modern algebra final

7/28/2019 Modern Algebra Final

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PLUMBING ARITHMETIC

CENTER FOR Philippines

TECHNICAL PROFESSIONS


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A. NUMBER SYSTEM

A number is an item that describes a magnitudeor position. Numbers are classified into twotypes, namely cardinal numbers and ordinalnumbers.

Cardinal numbers are numbers which allow usto count the objects or ideas in a givencollection (i.e. 1, 2, 3,... 1000, 100000. Ordinalnumbers state the position of individual objectsin a sequence (i.e. first, second, third,...)

The number system is divided into twocategories, namely, real numbers andimaginary numbers.


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Real numbers are classified as follows:

Natural numbersconsidered as the countingnumbers. Ex. 1, 2, 3, ...

Integers the natural numbers, the negative numbersand zero. Ex. -4, -1, 0, 3, 8

Rational numbers numbers which can be expressedas a quotient (ratio) of two integers. Ex. 0.5, , -3,

0.333... As a rule, a non-terminating but repeating (or

periodic) decimal is always a rational number. Also,all integers are rational numbers.

Irrational numbers numbers which cannot beexpressed as a quotient of two integers. Ex. , , e, ... They are in fact, non-terminating numbers with

non-repeating decimals.


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Past Board Exam:

The number 7 + 0i is a/an ________ number.

a. Irrational c. Real

b. Imaginary d. Complex

Past Board Exam:

The number 0.123123123... is a _______number.

a. Irrational c. Rationalb. Imaginary d. Complex


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A complex numberis an expression of bothreal and imaginary number combined. It

takes the form of a +bi, where a and bare real numbers.

If a = 0, then pure imaginary number isproduced while real number is obtained

when b = 0.An imaginary numberis denoted as i

which is equal to the square root of negativeone. In some other areas in mathematicalcomputation, especially in electronics andelectrical engineering it is denoted as j.


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Imaginary number and its equivalent:

Past Board Exam:

Find the value of

1i

2

1i

3

i i

4

i i

5

(1 ) .i


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B. SIGNIFICANT DIGITS

Significant figures or digits define thenumerical value of a number. A digit isconsidered significant unless it is used

to place a decimal point. The significant digit of a number begins

with the first non-zero digit and ends

with the final digit, whether zero or non-zero.


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Past Board Exam:

The number 0.004212 has how manysignificant digits?

Past Board Exam:

Round off 0.00432500 to 3 significant figures.

a. 0.00432000 c. 0.00433

b. 0.00433000 d. 0.00432


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C. LEAST COMMONMULTIPLEA common multiple is a number that two

other numbers will divide into evenly. Theleast common multiple (LCM) is the lowest

multiple of two numbers.

Past Board Exam:

What is the least common multiple of 15 and18?


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D. GREATEST COMMONFACTORA factor is a number that divides into a

larger number evenly. The greatestcommon factor (GCF) is the largest

number that divides into two or morenumbers evenly. Greatest commonfactor is the same as greatest commondivisor (GCD).

Past ECE Board Exam:

What is the GCF of 70 and 112?


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E. EXPONENTS

An exponent is a number that gives thepower to which a base is raised. Forexample, in 32, the base is 3 and the

exponent is 2. The exponential notation states that if a is a

real number, variable or algebraic

expression and n is a positive number,then:na a a a a


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Properties of Exponents


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Past Board Exam:

Solve for x in the given equation

a. 2 c. 4b. 3 d. 5

Past Board Exam:

Find the value of x in the equation (35

)(96

) = 32x

.

a. 8.5 c. 9.5

b. 9 d. 8

Past Board Exam:

Solve for x if

a. 0.723 c. 0.852

b. 0.618 d. 0.453

4 38 2 8 2.x

1 1 1 ...x


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F. LOGARITHMS

The logarithm of a number or variable x tobase b, is the exponent of b needed to give x.i.e. may be written as

The term logarithm comes from Greekwords, logus meaning ratio and arithmusmeaning number. John Napier(1550-1617)invented logarithm in 1614 using e = 2.718...for its base. In 1616, through the suggestion of

John Napier, Henry Briggs improved thelogarithm using 10 as the base. The logarithmwith base 10 is known as common logarithmor the Briggsian logarithm.

2log 16 442 16.


