Modul Syed Malek _repaired

8/10/2019 Modul Syed Malek _repaired

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MODUL GEMILANG SMKA SHEIKH MALEK 1 by CYY

MODUL KEM GEMILANG

SMKA SHEIKH MALEK

GROUP HALUS

2013

MATEMATIK TAMBAHAN

DISEDIAKAN OLEH : CIKGU YUSRI BIN YAAHMAT

yusriyaahmat.blogspot.com


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ALGEBRA

1. x =2 4

2

b b ac

a

2. ama

n = a

m+ n----pan-pen

3. ama

n = a

mn----pan-pen

4. (am)n = a

mn

5. logamn= loga m+ loga n-----pan-pen

6. logan

m= loga mloga n -----pan-pen

7. logamn = nlogam-----kuda pan

8.log

loglog

ca

c

bb

a= ---- kuda

9. ( 1)n

T a n d = + ------- kaki atok gbai

10. [2 ( 1) ]2

n

nS a n d = +

11.1n

nT ar

=

12. nS =1

)1(

r

ran

=r

ran

1

)1(, 1r

13. S =r

a

1, | r| < 1

KALKULUS

1. y= uv,dx

dy= u

dx

dv+ v

dx

du-----sida

2. y=v

u,

dx

dy =

2v

dx

dvu

dx

duv

3.dx

dy=

du

dy

dx

du----3p/u

4. Area under a curve

=

b

a

y dx or = b

a

dyx

5. Volume generated

= b

a

dxy2 or = b

a

dyx2

STATISTIK

1. x =N

x----tiada f

2. x =

f

fx

3. =

N

xx 2)(=

22

)(x

x

N

tiada f

4. =

f

xxf 2)( =

22)(

fxx

f

5. m = CLmf

FN

+

21

6. I =0

1

Q

Q100-2 thn

7. I =

i

ii

W

IW----fungsi gubahan

8. rn P =

!)(

!

rn

n

9. rnC =

!!)(

!

rrn

n

10. P(AB) = P(A) + P(B) P(AB)

11. )( rXp = = rnrrn

qpC , p+ q = 1

12. Mean /Min = np

13. = npq

14. Z =

X

=0

> 0

Paksi-y

Paksi-x

1

1

4Q N=

3

3

4

Q N=

DaTo TaBah


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GEOMETRI

1. Distance = 2212

21 )()( yyxx +

2. Midpoint

(x,y) =

++

2,

2

2121 yyxx

3. A point dividing a segment of a line

(x,y) = 1 2 1 2,nx mx ny my

m n m n

+ +

+ +

4. Area of triangle /Luas segi tiga

)()(3123121332212

1yxyxyxyxyxyx ++++

5. r = 22 yx +

6. r =22

yx

yx

+

+ ji

TRIGONOMETRI

1. Arc length, s = r

2. Area of sector =2

1 2r

3. AA 22 cossin + = 1

4. A2sec = A2tan1+

5. A2cosec = A2cot1+

6. sin 2A = 2 sinAcosA

7. cos 2A = cos2Asin

2A

= 2 cos2A1

= 1 2 sin2A

8. )(sin BA = sinAcosB cosAsinB

9. )(cos BA = cosAcosB m sinAsinB

10. )(tan BA =BA

BA

tantan1

tantan

m

11. tan 2A =A

A2

tan1

tan2

12. A

a

sin = B

b

sin = C

c

sin

13. a2 = b

2+ c22bccosA

14. Area of triangle =2

1absin C

FORMULA TAMBAHAN

1. x2 (SOR)x+ POR = 0 2

b cx

a a

+ = 0

2. f (x) =2 2

2 4

b ba x c

a a

+ +

3.100

X YZ

=

Z

Year C

X

YearA

Y

YearB

Sila t2

Vektor unit

SiLa

Anak

Panah


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MANUAL PENGGUNAAN-14 JAM

SLOT TAJUK PAGE

1 FORM 4 BAB 1- 4

FUNCTION

QUADRATICS EQUATION

QUADRATICS FUNCTIOND

SIMULTANEOUS EQUATION

5 - 17

2 FORM 4 BAB 5 6

INDICES N LOGARITME

COORDINATE GEOMETRY 18-26

3 FORM 4 BAB 7 8

STATISTICS

CIRCULAR MEASURE27-33

4 FORM 4 BAB 10 11

SOLUTIONS OF TRIAGLES

INDEX NUMBERS

34 -42

5 FORM 5 BAB 1 2

PROGRESSIONS

LINEAR LAW

43-50

6 FORM 5 BAB 4 5

VECTORS

TRIGONOMETRIC FUNCTIONS

51-63

7 FORM 4 FORM 5 BAB 3

INTEGRATION64-70


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FUNCTIONS

EXAMPLE 1:

Given that 43: xxf and xxg 2: ,

find fg(3).

Answer : f(x) = 3x - 4 , g(x) = 2x

g(3) = 2( )

= ( )

fg(3) = f [ g(3) ]

= f ( )

= 3 ( ) - 4

= ( )

EXAMPLE 2:

Given that xxf 23: and 2: xxg , find

gf(4).

Answer : f(x) = 3 2x , g(x) = x2.

f( ) = 3 2( )

= ( )

gf(4) = g ( )

= ( )2

= ( )

EXERCISES

1. Given that 12: + xxf and xxg 3: ,

find f g(1) .

2. Given that 92: xxf and xxg 31: + ,

find gf(3).

Given that 23: xxf , find f2(2).

Answer : f(x) = 3x - 2

f(2) = 3( ) 2 = ( )

f2(2) = f [ f(2) ]

= f ( )

= 3 ( ) 2

= ( )

Given that xxg 43: , evaluate gg(1).

Answer : g(x) = 3 4x

g(1) = 3 4( ) =

gg(1) = g [g(1)]

= g ( -1)

= 3 4 ( )

= ( )


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1.5 FINDING THE VALUE OF COMPOSITE FUNCTIONS 22 , gf

Notes : f2(a) = ff (a) =f [f(a) ]

1.7 FINDING COMPOSITE FUNCTIONS

EXAMPLE 1:

Given that 43: xxf and xxg 2: ,

find fg(x).

Answer : f(x) = 3x - 4 , g(x) = 2x

fg(x) = f [ g(x) ]

= f ( 2x)

= 3 (2x) - 4

= 6x 4

OR

fg(x) = f [ g(x) ]

= 3 [g(x)] - 4

= 3 (2x) 4

= 6 x - 4

EXAMPLE 2:

Given that xxf 23: and 2: xxg , find the

composite fuction gf.

Answer: f(x) = 3 2x , g(x) = x2.

gf(x) = g[f(x)]

= g (3 2x)

= (3- 2x)2

OR

gf(x) = g[f(x)]

= [f(x)]2

= (3- 2x)2

thus 2)23(: xxgf

EXERCISES


find fg(x) .

2. Given that 52: xxf and xxg 5: , find

the composite function gf .

3. Given the functions 4: +xxf and

12: xxg , find

(a) fg(x) (b) gf(x)


find

(a) f g (b) gf


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EXAMPLE 1 :

Given that f(x) = 4x 6 , find

x + ( )

f1

(x) =4

6+x

EXAMPLE 2 :

Given that f(x) = 2x + 3 , find f1

(x).

1. Given that f : x 4 + 8x , find f1

. 2. Given that g : x 5 + 6x , find g1

.

3. Given that g : x 10 2x , find g1

. 4. Given that f : x 4 3x , find f1

.

5. Given that f : x 5 + 2x , find f1

. 6. Given that g : x 3 2x , find g1

.

7. Given that f : x 6x - 15 , find f1

.8. Given that g : x 3

4

3x , find g

1.

9. Given that f(x) = 1 2x , find f 1(x). 10. Given that g(x) = 3x + 2 , find g1 (x).

11. Given that f(x) = 5 2x , find f

(x). 12. Given that g(x) = 5x + 2 , find g

(x).


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EXERCISES-QUESTIONS BASED ON SPM FORMAT

1.

