modul map 2010 item pack

7/28/2019 Modul MAP 2010 Item Pack

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MINIMUM ADEQUATE PRACTICE (MAP) ADDITIONAL MATHEMATICS SPM

1. FUNCTIONS

"IMPORTANT NOTES AND FORMULAE

RELATIONS

Relation is a connection between two sets.

The relation between two sets can be represented by :

(a) An arrow diagram

(b) Ordered pairs

{(2, 1), (3, 2), (5, 4)}

(c) A graph

subtract1from SetBSetA

1

2

4

2

3

5

0 1 2 3 4 5

2

3

1

4

SetA

SetB

Hak Cipta Cayzec Montoya SMK Sultan Sulaiman Kuala Terengganu 1


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Domain, codomain, object, image and range of a relation

The arrow diagram shows the relationsquares ofbetween setA and set B.

1 14

16

2

4 25

squares of

SetA SetB

Domain = {1, 2, 4}

Codomain = {1, 4. 16, 25}1, 2 and 4 are objects

1, 4, 16 are images

Range = {1, 4, 16}

Types of relations

(a) One-to-one relation

one half of SetBSetA

2

4

5

4

8

10

(b) Many-to-one relation

factor of SetBSetA

3

7

6

12

14

(c) One-to-many relation

multiple of SetBSetA

3

5

6

9

10



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(d) Many-to-many relation

prime factor of SetBSetA

6

9

2

3

FUNCTIONS

A function is a type of relation where each object in the domain has only one imagein the codomain.

A one-to-one relation and many-to-one relation can be a function.

Function notation


f SetBSetA

x y

x is the objecty is the image ofx underf.

In function notation,y is expressed in terms ofx.

Example. f:x 2x 1orf(x) = 2x 1

COMPOSITE FUNCTIONS

x y z

A B Cf g

gf

f(x) =y

g(y) =z

g[f(x)] =z


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INVERSE FUNCTIONS

o For a function f:xy, the inverse function is denoted asf 1 :yx.

x y

A B

f

f1

For example : Given thatf:x 2x + 1, findf 1(x).

Solution : Method 1

Let y = 2x + 1

2x =y 1

x =1

2

y

f 1(x) =1

2

x

Method 2

Let f 1(x) = a

f(a) =x

Since f(x) = 2x + 1

f(a) = 2a + 1 =x

a =1

2

x



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HHPAPER 1JJ

1. Diagram 1 shows the relation between setA and setB.

Rajah 1 menunjukkan hubungan di antara set A dan set B.

a

b

c

p

q

r

s

SetA SetB

Diagram 1 /Rajah1State

Nyatakan

(a) the range of the relation,julat hubungan itu,

(b) the type of the relation.

jenis hubungan itu.Answer : (a) ..

(b) ..

2. Diagram 2 shows the function : 2f x m x .

Rajah 2 menunjukkan fungsi : 2f x m x .

5

3 9

a

x 2mx

Diagram 2 /Rajah2

Find

Cari

Answer : (a) m = ....

(b) a = .....

(a) the value ofm,

nilai bagi m,

(b) the value ofa.nilai bagi a.



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3. In Diagram 3, the functionsgmapsx toy and the function h mapsy toz.

Dalam Rajah 3, fungsi g memetakan x kepada y dan fungsi h memetakan y kepada z

23

8

y zg h

Diagram 3 /Rajah3

Determine

Tentukan

(a) ,1(8)g

(b) h (8),

(c) .(3)hg

Answer : (a)

(b)

(c) .

4. The relation between setXand set Yis defined by the set of ordered pairs

{(4,2), (4,4), (9,3)}.

Hubungan antara set X dan set Y ditakrifkan oleh pasangan tertib {(4,2), (4,4), (9,3)}.

State

Nyatakan

(a) the range of the relation,

julat hubungan itu,

(b) the images of 4.

imej bagi 4.

Answer : (a) ..

(b) ..



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5. Diagram 5 shows the relation between two sets of elements.

Rajah 5 menunjukkan hubungan antara dua unsur set.

(1, 2)

(4, 3)

(4, 5)

(8, 3)

x

f(x)

2 4 6 8

2

4

0

Diagram 5 /Rajah5

State

Nyatakan(a) the type of relation,

jenis hubungan,

(b) .1(3)f

Answer : (a) ..

(b) ..

6. In Diagram 6, setB shows the images of certain elements of setA.Dalam Rajah 6, set B menunjukkan imej bagi unsur-unsur tertentu set A.

2

0 1

5

4

2 k

17

Set A SetB

Diagram 6 /Rajah6

(a) State the value ofk.

Nyatakan nilai k.

(b) Using the function notation, expressfin terms ofx.

Dengan menggunakan tatatanda fungsi, ungkapkan f dalam sebutan x.

Answer : (a) k= ...

Hak Cipta Cayzec Montoya SMK Sultan Sulaiman Kuala Terengganu 7(b) ..


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3, fi

3,

7. Given : 4f ndx x +

Diberi cari: 4f x x +

(a) the image of 3,

imej bagi3,

(b) the object which has the image of 5.objek yang mempunyai imej 5.

Answer : (a)

(b)

8. Diagram 8 shows the function :1

ah x ,

x

xk, where a and kare constant.

Rajah 8 menunjukkan fungsi :1

ah x ,

x

xk, dengan keadaan a dan k ialah pemalar.

1

a

xx

3

1

Diagram 8 /Rajah8

(a) Determine the value ofk.

Tentukan nilai k.(b) Find the value ofa.

Cari nilai bagi a.

Answer : (a) k= ......

(b) a = ..


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2. QUADRATIC EQUATIONS


Quadratic equations and their roots

o A quadratic equation is an algebraic equation which has only one unknown and thehighest power of the unknown is 2.

o The general form of quadratic equation is : ax2 + bx + c = 0, where a, b and c areconstants.

o The values which satisfy a quadratic equation are known as the roots.o The roots can be determined by

(a) factorization method.

(b) using formula,x =2 4

2

b b ac

a .

Conditions for different types of roots

o Given a quadratic equation ax2 + bx + c = 0, if(a) b

2 4ac > 0, the equation has two real and distinct roots,

(b) b2

4ac = 0, the equation has two real and equal roots,

(c) b2

4ac < 0, the equation has no roots.

(d) b2

4ac 0, the equation has real roots.

o The expression b2 4ac is known as the discriminant of a quadratic equation.

Forming a quadratic equation from given roots.

o Ifand are the roots of a quadratic equation, then

oSum of the root (SOR) + =b

a

o

Product of the roots (POR) =

c

a

Hence, the quadratic equation is :

x2

(SOR)x + POR = 0



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HHPAPER 1JJ

1. Express the quadratic equationx 3 = (2x + 5)(x + 2) in the general form.Ungkapkan persamaan kuadratikx 3 = (2x + 5)(x + 2) dalam bentuk am.

Answer : ..

2. One of the roots of the quadratic equation 2x2

+ kx 3 = 0 is 3, find the value of k.

Satu daripada punca-punca persamaan kuadratik2x2

+ kx 3 = 0 ialah 3, cari nilaik.

Answer : k= ..

3. Given that the roots of the quadratic equationx2

hx + 8 = 0 arep and 2p, find the

values of h.

Diberi punca-punca persamaan kuadratikx2

hx + 8 = 0 ialah p dan 2p, carinilai-nilai h.

Answer: h =


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4. A quadratic equationx2

+ kx + 9 = 2x has two equal roots. Find the possible values ofk.

Persamaan kuadratik x2

+ kx + 9 = 2xmempunyai dua punca sama. Cari nilai-nilai kyang mungkin.

Answer : k= ..

5. A quadratic equationpx

2

3px + 9 = 0 has two equal roots, wherep is positive.Determine the value ofp.Persamaan kuadratik px

2 3px + 9 = 0 mempunyai dua punca sama dengan keadaan p

positif. Tentukan nilai p.

Answer : p = ..

6. A quadratic equationx2

+ 3x +p + 5 = 0 has two distinct (different) roots.

Find the range of values ofp.

Persamaan kuadratik x2

+ 3x +p + 5 = 0 mempunyai dua punca berbeza.

Cari julat nilai p.

Answer : ..


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7. Form a quadratic equation which has roots 2 and 5. Give your answer in general form.Bentukkan persamaan kuadratik yang mempunyai punca-punca 2 dan 5.Berikanjawapan anda dalam bentuk am.

Answer : ..

8. A quadratic equation2x2 + 3x 2 = khas two equal roots. Find the value ofk.Suatu persamaan kuadratik2x

2+ 3x 2 = kmempunyai dua punca sama.

Cari nilai k.

Answer : k= ..

9. Given that the quadratic equation 4px2

3kx +p = 0 has two equal roots, find the ratio

k:p.Diberi persamaan kuadratik4px

2 3kx +p = 0 mempunyai dua punca sama, cari

nisbahk:p.

Answer : ..


