Algebra

COLLEGE ALGEBRA

TIP-QC 10 JUNE 2015 10:30 AM 1:30 PM Lecturer: Richard S Regidor

Quadratic Equation General Form: ax2 + bx + c = 0 ( a is non-zero) a = coefficient of the 2nd-degree term b = coefficient of the 1st-degree term c = constant term Quadratic Formula

a

acbbx

242

=

Sum of the roots

a

bxx =+ 21

COLLEGE ALGEBRA


Product of the roots

a

cxx =21

Remainder Theorem For any constant r , if a polynomial f(x) is divided by

x r , the remainder R is equal to f(r) Factor Theorem For any polynomial f(x) and any number r , x r is a factor of f(x) if and only if f(r) is equal to zero.

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Binomial Expansion Binomial Theorem

( ) ( ) nnnnn bbannbnaaba ++++=+ ...!2

1 221

rth Term

( )( ) ( )( )

11

!12...21

term +

+=

rrn bar

rnnnnrth

COLLEGE ALGEBRA


1. Find the 6th term of the expansion of

163

21

a.

a. 1112466339

a b. 11125

66339a

c. 1112366339

a d. 11128

66339a

2. Find the term involving x13 in the expansion of 142 )14(x

x +

a. 524812288 b. 524812286 c. 544812288 d. 624812288

3.Find the term involving x6y12 in the expansion of

( 3x2 4y3)7 a. 241920 b. 231920 c. 231224 d. 251234

4. Find the sum of the coefficients of ( a + 2b)6. a. 612 b. 711 c. 729 d. 741

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5. Find the sum of the coefficients of ( 2x -3)8.

a. -5 b. -7 c. -6 d. +1

6. Find the sum of the exponents of ( x + y)6. a. 40 b. 41 c. 42 d. 43

7. Find the sum of the exponents ( 3x3 + 2y4)10 a. 384 b. 385 c. 389 d. 390 8. Find the quotient and remainder when 4y3 + 18y2 +

8y 4 is divided by 2y + 3? a. 2y2 + 6y 5 remainder 11 b. 2y2 - 6y 5 remainder 12 c. 2y2 + 6y + 5 remainder 11 d. 2y2 + 3y 5 remainder 12

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9. What is the remainder when 3x4 + 2x3 5x2 + x + 7 is

divided by x 3? a. 234 b. 262 c. 311 d. 312

10. What is the remainder when 3x4 + 3x3 5x2 - 5x + 7 is divided by x + 2i ?

a. 12+3i b. 34 + 12i c. 8-25i d. 75 34i

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Arithmetic Progression A sequence of numbers such that successive numbers differ by a constant. nth term

dnaan )1(1 += Sum of terms

( ) ( )[ ]dnanaans nn 1222 11 +=+=

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Geometric Progression

A sequence of numbers such that the same quotient

is obtained by dividing a term by a preceding term.

nth term

11

=n

n raa Sum of terms

r

ras

n

n

=

1)1(1

Infinite Geometric Progression

r

asn

=

11

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Harmonic Progression A sequence of numbers such that the reciprocals of the numbers form an arithmetic progression Harmonic mean

HM = naaa

n

1...

1121

+++

11. Find the 30th term of an A.P. 4, 7, 10 . a. 88 b. 75 c. 91 d. 95 12. The 5th term of an A.P is 123 and the 30th term is 245. What is the 12th term? a. 3929/25 b. 3412/25 c. 3372/25 d. 3312/25

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13. The 3rd term of a harmonic progression is 15 and the 9th

term is 6. Find the 11th term a. 4 b. 5 c. 6 d. 7 14. Insert four geometric means between 2048 and 2 a. 512 , 128 , 32, 8 b. 236, 102 , 42 , 6 c. 625 , 324 , 152 , 24 d. 882 , 420, 212 , 84 15.The 5th term of a geometric progression is 162 and the

10th term is 39366. What is the 3rd term? a. 21 b. 18 c. 36 d. 24 16. a boy deposits 50 centavos in his piggy bank at the

beginning of the month, and doubles the amount of his deposit every month thereafter. How much will he have in his piggy bank at the end of the year? a. Php 3,048.45 b. Php 2,047.50 c. Php 1,568.25 d. Php 4,324.50

COLLEGE ALGEBRA


17. A golf ball is dropped from a height of 6 meters. On

each rebound it rises 2/3 of the height from which it last fell. What distance has it travelled at the instant it strikes the ground for the 7th time?

a. 32.27 m b. 28.89 m c. 27.89 m d. 30.81 m 18. Reduce the repeating decimal 0.123123123 to a rational fraction in lowest terms.

