Algebra Esplana Review Center

ALGEBRA CE Review May 2021 (EERC)

Page 1 of 6

ALGEBRA BINOMIAL/MULTINOMIAL EXPANSION 1.

In the expansion of (x + 4y)12, the numerical coefficient of the 5th term is:

2.

3.

4.

a.

253 440

c. 63 360

b.

126 720

d. 506 880

The term involving y4 in the expansion of (3x – y)11 is: a.

721 710 x7y4

c. – 721 710 x7y4

b.

727 110 x7y4

d. – 727 110 x7y4

The term involving x9 in (x2 +

2 12 ) x

is:

a.

32 168 x9

c. 6 987 x9

b.

25 344 x9

d. 512 x9

Find the term in (a + 2b – c)10 that involves a4c3 a.

33 600 a4b4c3

c. – 33 600 a4b4c3

b.

33 600 a4b3c3

d. – 33 600 a4b3c3

Solve for the value/s of x in the equation: 5x2 – 6x - 9 = 0

6.

a.

2.07, 0.87

c. 2.07, - 0.87

b.

– 2.07, 0.87

d. – 2.07, - 0.87

Solve for x in the equation: x=1+

1 1+

1

1 1+ 1+…

9.

a.

1.390

c. 1.521

b.

1.483

d. 1.618

What is the sum of the roots? a.

3

c. 6

b.

–3

d. – 6

a.

27

c. – 27/2

b.

– 27

d. 27/2

Find the quadratic equation if one of the roots is 8 – 3i. 2

x + 16x + 73 = 0

c.

x2 + 16x – 73 = 0

d.

x2 – 16x – 73 = 0

10. What is the value of k so that 4x + (4k – 3)x + 1 = 0 has two equal roots? a.

3/4

c. – 3/4

b.

– 1/4

d. 1/4

11. In a quadratic equation problem, one student made a mistake in copying the coefficient of x and got the roots 5 and – 3/2. Another student also made a mistake in copying the constant term and got the roots 2/3 and 1/4. Find the correct quadratic equation. a.

SITUATION 2: For log 1 456 12. Determine its characteristic. a.

2

c. 4

b.

3

d. 5

13. Determine its mantissa. a.

0.1632

c. 0.8368

b.

– 0.1632 d. – 0.8368

SITUATION 3: For log 0.0000564302 14. Determine its characteristic. a.

–2

c. - 4

b.

–3

d. – 5

a.

0.2485

c. 0.7515

b.

– 0.2485

d. – 0.7515

16. Solve for x if logx 36 = 2. a.

2

c. 6

b.

4

d. 8

a.

2

c. 6

b.

4

d. 8

a.

2

b.

4

c. 6

ex ) ex-2

d. 5 x

2x + 1

19. Solve the equation (3 )(5

) = 63x – 2

a.

4.9902

c. 4.9920

b.

4.9092

d. 4.2099

20. The polynomial x3 + 4x2 – 3x + 8 is divided by x – 5, then the remainder is: a.

175

c. 218

b.

140

d. 200

21. Find the remainder if we divide 4y3 + 16y2 + 8y – 4 by a.

10

c. 15

b.

11

d. 13

22. Given f(x) = (x + 3)(x – 4) + 4. When f(x) is divided 2

2

LOGARITHMS

(2y + 3)

x2 – 16x + 73 = 0

b.

d. 12x2 – 11x – 90 = 0

REMAINDER AND FACTOR THEOREM

What is the product of the roots?

a.

12x2 + 11x – 90 = 0

18. Find the value of y in the equation y = ln(

x2 + 6x – 27 = 0

8.

c.

17. Find a if log2(a + 2) + log2(a – 2) = 5.

SITUATION 1: Given the quadratic equation: 7.

12x2 – 11x + 90 = 0

15. Determine its mantissa

QUADRATIC EQUATIONS 5.

b.

12x + 11x + 90 = 0.

by (x – k), the remainder is k. What is the remainder? a.

2

c. 6

b.

4

d. 8

NUMBER/DIGIT PROBLEMS 23. The sum of five consecutive even integers is 1280. Find the product of the lowest and highest integer. a.

64 512

c. 65 532

b.

65 024

d. 65 520

JOHN REY M. PACTURANAN, CE, MP

ALGEBRA CE Review May 2021 (EERC)

Page 2 of 6

24. The sum of the digits of a two – digit number is 10.

a.

