Modul Add Maths 2012

Jabat anPel aj ar anKel ant an Jal anDokt or,15000Kot aBharu, Kel ant an.

ModulPecut anAkademi k

ADDI T I ONAL MAT HEMAT I CS SPM

KATA PENGANTAR

Puji dan syukur kita panjatkan kehadrat Allah Yang maha Esa atas segala rahmat dan kurnia-Nya sehingga Modul Pecutan Akademik, Jabatan Pelajaran Kelantan dapat dihasilkan pada tahun ini. Modul Pecutan Akademik ini diharapkan dapat menjadi panduan dan bimbingan kepada para guru dan para pelajar dalam membuat persediaan bagi menghadapi peperiksaan UPSR, PMR dan SPM. Setinggi-tinggi penghargaan dan terima kasih dirakamkan kepada semua guru dan semua pihak yang terlibat dalam menjayakan penghasilan Modul Pecutan Akademik ini. Semoga usaha murni ini dapat diteruskan pada masa hadapan bagi menghasilkan pelajar yang cemerlang, gemilang dan terbilang dan seterusnya menjayakan visi Jabatan Pelajaran Kelantan, “Cakna Pendidikan Kelantan Terbilang 2013”.

HJ MOHD ADNAN BIN MOHD NOOR Ketua Sektor Pengurusan Akademik Jabatan Pelajaran Kelantan

PANEL PENULIS Hj. Ahmad Zamri bin Aziz SMK Tumpat

Hjh. Zakiah bt. Zaid SM Sains Tengku Muhammad Faris Petra

Azmani bt. Daud SMK Kubang Kerian(1)

Noterzam bt. Jaafar SM Sains Tengku Muhammad Faris Petra

Rosli bin Ja’far SMK Dato’ Mahmud Paduka Raja (1)

Pahimi bt. Yaacob Maktab Sultan Ismail

Lee Sek Khon SM Sains Machang

Mohd Nasaruddin bin Deraman SM Sains Pasir Puteh

Hjh. Fatimah bt Mat SMK Long Yunus

Halizah bt Mohd Zain Sektor Pengurusan Akademik, JPN

CONTENT FORMULAE

i

PAPER I 1.0 Functions

1

2.0 Quadratic Equations

8

3.0 Quadratic Functions

12

5.0 Indices and Logarithm

15

6.0 Coordinate Geometry

19

7.0 Statistics

24

8.0 Circular Measure

28

9.0 Differentiations

34

12.0 Progression

38

13.0 Linear Laws

42

14.0 Integrations

47

15.0 Vectors

50

16.0 Trigonometry Function

57

17.0 Probability

60

18.0 Permutation and Combination

62

19.0 Probability Distribution

68

PAPER II 1.0 Functions

73

2.0 Quadratic Equations

75

3.0 Quadratic Functions

76

4.0 Simultaneous Equations

78

5.0 Indices and Logarithm

79

6.0 Coordinate Geometry

80

7.0 Statistics

83

8.0 Circular Measure

85

9.0 Differentiations

88

10.0 Index Number

90

11.0 Solution of Triangles

96

12.0 Progression

101

13.0 Linear Laws

105

14.0 Integrations

109

15.0 Vectors

112

16.0 Trigonometry Functions

114

19.0 Probability Distributions

117

20.0 Motion along a straight line

119

21.0 Linear Programming

122

Rumus-rumus berikut boleh membantu anda menjawab soalan. Simbol-simbol yang diberi adalah yang biasa digunakan. ALGEBRA b2 2a

b

4ac

1

x

2

am an

am

n

3

am an

am

n

4

n

am

a mn

5

log a mn

6

log a

7

loga m n

m n

log a m log a n

log a m log a n

8

log a x

log b x log b a

9

Tn

a

n 1d

10

Sn

n 2

11

Tn

ar n

12

Sn

a rn 1 r 1

13

S

2a

n 1d

1

a 1 r

,r

a 1 rn , (r 1 r

1)

1

n loga m KALKULUS (CALCULUS)

1 y

uv ,

dy dx

u

dv dx

v

4

du dx

Luas di bawah lengkung (Area under a curve) b

2 y

du v dx

u dy , v dx

ydx atau (or)

dv u dx v

a b

2

xdy a

5

Isipadu janaan (Volume of revolution) b

dy 3 dx

dy du

πy 2 dx atau (or)

du dx

a b

πx 2 dy

i a

STATISTIK (STATISTICS)

1x

x

fx

8

n

Pr

n! (n r )!

9

n

Cr

n! (n r )!r!

f

x

3

x

2

x

N

4

2

N

f x

x

2

f

M

I

Wi

N

2x

5

Wi I i

7

L

1 2

x2

fx 2 f

10

P( A

B)

11

p( X

r) nCr p r q n r , p q 1

x2

12 Min (Mean) np

N F C fm

npq

13 5

I

Q1 Q0

P( A) P(B) P( A

100 14

ii

z

x

B)

GEOMETRI (GEOMETRY) 1

Jarak (Distance) x1

x2

2

y1

2 y2

4

Titik tengah (Midpoint) x1 x2 y1 y2 ( x, y ) , 2 2

5

Titik yang membahagi suatu tembereng garis (A point dividing a segment of a line)

( x, y )

Luas segitiga (Area of triangle) = 1 x1 y 2 x2 y3 x3 y1 x2 y1 x3 y 2 2

2

nx1 mx2 ny1 my 2 , m n m n

iii

3

r

6

r

^

x

2

y2

xi

yi

x2

y2

x1 y3

TRIGONOMETRI (TRIGONOMETRY)

1.

2.

3.

Panjang lengkok, s Arc length, s rθ

jθ

1 2 j θ 2 1 2 Area of sector, A r θ 2

8.

sin( A B) sin AkosB kosAsin B sin( A B) sin AcosB cosAsin B

9.

kos( A B) kosAkosB  sinAsin B cos( A B) cosAcosB  sinAsin B

Luas sektor, L

sin2 A + kos2 A = 1 sin2 A + cos2 A = 1 2

2

4.

sek A = 1 + tan A sec 2 A = 1 + tan 2 A

5.

kosek 2 A = 1 + kot 2 A cosec 2 A = 1 + kot 2 A

6.

sin 2 A 2 sinA kosA sin 2 A 2 sinA cosA

7.

kos 2 A kos 2 A sin 2 A 2kos 2 A 1

B)

11. tan 2 A

2 tanA 1 tan 2 A

12.

1 2 sin 2 A 13. 8.

cos 2 A cos2 A sin 2 A 2cos2 A 1 1 2 sin 2 A 14.

i

tanA tan B 1  tan A tan B

10. tan ( A

a sin A

a2 a2

b sin B

c sin C

b 2 c 2 2bckos A b 2 c 2 2bccos A

Luas segitiga (Area of triangle) 1 2

ab sinC

1.0 FUNCTIONS 1.1

Given that f ( x) (a)

f

(b)

f 2 (x).

1

3x 1 , find

(x),

[ 2 marks ] Answer:

1.2

Given that g 1 ( x)

2x 5 , find 4

(a)

g (x),

(b)

the value of x if g(x) = 6. [ 3 marks ]

Answer:

1

1.3

A relation is given as {(1, 1), (-1, 1), (2, 4), (-2, 4), (3, 9), (-3, 9)}. State (a)

type of relation,

(b)

the domain of relation,

(c)

the range of relation. [ 3 marks ]

Answer:

1.4

Given that f ( x) (a) (b)

f

1

1 x 3 , find 2

(5),

the value of x if f

1

f ( 5) = 2x .

[ 3 marks ] Answer

2

1.5

Given that gh( x) 8x 7 , g ( x) 4x 5 , find (a)

h(x),

(b)

h 1 ( 2).

[ 4 marks ] Answer:

1.6

Given that gf ( x) 2 6 x and f ( x) 3x 1 , find (a)

g (x),

(b)

f

1

g ( 12 ).

[ 3 marks ] Answer:

3

1.7

Given that gf ( x) (a)

g (x),

(b)

fg (2).

4 x 2 1 , and f ( x)

2x2 5

[ 3 marks ] Answer:

1.8

It is given f ( x) 8 x , g ( x)

6 px 2q, where p and q are constants and

fg ( x) 3 12x . Find (a) (b)

f

1

(3),

the value of p and of q. [ 3 marks ]

Answer:

4

1.9

Two functions are defined by g ( x) Given that gh( x) (a)

m,

(b)

n

x2

x 5 and h( x) x 2 3 x 1 .

mx n , find the value of

[ 3 marks ] Answer:

1.10

7

The function g is defined by g(x) = g ( x)

3 x

(a)

g 1 ( x ),

(b)

the value of h such that gg 1 (h 1)

, x 3 . Find

4.