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3. QUADRATIC FUNCTIONS


Quadratic functions and their graphs

o A quadratic function in the general form isf(x) = ax2 + bx + c, where a, b and c are

constants and a 0.o The graph of a quadratic function is a parabola with an axis of symmetry passing

through the maximum or minimum point of the curve.

o Ifa > 0, then the graph is : Ifa < 0, then the graph is :

axis of symmetryaxis of symmetry

Hak Cipta Cayzec Montoya SMK Sultan Sulaiman Kuala Terengg

anu 13

x

a < 0

xa > 0

x

a < 0

xa > 0

x

a < 0

o The positions of quadratic functions and the types of roots are as follows :

Types of roots

m maximu

inimum graph m graph

Discriminant Position of graph

(a) Two different roots b2

4ac > 0

(b) Two equal roots b2

4ac = 0

(c) No roots b2

4ac < 0

x

a > 0


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(p, q)

x

c

y

0

x = p

y-intercept

Maximum and minimum values of quadratic functions

uadratic functions

2

GRAPH B

(a0) For graph B

- Function :f(x) = a(x +p + - Function :f(x) = a(x +p)2

+- a is positive. - a is negative.

p q nt is (p, q).- minimum point is ( , ). - maximum poi- minimum value off(x) is q. - maximm value off(x) is q.

- corresponding value ofx is p. - corresponding value ofx is - axis of symmetry isx = p. - axis of symmetry isx = p.

Methods of Completing the Square

f(x) = ax2 + bx + c (general form) convert to

(a

(i) Case

f(x) =

2 2

2 2

b bx c

a a

+

Example :

rt f(x) =x2 6x + 8 to CTS form.

+

Example :

rt f(x) =x2 6x + 8 to CTS form.

+

ConveConve

Solution :Solution :

f(x) = f(x) =

2

26( 3) 8

2(1)x +

x 3)2 1= (

(p, q)

x

c

0

x = p

y-intercept


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(ii) Casea 1

Hak Cipta Ca ngganu 15yzec Montoya SMK Sultan Sulaiman Kuala Tere

f(x) = 2a x

bx c

a+ +

=2 2

2 2b b

a x ca a

+ +

solve u xntil we get in the form f( ) = a(x +p)2

+ q

e the maximum or minimum point forf(x) = 2x2

4x + 5.

(x) = 2(x2

2x) + 5

2] + 5

(b) )

Example :

Determin

Solution :

f

= 2[(x 1)2 (1)

= 2(x 1)2

2 + 5

= 2(x 1)2 + 3

Method2(Using Formula2 24

2 4

b aca x

a a

bf(x) = + + OR f(x) =

2 2

2 4

b ba x c

a a

+ +

Example :

etermine the maximum or minimum point forf(x) = 3x2

12x + 13.

ch the graph off(x).

D

Hence, sket

Solution :2

f(x) =24

2 4

b ac bx

a a

+ + a

2 2( 12) 4(3)(13=

) ( 12)

2(3) 4(3)a x

+ +

(x 2)2

+ 1

value off(x) is 1 and the corresponding value ofx is 2.en .

= 3

Hence, the minimumH ce, the minimum point is (2, 1)

The graph off(x) is :Important facts on sketching the graph :

shown i.e. (0, 13) and (3, 4).

.

(2, 1)

x

y

13

- Minimum parabola shape.- Minimum point is (2, 1).- Minimumvalue ofy is 1.

e- The other 2 points must b - Axis of symmetry isx = 1

(3, 4)

0


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QUADRATIC INEQUALITIES

that satisfies quadratic inequalities can be determined

Ex of values ofx forx2

4x 12 > 0

Solution

x 4x 12 > 0

6) > 0

o The range of the values ofxby graphical method.

ample 1 : Find the range

:

2

(x + 2)(x

x = 2, 6

equalityx(2x 1) < 3.

(2x 1) < 3

< 0

x62

Thus,x < 2,x > 6

Example 2 : Solve the in

Solution :

x 2x

2x 3

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

3, 1

2

x

Thus, 1


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HHPAPER 1JJ

1. Find the range of values ofx if 2x2

9x + 4 > 0.Cari julat nilai x jika 2x2 9x + 4 > 0.

Answer : ..

2. Find the range of values ofx ifx 7x + 12 < 0.Cari julat nilai x jika x

2 7x + 12 < 0.

Answer : ..

Cari julat nilai x bagix(x 6) < 16.

Answer : ..

2

3. Find the range of values ofx for which x(x 6) < 16.


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Cari julat nilai x yang memuaskan ketaksamaan (x 3)2

< 9 8x.

Answer : ..

real and distinct roots.

Answer : ..

have real roots.

Answer : ..

4. Find the range of values ofx which satisfies the inequality(x 3)2

< 9 8x.

5. Find the range of values ofkif the quadratic equation (1 + k)x2

+ 4kx + 9 = 0 has two

Cari julat nilai k jika persamaan kuadratik(1 + k)x2

+ 4kx + 9 = 0 mempunyai dua

punca nyata yang berbeza.

6. Find the range of values ofp if the quadratic equationpx2

+ 8x +p 6 = 0 does not

Cari julat nilai p jika persamaan kuadratik px2

+ 8x +p 6 = 0 tidakmempunyaipunca nyata.


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ersamaan kuadratik(2 3p)x2 + (p 4)x + 2 = 0 mempunyai

Answer : ..

8. Find the range of values ofx for which (2x + 1)(x + 3) > (x + 3)(x 3).

Answer : ..

7. Find the range of values ofp if the quadratic equation (2 3p)x2 + (p 4)x + 2 = 0 hastwo distinct roots.

Cari julat nilai p jika pdua punca berbeza.

Cari julat nilai x untuk(2x + 1)(x + 3) > (x + 3)(x 3).


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0

19

(3, 1)

x

9. Find the range of values ofm such thatx2

+ 6x = mx 1 has two different roots.

za.

Answer : ..

10. Diagram 10 shows a curve y =p(x + q)2

+ rwhere (3, 1) is a turning point. Determine

lengkungy =p(x + q) + rdengan keadaan (3, 1) adalah titik

Diagram 10 /Rajah10

Answer : p = .. q = r=

Cari julat nilai m dengan keadaan x2

+ 6x = mx 1 mempunyai dua punca berbe

y

pusingan. Tentukan nilaip, q danr.

the value ofp, q and r.Rajah 10 menunjukkan

2


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HHPAPER 2JJ

11. Expressf(x) = 2x2

4x + 9 in the form a(x + h)2

+ kwhere a, h and kare constants.

k

a, h and k.

inimum value off(x) and the corresponding

i maksimum atau minimum bagif(x) dan nilai x yang sepadan.

x2

4x + 9 =p has two different real roots.

12. The quadratic function f(x) = 3[(x k)2

+ h], where h and kare constants, has a

k) + h], dengan keadaan h dan k adalah pemalar,

p.p.

uch thatf(x) = thas real roots.

13. Given that f(x) = 7 mxx2 = 16 (x + n)2 for all real values ofx.

mxx = 16 (x + n) untuk semua nilai x yang nyata.

,

e maximum point.

) and state the axis of symmetry.etrinya.

Ungkapkanf(x) = 2x2

4x + 9 dalam bentuka(x + h)2

+ kdengan keadaana, hdanadalah pemalar.

(a) State the values ofNyatakan nilai a, h dan k.

(b) Determine the maximum or mvalue ofx.

Tentukan nila

(c) Sketch the graph off(x) = 2x2

4x + 9.

Lakarkan graff(x) = 2x2

4x + 9.

(d) Find the range of values ofp such that 2 Cari julat nilai p dengan keadaan 2x

2 4x + 9 =pmempunyai dua punca nyata

yang berbeza.

minimum point at P(5p, 6p2).

Fungsi kuadratikf(x) = 3[(x 2

mempunyai titik minimum pada P(5p, 6p2).

h and ofkin terms of(a) State the value ofNyatakan nilai h dan nilai k dalam sebutan

(b) Given thatp = 1, find the range of values ofts Diberi bahawa p = 1, cari julat nilai t supayaf(x) = tmempunyai punca-punca

nyata.

Ifm > 0 and n > 0, find2 2

Diberi bahawaf(x) = 7 Jikam > 0 dann > 0, cari

(a) the value ofm and ofn

nilai m dan nilai n,(b) the coordinates of th

koordinat titik maksimum.

(c) Hence, sketch the graphf(x Seterusnya, lakar graf f(x) dan nyatakan persamaan paksi sim


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4. SIMULTANEOUS EQUATIONS


Linear equation = equation that involves two variables such asx andy where the

vesx andy other than linear equation.

Basic algebra in this topic :

o Expanding

Expand (2x 1)2.

ging subject

iven 2x 3y = 6, expressx as a subject.

=

maximum indices ofx andy is one.

Non linear equation = equation invol

2

32

x

.Example 1 :

Solution : (2x 1)(2x 1)

= 4x2

4x + 1

o Chan

e 3. G Exampl

Solution : 2x 3y + 6

x =3y 6

2

+

o Solving quadratic equation

5x 3 = 0.

n Method

0

Example 4. Solve 2x2

+

Solution :

Factorizatio

(2x 1)(x + 3) =

x =1

, 32

Example 2 : Expand

Solution :3 3

2 2

x x

=2 6 9

4

x x +


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Example 5. Solve 2x2 5x 4 = 0.

Solution :

Formula Method

(Usually used when the final answer is not an integer or fraction)

x =2

4

2

b b ac

a

2( 5) ( 5) 4(2)( 4)

2(2)

=

= 3137, 0637

Final answers can be calculated by using calculator, that is :

All early and final answers should be written in at least 4 significant figures.T

All simultaneous equations should be solved by substitution method.

