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Poor Class Standing

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Project in Finals
Project in Finals
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College algebra

Sample Problems on Matrices, Samples Problems on Determinants, and Logarithmic Functions
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College algebra

Sample Problems on Matrices, Samples Problems on Determinants, and Logarithmic Functions

Presented To:
Rev. Fr. Milo Mapaye, SVD
Instructor

Presented By:
Voc. Kenneth Etulle
Voc. John Rey Francisco
Voc. Dionisio Gayo, Jr.
Voc. Mark Angelou Labadan
Voc. Arlou Podador
Voc. Niño Anthony Umbina

October, 2014

Matrices
(Sample Problems with Solutions)

Sample Problem No. 01
Fr. Diony, the postulant master, asked our three postulants to record the eggs they gathered every day at the poultry.
Below is the table of the eggs gathered by our three postulants last week. Days | Post. Carlo | Post. Efren | Post. Marijon | | Morning | Afternoon | Morning | Afternoon | Morning | Afternoon | Sunday | 20 | 4 | 10 | 5 | 15 | 3 | Monday | 11 | 6 | 12 | 7 | 10 | 8 | Tuesday | 18 | 7 | 22 | 7 | 15 | 6 | Wednesday | 19 | 10 | 15 | 9 | 20 | 8 | Thursday | 21 | 10 | 18 | 8 | 19 | 11 | Friday | 8 | 6 | 19 | 5 | 9 | 4 | Saturday | 18 | 3 | 20 | 4 | 17 | 5 |
In matrices form:
Post. Carlo=2041161871910211086183 Post. Efren=105127227159188195204 Post. Marijun=153108156208191194175
How many eggs the postulants gathered per day last week?
Solution:
Post. Carlo+Post. Efren+Post. Marijun=?
2041161871910211086183+105127227159188195204 +153108156208191194175 =20+10+15 4+5+311+12+10 6+7+818+22+15 7+7+619+15+20 10+9+821+18+19 10+8+118+19+9 6+5+418+20+17 3+4+5=4512332155205427582936155712
Below is the table of the eggs gathered per day last week. Days | Morning | Afternoon | Sunday | 45 | 12 | Monday | 33 | 21 | Tuesday | 55 | 20 | Wednesday | 54 | 27 | Thursday | 58 | 29 | Friday | 36 | 15 | Saturday | 57 | 12 |

Sample Problem No. 02
If Post.Marijon forgot to record all his gathered eggs how many eggs were only recorded?
Solution:

Post. Carlo+Post. Efren=?
2041161871910211086183+105127227159188195204=20+10 4+511+12 6+718+22 7+7 19+15 10+9 21+18 10+8 8+19 6+518+20 3+4=30923134014341939182711387
Below is the table of the eggs gathered per day last week by Post. Carlo and Post. Efren. Days | Morning | Afternoon | Sunday | 30 | 9 | Monday | 23 | 13 | Tuesday | 40 | 14 | Wednesday | 34 | 19 | Thursday | 39 | 18 | Friday | 27 | 11 | Saturday | 38 | 7 |

Sample Problem No. 03
Fr. Allan, our Superior, asked Bro. Ignacio to get the no. Profess and Collegian seminarians who went out during their day off on the month of September. He wanted to know how many went to SM, NCCC and to the skyline. | First day-off of the month of September | Second day-off of the month of September | | Profess | Collegian | Profess | Collegian | SM | 4 | 3 | 6 | 4 | NCCC | 3 | 5 | 2 | 6 | Skyline | 2 | 6 | 1 | 4 |
Matrix Form:
A=First day-off of the month of September
B=Second day-off of the month of September
A=433526 B=642614

Solution:
A+B=? A+B=433526+642614=4+6 3+43+2 5+62+1 6+4=10 75 113 10
Therefore there were 10 profess & 7 collegian who went to SM. There were 5 profess & 11 collegian who went to NCCC. And there were 3 profess & 10 collegian who went to Skyline.

Sample Problem No. 04
Voc. Niño Anthony and Mark Angelou wanted to know their expenses from June to August, on their personal things, snacks and their expenses.
Here is the data of the expenses of Voc. Niño Anthony and Mark Angelou from the Month of June to August.
Let x be the expenses on personal needs, y be the expenses on snacks and z on other expenses. | June | July | August | | Voc.Niño | Voc.Mark | Voc.Niño | Voc.Mark | Voc.Niño | Voc.Mark | x | 200 | 170 | 150 | 100 | 210 | 150 | y | 82 | 164 | 82 | 82 | 82 | 164 | z | 520 | 300 | 500 | 100 | 200 | 50 |
Matrix form:
June=20017082164500300 July=1501008282500100 August=2101508216420050
Solution:
June+July+August=?

June+July+August=20017082164570300+1501008282500100+2101508216420050

=200+150+210 170+100+15082+82+82 164+82 +164170+500+200 300+100+50=5604202464101270450

Therefore Voc. Niño Anthony spent P560.00 for his personal needs and Voc. Mark Angelou spent P420.00. Voc. Niño Anthony spent P246.00 for the snacks and Voc. Mark Angelou spent P410.00. Voc. Niño Anthony spent P1270.00 for other expenses and Voc. Mark spent P.450.00.

