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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