[ 3 marks ] Answer

5

1.11 Given that f ( x)

(a)

f ( 1) 3,

(b)

f

1

(4)

px 5 , find the value of p if

6.

[ 3 marks ] Answer:

1.12.

Given that g 1 ( x)

2 x p

p , and h( x)

,x

(a)

g (x),

(b)

the value of p if g (4)

2 x, find

5.

[ 4 marks ] Answer:

6

1.13

Given that the function g ( x)

2x 5 ,x x 4

(a)

the objects that maps onto itself,

(b)

the value of g 2 (3).

4, find

[ 4 marks ] Answer:

ANSWER:

1.1 (a) 1.2 (a) 1.3

a) b) c)

1.4 (a) 1.5

1.8

x 1 (b) 9 x 4 3 4x 5 7 (b) 4 2 many to one { 1,-1,2,-2,3, -3 } {1,4,9 } 5 4 (b) 2

1.9 1.10

(a) 1.11

2x 3

(b)

5 2

1.12

1.6

(a) g ( x) 4 2x

(b)

2 3

1.13

1.7

(a) g ( x) 2x 9

(b) 55

(a) h( x)

(a) p=-2, q= -2.5 (b) 5 (a) m = 3 (b) n = - 4

7

3x 7 ,x 0 x

(a)

8

(b)

(a)

px 2 ;x x

(a) x 1,5

(b) 3

3 2 0 (b)

(b)

7 5

9 2

2.0 QUADRATIC EQUATIONS

2.1

Express the quadratic equation

in general form . [2 marks]

Answer:

2.2

Given 4 and

1 are the roots of a quadratic equation. State the quadratic equation 3

in the form

, where a, b and c are integers. [ 3 marks ]

Answer:

2.3

Solve the quadratic equation [2 marks] Answer:

8

2.4

Solve the quadratic equation correct to four significant figures.

Give your answer [3 marks]

Answer:

2.5

Find the values of roots.

if the quadratic equation

has two equal [2 marks]

Answer:

2.6

Find the range of values of k such that the quadratic equation has real roots. [ 3 marks ] Answer:

9

2.7

The quadratic equation values of k.

has no real root. Find the range of the [ 3 marks ]

Answer:

2.8

The quadratic equation where m and n are constants. Given that the sum of root is 1 and the product of root is -6. Find the value of and of . [ 3 marks ] Answer:

2.9

A straight line intersects with a curve points, find the range of values of m.

at two [ 3 marks ]

Answer:

10

2.10 If and are the roots of the quadratic equation quadratic equation that has the roots + 1 and + 1.

, form the [ 3 marks ]

Answer:

ANSWER:

2.1

2.6

2.2 2.3

2.7 2.8

2.4 2.5

2.9 2.10

11

3.0 QUADRATIC FUNCTIONS

3.1

Find the range of values of k if the graph of the quadratic functions intercept the x-axis at two distinct points. [2 marks] Answer:

3.2

Find the range of values of x for [ 3 marks ] Answer:

3.3

Find the range of values of x for [3 marks] Answer:

12

3.4

If the quadratic function

is expressed in the form where p and q are constants, find the value of p and of q. [3 marks]

Answer:

3.5

Diagram 3.5 shows the graph of the function constant.

, where h is a

f(x) k

x

3

Diagram 3.5 Find, (a) the value of h, (b) the value of k, (c) the equation of axis symmetry. [4 marks] Answer:

13

3.6

Diagram 3.6 shows the graph of the function f(x)

9 ● (h , k) x

Diagram 3.6 Find the value of, (a) h (b) k (c) a [ 3 marks ] Answer:

14

ANSWER:

3.1 3.2

3.4 3.5

3.3

3.6

(a)

(b)

(c)

(b)

(c)

5.0 INDICES AND LOGARITHM

5.1

Solve the equation 22 x

4

2 . 16x [3 marks]

Answer:

5.2

Solve the equation

3x (9 x 1 )

272 x 1. [3 marks]

Answer:

15

5.3 Solve the equation

6x

3

62 x 1

0 [2 marks]

Answer:

5.4

Solve the equation 5x

56 52 x

0 [3 marks]

Answer:

5.5 Solve the equation:

2 x (8 x 1 )

45 x [3 marks]

Answer:

16

5.6

Solve the equation log y 2

1 3 [3 marks]

Answer:

5.7 Solve the equation log 2 x log 2 ( x 3)

2

[3 marks] Answer:

5.8 Solve the equation log 3 (2 x 1) log 3 x log 3 (2 x 3) [4 marks] Answer:

17

5.9 Given that loga 2

p and log a 3 q , express loga 36 in terms of p and q

[3 marks] Answer:

5.10 Given that log 3 x

log 9 y

2 . Express y in terms of x. [4 marks]

Answer:

5.11 Given that log 3 x

p and log 3 y

q , express log9

Answer:

18

81x in terms of p and q. y [3 marks]

ANSWER:

5.1 5.2 5.3 5.4 5.5 5.6

x

1 2

5.8

1 3 x 4 x 2 x

x

5.7

5.9

1 2

y 8

5.10 5.11

x

4

x

3 2

2( p q) x 2

81 x2 p q 2 2

6.0 COORDINATE GEOMETRY

6.1

Given the equation of straight line AB is is 2, find the value of k .

. If the gradient of line AB [ 2 marks ]

Answer:

19

6.2. Diagram 6.2 shows a straight line 2x + y – 3 = 0 which intersects the x-axis and y-axis at point P and Q respectively. y P(0,h)

x Q

Find the value of h.

Diagram 6.2

[ 2 marks ] Answer:

6.3

Point M(4,5) divides line LN internally in the ratio of 2 : 3 . If L is the point (-2, 2), find the coordinates of point N . [ 3 marks ] Answer:

20

6.4

The straight line passes through point

is parallel to the straight line .Find the value of and .

and

[ 3 marks ] Answer:

6.5

Diagram 6.5 shows a straight line passes through point point . y

and

R(5,10)

x 0 P(-2,-4) Diagram 6.5

Find the equation of straight line which is perpendicular to line PR and passes through mid point of PR . [ 4 marks ] Answer:

21

6.6

Point C moves such that its distance from point and the equation . Find the equation of locus of point C.

is related by [ 3 marks ]

Answer:

6.7

The gradient of the straight line with equation is . If the straight line passes through the point , find the value of m and of n [ 4 marks ] Answer:

6.8

Diagram 6.8 shows the straight line PQ which is perpendicular to the straight line y QR at point Q. . P(0,4)

Q

x R Diagram 6.8 22

Given that the equation of the straight line QR is Find the coordinates of Q .

. [ 4 marks ]

Answer:

6.9

Diagram 6.9 shows a straight line RS where point R lies on the x-axis and point S lies on the y-axis. T y

S

R

x

Diagram 6.9

Given the equation of the straight line RS is x + 2y = 8 . Find the equation of the straight line ST. [ 3 marks ] Answer:

23

ANSWER:

6.1 k = 6 6.2 h = 3 6.3 N(13,12) 6.4 ,c=-3 6.5 4y + 2x = 15

6.6 6.7 m = 12, n = 4 6.8 6.9 y = 2x + 4

7.0

STATISTICS

7.1

A set of numbers x1 , x 2 , x3 , ... , x n has mean 25 and variance 64. Find (a) the mean of the set 2 x1 3, 2 x2 3, 2 x3 3, … , 2 x n 3, (b) the standard deviation of the set 3 x1 2 , 3x 2 2 , 3 x3 2 , ... , 3 x n Answer:

24

2 [ 4 marks ]

7.2

Table 7.2 shows the distribution of the scores acquired by 40 students in a quiz competition. Score 0 1 2 3 4 Number of students

2

9

16

12

1

Table 7.2 .Determine the interquartile range of the distribution. [ 3 marks ] Answer:

7.3

Table 7.3 shows the marks of a group of students in a test. Marks

1 – 20

Number of students

4

21 – 40 k Table 7.3

41 – 60

61 – 80

81 – 100

12

9

5

Given that the median of the marks is 50 5, find the value of k. [ 3 marks ] Answer:

25

7.4

A set of numbers a1 , a 2 , a3 , a 4 , a5 has a mean of 12 and a standard deviation of 4·5. Each number is multiply by 2.5 and then increase by 2. Find (a) the new mean, (b) the new variance. [ 4 marks ] Answer:

7.5

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

the value of m,

(b)

the variance of the set of numbers. [ 4 marks ]

Answer:

7.6

A set of data x1 , x2 ,...., x4 has a mean of 10 and a standard deviation of 2. (a) Calculate x and x2. (b) A few scores, y, with a sum of 540, a mean of 9 and a sum of squares of 4995 are added to set X. Calculate the mean and standard deviation when both the sets of data are combined. [ 4 marks ] Answer:

26

7.7

A set of game score x1 , x2 , x3 , x4 and x5 has the mean 6 and standard deviation 1·2 (a) Find the sum of the squares of the score, (b) Each score is multiplied by 3 and 2 is added to it, find the mean and variance of the new score [ 4 marks ] Answer:

7.8

Given that the mean of the numbers 2, 6, 7, x and 20 is 9 4. Find (a) the value of x, (b) the standard deviation of the set of the numbers [ 4 marks ] Answer:

27

ANSWER:

7.1 (a) 47

(b) 24

7.5 (a) m = 7

7.2 2

(b) 1579.83

7.6 (a) 400 4160 (b) mean = 9.4 S.d . = 1.786

7.3 k = 10

7.7 (a) x2 = 187.2

7.4 (a) 32

(b) min baru= 3(6) + 2=20 varians baru = 32 (1.2)2= 12.96 7.8 (a) x = 12 (b) 6.1838

(b) 126.56

8.0 CIRCULAR MEASURE

8.1

Diagram 8.1 shows sector OAB with centre O and radius 6.5 cm. A

O

660

B

Diagram 8.11 Find the length, in cm, the arc AB of the sector. [Use =3.142] [ 4 marks ] Answer:

28

8.2

Diagram 8.2 shows sector POQ with centre O . P

O

0.85 rad

Q

Diagram 8.2 It is given that the length of the arc PQ = 5.1 cm and Find (a) the length, in cm, of the radius of the sector, (b) the area, in cm2, of the sector.

POQ 0.85 radians.

[ 2 marks ] Answer:

29

8.3

Diagram 8.3 shows a circle with centre O. P

O Q

Diagram 8.3 It is given that the angle of the major sector POQ is 5.46 radians and radius 7 cm. [Use 3.142 ] Calculate (a) the value of , in radians, (b) the area, in cm2, of the minor sector POQ. [ 3 marks ] Answer:

30

8.4

Diagram 4 shows a quadrant OABC of a circle with centre O. A

B

C

O

Diagram 4 Given OB = OC = arc BC = 6.85 cm. [Use 3.142 ] Find (a) the angle of , in radians, (b) the area, in cm2, of the shaded region. [ 4 marks ] Answer:

31

8.5

Diagram 8.5 shows a sector BOC of a circle with centre O. A

O



B

C

Diagram 5 It is given that AOB = 0.6435 radians, and OC = 8 cm. Find (a) (b)

ACO 90 , OA = OB = 10 cm

the length, in cm, of arc AB, the area, in cm2, of the shaded region. [ 4 marks ]

Answer:

32

ANSWER:

1

7.488

2

(a)

6

(b)

15.3

(a)

0.824

(b)

20.188

(a)

1

(b)

13.4

(a)

6.435

(b)

8.175

3

4

5

33

9.0 DIFFERENTIATIONS 9.1

Given f (a)

4 3a , Find the limit of f (a) when a 5 2a

. [ 3 marks ]

Answer:

9.2

Given that y

x2

2 x , find

dy by using first principles. dx

Answer:

9.3

[ 3 marks ]

Differentiate x(1 x) 2 with respect to x. [ 3 marks ] Answer:

34

9.4

Given that g (r )

1 2r 2 , find the value of g ' (1) . r 1 [ 3 marks ]

Answer:

9.5

It is given that y

2 3 dy p , where p = 3x – 5. Find in terms of x. 3 dx [ 4 marks ]

Answer:

9.6

The gradient of the curve y

4 x2

kx 1 is 14 when x

1.

Find the values of k. [ 3 marks ] Answer:

35

9.7

A curve has a gradient function mx 2 - 4x, where m is a constant. The tangent to the curve at point (1, 3) is parallel to the straight line y x 5. Find the value of m. [ 3 marks ] Answer:

9.8

Given that h = 4p(5-p), calculate the value of p when h is a maximum. Hence, find the maximum value of h. [ 4 marks ] Answer:

9.9

Given that u 3 to 2.97.

v 2 5v .Find the small change in u when v changes from [ 3 marks ]

Answer:

36

9.10

Given that y

x(x 3) , find the value of x that satisfied the equation of d2y dy 2 x 2 2 (3 x) y 15 dx dx [ 4 marks ] Answer:

ANSWER: 9.1

9.6

6

9.2 9.3

3 2 2x 2 3x 2 4 x 1

9.7 9.8

3

9.4 9.5

-1 6 3x 5

9.9 9.10

2

5 h 2 0.33 x 2; 4 p

37

25

12.0 PROGRESSION 12.1

The second term and the fifth term of a geometric progression are 15 and respectively. Find (a) the common ratio, (b) the sum to infinity of the progression. [ 4 marks ] Answer:

12.2

The first three terms of an arithmetic progression are −5, −1 and 3. Find (a) the sixth term, (b) the sum of the first seven terms after the sixth term. [ 4 marks ] Answer:

38

12.3

The sum of the first three terms of a geometric progression is 96 and the common ratio is -2. Find (a) the first term, (b) the fourth term of the progression. [ 3 marks ] Answer:

12.4

The sum of the first n term, Sn , of a geometric progression is given by n = 81[1 – ]. Find (a) (b)

the common ratio, the sum to infinity of the progression. [ 3 marks ]

Answer:

39

12.5

Given the first three terms of an arithmetic progression are 3 . Find (a) the value of h, (b) the sum of the first 10 terms of the progression.

,

, and

[ 4 marks ] Answer:

12.6

The first term of the geometric progression is 6 and the fourth term of the progression is 9.375 , find (a) the common ratio, (b) the sum of the first three terms of the progression. [ 4 marks ] Answer:

12.7

The first three terms of a sequence are 2 , x and 18. Find the positive value of x so that the sequence is (a) an arithmetic progression, (b) a geometric progression. [ 3 marks ] Answer:

40

12.8

Given a geometric progression 3, 2m, p, …,express p in terms of m. [ 2 marks ] Answer:

12.9

Given = 0.16666… = 0.1 + h + k + … (a) (b)

Find the value of h and of k. Hence, find the value of p. [ 4 marks ]

Answer:

12.10

Find the sum to infinity of the geometric progression

3 , 8

3 3 , ,… 16 32 [ 3 marks ]

Answer:

41

ANSWER: 12.1 12.2 12.3 12.4 12.5

(a) r = (a)15 (a) a = 32

(b) 62.5 (b) 217 (b)-256

12.6

(a) r = 1.25

(b) 22.875

12.7 12.8

(a) r = (a) h = 4

(b) 81 (b) -55

12.9

(a) 6 (b) 10 p= (a) h = 0.06, k = 0.006

(b)p = 15

12.10

13.0 LINEAR LAWS

13.1 Variables x and y are related by the equation py 2

( x 1) 2

2q .

When the graph y 2 against . (x 1) 2 is plotted, a straight line passes through (0, 3) and (2, 7) Find the value of p and of q [ 3 marks ] Answer:

42

13.2

Diagram below shows a straight line is obtained by plotting

y against x x2

y x2 (3, 8)

(5, 2)

x

O

Given that y = sx3 + tx2, where s and Calculate the values of s and t.

t are constants. [4 marks]

Answer:

43

13.3 The diagram 13.3 shows the straight line obtained by plotting log10y against log10x .

log10 y (2, h)

(0, 2)

O

log10x Diagram 13.3

Given that y kx3 Find the value of (a) log10 k, (b) h. [ 3 marks ] Answer:

44

13.4

y The Diagram 13.4 shows part of the straight line obtained by plotting against x2 x y x (2, 7)

(6, 3)

.

x2

O Diagram 13.4

(a)

Express y in terms of x

(b)

Find the values of x when

y = 7. x [ 4 marks ]

Answer:

45

13.5 Diagram 5(a) shows the curve y = 4x2 + 27 x. y x

y y = 4x2 + 27x

3, m

(n, 3) O

Diagram 13.5(a)

x

O

Diagram 13.5(b)

x

Diagram 5(b) shows the straight line graph obtained when y = 4x2 + 27x is y expressed in the form = ax + b. x Find the values of m and n. [ 4 marks ] Answer:

ANSWER:

13.1 p =

1 2

13.2 s = 3

, q= t = 17

3

13.4 (a) y = x3 + 9x (b) 4, 4

4

13.5 m = 15

13.3 (a) 2 (b) 8

46

n= 6

14.0 INTEGRATIONS 3

14.1

x(4 x) dx .

Find the value of 0

[ 2 marks ] Answer:

14.2 The gradient function of a curve which passes through A(1, 12) is 3x2 6 x . Find the equation of the curve. [ 3 marks ] Answer:

4

4

14.3 Given that

f ( x) 1

kf ( x) 6 x

3 and

39 , find the value of k.