Base on the information above, a relation from P

into Q is defined by the set of ordered pairs

{ (1, 4), (1, 6), (2, 6), (2, 8) }.

State

(a) the images of 1, Ans:

(b)the object of 4, Ans:

(c) the domain, Ans:

(d)the codomain, Ans:

(e) the range, Ans:

(f) the type of relation Ans:

2.

The above diagram shows the relation between set

A and set B

State

(a) the images of b, Ans:

(b)the objects of p, Ans:

(c) the domain, Ans:

(d)the codomain, Ans:

(e) the range, Ans:

(f) the type of relation Ans:

3. Given the functions 12: + xxf and

3: 2 xxg , find

(a) f1

(5) ,

(b) gf (x) .

4. Given the functions 14: + xxg and

3: 2 xxh , find

(a) g1

(3) ,

(b) hg (x) .

5. Given the function xxf 32: and

2:2

+ xxxh , find(a) f

1(3) ,

(b) hf (x) ,

(c) f2(x) .

6. Given the functions 12: + xxf and2

2: xxh , find(a) f

1( 1) ,

(b) hf (x) ,

(c) f h(x).

P = { 1, 2, 3}

Q = {2, 4, 6, 8, 10}

P = { 1, 2}

Q = {2, 4, 6, 8, 10} a

b

c

p

q

r

s

A B


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QUADRATIC EQUATIONS/FUNCTIONS

Solve (x 3) = 1.

Ans : 2, 4

L4. Solve 1 + 2x = 5x + 4.

Ans : 1, 3/2

Solve (2x 1) = 2x 1 .

Ans : , 1

L4. Solve 5x 45 = 0.

Ans : 3 , 3

Selesaikan (x 3)(x + 3) = 16.

Ans : 5 , 5

L6. Selesaikan 3 + x 4x2= 0.

Ans : , 1


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Solve x 2x 9 = 0 by completing the square,

give your answers correct to 4 significant figures.

Ans : 2.212 , 4.162

L4. Solve x + 10x + 5 = 0 , give your

answers correct to 4 significant figures.

Ans : 0.5279, 9.472

Find the quadratic equation with roots 2 dan - 4.

x2+ 2x 8 = 0

L1. Find the quadratic equation with roots -

3 dan 5.

Ans : x2 2x 15 = 0

Given that the roots of the quadratic equation

x2+ (h 2)x + 2k = 0 are 4 and -2 . Find h and k.

(Ans : h = 0, k = -4)

L6. Given that the roots of the quadratic

equation 2x2+ (3 k)x + 8p = 0 are p and 2p

, p 0. Find k and p.

(Ans: p = 2, k = 15)

Given that the roots of the quadratic equation

2x2+ (p+1)x + q - 2 = 0 are -3 and . Find the

value of p and q.

x = -3 , x =

x + 3 = 0 or 2x 1 = 0

(x + 3) ( 2x 1) = 0

2x2+ 5x 3 = 0

Comparing with the original equation :

p + 1 = , q - 2 =

p = , q =

L4. Given that the roots of the quadratic

equation 3x2+ kx + p 2 = 0 are 4 and

- . Find k and p.

(Ans : k = -10 , p = -6)


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The roots of the quadratic equation

2x2+ px + q = 0 are - 6 and 3.

Find

(a) p and q,

(b) range of values of k such that

2x2+ px + q = k does not have real roots.

Answer :

(a) x = -6 , x = 3

(x + 6) (x 3) = 0

x2+ 3x - 18 = 0

2x2+ 6x 36 = 0

Comparing : p = , q =

(b) 2x2+ 6x 36 k = 0

a = 2, b = 6, c = -36 - k

62 4(2)(-36 k) < 0

k < 40.5

L1. The roots of the quadratic equation

2x2+ px + q = 0 are 2 and -3.

Find

(a) p and q,

(b) the range of values of k such that

2x2+ px + q = k does not have real roots.

Find the range of k if the quadratic equation

2x2 x = k has real and distinct roots.

( Ans : k > - 1/8 )

L3. The quadratic equation 9 + 4x2= px

has equal roots. Find the possible values of p.

( Ans : p = -12 atau 12)


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Find the range of p if the quadratic equation 2x +

4x + 5 + p = 0 has real roots.

(Ans : p - 3 )

L5. Find the range of p if the quadratic

equation x2+ px = 2p does not have real

roots.

( Ans : -8 < p < 0 )

The roots of the quadratic equation

2x2+ 8 = (k 3)x are real and different. Determine

the range of values of k.

( Ans : k < -5 , k > 11 )

L7. Find the range of values of k if the

quadratic equation x2+ 2kx + k + 6 = 0 has

equal roots.

( Ans : k -2 , 3 )


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(a) The equation x 6x + 7 = h(2x 3) has roots

which are equal. Find the values of h. [4]

(b) Given that and are roots of the equation

x2 2x + k = 0 , while 2 and 2 are the

roots of the equation x2 + mx + 9 = 0.

Determine the possible values of k and m [6]

( h = -1 , -2 ; k = 49

L2. One of the roots of the equation

2x2+ 6x = 2k 1 is twice the other.

Find the value of k and the roots of the

equation.

( x = -1 , x = -2 ; k = 2

3

)

(SPM 2003 , P1, S3). Solve the quadratic equation

2x(x 4) = (1 x)(x + 2). Give your answer correct to

4 significant sigures. [3]

( x = 2.591, - 0.2573)

L3. (SPM 2003, P1, S4) The quadratic

equation x (x+1) = px 4 has two distinct

roots.Find the range of values of p. [3]

( p , -3 , p > 5)

(SPM 2002) Find the range of k if the Q.E.

x2+ 3 = k (x 1), k constant, has two distinctroots. [3]

( k < -2 , k > 6)

(SPM 2001) Given 2 and m are roots of the

equation (2x 1)(x + 3) = k (x 1), with k as aconstant, find k and m. [4]

( k = 15 , m = 3 )


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FUNCTION VALUE MAX/MINAXIS OF

SYMMETRY

POINT

MAX/MINGRAPH SKETCHING

f(x) = (x 1)2+ 2 2 (x 1)

2

= 0x = 1

(1, 2)

g(x) = (x- 2)2+ 4 4

(x 2)2 = 0

x =( , )

f(x) = (x + 1) - 4

f(x) = 1 + 2 (x 3)

f(x) = 3x2

- 2

x

O

y

xO

y

(1,2)

x

O

y

x

O

x

y

xO

y


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Solve the inequality x + x - 6 0

x2 + x - 6 0

(x + 3) ( x 2) 0

Consider f(x) = 0. Then x = -3 , x = 2

Range of x is : x -3 atau x 2

L4. Solve the inequality x + 3x - 10 0.

x -5 , x 2

Solve the inequality 2x2 + x > 6.

x < -2 , x > 3/2

L6. Solve the inequality x(4 x) 0.

0 x 4

Diven y = h + 4kx 2x = q 2(x + p)

(a) Find p and q in terms of h and / or k.(b) If h = -10 and k = 3,

(i) State the equation of the axisof symmetry,

(ii) Sketch the graph of y = f(x)

(Ans : p = -k , q = 2k2+ h ; paksi simetri : x = 3)

L8. Sketch the graphs of

(a) y = x2+ 3

(b) y = 2 (x - 3)2 1

x

-3 2

y=f(x)


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SIMULTANEOUS EQUATIONS

1. Solve x + y = 3, xy = 10 .

x + y = 3 ........ (1)

xy = 10 ........ (2)

From (1), y = ( )......... (3)

Substitute (3) into (2),

x ( ) = 10

3x x2

= 10

x2

3x 10 = 0

(x ) (x ) = 0

x = ( ) atau x = ( )

From ( ), when x = ( ) , y = 3 ( ) =

x = ( ) , y = 3 ( ) =

2. Solve x + y = 5, xy = 4 .

(Ans : x = 1, y = 4 ; x = 4, y = 1)

3. Solve x + y = 2 , xy = 8 .

(Ans : x = 4 , y = 2 ; x = 2, y = 4 )

4. Solve 2x + y = 6, xy = 20 .

(Ans : x = 2 , y = 10 ; x = 5, y = 4 )

5.Solve the simultaneous equations

3x 5 = 2y

y(x + y) = x(x + y) 5

(Ans : x = 3 , y = 2 )

6. Solve the simultaneous equations :

x + y 3 = 0

x2

+ y2

xy = 21

(Ans : x = 1, y = 4 ; x = 4, y = 1 )


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6. Solve the simultaneous equations :

4x + y = 8

x2+ x y = 2

(Ans : x = 2 , y = 0 ; x = 3 , y = 4 )

7. Solve the simultaneous equations p m = 2

and p2+ 2m = 8. Give your answers correct

to three decimal places .

