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11:09:00 1 of 269
TABLE OF CONTENTS
ALGEBRA: SEQUENCE AND SERIES ................................................................................................... 2
ALGEBRA: SEQUENCE AND SERIES SOLUTIONS .............................................................................. 4
ALGEBRA: WORDED PROBLEMS ..................................................................................................... 14
ALGEBRA: WORDED PROBLEMS SOLUTIONS ................................................................................ 19
ALGEBRA: POLYNOMIALS, PARTIAL FRACTIONS AND INEQUALITIES ....................................... 35
ALGEBRA: POLYNOMIALS, PARTIAL FRACTIONS AND INEQUALITIES SOLUTIONS .................. 38
ALGEBRA: BINOMIAL EXPANSION .................................................................................................. 49
ALGEBRA: BINOMIAL EXPANSION SOLUTIONS ............................................................................. 50
ALGEBRA: COMPLEX NUMBERS ...................................................................................................... 54
ALGEBRA: COMPLEX NUMBERS SOLUTIONS ................................................................................. 57
MATRICES AND DETERMINANTS .................................................................................................... 68
MATRICES AND DETERMINANTS SOLUTIONS............................................................................... 71
PLANE AND SPHERICAL TRIGONOMETRY ..................................................................................... 77
PLANE AND SPHERICAL TRIGONOMETRY SOLUTIONS ................................................................ 84
PLANE ANALYTIC GEOMETRY ....................................................................................................... 105
PLANE ANALYTIC GEOMETRY SOLUTIONS .................................................................................. 110
SOLID MENSURATION .................................................................................................................... 123
SOLID MENSURATION SOLUTIONS ............................................................................................... 129
SPACE ANALYTIC GEOMETRY ....................................................................................................... 150
SPACE ANALYTIC GEOMETRY SOLUTIONS .................................................................................. 152
VECTORS .......................................................................................................................................... 162
VECTORS SOLUTIONS ..................................................................................................................... 166
DIFFERENTIAL CALCULUS ............................................................................................................. 176
DIFFERENTIAL CALCULUS SOLUTIONS ........................................................................................ 184
INTEGRAL CALCULUS ..................................................................................................................... 226
INTEGRAL CALCULUS SOLUTIONS ................................................................................................ 229
STATISTICS ...................................................................................................................................... 247
STATISTICS SOLUTIONS ................................................................................................................. 253
PROBABILITY .................................................................................................................................. 258
PROBABILITY SOLUTIONS ............................................................................................................. 263
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269 ENGINEERING MATHEMATICS BOARD
10/01/2026, EXAMINATION REVIEW BOOK
11:09:00 2 of 269
ALGEBRA: SEQUENCE AND SERIES
1. Find the nth term of the arithmetic progression 6, 10, 14, …
A. 2 + 4n C. 4 + 2n
B. 2 – 4n D. 4 – 2n
2. Find the sum up to the 10 term of the arithmetic progression 6, 10, 14, …
th
A. 250 C. 240
B. 120 D. 225
3. Find the n term of the arithmetic progression log 7, log 14, log 28, …
th
A. 0.845 + 0.301n C. -0.544 + 0.301n
B. 0.544 + 0.301n D. -0.544 – 0.301n
th
4. Find the sum up to the 10 term of the arithmetic progression log 7, log 14,
log 28, …
A. 18.442 C. 25.852
B. 26.397 D. 21.997
Situation: Logs are stacked so that there are 25 logs in the bottom row, 24 logs in
the second row, and so on, decreasing by 1 log each row.
5. How many logs are stacked in the sixth row?
A. 21 C. 19
B. 20 D. 18
6. How many logs are there in all six rows?
A. 136 C. 137
B. 134 D. 135
Situation: The distance a ball rolls down a ramp each second is given by the
arithmetic sequence whose nth term 2n – 1 in feet.
7. Find the distance the ball rolls during the 10th second.
A. 18 ft. C. 19 ft.
B. 20 ft. D. 21 ft.
8. Find the total distance the ball travels in 10 seconds.
A. 120 ft. C. 110 ft.
B. 100 ft. D. 90 ft.
Situation: A contest offers 15 prizes. The 1st prize is P 5000, and each successive
prize is P 250 less than the preceding prize.
9. What is the value of the 15th prize?
A. 1250 C. 1500
B. 1750 D. 1625
10.What is the total amount of money distributed in prizes?