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The natural logarithm can be converted into

a common logarithm and vice versa. Toobtain this, a factor known as the modulusof logarithm is necessary, such as:

The coefficients 0.4343 and 2.3026 are thereferred to as the modulus of logarithm.

log 0.4343lnx x ln 2.3026logx x


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Properties of Logarithm


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Past Board Exam:

Which of the following logarithmicequations is equivalent to

a. c.

b. d.

Past Board Exam:

If , what is the value of

Past Board Exam:Solve for x.

03 1?

3log 0 1 1log 3 0

3log 1 0

3log 1 1

2

3 729a 1

2 .a

log 27 log 3 2x x


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G. REMAINDER THEOREM

Remainder Theorem states that if apolynomial in an unknown quantity x isdivided by a first degree expression in

the same variable, (x-k), where k maybe any real number or complex number,the remainder to be expected will beequal to the sum obtained when thenumerical value of k is substituted for xin the polynomial.


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Past Board Exam:

What is the remainder if the polynomialis divided by (x 5)?

a. 281 c. 218

b. 812 d. 182

Past Board Exam:

Given: f(x) = (x + 3)(x 4) + 4 when dividedby (x k), the remainder is k. Find k.

a. 2 c. 4

b. 3 d. -3

3 24 3 8x x x


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H. FACTOR THEOREM

Factor theorem states that if apolynomial is divided by (x k) will resultto a remainder of zero, then the value

(x k) is a factor of the polynomial.

Past Board Exam:

What is the value of k if (x + 4) is afactor of 3 22 7 ?x x x k


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I. QUADRATIC EQUATION

Quadratic is an expression or an equationthat contains the variable squared, but notraised to any higher power. A quadratic

equation in x is also known as a second-degree polynomial equation.

The general quadratic equation isexpressed as:

Ax2 + Bx +C = 0

where A, B and C are real numbers andwith A 0.


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The quadratic formula:

The quantity B2 4AC in the above equationis known as the discriminant. It determines

the nature of the roots of the quadraticequation.

B2 4AC Nature of roots

0 Only one root

(real and equal)

>0 Real and unequal


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Sum of roots

Product of roots

Past Board Exam:

Two reviewees are solving a problem leading to a

quadratic equation. One student made a mistake inthe coefficient of the first degree term, got roots of 2and -3. The other made a mistake in the coefficient ofthe constant term and got roots of -1 and 4. What isthe correct equation?

Past Board Exam:

From the equationdetermine the value of k so that the sum and product ofthe roots are equal.

1 2

Br r

A

1 2

C

r r A

27 (2 1) 3 2 0x k x k


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J. BINOMIAL THEOREM

Binomial is an expression containing twoterms joined by either + or -.

Binomial theorem gives the result of raising

a binomial expression to a certain power.The expansion and the series it leads to arecalled the binomial expansion and thebinomial series, respectively. The binomialtheorem is expressed as follows:

2 2 11

......2!

n n n y n n nn nx y x nx y x y nxy y


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Properties of Binomial expansion of (x + y)n:

The number of terms in the resulting expansionis equal to n + 1.

The exponent of x decreases by 1 in succeedingterms, while that exponent of y increases by 1 in

succeeding terms. The sum of the exponents of each term is equal

to n.

The first term is xn and the last term is yn andeach of the terms has a coefficient of 1.

The coefficient increases and then decreases ina symmetric pattern.


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The Pascals Triangle:

Each number in the triangle is equal to thesum of the two numbers immediately above it.


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Another way to determine the coefficient of anyterm in the binomial expansion is to use the

following formula:

where C is the coefficient of any term and PT isthe preceding term

The rth term of the binomial expansion of (x + y)nmay be calculated using the following formulas:

.

1

Coeff ofPT ExponentofxofPTC

ExponentofyofPT

1 1

1 1

1

( 1) 2 ... 2

( 1)!

n rth r

n rth r

n r

n n n n r r x y

r

r C x y


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A term involving a variable with a specificexponent is obtained by using the followingformula:

Sum of the coefficients of the expansion of

(x + y)n: Sum = (Coeff. of x + coeff. of y)n

Sum of exponents of the expansion of

(x + y)n: Sum = n (n + 1)

1 2 ... 1'

!

n r rn n n n r

y x yr


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Past Board Exam:

Find the 6th term of the expansion of (3x + 4y)8.

