1. The terms of a sum may be grouped in any manner
without affecting the result. This law is known as:
a. Associative Law
b. Reflexive Law
c. Commutative Law
d. Distributive Law
the coefficient of the constant term got roots of -1 and 4.
What is the correct equation?
a. 𝑥 2 − 6𝑥 − 3 = 0
b. 𝑥 2 + 6𝑥 + 3 = 0
2
c. 𝑥 + 3𝑥 + 6 = 0
d. 𝑥 2 − 3𝑥 − 6 = 0
2. The expression (x – 3)3 is identical to:
a. x3 – 27
b. (x + 3) (x2 – 6x + 9)
2
c. (x – 3) (x – 3x + 9) d. x3 – 9x2 + 27x – 27
16. Find the remainder when (𝑥 12 + 2) is divided by
(𝑥 − √3).
a. 652
b. 851
c. 238
d. 731
3. Terms that differ only in numeric coefficients are
known as:
a. unlike terms
b. equal terms
c. unequal terms
d. like terms
17. Find the value of k so that x + 2 is a factor of 3𝑥 3 −
𝑘𝑥 2 + 2𝑥 − 4 = 0
a. -8
b. -4
c. 4
d. 10
4. Solve for z:
3x – 2y + z = 11
x + 5y – 2z = -9
2x + y – 3z = -6
a. 1
c. 3
18. If 𝑥 3 + 3𝑥 2 + (𝑘 + 5)𝑥 + 2 − 𝑘 is divided by x+1 and
the remainder is 3, then the value of k is
a. -2
b. -4
c. -3
d. -5
b. 2
d. 4
19. Which of the following has no middle term?
a. (𝑥 + 𝑦)3
b. (𝑎 − 𝑏)4
c.(𝑢 + 𝑣)6
d. (𝑥 − 𝑦)8
5. Solve for x if 8x = 2(y+2) and 16(3x-y) = 4y.
a. 1
b. 3
c. 2
d. 4
20. Find the 5th term in the expansion of (𝑥 2 − 3)8
a. −70𝑥 8
b. 70𝑥 8
8
c. −5670𝑥
d. 5670𝑥 8
8. Find x from the following equations:
and
a. 2.5
b. 2
c. 1
d. 1.5
9)!]2 / [(n!)(n-1)!] is
a. n(n+1)
b. n2
c. n+1
d. n(n+1)2
21. Find the term independent of y in the expansion of
(2y2 – 3y-1)9.
a. 217,728
b. -734,832
c. -326,592
d. 489,888
22. Find the sum of the coefficients in the expansion
(𝑥 + 2𝑦 − 𝑧)8
a. 256
b. 1024
c. 1
d. 6
10. Solve for U if U = √1 − √1 − √1 − ⋯
a. 0.723
b. 0.618
c. 0.852
d. 0.453
11. Solve for x: 𝑥 𝑥
a. 1.2589
c. 1.1745
…
𝑥𝑥
23. Find the 30th term of the A.P. 4, 7, 10 …
a. 88
b. 91
c. 75
d. 90
= 10
b. 2.4156
d. cannot be solved
12. Find k in the equation 4𝑥 2 + 𝑘𝑥 + 1 = 0 so that it
will only have one real root.
a. 1
b. 2
c. 3
d. 4
2
2
13. Find the value of constant “h” in the 2x – hx + 4x +
5h = 0 so that the sum of the roots is 2.
a. 4
b. 6
c. 12
d. 18
14. In the equation 3𝑥 2 + 4𝑥 + (2ℎ − 5) = 0, find h if
the product of the roots is 4.
a. -7/2
b. -10/2
c. 17/2
d. 7/2
15. Two engineering students are solving a problem
leading to a quadratic equation. One student made a
mistake in the coefficient of the first-degree term, got
roots of 2 and -3. The other student made a mistake in
24. Find the 100th term of the sequence 1.01, 1.00, 0.99,
0.98, 0.97, …..
a. 0.05
b. 0.04
c. 0.03
d. 0.02
25. The fourth term of a geometric progression is 189
and the sixth term is 1701, the 8th term is:
a. 5103
b. 1240029
c. 45927
d. 15309
26. Find the sum of the first 10 terms of the geometric
progression 2, 4, 8, 16 …
a. 510
b. 842
c. 1022
d. 2046
27. A rubber ball is dropped from a height of 15 meters.
On each rebound, it rises 2/3 of the height from which it
last fell. Find the distance traversed by the fall before it
comes to rest. The geometric progression occurs after
the first rebound.
a. 96 m
b. 100 m
c. 85 m
d. 75 m
28. Find the tenth term in the sequence 1, 1, ½, 1/6,
1/24…
a. 1/322560
b. 1/362880
c. 1/317520
d. 1/352800
29. The sum of two numbers is 68. If the larger is divided
by the smaller, the quotient is 4 and the remainder is 8.
