentric
with x2 + y2 – 7x + 9y = 25?
A. (x – 3)2 + (y + 5)2 = 25
B. (x – 7)2 + (y + 9)2 = 50
C. (x + 3.5)2 + (y – 4.5)2 = 100
D. (x – 3.5)2 + (y + 4.5)2 = 125
970. The distance D of
a projectile from the
ground t seconds
after launch is
given as
D = 8t – t2. How
many seconds
after launch does the projectile hit the ground?
A. 4
B. 6
C. 7
D. 8
Solution:
D = 8t – t2
Explanation:
The projectile will be on the ground when D = 0.
The center of x2 + y2 – 7x + 9y = 25 is ( , ) or
(3.5, -4.5).
................................................
0 = 8t – t2
0 = t(8 – t)
t=0
968. Find k such that (x + 2) is a factor of the
polynomial 5x3 + 11x2 – kx + 12.
A. 11
B. 7
C. -3
D. -8
The projectile will hit the ground 8 seconds after
launch.
................................................
Solution:
971. The distance D of a projectile from the
ground t seconds after launch is given as
D = 14t – t2. How many seconds after launch
does it attain its peak?
A. 4
B. 6
C. 7
D. 8
For (x + 2) to be a factor of the polynomial, the
value of f(-2) should be equal to 0. (See Factor
Theorem)
f(-2) = 5(-2)3 + 11(-2)2 – (-2)k + 12 = 0
f(-2) = -40 + 44 + 2k + 12 = 0
16 + 2k = 0
2k = -16
k = -8
................................................
969. If P = 4S and A =
A. A =
C. A = 16P2
S2,
express A in terms of P.
B. A =
D. A = 4P2
8–t=0
8=t
Solution:
D = 14t – t2
D’ = 14 – 2t
 first derivative
Max height or peak is attained when D’ = 0.
14 – 2t = 0
14 = 2t
7=t
This is a free reviewer. All rights reserved.
1000 MMR
972. The distance D of a projectile (in meters)
from the ground t seconds after launch is given
as D = 16t – 2t2. What is its maximum height?
A. 16 m
B. 24 m
C. 32 m
D. 40 m
Solution:
D = 16t – 2t2
D’ = 16 – 4t  first derivative
Max height or peak is attained when D’ = 0.
16 – 4t = 0
16 = 4t
4=t
Max height
= 16(4) – 2(4)2
= 64 – 32
= 32
................................................
973. Find the slope of the line tangent to the
graph of f(x) = x3 – 3x2 + 5x – 1 at x = 1.
A. -1
B. 0
C. 1
D. 2
Solution:
f ‘(x) = 3x2 – 6x + 5
f ‘(1) = 3 – 6 + 5
f ‘(1) = 2
................................................
974. Which of the following is a diagonal matrix?
9 0 0
0 1
4
A. [ 0 3 0]
B. [ 1 0
7]
0 0 6
2 3 0
7 6 9
4 2 9
C. [4
D. [ 1
3 8]
8 7]
5 4 5
2
4 6
Explanation:
A diagonal matrix is a square matrix where all its
non-diagonal elements are 0.
(See item #927)
Author: Victor A. Tondo Jr., LPT
975. Which of the following is a scalar matrix?
5 1
2
5 0
0
A. [ 4
]
B.
[
9 10
0
5 0]
8
5
3
0
0
5
7 1 1
0
2
5
C. [1
D. [ 9 0
5 1]
1]
1 1 6
1
8 0
Explanation:
A scalar matrix is a diagonal matrix where all its
leading diagonal elements are of equal value.
(See item #928)
................................................
976. Which of the following is an identity matrix?
4 1 1
0 6
3
A. [1 8 1 ]
B. [ 4 0 2 ]
1 1 12
7 9 0
1 0 0
0 1 1
B. [0 1 0]
D. [1 0 1]
0 0 1
1 1 0
Explanation:
An identity matrix is a scalar matrix where the
elements on its leading diagonal are 1 and the
rest are of value 0.
(See item #929)
................................................
977. Find the quotient when the polynomial
7x5 – 3x3 + 2x2 – 9x + 3 is divided by x – 1.
A. 7x4 + 7x3 + 4x2 + 6x – 3
B. 7x4 + 7x3 + 4x2 – 6x – 3
C. 7x4 + 7x3 – 4x2 – 6x – 3
D. 7x4 – 7x3 – 4x2 – 6x – 3
Solution:
Use synthetic division.
