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# Modules Linear Algebra Drills

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### Modules Linear Algebra Drills

1. 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. 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. 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. 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. 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. 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. 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. 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
9. 9. y 1 2 3 4 5
10. 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. 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. 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. 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. 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. 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. 16. C. EXPONENTIAL EQUATIONS LEADING TO QUADRATIC EQUATION 26. ( ) 27. ( ) 28. ( ) ( ) 29. ( ) 30. ( ) ( ) 31. = 4 y 32. 33. 34. 35.
17. 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. 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. 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
20. 20. 7. 8. 9. 10.