TO SOLVE SIMULTANEOUS EQUATIONS, FOLLOW THESE STEPS

o).

MODE/EQN/degree(2)/a?/b?/c?)

hen, conclude your final answer according to the question stated.

From linear equation, expressx ory as a subject.

o Substitute the result into the non-linear equation (notice that only one unknown remainso Solve the quadratic equation formed (by factorization or using formula).o Find the second unknown (by substituting in the third equation).


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HPAPER 2JJ

1. Solve the simultaneous equationx2

y +y2

= 2y + 2x = 10.

2. Solve the simultaneous equations

H

2 2x y+ = and 22 7y xy = . Give your answers

correct to two decimal places.

Selesaikan persamaan serentak 2 2x y+ = dan 22 7y xy = . Berikan jawapan anda

uhan.

3. Solve the simultaneous equation 1

betul sehingga dua tempat perpul

2 23 2 6x y x y = =

2 6 1x y x y Selesaikan persamaan serentak 3 2 2 = =

ltaneous equation x2

+y2

= 3x +y = 5..

5. Solve the simultaneous equations a b = 3 and a2

+ 2b = 10. Give your answers

4. Solve the simuSelesaikan persamaan serentak x

2+y

2= 3x +y = 5...

correct to two decimal places.

Selesaikan persamaan serentak 3a b = dan 2 2 10a b+ = . Berikan jawapan andauhan.

6. Solve the simultaneous equations

0

7. Solve the following simultaneous equations :

rect to two decimal place.

perpuluhan.

8. Solve the simultaneous equations x 5y = 2 and 6.

betul sehingga dua tempat perpul

Selesaikan persamaan serentak

2 3 4 0x y + = 2 5 1x xy =

Selesaikan persamaan serentak berikut:

3x + 2y + 1 = 02

x + 4xy + 4 = 0

Give your answer cor

Berikan jawapan anda betul kepada dua tempat

2 24 7x y xy =

Give your answers correct to two decimal places.

Selesaikan persamaan serentak x 5y = 2 dan 2 24 7 6.x y xy =

Berikan jawapan anda betul kepada dua tempat perpuluhan.


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Hak Cipta Cayzec Montoya SM

25

5. INDICES AND LOGARITHMS

Laws of Indices :

Zero Index :

:

Laws of L ogarithms

The properties of logarithms :

Change of base of logarithms

Method to change logarithmic for


am

an = a

m + na

m a

n = a

mn

(am

)n = a

mn (ab)n

= anb

n

K Sultan Sulaiman Kuala Terengganu

n na a

= nb b

a0 = 1

an

=1n

a Negative Index

Fractional Index :

:

:

m to index form and vice versa :

1

nna a= and ( )m

mnna a=

logam

n = logam loganlogamn = logam + logan logamn = n logam

Since a1

= a , logaa = 1

Since a0

= 1 , loga 1 = 0

loglog

log

c

a c

bb

a= and

1log

loga bb

a=

If log N = xa

then, ax

=


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HHPAPER 1JJ

1. Simp

lify2 3 2

13 9

27

p p

p

+

.

n Ringkaska 2 3 2

1

3 9

27

p p

p

+

.

Answer : ..

x x 1

Selesaikan persamaan 16 = 32 .

Answer : ..

2. Solve the equation 16 = 32 .x x 1


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3


3. Given that log10x = m and log10y = n, express 10log100

y

=

in terms ofm and n.

3

10log100

x ydalam sebutan m dan n Diberi log10x = mdan log10y n, ungkapkan .

Answer : ..

ri =x, ungkapkan setiap yang berikut dalam sebutan x,

,

.

Answer : (a) ..

(b) ..

4. Given that 3log N=x, express each of the following in terms ofx,

3log N Dibe

(a) 3log N2

9log N(b)


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5. Solve the equation 2 2log 2 log (1 3 )p p = 1.

Selesaikan persamaan 2 2log 2 log (1 3 )p p = 1.

Answer : ..

Selesaikan persamaan 3(9 ) = 27 .

Answer : ..

6. Solve the equation 3(9x + 4

) = 27x + 1

.x + 4 x + 1


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x 17. Solve the equation 5

125= 0.

1Selesaikan persamaan 5

x

125= 0.

Answer : ..

8. Solve the equation .2 1 24 32x x +=

Selesaikan persamaan 2 1 24 32x x +

= .

Answer : ..

9. Solve1

216

2x+

= .

Selesaikan1

216

2x+

=

Answer : ..


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Selesaikan persamaan 43x 1

2x = 16 x 5.

Answer : ..

10. Solve the equation 43x 1

2x = 16 x 5.

1 416 .

x = 11. Solve the equations52x

Selesaikan persamaan 15

416 .

2

x

x

=

Answer : ..


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"

Distance between two points.

6. COORDINATE GEOMETRY

IMPORTANT NOTES AND FORMULAE

2 22 1 2 1( ) ( )x x y y + DistanceAB =

Midpoint.

(x,y) = 1 2 1 2,2 2

x+ y y+

A point dividing a segment of a line with ratio m : n.

(x,y) = 1 2 1 2,nx mx ny mym n m n

+ + + +

Area of polygon.

Area =

1

2

1 2 3 1

1 2 3 1

x x

y y y

x

y

=1

1 2 2 3 3 1 2 1 3 2 1 3( ) ( )x y x y x y x y x y x y+ + + + 2

Gradient of a straight line.

2 1

2 1

y y

x x

Gradient, m =

ion of straight line.

Method 1

ient and (x1,y1) is any point on the straight line.

To form equat a

y y1 = m(x x1), where m is grad

Method 2

y = mx + c (by finding the value ofc)


32/111



Forms of equation of a straight line.

o Gradient form y = mx + c

General form ax + by + k= 0

o Intercept form

o

x y

a brespectively.

+ = 1, where a and b lie on thex-axis and y-axis

Parallel and perpendicular straight lines.

o For parallel straight lines m1 = m2For perpendicular straight lines m m2 = 1

en two points.

o Distance from a moving pointP(x,y) to a fixed point,A is a constant.

ual

atio m : n.

o 1

Equation of locus involving distance betwe

PA = k

o Distance from a moving pointP(x,y) to two fixed points,A andB is eq(equidistant).

PA =PB

o Distance from a moving pointP(x,y) to two fixed points,A andB is in the r

PA = kPB


33/111



HHPAPER 1JJ

through pointsA(k+ 3, 5k) andB(k 1, 2k+ 1) has a

gradient

1. A straight line that passes

of1

2. Find

(a) the value ofk,

Answer : (a) k= ..

(b) ....................................

perpendicular toACat pointA(0, 4).

dengan AC pada titikA(0, 4).

Diagram 2 /Rajah2

Find / Cari

(a) the value ofk,

(b) the equation ofAC.

an AC.

Answer : (a) k= ..

(b) ....................................

(b) the equation ofAB.

2. In Diagram 2, the equation of the straight lineAB is 2y =x + k. Given thatAB is

Dalam Rajah 2,persamaan garis lurus AB ialah 2y =x + k.Diberi AB berserenjang

nilai k,

persama

y

A (0, 4)

x

y =x + k

0

2

B

C


34/111



3. In Diagram 3,PQ and QR are two straight lines which are perpendicular to each other at

PQ isx 2y + 6 = 0.

Find / Cari

koordinat P dan Q,(b) the equation ofQR.

an bagi QR.

Answer : (a) P= ..

Q = .............................

point Q. PointsPand Q are on thex-axis and they-axis respectively. The equation of

Dalam Rajah 3,PQ dan QR adalah dua garis lurus yang berserenjang antara satusama lain pada titik Q. Titik P dan titik Q terletak pada paksi-x dan paksi-y masing-

masing.

Ry

P

Q

xO

x 2y + 6 = 0

Diagram 3 /Rajah3

(a) the coordinates ofPand Q,

persama

(b) ....................................


35/111



4. In Diagram 4, the straight linePQ intercepts thex-axis and they-axis at the points

Dalam Rajah 4,garis lurus PQ memintas paksi-x dan paksi-y pada titik P dan titik Q

masing-masing.

(a) Write down the equation ofPQ in the intercept form.

Tulis persamaan san.(b) Find the equation of a straight line which is parallel to the straight linePQ and

passes through the point S(8, 3).

lurus PQ dan melalui titik

Answer : (a) ............

(b) ....................................

Pand Q respectively.

y

P(0, 3)

O Q(4, 0)x

Diagram 4 /Rajah4

PQ dalam bentuk pinta

Cari persamaan garis lurus yang selari dengan garis

S(8, 3).


36/111



line 3x + 5y 15 = 0 is parallel to the straight line joining the pointsA andB.

Titik A dan titik B mempunyai koordinat(6, 3) dan (4, k) masing-masing. Garis lurus

k,ofAB,

jarak AB.

Answer : (a) k= ..

(b) ....................................

respectively. IfQ divides the straight linePR in the ratio 3 : 2, find the coordinates ofR.

PQR ialah garis lurus dengan koordinat P dan Q masing-masing ialah (2, 1) dan

Answer : ..

