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REO CELE ReviewREAL EXCELLENCE ONLINE CIVIL ENGINEERING REVIEW Effectiveness. Efficiency. ConvenienceMSTE 1 MSTE-Algebra------------------------------------------------2- TABLE OF CONTENTS 2 MSTE-Trigonometry---------------------------------------22- 3 MSTE-Plane Geometry----------------------------...

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REAL EXCELLENCE ONLINE CIVIL ENGINEERING REVIEW Effectiveness. Efficiency. Convenience

MSTE

TABLE OF CONTENTS 1.0 MSTE-Algebra------------------------------------------------2-21 2.0 MSTE-Trigonometry---------------------------------------22-31 3.0 MSTE-Plane Geometry-----------------------------------32-40 4.0 MSTE-Solid Geometry------------------------------------41-47 5.0 MSTE-Analytical Geometry------------------------------49-56 6.0 MSTE-Differential Calculus-------------------------------57-59 7.0 MSTE-Differential Calculus -Limits--------------------------60 8.0 MSTE-Differential Calculus -Derivatives ------------------61 9.0 MSTE-Differential Calculus -Applications -----------------62 10.0 MSTE-Integral Calculus-----------------------------------64-64 11.0 MSTE-Integral Calculus-Indefinite Integrals--------------65 12.0 MSTE-Integral Calculus -Applications -----------------66-67 13.0 MSTE-Differential Equations------------------------------68-69 14.0 MSTE-Differential Equations -Theories ----------------70-71 15.0 MSTE-Differential Equations -Applications -----------72-73 16.0 MSTE-Probability and Statistics--------------------------74-75 17.0 MSTE-Probability---------------------------------------------76-78 18.0 MSTE-Statistics-----------------------------------------------79-80 19.0 MSTE-Matrices and Vectors-------------------------------81-84 20.0 MSTE Matrices and Vectors--------------------------------85-89 21.0 MSTE-Physics--------------------------------------------------88-90 22.0 MSTE-Engineering Economics-----------------------------91-92 23.0 MSTE Engineering Economics-----------------------------93-95 24.0 Elementary and Higher Surveying-----------------------96-105 25.0 MSTE-Route Surveying-----------------------------------106-113 26.0 Transpo - Traffic Engineering)---------------------------------114 27.0 Transpo - Sight Distance)---------------------------------------115

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REAL EXCELLENCE ONLINE CIVIL ENGINEERING REVIEW Effectiveness. Efficiency. Convenience

Algebra ENGR. CARL EDGAR AMBRAY

ALGEBRA

i

prepared by: Engr. Carl Edgar C. Ambray SETS AND OTHER BASIC CONCEPTS Variable – a letter used to represent various numbers or another equation. Constant – letter used to represent a particular value. Algebraic exp expression ression – any combination of numbers, variables, exponents, mathematical symbols, and operations. Set – a collection of objects usually enclosed by curly brackets {} and separated by comma. Example: Set No. of ele elements ments 3 𝑨 = {𝒂, 𝒃, 𝒄} 4 𝑩 = {𝒚𝒆𝒍𝒍𝒐𝒘, 𝒈𝒓𝒆𝒆𝒏, 𝒃𝒍𝒖𝒆, 𝒓𝒆𝒅} C = {1, 2, 3, 4, 5} 5 ∈ – read as ‘is an element of’ Example: 2∈𝐶 “2 is an element of set C”

Finite Sets – Sets that have countable and enumerable elements Infinite Se Sets ts – sets that have countless and impractically enumerable elements Example: 𝑁 = {1,2,3,4,5, … } 𝐼 = {… , −4, −3, −2, −1, 0, 1, 2, 3, 4, … } Continuous Sets – sets having elements that are not infinite but are impractically enumerable one by one. Example: 𝐷 = {1, 2, 3, 4, 5, … . , 100} 𝐸 = {2, 4, 6, 8, 10, … , 100} Null Set – set that contains no element Example: 𝐺 = {} 𝑃=∅ Inequality symbols: Symbol Read as is greater than > is greater than or equal ≥ is less than < is less than or equal ≤ not equal ≠

Inequality notation scena scenarios rios Notation Means 𝑥>4 x is greater than 4 a 𝑥≥6 x is greater than or equal to 4 b 𝑥7

and less than or equal to 4 x is less than -5 and x is greater than 7

Illustration: Row a:

Row b:

Row c:

Row d: Row e: Row f:

