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 2 nd-degree term b = coefficient of the 1 st-degree term c = constant term Quadratic Formula

x =

−b±

2

b − 4ac

2a

Sum of the roots x1 + x 2 = −

b a

COLLEGE ALGEBRA


Product of the roots x1 x 2 =

c a

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.

COLLEGE ALGEBRA


Binomial Expansion Binomial Theorem n

(a + b )

=a

n

+ na

n −1

b+

(

)

n n −1 a

n−2

b

2!

2

+ ... + b

rth Term rth

term =

n(n − 1)(n − 2 )...(n − r +

(r − 1)!

2)

a

n − r +1

b

r −1

n

COLLEGE ALGEBRA


1. Find the 6th term of the expansion of a. c.

−

−

66339 124a

11

66339 11

123a

b. d.

−

16

 1  − 3   2a 

.

66339 125a 11

−

66339 128a 11

2. Find the term involving x13 in the expansion of a. 524812288 b. 524812286 c. 544812288 d. 624812288

( 4 x 2

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

+

1 x

)14

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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 ( 3x 3 + 2y4)10 a. 384 b. 385 c. 389 d. 390 8. Find the quotient and remainder when 4y 3 + 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

COLLEGE ALGEBRA


Arithmetic Progression A sequence of numbers such that successive numbers differ by a constant. nth term a n = a1 + ( n − 1) d

Sum of terms sn =

n

2

n

(a1 + a ) = [2a1 + (n − 1)d ] n

2

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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 n −1

an = a1r

Sum of terms n

sn =

a1 (1 − r

1 − r

Infinite Geometric Progression sn =

a1

1 − r

)

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

HM =

1 a1

+

1 a2

+ ... +

1 an

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 9 th term is 6. Find the 11 th 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 3 rd 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

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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 7 th 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. Juan’s 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 c. 2.25 hours

b. 2.50 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 75mm 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

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Case 4: The denominator contains irreducible quadratic factors, some of which are repeated 32. Resolve into partial fractions 2

2 x

+ 10 x +

2

( x + 1)( x − 2)( x + 3)

33. Resolve into partial fractions 5 x

2

− 25 x + 8

(3 x + 2)( x − 3)

2

34. Resolve into partial fractions 2 x ( x

2

3

− 4 x − 15

+ 3 x +

4)( x

2

+ x + 1)

35. Resolve into partial fractions 4 x

4

3

2

− 7 x + 5 x − x + 1

(2 x − 1)( x

2

− x + 1)

2

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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?

Algebra

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