(m = 0.606, p = 2.606 ; m = 6.606 , p = 4.606 )

7. Solve the simultaneous equations

11

2x y+ = and y

2 10 = 2x

(Ans : x = 4 , y = 3 ; x = , y = 3 )

8. Solve the simultaneous

x + 3y 1 = x2+ 3x + 6y 4 = 0

x = 1, y = 0 ; x = 2 , y = 1]

9 Solve the simultaneous equations 2x + y = 1 and 2x + y + xy = 5. Give your answers

correct to three decimal places .

(Ans : x = 1.618, y = 2.236 , x = 0.618, y = 0.236)


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INDICES N LOGARITME

Solve the equation log2(x+1) = 3.

Answers: log2(x+1) = 3

x + 1 = 23

x =

L1. Solve the equation log2(x 3 ) = 2.

Jawapan:

Ans : x = 7

Solve the equation log10(3x 2) = 1 .

Jawapan: 3x 2 = 10-1

3x 2 = 0.1

3x = 2.1

x = 0.7

L2. Solve the equation log5(4x 1 ) = 1 .

Ans : x = 0.3

Solve the equation log3 (2x 1)+ log24 = 5 .

Ans : x = 14

L6. Solve the equation

log4 (x 2)+ 3log2 8 = 10.

Ans : x = 6

Solve the equation

log2 (x + 5) = log2(x 2) + 3.

Ans : x = 3

L8. Solve the equation

log5 (4x 7) = log5(x 2) + 1.

Ans : x = 3

Solve log3 3(2x + 3)

= 4

Ans : x = 12

L10 . Solve log2 8(7 3x)

= 5

Ans : x = 1


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Given log 2 T - log4 V = 3, express T in terms of

V. [4]

(Ans: T = 8V

)

L8. Given log 4 T + log 2 V = 2, express

T in terms of V. [4]

(Ans: 16V-2

)

Solve 4x

= 7x. [4]

( Ans: x = 1.677 )

L10. Solve 4x

= 9x. [4]

( Ans: x = 2.409 )

Solve the equation 3 3log 9 log (2 1) 1x x + = .

[3]

(Ans: x = 1 )

6. Given that log 2m p= and log 3m r= ,

express227

log16

m

m

in terms ofpand r. [4]

(Ans: 3r 4p +2 )

Solve the equation 5 3 68 32x x += .

[3]

(Ans : x = 3.9 )

8. Given that 5log 2 m= and 5log 3 p= ,

express 5log 2.7 in terms of mandp.[4]

(Ans: 3p m 1 )


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PAPER 1- BAB 1/2/3/5

1 The relation is shown in the following set of ordered pairs.

{ (2, 4), (1, 1), (0, 0), (1, 1), (2, 4) }.

State

(a) the type of relation,

(b) the range of the relation.

[2 marks]

Satu hubungan ditunjukkan oleh set pasangan tertib berikut.

{ (2, 4), (1, 1), (0, 0), (1, 1), (2, 4) }.

Nyatakan

(a) jenis hubungan itu,

(b) julat hubungan itu.

2 The functions gis defined by : , 55

x pg x x

x

+

. Find the value ofpif g

1(3) = 12.

[2 marks]

Fungsi g ditakrifkan oleh : , 55

x pg x x

x

+

. Carikan nilai p jikag

1(3) = 12.

3 Form the quadratic equation which has the roots 3 and3

2. Write your answer in the

form 02 =++ cbxax where a, band care constants.

[2 marks]Bentukkan persamaan kuadratik yang mempunyai punca-punca 3 dan

3

2. Beri jawapan

anda dalam bentuk 02 =++ cbxax dengan keadaan a, b dan c adalah pemalar.


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4 Find the range of values ofxwhich satisfies the inequality (x1)(x+ 5) > 8(x+ 2). [3 marks]

Cari julat nilai x yang memuaskan ketaksamaan (x1)(x+ 5) > 8(x+ 2).

5 Solve the equation 2 13 5x x = . [4 marks]

Selesaikan persamaan2 13 5x x = .

6 Given that 3log 5 =p and 3log 2 = q. Express 3log 75 in terms of pand q. [4 marks]

Diberi 3log 5 = p dan 3log 2 = q. Ungkapkan 3log 75dalam sebutan p dan q.


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GEOMETRY COORDINATE

Eg. 1 Given two points A(2,3) and B(4,7)

Distance of AB =2 2

(4 2) (7 3) +

= 4 16+

= 20 unit.

E1. P(4,5) and Q(3,2)

PQ =

[ 10 ]

E1. The distance between two points A(1, 3) and

B(4, k) is 5. Find the possible vales of k.

7, -1

E2. The distance between two points P(-1, 3)

and Q(k, 9) is 10. Find the possible values of k.

7, -9

Eg. P(3, 2) and Q(5, 7) E1 P(-4, 6) and Q(8, 0)

(2, 3)

Eg1. The point P internally divides the line

segment joining the point M(3,7) and N(6,2) in

the ratio 2 : 1. Find the coordinates of point P.

P =

+

++

+12

)2(2)7(1,12

)6(2)3(1

E1. The point P internally divides the line

segment joining the point M (4,5) and N(-8,-5) in

the ratio

1 : 3. Find the coordinates of point P.

51,

2

1

2

N(6, 2)

M(3, 7)

P(x, y)


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E1. P(0, 1), Q(1, 3) and R(2,7)

Area of PQR =2

1 0 1 2 0

1 3 7 1

=

= 1 unit 2

1. P(2,3), Q(5,6) and R(-4,4)

Area of PQR =

17

2unit

2

1. P(1,5), Q(4,7), R(6,6) and S(3,1).

Area of PQRS =

= 8 unit 2

2. P(2, -1), Q(3,3), R(-1, 5) and S(-4, -1).

[27]

1. Given that the points P(5, 7), Q(4, 3) and

R(-5, k) are collinear, find the value of k.

k= 33

2. Show that the points K(4, 8), L(2, 2) and M(1, -

1) are collinear.

E1. Find the equation of a straight line that

passes through the point (5,2) and has a

gradient of -2.

y = -2x + 12

E2. Find the equation of a straight line that passes

through the point (-8,3) and has a gradient of4

3.

4y = 3x + 36


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P

Q

O x

y

EXERCISES-PAPER 1

1 Diagram shows a straight line PQwith the equation 3x+ 4y 12 = 0. The point Plies on thex-

axis and the point Qlies on they-axis. Find the equation of the straight line perpendicular to PQ

and passes through the point Q. [3 marks]

Rajah 2 menunjukkan garis lurus PQ yang mempunyai persamaan 3x + 4y 12 = 0.

Titik P terletak pada paksi-x dan titik Q terletak pada paksi-y. Carikan persamaan

garis lurus yang berserenjang dengan PQ dan melalui titik Q. [3 markah]

2 Given that P(0, 1), Q(k, s),R(6,1) and S(5, 6) are the vertices of a rhombus PQRS.

Find the values of kand s. [3 marks]

Diberi bahawa P(0, 1), Q(k, s), R(6, 1) dan S(5, 6) ialah bucu-bucu bagi rombus PQRS.