1

[ 3 marks ] Answer:

47

14.4 Given

dy dx

4 x 2 and y = 10 when x = – 1, find y in terms of x. [ 3 marks ]

Answer:

14.5 Diagram 14.5 shows part of the curve y = f(x). y 6 y = f(x) O

x 8

Diagram 14.5

Given that the area of the shaded region is 40 unit2, find the value of

8

f ( x)dx . 0

[ 4 marks ] Answer:

48

12 dx k (3x 2)n c , find the value of k and of n. 3 (3x 2)

14.6 Given that

[ 4 marks ] Answer:

ANSWER:

14.1 14.2 14.3

9

y k

x

3

2

3x

2

10

14.4

y

14.5

8

14.6

k

49

2x2

2x

2, n

2

15.0 VECTORS 15.1 Diagram 15.1 shows two vectors, PO and QP .

Diagram 15.1

(a) (b)

OP in the form

x . y

QP in the form xi

y j.

[2 marks] Answer:

15.2 Use the information below to find the values of h and k when r p

a 2b

q

3a 5b

r

( h k ) a k b,

Where h and k are constants.

4p q .

[3 marks]

Answer:

50

15.3 Diagram 15.3 shows a triangle OPQ.

Diagram 15.3

Given that OQ (a) QR (b)

6q , OR

p 2q and PR = 2RQ, express, in terms of p and q.

OP

[3 marks] Answer:

15.6 Diagram 15.6 shows vector PQ drawn on a Cartesian plane.

Diagram 15.6

51

(a) Express PQ in the form

x . y

(b) Find the unit vector in the direction of PQ . [3 marks] Answer:

52

15.5 Given that A(-3, 7), B(5, 2) and C(m, p), find the value of m and of p such that

AB 2CB 10i

j. [3 marks]

Answer:

15.4 Given that O(0, 0), A(5, -4) and B(11, 4), find in terms of the unit vectors, i and j , (a) AB , (b) the unit vector in the direction of AB. [3 marks] Answer:

53

15.7 Diagram 15.7 shows a parallelogram, OPQR, drawn on a Cartesian plane.

It is given that

PO

7i 5 j

and

OR

4i 3 j

. find OQ . [2 marks]

Answer:

54

  15.8 Diagram 15.8 shows two vectors, OA and AB .

Diagram 15.8

(a) Express AB in the form

x . y

(b) Find the magnitude of AB . [2 marks] Answer:

15.9 The points P, Q and R are collinear. It is given that PQ , where k is a constant. Find (a) The value of k, (b) The ratio of PQ : QR.

6a

2b and QR

(k 1)a 3b

[3 marks] Answer:

55

15.10 Diagram shows a rectangle OABC and D is the mid-point of OB. Express AD , in terms of x and y.

Diagram 15.10

[2 marks] Answer:

15.11 The following information refers to the vectors a and b. 6 9 a ,b 11 2 Find (a) the vector a 2b . (b) the unit vector in the direction of a 2b . [4 marks] Answer:

56

ANSWER:

15.1 (a) 15.2

15.7

5 6

(b)

9i

OQ

6j

h = 4and k = -3

15.8

3i 8 j

5 (b) 13 12 (a) 8 (b) 2:3

(a) 15.3

15.4 15.5

(a) QR

15.9

p 8q (b)

OP 3 p 18q (a) 6i 8 j (b) 53 i

4 5

15.10

j

3 2

15.11

m = 4, p = -1

(a) 15.6 (a)

6 8

(b)

3 5

i

4 5

x 2y 24 7

(b)

24 25

i

7 25

j

j

16. TRIGONOMETRY FUNCTION

16.1 Solve the equation

for

. [ 4 marks ]

Answer:

57

16.2 Given that (a) (b)

and A is an obtuse angle. Without using calculator, find

tan A sin 2A [3 marks]

Answer:

16.3 Given and such that angle A and B lie on the same quadrant. Without using calculator, find the value of [ 3 marks ] Answer:

16.4

Solve the equation

for

. [ 3 marks ]

Answer:

58

16.5 Given (a) tan (b) sin 2

, where

is an obtuse angle. Write in terms of k,

[3 marks ] Answer:

16.6 Solve the equation

for

. [ 3 marks ]

Answer:

ANSWER:

6.1 45o,146.31o,225o,326.31o 6.2 (a) (b) 6.3

6.4 110.91o,159.10o,290.91o,339.10o 6.5 (a) (b) o o 6.6 60 ,131.81 ,228.19o,300o

59

17.0 PROBABILITY

17.1 A basket contains four black pens and five blue pens. If two pens are randomly chosen from the basket, find the probability that (a)

both are the same colour

(b)

both are different colour [ 4marks ]

Answer

17.2 A gunny sack contains of 15 sweet mangoes and p sour mangoes. If a mango is 2 picked at random from the sack, probability of getting a sour mango is . Find 7 the value of p. [ 3 marks ] Answer:

60

17.3 The probability that Adlan and Ammar will score A+ for Additional Mathematics 3 2 in SPM is and respectively. Find the probability that 5 3 (a)

both of them will score A+

(b)

only one of them will score A+ [ 4 marks ]

Answer:

17.4 Table 3 shows the number of two different colour of pens placed in a box.

Colour of pen Black Read

Table 3

Number of pens 5 3

Two pens are drawn at random from the box. Find the probability that (a)

both are different colour

(b)

both are the same colour [ 3 marks ]

Answer:

61

ANSWER:

17.1

(a)

17.2 6 17.3 (a)

4 9

(b)

5 9

2 5

(b)

7 15

17.4

(a)

15 28

(b)

13 28

18.0 PERMUTATION AND COMBINATION

18.1 A code is to be formed using letters in Diagram 18.1. Find the possible number of codes that can be formed if all the three consonants are separated by a vowel.

ROUND Diagram 18.1

[ 3 marks ] Answer:

18.2 Five letters from the word V I C T O R Y is to be arranged in a row begins with a vowel. Find the probability that the arrangement begins with the letter O. [3 marks] Answer:

62

18.3 Diagram 18.3 shows five cards with different numbers.

1 2

3

4

6

8

Diagram 18.3

(a)

Find the number of different 4 digits that can be form from the given numbers above. (b) Find the number of different 4 digits that can be form if the first digit is the odd number and last with the even number. [4 marks] Answer:

18.4 A school’s debate team consisting of six students are to be selected randomly from 5 female and 8 male students. Find the number of ways that the team can be formed if (a) there are no restrictions, (b) more male student than female in the team. [4 marks] Answer:

63

18.5 Diagram 18.5 shows five cards of different letters.

S

M

I

L

E

Diagram 18.5

Find (a) the number of possible arrangements beginning with M. (b) the number of these arrangements in which the vowels are separated. [ 4 marks ] Answer:

18.6 A committee of five people is to be chosen from 5 men and 7 women. Find the number of ways the committee can be formed if (a) there is no restriction, (b) the committee must has at most 2 men [ 4 marks ] Answer:

64

18.7 Diagram 18.7 shows cards with two letters and five numbers. How many possible ways can all the letters and digits be arranged if

A

B

1

2

3

4

5

Diagram 18.7

(a) (b)

none of the letters and digits is repeated the letters are separated from each other. [ 3 marks ]

Answer:

18.8 A group of 6 prefect are to be selected randomly from 7 boys and 4 girls. Find the number ways the group can be formed if (a) there is no restriction, (b) the number of boys is less than the number of girls in the group [ 3 marks ] Answer:

65

18.9 The Mathematics Club of a school has 8 Form 5 students, 10 Form 4 students and 12 Form 3 students. (a) A teacher wants to choose Form 5 students to form a committee consisting a president, a vice president and a secretary, find the number of ways the committee can be formed. (b) A team is to be formed to take part in a Mathematics competition. How many different teams, each comprising 3 Form 5 students, 2 Form 4 students and 1 Form 3 student can be formed? [ 4 marks ] Answer:

18.10. There are 9 cup cakes, each with a different flavor, which are to be divided equally into three boxes. Find the number of different ways the division of the cup cakes can be done. [ 3 marks ] Answer:

66

ANSWER:

18.1 12

18.6

(a) 792

18.2 1/2

18.7

(a) 5040

(b) 525 (b) 3600

18.3 (a) 120

(b) 36

18.8

(a) 462

(b) 21

18.4 (a) 1716

(b) 998

18.9

(a) 336

(b) 30240

18.5 (a) 24

(b) 72

18.10 1680

67

19.0 PROBABILITY DISTRIBUTION 19.1 Diagram 19.1 shows a standard normal distribution graph. f(z)

m

z

Diagram 19.1 The probability represented by the area of the shaded region is 0.8944. (a)

Find the value of m

(b)

X is a continuous random variable which normally distributed with a mean of μ and a standard deviation of 4. Given that, when X = 35, the z-score is m, find the value of μ. [ 4marks ]

Answer:

68

19.2 Diagram 19.2 shows a standard normal distribution graph. f(z)

z

k

Diagram 19.2 Given that (a) Find

.

