Modul Jom Lulus Matematik

JOM LULUS MATEMATIK

Kaedah menyelesaikan soalan “silmultaneous linear equations” ,

2 langkah:

1) Kenal pasti anu atau huruf yang mudah diselesaikan dan darabkan mengikut nombor yang besar.

2) Kenal pasti tanda + Atau – dan pastikan tanda yang sama + dan + atau – dan - akan berlaku penolakan-tanda yang berbeza + dan – atau – dan + akan berlaku penambahan.

Contoh:

3 q + 5 r = 12q + 3 r = 4

Huruf q dipilih dan perlu didarabkan dengan nombor yang lebih besar, 33 q + 5 r = 24 ----------------1

x3 x3 x3( q + 3 r = 4 )-------------- 2

+3 q + 9 r = 12 ----------------3

Tanda sama-Penolakan - - - 3-1+3 q + 5 r = 24 ----------------1

9 r –(+5r ) = 12-24 gantikan r = -39 r – 5 r = -12 q + 3 r = 4 4 r = -12 q + 3 (-3) = 4

r = q -9 = 4

r = -3 q = 4 + 9 q = 13

Catatan- penyelesaian ini hanya untuk soalan “simultaneous linear equations” sahaja.

Tanda Penyelesaian

+ + Penolakan (-)

- - Penolakan (-)

+ - Penambahan (+)

- + Penambahan (+)

1

JOM LULUS MATEMATIK 1

Section A

1. Solve the following quadratic equation:Selesaikan persamaan kuadratik berikut:

5 x2 + 5 x = 2(3 –x) [ 4 marks]

[ 4 markah]

2. Calculate the value of x and y that satisfy the following simultaneous linear equations:Hitung nilai x dan nilai yang memuaskan persamaan linear serentak berikut:

2 a – b = 9a +3 b = -6

[ 4 marks]

[ 4 markah]

2

3. . On the graph provided, shade the region which satisfies the three inequalities

, ,

[ 4 marks]

[ 4 markah]

4. a) Write down two implications based on the following compound statement: Tulis dua implikasi berdasarkan pernyataan berikut:

x 3 = 64 if and only if x = 4 x 3 = 64 jika dan hanya jika x = 4

Implication 1/Implikasi 1:...............................................................................................Implication 2/Implikasi 2:...............................................................................................

[2 marks][2 markah]

b) Make a general conclusion by induction for the sequence of numbers 4, 6, 20, 58... which folloes the following pattern. Buat satu kesimpulan umum secara aruhan bagi urutan nombor 4, 6, 20, 58...yang mengikut pola berikut.

4= 3(0)3 + 56= 3(1)3 + 520= 3(2)3 + 558= 3(3)3 + 5...=...............

[2 marks][2 markah]

3

xy21

5 xy

y

x

5

5

5) Write the following simultaneous linear equations as matrix equation.

Tulis persamaan linear serentak berikut dalam bentuk persamaan matriks.

3 x – 2 y = 8

6x - 5 y = 21

Hence, using matrix method, calculate the value of x and y.

Seterusnya, dengan menggunakan kaedah matriks, hitung nilai x dan nilai y.

[4 marks][4 markah]

4

Section B

6 . a) Complete the table in the answer space for the equation y = 8-3x -2x2 by writing down

the values of y when x =-4 and x = 1.

Lengkapkan jadual di ruang jawapan bagi persamaan y = 8-3x -2x2 dengan menulis

nilai-nilai y apabila x =-4 dan x = 1.

[2 marks]

[2 markah]

b) By using a scale of 2 cm to 1 unit on the x –axis and 2 cm to 5 units on the y- axis, draw

the graph of y = 8-3x -2x2 for -5 ≤ x ≤ 3 and -27 ≤ y ≤ 9

Dengan menggunakan skala 2 cm kepada 1 unit pada paksi x dan 2 cm kepada 5 unit

pada paksi y,lukis graf y = 8-3x -2x2 for -5 ≤ x ≤ 3 and -27 ≤ y ≤ 9.