26 y.o.

c. 42 y.o.

If the number is divided by the units’ digit, the

b.

30 y.o.

d. 36 y.o.

quotient is 3 and the remainder is 4. Find the

32. The sum of the parents’ ages is twice the sum of the children’s ages. Four years ago, the sum of the

number. a.

37

c. 46

parents’ ages was thrice the sum of the children’s

b.

28

d. 19

ages. In 16 years, the sum of the ages of the

25. The sum of the digits of a three – digit number is 17.

parents and children will be equal. How many children

If the digits are reversed and the resulting number

are there?

is added to the original number, the result is 1 474.

a.

4

c. 5

b.

7

d. 6

If the resulting number is subtracted from the original number, the result is 396. Find the original

33. When John was as old as Paul is now, the sum of their ages was 51. When Paul will be as old as John is

number. a.

935

c. 854

now, the sum of their ages will be 103. John is older

b.

845

d. 953

than Paul by how many years?

PROPORTION/VARIATION PROBLEMS 26. z varies directly as x and inversely as the square

a.

25 years

c. 13 years

b.

19 years

d. 32 years

CLOCK PROBLEM

root of y. It is equal to 3 when x is 13 and y is 16. Find the value of z when x is 52 and y is 9.

34. At what time between 4:00 PM and 5:00 PM will the hands be coincident?

a.

24

c. 25

b.

20

d. 16

27. At constant temperature, the resistance of a wire

a.

4:21.82

c. 4:23.64

b.

4:22.27

d. 4:21.64

varies directly as its length and inversely as the

35. In how many minutes after 2 o’clock will the hands

square of its diameter. If a piece of wire 0.10 in in

of the clock extend in opposite directions for the

diameter and 50 ft long has a resistance of 0.10

first time?

ohm, what is the resistance of another piece of wire

a.

40.64 min

c. 42.64 min

b.

41.64 min

d. 43.64 min

of the same material, 2 000 ft long, 0.20 in in

36. Find the angle between the hands of the clock at

diameter? a.

0.50 ohm

c. 1 ohm

b.

4 ohms

d. 2 ohms

28. If three cats can kill three mice in three minutes, in how many minutes can twelve cats kill twelve mice?

3:43 PM. a.

148.500

b.

0

147.50

c. 145.500 d. 146.500

37. A man left his home at past 3:00 PM as indicated in

a.

12 min

c. 4 min

his wall clock. Between two or three hours after, he

b.

6 min

d. 3 min

returned home and noticed that the hands of the

29. Ten students from Pampanga decided to stay in

clock interchanged. At what time did he leave his

Manila for a regular review in preparation for the CE

home?

Board Exam. To minimize their expense, they agreed

a.

3:33.47

c. 3:32.47

to bring 18 sacks of rice that will last for 4 months.

b.

3:31.47

d. 3:34.47

After their regular review, 4 students went back to Pampanga. Only the remaining students enrolled in

MIXTURE PROBLEMS

the refresher for 1 month. How much more rice will the remaining students need for their extended stay in Manila? (1 sack = 50 kg)

38. A 50 mL 40% acid solution is added to 150 mL 30% acid solution. What will be the concentration of the

a.

120 kg

c. 150 kg

b.

135 kg

d. 175 kg

AGE PROBLEMS

resulting mixture? a.

25 %

c. 30 %

b.

27.5 %

d. 32.5 %

39. How much of a 90 % solution of insect spray must a farmer add to 200 cc of 40% insect spray to make a

30. Elmer is 36 years old and his daughter is 8 years old.

50% solution of insect spray?

In how many years will Elmer’s age be twice his

a.

30 cc

c. 50 cc

daughter’s age?

b.

40 cc

d. 60 cc

a.

20 years

c. 18 years

40. A container is filled with 70 L which is 40 % alcohol

b.

24 years

d. 22 years

by volume. How much of a mixture must be taken and

31. Two years ago, a father was four times as old as his son. In 3 years, the father will only be three times

then replaced with equal amount of water so that the resulting solution is 30 % alcohol by volume?

as old as his son. How old was the father when his

a.

17.50 L

c. 20 L

son was born?

b.