A. P 49,000 C. P 50,750
B. P 48,750 D. P 47,250
Situation: The 4th and 7th terms of an arithmetic sequence are 13 and 25.
11.Find the first term.
A. 1 C. 2
B. 3 D. 5
12.Find the common difference.
A. 1 C. 5
B. 4 D. 2
13.Find the 20th term.
A. 81 C. 73
B. 77 D. 83
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14.Insert 5 arithmetic means between -1 and 23.
A. 3, 7, 11, 15, 19 C. -1, 3, 7, 11, 15
B. 7, 10, 13, 16, 19 D. 1, 5, 9, 13, 17
15.An object dropped from a cliff will fall 16 feet the first second, 48 feet the
second, 80 feet the third, and so on, increasing by 32 feet each second. What
does the total distance the object will fall in 7 seconds?
A. 874 ft. C. 748 ft.
B. 847 ft. D. 784 ft.
16.A besieged fortress is held by 5700 men who have provisions for 66 days. If
the garrison loses 20 men each day, how many days can the provision hold
out?
A. 70 days C. 67 days
B. 76 days D. 80 days
17.In a racing contest, there are 240 cars with fuel provision for 15 hours each.
Assuming a constantly hourly consumption for each car, how long will the
fuel provision last if 8 cars withdraw from the race every hour?
A. 72 hours C. 25 hours
B. 23 hours D. 20 hours
18.How many numbers divisible by 4 lie between 70 and 203?
A. 33 C. 34
B. 35 D. 36
19.Find the sum of the numbers divisible by between 70 and 203.
A. 4848 C. 8484
B. 4488 D. 8844
20.Two men set out from a certain place going in the same direction. The first
travels at a constant rate of 8 kilometers per hour, while the second goes 4
km for the first hour, 4.5 km the second hour, 5 km the third hour, and so
on. After how many hours will the second man overtake the first?
A. 18 hours C. 17 hours
B. 20 hours D. 19 hours
21.Ten balls are placed in a straight line on the ground at intervals of 2 meters.
Six meters from the end of the row a basket is placed. A boy starts from the
basket and picks up the balls and carries them, one at a time to the basket.
How far did he walk all in all?
A. 120 m C. 250 m
B. 130 m D. 300 m
22.Find the third term of a geometric sequence whose first term is 2 and whose
fifth term is 162.
A. 18 C. 9
B. 6 D. 27
23.A basketball is dropped from a height of 10m. On each rebound it rises 2/3
of the height from which it last fell. Determine the total distance travelled
until it comes to rest.
A. 45 m C. 50 m
B. 60 m D. 75 m
24.Find the sum of the geometric progression 2, 6, 18, … up to the 10th term.
A. 10,682 C. 59,048
B. 177,146 D. 6,560
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269 ENGINEERING MATHEMATICS BOARD
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ALGEBRA: SEQUENCE AND SERIES SOLUTIONS
1. Find the nth term of the arithmetic progression 6, 10, 14, …
First Solution:
Using the formula:
an = a1 + (n − 1)d
a1 = 6 → first term
d = 4 → common difference
We have:
an = 6 + (n − 1)(4)
an = 6 + 4n − 4
an = 2 + 4n
Second Solution:
Since the relationship between an and n is linear, hence we can use here the
STAT Mode 3-2 → A + Bx
For Linear Mathematical Model, we only need 2 points to define the function
in the form of y = A + Bx.
Input:
x y
1 6
2 10
Press AC.
Then press Shift 1 – 5 (Reg – Regression)
Then select 1: A and then select 2: B
A = 2; B = 4
Therefore, y = A + Bx
an = 2 + 4n
2. Find the sum up to the 10th term of the arithmetic progression 6, 10, 14, …
First Solution:
By using the formula of the sum of an arithmetic progression of n terms:
[2a1 + (n − 1)d]n
sn =
2
[2(6) + (10 − 1)(4)](10)
sn =
2
sn = 240
Second Solution:
From the formula, it shows that the relationship between n and sn is in
quadratic form, so we can use STAT MODE 3-3 → A + Bx + Cx2
For Quadratic Mathematical Model, we need 3 points to define the function
in the form of y = A + Bx + Cx 2 .
Input:
x y
1 6
2 6 +10 = 16
3 6 + 10 +14 = 30
For the sum of the first 10 terms:
Find the value of y which corresponds to the value of x = 10.
Hence 10ŷ = 240.