[4 marks]

[4 markah]

c) From your graph, find

Dari graf, cari

i) the value of y when x = -2.4nilai y apabila x = -2.4

ii) the value of x when y = -11nilai x apabila y = -11

[2 marks] [2 markah]

Answer:

a) y = 8-3x -2x2

x -5 -4 -3.5 -2 -1 0 1 2 3

y -27 -6 6 9 8 -6 -19

b) Refer graph

5

c) i) y =.....................................

ii)x = ....................

6

Graph for Question 6

7) Transformation R is a 90° clockwise rotation at centre (2, 2).

Transformation T is a translation .

State the coordinate of the image for coordinate (6 , 4) under the following

transformations:

(i) R2.

(ii) TR.

Answer:

(a) (i)

(ii)

(b) Diagram 9 shows quadrilateral , ABCD, PQRS and EFGH, drawn on a

Cartesian plane.

DIAGRAM 9

PQRS is the image of ABCD under transformation S and EFGH is the image of PQRS under transformation Q. (i) Describe in full transformation :

(a)Transformation S

(b)Transformation Q

(ii) Given the area of ABCD is 64 unit2, calculate the area of shaded region.

[12 marks]

7

-12 -10 -8 -6 -4 -2 2 4 6 8 10 12F

E

GH

S

QP

DC

B A

y

xO

R

6

4

2

- 2

- 4

Answer:

(b) (i) (a)

(b)

(ii)

8. The data in Diagram 5 shows the ages of members of a golf club.

45 53 48 54 46 53 55 43 47 52

63 57 50 40 52 45 49 61 54 56

51 41 56 51 61 50 53 48 51 44

57 53 47 55 46 54 42 57 58 63

42 64 50 49 52 47 55 52 45 51

Diagram 5

a) Based on the data in Diagram 5 and using a class interval of 5 years,

complete Table 4 in the answer space. [4 Marks]

b) Based on the table in a), calculate the estimated mean age of the members of the golf club. [4 Marks]

c) By using a scale of 2 cm to 5 years on the x-axis and 2 cm to 2 persons on the y-axis, draw a histogram based on the data. [4 Marks

Answer

a)

Age Frequency Midpoint

40-44

45-49

8

Table

b)

c)Refer graph

9


9.

(a) Diagram 15 (i) shows a solid right prism with rectangular base ABHG on a horizontal plane. The surface

ABCDE is the uniform cross-section of the prism EA and CB are vertical edges. Rectangle EDFK and

rectangle DCJK are inclined planes. The height of D from AB is 5 cm and ED = DC.

Rajah 15 (i) menunjukkan sebuah pepejal berbentuk prisma tegak dengan tapak segiempat tepat ABHG

terletak di atas satah mengufuk. Permukaan ABCDE ialah keratan rentas seragam prisma itu. Tepi EA dan

CB adalah tegak. Segiempat tepat EDFK dan segi empat tepat DCJK ialah satah condong. Tinggi D dari AB

ialah 5 cm dan ED = DC.

Rajah 15 (i)

Lukis dengan skala penuh, pelan pepejal itu.

Sebuah pepejal lain berbentuk prisma tegak dengan trapezium PQRS sebagai ketatan rentas seragam dicantumkan kepada prisma dalam Rajah 15.1 pada satah mencancang PHTS. Gabungan pepejal adalah seperti yang ditunjukkan dalam Rajah 15.2. Tapak ABPQVHG terletak di atas satah mengufuk dan segi empat tepat SRUT ialah satah condong.

10

A

H

EC

F

K

J

7 cm

B

4 cm

D

4 cm

G

KA

H

EC

F

J

3 cm

B

4 cm

D

4 cm

G

4 cm

4 cm

4 cmT

S

PQ

U

VR

X

Y

Draw to full scale,

Lukis dengan skala penuh,

(i) the elevation of the composite solid on a vertical plane parallel to AB as viewed from X,

Dongakan gabungan pepejal itu pada satah mencancang yang selari dengan AB sebagaimana dilihat dari X, [4 markah]

(ii) the elevation of the composite solid on a vertical plane parallel to QV as viewed from Y.