1
7
0
7
7
7
x4
x3
This is a free reviewer. All rights reserved.
-3
7
4
x2
2
4
6
x
-9
6
-3
c
3
-3
0
rem
1000 MMR
978. Find k such that 9x2 + 36x + k = 0 has only
one unique root.
A. 4
B. 16
C. 18
D. 36
Solution:
B2 – 4AC = 0
362 – 4(9)(k) = 0
362 – 36k = 0
362 = 36k
36 = k
................................................
979. There are 210 candies in a jar. The ratio of
red candies to blue is 2:3, and the ratio of blue to
yellow is 4:5. How many yellow candies are
there?
A. 35
B. 45
C. 60
D. 90
Solution:
First, make a common ratio.
Red Blue Yellow
2
3
4
5
8
12
15
Author: Victor A. Tondo Jr., LPT
981. A square picture was framed and given 3 cm
margins. If the total area of the margin is 324
cm2, what is the area of the picture?
A. 441 cm2
B. 576 cm2
2
C. 729 cm
D. 900 cm2
Solution:
(x + 3 + 3)2 – x2 = 324
(x + 6)2 – x2 = 324
2
x + 12x + 36 – x2 = 324
12x + 36 = 324
12x = 288
x = 24
.: x2 = 576
................................................
982. Given f(x) = x2 + 10x + 25 and
g(x) = 3x + 1, find f(g(x)).
A. 3x2 + 30x + 26
B. 9x2 + 36x + 36
2
C. 9x + 30x + 26
D. 3x2 + 36x + 36
Solution:
f(g(x)) = (g(x))2 + 10(g(x)) + 25
f(g(x)) = (3x + 1)2 + 10(3x + 1) + 25
f(g(x)) = (9x2 + 6x + 1) + (30x + 10) + 25
f(g(x)) = 9x2 + 36x + 36
................................................
8x + 12x + 15x = 210
35x = 210
x=6
Number of yellow candies = 15x = 15(6) = 90
................................................
980. A certain bacteria doubles its population
after 3 minutes. If the bacteria in a petri dish is
512,000 at 8:45 AM, at what time was its
population count 125?
A. 6:30 AM
B. 7:15 AM
C. 8:09 AM
D. 9:21 AM
Solution:
125 x 2n = 512,000
2n = 4096
2n = 212
n = 12
12(3) = 36
36 minutes before 8:45 AM is 8:09 AM.
983. A whole number is 3 more than another
number. The sum of their squares is 4145. Find
the larger number.
A. 47
B. 46
C. 45
D. 44
Solution:
Let x = larger number
x – 3 = smaller number
x2 + (x – 3)2 = 4145
x2 + (x2 – 6x + 9) = 4145
2x2 – 6x + 9 = 4145
2x2 – 6x = 4136
x2 – 3x = 2068
x2 – 3x + 2.25 = 2068 + 2.25
(x – 1.5)2 = 2070.25
x – 1.5 = 45.5
x = 47
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1000 MMR
984. The numbers x, y, z, and w have an average
of 30. If x, y and z have an average 35, what is w?
A. 5
B. 10
C. 15
D. 20
Solution:
x + y + z + w = 4(30)
x + y + z + w = 120
x + y + z = 3(35)
x + y + z = 105
Solution:
120 – 105 = 15
................................................
985. A is a constant. Find A such that the
equation 2x + 1 = 2A + 3(x + A) has a solution
at x = 2.
A. -0.4
B. -0.2
C. 0
D. 0.2
Solution:
2(2) + 1 = 2A + 3(2 + A)
5 = 2A + 6 + 3A
-1 = 5A
-0.2 = A
................................................
986. It takes 4 men 9 days to build 2 houses. How
many days will it take 6 men to build 5 houses?
A. 12
B. 13
C. 14
D. 15
Solution:
4m (9d) = 2h
36md = 2h
18md = h
6m (kd) = 5h
6m (kd) = 5(18md)
6m (kd) = 90 md
kd = 15d
k = 15
................................................