Cari nilai bagi k dan s. [3 markah]


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P

Q

O x

y

EXERCISES-PAPER2-SECTION A

1 Diagram 1 shows a parallelogram PQRSon a Cartesian plane.

(a) Find the value of t.

Hence, state the equation of a straight line PQin the intercept form. [3 marks]

(b) Nis a moving point such that its distance is in the ratioRQ: QN= 2 : 3.

Find the equation of the locus ofN. [2 marks]

(c) Calculate the area of PQRS. [2 marks]

Rajah 1 menunjukkan segi empat selari PQRSpada satah Cartesan.

(a) Cari nilai t.

Seterusnya, nyatakan persamaan garis lurus PQdalam bentuk pintasan. [3 markah]

(b) Nialah titik bergerak di mana jaraknya adalah dalam keadaanRQ: QN= 2 : 3.

Cari persamaan lokusN. [2 markah]

(c) Hitung luas PQRS. [2 markah]


26/70


EXERCISES-PAPER2-SECTION B

1 In Diagram 3, the straight lineABCintersects the line 5y+x+ 35 = 0 at the point C.

(a) Write the equation of AC in intercept form. [1 mark]

(b) Find the coordinates of C. [2 marks]

(c) Given the pointRmoves such that the ratioRA:RC = 1 : 2, find the equation

of the locusR. [3 marks]

(d) Find the points of intersection of the locusRwith they-axis. [2 marks]

(e) Find the area of ACO. [2 marks]

Dalam Rajah 3, garis lurusABCbersilang dengan garis 5y+x+ 35 = 0 pada titik C.

(a) Tuliskan persamaanACdalam bentuk pintasan. [1 markah]

(b) Cari koordinat titik C. [2 markah]

(c) Diberi titikRyang bergerak dengan keadaan nisbah RA:RC= 1 : 2, carikan

persamaan lokusR. [3 markah]

(d) Cari titik persilangan lokusRdengan paksi-y. [2 markah]

(e) Cari luas ACO. [2 markah]

x

y

C

A

B5


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STATISTICS

(a) Find the mode, median and mean for

2, 3, 1, 2, 6, 8, 9, 3, 2, 3.

[Mode = 2, Median = 3, Mean = 4]

(b) Find the mode, median and mean for the data

in the table below.

[Mode = 4, Median = 6, Mean = 7.2]

Score 2 4 6 9 12 13

Frequency 1 3 2 1 2 1

(a) Find the range and the interquartile for

5, 1, 2, 3, 4, 6, 3, 8, 2, 5, 9.

[Range = 8, Interquartile range = 4]

(b) Find the range and the interquartile range for

12, 17, 13, 19, 15, 8, 12, 11.

[Range = 11, Interquartile range = 4.5]

(c) Find the range and the interquartile for

Score 1 4 5 6 8 9Frequency 1 3 1 1 2 1


(d) Find the range and the interquartile range for

Points 2 4 6 8No. of person 3 5 2 2


(a) Find the mean, variance and the standard

deviation for the data below.

5, 12, 6, 3, 6, 10.

[Mean = 7,92= 9.333,9= 3.055]

(b) Find the mean, variance and the standard

deviation for the data below.

18, 12, 16, 11, 19, 18, 12, 14.

[Mean = 15,92= 7.5,9= 2.739]


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EXERCISES-PAPER 1

1 Table 1 shows the distribution of the scores acquired by 40 students in a quiz competition.

Determine the inter-quartile range of the distribution.

[3 marks]

Markah 0 1 2 3 4

Bilangan pelajar 2 9 16 12 1

JADUAL 1

Jadual 1 menunjukkan taburan markah yang diperolehi 40 pelajar dalam suatu pertandingan

kuiz. Tentukan julat antara kuartil bagi taburan tersebut. [3 markah]

2 The marks for students who sat for a test is shown in Table 1.

Marks 1 20 21 40 41 60 61 80 81 100

Number of students 4 k 12 9 5

TABLE 1

If the median of the marks is 505, find the value of k. [4 marks]

Jadual 1 menunjukkan markah pelajar-pelajar yang menduduki suatu ujian.

Marks 1 20 21 40 41 60 61 80 81 100

Number of students 4 k 12 9 5

JADUAL 1

Jika markah median ialah 505, cari nilai k. [4 markah]


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EXERCISES-PAPER2-SECTION A

3 (a) The mean and standard deviation of a set of numbers 1 2 3, , , ..., nx x x x are 20 and 7

respectively. Calculate the mean and standard deviation of the following set of data :

1 2 35 2, 5 2, 5 2,..., 5 2.nx x x x [4marks]

Min dan sisihan piawai bagi satu set nombor 1 2 3, , ,..., nx x x x adalah masing-masing

20 dan 7. Hitungkan min dan sisihan piawai bagi set data berikut :

1 2 35 2,5 2,5 2,...,5 2.nx x x x [4markah]

Marks Number of students

0 19 5

20 39 8

40 59 11

60 79 10

80 99 6

TABLE 1

(b) Table 1 shows the distribution of Chemistry marks obtained by 40 students.

Calculate the median of the distribution. [3marks]

Jadual 1 menunjukkan taburan markah Kimia yang diperolehi 40 orang pelajar.

Hitungkan median bagi taburan itu. [3markah]


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CIRCULAR MEASURE

Example: 25.8o

Since 180o= rad

rad4504.0

1808.258.25

=

=

(a) 30o=

[0.5237 rad]

(b) 1.54 rad =

[88.22o]

(c) =rad5

18

[206.24o]

Example:

r= 10 cm and = 1.5 rad

Since S= r

= 10 1.5

= 15 cm

(a) r= 20 cm and = 0.6 rad

[12 cm]

Example:r= 3 cm and = 1.8 rad

Since 2

2

1rA =

8.132

1 2=

= 8.1 cm2

(a) r= 5 cm and = 0.8 rad

[10 cm2]

(c)

[Perimeter = 10.19 cm, Area = 1.145 cm2]

(d)

[Perimeter = 8.599 cm, Area = 4.827 cm2]

25.8o

O

A

B

O

P

Q

1.8 rad

3cm

3cm

O

P

Q

3.8

cm

3.8cm

1 rad

OA

B

4.2cm

4.2cm

60o


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P O R

Q

EXERCISES

1 Diagram 1 shows a semicircle OPQRwith centre O. Given that the length of arc PQis

64 cm, calculate the area of the sector OQR.

(Use = 3142)[3 marks]

Rajah 1 menunjukkan sebuah semibulatan OPQR berpusat O. Diberi panjang lengkok PQ

ialah 64 cm. Hitungkan luas sektor OQR.(Gunakan = 3142)

[3markah]

DIAGRAM 2 / RAJAH 2

2 Diagram 2 shows a sector OABof a circle with centre O and radius r. If the radius is

35 cm and the perimeter of sector OABis 122 cm, find AOB in radian.

[3 marks]

Rajah 2 menunjukkan sektor OAB bagi bulatan dengan pusat O dan jejari r. Jika jejarinya

ialah 35 cm dan perimeter sektor OAB ialah 122 cm, cari AOB dalam radian.

[3 markah]

O

A

B


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3 The diagram above shows a circle of radius of 6 cm and the arcEFwhich subtends an angle of 60

at the centre, O. Calculate the perimeter of the shaded segment.

[3 marks]

Rajah di atas menunjukkan bulatan berjejari6 cm dan lengkokEFmencangkup sudut60pada pusatO. Hitungkan perimeter kawasan yang berlorek.

Diagram 3 / Rajah 3

4 Diagram 3 shows an equilateral triangle PQR and a sector PTS with centre P. Given that

QR = 18 cm and the area of the shaded region is 225 cm2. Find the radius PT. [3 marks]

Rajah3 menunjukkan segitiga sama PQR dan sektor PTS berpusatP.Diberi QR = 18 cm dan luas

kawasan berlorek ialah225 cm2. Cari panjang jejariPT.