(b) X is continuous random variable which is normally distributed with a mean of 54 and the standard deviation of 12. Find the value of k. [ 3 marks ] Answer:

69

19.3 A continuous random variable X has a normal distribution with mean 5 and standard deviation σ. Find, (a)

the value of σ if the z-score is 1.25 when x = 7.5

(b)

the value of k if

5 [ 3 marks ]

Answer:

19.4 A survey in a school found that 2 out of 5 students are going to school by bus. If a sample of 10 students is randomly selected, find the probability that 3 of them are going to school by bus. [ 4 marks ] Answer:

70

19.5 X is a continuous random variable which normally distributed with a mean of 6.2 and a variance of 0.64. Find, (a)

Z score when X = 5.8

(b) [ 4 marks ] Answer :

ANSWER: 19.1 (a) 19.2 (a) 0.7734 19.3 (a) 2

(b) (b) 56.72 (b) 5.9

19.4 0.2150 19.5 (a) – 0.5

71

(b) 0.3829

72

1.0 FUNCTIONS 1.1 Shows the function f : x

px q , where p and q are constants.

3

7

5

11

Diagram 1

(a) Find (i)

the value of p and of q. [ 3 marks ]

(ii)

f

1

( x)

[ 1 marks ] (b) Given that g ( x)

x 5 , find the value of x such that f 1 ( x)

g ( x) .

[ 2 marks ]

1.2

Given that f : x

x 2 and g : x 3

4x 1 ,

Find (a)

g 1 ( x),

[ 1 marks ] (b)

g 1 f ( x),

[ 2 marks ] (c)

h( x) such that hg ( x) 7 6x. [ 3 marks ]

73

1.3

A function f : x (a)

h , x k where h is a constant. 2x 1

State the value of k, [ 1 marks ]

(b)

Given that x = 2 maps the function onto itself, find the value of h. [ 2 marks ]

(c)

Given that fg ( x)

3x , find g (x) . [ 3 marks ]

ANSWER:

1.1

(a)

1.2

(b) 9 (a) g 1 ( x) (b) (c)

1.3

(a)

(ii) f 1 ( x)

(i) p = 2 and q = 1

x 1 4

x 5 12 h( x )

k

5 2 1 2

3 x 2

(b) 10 (c) 10 3x g ( x) 6x

,x 0

74

x 1 2

2.0 QUADRATIC EQUATIONS

2.1

(a)

Show that the quadratic equation values of

(b)

Find the values of p if the straight line

has real roots for all

is a tangent to the curve

[ 6 marks ]

ANSWER:

2.1

(a) (b)

75

3.0 QUADRATIC FUNCTIONS

3.1

x 2 2kx 3k has a maximum value of 4 and The quadratic function f ( x) symmetrical about x 2p 5, where k and p are constants. (a) By using the method of completing the square, find the possible values of k. [ 5 marks ] (b) Hence, by using the positive value of k, find the value of p. [ 2 marks ]

3.2

Diagram 3.2 shows the graph of a quadratic function, c is the constant. Graph of the quadratic function has a minimum point at and intersect the x-axis at point M and point N.

where

f(x)

N

M

x

● L(p, –9)

Diagram 3.8 (a) (b)

By using the method of completing the square, find the value of c and of p. [ 4 marks ] Find the coordinates of point M and of point N. [ 2 marks ]

76

3.3

Diagram 3.9 shows the graph of a quadratic function, where b is a constant. Graph of the quadratic function has a maximum point at and intersect the y-axis at point Q. f(x)

P(k, 0) ●

Q

x

Diagram 3.9

(a)

State the coordinates of Q

(b)

[ 3 marks ] By using the method of completing the square, find the value of b and of k [ 4 marks ]

ANSWER:

3.1

3.2 3.3

(a) (b)

p

1 2

(a) (b) (a) (b)

77

4.0 SIMULTENOUS EQUATIONS

1

Solve the simultaneous equations 2x y 8 0 and x 2 3 x

y

2

[5 marks] 2

Solve the simultaneous equations x y 3 and x 2 Give your answers correct to three decimal places.

xy 4 x

7.

[5 marks] 3

Solve the simultaneous equations x 4 y 12 and

2x 3

3 10 . y [5 marks]

4

Solve the simultaneous equations 4 x

y 8

x2

x

y

2.

[6 marks] 5

Solve the simultaneous equations 2( x y ) x y 1 2 x 2 11y 2 . Give your answers correct to three decimal places. [6 marks]

ANSWER:

4.1 4.2 4.3 4.4 4.5

x = 2 , -3 y = 12 ,2 x = -2.137 , 1.637

y = 5.137 , 1.363 3 3 y , x = 9 , 18 4 2 x = -1 , - 4 y = -2 , 10 x = -0.734 , 5.459 y = 0.133 , 2.153

78

5.0 INDICES AND LOGARITHM

5.1 (a) Solve the equation 32 x

1

9x 1 28 [3 marks]

(b) Given that log 3 5 1.465 , solve equation 3x

2

1 5 [4 marks]

ANSWER:

5.1

(a) x 1 (b) 0.535

79

6.0 COORDINATE GEOMETRY

6.1

Diagram 6.1 shows a rectangle PQRS. y R

Q(4,3)

S

P(3,0)

0

x

Diagram 6.1

Given the equation of the line PR is 4y + 3x = 9 , find (a) the equation of the line RQ, [3 marks] (b) the coordinates of point R and point S, [5 marks ] (c) the area of rectangle PQRS. [2 marks]

80

6.2

In Diagram 6.2, the straight line AB intersects the straight line BC at point B such that AB is perpendicular to BC . . C

y A(- 5,2)

D x

0 B Diagram 6.2

(a) Given that the equation for straight line BC is y – 2x + 3 = 0. Find (i) the equation of straight line AB, (ii) the coordinates of B, [4 marks] (b) Point D lies on AB such that 2AD = 3DB. Find the coordinates of D. [2 marks] (c) A point P moves such that its distance from point A is always 3 units. Find the equation of the locus of point P. [2 marks]

81

6.3

Solution by scale drawing is not accepted. y A

B

O

x

D C Diagram 6.3

Diagram 6.3 shows a trapezium OABC. The line OC is perpendicular to the line BC, which intersects the x-axis at point D. It is given that the equation of OC is and the equation of BC is 2y = hx – 10. (a) Find (i)

the value of h. [2 marks]

(ii)

the coordinates of C, [3 marks]

(b) Given CD : DB = 1 : 2 , find [ 2 marks ] (i)

the coordinates of B.

(ii)

the equation of the straight line AB.

[3 marks] [2 marks] Answer: 1. (a) 3y + x = 13 (b) R(-5,6) , S(-6,3) (c) 30

2. (a) (i) 2y + x = - 1 (ii) B(1,-1) (b) (c)

3. (a) (i) h = 4 (ii) C(2, -1) (b) (i) (ii) 2y = - x + 6

82

7.0 STATISTICS

7.1

(a) Table 7.1 shows the marks obtained by a group of students in a particular test Marks

1 – 20

Number of students

4

21 – 40

41 – 60

61 – 80 81 – 100

.(i)

10 13 8 Table 7.1 Without drawing an ogive, find the median marks.

(ii)

Calculate the mean marks.

5

[3 marks] [2 marks] (b) Given a set of 10 numbers x1, x2, x3 ,........x10 has a mean 5 and variance 4·8. A set of another 15 numbers y1, y2, y3 ,........y15 has a mean 10 and variance 5·8. Find (i) the values of x, y, x2 and y2. [ 3 marks ] (ii) the mean and the standard deviation for the combination of both sets of numbers. [2 marks] 7.2

Table 2 shows the range of worker’s age in a particular company Age

25 – 29

30 – 34

35 – 39

40 – 44

45 – 49

50 – 54

Number of Workers

5

8

10

15

8

4

Table 2 (a) Calculate (i) the mean, (ii)

the standard deviation of the worker’s age. [5 marks]

(b) Without drawing the ogive, calculate (i) the median, (ii)

the first quartile of the worker’s age. [5 marks]

83

7.3

(a) The mean of data 2, m, 3m, 8, 13 and 17 which has arrange in ascending order is n. If each elements is decreases by 2, a new median for new data is 5n . 8 Find the values of m and n. [4 marks] (b) Table 7.3(i) shows the score of 100 students in a test. Marks Markah

< 10

< 20

< 30

< 40

< 50

< 60

< 70

< 80

Number of Students Bilangan Pelajar

3

10

34

48

75

80

94

100

Table 7.3(i) Marks Markah

0–9

Frequency Kekerapan Table 7.3(ii) (i)

Based on Table 7.3(i), copy and complete Table 7.3(ii)