Dongakan gabungan pepejal itu pada satah mencancang yang selari dengan QV sebagaimana dilihat dari Y. [5 markah]

11

Diagram 15 (ii)

Rajah 15 (ii)


Section A

1. Using factorization, solve the following quadratic equation:Menggunakan pemfaktoran, selesaikan persamaan kuadratik berikut:

= m+3

[ 4 marks/4 markah]


x + 2 y = -13 x – y = 11

[ 4 marks/4 markah]

12

.3. On the graph in the answer space, shade the region which satisfies the three inequalities , y < x and y > 1. [3 marks]

Pada graf di ruang jawapan, lorek rantau yang memuaskan ketiga-tiga ketaksamaan , y < x and y > 2. [3 markah]

Answer/ Jawapan:

4(a) State whether the following compound statement is true or false:

13

y

y = 2

y = – x + 20

x O

Nyatakan sama ada pernyataan berikut adalah benar atau palsu:

(b) Write down the converse of the following implication.

Hence, state whether the converse is true or false.

Tulis akas untuk implikasi berikut.

Seterusnya, nyatakan sama ada akas tersebut adalah benar atau palsu.

(c) Write down Premise 2 to complete the following argument:

Tulis Premis 2 untuk melengkapkan hujah berikut:

Premise 1: If the radius of a circle is 14 cm, then the circumference of the circle is cm.

Premis 1: Jika jejari sebuah bulatan ialah 14 cm, maka lilitan bulatan itu ialah cm.

Premise 2:/ Premis 2: ……………………………………………………………………………..

Conclusion: The radius of the circle is not 14 cm.

Kesimpulan: Jejari bulatan itu bukan 14 cm.

[4 marks]

[4 markah]

5. It is given that matrix is

Diberi bahawa matriks

(a) Find the inverse matrix of A.

Cari matriks songsang bagi A.

14

Some even numbers are divisible by 3.

Sebilangan nombor genap boleh dibahagikan dengan 3.

If x is a multiple of 12, then x is a multiple of 3.

Jika x adalah gandaan bagi 12, maka x adalah gandaan bagi 3.

(b) Write the following simultaneous linear equations as matrix equation:

Tulis persamaan linear serentak berikut dalam bentuk persamaan matriks:

Section B

6 . a) Complete the table in the answer space for the equation y = 2x2 – 5x -3 by writing down

the values of y when x =-2 and x = 0.5

Lengkapkan jadual di ruang jawapan bagi persamaan y = 2x2 – 5x -3 dengan menulis

nilai-nilai y apabila x =-2 dan x = 0.5

[2 marks]

[2 markah]


the graph of y = 2x2 – 5x -3 for -3 ≤ x ≤ 5


pada paksi y,lukis graf y = 2x2 – 5x -3 for -3 ≤ x ≤ 5

[4 marks]

[4 markah}


Dari graf, cari

iii) the value of y when x = -2.4nilai y apabila x = -2.4

iv) the value of x when y = 4nilai x apabila y = 4

[2 marks] [2 markah]

d) Answer: y = 2x2 – 5x -3

15

x -3 -2 -1 0 0.5 1 2 3 4 5

y 30 4 -3 -6 -5 0 9 22

e) Refer graphf) i) y =.....................................

ii)x = ........................

16


7) Diagram 13.1 shows the point (-2, 4) and the straight line x = 1 drawn on a Cartesian plane.

Rajah 13.1 menunjukkan titik (-2, 4) dan garis lurus x = 1 dilukis pada suatu satah Cartesan.

Diagram 13.1

17

y

x– 4

– 2

2 4

2

4

– 2

– 4

O

(- 2, 4)

6 8 10

6

x = 1

Rajah 13.1

(a) Transformation T is a translation

Transformation P is a reflection in the straight line x = 1.