Past Board Exam:

What is the sum of the numerical coefficients of theexpansion of (x + 4y)12?

Past Board Exam:

Find the sum of the coefficient of the variables inthe expansion of (3x 5)6.

Past Board Exam:

Find the term involving x3 in the expansion of

(x 3x-1)9.


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K. WORD PROBLEMS

Most word problems in algebra areexpressed in sentences rather than puremathematical expressions or equations.

To solve a word problem, one must firstinterpret the sentences into its accuratemathematical expression and then apply

algebraic operations to the expressionobtained.


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AGE PROBLEMS

One of the most common problems inAlgebra is the age problem. This type ofproblems must be solved meticulously bygiving emphasis to the tenses (i.e. past,

present or future) of the statement.

Past Board Exam:

Anna is 3 times as old as Mae. In 3 years,she will be 2 years more than twice the ageof Mae will be then. What are their ages?


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Past Board Exam:

Mary is 24 years old. She is twice as old asAnn was when she was as old as Ann is now.How old is Ann?

Past Board Exam:

The sum of the parents ages is twice the sumof their childrens ages. Five years ago, the

sum of the parents ages is four times the sum

of their childrens ages. In fifteen years, the

sum of the parents ages will be equal to the

sum of their childrens ages. How many

children were in the family?


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WORK PROBLEMS

Work done = rate of work x timeFor a complete job.

Rate x time = 1

When there is a specific work and specifictime and manpower, the rate of doing thework may be computed using the number ofman-hours.


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Past Board Exam:

A tank is fitted with an intake pipe that canfill it in 4 hours and an outlet pipe that canempty it in 9 hours. If both pipes areopened, how long will it take to fill the empty

tank? Past Board Exam:

If 20 bakers can bake 40 pizzas in 8 hours,

how many bakers can bake 10 pizzas in 2hours?


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Past Board Exam:

A father can do a job in 9 days and hisson can do same job in 16 days. Theyworked together. After 4 days, the sonleave and the father finished the job

alone. How many more days did it takefor the father to finish the job alone?


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MIXTURE PROBLEMS

A basic idea in solving mixture problems is the

fact that the amount of the substances in themixture components is equal to the amount ofthe final mixture.

Past Board Exam:A 100 liter solution is 75% alcohol and 25%gasoline. How much gasoline must be added toproduce a 50-50 mixture?


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Digit Problems

Let h, t and u be the hundreds, tens, and

units digit, respectively. A three-digitnumber must be represented in thefollowing manner.

Number = (h)(100) + (t)(10) + (u)

A two-digit number is represented by;

Number = (t)(10) + (u)


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Past Board Exam:

The sum of the digits of a two-digit numbersis 11. If the digits are reversed, the resultingnumber is seven more than twice theoriginal number. What is the original

number?

Past Board Exam:

The sum of the digits of a 2-digit number is

10. If the number is divided by the unitsdigit, the quotient is 3 and the remainder is4. Find the number.


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Past Board Exam:

The denominator of a certain fraction isthree more than twice the numerator. If 7 isadded to both terms of the fraction, theresulting fraction is 3/5. Find the originalfraction.


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Motion Problems

In Algebra, the problems pertaining tomotion deals only with a uniform velocity,i.e., no acceleration or deceleration in theprocess. The following is the relationshipbetween the distance, time and velocity.

D = Vt V = D/t t = D/V


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Past Board Exam:

A man rows downstream at the rate of 5

mph and upstream at the rate of 2 mph.How far downstream should he go if he isto return in 7/4 hours after leaving?

Past Board Exam:An airplane travels from point A to point Bwith a distance of 1500 km and a wind

along its flight line. If it takes the airplane 2hours from A to B with the tailwind and 2.5hours from B to A with the headwind, whatis the velocity of the plane?


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Past Board Exam

Pedro started running at a speed of 10 kph.Five minutes later, Mario started running inthe same direction and catches up withPedro in 20 minutes. What is the speed of

Mario?


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Coin Problems

In solving coin problems, take note of thedenominations of the currency being used.

For US dollar currency, the following are the

equivalent value for each coin.Penny 1 cent

Nickel 5 cents

Dime 10 centsQuarter 25 cents

Half 50 cents


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Past Board Exam:

A purse contains $11.65 in quarters anddimes. If the total number of coins is 70,find how many dimes are there.