(ii) Without drawing an ogive, estimate the inter quartile range for the distribution. [6 marks] ANSWER:

7.1 (a) (i) median , m = 49.73 (ii) min , x = 50.5

7.2 (a) (i) x = 39 5 3 (b) (i)

(ii)

y 2 1587 (b) (i) x 2 298 dan (ii) min gabungan = 8 s.d = 3 376

(b) (i) m = 40 17

= 7 018

(ii) Q1 = 34 19

(a) m = 2, n = 8 Markah 0–9 Kekerapan 3

(ii) Q1 =25.75

10 – 19 7

20 – 29 24

30 – 39 40 – 49 50 – 59 60 – 69 70 – 79 14 27 5 14 6

Q3 = 49.5

84

8.0 CIRCULAR MEASURE

8.1 Diagram 8.1 shows a sector OABC with centre O and has a radius of 6 cm.

It is given that the length of arc ABC is 7.68 cm. Calculate (a) AOC , in radians, [2 marks] 2

(b) the area, in cm , of the shaded region. [4 marks]

85

8.2 Diagram 8.2 shows a sector OAB and a sector OCD with centre O. E

C

O



A

B F

D

Diagram 8.2

Given that AOB 0.6 radians, the length of arc AB = 1.2 cm, OC = 4 OA, OE = EC and EF = 3 cm. Find (a) the length, in cm, radius of the sector OAB, [2 marks] (b)

EOF , in radians,

[3 marks] (c)

the area, in cm2, of the shaded region. [3 marks]

8.3 Diagram 8.3 shows a semi circle OACB with centre O and a sector BCD with centre B. C

A



O

D

Diagram 8.3 BOC 90 .

B

Given that diameter AB = 20 cm and Calculate (a) the radius, in cm, of the sector BCD,

[1 marks] (b) the area, in cm2, of the sector BCD, [3 marks] 2

(c) the area, in cm , of the shaded region [6 marks]

86

8.4 Diagram 8.4 shows a circle PQRS, centre O with radius 13 cm. PQTS is a sector, centre P with radius 20 cm. P

O

600 T

Q

S

R

Diagram 8.4 (Use 3.142) Calculate (a) the length, in cm, of the arc QRS, [3 marks] (b) the perimeter of sector PQTS, [3 marks] (c) the area, in cm2 , of shaded region. [4 marks]

ANSWER:

1 2

3

4

(a) (b) (a) (b) (c) (a) (b) (c) (a) (b) (c)

1.28 5.8 2 0.6435 14.592 200 or 14.14 78.55 50 13.62 cm 53.61 426.26

87

9.0 DIFFERENTIATIONS

9.1

Given that y 4 x 1 (a) d2y find . dx 2

3

[ 2 marks ] (b)

the value of x if . if x

dy dx

2

d y dx 2

0 [ 4 marks ]

9.2 The curve y x 3 6 x 2 9 x 2 passés through the point P(2 , -4) and has two turning point A(1 , -6) and B. Find (a)

the gradient of the curve at P. [ 2 marks ]

(b)

the equation of the normal to the curve at P. [ 3 marks ]

9.3

(c)

the coordinate of B and determine whether maximum or the minimum point.

(a)

Given that k (i)

2x 3 and y

B

is the

[ 5 marks ]

3

. Find k the rate of change of x, given that the rate of change of k is 3 unit per second. 2

[ 2 marks ] (ii)

dy in term of x dx

(iii)

the small change in y, given that the value of x decreases from 2 to 1.99.

[ 2 marks ]

[ 2 marks ] (b)

dy 8 , find the value of when x = 2.Hence, find 4 dx x 8 the value of . (2.03) 4 Given that y

[ 4 marks ] 88

ANSWER: 1

(b)

96 4 x 1

2

(a)

3

(a)

(i)

3

(b)

(b)

1 7 , 4 4

(b) 3y = x – 10

3 12 (ii) 3 2 2x 3 1 ; 0.47

(c) (3 , -2) maximum (iii)

0.12

89

10.0 INDEX NUMBER 10.1 Table 10.1 shows the prices and price indices for the four ingredients, A, B, C and D used in making a type of biscuit. Diagram 10.1 is a pie chart which represents the usage of four ingredients, A, B, C and D used in the production of this biscuit. Price (RM) for the year Ingredients

Price index for the year 2008 based on the year 2006

2006

2008

A

2.00

2.50

x

B

5.00

y

140

C

1.40

2.10

150

D

z

4.00

125

Table 10.1

Diagram 10.1 (a) Find the value (i) x, (ii) y, (iii) z. [ 3 marks ]

(b) (i)

Calculate the composite index for the cost of production of this biscuit in the year 2008 based on the year 2006. (ii) Hence, calculate the cost of the production of this biscuit in the year 2006 if the cost of the production in the year 2008 was RM20 000. [ 5 marks ]

(c) The cost of production of this biscuit is expected to decrease by 10% from the year 2008 to the year 2010. Find the expected composite index for the year 2010 based on the year 2006. [ 2 marks ] 90

10.2 Table 2 shows the price indices and respective weightages, in the year 2009 based on the year 2007, of four materials, P, Q, R and S in the production of a type of moisturizing cream. Material

Price index in the year 2009 based on year 2007

Weightage

P

125

4

Q

120

m

R

80

5

S

150

m+3

(a)

(b)

(c)

Table 2 If the price of material P in the year 2007 was RM60.00, calculate its price in the year 2009. [ 2 marks ] Given that the composite index for the production cost of the moisturizing cream in the year 2009 based on the year 2007 is 120. Find (i) the value of m. (ii) the price of the moisturizing cream in the year 2007 if its price in the year 2009 is RM30.00. [ 6 marks ] Given that the price of material Q is estimated to increase by 15 % from the year 2009 to the year 2010, while the others remain unchanged. Calculate the composite index of the moisturizing cream in the year 2010 based on the year 2007. [ 2 marks ]

91

10.3 (a)

Table 3 shows the prices and the price indices for the four ingredients, J, K, L and M, used in making particular kind of cakes. Diagram 3 is bar chart which represents the relative amount of the ingredients J, K, L and M, used in making these cakes. Prices per kg (RM) for the year Prices index for the year 2007 based on the year 2003

Ingredients

2003

2007

J

0.40

0.60

150

K

a

0.40

80

L

2.00

b

140

M

0.80

1.00

c Table 10.3

120 100 80 60 40 0 20 0

J

K

L

M Ingredients

(a)

Diagram 10.3 Find the values of a, of b and of c.

Bahan

[ 3 marks ] (b)

(i)

Calculate the composite index for the cost of making these cakes in the year 2007 based on the year 2003. 92

(iii) (c)

Hence, calculate the corresponding cost of making year 2003 if the cost in the year 2007 was RM 3125.

[ 2 marks ] these cakes in the

[ 5 marks ] The cost of making these cakes is expected to increase by 40% from the year 2007 to the year 2008. Find the expected composite index for the year 2008 based on the year 2003. [ 2 marks ]

10.4 Table 10.4 shows the price indices and percentage of usage of four Q, R and S which are used to make a type of cake.

ingredients, P,

Percentage of usage(%)

Ingredient

Price index for the year 2010 base on the year 2005

P

104

15

Q

140

45

R

x

30

S

125

10 Table 10.4

(a) Calculate (i) the price of ingredient Q in the year 2010 if its price in the year 2005 is RM 15, (ii) the price index of ingredient S in the year 2010 based on the year 2008 if its price index in the year 2005 based on the year 2008 is 110. [ 5 marks ] (b) The composite index of the cost of cake production for the year 2010 base on the year 2005 is 125.6. Calculate (i) the value of x, (ii) the price of a cake in the year 2005 if the corresponding price in the year 2010 is RM54.20. [5 marks ]

93

10.5 Table 10.5 shows the prices, the price indices and the weightages for four items A, B, C and D.

Item

Price (RM) per kg

Price index in the year

in the year

2009 base on the year 2007

Weightage

2007

2009

A

15.00

16.50

110

4

B

6.00

7.50

x

6

C

y

1.80

120

7

D

1.00

z

130

3

Table 10.5 (a) Find the value (i) x, (ii) y, (iii) z. [ 3 marks ] (b) Calculate the composite index for the cost of production of this biscuit in the year 2009 based on the year 2007. [ 3 marks ] (c) The price index for item A in the year 2007 base on the year 2005 is 120. [ 3 marks ]

94

ANSWER: 1 (a)

2

3

4

5

x = 125

y = 7 z = 3.20

(b)

(i) 137.64

(ii)14530.66

(c) (a)

123.88

(b)

(i) 3

(c)

123

(a)

a = 0.50 b = 2.80 c = 125

(b)

(i) 123.33

(c)

172.66

(a)

(i) 21

(ii) 137.5

(b)

(i) 115

(ii) 43.15

(a)

x = 125 y = 1.50 z = 1.30

(b)

(i) 121

(c)

(i) 12.05

75 (ii) 25

(ii) 2533.78

(ii) 132

95

11.0 SOLUTION OF TRIANGLES

11.1

Diagram shows a triangle ABC. AB = 14 cm, AD = 6 cm, BC = 12 cm and

ABD = 20 .