5. The pointsA andB have coordinates (6, 3) and (4, k) respectively. The straight

3x + 5y 15 = 0 adalah selari dengan garis lurus yang menyambungkan titik A dan

titik B.Find / Cari

alue ofk,(a) the v

nilai(b) the distance

6. PQR is a straight line where the coordinates ofPand Q are(2, 1) and (1, 2)

(1, 2). Jika Q membahagi garis lurus PR dengan nisbah 3:2, cari koordinat R.


37/111



7. Given that the coordinates ofA andB are (4, 1) and (1, 2) respectively. Find the

: ..

Titik-titik R, S dan T masing-masing mempunyai koordinat(5, 2), (1, 4) dan (3, 6).

Answer : m = ..

equation of the straight line that passes through the point T(1, 3) and parallel withto the straight lineAB.

Diberi koordinat A dan B masing-masing ialah (4, 1) dan (1, 2). Cari persamaangaris lurus yang melalui titik T(1, 3) dan selari dengan garis lurus AB.

Answer

8. The coordinates ofR, Sand Tare (5, 2), (1, 4) and (3, 6) respectively. If point Sdivides the straight lineRSTin the ratio m : 1, find the value ofm.

Jika titik S membahagi garis lurus RST dengan nisbah m :1, cari nilai m.


38/111



9. Find the equation of the straight line perpendicular to the straight line 6x + 3y 2 = 0

Answer : ..

passes through the point (12, 5), find the value ofa and ofb.

Answer : a = b =

and passes through the point (6, 1).Cari persamaan garis lurus yang berserenjang dengan garis lurus 6x + 3y 2 = 0 dan

melalui titik(6, 1).

10. Given the straight line 4x + 3y 6 = 0 is parallel to the straight liney = ax + b which

Diberi garis lurus 4x + 3y 6 = 0 adalah selari dengan garis lurus y = ax + b yang

melalui titik(12, 5), cari nilai a dan nilai b.


39/111



HHPAPER 2JJ

y scale drawing will not be accepted.

Penyelesaian secara lukisan berskala tidak diterima.

4 = 0.

x y + 4 = 0.

Diagram 11 /Rajah11

Diberi PQ : QR = 2 : 3, cari

hR.

is rus yang berserenjang dengan PR dan melaluiR.

11. Solution to this question b

In Diagram 11, the equation of the straight linePQR is 2x y +

Dalam Rajah 11, persamaan garis lurusPQRialah 2

P

Q

yR

O x

Given thatPQ : QR = 2 : 3, find

(a) the coordinates ofR,

koordinatR,

(b) the equation of a straight line perpendicular toPR and passes throug

persamaan gar lu


40/111



Solution to this question by scale drawing will not be accepted.


=x + 4 such that

4AB : BC = 1 : 4.

Diagram 12 /Rajah12

Find

Cari

e coordinates ofA,

inat A,

A.

12.

Diagram 12 shows three pointsA, B and Con the straight line 2y

AB :BC= 1 : 4.

Rajah 20 menunjukkan tiga titik, A, B dan C yang berada pada garis lurus 2y = x +dengan keadaan

y

x

A

B(2, 3)

C

O

(a) th

koord

(b) the coordinates ofC,koordinat C,

(c) the area of triangle CO luas segitigaCOA.


41/111



13. Solution to this question by scale drawing will not be accepted.


-axis respectively. Given that the

(a) Find the equation

Cari persamaan garis lurus QR.

ended to a point Swhich lies on thex-axis andfR.

= 2QS.

In Diagram 13,PQ and QR are two straight lines whic

at point Q. PointPand point Q lie on thex-axis andy

h are perpendicular to each other

equation ofPQ is 3y + 2x 9 = 0.

Dalam Rajah 13,PQdanQRialah dua garis lurus yang berserenjang di titikQ. Titik Pdan titik Q masing-masing terletak di atas paksi-xdan paksi-y.Diberi persamaan garis

lurus PQ ialah 3y + 2x 9 = 0.

x

y

O P

Q

R

3y + 2x 9 = 0

Diagram 13 /Rajah13

ofQR.

(b) The straight lineRQ is ext

RQ = 2QS. Find the coordinates o Garis lurus RQ dipanjangkan ke titik S yang terletak di atas paksi -x danRQ

Cari koordinat R.

(c) Calculate the area ofPQR.Hitung luasPQR.


42/111



7. STATISTICS

ND FORMULAE"IMPORTANT NOTES A

For Ungrouped Data

Mean, xN

=

e,Varianc2

2 2( )x

xN

=

Standard deviation,2

2( )x

xN

=

For Grouped Data

fxx

f

=

Mean,

Variance,2

2 2( )fx

f =

eviation,2

2( )fx

xf

= Standard d

The mode of a data set can be obtained from a histogram :

Mode

Mode

Frequency

Variable

Modal

class


43/111



Median

m = 2L

m

NF

C

f

+

, where

First Quartile

Q1 = 14

Q

F

L

1Q

N

Cf

+

, where

Third Quartile

Q3 = 34

Q

F

L

3

3

Q

N

Cf

+

, where

L = lower boundry of the median class

N= total frequency

C= class size

class

LQ1 = lower boundry of the 1

F= cumulative frequency before mediandian classfm = frequency ofme

stquartile class

N= total frequency

C= class size

lass

LQ3 = lower boundry of the 3rdquartile class

N= total frequency

C= class size

lass

F= cumulative frequency before 1stquartile c

quartile classfQ1 = frequency of 1st

F= cumulative frequency before 3rdquartile crdquartile classfQ3 = frequency of 3


44/111



Marks/Markah 30-39 40-49 50-59 60-69

Number of students

r4 9 16 11

Bilangan pelaja

HHPAPER 1JJ

marks acquired by 40 students in a test.

Jadual1 menunjukkan taburan markah yang diperolehi 40pelajar dalam suatu ujian.

1

Without drawing an ogive, calculate the median mark.

Answer : ..

1. Table 1 shows the distribution of the

Table 1 /J adual

Tanpa melukis ogif, hitung markah median.

2. Table 2 shows the distribution of age of 100 residents in a particular housing area.

Jadual2 menunjukkan taburan umur100 penduduk di suatu taman perumahan.

Age (in years)

Umur(dalam tahun)1 15 16 30 31 45 46 60

No. of residents

Bilangan penduduk28 52 14 6

T le 2 / ual 2

Without drawing the ogive, find the third quartile age.

Answer : ..

ab J ad

Tanpamelukis ogif, cari umur kuartil ketiga.


45/111



3. The sum of the 10 numbers is 170 and the sum of the squares of the numbers is 2930.

bor itu.

Answer : ..

Diberi bahawa min bagi set data 5, 3, 5, 10 dan xialah 6. Cari nilai

variance.ians.

Answer : (a)x = ....

Min bagi set data 12 5a, 4a, 3a dan a2

ialah 5. Cari nilai-nilai a yang mungkin.

Answer : a = ..

Find the variance of the 10 numbers.

Hasil tambah 10 nombor ialah 170 dan hasil tambah kuasa dua nombor-nombor itu

ialah 2930. Cari varians bagi 10 nom

4. Given that the mean of a set of data 5, 3, 5, 10 andx is 6. Find the value of

(a) x,

(b) the var

(b) ..

5. The mean of the set of data 12 5a, 4a, 3a and a2

is 5. Find the possible values ofa.


46/111



6. Given that the mean of a set of six numbers, 11, 13, 19, 20, m and 2m is 14, find

umbers.

Answer : (a) m = ..

1 2 3 4 5

ri

x,

Answer : (a) ..

Diberi min bagi satu set enam nombor, 11, 13, 19, 20, m dan 2m ialah 14, cari

(a) the value ofm,

nilai m,

(b) the standard deviation of the set of n sisihan piawai bagi set nombor itu.

(b) ..

7. A set of data x ,x ,x ,x ,x has mean 7 and standard deviation 3. FindSatu set datax1,x2,x3,x4,x5 mempunyai min 7 dan sisihan piawai 3. Ca

(a) the sum of the data, x,

hasil tambah bagi data,

(b) the sum of the squares of the data, x2hasil tambah bagi kuasa dua data, x2 .

(b) ..


47/111



the score.

i skor min.

8. Table 8 shows the score of a group of students in a quiz, calculate the mean of

Jadual8 menunjukkan skor sekumpulan pelajar dalam suatu pertandingan kuiz,

hitung nila

Score1 2 3 4

SkorNumber of students

an pelajarBilang3 9 13 5

Table 8 / 8

Answer : ..

9. A set of six numbers has a mean of 13 and the sum of squares of these numbers is1030. Find the variance of the set of data.

bortu.

Answer : ..

Rajah

Satu set enam nombor mempunyai min 13 dan hasil tambah kuasa dua nombor-nomitu ialah 1030. Cari varians bagi set data i


48/111



HHPAPER 2JJ

x4,x5 andx6 has a mean of 4 and a standard deviation of 3.

Satu set skor,x1,x2,x3,x4,x5danx6mempunyai min 4 dan sisihan piawai 3.

e is multiplied by 4 and then 3 is added to it. For the new set of scores,

ean,

viation.n piawai.

. The mean of a set of numbers 2, 6,x,x + 1, 6, 10, 7, 8 is 6.

