maths 1 28 pages

CIIAPTER 1 . MATHSI INTRODUCTION

Lecture Sussested Outline Chanter**I @!g: Basic algebra, Sets, functions and graphs,

Factorisation (including cubics); Inverse and compositefunctions; Exponential and logarithm functions,Trieonometrical functions

MyNotesChapter 2

l31hruJdrf t0l:.

2 Differentiationlhe meaning olthe derivative; Standardderivatives, Product Rule; Quotient Rule and Chain Rule.

MyNotesChapter 3

3 Differentiation: Optimisation; Curve sketching,Economics applications of the derivative; marginal andnrofit maximisation.

MyNotesChapter 3.

4 Differentiation: Optimisation; Curve sketching,Economics applications of the derivative; marginal andprofit maximisation.

MyNotesChapter 3.

5 Integration: Economic applications of integration;determination of total cost from marginal cost andcumulative chanees.

MyNotesChapter 4

6 Intesration: Economic applications of integration;determination of total cost from marginal cost andcumulative changes. Plus Test L

MyNotesChapter 4

rll"*

Optimisation of functions of more than L variable MyNotesChapter 5.Partial differentiation; Implicit partial differentiation'

Critical points and their natures; Optimisation; Economicapplications of optimisation and the Lagrange multipliermethod; The meaning of the Lagrange multiplier;Economic applications of constrained optimisation.PlusTest 2

8 Optimisation of functions of more than 1 variablePartial differentiation; Implicit pdial differentiation'Critical points and their natures; Optimisation; Economicapplications of optimisation and the Lagrange multipliermethod; The meaning of the Lagrange multiplier;Economic applications of consffained optimisation.

My NotesChapter 5.

\9DrfRYr'rrqPl/oq(

MatricesElementary Row Operations. Applications of matricesand linear equations.

MyNotesChapter 6.

10 MatricesElementary Row Operations. Applications of matricesand linear equations.

MyNotesChapter 6.

tlAf&

GP

Sequences and Series: Arithmetic and GeometricProgressions; Some Financial application of sequencesand series

MyNotesChapter 7

** Subject Guide Clnpter Reference may change due to new updates fum UOL.The above are tentative lesson plan. It may change due ta the leaming speed ofstudents.

Bemard Ong Page? of4 17 June20l2

CIIAPTER 1 . MATHS 1INTRODUCTION

3. Please ask your friend @) to pick up notes for you ifyou are unable to attend class as I do not keep spare copies and the copiesleft in the lecture theatre mav be misplaced..

If you do have any problems or questions, please feel free to let me know. Ipractice open communication. Do not let your problem grows!

If you need me to wait for you to copy or clarify any doubts etc., B@!sRAISE YOUR HANDS or ask me l0 to 15 minutes before the end of eachlecture. I will try to make it a point to end l0 to 15 minutes earlier.

Candidates should answer all EIGHT questions: all SIX questions in Section A (60mar*s in total) and BOTII questions in Section B (20 marks each).

Examination Techniques

a. Always show your workings. Marks are given for steps and the final answerscount very little to the overall marks. For example, a wrong answer with thecorrect workings will still get the most marks.

b. How many decimal places? @gE: Depends, there isn't any hard and fastrule. Common sense prevails.

If you answer is 0.0000023, are you going to give, say, 3 decimal places like0.000, which is absurd.

However, if you answer is $1000345 .5677, then an answer like $10000345.57or even $10000346 will be fine.

c. Label all diagrams used like including the x-axis and y-axis, title of the graphetc..

We will strive to achieve a better percentage pass than last year!

Good Luck!!!

Bernard Ong17 Jane2012@@o

4.

5.

