1.
SUMMER COURSE OUTLINE IN LINEAR ALGEBRA
1. SETS AND ITS OPERATION
2. VENN-EULER DIAGRAM
3. APPLICATION OF SET
4. RELATIONS AND FUNCTIONS
5. COMPOSITION OF FUNCTIONS
6. REVIEW OF LONG METHOD DIVISION OF POLYNOMIALS
7. POLYNOMIAL THEOREMS
8. EXPONENTIAL EQUATIONS
9. LOGARITHMIC EQUATION
10. SOLVING LINEAR EQUATIONS IN TWO VARIABLES UNKNOWN
USING ELIMINATION AND SUBSTITUTION
11.WORDED PROBLEMS
A. INTEGER PROBLEM
B. AGE PROBLEM
C. MOTION PROBLEM
D. CLOCK PROBLEM
E. MIXTURE PROBLEM
F. MONEY PROBLEM
G. INVESTMENT PROBLEM
H. WORK PROBLEM
12.EXPONENTIAL GROWTH AND DECAY
13.MATRICES
A. GAUSSIAN ELIMINATION METHOD
B. DETERMINANTS AND CRAMERS‟ RULE
2.
Module 1 : SETS and OPERATION
Set- a well-defined collection of objects, concrete or abstract of any kind.
TWO METHODS OF WRITING A SET
1. ROSTER METHOD
2. SET- BUILDER NOTATION
2 Major Type
1. Finite set- a set whose elements are limited or countable and the last element is
identifiable.
2. Infinite Set- a set whose elements are unlimited.
OTHER TYPE OF SET
1. EMPTY SET/ NULL SET or { }
2. UNIT SET {1}
3. UNIVERSAL SET U= {0,1,2,3,4,5,…20.}
4. SUBSET A= { 1,2,3,4,5,6,7,8,9,10.} B= { 2,3,4,6,8.} B A
5. EQUAL SET A= { L,I,S,T,E,N.} B= { S,I,L,E,N,T} B A
6. EQUIVALENT SET A= { 1,2,3,4} B= { A,B,C,D}
7. JOINT SET A={ 1,2,3,4,5,6,7,8,9,10.} B= { 2,3,4,6,8}
8. DISJOINT SET A={ 1,2,3,4,5,6,7,8,9,10.} B = { 0,11, 12,13,14…}
BASIC NOTATION
1. - INTERSECTION
2. - UNION
3. - NOT AN ELEMENT OF
4. - AN ELEMENT OF
5. - PROPER SUBSET OF
6. - IS EQUIVALENT TO
7. NULL SET
8. „ PRIME SYMBOL
3.
NAME: _______________________________________ SCORE_________
DATE: _______________
EXERCISE 1.1: SET AND BASIC NOTATION
DIRECTIONS: IDENTIFY EACH STATEMENT AS TRUE OR FALSE
GIVEN A={ 2,5,8,11} and B={ 8,11,14,2}
1.
2.
3.
4.
5.
6. * +
7. * +
8. * +
9. {14,2}
10.* +
4.
MODULE 2: OPERATIONS ON SET
THERE ARE 5 OPERATIONS ON SET SYMBOL USED
1. UNION OF SET
2. INTERSECTION OF SET
3. CARTESIAN PRODUCT AXB
4. COMPLEMENT OF A SET A‟( PRIME)
5. DIFFERENCE OF A SET A-B
UNION OF SET
Combined elements found in a given set with the other set.
A= { 1,2,3,4,5,6,7,8,9,10} B= { 2,5,7,9} * +
ITERSECTION OF SET
Element(s) which that are common to both given sets.
A= { 1,2,3,4,5,6,7,8,9,10} B= { 2,5,7,9} * +
CARTESIAN PRODUCT
Each Element of a given set A are being paired with each elements found in Set B.
A= { 1,2,3} B= { 2,5,7,9}
AXB= { (1,2), (1,5) ,(1,7), (1,9),( 2,2).(2,5),(2,7),(2,9), (3,2),(3,5),(3,7),(3,9)}
COMPLEMENT OF A SET
Elements which are found in the universal set but not in the given set.
U= { 0,1,2,3,4,5,6,7,8,9,10…15} A‟ = { 2,4,5,7,8,10,11,12,13,14,15}
A= { 0,1,3,6,9}
5.