E

F

O

60

R

P

Q

ST


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5 Diagram 4 shows a circle with centre Oand radius 10 cm. PTQ is a tangent to the circle

at T. PSO and QRO are straight lines. Given that OP = OQ and S is a mid-point of OP.

Calculate

Rajah4 menunjukkan bulatan berpusatOdengan jejari10 cm. PTQ ialah tangen kepada bulatan

itu di T. PSO dan QRO adalah garis lurus.Diberi bahawa OP = OQ dan S adalah titik

tengah OP. Hitungkan

(a) SOT, [2 marks]

(b) the length of the minor arc ST, [2 marks]panjang lengkok minorST,

(c) the area of the minor sector OST, [2 marks]

luas sektor minorOST,

(d) the area of the shaded region. [4 marks]

luas kawasan berlorek.

[Use / Gunakan = 3142 ]

S

O

P T Q

R

Diagram 4 / Rajah 4


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INDEX NUMBER

Notes :

a. Price Index,I= 1

0

100P

P , P1= price at a specific time, P0= price at the base year.

b. Composite Index ,IW

I

W

=

,I =price index, W= weightage

1. Table below shows the price indices and percentage of usage of four main

ingredients ,P,Q,Rand S, in the production of a type of cake.

Ingredients

Price index for the

year 2012

(2010=100)

Percentage of

usage

P m 20

Q 105 30

R 108 10

S 120 40

(a) Calculate

(i) the price of ingredient Qin the year 2010 if its price in the year 2012 is RM 50.00,

(ii) the price index ofRin the year 2012, based on the year 2008, given that its price

index in the year 2010, based on the year 2008 is 110.

(b) The composite index number of the cost of production of this type of cake in the year

2012,based on the year 2010 is 112.8. Calculate.

(i) the value of m,

(ii) the cost of these ingredients for the production of this type of cake in

the year 2012 if the corresponding cost in year 2010 is RM60.00.

Guided Solutions

a (i)

,

(ii)( ) ( )

( )1 0 0

=

b (i)

(ii)

5 0( ) 1 0 5 , ( ) = =x

x

( )( ) 105 ( 3 0 ) 1 08 (1 0 ) 120 ( 40 )( ) , ( )

10 0

+ + += =

mm

2012100 ( ) , ( )2012

60 = =

QQ


35/70


2. The bar chart in Diagram shows the monthly cost of ingredients P, Q,Rand Sfor making biscuit

in year 2009. Table shows the prices and the price indices used to make the biscuit.

a) Find the values ofxandy.

b) Calculate the composite index for the cost of making the biscuit in year 2011 based on the

year 2009.

c) If the price for making biscuit in 2009 is RM 1200, find the corresponding price in year 2011.

d) The cost of these ingredients increases by 15% from the year 2011 to 2012. Find the

composite index for the year 2012 based on 2009.

Ingredient Price in 2009 Price in 2011 Price index in 2011

(2009=100)

P 0.90 1.35 150

Q x 3.00 120

R 3.20 y 125

S 1.25 1.75 140

15

20

25

40

P Q R S

Monthly cost (RM)


36/70


1 Table 3 shows the price indices and percentage of usage of four main ingredients, P,

Q,Rand S, in the production of a type of cake.

Jadual 3 menunjukkan indeks harga dan peratus penggunaan empat bahan utama, P, Q,R

dan Sdalam penghasilan sejenis kek.

Ingredients /

Bahan

Price index for the year 2006Indeks harga pada tahun 2006

(2003 = 100)

Percentage of usage /

Peratus penggunaan

P m 20

Q 105 30

R 108 10

S 120 40

Table 3 / Jadual 3

(a) Calculate

Hitungkan

(i) the price of ingredient Qin the year 2003 if its price in the year 2006 is RM 5000,

harga bahanQpada tahun2003jika harganya pada tahun2006 ialahRM 5000,

(ii) the price index ofRin the year 2006, based on the year 2000, given that its price

index in the year 2003, based on the year 2000 is 110.indeks hargaRpada tahun2006, berasaskan tahun2000jika indeks harganya

pada tahun2003, berasaskan tahun2000 ialah 110.

(b) The composite index number of the cost of production of this type of cake in the year

2006, based on the year 2003 is 1128.

Calculate

Nombor indeks gubahan bagi kos bahan-bahan untuk menghasilkan kek tersebut pada

tahun2006 berasaskan tahun2003 ialah1128 .

Hitungkan

(i) the value of m,

nilaim,

(ii) the cost of these ingredients for the production of this type of cake in the year 2006

if the corresponding cost in year 2003 is RM 6000.

kos bahan-bahan untuk menghasilkan kek itu pada tahun2006jika kos yang

sepadan pada tahun 2003 ialahRM6000.


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Expenditure /

Barangan Keperluan

Price index /

Indeks Harga

Weightage /

Pemberat

Shoe / kasut

Bag / beg

Shirt / bajuTrouse / seluar

125

120

n108

m

2

7 m4

TABLE 3 / JADUAL 3

2 Table 3 showsthe price indices and the weightages for 4 types of goods for En. Ahmad in the

year 2004 based on the year 2002.

Jadual3 menunjukkan indeks harga dan pemberat untuk empat jenis barangan keperluan tahunan

Encik Ahmad bagi tahun2004 berdasarkan tahun2002 sebagai tahun asas.

(a) Calculate the value of nif the expenditure for a shirt in the year 2002 is RM 35 and

increases to RM 3920 in the year 2004

Hitungkan nilainjika harga sehelai baju pada tahun2002 ialahRM35 dan meningkat

menjadiRM 3920pada tahun2004. [2 markah]

(b) The composite index number for the expenditures in the year 2004 based on the year 2002

is 115. Calculate the value of m.

Nombor indeks gubahan barangan keperluan itu pada tahun2004 berasaskan tahun2002

ialah115.Hitungkan nilai m. [3 markah]

(c) Calculate En. Ahmads total yearly expenses for the expenditures in the year 2002 if the

corresponding expenses for the year 2004 is RM1380.

Hitung jumlah perbelanjaan tahunan Encik Ahmad untuk barangan keperluan tersebut

pada tahun2002jika perbelanjaan yang sepadan pada tahun2004 ialah RM 1380.[2 markah]

(d) The expenses increases 25% from the year 2004 to 2006. Find the composite index number

in the year 2006 based on the year 2002.

Kos barangan itu meningkat25% dari tahun2004 ke tahun2006.

Cari nombor indeks gubahan tahun2006 berasaskan tahun2002.

[3 markah]


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SOLUTION OF TRIANGLES

(1) Diagram 1 shows the triangle ABC.

Calculate the length of BC.

Answer :

8.2

( ) ( )

BC=

( ) ( )( )

BC =

Using the scientific calculator,

BC= ( )

(2) Diagram 2 shows the triangle PQR

Calculate the length of PQ.

( ) ( )

( ) ( )=

[ 8.794 cm ]

(3) Diagram 3 shows the triangle DEF.

Calculate the length of DE.

( ) ( )( ) ( )

=

[ 10.00 cm ]

(4) Diagram 4 shows the triangle KLM.

Calculate the length of KM.

( ) ( )

( ) ( )=

[ 11.26 cm ]

Diagram2

D

E F

600 35

016

15 cm

Diagram 3

LK

M

420

630

15 cm

Diagram 4

Diagram 1


39/70


(1) Diagram 1 shows the triangle ABC.

Find ACB.

( ) ( )

( ) ( )=

(2) Diagram 2 shows the triangle KLM

Find KLM

( ) ( )

( ) ( )=

[ 27.360]

(3) Diagram 3 shows the triangle DEF.

Find DFE.

( ) ( )

( ) ( )=

[ 11.090]

(4) Diagram 4 shows the triangle PQR.

Find QPR.