Bernard Ong Page 4 of4 17 lnne2frl2

EXAM CIIECKLIST - Some Usefirl tr'ormulas and Techniques for Matls l/Bridging Maths(Ihis checklist may not be exhaustive to some students - to be read in conjunction with my notes)

.Chaoter 2+

1. FACTORIZATION (the opposite of "exnansiorf') - You have tomemorise!

1. * - ab = a(a - b) (by taking out the common factor'h")

2. *-a' = (a-b)(a+b\

3. a2+2ab+b2 = (a+b)(a+b) = (a+b)2

4. *-ZaA+b2 = (a-b)(a-b) = (a-b)'

5. u'- b' = (a- b) 1a2 + ab + b2)

G. a3+b3 = (a+b)1a2-ab+b2)

7. ao -ba = (* -a\ @2 + b2) (Using (2) above)

= (a - b) (a + b) 1a2 +*7 (Using (2) above)

NOTE:. (5) and (6) are hadly ot never used in Bxam

2. EXPOI\ENTS (sometimes call Indices)

1, a^ *an = em+n

2. a*+a'o, #= a'-n

3. (a')' = 1'o*"

4. (a*b)" = a"*b"

5. ao= 1

I6.^=a-n

en

7. \[; = ottn

/ \n n

8. I +l = + (this is actually similar to (4))\b) b"

Copyright - Bernard Ong Page l of 28 13 August 2012

EXAM CIDCKLIST - Some Useful Forrnulas and Techniques for Maths l/Bridging Maths(This checklist may not be exhaustive to some students - to be read in conjunc'tion with my notes)

3. LOG

ffi"r = rosox+ tosay t'*d = a

2. bg" L = bgox- Iogoyv

3. Iogoa = I

(eg: Iogn 10 = 1 and ln e = 1)

4. Iosof = ntosox lE Xn * (fgX)o

5. toso di = L ^r,,n

6. "tna = a ( a verv imnortant pronertv )(Note: e = 2.7 1828, approximately)

7. ! =logtx if and only if d = x.

Forexample:2 = log;1olC[ if andonlyif 102 = 100

g. los., m-logomeo log"b

For example: log,5 = log: = g = 1o#- s1g.log2 ln2 logr2

(You ttuy ignore Propefi 7 and 8, knowlcdge of Propefi I to 6 willbe suffuient)

Copyright - Bernard Ong Page2 fi?S 13 August 2012

EXAM CIIECKLIST - Some Useful Formulas and Techniques for Maths l/Bridging l\{aths(This checklist may not be exhaustive to some students - to be read in conjunction with my notes)

2.

3.

4a. Graph of Ouadratic Function

The graph of the quadratic function y = f (x) = ax2 +bx+c is a parabola.

1. H a > 0, the parabola opens upward ( "U " shape). If a < 0, it opens downward("fl " shape).

Theverrexis (-r b \[ 2a' I(-za))'

The y-intercept is c. This is when x = 0.

-b+2e

a,b, and c are constants and a + 0. Wefactorization or using the formula above.

can solve a quadratic equation either by

3 possible cases:

(a) V b2 - 4 at> 0, the equation has two roots or solutions.(b) If b2 - 4 ac < 0, the equation has no real roots or solutions(c) If b2 - 4 ac = 0, the equation has repeated roots, i.e. only one solution.

Sometimes b2 - 4ac is written as D, in short.

The vertex is also known as turning point (maximum or minimum point) whichwill cover extensively in Chapter 3. The corresponding value of y

)* or ),,i, o, f e*).

4b. Solvine Ouadratic Eouations

Bysetting 1l = 0,wehave f(x)=y =axz*bxtc = 0,then

weis

- 4ac

Copyright - Bernard Ong Page3 of28 13 August 2012

EXAM CIIECKLIST - Some Usefirl Formulas and Techniques for Maths UBridging Maths

(This checklist may not be exhaustive to some students - to be read in conjunction with my notes)

sraph)

Graphofy=(a)* fora>l

-.1.00,0.50--2.00. 0.25

Graphofy-(a)* 0<acl,

4.0#

50, 2.83

.25.234

.00,0.41\re.oo.o.ze

--.-g.oo, 0.17_a to.oo. o.tt i

6.(x) 8.00 10.00 12.()()

Copyright - Bernard Ong Page4 of28 13 August 2012

EXAM CIIECKLIST - Some Useful Formulas and Techniques for Maths l/Bridging Maths(This checklist may not be exhaustive to some students - to be read in coqiunction with my notes)

Graph of y = ax2 + bx + c for a > 0 (the quadratic will have a minimum)

Curve ofy vs x

d. Graphof y= ax2 + bx + c for a< 0 (the quadratic will have a maximum)

Curve ofy vs x

wu shoald iustknow how to SkgtCh.