NAME: _________________________________________SCORE_____________
DATE: _________________
Exercise 1.2 OPERATIONS ON SET
DIRECTIONS: Given the sets below, give the elements of the following operations on set.
U= {1,2,3,4,5,6,7} A= {1,3,5,7} B= { 2,4,6} C= {1,3,5}
1. = {2,4,6} or B
2. = A
3. ‟= {2,4,6,7}
4. U
5. B
6. { }
7. ( ) U
8. ( ) { }
9. { }
10. C‟
6.
NAME: _________________________________________
SCORE_____________
DATE: _________________
Exercises 1.3A VISUAL REPRESENTATION OF OPERATIONS ON SET
TAKE HOME QUIZ
DIRECTIONS: Given the sets below, give the elements of the following operations on set
using
Venn- Euler diagram and Shade the corresponding region where the
elements are found.
Given; U= {1,2,3,4,5,6,7} A= {1,3,5,7} B= { 2,4,6} C= {1,3,5}
1.
2. =
3. * + =
4.
5.
6. ( )
7. ( )
8.
9.
10.( )
EXAMPLE
U= { 1,2,3,4,5,…20}
A= {1,2,3,4,5,6}
B= { 4,6,9}
4
6
9
1
2
3
7 8 20
A B
10
11 5 19
12 13 14 15 16 17 18
U
7.
NAME: _________________________________________ SCORE_____________
DATE: _________________
EXERCISES 1.3B APPLICATION OF SETS
VENN- EULER DIAGRAM
DIRECTIONS: Read and analyze the following problem s below and represent the
following set to solve
using the Venn- Euler diagram. Show your solution. (30 points)
1. A survey on subjects being taken by 250 college students in Metro Manila revealed
the following information; 90 likes Math; 88 English; 145 Filipino; 25 Math &
Filipino; 38 Filipino & English; 59 Math and English; 15 All. How many did not
take the survey?
2. There are 20 seniors serving the student council of CDNSHS. Of these, 3 have not
served before. 10 served on the council in their junior years, 9 in their sophomore
years, and 11 in their freshmen years. There are 5 who served during both their
sophomore and junior years, 6 during both freshman & junior years and 4 during
both freshman and sophomore years. How many seniors served in the student
council during each of the 4 years in high school?
3. In a survey involving 800 employees, it was found that 485 employees saves in
BDO , 550 save in Metro Bank , 540 save in BPI, 255 save in BDO and Metro
Bank. 270 save in BDO and BPI, and 325 save in Metro bank and BPI. All
employees save with at least one of these banks.
a. How many employees save in BPI but not in Metro Bank?
b. How many employees save in BDO but not in Metro Bank or BPI?
c. How many employees save in all three banks?
8.
MODULE 3 RELATIONS AND FUNCTIONS
Definition of terms
A relation is any set of ordered pairs ( x,y), ( domain, range)
Relation = {x/x is any ( x,y)} i.e., { (1,2), (2,3), ( 4,5), ( 6,7)}
A relation S is a function if and only if (a,b) S and ( a,c) S implies that b=c
A function is a set of ordered pairs ( x,y) such that for each first component, there is
at most one value of the second component.
A function is a set of ordered pairs having the property that no two distinct
ordered pairs have the same first entry or domain.
NOTE:
All functions are relations but not all relations are functions.
A graph represents a function if and only if no vertical line intersects the graph
more than once.
y
0
1
2
3
3
3
10.
NAME: _________________________________________
SCORE_____________
DATE: _________________
EXERCISES 1.4 RELATIONS AND FUNCTIONS
A. Directions: Determine whether the following relations are function or not. Write
F if it is a function and N if it is not a function.
______1. { ( 2,3), ( 4,4), ( 2,4), ( 3,2)}
______ 2. { ( 1,2), (2,3), (-1,2),( -2,3)}
______3. {0.2, 0.003), (0.002, .005), (2/10, 1/3)}
______4. { ( ), ( , ), ( ) }
_____5. { (1.5, -1.5), ( 2.5, -2.5), ( 3,3), ( -2,3)}
B. Directions: Determine the domain and range. Use the vertical line test to
determine whether the relation is a function or not.