Determinants
(Sample Problems with Solutions)

Sample Problem No. 1
There are 15 non professed vocationarians in our seminary consists of the first years and second years. The first years stay almost one year and the second years almost two years. If the sum of the non-professed vocationarian times their corresponding years of staying is 24, how many are the first years? How about the second years?
Solution:
Let x be the number of first years, and y be the second years. x+y=15 x+2y=24

a1=1 a2=1 b1=1 b2=2 c1=15 c2=24 Using the Cramer’s Rule, we get. x=c1b1c2b2a1b1a2b2=1512421112=152-(24)(1)12-(1)(1)=30-242-1=61=6 x=6 y=a1c1a2c2a1b1a2b2=1151241112=124-(1)(15)12-(1)(1)=24-152-1=91=9 y=9
Therefore, there are 6 first year vocationarians and 9 second year vocationarians.
To Check: x+y=15 by substitution:
6+9=15
15=15 x+2y=24 substituting the values of x and y:
6+2(9)=24
24=24

Sample Problem No. 2
I am looking for yarn to be used in making ‘bantings’. John Rey, Dionisio and Arlou give me their yarns. If I’m going to put them together I have a 5 merters yarn. Find the contribution of each. If Arlou’s yarn is twice the yarn of John Rey and Dionisio’s less than 27 meters. And if thrice the yarn of Dionisio plus john Rey’s less than 43 meters twice the yarn of Arlou’s. Standard Form: x+y+z=50 x+y+z=50 z=2x+y-27 2x+y-z=27 3y+x-43=2z x+3y-2z=43

a1=1 a2=2 a3=1 b1=1 b2=1 b3=3 c1=1 c2=-1 c3=-2 d1=50 d2=27 d3=43 Solving for x using Cramer’s Rule, we get:

x=d1b1c1d2b2c2d3b3c3a1b1c1a2b2c2a3b3c3=d1b1c2b3c3-d2b1c1b3c3+d3b1c1b2c2a1b2c2b3c3-a2b1c1b3c3+a3b1c1b2c2

x=5011271-1433-211121-113-2=501-13-2-27113-2+43111-111-13-2-2111-2+1111-1 x=501-2-3-1-27 1-2-31+43 [1-1-11]1 1-2-3-1-2 1-2-31+1 [1-1-11] x=50+135-861+10-2 x=185-869 x=999 x=11

Solving for y using Cramer’s Rule, we get:

y=a1d1c1a2d2c2a3d3c3a1b1c1a2b2c2a3b3c3=a1d2c2d3c3-a2d1c1d3c3+a3d1c1d2c2a1b2c2b3c3-a2b1c1b3c3+a3b1c1b2c2 y=1501227-1143-211121-113-2=127-143-2-250143-2+150127-19 (Since the denominator is the same in solving for x and we already solved the x, the denominator is 9.)

y=1 27-2 - 43-1 -2 50-2 - 431+1 [50-1 - 271]9 y=1 54+43 -2 -100 - 43 + (-50-27)9 y=1 -11 -2 (-143) +1 (-77)9 y=-11+286 -779 y=1989 y=22

Solving for z using Cramer’s Rule, we get: z= a1b1d1a2b2d2a3b3d3a1b1c1a2b2c2a3b3c3=a1b2d2b3d3-a2b1d1b3d3+a3b1d1b2d2a1b2c2b3c3-a2b1c1b3c3+a3b1c1b2c2 z=11502127134311121-113-2=127343-2150343+115012711-13-2-2111-2+1111-1 z=1143-327-2 143-350+1 [127-150]9 z=-38+214-239 z=214-619 z=1539 z=17

Therefore John Rey contributed 11 meters of yarn, Dionisio 22 meters and Arlou 17 meters.

To check : x+y+z=50 11+22+17=50
50=50

2x+y-z=27
2 11+22-17=27
22+22-17=27
44-17=27
27=27

x+3y-27=43
11+322-2 17=43
11+66-34=43
77-34=43
43=43

Logarithmic Functions

LOGARITHM

Definition: The logarithm of a number M is the exponent x, indicating the power to which some base number a must be raised in order to equal M. The base number a must be positive and different from unity. This restriction is due to the fact that all powers of unity equal unity.
Example:
1. 22=4 log24=2 2. 33=27 log327=3 3. 23=8 log28=3 4. 32=9 log39=2 5. 52=25 log525=2 6. 62=36 log636=2 7. 42=16 log416=2 8. 43=64 log464=2 9. 81/3=2 log82=1/3 10. 41/2=2 log42=1/2

LAWS OF LOGARITHM: 1. Log2 PQ=log2 P + log2 Q 2. Log2 (P/Q)=log2 P – log2 Q 3. Log2 Pn=n log2 P

Example: 1) Log33P2Q
=log3P2/3+log3Q1/3
=2/3log3P+1/2log3Q

Problems to solve: 1. Log 36
Solution:
Log 36 =log (9) (4)
Log 36 =log (32) (22)
Log 36 =log 32 +log 22
Log 36 =2log3+2log2
Log 36 =2(0.477+0.301)
Log 36 =2(.778)
Log 36 =1.556

2. Log 312 =log 121/3
Log 312 =1/3 log 12
Log 312 =1/3 log 4(3)
Log 312 =1/3 log 22 +1/3 log 3
Log 312 =2/3 log 2 + 1/3 log 3
Log 312 =2log2+log33
Log 312 =2.301+.4773
Log 312 = .602+ .4773
Log 312 = 1.0793
Log 312 = .3596

3. log 322

Sol. log 322 = log 32- log 3 = log (8) (4) - log 3 = log (23) (22) – log 3 = 3 log 2 + 2 log 2 – log 3 = 3 (0.301) + 2 (0.301) – 0.477 = 0.903 + 0.602 – 0.477 = 1.028

4. log327 = 5

Sol. log327=5 3x= 27 3x= 33

5. log216 = 4

Sol. log216=4 2x= 16 2x= 24

6. log53400

Sol. log53400 53400 = 5.34 × 104 = log5.54 + log 104 = log5.54 + 4 log10 = .7275 + 4(1) = 4.7275

7. log0.000534

Sol. log0.000534 0.000534 = 5.34 × 10-4 = log5.54 + log 10-4 = log5.54 - 4 log10 = .7275 - 4(1) = -3.2725

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