."i

Copyright - Bernard Ong Page 5 of28 13 August m12

EXAM CIIECKLIST - Some Useful Formulas and Techniques for Matlu llBridging Maths(This checklist may not be exhaustive to some students - to be read in corqiunction with my notes)

6. Inverse functions

If f(x) is the function of x, then f -r (x) is the inverse function of x.

Procedure:

Step l: Find x in terms of y

Step 2: Replace "y'' with "x" to find f -l (x) - remember to express this in terms ofx.

7. Composition of functions

ff we are gSven 2 functions, f and g, we can apply them consecutively to obtain whatis known as.gg!Ipgi!g.function, given as below:

f (g(x)) + you start with "g" first then "f'.

g (f(x)) --+ you start with "f' first then "g".

Note: f (g(x)) + g (f(x)), in general.

8. .Techniques for Curve Sketchins

General Technioues

You have seen so far, many graphs. Irt us summarise the general techniques forcurve sketching.

l. whenx = 0, y = ?

2. wheny = 0, x = ?

3. Finding critical values (maximum, minimum or stationary points) -Chapter 3

4. wheny -+ 0or*-, x + ?

5. whenx -+Oort-, y -+?

Page 6 of2E

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Copyright - Bernard Ong 13 August 2012

EXAM CIIECKLIST - Some Useful Fonnulas and fschniques for Maths l/Bridging l\{aths(This checklist may not be exhaustive to some students - to be read in coqiunction with my notes)

9a. Trisonometrv - An Introduction

Radians and Desrees

1800 = a radians (n =2217)

lo = fr mdians

180

Degrees or Radians (note: a radians = 1800)

00 30oor 36

45o or !4

6O0 or L3

g0o or L2

sln 0 12

I:Jz

Ji2

I

cos I Ji2

1

JzI2

0

tan 0I:

JzI s undefined

PageT oI2E

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EXAM CIIECKLIST - Some Useful tr'onnulas and Techniques for Maths UBridging Maths(TNs checldist may not be exhaustive to some students - to be read in conJunctlon with my notes)

Trisonometric functions of Comnlementarv Ansles

a

(1)

(2)

sind =

cos'd =

b

opposite / hypotenuse

adjacent / hlpotenuse

a

c

Lc

(3) tan? = opposite / adjacenta

b

("sin" is the short form for "sine", "cos" is the short form for'tosine" and 'tan" is the

short form for "tangent")

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203


EXAM CHECI(LIST -:Sme,Use&d I'umulas and -Trchniqnes forMatb l/Brldghg llfleths(lbis checkHst may, not be cxhausive to sme studsnts + to be,rcad in eoqiunction,*ff my nohs)

@,I*t 9be any angle, then

(4) sin(- e) = '- sin d

(5) cos(- a) = + cosd

(6) tan(-a) = -ua,ne

For example (pu may check with pur scientific calculator!),

sint+so) = -sin45o,

"orf +So) = + cos45o ,

tun(-+St) = -tan45o,

Copyright-BernandOng Page9of2E 13 Augurt 2012

EXAM CIIECKLIST - Some Useful X'ormulas and Techniques for Matlrs VBridging Maths(This checklist may not be exhaustive to some students - to be read in codunction with my notes)

Reciorocal Relationships

(7) I = cosec 0 (read as "cosecant")

sin d

I(8) = secd (read as "secant")cosd

1(9) = cot? (read as "cotangent')tan0

Ouotient Relationships

(10) tanl = #

(ll)cotl=#(becauseof(9))

Hence, we have the following Pyhagorean relationship.

hrthaeorean Relationships

(12\ cos'e+sin2d = I

(13) 1.+tan20 = sec'?