Domain Range
1. _______ __________
2. _______ __________
3. _______ __________
4. _______ __________
5. _______ __________
C. Directions: In each of the following, indicate the values of x must be excluded from
the domain.
1. ( )
( )
2. ( )
( )
3. √
4. √
5. ( )
11.
OPERATIONS ON FUNCTIONS (MDAS)
Let f and g be two functions with domains Df and Dg respectively. Then
1. ( )( ) ( ) ( )
2. ( )( ) ( ) ( )
3. ( )( ) ( ) ( )
4. ( ) ( )
( )
( )
COMPOSITE FUNCTIONS
Let f and g be functions. The composition of f on g is the function defined by ( )( )
( ( )) where its domain is dom ( ) * ( ) ( ) ( ) +
( ) ( ( ))
( ) ( ( ))
OPERATIONS ON FONCTIONS
LET f(x) = g(x) =
1. ( )( ) ( ) ( )
=
=
=
2. ( )( ) ( ) ( )
= ( )
=
3. ( )( ) ( ) ( )
= ( )( )
=
4. ( ) ( )
( )
( )
=
( )( )
=
12.
MODULE 4: POLYNOMIAL THEOREMS
REMAINDER THEOREM
If a polynomial f(x) is divided by (x-r) until a remainder independent of x is
obtained then, the remainder is equal to P(r).
FACTOR THEOREM
If (x-r) is a factor of P(x), then r is a root of the equation P(x) =0.
FUNDAMENTAL THEOREM OF ALGEBRA
If P(x) is a polynomial with positive degree, then P(x) has at least one zero.
THE NUMBER OF ROOTS THEOREM
IF P(x) is a polynomial of degree n, then P(x) =0 will have n roots.
Descarte‟s rule of sign
This states that for every change in the sign in a given polynomial this corresponds to
the number of positive and negative roots.
Example
FUNDAMENTAL THEOREM OF ALGEBRA
Since the equation consist of a positive degree of n, therefore it has at least one zero.
THE NUMBER OF ROOTS THEOREM
In the above equation the highest degree of exponent is 2, then there exist 2 roots of
P(x)
Descarte‟s rule of sign
Consider the change in the sign of the first term and second term.
last term.
( ) There are 2 positive roots taken from the original equation
FACTOR THEOREM
( )( ) ?
( ) ( )
40-77+28=0 16-44+28=0
-28+28=0 -28+28=0
0=0 0=0
Then, 4 and 7 are the roots of
13.
NAME: _________________________________________ SCORE_____________
DATE: _________________
EXERCISES 1.5 FINDING THE ROOTS OF A POLYNOMIAL
using the different Polynomial Theorems
Example
P(+) = 2 roots factor( x-7)(x-4)=0 then 4 and 7 are the roots.
P(-) 0
#of roots positive negative factors
roots
1.
2.
3. –
4.
5.
6.
7.
8.
9.
10.
11.If Is a factor of the given
equation?
12.Find the value of k such that when
13.Find the value of k such that when e
14.If (x+1) is a factor of ____.
15.Find the value of k such that when
( )
14.
MODULE 5: EXPONENTIAL AND LOGARITHMIC EQUATION
PRE- REQUISITE TOPICS
LAWS OF RATIONAL EXPONENT
FACTORING
SOLVING POLYNOMIAL EQUATION
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
EXPONENTIAL FUNCTIONS
If b is any positive number, then the expression b
x
=designates exactly one real
number for every real value of x. where b is no equal to zero.
f(x)= b
x
Properties of
Logarithms
1. Product rule:
2. Quotient rule: ( )
3. Power rule:
4. If and and = x
x -2 -1 0 1 2
y 0.25 0.5 1 2 4
f(x)=
2x
y
-2
-1
0
1
2
15.
NAME: _________________________________________ SCORE_____________
DATE: _________________
EXERCIES 1.6 EXPONENTIAL AND LOGARITHMIC EQUATIONS
DIRECTIONS: Solve for x in the following equation below. Match Column A with the
correct value of x in
Column B. If the answer is not found in the selection, write the correct
answer.
Note: Calculator is allowed.