Find the smaller number.
a. 13
b. 12
c. 15
d. 4
30. There are two numbers whose sum is 53. Three times
the smaller number is equal to 19 more than the larger
number. What are the numbers?
a. 16, 37
b. 18, 35
c. 20, 33
d. 24, 29
31. A father is twice older than his son and the sum of
their ages is 48. How old is each?
a. 8, 40
b. 12, 36
c. 16, 32
d. 15, 33
32. Maria is 36 years old, Maria was twice as old as Anna
was when Maria was as old as Anna is now. How old is
Anna now?
a. 24
b. 16
c. 18
d. 20
33. How much of a 7% solution should be mixed with
appropriate amount of 12% solution to get 5 liters of a
10% solution?
a. 2 L
b. 3 L
c. 2.5 L
d. 4 L
34. The tank of a car contains 50 liters of alcogas 25% of
which is pure alcohol. How much of the mixture must be
drawn off which when replaced by pure alcohol will yield
a 50 – 50% alcogas?
a. 16 2/3
b. 14
c. 15 1/3
d. 20
35. What time between the hours of 12:00 noon and 1:00
pm would the hour-hand and the minute-hand of a
continuously driven clock be in straight line?
a. 12:33 pm
b. 12:30 pm
c. 12:37 pm
d. 12:28 pm
36. How soon after seven o’clock will the hands of a clock
be together?
a. 38.18 mins
b. 37.17 mins
c. 39.19 mins
d. 36.16 mins
37. How many minutes after 3:00 PM will the minute
hand of the clock overtakes the hour hand?
a. 14/12 minutes
b. 16 11/12 minutes
c. 16 4/11 minutes
d. 14/11 minutes
38. A tank can be filled by three pipes separately in 20,
30 and 40 minutes respectively. In how many minutes
can it be filled by the three pipes acting together?
a. 10.69 mins
b. 1.23 mins
c. 9.23 mins
d. 8.19 mins
39. Ana can finish her differential equations homework
in 30 minutes while Annie can do the same homework
for 26 minutes. If Ana did the homework for 12 minutes
until Annie helped her, after how many minutes will they
finish the homework?
a. 12.36
b. 7.64
c. 8.36
d. 10.72
40. Two cars A and B start at the same point and at the
same time and travel in opposite directions, car B
travelling 20 km/hr slower than A. If they are 420
kilometers apart after 3 hours, find the rate of each.
a. 60 kph, 80 kph
b. 80 kph, 100 kph
c. 70 kph, 90 kph
d. 90 kph, 110 kph
Supplementary Problems
41. Foghorn A sounds every 18 seconds, and Foghorn B
sounds every 24 seconds. They sound together at noon.
How many seconds later will they next sound together?
a. 70 seconds
b. 72 seconds
c. 74 seconds
d. 76 seconds
42. A cask containing 20 gallons of wine was emptied on
one-fifth of its content and then is filled with water. If
this is done 6 times, how many gallons of wine remain in
the cask?
a. 5.121
b. 5.010
c. 5.243
d. 5.343
43. A man wishes to buy a piece of land worth 15 million
pesos. If it were possible for him to save one peso for the
first day, two pesos on the second day, 4 pesos on the
third day and so on. In how many days would he save
enough money to buy the land?
a. 20
b. 24
c. 23
d. 27
44. Pure tin and pure iron was added to a 50 kg of an alloy
containing 10% tin and 20 % iron. The process produced
a new alloy containing 20% tin and 50% iron. What is the
weight of the new alloy?
a. 66.67 kg
b. 116.67 e
c. 86.25 kg
d. 153.33 kg
45. A can do a job in 4 days, B can do the job in 6 days
and C can do the job in 8 days. How long will it take to do
the job if A and B work for 1 day then B and C finish the
job?
a. 1
b. 4
b. 2
d. 3