(dividing (Iz)bycos'd andsince tarrl = !A{ -6 I

= = secd)cosO cosd

(14) l+cot20 = cosec'd

(dividing (12)by sin2d andsince cotd = Tt9 *A I ^ = cotl)

sin d tan4

Copynght - Bernard Ong Page L0 of28 13 August 2012

EXAM CIIECKLIST - Some Useful Forrnulas and Techniques for Maths l/Bridging Maths(This checklist may not be exhaustive to some students - to be read in coqiunction with my notes)

Addition Formulas (sometimes called Sum Formulas)

(15) sin(a+B) = sinacosp + cosasinp

(16) cos(a + B) = cosacosB - sin a sinB

(17) tan(a+ B) = 'tana+tanf| - tanatanB

Subtraction Formulas (sometimes called Difference Forrnulas)

(18) sn(a-fl = sinacosB- cosasnp

(19) cos(a-p) = cosacosB + sinasinB

(20) tan(a- B) = .tana-tanf"I+tanatanB

Double Anele Formulas (which is a derivation from the above AdditionFormulas)

(21) sn20 = 2sn0cos0 (derived from (15))

(22) cos20 = cos'? - srn20 (derived from (16))

(23) tan20 = 2tan0- (derived from (17))I- tanz0

Copyright - Bernard Ong Page 11 of28 13 August 2012

EXAM CIIECKLIST - Some Useful X'ormulas and Techniques for Maths UBridging nlaths(This checklist may not be exhaustive to some students - to be read in co4iunction with my notes)

Pro duct of Sines and Cosines, ftnn examfuub le. fo r knowlede el

(24) sina cosB = |t*t"* f)+ sin(a- f)j(using (15) + (18))

(zs) cosasinB = |t'tto* f)- sn(a-f)l(using (1s) - (18))

(26, sna sinB = - jh"do*p)+ cos (o- f)l(usrng (16) - (19))

(27) cosa cosB = !I*"@* O)+ cos(a-l)]

(using (16) + (19))

Copynght - Bernard Ong Page 12of2E 13 August 2012

EXAM CIIECKLIST - Some Useful Fonnulas and Techniques for Matls l/Bridging Maths(This checklist may not be exhaustive to some students - to be read in coqjunction with my notes)

We can derive the below using (28) to (31) by letting a+ P 'A and

a-f =B

Prod\ct of Sines and, Cosins hon examinable. for knowledee\

(28) sinA+ sinB = zsin|(e+r)cosj(a-r)

(2s) sinA- sinB = z"or|6+B)sin f,U-nl

(30) cosA+ cosB = z"orLr(e+r)cosj(a-r)

(3r) cosA- cosB = -2sin f,to*r)rir|(a-r)

Deriving from (22) and (12),

(32) cos20 = 2cos20-l

(33) cos20 = 7-2sn20

Copynght -Bernard Ong Page 13 of2E 13 August 2012

EXAM CHECKLIST - Some Useful Formulas and Techniques for Maths l/Bridging Maths(This checklist may not be exhaustive to some students - to be read in conjunction with my notes)

Chapter 3

10. Standard Derivatives

12. .Sum Rule

ditf(x)+g(x)l=f(x)+g'(x)dx

ln words, the derivative of a sum is the sum of the separate derivatives.