Row g: Row h:

Set Builder Notation second method of describing a set Example: 𝐸 = {𝑥|𝑥 𝑖𝑠 𝑎 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 5} “set E is a set of all element x such that x is a natural number greater than 5”

Union and Intersection of Sets

The union of set A and set B, written 𝑨 ∪ 𝑩, is the set of elements that belong to either se sett A or set B B. Set builde builderr notation: 𝐴 ∪ 𝐵 = {𝑥|𝑥 ∈ 𝐴 or 𝑥 ∈ 𝐵} Example: 𝑨 𝑩 𝑨∪𝑩 𝑨 𝐵 𝐴∪𝐵 = {𝟏, 𝟐, 𝟑, 𝟒, 𝟓} = {3, 4, 5, 6, 7} = {1,2,3,4,5,6,7} 𝑨 = {𝒂, 𝒃, 𝒄, 𝒅, 𝒆}

𝐵 = {𝑥, 𝑦, 𝑧}

Venn Diagram Diagram::

A

𝐴∪𝐵 = {𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑥, 𝑦, 𝑧}

B

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Algebra ENGR. CARL EDGAR AMBRAY

The interse intersection ction of set A and set B, written 𝑨 ∩ 𝑩, is the set of all elements that are common to bo both th set A and set B B. Set builde builderr notation: 𝐴 ∩ 𝐵 = {𝑥|𝑥 ∈ 𝐴 and 𝑥 ∈ 𝐵} Example: 𝑨 𝑩 𝑨∪𝑩 𝑨 = {𝟏, 𝟐, 𝟑, 𝟒, 𝟓} 𝑨 = {𝒂, 𝒃, 𝒄, 𝒅, 𝒆}

Venn Diagram Diagram::

𝐵 = {3, 4, 5, 6, 7}

B

Sample P Problems: roblems: 1. Find the resulting set of {1, 3, 7} ∩ {2, 3, 8} a. {1, 2, 7, 8} c. {1, 2, 3, 7, 8} ∅ d. {3} b. 2.

If the set of A is a set of all whole numbers and set B is a set of all whole numbers divisible by three less than 12. What is 𝐴 ∩ 𝐵? c. {3} a. {3, 6, 9} b.

3.

4.

5.

6.

{3, 6, 9, 12}

d.

{3, 6}

Find the resulting set of {4, 6, 9} ∪ {6, 8, 11} a. {4, 6, 8} c. {4, 6, 6,8, 9, 11} b. {4, 6, 8, 9, 1 11} 1} d. {8, 11}

If set A is a set of all whole numbers divisible by 3 less than or equal to 9 and set B is a set of all whole numbers divisible by 8 less than or equal to 24. What is 𝐴 ∪ 𝐵 a. {3,6,8,9,16,24 {3,6,8,9,16,24}} c. {3,6,9,24} b. ∅ d. {16,24} Out of 51 students, 9 are taking Algebra and 13 are taking Geometry if 2 students are in both classes, how many students are in neither class? a. 30 c. 31 b. 28 d. 29

c. d.

40 41

In a survey of University students,106 had taken Algebra course, 92 had taken Arts course, 89 had taken Chemistry course, 52 had taken Algebra and Chemistry courses, 54 had taken Algebra and Arts courses, 21 had taken Arts and Chemistry , 19 taken all courses. How many taken only one course? c. 87 a. 90 b. 89 d. 86

8.

In a survey of University students,42 had taken Chemistry course, 86 had taken Trigonometry course, 118 had taken Geometry course, 30 had taken Chemistry and Geometry courses, 5 had taken Chemistry and Trigonometry courses, 50 had taken Trigonometry and Geometry, and 2 taken all courses. How many taken only Trigonometry? a. 30 c. 31 d. 33 b. 35

𝐴 ∩ 𝐵 = { }or ∅

𝐴∩𝐵

45 47

7.

𝐴 ∩ 𝐵 = {3, 4, 5}

𝐵 = {𝑥, 𝑦, 𝑧}

A

a. b.