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Page 1 of 269CIVIL ENGINEERING MATHEMATICS BOARD
10/01/2026, EXAMINATION REVIEW BOOK
11:09:00 1 of 269
TABLE OF CONTENTS
ALGEBRA: SEQUENCE AND SERIES ................................................................................................... 2
ALGEBRA: SEQUENCE AND SERIES SOLUTIONS .............................................................................. 4
ALGEBRA: WORDED PROBLEMS ..................................................................................................... 14
ALGEBRA: WORDED PROBLEMS SOLUTIONS ................................................................................ 19
ALGEBRA: POLYNOMIALS, PARTIAL FRACTIONS AND INEQUALITIES ....................................... 35
ALGEBRA: POLYNOMIALS, PARTIAL FRACTIONS AND INEQUALITIES SOLUTIONS .................. 38
ALGEBRA: BINOMIAL EXPANSION .................................................................................................. 49
ALGEBRA: BINOMIAL EXPANSION SOLUTIONS ............................................................................. 50
ALGEBRA: COMPLEX NUMBERS ...................................................................................................... 54
ALGEBRA: COMPLEX NUMBERS SOLUTIONS ................................................................................. 57
MATRICES AND DETERMINANTS .................................................................................................... 68
MATRICES AND DETERMINANTS SOLUTIONS............................................................................... 71
PLANE AND SPHERICAL TRIGONOMETRY ..................................................................................... 77
PLANE AND SPHERICAL TRIGONOMETRY SOLUTIONS ................................................................ 84
PLANE ANALYTIC GEOMETRY ....................................................................................................... 105
PLANE ANALYTIC GEOMETRY SOLUTIONS .................................................................................. 110
SOLID MENSURATION .................................................................................................................... 123
SOLID MENSURATION SOLUTIONS ............................................................................................... 129
SPACE ANALYTIC GEOMETRY ....................................................................................................... 150
SPACE ANALYTIC GEOMETRY SOLUTIONS .................................................................................. 152
VECTORS .......................................................................................................................................... 162
VECTORS SOLUTIONS ..................................................................................................................... 166
DIFFERENTIAL CALCULUS ............................................................................................................. 176
DIFFERENTIAL CALCULUS SOLUTIONS ........................................................................................ 184
INTEGRAL CALCULUS ..................................................................................................................... 226
INTEGRAL CALCULUS SOLUTIONS ................................................................................................ 229
STATISTICS ...................................................................................................................................... 247
STATISTICS SOLUTIONS ................................................................................................................. 253
PROBABILITY .................................................................................................................................. 258
PROBABILITY SOLUTIONS ............................................................................................................. 263
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269 ENGINEERING MATHEMATICS BOARD
10/01/2026, EXAMINATION REVIEW BOOK
11:09:00 2 of 269
ALGEBRA: SEQUENCE AND SERIES
1. Find the nth term of the arithmetic progression 6, 10, 14, …
A. 2 + 4n C. 4 + 2n
B. 2 – 4n D. 4 – 2n
2. Find the sum up to the 10 term of the arithmetic progression 6, 10, 14, …
th
A. 250 C. 240
B. 120 D. 225
3. Find the n term of the arithmetic progression log 7, log 14, log 28, …
th
A. 0.845 + 0.301n C. -0.544 + 0.301n
B. 0.544 + 0.301n D. -0.544 – 0.301n
th
4. Find the sum up to the 10 term of the arithmetic progression log 7, log 14,
log 28, …
A. 18.442 C. 25.852
B. 26.397 D. 21.997
Situation: Logs are stacked so that there are 25 logs in the bottom row, 24 logs in
the second row, and so on, decreasing by 1 log each row.
5. How many logs are stacked in the sixth row?
A. 21 C. 19
B. 20 D. 18
6. How many logs are there in all six rows?
A. 136 C. 137
B. 134 D. 135
Situation: The distance a ball rolls down a ramp each second is given by the
arithmetic sequence whose nth term 2n – 1 in feet.
7. Find the distance the ball rolls during the 10th second.
A. 18 ft. C. 19 ft.
B. 20 ft. D. 21 ft.
8. Find the total distance the ball travels in 10 seconds.
A. 120 ft. C. 110 ft.
B. 100 ft. D. 90 ft.
Situation: A contest offers 15 prizes. The 1st prize is P 5000, and each successive
prize is P 250 less than the preceding prize.
9. What is the value of the 15th prize?
A. 1250 C. 1500
B. 1750 D. 1625
10.What is the total amount of money distributed in prizes?