11. Power Rule

9 ( *') = nax n-l

dx

13. Product Rde

ddvdu-luvl = u-+v-dx' dx dx

f(x) or yf '(x) or Q Comments

a{ n-lnax, Known as Power Rule.

a and n are constants.r

e gt No chaneeInx 1

xx must be positive

smr cos J t{ardlv tested in examcos.r -sln .r Hardlv tested in exam

Copyright - Bernard Ong Page 14 of28 13 August 2012

EXAM CIIECKLIST - Some Useful Formulas and Techniques for Maths llBridging Maths(This checklist may not be exhaustive to some students - to be read in coqiunction with my notes)

14. .Ouotient Rule

du dva("\ 'ar-"a*d.l;) ,'

15. .Chain Rule (or Composite Function RuIe)

dy = 4 *dp whereycanbewrittenas /(.r)dx dp dx

Ihe below can easily be derived using the chain rule:

ft<r<o"l = n f(x)^-' .ft{r e>)

4@t,o, = "r,n*4(/r"l)dx dx

ftonru>t = #.*kr"l)L*u f @)) = cos/(x) .ft{r a>l

fr<ro, f (x)) = -sin /(.r) .fttf A>) etc..

Note: frVAtl can bewrittenas f'(x)

Copyright-BernardOng Pagel5of2S 13 August 2012

EXAM CIIECKLIST - Some Usefirl f,'ormulas and Techniques for Matls l/Bridging l\{aths(This checklist may not be exhaustive to some students - to be read in conjunction with my notes)

16. .Optimisation

The derivative is useful for finding the maximum or minimum value (or wesometimes callrelative extrenu) of a function.

Procedare

l. Find 4dx

2. S"t 4 = 0 and solve for x and these values of x are known as thedx

critical noints or stationarv noints.

3. Find the second derivative. d I .' dxz

Then substitute the values or values of -r, if any, into d='Y.

.dx'

rf+ox

bdxz

4 = 0, x is a neither a maximum or minimum pointdx2

(orpoint of inflexion)

You maybe asked to frndy* or!^in

Copyright. Bernard Ong Page 16 of2E 13 August 2012

EXAM CIIECKLIST - Some Useful Formulas and Techniques for Maths l/Bridging Maths(This checklist may not be exhaustive to some students - to be read in conjunction with my notes)

17. Mareinal Concepts

You have probably learnt that in Economics:

!{arginal eost (MC) is defined as the change in total cost incurred from theproduction of an additional unit.

!{arginal levenue (MR) is defined as the change in revenue brought about bythe sale of an extra good.

Mathematically,

1. uc(q) = 4t rl (or TC',in short)clq

2. mn(q) = hri ( or zR", in shon)

where

rC(q) = Total Cost

rn(q) = Total Revenue

Some useful formulas (or identities),

3. TC = VC+FCwhere VC = Variable Cost, FC = Fixed Cost (a constant)

4.AC=Tcq

where AC = Average Cost

5. AVC = vCq

where AVC = Average Variable Cost

6. n = rR(q)-rc(q)where n = Profit.

7. TC = FCwheng=0@Iy)

8. TR = 0whenq=0(@ggy)

9. For break even, fR(4) = fC(q) (or fI = 0)

10. For monopoly, fn(q) = p* q

Copyright - Bernard Ong PagelT of2E 13 August 2012

EXAM CIIECKLIST - Some Useful Formulas and Techniques for Maths l/Bridging Maths(This checklist may not be exhaustive to some students - to be read in co4iunction with my notes)

.Chapter 4

lS. .Standard Inteerals

19. .Integration usins substitution

What to look out for:

This is similar to the chain rule of differentiation where we usually

(1) substitute the more complicated part with n.

(2) then convert all to a.

(3) after integration, convert all back to the original variable.

20. .Integration bv parts

.What to look out for:

A dffirentiable part and an integrable part.

Formula:

ludv = uv -tvdu

where u is the differentiable part and v is the integrable part.

f(x) If<oa* Comments

.ro(n+-l) n+lx

-+c(n +l)

Reverse to Power Ruleof differentiation

et et+c No chanse

1x

h l,rl +c r must be positive

SIN J -cosx+ccos.r $nr+c

Copyright - Bernard Ong Page 18 of2E 13 August 2012

EXAM CIIECKLIST - Some Useful X'onnulas and Techniques for Maths UBridging Maths(This checklist may not be exhaustive to some students - to be read in co4iunction with my notes)

21. .Inteeration usins partial fractions

What to do look out for.:

The denominator of the integrand can be factorised and make sure that theintegrand is a proper function.