( ) ( )

( ) ( )=

[ 36.110]

D

E F

3.5 cm

43024

12.5 cm

Diagram 3

LK

M

9 cm 500

15 cm

Diagram 2

RP

Q

10 cm

1300

13 cm

Diagram 4

A

B C60

0

15 cm

Diagram 1

10 cm


40/70


a = b + c 2bc cos A

b2= a

2+ c

2 2ac cos B

c2= a

2+ b

2 2ab cos C

SOLUTION OF TRIANGLES

2.2 Use Cosine Rule to find the unknown sides or angles of a triangle.

(1)

Find the value ofx.

2 2 2( ) ( ) 2( )( )cos( )x = +

x=

(2)

Find the value ofx.

2 2 2( ) ( ) 2( )( )cos( )x = +

[ 7.475 ]

(3)

Find the value of x.

2 2 2( ) ( ) 2( )( )cos( )x = +

[ 9.946 ]

cm

E13 cm

430

5 cm

Diagram

12.3

P

cm

Q R

670

x cm

7PQ

R

750

5

bc

acbA

2cos

222+

=

ac

bcaB

2

cos222

+=

ab

cbaC

2cos

222+

=

Diagram

12.3

P

16.4 cm

cm

Q R67

0


41/70


(1) In

Find BAC .

2 2 215 ( ) ( ) 2( )( )cos BAC= +

)14)(13(2

151413cos

222+

=BAC

(2)

Find BAC .

2 2 2( ) ( ) ( ) 2( )( )cos BAC= +

[ 83.17]

(3)

Calculate BAC

2 2 2( ) ( ) ( ) 2( )( )cos BAC= +

[ 73.41]

(4)

Calculate BCA

2 2 2( ) ( ) ( ) 2( )( )cos BAC= +

[39.17]

5)

Find RQP .

2 2 2( ) ( ) ( ) 2( )( )cos BAC= +

[108.07]

C

A

B

13cm 14 cm

15 cm

Diagram 1

C

A

B

11cm 13 cm

16 cm

Diagram 2

C

A

B

13cm 16 cm

17.5 cmDiagram 3

A

B

12.67cm 16.78 cm

19.97 cm

Diagram 4

C

2.23 cm6.45 cm

5.40 cmP

Q

R


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1 (a) In Diagram 6, PQRand QRS are two triangles. Given that QPR= 65, PQ= QR= 5 cm

and SR= 4 cm, calculate

Dalam Rajah 6, PQR dan QRS ialah dua buah segi tiga. Diberi QPR = 65, PQ = QR = 5cm

dan SR = 4 cm, hitungkan

(i) the length of SQ,

panjangSQ,

(ii) QSR,

(iii) the area of PQS.

luas PQS.

[8 marks]

(b) A triangle QRShas the same measurement the triangle QRSas in the Diagram 6, that is

QR= 5 cm, RS= 4 cm and RQS = RQS, but which is different in shape to triangle

QRS.

Sebuah segi tiga QRS mempunyai ukuran-ukuran yang sama sebagaimana segi tiga

QRS seperti dalam Rajah 6, iaitu QR = 5 cm, RS = 4 cmdan RQS = RQS,

tetapi bentuk yang berbeza daripada segi tiga QRS.

(i) Sketch the triangle QRS,

Lakarkan segi tigaQRS,

(ii) State the size of QSR.

Nyatakan saizQSR.

[2 marks]

P

Q

RS

5 cm

4 cm

65


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PROGRESSION

(a) State the common difference of each of the following arithmetic progressions

1. 2, 6, 10, 14,

[4]

2. 21, 18, 15, 12,

[3]

(b) Find the tenth term and the twentieth term of the following arithmetic progressions

1. 2, 6, 10, 14,

[38 ; 78]

2. 21, 18, 15, 12,

[-6 ; -36]

(c) Calculate the number of terms in each of the following arithmetic progressions

1. 2, 6, 10, , 82

[21]

2. 21, 18, 15, , -66.

[30]

(a) Find the sum of the first 20 terms of each of the following arithmetic progressions

1. 2, 6, 10, 14,

[800]

2. 21, 18, 15, 12,

[-150]

(b) Find the sum of the following arithmetic progressions

1. 2, 6, 10, 14, , 54

[1456]

2. 21, 18, 15, 12, , -30

[-81]

(c) Sum of a specific number of consecutive terms

1. Given an arithmetic progression 2, 6, 10, 14,

find the sum from fifth term to the sixteenth

term.

[504]

2. Given an arithmetic progression21, 18, 15, 12, find the sum from seventh

term to the eighteenth term.

[-180]

(d) (i) Find the sum to infinity of each of the following geometric progressions

1. 8, 4, 2, 1,

[16]

2. 27, 9, 3, 1,

[40.5]


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PAPER 1

1 The first three terms of an arithmetic progression are 2k3, 4k+ 1 and 5k+ 6.

Find

(a) the value of k, [2 marks]

(b) the eighth term. [1 mark]

Tiga sebutan pertama suatu janjang aritmetik adalah 2k 3, 4k + 1 dan 5k + 6.

Carikan

(a) nilai k, [2 markah]

(b) sebutan ke lapan. [1 markah]

2 The sum of the first nterms of the geometric progression 128, 64, 32, 16 . is 255875.

Find

(a) the common ratio of the progression, [1 mark]

(b) the value of n. [3 marks]

Hasil tambah n sebutan pertama suatu janjang geometri 128, 64, 32, 16 ialah

255875. Carikan

(a) nisbah sepunya, [1 markah]

(b) nilai n. [3 markah]

3 Given that 25, 22, 19,. are the first three terms in an arithmetic progression, find the

sixteenth term. [2 marks]

Diberi 25, 22, 19,. adalah tiga sebutan pertama bagi suatu janjang aritmetik, cari sebutan

keenam belas. [2 markah]

4 A geometric progression has a first term of 27 and a sixth term of9

1. Find

(a) the common ratio, [2marks]

(b) the sum to infinity. [2marks]

Sebutan pertama suatu janjang geometri ialah 27 dan sebutan keenam ialah9

1.

Cari

(a) nisbah sepunya, [2markah]

(b) hasil tambah sehingga ketakterhinggaan. [2markah]


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PAPER 2-SECTION A

1 Diagram 1 shows several sectors of circles, centre O, which are the results of electromagnetic

wave emission from the radio station at O.

Rajah 1menunjukkan beberapa sektor bulatan berpusat di O, yang merupakan hasil

pancaran gelombang elektromagnet dari sebuah stesen radio di O.

The radius of each sector increases by 3 cm compared to that of the previous sector.

Given that the area of the nth

sector is 512cm2, the radius of the first sector, OA = 10 cm dan

'4

AOA

= rad. Find

Panjang jejari sektor bulatan bertambah sebanyak 3 cmdaripada bulatan sebelumnya. Diberi

luas sektor bulatan ke-n ialah 512cm2, jejari sektor bulatan awal, OA = 10 cm dan

'

4AOA

= rad. Cari

(a) the radius of the nth

sector,

panjang jejari sektor bulatan ke-n, [2 marks]

(b) the value of n

nilai n, [3 marks]

(c) the sum of the arc lengths of the first 20 sectors.

hasil tambah 20panjang lengkok pertama gelombang tersebut. [2 marks]

rad4 O

A

CB

D

AB

CD

10 cm

3 cm

3 cm

Diagram 1 / Rajah 1


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LINEAR LAW

1.1 Draw the line of best fit

1 2

1.2 Write the equation for the line of best fit of the following graphs.

1

[ 53

3y x= +

]

2

[ 5 52

y x= + ]

P(0,5)

Q(2,0)

x

.

.

X

X

X

X

X

y

x2

y

x

X

X

X

X

y

x

P(0,3)

Q(6,13)


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2.2 Determine values of constants of non-linear relations given lines of best fit

1 The diagram below shows the line of best fit

for the graph of y2against x. Determine the

non-linear equation connecting y and x.

[y2=-2x+4]

2 The diagram below shows the line of best fit for the

graph of y2against

x

1 . Determine the non-linear

equation connecting y and x.

[ 21

5 2yx

= +

]


for the graph of 2x

y

against x. Determine the

non-linear equation connecting y and x.