Min bagi satu set nombor-nombor2, 6,x,x + 1, 6, 10, 7, 8 ialah 6.

e ofx,

viation of the numbers above.

iawai bagi nombor-nombor di atas.

d then 3 is added it, find

3, cari

ns

umbers.

bor-nombor baru itu.

10. A set of scores,x1,x2,x3,

(a) Find / Cari(i) x,(ii) x2.

(b) Each scor

find

Setiap skor didarab 4 dan kemudian ditambah 3. Untuk set skor yang baru, cari

(i) the m

min,

(ii) the standard de sisiha

11

Find / Cari

(a) the valu

nilai x,(b) the standard de

sisihan p

If each of the numbers above is multiplied by 2 an

Jika setiap nombor di atas didarab 2 dan kemudian ditambah

(c) the mean,

min,

(d) the variance

varia

f th t of no e new se

bagi set nom


49/111



a) A set of positive integers consists of 1, 4 andp. The variance for this set of

integers is 6. Find the value ofp.

3 4 5 6 7

bers is 3080.

kuasa dua nombor-nombor itu ialah

in dan varians bagi 7 nombor itu.

viation of the set of 8 numbers.iawai bagi set8 nombor itu.

12. (

Satu set integer positif terdiri daripada 1, 4 dan p.Nilai varians untuk set iniialah 6. Cari nilaip.

,x ,x ,x ,x ,x }. The sum of the numbers is 140 and the(b) Given that setP= {x1,x2sum of the squares of the num

Diberi bahawa setP= {x1,x2,x3,x4,x5,x6,x7}.Hasil tambah bagi nombor-nombor itu ialah 140 dan hasil tambah bagi3080.

(i) Find the mean and the variance for the 7 numbers.Cari m

(ii) When kis added to setP, the mean increased by 2.Apabila k ditambah kepada set P, min bertambah sebanyak2.

Find / Cari

(a) the value ofk,nilai k,

(b) the standard de sisihan p


50/111



8. CIRCULAR MEASURE

rad = 180

adians to degrees and vice versa


Conversion of r

(a) Convert from radians to degrees.

180 rad =

o

(b) Convert from degrees to radians

= 180

o

Length of arcAB,s = r, where must be in radian.

=Area of a sector of a circle,A 21

2r , where must be in radian.

2

can be in degrees or radians.

s (Length of arcAB)

B

r

r

A

r

r

O

P

Q

major sector

minor sector

Area of triangle,A = ab sin C

Hence,A = r sin , where


51/111



HHPAPER 1JJ

1. Convert / Tukar

(a) 3925 to radians,ian,

rees and minutes.

a darjah dan minit.

Answer : (a) ..

Rajah 2 menunjukkan sektorOAB.

s.

TukarBOA ke radian.

Cari panjang lengkok AB.

Answer : (a) ..

3925ke rad(b) 0428 radian to deg

0428 radian kepad

(Use / Guna= 3142)

(b) ..

2. Diagram 2 shows a sectorOAB.

B

125 cm

82O

A

Diagram 2 /Rajah2

(a) Convert BOA to radian

(b) Find the length of arcAB.

(Use / Guna= 3142)

(b) ..


52/111



3. Diagram 3 shows a sectorPOQ with centre O.

Gi m and the perimeter of sector POQ is 425 cm.Find the value ofin radians.

.

Answer : = ..

Rajah 4 menunjukkan bulatan berpusat O.

Given that th cEFis 4187 cm, find the length, in cm, of theradius.

.

Answer : ..

Rajah 3 menunjukkan sektor POQ berpusat O.

P

QO

Diagram 3 /Rajah3

ven the length of arcPQ is 125 c

Diberi panjang lengkok PQ ialah 125 cm dan perimeter sektor POQ ialah 425 cm

Cari nilaidalam radian.

4. Diagram 4 shows a circle with centre O.

O

E F

105 rad.

Diagram 4 /Rajah4

e length of the major ar

Diberi bahawa panjang lengkok major EF ialah 4187 cm, cari panjang, dalam cm,bagi jejari

(Use / Guna= 3142)


53/111



O

A

B

20

5. In Diagram 5, AOB = 20 and the length of arcAB is 8 cm.

Diagram 5 /Rajah5

Find / Cari

(a) the length ofOB,

OB,orOAB.

B.

Answer : (a) ..

OQ = 5 cm and QR = 13 cm.

13 cm.

n.

Cari perimeter kawasan berlorek.

Answer : ..

Dalam Rajah 5, AOB = 20dan panjang lengkok AB ialah 8 cm.

panjang(b) the area of the sect

luas sektor OA(Use / Guna= 3142)

(b) ..

6. In Diagram 6, OPR is a quadrant of a circle with centre O. OQR is a triangle where

Dalam Rajah 6, OPRialah sukuan bulatan berpusatO. OQRialah segitiga dengankeadaanOQ = 5 cm danQR =

P

O

Q

R

Diagram 6 /Rajah6

Find the perimeter of the shaded regio

(Use / Guna= 3142)


54/111



C D O A

B

HHPAPER 2JJ

7. In Diagram 7,ABCD is a semicircle with centre O and diameter 10 cm.

.

Diagram 7 /Rajah7

iven thatBD is an arc with centreA.

Diberi bahawa BD ialah lengkok bulatan berpusat A.

Nyatakan BADdalam radian.

e length of arcBD,ng lengkok BD,

region.

.

Dalam Rajah 7,ABCD ialah semibulatan berpusat O dan diameter10 cm

G

(a) State BAD in radian.

(b) Find

Cari

(i) th panja

(ii) the area of the shaded

luas kawasan berlorek


55/111



8. In Diagram 8,AB is an arc of a circle, with centre O whileBCis an arc of a circle

8, AB adalah lengkok suatu bulatan berpusat di O manakala BC adalah

Diagram 8 /Rajah8

e of, in radians,

with centreD. Given thatD is the mid point of OB where OB = 12 cm and CDB =18 rad.

Dalam Rajah

lengkok suatu bulatan berpusat di D. Diberi bahawa D ialah titik tengah OB dengankeadaan OB = 12cm dan CDB = 18rad.

18 rad

A

B

C

DO

Find / Cari

(a) the valu

nilai , dalam radian,

(b) the area of the sectorOAB,

luas sektor OAB,

(c) the perimeter of the shaded region.perimeter bagi rantau berlorek.

[Use / Guna= 3142]


56/111


Hak Cipta Ca

56

9. DIFFERENTIATION


Techniques of differentiation

1. The first derivative ofy = axn

using formula.

is an integer.

Ify = k

Given thaty = ax n where a is a constant and n

dy

= naxn 1

dx

, where kis constant

yzec Montoya SMK Sultan Sulaiman Kuala Terengganu

dx

dy= 0

2. The first derivative of a product of two polynomials.

3. of two polynomials.

Given that y =

iven that y = uvG

The first derivative of a quotient

v

u

posite function.4. The first derivative of the com

Given that y = k(ax + b)n

dx= u

dy

dx

dv+ v

dx

du

dy

dx =

2

du dvv u

dx dx

v

1( )

ndynka ax b

dx

= +

the value ofa is the differentiationof the expression in the brackett


57/111



HHPAPER 1JJ

Giveny = 4(1 2x)3, find

dy

dx

.1.

Diberi y = 4(1 2x)3, cari

dy

dx.

Answer :

2. Differentiate x2(2x 5)

4with respect tox.

Answer :

3. Differentiate

Bezakan x2(2x 5)

4terhadap x.

2

1

(3 5)x with respect tox.

Bezakan2

1

(3 5)x terhadap x.

Answer :


58/111


4. Given a curve with an equationy = (2x + 1)5, find the gradient of the curve at the point

x = 1.

Diberi suatu lengkung dengan persamaany = (2x + 1)5, cari kecerunan lengkung itu

pada titik di manax = 1.

Answer:

5. Given thaty = 3x2

4

x

+ 4, find the value ofdy

dx

whenx = 2.

Diberiy = 3x2

4

x+ 4, cari nilai

dy

dxapabila x = 2.

Answer : ..

6. Given y =35

6

x 1, evaluate

2

2

d y

dxwhenx = 3.

Diberiy =35

6

x 1, nilaikan

2

2

d y

dxapabilax = 3.

Answer :



59/111



57. Find the equation of the tangent to the curve 23y x= + at the point (1, 2).

Cari persamaan tangen kepada lengkung 23y x 5= + pada titik(1, 2).

Answer :

8. Find the equation of the normal to the curve y = 23 8 1x x + at the point (1, 4).

Cari persamaan normal kepada lengkung y =

2

3 8 1x + pada titik(1, 4).

Answer : ..

9. Giveny =2

3

(2 3)x , find the value of

dy

dxwhenx = 1.

Diberiy =2

3

(2 3)x ,cari nilai bagi

dy

dxapabila x = 1.

Answer :


60/111


HHPAPER 2JJ

10. The gradient of the curve 4k

y x= at the point (2, 7) is 41

2 ,

Kecerunan lengkung 4k

y x= pada titik(2, 7) ialah 41

2,

Find / Cari

(a) value of k,

nilai k,

(b) the equation of the normal at the point (2, 7),

persamaan normal pada titik(2, 7),

(c) small change in y whenx increases from 2 to 202.

perubahan kecil bagi y bila x menokok dari 2 kepada 202.