Diagram 11.1

Find (a)

ADB [2 marks]

(b)

the length of BD [2 marks]

(c)

the length of DC [2 marks]

(d)

the area of triangle ABC [2 marks]

96

11.2

Diagram shows a quadrilateral ABCD such that

ABC is acute.

Diagram 11.2

(a)

Calculate (i)

the length, in cm, of AC, [2 marks]

(ii)

ABC, [2 marks]

(iii)

the area, in cm2, of quadrilateral ABCD. [3 marks]

(b)

Point B lies on AB such that B’C = BC . (i)

Sketch the triangle AB’C, [1 marks]

(ii)

Find

AB’C. [2 marks]

97

11.3

(a)

Diagram 11.3 shows a triangle ABC.

Diagram 11.3 Find the area of the triangle ABC. [4 marks] (b)

Given a triangle PQR, PQ = 9.3 cm, QR = 7.1 cm and (i)

QPR = 42 .

Sketch two different triangles with above given measurements. [2 marks]

(ii)

Find the length of shorter PR. [4 marks]

98

11.4

Diagram 11.4 shows a pyramid with rectangular base ABCD and vertex V. VA VD 7.4 cm , VB VC , and VBA = 38 .

Diagram 11.4 Find (a)

the length VB, [3 marks]

(b)

the angle between the plane VAD and the plane VBC, [4 marks]

(c)

the height of V above the base ABCD [3 marks]

99

11.5

Diagram 11.5 shows quadrilateral ABCD. The area of triangle ABC is 20 cm2 and ABC is acute angle.

Diagram 11.5 Calculate (a)

ABC, [3 marks]

(b)

the length, in cm, of AC, [2 marks]

(c)

ACD, given

ACD is obtuse, [3 marks]

(d)

the area, in cm2, of quadrilateral ABCD. [2 marks]

ANSWER:

11.1 (a) 127.06 (b) 9.54 (c) 15.03 cm (d) 80.03 cm2 11.2 (a) (i)14.18 (ii) 52.25 (iii) 111.73 (b) (ii) 122.68 11.3 (a) 17.16 cm2 (b) (ii) 3.84

11.4 (a) (b) (c)

9.87 cm 102.15 4.581 cm

11.5 (a) (b) (c) (d)

48 6,035 108.12 38.11

100

12.0 PROGRESSION

12.1

Diagram 1 shows the arrangement of the three of an infinite series of rectangles. The first rectangle has a length of b cm and the breadth of h cm. The measurements of the length and breadth of each subsequent rectangle are half of those of its previous one.

Rajah 1 menunjukkan susunan tiga buah segi empat tepat bagi satu siri segi empa tepat serupa yang tak terhingga. Segi empat tepat yang pertama mempunyai panjang berukuran b cm dan lebar h cm. Ukuran bagi panjang dan lebar segi empat tepat yang berikutnya adalah separuh ukuran panjang dan lebar segi empat tepat yang berikutnya.

h cm b cm

Diagram 1 Rajah 1

(a)

Shows that the areas of the rectangle form an arithmetic progression or a geometric progression. Tunjukkan bahawa luas segi empat itu membentuk satu janjang aritmetik atau satu janjang geometri. [3 marks]

101

(b)

Given b = 80 and h = 50. Di beri b = 80 dan h = 50. (i)

Determine which rectangle has an area of

cm2

Tentukan segi empat tepat yang keberapakah yang mempunyai luas cm2 . Find the sum to infinity of areas, in cm2, of the rectangles.

(ii)

Cari hasil tambah hingga ketakterhinggaan bagi luas, dalam cm2, segiempat tepat itu. [5 marks]

12.2

Diagram 2 shows a coil of wire is bent into a few semicircles, such that the radius of each subsequent semicircle increases by 1 unit.

Rajah 2 menunjukkan seutas wayar yang dibentuk kepada beberapa separa bulatan dengan keadaan jejari separa bulatan yang berkutnya bertambah sebanyak 1 unit.

r

r+1

r+2

Diagram 2 Rajah 2 (a)

Show that the perimeter of each semicircle form an arithmetic progression and state the common difference. Tunjukkan bahawa perimeter-perimeter separa bulatan itu membentuk satu

102

janjang arithmetik dan nyatakan beza sepunya janjang itu. [3 marks] (b)

[Use/Guna π = 3.142] Given that r = 5 cm, Diberi bahawa r = 5 cm, (i)

determine which semicircle has the perimeter of 51.42, tentukan separa bulatan yang keberapakah mempunyai perimeter 51.42,

(ii)

find the sum of the perimeter of the first 15 semicircles. cari hasil tambah perimeter bagi 15 separa bulatan pertama. [3 marks]

12.3

Tini and Rudy start working on the same day. Tini earns RM 14 on the first day. Her earning increases constantly by RM x for every subsequent day. She earns RM 50 on her 25th day of working. Rudy earns a fixed salary of RM 30 per day.

Tini dan Rudy mula bekerja pada hari yang sama. Tini mendapat gaji RM 14 pada hari pertama. Pendapatannya pada setiap hari yang berturutan yang berikutnya bertambah RM x. Pendapatannya pada hari yang ke 25 adalah RM 50. Rudy mendapat pendapatan tetap RM 30 sehari.

Find Cari (a)

the value of x, nilai x, [2 marks]

103

(b)

the sum of Tini’s earning after 25 days of working, jumlah pendapatan Tini selepas 25 hari bekerja, [2 marks]

(c)

the number of days working such that Tini’s total earning is more than Rudy’s total earning. bilangan hari bekerja yang mana jumlah pendapatan Tini melebihi jumlah pendapatan Rudy.

[4 marks] ANSWER:

12.1 (a)

1 4 (b) (i) n = 6 r

(ii) 5333 or 5333.33 or 12.2 (a) d = + 2 // 5.142 (b) (i) n = 6 (ii) 925.56 12.3 (a) x = 1.50 (b) 500 n = 23

104

13.0 LINEAR LAWS

13.1

Table 13.1 shows the variables x and y obtained from an experiment. Variables x q and y are related by the equation y px 2 , where p and q are constants. x x

0·8

1

1·3

1·4

1·5

1·7

y

108·75

79

45·38

36·5

26·67

8·19

Table 13.1 (a) Plot the graph of xy against x3 using the scale of 2 cm to 1 unit on the x3-axis and 2 cm to 10 units on the xy-axis. Hence, draw the line of best fit. [5 marks] (b) Using your graph, find the value of (i) p and q, 45 (ii) x when y . x [ 5 marks ] 13.2 Use the graph paper provided to answer this question. Table 2 shows the variables u and v, obtained from an experiment. Variables u and b v are related by the equation v = au + where a and b are constants. u u 05 10 15 20 25 30 v

14 6

68

40

24

12

04

Table 13.2 (a) Using the scale of 2 cm to 1 unit for both axes, plot uv against u2. Hence, draw the line of best fit. [5 marks] (b) Use the graph from (a) to (i) the value of a and of b, (ii) the value of v when u = 5 ; [5 marks]

105

13.3

Use the graph paper provided to answer this question. Table 13.3 shows the values of two variables x and T , obtained from an experiment. Variables x and T are related by the equation T 8 Kn x , where K and n are constants. x (mols 1) T( C)

08

25

40

58

75

91

–5 23

–2 49

2 13

12 89

33 69

71 43

Table

13.3

(a) Plot the graph log10 (T + 8) against x using the scale of 2 cm to 1 unit on the x-axis and 2 cm to 0 2 unit on the log10 (T + 8)-axis Hence, draw a line of best. [4 marks] (b) Use your graph from (a) to find the value of (i) K. (ii) n. [4 marks] (c) Hence, find the value of T when x = 6 5. [2 marks] 13.4 Table 13.4 shows the values of two variables, x and y, obtained from an m n experiment. It is known that x and y are related by the equation 1 , where y x m and n are constant. x

15

20

25

40

50

10 0

y

10

12

14

20

22

30

Table 13.4 (a)

1 1 against . x y Hence, draw the line of best fit. Plot

[5 marks] (b)

Use the graph in (a) to find the value of (i) m. (ii) n, (iii) y when x = 3. [ 5 marks ]

106

13.5 Table 13.5 shows the values of two variables, x and y, that are obtained experimentally. It is known that the values of x and y are related by the equation y = pk 2 x , where p and k are constants. x