State the coordinates of the image if point (- 2, 4) under the following transformation:

Penjelmaan T ialah translasi

Penjelmaan P ialah satu pantulan pada garis lurus x=1.

Nyatakan koordinat imej bagi titik (-2, 4) di bawah penjelmaan berikut:

(i) T

(ii) TP [4 marks/4 markah]

(b) Diagram 13.2 shows pentagons, ABCDE, JQRSU and JKLMN, drawn on a Cartesian plane.

Rajah 13.2 menunjukkan pentagon, ABCDE, JQRSU dan JKLMN, dilukis pada suatu satah Cartesan.

Diagram 13.2

Rajah 13.2

18

N

842

y

2

4

- 2

10

6

O 6 x

B

D

A

J

U

- 4

C

E

ML

K

Q

RS

- 6

(i) JKLMN is the image of ABCDE under a combined transformation HG.

Describe in full the transformation:

JKLMN ialah imej bagi ABCDE di bawah gabungan penjelmaan HG.

Huraikan selengkapnya penjelmaan

(a) G,(b) H.

(ii) It is given that pentagon ABCDE represents a region of area 80 m2. Calculate the area in m2, of the region represented by the coloured region.

Diberi bahawa pentagon ABCDE mewakili luas 80 m2.

Hitung luas, dalam m2, kawasan yang diwakili oleh kawasan yang berwarna.

[8 marks/8 markah]

8. The data in Diagram 7 shows the height, in cm, of 40 students in a class

(a) Based on the data in Diagram 7, complete Table in the answer space. [3 marks]

Heights Frequency Midpoint

141-145 2

145 174 150 153 153 148 175 154

168 171 166 159 169 175 165 169

160 167 155 160 160 143 176 153

161 167 170 148 177 165 167 165

170 156 162 162 161 157 179 165

19

(b) Using the completed table, in (a)

i) state the modal class

ii) estimate the mean mark [4 marks]

(c) By using the scale of 2cm to 5 cm on the x- axis and 2 cm to 1 student on the y- axis, draw a

frequency polygon for the data. [4 marks]

20


9 )Diagram 9 shows a solid right prism with rectangular base FGPN on a horizontal table . EFGHJK is

the uniform cross-section of the prism . Rectangle EKLM is an inclined plane. Rectangle JHQR is a

horizontal plane. EF , KJ and HG are vertical edges.

Draw a full plan of the solid. [3 marks]

21

F

G

P

Q

R

N

H

J

EK

LM

5 cm

7 cm 5 cm

5 cm

2 cm

2 cm

Diagram 9(i)

( b ) A solid cuboid is joined to the prism in Diagram9(i) at the vertical plane PQRLMN. The

combined solid is as shown in Diagram 9(ii).The square base FGSW is a horizontal.

Draw full scale

i)the elevation of the combined solid on a vertical plane parallel to FG as viewed from C, [4 marks]

ii)the elevation of the combined solid on a vertical plane parallel to GPS as viewed from D.[5 marks]

22

F

G

P

Q

R

N

H

J

EK

LM

5 cm

7 cm 5 cm

5 cm

2 cm

2 cm

Diagram 9(ii)

SS

T

X

V U

W

7 cm

CD

2 cm

7 cm


Section A

1.Solve the following quadratic equation: Selesaikan persamaan kuadratik berikut:

x2 - 2 x + 3 = 2(x-1) [ 4 marks/4 markah]

23


p - q = 6

2 p +3 q = -5

[ 4 marks / 4 markah]

]


, ,

[ 4 marks]

[ 4 markah]


x 3 = 64 if and only if x = 4

24

xy21

5 xy

y

x

5

5

x 3 = 64 jika dan hanya jika x = 4


[2 marks][2 markah]


4= 3(0)3 + 56= 3(1)3 + 520= 3(2)3 + 558= 3(3)3 + 5...=...............