Set of Numbers Set Builder N Notation otation Real ℝ = {𝑥|𝑥 is a point on the number line} Numbers Natural Natural/ / 𝑁 = {1, 2, 3, 4, 5, … } Counting Numbers Whole 𝑊 = {0, 1, 2, 3, 4, 5, … } Numbers Integers 𝐼 = {… , −3, −2, −1, 0, 1, 2, 3, … } 𝑞 Rational 𝑄 = { | 𝑝 and 𝑞 are integers, 𝑞 ≠ 0} Numbers 𝑝 Irrational 𝐻 = {𝑥|𝑥 𝑖𝑠 𝑎 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑡ℎ𝑎𝑡 𝑖𝑠 𝑛𝑜𝑡 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙} Numbers Real Numbers

Rational Numbers

Positive Integers

Irrational Numbers

Integers

Nonintegers

Zero

Negative Integers

PROPERTIES AND OPERATION OF REAL NUMBERS Absolute Val Value ue If a represents any real number, then

Out of 100 students, 28 are taking Physics and 33 are taking Algebra if 14 students are in both classes, how many students are in either class? 𝑎 if 𝑎 ≥ 0 |𝑎| = { −𝑎 if 𝑎 < 0 Effectiveness. Efficiency. Convenience

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For Real Numbers a, b, c Commutative Associative Identity Inverse Distributive

Sample P Problems: roblems: ADDITION

SUBTRACTI SUBTRACTION ON

𝑎+𝑏 =𝑏+𝑎

𝑎𝑏 = 𝑏𝑎

(𝑎 + 𝑏) + 𝑐 = 𝑎 + (𝑏 + 𝑐)

𝑎+0 =0+𝑎 =𝑎 𝑎 + (−𝑎 ) = (−𝑎) + 𝑎 = 0

9.

(𝑎𝑏) ∙ 𝑐 = 𝑎 ∙ (𝑏𝑐) 𝑎∙1=1∙𝑎 =𝑎

𝑎∙

1 1 = ∙𝑎 =1 𝑎 𝑎

a. b.

10.

𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐

EXPONENTS The number that gives the power to which a base is raised. Number of times a base number to multiply to itself. 𝑏𝑛 = 𝑏 ∙ 𝑏 ∙ 𝑏 ∙ 𝑏 ∙ … ∙ 𝑏 Rules Product Rul Rule e Quotie Quotient nt Rule

Zero Exponen Exponentt Rule Negative Ex Exponent ponent Rule Power Rul Rule e ProductProduct-Power Power Rule Quotie Quotient-Power nt-Power Rule

Square Roots

Formula 𝑎𝑚 ∙ 𝑎 𝑛 = 𝑎 𝑚+𝑛 𝑎𝑚 = 𝑎𝑚−𝑛 𝑎𝑛 𝑎0 = 1 1 𝑎−𝑚 = 𝑚 𝑎 (𝑎𝑚 )𝑛 = 𝑎𝑚𝑛 (𝑎𝑏)𝑚 = 𝑎 𝑚 𝑏 𝑚 𝑎 𝑚 𝑎𝑚 ( ) = 𝑚 𝑏 𝑏

√𝑎 = 𝑏 if 𝑏 2 = 𝑎

Cube Roots

11.

nth roots

𝑛

√𝑎 = 𝑏 if 𝑏 𝑛 = 𝑎

To Write a Number in Scientific Notation 1. Move the decimal point in the number to the right of the first nonzero digit. 2. Count the number of places you moved the decimal point in step 1. If the original number is 10 or greater, the count is positive. If the original number is less than 1, the count is negative. 3. Multiply the number obtained in step 1 by 10 raised to the count (power) obtained in step To Convert a Number in Scientific Notation to Decimal Form 1. Observe the exponent on the base 10. 2. a) If the exponent is positive, move the decimal point in the number to the right the same number of places as the exponent. b) If the exponent is negative, move the decimal point in the number to the

- 3y30 / x10 - 27y30 / x10

Simplify

a. b.

12.

- 27y15 / x18 20y15 / x18

5 × 104 4 × 103

Simplify 1 × 10−6 2 × 10−7

a. b.

13.

14.

15.

(

3

−15𝑥 4 𝑦 2 ) 5𝑥 10 𝑦 −3 c. - 3y15 / x18 d. - 27y15x18

Simplify

a. b.

3

√𝑎 = 𝑏 if 𝑏 3 = 𝑎

Simplfy

(

3

−30𝑥14 𝑦 8 ) 10𝑥17 𝑦 −2 c. 27y30 x10 d. - y20 / x10

1.5 × 10−2 3 × 10−6 c. 3 × 103 d. 𝟓 × 𝟏𝟎𝟑 4.8 × 10−2 2.4 × 106 c. d.

