Day 02 - Algebra 2

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MATHEMATICS(ALGEBRA)

DAY 2

Engineering Review Center4thfloor, Coast Pacific Downtown Center, Sanciangko St., Cebu CityTel. 254-3384Manila: 4thFlr. Anacleta Bldg. 891 Galicia St. cor. Espana Ave. Sampaloc Manila (Tel. 742-56-00)

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Engineering Review Center

PROPERTIES OF INEQUALITY

Let a, b, and c be any real numbers.

1. Comparison PropertyExactly one of the following statements is true:

a < b, a = b, a > b

2. Transitive Property

If a < b and b < c, then a < c.

3. Additive or Subtraction Property

If a < b, then a + c < b + c

4. Multiplication or Division Property

If a < b and c is positive, then ac < bc or a/c < b/c.

If a < b and c is negative, then ac > b/c or ac > b/c.

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FUNCTIONS

A. Basic Operations:

Addition: (f + g)(x) = f(x) + g(x)

Subtraction: (fg)(x) = f(x) - g(x)

Multiplication: (f . g)(x) = f(x) . g(x)

Division: (f/g)(x) =

B. Composition of two functions f and g:

(f g)(x) = f(g(x)) and. (g f)(x) = g(f(x))

0)(,)(

)(

xgprovidedxg

xf

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COMPLEX OPERATIONS

Rectangular form Polar form Exponential form

a + bi Z < Z ei

where:

= tan-1b/a a = Z cos; b = Z sin22 ba Z =

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PROBLEMS

1. The solution to the inequality 3x4 10 + x.

A. x -7 B. x -7 C. x 7 D. x 7

2. Solve the inequality.

3. Given: f(x) = 3x + 2 and g(x) = 2x - 5Compute: A. g(f(x) B. f(f(1)) C. g(f(2))

4. What is the inverse function y = 4x2?

A. B. C. D. y = x2- 4

44x5

4xy x4y 4xy 2

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PROBLEMS

5. Given: A = 3i4j + 5k; B = 2i + 3j6k; C = -4i + 7j3k.

Determine (A x B).C.A. 85 B. 93 C. 101 D. 109

6. Determine the absolute value of resultant vector of the

following vectors:

F1= 4j + 7j + 6k; F2= 9i + 2j + 11k; F3 = 5i3j8k

A. 21 B. 18 C. 25 D. 9

7. Find the product of two complex numbers 3 + 4i and 72i.

A. 10 + 2i B. 13 + 22i C. 13 + 34i D. 29 + 22i

8. Evaluate (23i)6

9. Determine the absolute value of the complex number 3 + 4i.A. 4 B. 5 C. 8 D. 6

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PROBLEM SOLVING IN ALGEBRA

1. NUMBER PROBLEMS

Consecutive numbers:

x = first number

x + 1 = second number

x + 2 = third number

Consecutive odd numbers:

x = first number



Consecutive even numbers

x = first number



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1. NUMBER PROBLEMS

Digit numbers:

Let: u = unit digit t = tens digit h = hundreds digit

For two-digit number:The number = 10t + u

If digits are reversed = 10u + t

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1. NUMBER PROBLEMS

Digit numbers:

Let: u = unit digit t = tens digit h = hundreds digit

For three-digit numberThe number = 100h + 10t + u

If digits are reversed = 100u +10t + h

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2. RATE PROBLEMS

where:

S = distance traveled

v = constant velocity

t = time

t

Sv

v

St ST = v t

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A. For horizontal straight path:

Car moving in continuous path with different speed.

1. S = total distance =

2. t = total time = t1 + t2 =

3. Average velocity, vave=

221121 tvtvSS

2

2

1

1

v

S

v

S

2211

21

21

21

v/Sv/S

SS

tt

SS

TimeTotal

cetanDisTotal

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B. Cars approaching each other moving at different speed.

S = total distance = 221121 tvtvSS

t1 = t2 or2

2

1

1v

S

v

S

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C. Moving on a circular path

a. For the opposite direction at same

point and same time.

S1+ S2 = Circumference

S1+ S2 = D

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C. Moving on a circular path

b. For the same direction at same point

S1 = S2 + D

S1S2 = D = Circumference

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3. WORK PROBLEMS

A. If both Aand Bworking

together could finish the

work in Tdays.