[2 marks][2 markah]



4 x – 2 y = 7

2x + 5 y = -2



[4 marks][4 markah]

25

Section B





[2 marks]

[2 markah]





[4 marks]

[4 markah]

26


Dari graf, cari

v) the value of y when x = -2.4nilai y apabila x = -2.4

vi) the value of x when y = -11nilai x apabila y = -11

[2 marks]

[2 markah]

Answer:

g) y = 8-3x -2x2

x -5 -4 -3.5 -2 -1 0 1 2 3

y -27 -6 6 9 8 -6 -19

h) Refer graphi) i) y =.....................................

ii)x = ......................

27





transformations:

(i) R2.

(ii) TR.

Answer:

(a) (i)

(ii)


Cartesian plane.

28 -12 -10 -8 -6 -4 -2 2 4 6 8 10 12F

E

GH

S

QP

DC

B A

y

xO

R

6

4

2

- 2

- 4

DIAGRAM 9

PQRS is the image of ABCD under transformation S and EFGH is the image of PQRS under transformation Q.

(i) Describe in full transformation :

(a)Transformation S

(b)Transformation Q


[12 marks]

Answer:(b) (i) (a)

(b (ii)


45 53 48 54 46 53 55 43 47 52

63 57 50 40 52 45 49 61 54 56

51 41 56 51 61 50 53 48 51 44

57 53 47 55 46 54 42 57 58 63

42 64 50 49 52 47 55 52 45 51

Diagram 5



29

d) Based on the table in a), calculate the estimated mean age of the members of the golf club. [4 Marks]

e) By using a scale of 2 cm to 5 years on the x-axis and 2 cm to 2 persons on the y-axis, draw a histogram based on the data. [4 Marks]

Answer

a)


40-44

45-49

Table

30



Section A


5 x2 + 4 x = 2(3 –x) [ 4 marks]

[ 4 markah]

31


2 a – b = 6a +3 b = -6

[ 4 marks]

[ 4 markah]


, ,

[ 4 marks]

[ 4 markah]

32

xy21

5 xy

y

x

5

5




[2 marks][2 markah]


4= 3(0)3 + 56= 3(1)3 + 520= 3(2)3 + 558= 3(3)3 + 5...=...............

[2 marks][2 markah]



4 x – 2 y = 7

2x + 5 y = -2



33

[4 marks][4 markah]

Section B





[2 marks]

[2 markah]





[4 marks]

[4 markah]


Dari graf, cari

vii) the value of y when x = -2.4nilai y apabila x = -2.4

viii) the value of x when y = -11nilai x apabila y = -11

34

[2 marks]

[2 markah]

Answer:

j) y = 8-3x -2x2

x -5 -4 -3.5 -2 -1 0 1 2 3

y -27 -6 6 9 8 -6 -19

k) Refer graphl) i) y =.....................................

ii)x = ........................

35





transformations:

(i) R2.

(ii) TR.

Answer:

36

(a) (i)

(ii)


Cartesian plane.



(a)Transformation S

(b)Transformation Q


[12 marks

Answer:

(b) (i) (a)

(b) (ii)


45 53 48 54 46 53 55 43 47 52

63 57 50 40 52 45 49 61 54 56

51 41 56 51 61 50 53 48 51 44

57 53 47 55 46 54 42 57 58 63

42 64 50 49 52 47 55 52 45 51

37

-12 -10 -8 -6 -4 -2 2 4 6 8 10 12F

E

GH

S

QP

DC

B A

y

xO

R

6

4

2

- 2

- 4

Diagram 5



f) Based on the table in a), calculate the estimated mean age of the members of the golf club. [4 Marks]

g) By using a scale of 2 cm to 5 years on the x-axis and 2 cm to 2 persons on the y-axis, draw a histogram based on the data. [4 Marks]

Answer

a)


40-44

45-49

b)

38


9.


Section A


39

5 x2 + 4 x = 2(3 –x) [ 4 marks]

[ 4 markah]


2 a – b = 6a +3 b = -6

[ 4 marks]

[ 4 markah]


, ,

[ 4 marks]

[ 4 markah]

40

xy21

5 xy

y

x

5

5




[2 marks][2 markah]


4= 3(0)3 + 56= 3(1)3 + 520= 3(2)3 + 558= 3(3)3 + 5...=...............