𝟐 × 𝟏𝟎−𝟖 3 × 10−5

Simplify (−5𝑎4 𝑏)(−6𝑎 7 𝑏11 ) a. -30a3b10 c. d. b. 30a11b12

30a28b11 -3ab12

Simplify (−9𝑎3 𝑏 )(−2𝑎 6 𝑏 4 ) c. a. 18a9b5 b. -18a18b4 d.

-18a9b 18a3b3

Simplify

a. b.

y12/3x8 y12/3x

10𝑥 4 𝑦 9 30𝑥12 𝑦 −3 c. y/3x8 d. y12/x8

SOLVING LINEAR EQUATIONS terms – parts being added in an algebraic expression.

coefficient – the numerical part of a term that precedes the variable. e.g. Term Coefficient 16

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degree of a term with whole number exponents is the sum of the exponents on the variables. e.g. Term Degree 1+5=6 𝟏𝟖𝒙𝒚𝟓 equation – mathematical statement of equality.

solution of aan n equation – number(s) that make the equation a true statement. To Solve Linear Equations 1. Clear fractions. 2. Simplify each side separately. 3. Isolate the variable term on one side. 4. Solve for the variable. 5. Check.

Quantity Concentration Ratio

A+B 20 units 40%

3:7

1:1 (5:5)

2:3 (8:12)

16.

Solve the value of x 3𝑥 𝑥 − 4 𝑥 + 3 = − 3 5 4 c. 30 a. -35 b. 29 d. 35

17.

How many liters of a 12% salt solution must be added to 10 liters of a 25% salt solution to get a 20% salt solution? a. 6.53 L c. 7.25 L d. 5.35 L b. 6.25 L

18.

A clothier has two blue dye solutions, both made from the same concentrate. One solution is 6% blue dye and the other is 20% blue dye. How many ounces of the 20% solution must be mixed with 10 ounces of the 6% solution to result in the mixture being a 12% blue dye solution? a. 6.3 oz c. 8.1 oz b. 6.9 oz d. 7.5 oz

19.

Dale has liquid fertilizer solutions that are 20% and 60% nitrogen. How many gallons of each of these solutions should Dale mix to obtain 250 gallons of a solution that is 30% nitrogen? a. 108.5 gal & 56.5 c. 190.5 gal & 54.5 gal gal

identity – an equation that has an infinite number of solutions (solution set is ℝ).

mathematic mathematical al model is a real-life application expressed mathematically. formula is an equation that is a mathematical model for a real-life situation. Guidelines for Problem Solving 1. Understand the problem. 2. Translate the problem into mathematical language. 3. Carry out the mathematical calculations necessary to solve the problem. 4. Check the answer found in step 3. 5. Answer the question.

B 10 units 50%

Sample P Problems: roblems:

conditional eq equation uation – an equation that is true only for specific values of the variable. contradictio contradiction n – an equation that has no solution (solution set is ∅).

A 10 units 30%

b.

Mixture Problems Principles of Mixture Problems:

179.5 gal & 75.5 gal

d.

187.5 gal & 62.5 gal

20.

Club members equally share the cost of ₱1,500,000 to charter a fishing boat. Shortly before the boat is to leave, four people decide not to go due to rough seas. As a result, the cost per person is increased by ₱100,000. How many people originally intended to go on the fishing trip? a. 11 c. 9 b. 8 d. 10

21.

David invests ₱12,000 in two savings accounts. One account is paying 10% simple interest and the other account is paying 6% simple interest. If in one year, the same interest is earned on each account, how much was invested at each rate? a. c. b. d.

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22.

23.

24.

June has ₱12,000 to invest. She places part of her money in a savings account paying 8% simple interest and the balance in a savings account paying 7% simple interest. If the total interest from the two accounts at the end of 1 year is ₱910, determine the amount placed in the 8% account. c. ₱8000 a. ₱7000 b. ₱5000 d. ₱6000 Two hose are filling a swimming pool. The hose with the larger diameter supplies 1.5 times as much water as the hose with the smaller diameter. The larger hose is on for 2 hours before the smaller hose is turned on. If 5 hours after the larger hose is turned on there are 3150 gallons of water in the pool, determine the rate of flow from the smaller hose. a. 400 c. 500 d. 600 b. 300

Tom, the owner of a gourmet coffee shop, has two coffees, one selling for ₱16.00 per grams and the other for ₱16.80 per grams. How many grams of each type of coffee should he mix to make 1.4 kg of coffee to sell for ₱16.50 per grams? c. 255g & 1145g a. 525g & 875 875g g b. 225g & 1175g d. 555g & 845g

Clock Pro Problem blem 25. In how many minutes after 2:00 PM will the hands of the clock extend in opposite directions for the first time? c. 42.5 mins a. 43.6 mins b. 40.5 mins d. 41.4 mins

26.