COLUMN A COLUMN B
1. A. x=4 P. x = 49/4
2. B. x= 5 Q. x = -8/3
3. C. x = -1/2 R. x = 6
4. ( )
D. x = 3 S. x = 0
5. E. x = 3/2 T. x = 4/49
6. F. x = ½
7. G.
8. H. x = 81
9. I. x = 16/9
10. J. x = 5/2
11. K. x = 19/6
12. L. x = -1
13. M. x = 2/21
14. N. x = 21/2
15. O. √
B. Directions: Solve each equation applying the different properties of Logarithm.
16.
17.
18.
19.
20. ( )
21. =
22.
23.
24.
25.
16.
C. EXPONENTIAL EQUATIONS LEADING TO QUADRATIC EQUATION
26. ( )
27. ( )
28. ( ) ( )
29. ( )
30. ( ) ( )
31. = 4
y
32.
33.
34.
35.
17.
INTEGER PROBLEM
Let x, x+1, x+2, x+3, x+4,… for a consecutive integer
Let x,x+2, x+4, x+6, x+8,… for a consecutive even/ odd integer
Age Problem
Past & ago implies subtraction from the present age.
Ex. A is one year younger than B means A‟s age is x-1 and B‟s age is x.
In n
th
year means that one needs to add the particular number to the
present age.
In a given problem sentence, when one sees the word “is” this implies
equal.
Mixture
The word” is/ must be added to “means add the two mixtures.
The word “must be reduced to” means deduct the amount of mixture from
the other.
The word “to make a mixture of” or “to have a result of” means add up the
two mixtures.
Ex.
A 5 gram of 20% SALT SOLUTION must be added to a pure concentration
of SALT solution must be added to make a 75% Salt solution.
5g( 20%) + x g ( 100%) = 75%(5g + x g)
WORK PROBLEM
SAME JOB TO BE DONE
X+Y = 1
RATE OF WORK= 1/X
MOTION PROBLEM
DISTANCE = RATE X TIME or d = rt
18.
Over take problem
d1 = d2
Opposite Direction
d1+ d2= dt
d1-d2= difference on distance
Worded Problems
1. Trixie has 10 pieces of P100- bills and 35 pieces of P20-bills. How much money does
she have?
2. Mollie has a total of P 4,800 consisting of P50 and P100-bills. The number of P50-
bills is 16 less than twice the number of P 100-bills. How many P50-bills does she
have?
3. Jonas has a jar in his office that contains 39 coins. Some are 5 cents and the rest
are 10 cents. If the total value of the coins is P2.55, how many 5 cents does he
have?
4. In problem number 3, how many 10 cents does Jonas have in the jar?
5. Erick has a box of coins. The box currently contains 40 coins, consisting of 5 cents,
10 cents, and 25 cents. The number of 5 cents is equal to the number of 25 cents,
and the total value is P5.80 How many of 25 cents of coin does he have in the box?
6. Ruth has P1900 consisting of P50 and P10 bills. The number of P10 bills is 5 less
than the number of P 50 bills. How many P10 bill does she have?
7. Mildred can sew a dress in ten days. What part of the dress is finished after 6 days?
8. Glen is 3 years older than his brother. Three years ago, Four years from now, the
sum of their ages will be 33 years, how old are they now?
9. Mr. Sta. Maria is five years older than his wife. Five years ago, his age was 4/3 her
age. What will be their ages 8 years from now?
10.A man can wash the car in 120 minutes if he works alone. His son, working alone
can do the same job in 3 hours. How long will it take to take them to wash the car if
they work together?
19.
11. An inlet pipe can fill a tank in 9 minutes. A drain pipe can empty the tank in ten
minutes. If the tank is empty and both pipes are open, how long will it take before
the tank overflows?
12. Find three consecutive even integers such that four times the first less the
third is six more than twice the second.
13. Find three consecutive integers such that the sum of the first plus one-third
of the second plus three eights of the third is 25.
14. A paint that contains 21% green dye is mixed with a paint that contains
15% green dye. How many gallons of each must be used to make 60 gallons
of paint that is 19% green dye.
15. A chemist has 10 milliliters of a solution that contains a 30% solution of
acid. How many milliliters of pure acid must be added in order to increase the
concentration to 50%?
1. ( )
2. ( )
3. ( ) ( )
4. ( )
5. ( ) ( )
6. = 4
y
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