psae math reviewer

7/28/2019 PSAE Math Reviewer

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PSAE

Math Review

Prof. Alleli C. Domingo

IMSP CAS

UP Los Baos

Re-VIEW20 June 2011 @ AMTEC

Alleli Ester C. Domingo

Institute of Mathematical Sciences and PhysicsCollege of Arts and Sciences

Universityof the Philippines Los Baos

What is mathematics?

MATH:

Science of

patterns

Branches of Math

*Arithmetic

*Algebra

*Geometry

*Trigonometry

*Logic

*Calculus

*Statistics

Board Exam

Speed

AccuracyAlbert Einstein

Man of the Centur y

If I were given an hour to do

a prob lem up on wh ich my l i fe

depends, I wou ld spend40 minu tes study ing i t ,

15 minu tes reviewing i t ,

an d 5 minutes solving it.


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George Plyalived from 1887 to 1985

Prof. George Polyaa Hungarian who

immigrated to the United

States in 1940.

major contribution: his work

in problem solving.

Prof. GEORGE POLYAPlya worked on

analysis

number theory

geometry

combinatorics

mathematical physics.

How To Solve ItGeorge Plya's most famous book

Before going to the United States he hada draft of the book written in German butit was the English version which hepublished.

Four publishers turned down the chanceto publish How To Solve It beforePrinceton University Press agreed topublish the book.

The first edition appeared in 1945 andwas a best seller!!!

Quotations

by George Plya

If there is a problem youcan't solve, then there is aneasier problem you can solve:find it.

"What is the dif ference

betweenmethodan ddevice?

A m ethod is a device which

you u sed twice."

A GREAT discovery solves agreat problem, but there is a

grain of discovery in the solutionof any problem. Your problem maybe modest, butif it challengesyour curiosityand brings into playyourinventive faculties, and if yousolve it by your own means, youmay experience the tension andenjoythe triumph of discovery.


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Mathematics is thecheapestscience.Unlike physics or chemistry,it does not require any expensiveequipment. All one needs formathematics is a pencil and paper.

Look around when you have got yourf i r s t mushroom o r made your f i rs t

discovery: they grow in clusters.

Polyas Hueristics

STEP 1. Understand the problem.

*What is the unknown?

STEP 2. Devise a plan.

STEP 3. Implement the plan.

STEP 4. Look back.

*Check your answer.

Even NumbersLinear Equation

Answer:126,128,130

1) What three

consecutiveeven numbers

have theirsum equal to

384?Odd NumbersLinear Equation

Answer:

439, 441, 443

2) Find three consecutiveodd numbers whose sum is1,323.

Arithmetic ProgressionPythagorean Theorem

ANSWER: 18, 24, 30

3) The sides of a right

triangle form an arithmetic

progression with a

commondifference of6.

Find the sides of the

triangle .

Arithmetic ProgressionCommon Difference: P250

4) A student takes a part-

time job paying a starting

salary of P5,000 a month

and is promised a fixedraise each month. How

much is his monthly raise

if he receives P7,000 on

the ninth month?


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Arithmetic ProgressionSum of the first n terms

S7 = 490 pesos

5) A well driller chargesP100 for the first 50 feetand P10 less for every 50feet thereafter. How much

would a 350-ft deep wellcost?

Arithmetic ProgressionSum of the first n terms

S10 = 55 blocks

6) Daniel is constructing a

ten-level tower. He puts ten

blocks on the bottom level.

If each level after that, thereis one block less than the

level below it, the completed

tower has how many blocks?

Work Problem

Answer:3 hours for the faster computer ;6 hours for the slower computer

7) One computer can do ajob twice as fast as theother. Working together,both computers can do thejob in 2 hours. How longwould it take eachcomputer, working alone,to do the job?

Subukan natin ito

Ulo, abaga, hawak dapi-dapi

Tuhod, tiil, tuhod, tiil.

**********************************************

Cloga, mahetna, centuga, leletaka

Melwene lelewana, melwene, lelewana

Work ProblemAnswer: 6 and 2/3 hours

or6 hrs and 40 minutes

8) Apipe can fill aswimming pool in 10

hours. If a second pipe is

open, the two pipestogethercan fill the pool in

4 hours. How long would it

take the second pipe alone

to fill the pool?

InvestmentAnswer:

P= 750,000/(1.03)10

9) How much money do wehave to invest at 3 percent

compounded annually if wewant to have 750 thousand

pesos in the bank after tenyears?


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Age Problem

ANSWER: In 9 years,Immaya will be 20

and her aunt will be 40.

10) Immaya is 11 yearsold today. If her favorite

aunt is 31 years old,

how many years from

now will the aunt betwice as old as she?

Mixture ProblemAnswer: 6liters

11) How many liters of a15% solution of alcoholshould be added to 3 liters

of the 30% solution to get a20% solution?

Investment as aMixture Problem

Answer: 2,500 pesos at 3%1,500 pesos at 4%

12) Ms. Sison invests part

of P4,000 at 3% and the

balance at 4% per annum .