[ 2 4 12y xx= ]

4 The diagram below shows the line of best fit for the

graph of 2

y

x against x. Determine the non-linear

equation connecting y and x.

[2

1 32

y xx

= + ]

P(0,4)

Q(2,0)

y

x

X

X P(0,2)

Q(2,12)y

x

X

P(3,0)

Q(6,12)2x

y

x

X

X

(2,4)

2x

y

(4, 5)

x0


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5 The diagram below shows the line wheny

x

against x is drawn. Determine the non-linear

equation connecting y and x

[3+

=x

xy ]


for the graph of log10y against x. Determine

the relation between y and x.

[ y = 103x

]

7 The diagram below shows part the graph oflog10y against x. Form the equation that

connecting y and x.

[ 16410 += xy ]

8 The diagram below shows the line of best fitfor the graph of log10y against log10x.

Determine the relation between y and x.

[y= 100x2]

9 The diagram below shows part the graph of

log10y against log10x. Form the equation that

connecting y and x.

[1000

xy= ]

10 The diagram below shows part the graph of

log 2y against log2x. Determine the relation

between y and x.

[16

2

xy= ]

(3,6)

(0,3)

y

x

x0(0,0)

(2,6)y10log

x

X

X

log10y

(3,4)

(4,0) x

(0,2)

(2,6)

y10log

x10og

X

X

6

-3

log10x

log10y

2

(5,6)

log2x

log2y

0


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1. Use graph paper to answer this question.

The table below records the values of an experiment for two variables x and y which are related

byx

qpxy +=

2 where p and q are constants.

x 0.8 1 1.3 1.4 1.5 1.7

y 108.75 79 45.38 36.5 26.67 8.19

(a) Plot xy against x3using scale 2 cm represents 1 unit in x-axis and 2 cm represents

10 units for y-axis.

Hence, draw the line of best fit

[5marks]

(b) From the graph, estimate the value of

(i) p and q

(ii) x when y=x

45 [5marks]

[Answer:p=-16.67, q=95, x=1.458]



by kxx

p

x

y+= where p and k are constants.

x 3 5 6 7 8 9

y 4.7 4.0 3.6 3.0 2.5 1.8

(a)Plot the graph y against x2 [4 marks]

(b)use the graph to estimate the values of

(i) p

(ii) k.

(iii) x which satisfy the simultaneous equation kxx

p

x

y+= and y = 2 [6 marks]

[answer: p=5, k= -0.04, x= 8.60 - 8.75]


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by y=p qx where p and q are constants.

x 3 4 5 6 7

y 5 10 20 40 80

(a) Plot the graph log 10y against x [4 marks]

(b)Use the graph to estimate the values of

(i) p

(ii) q.

(iii) ywhenx=4.8 [6marks]

[answer: 1.995, 0.6166, 17.38]

4. SPM 2003 Paper 2 Question 7

Use graph paper to answer this question.

Table below shows the value of two variables, x and y, obtained from an experiment. It is known that x

and y are related by the equation2

xy pk= ,wherep and kare constants

x 1.5 2.0 2.5 3.0 3.5 4.0

y 1.59 1.86 2.40 3.17 4.36 6.76

(a)Plot log10y against x2

Hence, draw the line of best fit. [5 marks]

(b)Use the graph in (a) to find the value of

(i) p

(ii) k [5 marks]

[Answer: p=1.259 , k =1.109 ]


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VECTOR

Task 1 : Express the following vectors in terms of x%

and y%

.

(1)

ABCD is a parallelogram.

ACuuur

=

(2)

QRuuur

=

(3)

TQuuur

=

PRuuur

=

(4)

EFGH is a parallelogram.

EG

uuur

=

(5)

FGuuur

=

(6)

BC

uuur

=

ADuuur

=

(1) PQuuur

=

6

8

magnitude of PQuuur

PQuuur

=

Unit vector in the direction

of PQuuur

PQuuur

=

10 ;

4

535

(2) STuuur

=

8

15

magnitude of STuuur

STuuur

=


of STuuur

STuuur

=

17 ;1517

817

(3) ) CDuuur

=

8

6

magnitude of CDuuur

CDuuur

=


of CDuuur

CDuuur

=

10 ;35

45


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VECTORS

3.4 3.7 Addition, Subtraction and Multiplication of Vectors

Task 1 : Given 2 5a i j= +% % %

, 4b i j= % % %

and 3 7c i j= +% % %

, find the following vectors in terms of i%

and j%

.

(1) 2b%

=

2 8i j% %

(2) 3a%

=

6 15i j+% %

(3) 4c%

=

12 28i j +% %

(4) 12

a

%

=

52

i j+

% %

(5) 3a b+

% %

=

5 7i j% %

(6) 2a b+

% %

=

5 6i j+% %

(7) b c+% %

=

2 3i j % %

(8) 12b a+% %

=

52

3i j+% %

(9) 3a b% %

=

17i j +% %

(10) 4b c% %

=

7 23i j% %

(11) 3 2c a% %

=

13 11i j +% %

(12) 12

b a% %

=

132

j%


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Task 2 : Given3

4a

=

%,

2

5b

=

%and

6

1c

=

%, express the following in the form of

y

x.

(1) 4b%

=

8

20

(2) 2c%

=

12

2

(3) 12

a%

=

32

2

(4) 2a b+% %

=

4

13

(5) 3b c+% %

=

16

2

(6) 2c a+% %

=

15

2

(7) 3a b% %

=

11

7

(8) 2b c% %

=

10

11

(9) 2c a% %

=

0

9

(10) 3 2c b% %

=

2213

(11) 3b a+% %

=

3

19

(12) 4a c% %

=

617


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Example:

Given that a%

= 13i+jand b%

= 7i kj, find

(a) a%

b%

in the formxi +yj,

(b) the value of kif | a%

b%

| = 10.

Solution:

(a) a%

b%

=

(b) | a%

b%

| = 10

2 26 (1 ) 10k+ + =

1. Given that b%

= 3i 2j, c%

= 5i+ mj, where m is a constant. Find the value of m.

(a) if b%

c%

is parallel to b%

(b) | 2 b%

c%

| = 125

2. Given u= 2i+ 3j dan v= 2i+ kj, find the values of k if 2 +u v = 10.

[3 marks]

Diberi u= 2i+ 3j dan v= 2i+ kj, cari nilai-nilai k jika 2 +u v = 10.

[3 markah]


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1. SPM 2003 P1 Q12

Diagram shows two vectors, OP andQOuuur uuur

.

Express

(a) OPuuur

in the formx

y

(b) OQuuur

in the form xi yj+% %

[2marks]

(5

3

, -8i+4j)

2. SPM2003 P1Q13

Use the information given to find the values of h

and kwhen r= 3p-2q

[3marks]

( h= 2 , k= 13)

3. SPM 2004 P1 Q16Given that O(0,0)A(-3,4) andB(2,16), find in terms

of unit vectors, i and j% %,

(a) ABuuur

(b) the unit vector in the direction ofABuuur

[4marks]

(a)

12

5(b)

12

5

13

1

4. SPM 2004 P1 Q17Given thatA(-2,6),B(4,2) and C(m,p), find the

value of mand ofpsuch that2 10 12AB BC i j+ =

uuur uuur

% %

[4 marks]

m= 6 ,p= 2

p =2a +3 b

q = 4a b

r = ha + (h-k) b, where h and k are

constants


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1 In Diagram 2,APC and BPD are straight lines.

DIAGRAM 2

Given that AB

= 2x, AD

= y and BC

= 3AD

.

(a) Express in terms of x and/or y ,

(i) AC

,

(ii) BD

.

[2 marks]

(b) Given that AP

= m AC

and BP

= n BD

.

Express AP

(i) in terms of m, x and y.

(ii) in terms of n, x and y.

Hence, show that m+ n = 1. [5 marks]

(c) If AQ

=4

3x y, prove thatAC and QB are parallel. [3 marks]

D

A B

C

P

Q


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DIAGRAM 6

2 In Diagram 6, OP

= p%

, OQ

= q%

, PR

= 2q%

. Sis the mid point of OPand QU

=1

2UR

.