[2 marks][2 markah]



4 x – 2 y = 7

41

2x + 5 y = -2



[4 marks][4 markah]

Section B





[2 marks]

[2 markah]





[4 marks]

[4 markah]


Dari graf, cari

42

ix) the value of y when x = -2.4nilai y apabila x = -2.4

x) the value of x when y = -11nilai x apabila y = -11

[2 marks]

[2 markah]

Answer:

m) y = 8-3x -2x2

x -5 -4 -3.5 -2 -1 0 1 2 3

y -27 -6 6 9 8 -6 -19

n) Refer grapho) i) y =.....................................

ii)x = ........................

43




44


transformations:

(i) R2.

(ii) TR.

Answer:

(a) (i)

(ii)


Cartesian plane.

DIAGRAM 9



(a)Transformation S

(b)Transformation Q


[12 marks]

Answer:

(b) (i) (a)

45

-12 -10 -8 -6 -4 -2 2 4 6 8 10 12F

E

GH

S

QP

DC

B A

y

xO

R

6

4

2

- 2

- 4

(b)

(ii)


45 53 48 54 46 53 55 43 47 52

63 57 50 40 52 45 49 61 54 56

51 41 56 51 61 50 53 48 51 44

57 53 47 55 46 54 42 57 58 63

42 64 50 49 52 47 55 52 45 51

Diagram 5



h) Based on the table in a), calculate the estimated mean age of the members of the golf club. [4 Marks]

i) By using a scale of 2 cm to 5 years on the x-axis and 2 cm to 2 persons on the y-axis, draw a histogram based on the data. [4 Marks]

Answer

a)


46

40-44

45-49

Table

b)

c)Refer graph

47


9.

48


Section A


5 x2 + 4 x = 2(3 –x)

50

[ 4 marks]

[ 4 markah]


2 a – b = 6a +3 b = -6

[ 4 marks]

[ 4 markah]


, ,

[ 4 marks]

[ 4 markah]

51

xy21

5 xy

y

x

5

5




[2 marks][2 markah]


4= 3(0)3 + 56= 3(1)3 + 520= 3(2)3 + 558= 3(3)3 + 5...=...............

[2 marks][2 markah]



4 x – 2 y = 7

2x + 5 y = -2



52

[4 marks][4 markah]

Section B





[2 marks]

[2 markah]





[4 marks]

[4 markah]


Dari graf, cari

xi) the value of y when x = -2.4nilai y apabila x = -2.4

xii) the value of x when y = -11nilai x apabila y = -11

[2 marks]

[2 markah]

Answer:

p) y = 8-3x -2x2

53

x -5 -4 -3.5 -2 -1 0 1 2 3

y -27 -6 6 9 8 -6 -19

q) Refer graphr) i) y =.....................................

ii)x = ........................

54





transformations:

(i) R2.

(ii) TR.

Answer:

(a) (i)

55

(ii)


Cartesian plane.

DIAGRAM 9



(a)Transformation S

(b)Transformation Q


[12 marks]

Answer:

(b) (i) (a)

(b)

(ii)


56

-12 -10 -8 -6 -4 -2 2 4 6 8 10 12F

E

GH

S

QP

DC

B A

y

xO

R

6

4

2

- 2

- 4

45 53 48 54 46 53 55 43 47 52

63 57 50 40 52 45 49 61 54 56

51 41 56 51 61 50 53 48 51 44

57 53 47 55 46 54 42 57 58 63

42 64 50 49 52 47 55 52 45 51

Diagram 5



j) Based on the table in a), calculate the estimated mean age of the members of the golf club. [4 Marks]

k) By using a scale of 2 cm to 5 years on the x-axis and 2 cm to 2 persons on the y-axis, draw a histogram based on the data. [4 Marks]

Answer

a)


40-44

45-49

57

Table

b)

c)Refer graph

58


9.

59

Modul Jom Lulus Matematik

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