A. P 49,000 C. P 50,750
B. P 48,750 D. P 47,250
Situation: The 4th and 7th terms of an arithmetic sequence are 13 and 25.
11.Find the first term.
A. 1 C. 2
B. 3 D. 5
12.Find the common difference.
A. 1 C. 5
B. 4 D. 2
13.Find the 20th term.
A. 81 C. 73
B. 77 D. 83
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14.Insert 5 arithmetic means between -1 and 23.
A. 3, 7, 11, 15, 19 C. -1, 3, 7, 11, 15
B. 7, 10, 13, 16, 19 D. 1, 5, 9, 13, 17
15.An object dropped from a cliff will fall 16 feet the first second, 48 feet the
second, 80 feet the third, and so on, increasing by 32 feet each second. What
does the total distance the object will fall in 7 seconds?
A. 874 ft. C. 748 ft.
B. 847 ft. D. 784 ft.
16.A besieged fortress is held by 5700 men who have provisions for 66 days. If
the garrison loses 20 men each day, how many days can the provision hold
out?
A. 70 days C. 67 days
B. 76 days D. 80 days
17.In a racing contest, there are 240 cars with fuel provision for 15 hours each.
Assuming a constantly hourly consumption for each car, how long will the
fuel provision last if 8 cars withdraw from the race every hour?
A. 72 hours C. 25 hours
B. 23 hours D. 20 hours
18.How many numbers divisible by 4 lie between 70 and 203?
A. 33 C. 34
B. 35 D. 36
19.Find the sum of the numbers divisible by between 70 and 203.
A. 4848 C. 8484
B. 4488 D. 8844
20.Two men set out from a certain place going in the same direction. The first
travels at a constant rate of 8 kilometers per hour, while the second goes 4
km for the first hour, 4.5 km the second hour, 5 km the third hour, and so
on. After how many hours will the second man overtake the first?
A. 18 hours C. 17 hours
B. 20 hours D. 19 hours
21.Ten balls are placed in a straight line on the ground at intervals of 2 meters.
Six meters from the end of the row a basket is placed. A boy starts from the
basket and picks up the balls and carries them, one at a time to the basket.
How far did he walk all in all?
A. 120 m C. 250 m
B. 130 m D. 300 m
22.Find the third term of a geometric sequence whose first term is 2 and whose
fifth term is 162.
A. 18 C. 9
B. 6 D. 27
23.A basketball is dropped from a height of 10m. On each rebound it rises 2/3
of the height from which it last fell. Determine the total distance travelled
until it comes to rest.
A. 45 m C. 50 m
B. 60 m D. 75 m
24.Find the sum of the geometric progression 2, 6, 18, … up to the 10th term.
A. 10,682 C. 59,048
B. 177,146 D. 6,560
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269 ENGINEERING MATHEMATICS BOARD
10/01/2026, EXAMINATION REVIEW BOOK
11:09:00 4 of 269
ALGEBRA: SEQUENCE AND SERIES SOLUTIONS
1. Find the nth term of the arithmetic progression 6, 10, 14, …
First Solution:
Using the formula:
an = a1 + (n − 1)d
a1 = 6 → first term
d = 4 → common difference
We have:
an = 6 + (n − 1)(4)
an = 6 + 4n − 4
an = 2 + 4n
Second Solution:
Since the relationship between an and n is linear, hence we can use here the
STAT Mode 3-2 → A + Bx
For Linear Mathematical Model, we only need 2 points to define the function
in the form of y = A + Bx.
Input:
x y
1 6
2 10
Press AC.
Then press Shift 1 – 5 (Reg – Regression)
Then select 1: A and then select 2: B
A = 2; B = 4
Therefore, y = A + Bx
an = 2 + 4n
2. Find the sum up to the 10th term of the arithmetic progression 6, 10, 14, …
First Solution:
By using the formula of the sum of an arithmetic progression of n terms:
[2a1 + (n − 1)d]n
sn =
2
[2(6) + (10 − 1)(4)](10)
sn =
2
sn = 240
Second Solution:
From the formula, it shows that the relationship between n and sn is in
quadratic form, so we can use STAT MODE 3-3 → A + Bx + Cx2
For Quadratic Mathematical Model, we need 3 points to define the function
in the form of y = A + Bx + Cx 2 .
Input:
x y
1 6
2 6 +10 = 16
3 6 + 10 +14 = 30
For the sum of the first 10 terms:
Find the value of y which corresponds to the value of x = 10.
Hence 10ŷ = 240.
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