15 L

d. 22.50 L

JOHN REY M. PACTURANAN, CE, MP

ALGEBRA CE Review May 2021 (EERC)

Page 3 of 6

41. A vat contains a mixture of acid and water. If 25

49. Twenty – eight men can finish the job in 60 days. At

gallons of acid are added, the mixture will be 80 %

the start of the 16th day, 5 men were laid off and

acid. If 25 gallons of water are added, the mixture

after the 45th day, 10 more men were hired. How

will be 60 % acid. Find the percentage of acid in the

many days were they delayed in finishing the job?

mixture. a.

65 %

c. 75 %

b.

70 %

d. 72 %

a.

2.27 days

c. 2.97 days

b.

2.45 days

d. 3.67 days

VENN DIAGRAM MOTION PROBLEM 42. Two airplanes left airports which are 960 km apart

50. In a certain group of consumers, each one may drink

and flew toward each other. One plane flew 32 kph

beer, and/or brandy, and/or whisky, or all. Also, 155

faster than the other. If they passed each other at

drink brandy, 173 drink beer, 153 drink whisky, 53

the end of an hour and 12 minutes, what was the

drink beer and brandy, 79 drink beer and whisky, 66

rate of faster plane?

drink brandy and whisky, 21 of them drink beer,

a.

352 kph

c. 416 kph

brandy and whisky. How many are there in the

b.

384 kph

d. 448 kph

group?

43. The boat travels downstream in two – thirds the time as it does upstream. If the speed of the river current is 8 kph, determine the velocity of the boat in still water.

a.

302

c. 304

b.

303

d. 305

51. The probability of the students passing Chemistry and Physics are 40 % and 55 %, respectively. If

a.

30 kph

c. 50 kph

there are 260 students who took the exam and 37

b.

40 kph

d. 60 kph

of them failed in both subjects, how many students

44. A man walks from his house to the office. If he

passed both subjects?

leaves at 8:00 AM and walks at the rate of 2 kph, he will have arrived 3 minutes earlier, but if he leaves

a.

20

c. 27

b.

28

d. 24

at 8:30 AM and walks at 3 kph, he will have arrived 6

52. CSPC CE 5 students took an examination on

minutes late. What time should he arrive at the

Geometry, Calculus, Probability and Engineering

office?

Economy. The results are as follows:

a.

9:06 AM

c. 8:54 AM

b.

9:12 AM

d. 8:43 AM

45. A man started on his mountain bike for Manila, a

178 Passed Engineering Economy 172 Passed Probability

distance of 30 km, intending to arrive at a certain

177 Passed Calculus

time. After biking for 10 km, he was detained due to

161 Passed Geometry

the bad weather for half an hour. As a result, he had

65 Passed Probability and Engineering Economy

to speed up 2 km faster. What was his original

63 Passed Calculus and Engineering Economy

speed?

56 Passed Calculus and Probability

a.

8.50 kph

c. 7.50 kph

49 Passed Geometry and Engineering Economy

b.

8 kph

d. 9 kph

60 Passed Geometry and Probability 51 Passed Geometry and Calculus

WORK PROBLEMS

19 Passed Calculus, Probability and Engineering Economy

46. A can do a piece of work alone in 30 days, B in 20

20 Passed Geometry, Probability, and Engineering

days, and C in 60 days. If they work together, how many days would it take them to finish the work? a.

15 days

c. 10 days

b.

8 days

d. 12 days

47. A and B working together can finish the job in 10

Economy 15 Passed Geometry, Calculus, and Engineering Economy 16 Passed Geometry, Calculus and Probability 6 Passed all subjects

days. If A works 4 days and B works 3 days, onethird of the job shall be finished. How many days will

If there were 93 students who did not pass in any

it take A to finish the job alone?

subject, how many students took the examination?

a.

30 days

c. 20 days

a.

498

c. 500

b.

15 days

d. 45 days

b.

499

d. 501

48. Eight men can excavate 50 m of canal in 7 hours. Three men can backfill 30 m of the excavated canal

ARITHMETIC SEQUENCE AND SERIES

in 4 hours. How long would it take 10 men to dig and backfill 100 m of canal?

SITUATION 4: Given the sequence 15, 18, 21, …

a.

12.50 hrs

c. 21.50 hrs

b.

15.20 hrs

d. 25.10 hrs

53. Determine the 30th term. a.

153

JOHN REY M. PACTURANAN, CE, MP

c. 120

ALGEBRA CE Review May 2021 (EERC) b.

Page 4 of 6

102

d. 135

54. What is the order of “93” in the sequence?

66. Find the sum of the geometric progression 2x, 4x + 14, 20x – 14, … up to the 10th term.

a.

