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PAPER 2 FORMAT

Subjective Questions

No. of Questions :A (6), B (4/5), C (2/4)

Total Marks : 100 Duration : 2 hours 30 minutes

L.O.D : L(6) , M(4-5), H(4-5) Additional Materials: Scientific Calculators,

Mathematical Tables (Given),

Geometrical sets

(+ Pencils & LONG RULER !)

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Key towards scoring MAXIMUM MARKS

Start with your favourite question !

CUBA ! DONTGIVE UP!

Try as many questions as possible!

Choose and write the correct

formula +gan tikan ni lai!Show your working!

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Strategic Revision :

Top ic 1

Simultaneous

equations

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4. Simultaneous Equations (20032007)

1. (03) Solve the simultaneous equations

4x + y = 8 and x2

+ xy = 2 [5m]


pm = 2 and p2+ 2m = 8 [5m]

Give your answers cor rect to 3 d.p.


x + y = 1 and y210 = 2x [5m]

4. (06) Solve the simultaneous equations 2x + y = 1

and 2x2+y2+ xy = 5.Give you r answer co rrect to three decimal

places. [5 m]

5. (07) Solve : 2x y 3 = 0 , 2x210x + y + 9 = 0

F4

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Simultaneous Equations

[Form 4, Topic 4]

Usually asked in PAPER 2, Question 1

Characteristics of simultaneousequations:

Involves TWO variables,usually inxand y.

Involves TWO equations: one linearand the other non- linear.

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[Form 4, Topic 4]

Solving simultaneous

equations means findingthe values of x andcorresponding values of ywhichsatisfy BOTH the equations.

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Methods of solving : Starting from the LINEAR equation, express y

in terms of x(orx in terms of y).

Substitute y ( or x ) into the second equation(which is non-linear) to obtain a quadratic

equation in the formax2+ bx + c = 0.

Solve the quadratic equation by factorisationorby using the FORMULA.

Make sure you use the formula correctly !

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[Form 4, Topic 4]

Methods of solving :

If the equations are given in the form

a = b = c, you are advised to writeTWO equations : one LINEAR and

the other NON-LINEAR :

Write either a = b, b = c or a = c

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4. Simultaneous Equations

F4

Please note thatNOT ALL questions

can be solved byfactorisation

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[Form 4, Topic 4]

If the quadratic equation CANNOT

be factorised, candidates will be

asked to give your answer

correct to 4 significant figures

or give your answer correctto three decimalplaces.

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Some common mistakes

Careless..

(P2, Q1): Simultaneous Equations

After finding the values of x (or y),candidates did not proceed to findthe corresponding values of y(or x),

thinking that he/she had answeredthe whole question!

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Some common mistakes

Tak Peka

If you obtain a quadratic

equation which cannot befactorised, you should stopawhileand think :

Did I make a mistake somewhere ?

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F4

Now, let us see

someexamples

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Solve the simultaneous equation

4x + y = - 8 and x2+ xy = 2.(5marks )

SPM 2003 Paper 2, Question 1

Answer (In your answer sc r ip t):

STEP 1 : Write down the TWOequations :

4x + y = -8 ..... (1)x2+ xy = 2 ..... (2)

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Solve the simultaneous equation 4x + y = - 8

and x2

+ xy = 2 (5marks)


Answer (Cont...)

STEP 1 : Write down TWOequations :

4x + y = -8 ..... (1)x2+ xy = 2 ..... (2)

STEP 2 : From the linear equation (1),

make x or y the subject :

Can you tell me : which should be thesubject , x or y??? WHY ???

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Answer (cont....)

P1

K1

STEP 1 : Write down TWOequations :

4x+y = -8 ..... (1)x2+ xy = 2 ..... (2)

STEP 2 : From the linear equation (1),makeythe subject :

STEP 3 : Substitute (3) into (2),

x2+ x(-84x) = 2

y = -84x..... (3)

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Answer (cont...)

x2+ x(-84x) = 2

x2+ x + 8 + 4x = 2

Simplifying,

x2+ 5x + 6 = 0

x2+x + 4x+ 82 = 0

Factorising,

(x + 2) (x + 3) = 0K1

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Answer (Cont.....)