Past ECE Board Exam:In a box there are 25 coins consisting ofquarters, nickels and dimes with a total

amount of $2.75. If the nickels were dimes,the dimes were quarters and the quarterswere nickels, the total amount would be$3.75. How many quarters are there?


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Clock Problems

Clock problems focus on the relationship ofthe movements of the hands (hour hand,minute hand, second hand) of the clock.

The longest hand is the second hand while

the shortest hand is the hour hand. Byprinciple, the second hand (SH) alwaysmoves faster than the minute hand (MH)

and the minute hand always moves fasterthan the (HH).


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The relation between the three hands of theclock are as follows:

Past Board Exam:

In how many minutes after 2 oclock will the

hands of the clock extends opposite

direction for the first time?

12

MHHH

720

SHHH

60

SHMH


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Past Board Exam:

At how many minutes after 3pm will theminute hand coincide with the hour hand?

Past Board Exam;

At what time after 12 noon will the hourhand and minute hand of the clock forms anangle of 120o?


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Variation Problems

Variation problems are problems in algebrawhich show the relationship between thevariables in terms of expressions such asdirectly proportional or inversely proportion

or simply proportional. The expression x varies directly as y is

expressed as follows: x y

The symbol varies () is replaced by anequality symbol and a constant ofproportionality, k, hence: x = ky


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The expression x varies inversely as thesquare of y is expressed as follows:


If x varies directly as y and inversely as z,and x = 14 when y = 7 and z = 2, find thevalue of x when y = 16 and z = 4.

2

1x k

y


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The resistance of a wire varies directly withits length and inversely with its area. If acertain piece of wire 10m long and 0.10cmin diameter has a resistance of 100 ohms,

what will its resistance be if it is uniformlystretched so that its length becomes 12m?


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L. PROGRESSION

A sequence is a set or collection ofnumbers arranged in an orderly mannersuch that the preceding and the following

numbers are completely specified.An infinite sequence is a function whose

domain is the set of positive integer. If the

domain of the function consists of the first npositive integers only, then it is said to be afinite sequence.


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Elements are the term used to describe thenumbers in a given sequence. An element

is sometimes called a term. Series is the sum of the terms in a

sequence. An alternating series has

positive and negative terms arrangedalternately.

If an infinite series has a finite sum, it isreferred to as convergent series anddivergent series if it has no sum at all.

A progression is simply another term for asequence.


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Arithmetic Progression

A sequence is said to be in arithmeticprogression if its succeeding terms have acommon difference.

The corresponding sum of all the terms inarithmetic progression is called asarithmetic series.


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Last term (nth term):

Sum of all terms:

or

Where:

= first term

= last term (nth term)

n = number of terms

d = common difference

d = a2 a1 = a3 a2 = ...

1 1na a n d

12

n

nS a a 12 1

2

nS a n d

1a

na


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Geometric Progression

A sequence is said to be a geometricprogression if its succeeding terms have acommon ratio.

The corresponding sum of all the terms ingeometric progression is called asgeometric series.


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Last term (nth term): an = a1rn-1

Sum of all terms:

or

Where:

= first term

= last term (nth term)

n = number of terms

r = common ratio =

1a

na

1 11

na rS

r

1 11

na rS

r

32

1 2

aa

a a


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Infinite Geometric Progression

This type of progression is a geometricprogression only that the number of terms(n) is extremely large or infinity.

If r>1, sum of all terms is infinite

If r


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Harmonic Progression

A sequence of numbers whose reciprocalsform an arithmetic progression is known asharmonic progression. In solving aproblem, it would be wise to convert all

given terms into arithmetic sequence bygetting its reciprocals. Use the formulas inarithmetic sequence and take the reciprocal

of resulting value to obtain the equivalentharmonic term for an answer.


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Past Board Exam:

How many terms of the sequence -9, -6, -3,

must be taken so that the sum is 66?

Past Board Exam:

There are seven arithmetic means between 3 and

35. Find the sum of all the terms. Past Board Exam:

Determine the sum of the infinite geometric seriesof 1, -1/5, 1/25,

Past Board Exam:

Find the sum of the first 10 terms of the GP 2, 4,8, 16,

modern algebra final

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