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WORDED PROBLEMS 19. The sum of the digits of a 3 digit number is 17, the hundreds digit is twice the unit digit. If 495 is subtracted from the number, the order of the digits will be reversed. Find the units digit. a. 2 b. 3 c. 4 d. 5 20. The denominator of a certain fraction is 3 more than twice the numerator. If 7 is added to both terms of the fraction, the resulting fraction is 3/5. Find the original fraction a. 4/13 b. 5/13 c. 6/13 d. 7/13

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21. Juans age on his birthday in 1989 is equal to the sum of the digits of the year 19xy in which he was born. If x and y satisfy the equation x y 6 = 0, find the age of Juan in 1990. a. 18 b. 19 c. 20 d. 21 22. In how many minutes after 2 PM will the hands of the clock be at extend in opposite direction for the first time? a. 45.6 min b. 43.6 min c. 47.8 min d. 44.3 min 23. At what time between 3 PM to 4 PM will the hands of the clock be at right angle ? a. 3: 32 8/11 PM b. 3: 33 7/11 PM c. 3: 32 5/11 PM d. 3:33 5/11 PM

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24. Two runners A and B complete for a race of 1000 m

long. It took 130 sec for A to reach the finish line and for B 138 sec. How far was B behind A when A reaches the finish line?

a. 58.78 m b. 57.97 m c. 59.21 m d. 53.43 m 25. A cat is now 50 of her own leaps ahead of a dog which

is pursuing her. How many more leaps will the cat take before it is overtaken if she takes 5 leaps while the dog takes 4, but 2 of the dogs leap are equivalent to 3 of the cats leap.

a. 200 b. 220 c. 230 d. 250

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26.A can do a piece of work in 10 days. After he has

worked 2 days, B came to help him and together, they finish the job in 3 days. In how many days could B alone do the work?

a. 5 days b. 6 days c. 7 days d. 8 days 27. A 400-mm diameter pipe can fill the tank alone in 5

hours and another 600 mm diameter pipe can fill the tank alone in 4 hours. A drain pipe 300-mm diameter can empty the tank in 20 hours. With all the three pipes open, how long will it take to fill the tank?

a. 2.00 hours b. 2.50 hours c. 2.25 hours d. 2.75 hours

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28. A goldsmith has two alloys of gold, the first being 70%

pure gold, and the second 60% pure gold. How many grams of each must be used to make 100 grams of an alloy which will be 66% pure gold?

29. Determine how much water should be evaporated from

50 kg of a 30% salt solution to produce a 60% salt solution. All percentages are by weight.

Variation 30. The horsepower which a shaft can transmit varies as

the cube of the diameter and the angular speed. If a 75-mm diameter shaft transmits 270 hp when turning 1000 rpm. Find the rpm of a 50-mm shaft if it transmits 100 hp.

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31. the time required for an elevator to lift a weight varies

directly with the weight and the distance through which it is to be lifted, and inversely as the power of the motor. If it takes 20 seconds for a 5-hp motor to lift 220 N through 12 meters, what size of motor is required to lift 5340 N in 30 seconds through 12 meters?

Partial Fractions Case 1: All factors of the denominator are linear, none

of them repeated. Case 2: All factors of the denominator are linear, but some are repeated. Case 3: The denominator contains irreducible quadratic

factors, none of them repeated

COLLEGE ALGEBRA


Case 4: The denominator contains irreducible quadratic factors, some of which are repeated 32. Resolve into partial fractions

)3)(2)(1(2102 2

++

++

xxx

xx

33. Resolve into partial fractions

2

2

)3)(23(8255

+

+

xx

xx


)1)(43(1542

22

3

++++

xxxx

xx


22

234

)1)(12(1574

+

++

xxx

xxxx

COLLEGE ALGEBRA


Miscellaneous Problems 36. if 2x = 4y and 8y = 16z , find x/z. a. 1/3 b. 2/3 c. 8/3 d. 5/3 37. In a commercial survey involving 1000 persons on

brand preference, 120 were found to prefer brand x only, 200 prefer brand y only, 150 prefer brand z only, 370 prefer either brand x or y but not z, 450 prefer brand y or z but not x and 370 prefer either brand z or x but not y. How many persons have no brand preference, satisfied with any of the three brands?

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Algebra