B. If the time of Ais twice that

of Band they could finish

working together in Tdays,

then A = 2B

TBA

111

TBB

11

2

1

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3. WORK PROBLEMS

C. For pumping a reservoir.

If A and Care inlet pipes

and Bis the outlet pipe,

then if they are workingtogether they could finish

the work in time T.

TBCA

1111

A C

B

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3. WORK PROBLEMS

D. Work replacement.

If Aand Bworking together for xdays and if C

replaces Bthen Aand Bcan finished the work in

remaining y days .

11111

CAy

BAx

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4. AGE PROBLEM

Note:

A. The difference of their ages at any time are equal

B. The sum of their ages 5 years ago = (M - 5) + (J - 5)C. The sum of their ages 6 years from now = (M + 6) + (J + 6)

Let: M = age of Maria now J = age of Jose now

PAST

Five years agoPRESENT

FUTURE

Six years from now

M - 5 M M + 6J - 5 J J + 6

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5. MIXTURE PROBLEM

A. If two substances added:

m1 m2 m1 + m2

%x1 %x2 %x3

+ =

m1(%x1) + m2(%x2) = (m1 + m2)(%x3)

Note: For a mixture of water and NaCl, if water is evaporated,then 100% water is removed and 0% NaCl is removed.

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5. MIXTURE PROBLEM

B. If certain amount removed from the original amount:

m1 m2 m1 - m2

%x1 %x2 %x3

=-

Note: For a mixture of water and NaCl, if water is evaporated,then 100% water is removed and 0% NaCl is removed.

m1(%x1) - m2(%x2) = (m1 - m2)(%x3)

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PROBLEMS

NUMBER PROBLEMS

1. The numbers has a ratio of 2 : 5 : 8. If the sum of the

numbers is 60, what is the larger number?

A. 36 B. 32 C. 24 D. 22

2. The difference of the cubes of two positive numbers is 2402

and the cube of their difference is 8.

Find the larger number.

A. 21 B. 23 C. 25 D. 27

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PROBLEMS

RATE PROBLEMS

1. A man left Sta. Rosa City to drive to Lopez, Quezon at 6:15

pm and arrived at 11:45 pm. If he averaged 50 kph and

stopped 1 hour for dinner, how far is Lopez, Quezon from

Sta Rosa City?

A. 225 km B. 522 km C. 252 km D. 215 km

2. A boat can go 15 km upstream in the same time that it

takes to go 27 km downstream. The speed of the current is2 km per hour. Find the speed of the boat in still water.

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PROBLEMS

WORK PROBLEMS

1. If John can paint a room in 30 minutes and Tom can paint it

in 1 hour, how many minutes will it take them to paint the

room if they work together?

2. Tukmol can paint a fence of 50% faster than Kikoy and

20% faster than Tiburcio and together they can paint a

given fence in 4 hours. How long will it take Tukmol to

paint the same fence if he had to work alone?

A. 11 B. 8 C. 9 D. 10

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PROBLEMS

WORK PROBLEMS

3. Crew A can clean the Mega dome in 8 hours and crew B

can clean it in 12 hours. After a night game, Crew A began

a cleanup at midnight and was joined by crew B at 2:00

am. When was the job completed?

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PROBLEMS

AGE PROBLEMS

1. John is four times as old as Harry. In six years, John will be

twice as old as Harry. What is the age of Harry now?

A. 2 B. 3 C. 4 D. 5

2. Two times the mothers age is 8 more than six times her

daughters age. Ten years ago, the sum of their ages was

44. What is the daughters age?

A. 15 yrs old B. 18 yrs old C. 12 yrs old D. 16 yrs old

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PROBLEMS

MIXTURE PROBLEMS

1. Two thousand (2000) kg of steel containing 8% nickel is to

be made by mixing a steel containing 14% nickel with

another containing 6% nickel. How much of each is

needed?

2. A high concentrated solution having a volume of 100 liters

is 25% gasoline. How much gasoline should be added to

the solution to produce a 50-50% mixture?A. 50L B. 75L C. 40L D. 60L

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Day 02 - Algebra 2