11. Given that2

4y

h= and , find2 5h x=

Diberi2

4y

h= dan , cari2 5h x=

(a)dy

dxin terms ofx,

dy

dx dalam sebutanx,

(b) the rate of change ofx when h changes at a rate of 4 units s1

,

kadar perubahan bagi x apabila kadar perubahan bagi h ialah 4unit s1

,

(c) the small change iny whenx decreases from 2 to 198.

perubahan kecil dalam y apabila x menyusut dari 2 kepada 198.



61/111



10. SOLUTIONS OF TRIANGLES


SINE RULE

For any triangleABC, the sine rule is applicable.

c

b

B

A C

a

c

b

B

A C

a

A C

B

Ca

a

b

c

sin

a

A=

sin

b

B=

sin

c

C

Sine Rule

The sine rule can be applied in the following cases :(a) Two angles and the length of a side are given.

(b) Two sides and a non-included angle are given.

COSINE RULE

For any triangleABC, the cosine rule is applicable.

a2 = b2 + c2 2bc cosA

b2 = a2 + c2 2ac cosB

c2 = a2 + b2 2ab cos C

Cosine Rule

The cosine rule can be applied in the following cases :

(a) Two sides and an included angle are given.(b) Three sides are given.

AMBIGUOUS CASE

Ambiguous case arises when :(a) two sides and a non-included angle are given and(b) the non-included angle is an acute angle that is

opposite the shorter side given.


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AREA OF A TRIANGLE

For any triangleABC, the area can be calculated by the formula :

Area ofABC= 12

ab sin C

or1

2ac sinB

or1

2bc sinA

c

b

B

A C

a

HHPAPER 2JJ

SECTION C

1. Diagram 1 shows a triangleABCwhereADCis a straight line and BDCis obtuse.Rajah 1 menunjukkan segi tiga ABC dengan keadaan ADC adalah garis lurus dan

BDCcakah.

5 cm6 cm

x cm

54

2 cmA

DC

B

Diagram 1 /Rajah1

Find

Cari

(a) BDA,(b) the value ofx.

nilai x.


63/111


2. In Diagram 2,ADB is a straight line.

Dalam Rajah 2,ADB ialah garis lurus.

44

34 cm

62 cm38 cm

A B

C

D

Diagram 2 /Rajah2

Calculate the length ofDB.

Hitung panjang DB.

3. In Diagram 3, SUTis a straight line whereRU= 5 cm, SU= 3 cm and UT= 7 cm.

Dalam Rajah 3, SUT ialah garis lurus dengan keadaan RU= 5 cm, SU= 3 cm

danUT= 7 cm.

65

3 cm 7 cm

5 cm

R

SU T

Diagram 3 /Rajah3

Given that RUS= 65, calculateDiberi bahawaRUS= 65, hitung

(a) the length ofRS,

panjang RS,

(b) the area of the triangleRTU.

luas segitiga RTU.



64/111


4. Diagram 4 shows a quadrilateralPQRS.

Rajah 4 menunjukkan sebuah sisiempat PQRS.

58 cm

62 cm

167 cm106

P

Q

R S48

Diagram 4 /Rajah4

(a) Calculate

Hitung

(i) the length, in cm, ofPR,

panjang, dalam cm, bagi PR,

(ii) PRQ.

(b) Calculate the area, in cm2, ofPQR.

Hitung luas, dalam cm2, bagiPQR.



65/111


11. INDEX NUMBERS


Index number or price index.

Q0

= Quality or price of the item at base time

Q1

= Quality or price of the item at the specific time1

0

100Q

I= Q

Composite index.

W= weightageI= index number or price index


To find corresponding price.

Since,1

0

100QIQ

=

01Corresponding price, =

100

I QQ

To find composite index involving three years.

100

P QFormula R

=

I = i i

i

W IW

YearB

Price = Q1

Index Number

I

YearA

Price = Q0

YearB

Index or

Composite Index

P

YearA YearC

Index or

Composite Index

Q

Index or

Composite Index

R


66/111


HHPAPER 2JJ

SECTION C

1. Table 1 shows the prices and the price indices of four types of food items in the year2006 based on the year 2004. Diagram 1 shows the pie chart that reflects the proportion

of expenditure of Puan Aminah in the year 2004.

Jadual1 menunjukkan harga dan indeks harga bagi empat jenis makanan dalam tahun

2006 berasaskan tahun 2004. Rajah 1 menunjukkan carta pai bagi perbandingansebahagian perbelanjaan Puan Aminah pada tahun 2004.

Price per kg

Harga se kgFood item

Makanan Year 2004

Tahun 2004

Year 2006

Tahun 2006

Price index in 2006

based on 2004

Indeks harga tahun2006 berasaskan

tahun 2004

P x RM 720 120

Q RM 500 y 110

R RM 400 RM 520 130

S RM 700 RM 910 z

Table 1 /J adual 1

70

80120P

Q

R

S

Diagram 1 /Rajah1

(a) Find the values ofx,y andz.Cari nilai-nilai x, y dan z.

(b) Calculate the price index ofPin the year 2004 based on the year 2002 if the

price ofPin the year 2002 is RM 545.Hitung indeks harga bagi P pada tahun 2004 berasaskan tahun 2002jika

harganya pada tahun 2002 ialah RM 545.

(c) Find the composite index for the four types of food in the year 2006 based on

the year 2004.

Cari nombor indeks gubahan bagi empat jenis makanan itu pada tahun 2006

berasaskan tahun 2004.



67/111


2. Diagram 2 is a bar chart indicating the weekly cost of the itemsP, Q,R, Sand T for the

year 1990. Table 2 shows the prices and the price indices for the items.Rajah 2 menunjukkan carta bar bagi kos mingguan bagi item P, Q, R, S dan T pada

tahun 1990. Jadual2 menunjukkan harga dan indeks harga bagi barangan-barangan

itu.


Diagram 2 /Rajah2

P Q R S T0

1215

24

3033

Items

Barangan

Weekly cost (RM)

Kos mingguan

Items

Barangan

Price inyear 1990

Harga padatahun 1990

Price inyear 1995

Harga padatahun 1995

Price index in 1995based on 1990

Indeks harga tahun 1995

beasaskan tahun 1990

P x RM 070 175Q RM 200 RM 250 125R RM 400 RM 550 yS RM 600 RM 900 150T RM 250 z 120

Table 2 /J adual 2

(a) Find the value of / Cari nilai(i) x (ii) y (iii) z

(b) Calculate the composite index for items in the year 1995 based on the year

1990.

Hitung indeks komposit bagi item-item ini pada tahun 1995 berasaskan tahun1990.

(c) The total weekly cost of the items in the year 1990 is RM456.

Calculate the corresponding total weekly cost for the year 1995.

Jumlah kos mingguan bagi item-item ini pada tahun 1990 ialah RM456.

Hitung jumlah kos mingguan yang sepadan bagi tahun 1995.

(d) The cost of the items increases by 20% from the year 1995 to the year 2000.

Find the composite index for the year 2000 based on the year 1990.

Kos bagi item-item meningkat sebanyak20% dari tahun 1995 kepada tahun2000. Cari indeks komposit bagi tahun 2000 berasaskan tahun 1990.


68/111



3. A particular kind of cake is made by using four ingredients,P, Q, R and S.Table 3 shows the prices of the ingredients.

Sejenis kek dibuat dengan menggunakan empat jenis bahan, P, Q, R dan S.

Jadual3 menunjukkan harga bagi bahan-bahan tersebut.

Price per kilogramHarga sekilogram (RM)Ingredient

Bahan Year 2004Tahun 2004

Year 2005Tahun 2005

P 500 xQ 250 400R y 1000S 400 440

Table 3 /J adual 3

(a) Given that the index number of ingredientPin the year 2005 based on the year2004 is 120, calculate the value ofx.

Diberi bahawa nombor indeks bagi bahan P dalam tahun 2005 berasaskan

tahun 2004 ialah 120, hitung nilai x.

(b) The index number of ingredientR in the year 2005 based on the year 2004 is 125.

Nombor indeks bagi bahan R dalam tahun 2005 berasaskan tahun 2004 ialah 125.Calculate the value ofy.

Hitung nilai y.

(c) The composite index for the cost of making the cake in the year 2005 based on the

year 2004 is 1275.Indeks komposit bagi kos untuk membuat kek dalam tahun 2005 berasaskan tahun

2004 ialah 1275.

Calculate /Hitung

(i) the price of a cake in the year 2004 if its corresponding price in the year 2005

is RM3060,harga kek bagi tahun 2004jika harganya yang sepadan dalam tahun 2005

ialah RM3060,

(ii) the value ofm if the quantities of ingredientsP, Q, R and Sused are in the

ratio of 7 : 3 : m : 2.

nilai m jika kuantiti bahan-bahan P, Q, RdanSyang digunakan adalahmengikut nisbah 7 : 3 : m : 2.


69/111


12. PROGRESSIONS


ARITHMETIC PROGRESSIONS (AP)

Is a sequence of the terms in which each term is obtained by adding a constantto the previous term.

The difference between two consecutive terms is known as the commondifference, d.

o 1n nd T T=

The n-th term of theAPo

where a is the first term( 1)nT a n d = +

dis the common difference.

n is the number of terms.