Given that SUand OR intersects at T.

(a) Express in terms of p%

and / or q%

,

(i) OR

(ii) QR

(iii) QU

[3marks]

(b) Given that ST

= h SU

and OT

= kOR

. State OT

(i) in terms of h, p%

and q%

.

(ii) in terms of k, p%

and q%

.

[4marks]

(c) Hence, find the values of hand k.

[3marks]

R

P

Q

O

T

S

U


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TRIGONOMETRIC FUNCTIONS

5.2 Six Trigonometric Functions of any Angle (1)

1. Given that px=sin and 00


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Solve each of the following trigonometric equations for 00


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5.3 Graphs of Trigonometric Functions- (2) Graphs of cosine

Sketching the graphs of each of the following trigonometric functions for 20 x .

(a). xy cos= (b). xy cos2=

(c). xy cos3= (d). xy cos

2

1=

(e) xy cos= (f) 1cos += xy

(g) 1cos2 = xy (h) 1cos2 += xy

(i) xy cos= (j) 1cos2 += xy

(k) xy 2cos= (l) 12cos += xy


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Notes :

Using :y =asinpx or y =acospx or y =atanpx

STEPS TO SKETCH GRAPH

1. Shape of graph : look at basic sin, cos and tan graph.

2. Constant of angle : periodic/cycle (repeat) in 360or 2.

3. Constant of trigonometric equation : amplitude (maximum/minimum).

i. Negative shape : upside down/reverse shape.

ii. Shifting upward and downward.

iii. Modulus/absolute sign

4. Find a suitable lineby solving simultaneous equations using substitution method.5. Draw the straight line.

6. State the number of solutions.

GRAPHS OF TRIGONOMETRIC FUNCTIONS

1. (a) Sketch the graph of y= 3 sin 2xfor 0 x2.

(b) Hence, using the same axes, sketch a suitable straight line to find the number of solutions to the

equation 2x+ 3sin 2x= 0 for 0 x2.

2. (a) Sketch the graph ofy= |cos 3x|for 0 x.

(b) Hence, using the same axes, sketch a suitable straight line to solve the equation

x2|cos 3x|= 0.

Hence, state the number of solutions to the equation for 0 x.

ANSWERS

1 b) 3 solutions 2. b) 6 solutions

Period/cycle in 360

ShapeAmplitude

1

y

0

Cosinus graph

0

y

x


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Paper 1

1. Given that t=tan ,00

900


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2. (a) Sketch the graphy= cos 2xfor 00 1800 x . [ 3 marks ]

(b) Hence, by drawing a suitable straight line on the same axes, find the number of solutions

satisfying the equation180

-2sin22 x

x = for 01800 xo .

[ 3 marks ]

( SPM P2 No. 3 )

3. (a) Prove that cosec

2

x 2 sin

2

x cot

2

x= cos 2x. [ 2 marks ]

(b) (i) Sketch that graph of y = cos 2xfor 2x0 .

(ii) Hence, using the same axes, draw a suitable straight line to find the number of

solutions to the equation 3( cosec2

x 2 sin2x cot

2x) = 1-

xfor 20 x .

State the number of solutions. [ 6 marks ]

(SPM P2 No.5)

4. (a) Sketch the graph y = -2cos x for 20 x . [ 4 marks ]

(b) Hence, using the same axis, sketch a suitable graph to find the number of solutions

to the equation 0cos2 =+ xx

for 20 x . State the number of solutions.

[ 3 marks ]( SPM P2 No. 4 )


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INTEGRATION

1. 43x dx 2. 523

x dx

3.6

2

3dx

x 4.

4

7dx

x

5. 3(2 3)x dx 6. ( )42 5 43

x dx

4.

2

1

8x dx =

=

[12]

5.4

3

2

x dx =

=

[60]

13.

2

1

(2 1)(2 1)x x dx +

=

[ 253

]

14.

3

2

1

(3 2)x dx

=

[38]


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7.

2

2

1

3x dx =

=

[ 32

]

8.

3

3

1

2( ) dxx

=

[ 8

9]

10.

3

0

(2 6 )x dx =

[-21]

11.

3

2

1

(4 3 )x x dx =

[-10]

y = 3x + 2

pintasan y=

pintasan x=

x dalam sebutan y =

y2dalam sebutan x =

y = 3x23

pintasan y=

pintasan x=

x2dalam sebutan y =

y2dalam sebutan x =

y = 3x + 2 , y = 3x 3

cari titik persilanganantara garis dan lengkung di atas


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Notes :

a. To integrate, use : nax dx =1

1

nax

n

+

++ c

b. Gradient function, m =dy

dx

c.dy

ydx

=

d. Area under the curve :

(a) atx-axis, Ax=b

a

y dx (b) aty-axis, Ay=b

a

x dy

e. Volume of revolution :

(a) aboutx-axis, Vx= 2

b

a

y dx (b) abouty-axis, Vy= 2

b

a

x dy

PAPER 1

1.5

4

(2 3)dx

x ,

Answer :

2. 2 (4 1)x x dx

Answer :

3. Given that3

1( )g x dx = 6, find

a) the value of3

1

( )

2dx

g x ,

b) the value of1

35 ( ) dxg x ,

c) the value of ksuch that3

1[ ( ) ]g x k dx+ = 10.


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Guided Solutions

a)3

1

( )

2dx

g x

=3

1

( )1

2g x dx

=1

( )2

= ( )

c)3 3

1 1( )g x dx k dx+ = 10

( ) + [ ]3

1kx

= 10

( ) ( ) = 4

k= ( )

4. Given that5

2( )f x dx = 9, find

a) the value of5

2

2 ( )

3dx

f x ,

b) the value of2

54 ( )f x dx ,

c) the value of ksuch that5

2[ ( ) ]f x kx dx+ = 30.

b)1

35 ( )g x dx

5( ) = ( )


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5. Diagram shows the curvey = 5x4and the straight linex=p.

If the area of the shaded region is 32 unit2, find the value ofp.

Guided Solutions

Area =b

ay dx =

4

0

5p

x dx= ( )

[ ] 0p

= ( )

( ) ( ) = 32

p = ( )

6. Diagram shows the shaded region bounded byy-axis, the curvey2= 4xand

a straight liney= k.

Given that the area of the shaded region is9

4unit

2, find the value of k.

Answer :

y2= 4x

y= k

0 x

y

y= 5x4

x=p

0 x

y


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PAPER 2

7. Diagram shows the straight line PQis normal to the curve2

13

xy= + atM(3, 4). The straight line

MNis parallel toy-axis.

Find

a) the value of h,

b) the area of the shaded region,

c) the volume of revolution, in terms of , when the region bounded by the curve, the

y-axis and straight liney= 4 is rotated through 360abouty-axis.

Guided Solutions

a) m1 =

dy

dx =

2

3

x

=

2( )

3 = ( )

m2 =1

1

m

= ( ) =

4 0

3 h

(m2 is gradient of PQ)

h = ( )

b) Area of regionA =23

0

( 1)3

xdx+ =

Area of regionB =

Hence, the area of the shaded region =A+B=

c) The volume of revolution = 4

2

1

x dy = 4

1

( ) dy =

.

BA

N Q(h, 0)Ox

y

y=

2

13

x +

M (3, 4)

P


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AO

B

y

P

C

xk

Q

32 +=xy

9=+xy

8. Diagram shows the curve 32 +=xy intersects the straight lineAC at point B.

It is given that the equation of straight lineACisy + x = 9and the gradient of the curve at pointBis

4. Find

a) the value of k,

b) the area of the shaded region P,

c) the volume of revolution, in terms of , when the shaded region Q is rotated through

360 about the xaxis .

9. Diagram shows the curve 22 1y x= + intersects the straight line 9 9y x= at point (2, k).

Find,

(2,k)

y= 2x2+ 1

y= 9x9

x

Modul Syed Malek _repaired

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