Note:. Usually, most questions can be done by using substitution.

Method:

Step I

Factorise the denominator.

Step 2

Express the integrand as:

ABx*C xID

Step 3

Integate as per normal using below:

f(x)J 11*10^

1x

hl.rl + c

IxlC

hlx+Cl + c

Ix+D

ml.r+al + cetc..

Copyright - Bernard Ong Page 19 of2E 13 August 2012


?2. Economic Annlications

Calculatine functions from their marqinals

Formulas:

1. ttc(q) = hOO (orrc',inshort)

z. mn(q) = hrO ( or 7R", in shorr)

where

rC(q) = Total Cost

rn(q) = Total Revenue

Similarly, in order to find fc(q), we integrate UC(q) (to reverse the process)

Formulas:

1. rc(q) = ! uc aq

2. rn(q) = !unaq

NOTts

1. rC(0) = FC (since no variable cost)

(which I alreadymentioned in Chapter 3, differentiation)

2. rn(o)l = 0 (since no production, no revenue)

(which I alreadymentioned in Chapter 3, differentiation)

3. For profit maximisation,

mn(q) = uc(q)(this is not mentioned in the Subject Guide)

Copyright -Bernard Ong Page20 of28 13 August il12


.Chanter 5

23a.@af af=-, -dx dy

23b. 2nd Order Partial Derivatives

a'f a'f a'f a'farl ay" W' W

Note: a'f = d'faxEy Ayd,

24a. Inplicit Partial Differentiation (normallv NOT TESTED)

q- = -!f!,!. where f (x, y) = c, where c is a constantdx df /dy

(Take note of the negative sign)

24b. .Chain Rule (normallv NOT TESTED)

Formula:

df df dx df dy

dt dx dt dy dt

The above formula is rather difficult to remember. It is best that youreplace x and y with t and then differentiate directly with respect to t. Noneed to memorize formula!

Copyright - Bernard Ong PageZl of28 13 August 2012


25. .Optimization WITHOUT constraint

Optimize: a function of x and y, say f(ny)

Method (similar to Chanter 3 - findins critical ooints usins differentiation)

Step IFind: af .a! .u-.y!.u-.9:!-

dx ' dy' ax' ' Oy2 'dxdy 'EyEr

Step 2Set both

af =odx

af=oEy

and solve the 2 equations to find x and y.

Step 3

Test for critical points(or stationary points):

.Case I

t#)(#) [#)'= it is maximum.

.Case 2

t#)t#) [#)'3 it is minimum

Case 3

t#)t#) [#)'

Copynght - Bernard Ong Page22 d?3 13 August 2012

EXAM CIIECKLIST - Some Useful Formulas and Techniques for Matlrs UBridging Maths(This checklist may not be exharutive to some students - to be read in coqiunction with my notes)

26. Ontimization WITH onstraint

Optimize f (x,y) subject to g(x, y)

Method

Step L

Express l, the Langrangean as:

L = f(x,y)-Ag(x,y)

where/is the function to be optimized and g is the constraint which is .equal to 0.

Step 2

Find: +, +, * *asetallofthemequalto0.dx dy' U"

We eliminate,l.using equation (1) and (2) so that we have an equation (orrelationship) involvingx andy which we can substitute this relationship intoequation (3).

Sten 3

No step 3 because you need not test for max or min.

Note

The Lasranse Multiplier

2 is called the Lagrange Multiplier. suppose M is the maximum or minimum off(x, y), subject to the constraint 56, , = k The Lagrange multiplier, l, is therate of change of M with respect to &. That is,

^aM/v=&

Hence,

2is approximately the change in M resulting from a l-unit increase ink.