Sum of the first n terms ofAP

o [2 ( 1) ]2

n

nS a n= + d where a is the first term

dis the common difference

n is the number of terms.

GEOMETRIC PROGRESSIONS (GP)

Is a sequence of terms in which each term is obtained by multiplying the precedingterm by a constant.

This constant is known as common ratio, r.

o1

n

n

Tr

T=

The n-th term of the GP, 1nnT ar=

Sum of the first n terms of GP

or

(1 ), 1

1

( 1), 1

1

n

n

n

n

a rS r

r

a rS r

r

=



70/111


The sum to infinity of the GP, where 1< r< 1

1

aS

r =

T1 = S1T2 = S2 S1T3 = S3 S2and so on

Relation between Tn and Sn

Tn= Sn Sn 1

HHPAPER 1JJ

1. The first three terms of an arithmetic progression are 2k 3, 4k+ 1 and 5k+ 6.Tiga sebutan pertama suatu janjang aritmetik adalah 2k 3, 4k+ 1 dan 5k+ 6.

Find / Cari(a) the value ofk,

nilai k,(b) the eighth term.

sebutan kelapan.

Answer : (a) k= ....

(b) ..

2. The first three terms of an arithmetic progression are m 3, m + 3, 2m + 2.Tiga sebutan pertama suatu janjang aritmetik adalah m 3, m + 3, 2m + 2.

Find / Cari

(a) the value ofm,nilai m,

(b) the sum of the first 9 terms of the progression.

hasil tambah 9 sebutan pertama janjang ini.

Answer : (a) m = ..

(b) ..



71/111


3. Given that 25, 22, 19,. are the first three terms in an arithmetic progression, findthe sixteenth term.Diberi 25, 22, 19,. adalah tiga sebutan pertama bagi suatu janjang aritmetik, cari

sebutan keenam belas.

Answer :

4. The first three terms of an arithmetic progression are 3, 7, 11. FindTiga sebutan pertama suatu janjang arithmetik adalah 3, 7, 11. Cari

(a) the common difference of the progression,

beza sepunya janjang itu,

(b) the sum of the first 10 terms after the third term.

hasil tambah sebutan sebutan pertama selepas sebutan ketiga.

Answer : (a) ..

(b) ..

5. Given a geometric progression k, 3,9

k, m,. express m in terms ofk.

Diberi janjang geometrik, 3,9

k, m, ungkapkan m dalam sebutank.

Answer :



72/111


13. LINEAR LAW


Reducing non-linear relations to linear form.

y Y


x XOO

c

Y= mX+ c

y =f(x)

DiagramA DiagramB

Non-linear graphs refer to any graph Linear graph refers to a straight line graph

with gradient, m andy-intercept, c. DiagramB

shows a linear graph.

which is not linear such as quadratic,

cubic, reciprocal etc. DiagramA shows

a quadratic graph.

o A non-linear function which has variablesx andy can be reduced to the linearfunction :

Y= mX+ c

y only or bothx andy

but without constantx and a constant

but withouty

a constant only

IMPORTANT STEPS IN PAPER 2

Construct a table value. Plot all points on the correct axis. Draw a line of best fit.

Reduce the equation to linear form. From the graph obtained,

(a) read the value ofc, that isy-intercept

(b) find the gradient, m by using m = 2 1

2 1

y y

x x

(c) find the values of constants.

(d) find the value of variables.


73/111


HHPAPER 1JJ

1. Diagram 1 shows the linear graph ofy against1

x.

Rajah 1 menunjukkan graf linear y melawan 1x

.

O

y

1

x(3, 0)

(0, 6)

Diagram 1 /Rajah1

Expressy in terms ofx

Ungkapkan y dalam sebutan x.Answer :

2. Diagram 2 shows the linear graph ofy

xagainstx

2. The variablesx andy are related by

the equationy = ax3

bx, where a and b are constants.

Rajah 2 menunjukkan graf lineary

xmelawanx

2.Pembolehubah x dan y dihubungkan

oleh persamaany = ax3 bx, dengan keadaan a dan b adalah pemalar.

x2

y

x

(12, 2)

4

O

Diagram 2 /Rajah2

Determine the value ofa and ofb.

Tentukan nilai a dan nilai b.

Answer : a = b =



74/111


3. The variablesx andy are related by the equationx

ky

5= ,where kis a constant.

Diagram 3 shows the line graph obtained by plotting log10y againstx.

Pembolehubah x dan y dihubungkan oleh persamaanx

ky

5= , dengan keadaan k

adalah pemalar.Rajah 3 menunjukkan graf yang diperoleh dengan memplotlog10 ymelawan x.

x

log10y

(0, 2)

O

Diagram 3 /Rajah3

(a) Express the equationx

ky

5= in its linear form used to obtain the straight line graph

shown in Diagram 5.

Ungkapkan persamaanx

ky

5= dalam bentuk linear yang digunakan untuk

memperoleh graf garis lurus seperti yang ditunjukkan dalam Rajah 5.

(b) Find the value ofk.

Cari nilai k.

Answer : (a) ..

(b) k= ..



75/111


HHPAPER 2JJ

4. Table 4 shows the values of two variables,x andy, obtained from an experiment.

Variablesx andy are related by the equation y = hx +k

hx , where h and kare constants.

Jadual4 menunjukkan nilai-nilai bagi dua pembolehubah,x dan y,yang diperoleh

daripada satu eksperimen.Pembolehubah x dan y dihubungkan oleh persamaan

y = hx +k

hx, dengan keadaan h dan k adalah pemalar.

x 10 20 30 40 50 55

y 55 47 50 65 77 84

Table 4 /J adual 4

(a) Plotxy againstx2, by using a scale of 2 cm to 5 units on both axes.

Hence, draw the line of best fit.

Plotkan xy melawan x2, dengan menggunakan skala 2 cm kepada 5 unit pada

kedua-dua paksi.

Seterusnya, lukiskan garis lurus penyuaian terbaik.

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

Gunakan grafanda dari (a) untuk mencari nilai

(i) h,

(ii) k.



76/111


5. Table 5 shows the values of two variables,x andy, obtained from an experiment.

Variablesx andy are related by the equation y = 2kx2

+px

k, wherep and kare

constants.

Jadual5 menunjukkan nilai-nilai bagi dua pembolehubah,x dan y,yang diperoleh

daripada satu eksperimen.Pembolehubah x dan y dihubungkan oleh persamaan

y = 2kx2

+px

k, dengan keadaan p dan k adalah pemalar.

x 2 3 4 5 6 7

y 8 132 20 275 366 455

Table 5 /J adual 5

(a) Ploty

againstx, by using a scale of 2 cm to 1 unit on both axes.

Hence, draw the line of best fit.

Plotkany

melawan x, dengan menggunakan skala 2 cm kepada 1unit pada

kedua-dua paksi.

Seterusnya, lukiskan garis lurus penyuaian terbaik.

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

Gunakan grafanda dari (a) untuk mencari nilai

(i) p,

(ii) k,

(iii)y whenx = 12.y apabila x = 12.



77/111


14. INTEGRATION


INDEFINITE INTEGRAL

Integration is the reverse process of differentiation.

Ifdy

dx=f(x), theny = ( )f x dx

Integration of algebraic functions.

o =nax dx

1

1

n

ax cn

+

++, where a and c are constants.

o nx dx =1

1

nx

cn

+

++

, where c is a constant.

o = kx + c, where kand c are constants.k dx

o =( )nax b dx+1

( )

( 1)( )

nax b

cn a

+++

+, where a, b and c are constants.

differentiation in

the brackett

To obtain equation of a curve from given gradient function.

If the gradient function isdy

dx=f(x), then its equation isy = ( )f x dx (where c must

be found)

DEFINITE INTEGRAL

( )b

a

f x dx = [ ] ( )b

ag x

= g(b) g(a).

Example : Evaluate

2

2

1

3x dx .

Solution :2

3

1x

= 23

13

= 7



78/111


Characteristics of Integral

o = , where kis a constant.( )b

a

k f x dx ( )b

a

k f x dx

o ( )b

a

f x dx = ( )a

b

f x dx .

o [ ( ) ( )]b

a

f x g x dx = ( ) ( )b b

a a

f x dx g x dx .

o ( ) ( )b c

a b

f x dx f x dx+ = ( )c

a

f x dx .

AREA UNDER A CURVE

y =f(x)

a b x

y

0 x

y

0

a

b

y =f(x)


(a) The area under a curve which is

enclosed byx = a,x = b andx-axis is(b) The area under a curve which is

enclosed byy = a,y = b andy-axis is

Ay=

b

a

Ax =

b

a

x dy dx

The limits are on

y-axis and write

x in terms ofy.

The limits are on

x-axis and write

y in terms ofx.


79/111



VOLUMES OF REVOLUTIONS

x

y

a

b

y =f(x)

0

x

y

0

y =f(x)

a b

(a) The volume generated when the

region bounded by the curve y =f(x) ,

x = a andx = b is rotated through 360

about thex-axisis

(b) The volume generated when the

region bounded by the curve y =f(x) ,

y = a andy = b is rotated through 360

about they-axisis

Vy=2

b

a

x dy Vx = 2b

a

dx

The limits are ony-axis andThe limits are onx-axis andwritex2in terms ofy.writey2in terms ofx.