How much did she invest at

each of these rates if she

earns P135 in one year?

Distance = Rate x TimeAnswer:Andre 55 kphChris 55 +15 = 70 kph

13) Andre and Chris aretraveling to a businessconference. Andre travels110 km in the same timethat Chris travels 140 km.Chris travels 15km perhourfasterthan Andre. Find theaverage rate of eachperson.

Dimensions; Area

Ans: Cut a square of side = 2 inches

14) A man wants to make anopen box from a piece of metalthat is 12 inches wide and 14inches long by cutting equalsquares out of the corners andfolding up the sides. How large asquare must be cut out of eachcorner if the area of the bottomof the box is 80 square inches?

Binomial TheoremPascals Triangle

Answer: -240 x7y3

15) Find the 4th term of

(2xy)10


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ANALYTICGEOMETRY

Distance FormulaQuadratic Equation

1) Find two values

of x if the distance

between (x, 2) and(6, 6) is 5.

2) Determine if the

three points (0,3) ,

(1, 4) and (2 , 1) are

collinear.

Collinear Points

Slope of Line Segments

3) Prove that the three

points(2, 4) , (1,4) and

(5,2) are the vertices

of a right triangle.

Slope

Perpendicular Lines

4) Find the slope and

y intercept of the line

having the equation

2x5y10 = 0

SlopeIntercept Form

5) Find an equation of

the line passing throughthe point (1, -3) and

having a slope of .

PointSlope Form


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6) Find an equation of

the circle with centerat

(3,5) and radius 2.

CenterRadius Form

7) Find the equation

of the circle having a

diameterwithendpoints at (3,4)

and (1, 2).

Midpoint of a Diameter

CenterRadius Form

9) Draw a sketch

of the graph of the

equation y= x2 +3.

2nd Degree EquationQuadratic inx, linear iny

Parabola

10) Locate the vertex

and focus of the

parabola

9y2 + 2x24y 96 = 0 .

Completing the Square

Vertex and Focus

11) Find the equation

of the line tangent tothe curve

y22x4y1= 0

at (2 ,1).

Slope of Tangent Line

PointSlope Form

12) Find the

equation of the line

normal to the curve

xy + 2x5y2 = 0

at (3 , 2) .Implicit Differentiation

Slope of Normal LinePointSlope Form


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13) Find the equationsof the lines tangent tothe curve

y = x36x + 2

and parallel to the liney = 6x2 .

Slope of Tangent Line

Slope of Parallel Lines

14) Find the equationof the line normal tothe curve

xy + 2xy = 0

and parallel to the line

2x + y = 0 .

Implicit Differentiation

Normal,Parallel Lines

15) Does the line

tangent to the curve

y = x3 at the point (1,1)

intersect the curve at

any other point?

If so, find the point.

Tangent LineIntersection Point

16) Sketch the ellipse

9x2 + y2 = 9 .

Central Conic: EllipseMajor Axis; Minor Axis

17) Sketch

the hyperbola

9x2 4y2 = 36 .

Auxiliary RectangleDiagonals: Asymptotes

Exponential GrowthExponential Decay


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1) In a particular bacterialculture, if f(t) bacteria arepresent at t minutes, then

f(t) = Be0.04t where B is aconstant. If there are 1,500bacteria present initially, howmany bacteria will be presentafter 1hour?

Exponential Growth

2) Iff(t) grams of radioactivesubstance are present after

t seconds, then f(t) =ke-0.3 t

where kis a constant. If100grams of the substance arepresent initially, how much ispresent after 5 seconds?

Exponential Decay

3) If V(t) pesos is the value of

a certain equipment t years afterits purchase, thenV(t) = Be0.20twhere B is a constant, If theequipment was purchased for8,000 pesos, what will be its valuein 2 years?

DepreciationExponential Decay

4) If P(h) pounds per square foots the atmospheric pressure at aheight h feet above sea level, thenP(h) = ke-0.00003h, where k is aconstant. Given that the atmosphericpressure at sea level is 2116 ft/lb2,

find the atmospheric pressure outsideof an airplane that is 10,000 ft high.

Exponential Decay

5) The population of a particular

town is increasing at a rate

proportional to its size. If the rate is 6percent , and the population after tyears is P(t), then P(t) = ke0.06t, wherek is a constant. If the currentpopulation is 10,000, what is theexpected population

a) after 10 years? b) after 20 years?

Exponential Growth

6) Carbon 14, also known as radiocarbon, is aradioactive form of carbon that is found in all livingplants and animals. After a plant or animal dies the

radiocarbon disintegrates. Scientists can determinethe age of the remains by comparing theamount of radiocarbon with the amount presentin living plants and animals. This technique isknown as carbon dating. The amount ofradiocarbon present after t years is given by

y =y0e-(ln2) (1/5700)t, where y0 is the amount

present inliving plants and animals.Find the half-life of radiocarbon.

Half-life; Exponential Decay

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