27.

In how many minutes after 7:00 PM will the hands the hands be directly opposite to each other for the first time? c. 5.67mins a. 5. 5.45mins 45mins b. 5.33mins d. 6.65mins

What time after 3:00 PM will the hands of the clock be together for the first time? a. 3:17.56 c. 3:16.36 b. 3:18.36 d. 3:17.67

𝑏

𝑐

𝑐

compound inequality – formed by joining two inequalities and d or or. with the word an e.g. 𝑥 ≤ 7 and 𝑥 > 5 𝑥 < −1 or 𝑥 ≥ 4 To Solve Equations of the Form |𝒙| = 𝒂 If |𝑥| = 𝑎 and 𝑎 > 0, then 𝑥 = 𝑎 or 𝑥 = −𝑎 e.g. Solve |𝑥| = 6 |𝑥| = 6 𝑥 = 6 or 𝑥 = − 6 To Solve Inequalities of the Form |𝒙| < 𝒂 If |𝑥| < 𝑎 and 𝑎 > 0, then −𝑎 < 𝑥 < 𝑎 e.g. Solve |4𝑥 + 1| < 13 similarly:

4𝑥 + 1 < 13

;

4𝑥 + 1 > −13

−13 < 4𝑥 + 1 < 13 7 − 𝑎 and 𝑎 > 0, then 𝑥 < −𝑎 , 𝑥 > 𝑎. e.g. Solve |4𝑥 + 1| > 13 4𝑥 + 1 > 13 Solve simultaneously 4𝑥 > 12 𝑥>3

;

;

;

4𝑥 + 1 < −13

4𝑥 < −14 7 𝑥 𝑎 and 𝑎 < 0, then the solution are any real numbers If |𝑥| < 𝑎 and 𝑎 < 0, then the solution is null.

Sample P Problems: roblems: 28.

SOLVING LINEAR INEQUALITIES Properties Used to Solve Linear Inequalities Addition/Subtraction at both sides:

𝑎

• If 𝑎 > 𝑏 , and 𝑐 > 0, then > 𝑐 𝑐 Multiplying/Dividing with negative integer at both sides: • If 𝑎 > 𝑏 , and 𝑐 < 0, then 𝑎𝑐 < 𝑏𝑐 𝑏 𝑎 • If 𝑎 > 𝑏 , and 𝑐 < 0, then <

29.

Solve for x; 2𝑥 + 1 > 6 a. 𝒙 > 𝟓/𝟐 b. 𝑥 < 3/2 Solve for x; a. b.

4𝑥+3 3

> −5

c. d.

𝑥>5 𝑥 < 6/2

c. d.

Multiplying/Dividing with positive integer at both sides: • • •

If 𝑎 > 𝑏 , then 𝑎 + 𝑐 > 𝑏 + 𝑐 If 𝑎 > 𝑏 , then 𝑎 − 𝑐 > 𝑏 − 𝑐

If 𝑎 > 𝑏 , and 𝑐 > 0, then 𝑎𝑐 > 𝑏𝑐

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30.

Solve for the inequality 8 − 2𝑥 greater than or equal to −4. a. 𝑥 > 6 c. 𝑥 < 6 d. 𝒙 ≤ 𝟔 b. 𝑥 ≥ 6

31.

Find the solution for |4 − 2𝑥| = 5 a. 1/2 & -9/2 c. -1/2 & 9/2 b. -2 & 2/9 d. -1/9 & 9

32.

40.

b.

Solve the equation |𝑥 + 5| = 13 c. 8 & -18 a. -8 & 18 b. -8 & -18 d. 8 & 18

41.

𝑥

33.

Solve for the equation 4 + |3 − | = 9 3 c. 24 & 6 a. 24 & -6 b. -24 & -6 d. -24 & 6

34.

Solve the inequality 3𝑥 + 9 less than or equal to 15. c. 𝑥 ≥ 2 a. 𝒙 ≤ 𝟐 b. 𝑥 > 2 d. 𝑥 < 2

35.

1 a. b.

36.

𝑥

Solve for the inequality + 𝑥 > −1/3 𝑥...