80/111


HHPAPER 1JJ

1. Find 32(1 4 ) dx .

Cari 32(1 4 ) dx

Answer :

2. Find2

16

(4 1)x dx.

Cari2

16(4 1)x

dx.

Answer :

3. Integrate2

2

3 2x

x

with respect tox.

Kamirkan2

2

3 2x terhadap x.

Answer :

Hak Cipta Cayzec Montoya SMK Sultan Sulaiman Kuala Terengganu 804. Evaluate

2

21

(2 )(2 )x

x

+ dx.


81/111


Nilaikan2

21

(2 )(2 )x

x

+ dx.

Answer :

5. Given that

4

0

( )f x dx = 12, evaluate4

0

1( ) 2

2f x d

+

x .

Diberi bahawa

4

0

( )f x dx = 12, nilaikan4

0

1( ) 2

2f x d

+

x .

Answer :

6. Given that3

1

( )f x dx = 6, evaluate3

1

[2 ( ) 5]f x d x .



82/111


Diberi bahawa3

1

( )f x dx = 6, nilaikan3

1

[2 ( ) 5]f x d x .

Answer :

7. Given that4

0

( )f x dx = 5 and5

4

( )f x dx = 2, find the value of

Diberi bahawa

4

0

( )f x dx = 5 dan5

4

( )f x dx = 2, cari nilai bagi

(a)

4

5

2 ( )f x dx ,

(b)

5

0

3 ( )f x dx .

Answer : (a) ..

(b) ..

HHPAPER 2JJ



83/111



8. Given that kx2

x is a gradient function of a curve where kis a constant.

y 5x + 7 = 0 is the equation of tangent to the curve at the point (1, 2). FindDiberikx

2xadalah fungsi kecerunan bagi suatu lengkung di mana k ialah pemalar.

y 5x + 7 = 0 ialah persamaan tangen kepada lengkung itu pada titik(1, 2). Cari(a) the value ofk,

nilai k,

(b) the equation of the curve.

persamaan lengkung itu.

9. In Diagram 9, the straight linePQ is normal to the curve2

12

xy = + atA(2, 3).

The straight lineAR is parallel to they-axis.

Dalam Rajah 9,garis lurus PQ adalah normal kepada lengkung2

12

xy = + pada titik

A(2, 3).garis lurus AR adalah selari dengan paksi-y.

x

A(2, 3)

y

Q(k, 0)0

P2

12

xy = +

R

Diagram 9 /Rajah9

Find / Cari

(a) the value ofk,

nilai k,

(b) the area of the shaded region,

luas rantau berlorek,

(c) the volume generated, in terms of, when the region bounded by the curve, they-axis and the straight liney = 3 is revolved through 360 about they-axis.

isipadu yang dijanakan, dalam sebutan , apabila rantau yang dibatasi oleh

lengkung,paksi-y dan garis lurus y = 3 diputarkanmelalui 360pada paksi-y.

15. VECTOR


84/111



A vector quantity is a quantity that has magnitudeand direction.

A vector in the direction fromA toB can be denoted as or aor u or any other

suitable letters.

AB

%


=AB

BA

A

B

a u%

The magnitude of a vector is denoted as | | or | a |.AB

AB

A vector can be multiplied by a scalar, k .AB

Triangle Law of Vector Parallelogram Law of Vector

P

Q

R

PQ QR

+ = PR

a

b

a+ ba

b

Vectors in a Cartesian Plane

In aCartesian plane, a vector can be expressed in the form :

o xi yj+% %

orx

where is a unit vector which is parallel tox-axis andi%

j%

is a

unit vector parallel toy-axis.

o In the diagram, = 3PQ

4i j+% %

or = .PQ

4

3

P

Q

o The magnitudeof ,PQ

|PQ | =

2 2x y+

| | =PQ

2 23 4+

= 5 unit

o For a vector =r%

xi yj+% %

, the unit vector in the direction of isr%


85/111


=r% | |

r

r%

%

=2 2

xi yj

y

+

+% %

To prove collinear and parallel vector

A

B

P

A

B

Q

R

(a) To prove thatA,PandB are collinear,

you have to show that AB = or = (must find the value of).

AP

AB

PB

(b) To prove thatAB is parallel to QR,

you have to show that AB = (must find the value of).

QR

Problems on non-parallel vector

%

x%

o Ifx%

and are non-parallel vectors where hy% %

+ k = my% %

+ n , theny%

(a) h = m and k= n (by comparing the coefficient of each vector).then solve the simultaneous equations.

OR

(b) rearrange : h mx%

= n k (such thaty


% % %

y%

and are separated)y%

then factorize : (hm)%

= (n k)y%

and then solve the simultaneous equations : h m = 0

n k= 0


86/111


HHPAPER 1JJ

1. Diagram 1 shows vectorOPdrawn on a Cartesian plane.

Rajah 1 menunjukkan vektor yang dilukis pada satah Cartesan.OP

O 2 4 6 8

2

4

6

y

x

P

Diagram 1 /Rajah1

(a) Express OP in the form x

y

.

Ungkapkan OPdalam sebutan x

y

.

(b) Find |OP|.

Cari |OP|.

.

Answer : (a) OP= ..

(b) |OP| = ..



87/111


2. Diagram 2 shows two vectors, and QO .OP

Rajah 2 menunjukkan dua vektor, danQO .OP

x

y

P(7, 5)Q(3, 4)

O

Diagram 2 /Rajah2

Express

Ungkapkan

(a) OP in the form xy

.

dalam bentukOP x

y

.

(b) QP in the formxi +yj.

dalam bentukxi +yj.QP

Answer : (a) ..

(b) ..

3. Given that O(0, 0),P(3, 3) and Q(2, 12), find in terms of the unit vectors, i andj,DiberiO(0, 0),P(3, 3) danQ(2, 12), cari dalam sebutan vektor unit, idanj,

(a) ,PQ

(b) the unit vector in the direction of .PQ

vektor unit dalam arah .PQ

Answer : (a) PQ = ..

(b) ..



88/111


4. Diagram 4 shows vector .AB

Rajah 4 menunjukkan vektor .AB

A

B

Diagram 4 /Rajah4

(a) Express AB in the formxi +yj.

Ungkapkan dalam bentukxi +yj.AB

(b) Find the unit vector in the direction of .AB

Cari vektor unit dalam arah .AB

Answer : (a) AB = ..

(b) ..

5. Diagram 5 shows a rectangle OPQR and the point Tlies on the straight line OQ.

Rajah 5 menunjukkan sebuah segiempat tepatOPQRdan titik T terletak pada garis

lurusOQ.

PO

QR

T

12%

9y%

Diagram 5 /Rajah5

It is given that OT= 2TQ. Express in terms ofOT

%and .y

%

Diberi bahawaOT= 2TQ. Ungkapkan OTdalam sebutan

x% %

dan .y

Answer : OT=



89/111


HHPAPER 2JJ

6. Diagram 6 shows a trapeziumABCD.

Rajah 6 menunjukkan sebuah trapeziumABCD.

A B

CD T

S

Diagram 6 /Rajah6

It is given that DA =

4%

, = 12 , =DC

y%

DT 3

4DC

, =DC

2

3AB

and TS= 1

2DA

.

Diberi bahawa D =A

4%

, = 12 , =DC

y%

DT 3

4DC

, =DC

2

3AB

danTS= 1

2DA

.

(a) Express in terms ofx%

and y%

Ungkapkan dalam sebutanx%

dany%

(i) DB ,

(ii) DS.

*(b) Hence, prove thatD, Sand B are collinear

Seterusnya, buktikan bahawa D, S dan B adalah segaris.



90/111


7. In Diagram 7, = 4a, = 2b and = a.PQ

PS

SR

Dalam Rajah 7, = 4a, = 2bdan = a.PQ

PS

SR

P Q

RS

T

Diagram 7 /Rajah7

(a) Express in terms ofa and b,Ungkapkan dalam sebutanadanb,

(i) PR (ii)

QS

(b) Given that PT= h and QT= k , express

PR

QS

PT

Diberi bahawa = h dan = kQ , ungkapkanPT

PR

QT

S

PT

(i) in terms of h, a and b,dalam sebutanh, adanb,

(ii) in terms of k, a and b.dalam sebutank, adanb,

*(c) Hence, find the value ofh and k.Seterusnya, cari nilai h dan nilai k.



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16. TRIGONOMETRIC FUNCTIONS


Six Trigonometric Functions


P(x,y)

y

x

r

cot = cossin

cosec =sin

1

sec =cos

1

cot =tan

1

sin =r

y

cos =r

x tan = sin

cos

tan =y

.

Relation between complementary angles

x

yr

90

sin =r

yand cos (90) =

r

y sin = cos (90)

cos =r

xand sin (90) =

r

x sin = cos (90)


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Negative angle

sin

costan

all (positive)

(clockwise)

x

y

sin () = sin cos () = cos tan () = tan

GRAPHS OF TRIGONOMETRIC FUNCTIONS

Sine graph


y1

xO

y = sinx

-1

36027018090

Cosine graph

Tangent graph

xO

1

-1

y

y = cosx

36027018090

y = tanx

O 36027018

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