987. If 860 = 32x, what is x?
A. 18
B. 24
C. 20
Solution:
860 = 32x
(23)60 = (25)x
2180 = 25x
180 = 5x
36 = x
Author: Victor A. Tondo Jr., LPT
988. In a certain university, a student’s grade is
computed as the sum of 25% of his prelims
grade, 30% of his midterms grade, and 45% of
his finals grade. He knows that his prelims grade
is 72 and his midterms grade is 70. What is his
grade for the finals if he got a grade of 75 in
Calculus?
A. 75
B. 77
C. 79
D. 80
D. 36
0.25(72) + 0.3(70) + 0.45(X) = 75
18 + 21 + 0.45X = 75
39 + 0.45X = 75
0.45X = 36
X = 80
................................................
989. Mr. Lazada bought 75 pieces of Elunium for
8,000 per piece and sold them for a total of
720,000. What is his mark-up rate?
A. 20%
B. 25%
C. 32.5%
D. 40%
Solution:
MUR =
(
(
)
)
x 100%
MUR =
x 100%
MUR = 20%
................................................
990. Triangular numbers are numbers that can
be shown by triangular arrangements of dots.
The triangular numbers are 1, 3, 6, 10, 15, 21, …
What is the 20th triangular number?
A. 200
B. 205
C. 210
D. 220
Explanation:
The Nth triangular number is
.
................................................
991. What is a pattern of shapes that covers a
surface completely without overlaps or gaps?
A. translation
B. tessellation
C. constellation
D. transposition
................................................
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1000 MMR
992. Find the equation of the line perpendicular
to 5x + 3y = 12, passing through (3, -7).
A. 3x – 5y = -44
B. 3x + 5y = 44
C. 3x – 5y = 44
D. 3x + 5y = -44
Solution:
The perpendicular line for Ax + By = C is given
as Bx – Ay = B(xp) – A(yp).
3x – 5y = 3(3) – 5(-7)
3x – 5y = 9 – (-35)
3x – 5y = 44
................................................
993. Point V is two-fifths the way from A(9, -7)
to B(-6, 13). Find the coordinates of point V.
A. (2, 2)
B. (2, 0)
C. (3, 1)
D. (3, 2)
Solution:
x = 9 + ((-6) – 9)
y = (-7)+ (20)
y = (-7)+ 8
y=1
................................................
994. Find the angle of inclination of the line
passing through V(5, -5) and T(2, 4)
A. 88.145o
B. 92.429o
o
C. 97.593
D. 108.435o
Solution:
)
d=
d=
=5
A25 = A1 + (n – 1) d
A25 = 3 + (25 – 1)(5)
A25 = 123
................................................
996. An educational psychologist classifies
students as high, medium and low intelligence.
What kind of scale is being used?
A. nominal scale
B. ordinal scale
C. interval scale
D. ratio scale
................................................
Explanation:
y = (-7)+ (13 – (-7))
(
Solution:
997. The GCF of two numbers is 8 and their LCM
is 80. What is their product?
A. 80
B. 160
D. 320
D. 640
x = 9 + (-15)
x = 9 + (-6)
x=3
Slope =
Author: Victor A. Tondo Jr., LPT
995. The first term of an arithmetic sequence is 3
and the 20th term is 98. What is the 25th term?
A. 108
B. 115
C. 118
D. 123
= -3
Angle of inclination = tan-1 (-3)
Angle of inclination = -71.565o or 108.435o
The product of two numbers is equal to the
product of their GCF and LCM.
................................................
998. If c1.5 – 4 = 7, what is c3?
A. 784
B. 121
C. 49
D. 16
Solution:
c1.5 – 4 = 7
c1.5 = 11
(c1.5)2 = 112
c3 = 121
................................................
999. A certain Math challenge gives the
competitors a score of 4 for each correct answer,
and a deduction of 1 point for each wrong
answer. If a contestant answered all 100 items
and got a score of 200, how many items did the
contestant answer correctly?
A. 45
B. 50
C. 60
D. 35
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1000 MMR
Solution:
Author: Victor A. Tondo Jr., LPT
Let C = number of correct answers
.: 100 – C = number of wrong answers
(4)C + (-1)(100 – C) = 200
4C – 100 + C = 200
5C = 300
C = 60
1000. What is 111011012 in decimal?
A. 257
B. 237
C. 217
D. 197
Solution:
101011012  ____10
1 x 27 = 128
1 x 26 = 64
1 x 25 = 32
0 x 24 = 0
1 x 23 = 8
1 x 22 = 4
0 x 21 = 0
1x 20 = 1
237
End of 1000MMR.
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