27

c. 29

a.

566 579

c. 617 774

b.

28

d. 30

b.

312 228

d. 413 336

55. Determine the sum of first 20 terms. a.

840

c. 870

b.

900

d. 930

56. How many terms of the sequence are needed in order to obtain a sum of 1188?

67. Find the geometric mean of 2.138, 6.414, 19.242, and 57.726. a.

10.367

c. 11.109

b.

10.995

d. 12.607

68. A container is filled with 20 gallons of pure water.

a.

21

c. 23

Five gallons of water is taken from the container and

b.

22

d. 24

is replaced by 5 gallons of pure acid and then thoroughly mixed. Another 5 gallons is taken from

57. Find the sum of all odd numbers between 100 and 1000.

the mixture and is replaced again by 5 gallons of pure acid. If this process is done repeatedly, find

a.

472 500

c. 247 500

the amount of water in the container after doing the

b.

427 500

d. 274 500

process 15 times.

2

2

58. If the term 3(x -1), x – 4x + 5, 11 – 9x form an arithmetic progression, find the sum of the first 8

a.

0.672 gal

c. 0.276 gal

b.

0.267 gal

d. 0.627 gal

69. A ball is dropped from a height of 48 ft and

terms. a.

– 144

c. - 160

rebounds two-thirds of the distance it falls. If it

b.

– 168

d. – 152

continues to fall and rebound in this way, how far will

59. A particle moves along a straight path. For the first

it travel before coming to rest?

second, it travels 16 m. In every second after the

a.

120 ft.

c. 240 ft.

first, it travels 2 m farther than it did in the

b.

192 ft.

d. 200 ft.

preceding second. How far will it travel after 10 HARMONIC SEQUENCE AND SERIES

seconds? a.

250 m

c. 180 m

b.

34 m

d. 92 m

60. In the recent “Gulf War” in the Middle East, the Allied Forces captured 6 390 of the Saddam’s soldier with provisions on hand that will last for 216 meals taking 3 meals a day. The provisions lasted 9 more days because of daily deaths. At an average, how many captured soldiers died per day? a.

15

c. 17

b.

16

d. 18

70. Find the 20th term of the harmonic progression 1/2, 1/5, 1/8, … a.

1/50

c. 1/56

b.

1/53

d. 1/59

71. The 3rd term of a harmonic progression is 15 and the 9th term is 6. Find the 11th term. a.

4

c. 6

b.

5

d. 7

72. The geometric mean and the arithmetic mean of two numbers are 8 and 17, respectively. Find the

GEOMETRIC SEQUENCE AND SERIES SITUATION 5: Given the sequence 2, 6, 18, … 61. What is the 15

th

harmonic mean. a.

3.45

c. 3.54

b.

3.76

d. 3.67

73. A race is scheduled for four laps. The velocities of a

term?

a.

9 565 938

c. 9 656 938

car for these laps are 60 kph, 56 kph, 52 kph, and

b.

9 556 938

d. 9 665 938

63 kph, consecutively. Find its average velocity for

62. In what order of the sequence is 354 294?

the whole race.

a.

11

c. 13

a.

57.30 kph

c. 57.60 kph

b.

12

d. 14

b.

57.45 kph

d. 57.75 kph

63. What is the sum of the first 10 terms? a.

59 408

c. 59 048

b.

59 480

d. 59 084

64. How many terms of the sequence are needed in

SUMMATION 74. Evaluate: 10

order to obtain a sum of 6560? a.

7

c. 9

b.

8

d. 10

65. Find the sum of the infinite geometric progression 36. 24, 16, … a.

108

c. 160

b.

180

d. 116

a.

1.08

b.

1.18

� 1

75. Determine the value of:

JOHN REY M. PACTURANAN, CE, MP

1 2x+1 c. 1.28 d. 1.38

ALGEBRA CE Review May 2021 (EERC)

Page 5 of 6

15

a.

-0.934

b.

-0.439

88. What is the cofactor of A23?

2x � 6-3x2 8

c. -0.943

d. -0.493

MATRICES AND DETERMINANTS

a.

31

c. 13

b.

– 31

d. – 13

89. What is the cofactor of A31? a.

36

c. 63

b.