N1

N1

(x + 2) (x + 3) = 0

x + 2 = 0 , x + 3 = 0

x = -2 , x = -3

Last STEP: Dont forget to find values of y !

Using y = -84x ..... from (3)

x = -2, y = -84(-2)= 0

x = -3, y = -84(-3)= 4

5

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Solve the simultaneous equation 4x + y = - 8

and x

2

+ x

y = 2 (5marks)

SPM 2003 P2, Q1 [MARKING SCHEME]

Answer

Make x or y the subject

P1

N1x = -2, -3 or y = 0 , 4

Eliminating xory

88 4

4

y

x or y x

2

28 82 ( 8 4 ) 2

4 4

y yy or x x x

Solving the quadratic equation :

y = 0 , 4 or x = -2, -3N1

K1

K1

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F4

Do you still rememberthat NOT ALL questionscan be solved by

factorisation ?

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1. Solve the simultaneous equations

x + y = 1

x2

+ y2

= 5

2. Solve the simultaneous equations, give your

answer correct to foursignificant figures.

x + y = 1

x2+ y2= 4

Factorisation

a

acbb

2

42

4. Simultaneous EquationsF4

Factorisation orFormula ?

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Q1. x + y = 1 (1)x2+ y2 = 5 (2)


x2+ (1 x )2 = 5

x2+ 12x + x2 = 5Simplifying,

x2

x 2 = 0

2x22x 4 = 0

Factorising, (x + 1) (x2) = 0

From (1) : y = 1 x

x = -1 , x = 2

y = 2 , y = -1

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Q2. x + y = 1 (1)x2+ y2 = 4 (2)


x2

+ (1 x )2

= 4

x2+ 12x + x2 = 4Simplifying,

2x22x 3 = 0

So, use formula to find x !

From (1) : y = 1 x

Cannot befactorised !

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2 4

2

b b acx

a

Q2. x + y = 1 (1)x2+ y2 = 4 (2)

F4

2x22x 3 = 0

Using formula with a = 2, b = -2, c = -3

x = - 0.8229 , x = 1.823

y = 1.823 , y = -0.8229

Cannot befactorised !

2( 2) ( 2) 4(2)( 3)

2(2)x

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(SPM05) Solve the simultaneous equations

x + y = 1 and y210 = 2x [5m]

x + y = 1

x = 1 y 1 m

y210 = 2( 1 y) 1 m= 2y

y2 + y12 = 0

(y + 4) (y3) = 0 1 m

y =4 , y = 3 1 m

x = 3 , x = - 1 m

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Common mistakes

Forgotten the basics..


eg. Solve the simultaneous equations

4x = x2

y + 3 = 10 +2y

x2y -3 = 10 +2y (2)

4x = x2y + 3 (1)

You should write one linear&

one quadraticequation!

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Common mistakes



eg. Solve the simultaneous equations

4x = x2

y + 3 = 10 +2y

4x = 10 + 2y (1)

4x = x2y + 3 (2)

And, Always simplify first !

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Common mistakes



eg. Solve 4x = x2y +3 = 10 + 2y

4x = 10 + 2y

2y = 4x10

y =4 10

2

x

4x = 10 + 2y

x = 10 2

4

y

Please SIMPLIFYfirst &

AVOID fractionsif possible !!!

Answers : x = 2, y = -1 , x = 4, y = 3

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Kunci Mencapai KeputusanCemerlang

Belajar secara konsisten

Tabahkan hatiRajin tanya cikgu

Rajin buat latihan

Jaga kesihatan anda !!!

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THANK Q &Selamat maju jaya !

F4_4 SEqn

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