Copyright -Bernard Ong Page23 of2E 13 August 2012

EXAM CHECKLIST - Some Useful Formulas and Techniques for Maths l/Sridging lMaths(This checklist may not be exhaustive to some students - to be read in co4iunction with my notes)

26a. Modification to the.Optimization WITH constraint

Here we added another constraint, i.e. h(x, y)

Optimize f (x,y) subjectto g(x,y) and h(x, y)

Method

Step I

Express L, the Langrangean as:

L = f(x,y)- hg@,9-kh(x,y)

where/is the function to be optimized andg andft are the mnstraints which areeoual to 0.

Sten 2

Find: +, +, *, .pandsetallofthemequalto0.Ex' ay' a4' aL,

We eliminate Xa and, Tztofindthe values of r andy.

Copyright - Bernard Ong Page24 d?8 13 August 2012

EXAM CIIECKLIST - Some Useful Formulss and Techniques for Maths UBridging n[aths(This checklist may not be exhaustive to some students - io be read in conjunction with my notes)

26b. Modification to the.Ontimization \ilITH constraint

Here we added another variable, z.

Optimize f (x, y, z ) subject to g(x, y, z ) and h(x, y, z)

Method

Steo I

Express L, the Langrangean as:

L = f (x, y,z)- h g(x, y,z)- kh(x, y,z)

where/is the function to be optimized and g and lr are the onstraints which areequal to 0.

Step 2

Find: +, +, +, g , +and set all of them equat to 0.dx cty dz drI1 dr,

IVe eliminate Xa and, ktofrnd the values of r,y and z.

Copyright-BernardOng Page25of28 13 August 2012


Chapter 6

n. Row Onerations

Rules

The 3 permissible row operations '?ules" are:

1) ri <+ rj. .r,.: , ,. , tworows.

2) r; -> ariN{ultiplying (or dividing) of one row by a constant.

3) ri -+ ri + arjAdding ( tir strhtraeting i a multiple of another row.

Note: a is a non-zero constant.

Reference: Subject Guide, Page 111 - 112

Outcome

To arrive at the final matrix (with the entries in the lower friangle are zeros)

(, x r I r )Io * " |" I r*oo, The'1"canbeanynumber)

[o o ,1.)Proceed to use back-substitution to find the values of the variables.

4J!er@4,

you can proceed to obtain the below matrix (which is the 3 by 3 unit matrix).

(t o o

lo 1 o

[oo 1

: I where b, c and d are thesotutions.

d)

You will, of course, need to perform a few more row operations to reach this 3 by 3unit matrix.

Copyright - Bernard Ong Page26 of 2E 13 August 2012

EXAM CIIECKLIST - Some Useful Formulas and Techniques for Maths l/Bridging l\tlaths(Ihis checklist may not be exhaustive to some students - to be read in conjunction with my notes)

.Chanter 7

28. AP (ARITHMETIC PROGRESSION)

Tn = a+(n-l)d

sn = ;12" + (n -Ddl or | {" */) (usually not used)

where g is the I!$[ term, ! is the common {ifference and / is the [ast term.

29. GP (GEOMETRIC PROGRESSION)

n-lln = Af

where g is the first term and g is the common ratio.

a(l- r")Sn= l-rSnecial Case

Inparticular,if n 1 @, xn + 0, ltl . 1

So,Sn=*

30. First-Order Difference Eouations (Short cut to GP ouestions- anolvinsfonnulas without understandins "lryhat is goins oft) - "Maths 2 Method"

The linear frst-order difference equation:

lw = aY*-r*b

then

ln = y* * (yo -y*)a*

where: Y{< - b

' | -a

Page27 ofZECopyright - Bernard Ong 13 August 2012

EXAM CHECKLIST - Some Usefd Formulas and Techniques fm Math UBridging l[aths(This checHist may not be exhaustive to sune students - to be read in conJunc'tion with my notes)

However, if y, is the initial term,

then

lu = y* * (yr -y*)an-t

where: Y* =h

However, if y, is the initial term,

then

lu = y* t (yz -y*)a'-'

where: y* = b

l-a

Good Luck! ! !

Copyright - Bernard Ong Page 28 of 2E 13 August 2012

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