– 36

d. – 63

-1

SITUATION 6: Given the following matrices: 3 7 0 9 -2 5 A = -2 6 -9 B = 2 -3 7 1 -8 -3 -4 0 1 76. Find A + B 12 5 5 A + B = 0 3 -2 -3 -8 -2 77. Find A – B -6 9 -5 A – B = -4 9 -16 5 -8 -4 78. Find 2B – 3A 9 -25 10 2B – 3A = 10 -24 41 -11 24 11 79. Find AB 41 -27 64 AB = 30 -14 23 5 22 -54 80. Find BA 36 11 3 BA = 19 -60 6 -11 -36 -3 81. Find |A| a.

325

c. - 325

b.

375

d. - 375

82. Find |B| a.

27

c. - 72

b.

– 27

d. 72

83. Find AT (or A’) 3 -2 1 AT = 7 6 -8 0 -9 -3 84. Find BT (or B’) 9 2 -4 BT = -2 -3 0 5 7 1

90. What is A ? 6/25 -7/125 A-1 = 1/25 3/125 -2/75 -31/375

21/125 -9/125 -32/375

91. What is the value of: 8

9

0

-2

4

1

-2

3

0

3

2

-4

1

-9

5

3

a.

160

c. 106

b.

– 160

d. – 106

92. What is the value of: -7

0

5

3

10

9

-1

4

6

-3

9

1

4

0

12

3

a.

1374

c. 1734

b.

– 1374

d. – 1734

COMPLEX NUMBER 93. What is the value of i3? a.

i

c. 1

b.

– i

d. – 1

94. What is the value of i12 322 a.

i

c. 1

b.

–i

d. – 1

95. What is the value of i-243 a.

i

c. 1

b.

–i

d. – 1

SITUATION: Given that z1 = 6 – 3i and z2 = 5 + 4i 96. Determine z1 + z2

SITUATION 7: Given the following matrices: 4 0 2 5 7 B = -6 5 A= -1 0 8 1 0 85. Find AB AB = - 15 25 4 0 86. Find BA 8 20 28 BA = -17 -30 -2 2 5 7 SITUATION 8: Given the matrix: 3 7 0 A = -2 6 -9 1 -8 -3 87. What is the minor of A23? 3 1� 3 7� a. � c. � 1 8 7 -8 3 7� -3 1 � b. � d. � 1 -8 7 8

a.

11 + i

c. – 11 + i

b.

11 – i

d. – 11 - i

97. Determine 4z2 - 3z1 a.

25 + 2i

c. 2 + 25i

b.

25 – 2i

d. 2 – 25i

98. Determine z1z2 a.

9 – 42i

c. 42 – 9i

b.

9 + 42i

d. 42 + 9i

99. Determine z1/z2 a. b.

1 41 1 41

(18 + 39i)

c.

(18 - 39i)

d.

1 39 1 39

(18 + 41i) (18 - 41i)

SITUATION: Given that z = 6 – 3i 100. Determine |z| a.

6.780

c. 7.608

b.

6.708

d. 7.806

101. Determine z3 a.

54 + 297i

JOHN REY M. PACTURANAN, CE, MP

c. 297 + 54i

ALGEBRA CE Review May 2021 (EERC) b.

54 – 297i

Page 6 of 6 d. 297 – 54i

102. Determine √z a.

0.595 + 2.521i

b.

0.595 – 2.521i

c.

2.521 + 0.595i

d. 2.521 – 0.595i 103. Determine ln z a.

0.464 – 1.903i

b.

0.464 + 1.903i

c.

1.903 – 0.464i

d.

1.903 + 0.464i

SITUATION 9: Given that z = 5 + 4i 104. Determine |z| a.

6.304

c. 6.403

b.

6.340

d. 6.430

5

105. Determine z a.

-10475 – 2476i

b.

10475 – 2476i

c.

10475 + 2476i

d.

-10475 + 2476i 3

106. Determine �z2 a.

3.105 – 1.499i

b.

-3.105 – 1.499i

c.

-3.105 + 1.499i

d. 3.105 + 1.499i 107. Determine log5z a.

1.154 – 0.419i

b. 1.154 + 0.419i c.

-1.154 – 0.419i

d.

-1.154 + 0.419i

108. Determine ln(-6) a.

1.791 + 3.142i

b.

1.971 – 3.142i

c.

1.791 + 6.232i

d.

1.971 – 6.232i

109. Determine log7(-52) a.

2.031 + 3.142i

b. 2.031 + 1.614i c.

2.031 + 2.231i

d.

2.031 + 2.748i

JOHN REY M. PACTURANAN, CE, MP