05

y

10

15

33 33 22 22 14 82

20

25

30

35

9 88

6 58

4 39

2 93

Table 13.5 (a) Reduce the equation y = pk

2x

to the linear form. [1 mark]

(b) Draw the graph of log 10 y against x. [ 4 marks ] (c) From the graph, find (i)

the values of p and of k,

(ii)

the value of y when x = 2 1. [ 5 marks ]

ANSWER: 13.1

(b)

px 3

yx

x3

0.51

1.0

2.2

2.7

3.4

4.9

xy

87

79

59

51.1

40

13.9

q

(i)

p = − 16.67

(ii)

xy = 45 , x = 1.458

,

q = 95

13.2 u2

0 25

1 00

2 25

4 00

6 25

9 00

uv

73

68

60

48

30

12

07

b = 75

uv = au2 + b

(b) (i)

a =

(ii)

v

= 1 789

107

13.3 x

08

log10(T+8) 0 4425 (b)

25

40

58

75

91

0 7412

1 006

1 320

1 62

19

log10 (T + 8) = log10 K + xlog10 n K = 1 995

(c)

n = 1 489

log10 (T + 8) = 1 44 T = 19 54

13.4

1 x

0 6667

05

04

0 25

02

01

10

0 8333

0 7143

05

0 4545

0 3333

1 y

1

(b)

=

y

n 1 m x

m=5 (c)

1 m

n = 6 12

y = 1 653

13.5 x

05

10

15

20

25

30

35

log 10 y 1 5228 1 3467 1 1708 0 9948 0 8182 0 6425 0 4669 (a) log 10 y = log 10 p (c) (i)

2x log 10 k

k = 1 498 ( 0 005) p = 50 12

(ii)

y = 9 12

108

14.0 INTEGRATIONS

4

g ( x)dx 10 , find

14.1 Given that 1

1

2 g ( x)dx,

(a) the value of 4

[ 2 marks ] 4

k

(b) the value of k if

2 g ( x) dx = 50.

1

[ 3 marks ]

14.2 (a) A curve passing through point (1, 2) has a gradient of 3x2 2 . Find the equation of the curve [3 marks] 2 y 3 x x . (b) Diagram 2 shows part of the curve y

y = 3x – x2 P

0

2

x

Diagram 2

Find the volume generated, in terms of π , when the region P is revolved through 360o about the x-axis. [ 3 marks ]

109

14.3 (a) Diagram 14.3(a) shows the straight line y at point (k , 8).

f ( x) which intersect the curve y 2 x 2

y y = f (x) y = 2x

2

(k, 8)

x

O Diagram 14.3(a) k

f ( x)dx 30, find

Given that 0

(i) the value of k, (ii) the area of the shaded region. [3 marks] (b) Diagram 14.3(b) shows the curve y 2 x 3 which intersects the straight line y 1 at point A and intersects the x-axis at point B. y y2 = x - 3 A

O

y=1

x

B Diagram 14.3(b)

Calculate the volume of revolution when the shaded region is rotated completely about the x- axis. [ 3 marks ]

110

ANSWER:

14.1 (a) 20 (b) k 10 14.2 (a) y x 3 2 x 3 (b) 6.4π unit3 14.3 (a) (i) 2 (ii) 24 23 unit2 (b) 7 unit3 2

111

15.0 VECTORS

15.1 Diagram 15.1 shows OP = 5p, OR = 4r and RQ = kp – r , where k is a constant. M is the midpoint of PR.

Diagram 15.1 (a) Express in terms of p and/or r , (i) PR (ii) OM [ 3 marks ] (b) Express OQ in terms of k , p and r. [ 3 marks ] (c) If Q is lie on OM that has extended, find the value of k. Hence find the ratio of OM:OQ. [ 4 marks ] 15.2 In the Diagram 15.2, OABC is a parallelogram.

Diagram 15.2 Given P and Q lie on BC and OC respectively. AQ and OP intersect at R. OA = 6a , OC = 4c, BP : PC = 1 : 2 and OQ : QC = 1 : 3 (a) Express in terms of a and/or c, (i) AQ (ii) OP [ 2 marks ] 112

(b) Given AR

h AQ , express BR in terms of h, a and / or c.

[ 3 marks ] (c) Given RP

k OP , express BR in terms of k, a and / or c.

[ 3 marks ] (d) Hence, find the value of h and k. [ 2 marks ] 15.3 Diagram 15.3 shows a triangle OAB.

Diagram 15.3 Two straight lines OM and AN intersect at T. Given that OA

2 p , OB

4q , 2BM = MA

1 OB. 4 (a) Express in terms of p and q (i) AN (ii) OM and NB =

[ 3 marks ] (b) (i)

Given that OT

(ii) Given that AT

hOM , state OT in terms of h, p and q . k AN , state OT in terms of k, p and q .

[ 3 marks ] (c) Hence, find the value of h and of k. [ 4 marks ]

113

ANSWER:

15.1 (a) (i) 5 p 4r (ii) 52 p 2r (b) k p 3r (c) 15 , 2:3 4 15.2 (a) (i) 6a c (ii) 4a 4c (b) (h 4)a 6hc (c) (2 4k )a 4k c (d)

3 1 ,h 8 4 15.3 (a) (i) 2 p 3q (ii) (c) 8 9 k ,h 11 11 k

2 3

p

8 3

q

16.0 TRIGONOMETRY FUNCTIONS

16.1 (a) Sketch the graph of

for

.

[4 marks] (b) Hence, using the same axes, sketch a suitable straight line to find the number of solutions for the equation for . State the number of solution. [3 marks] 16.2 (a) Prove that (b)

.

[2 marks] (i) Sketch the graph of for . (ii) Hence, using the same axes, draw a suitable straight line to find the number of solutions to the equation for . [5 marks]

114

16.3 (a)

Prove the identity cosec x

sin x 1 cos x

cot x . [3 marks]

(b) Given that (i) sin 2A (ii) tan(A – 45o )

and

, find the value of

[4 marks] 16.4 (a) Sketch the graph of

for

.

[3 marks] (b) Hence, using the same axes, sketch a suitable straight line to find the number 4x 2 sin 2 x of solutions to the equation for . 9π 3 State the number of solutions. [3 marks]

16.5 (a) Prove the identity tan 2 x(cosec 2 x 1) 1 . [2 marks] (b) On the same axes, sketch the graph of for and the straight line . Hence, state the number of solutions for the equation . [5 marks]

ANSWER: 3

1. -1 No. of solutions = 2 1 x

x

x

2. -1 No. of solutions = 4 115

x

3. (b) (i)

(ii)

4. x

x

x

x

No of solutions = 4

5. 2

x x xx

-2 No of solutions = 4

116

19.0 PROBABILITY DISTRIBUTIONS

19.1 (a) A survey conducted in a certain school shows that two out of three have a face book account. If 8 students selected at random, calculate the probability that, (i) exactly 3 students have a face book account. (ii) at least 3 students have a face book account. [5 marks] (b) The height of the students in a class is normally distributed with the mean of 130 cm and the variance of 100 cm (i) A student from the class is chosen at random, find the probability that the height of the student is more 135 cm. (ii) It is found that 88.5% of the student have height less than k cm, find the value of k. [5 marks] 19.2 The masses of the mangoes from a farm have a normal distribution, with a mean of 170 g and a standard deviation of 25 g. (a) Find the probability that a mango chosen randomly from the farm has a mass of more than 150 g. [3 marks] (b) Table 19.2 shows the grade of the mangoes. Grade A B C Mass, x (g) > 190 m< m 190 Table 19.2 Every season the farm produces 8 000 mangoes. Find (i) the number of mangoes are classified in the grade A. (ii) find the value of m, if given that 2 302 of the mangoes are classified in the grade C. [7 marks] 19.3 (a) In a school 80% of the students are passed in the “SEGAK” test. (i) A sample of 10 students is chosen at random, find the probability that more than 8 students are passed the minimum standard of the test. (ii) Given that the number of students in the school is 1250, find the mean of the number of student who passed the test. [5 marks]

117

(b) The height of the students in a school has a normal distribution with a mean of 125 cm and standard deviation of 20 cm. (i) Find the probability that the height of students chosen at random from the school is more that 140 cm. (ii) Find the percentage of students with the height between 115 g to 140 kg [5 marks] 19.4 (a) In a survey carried out in a school. It is found that 2 out of 5 students are interested in mathematics. If the sample of 10 students are chosen at random, calculate the probability that, (i) exactly 6 students are interested in mathematics (ii) at most 2 students are interested mathematics [5 marks] (b) The volume of a bottles of sauce produced by a factory is normally distributed with a mean of 340 ml and a variance of 64 ml. Find, (i)

the probability that the volume of a bottle of sauce chosen at random from the factory will be more than 348 g (ii) the number of bottle of budu whose volume less than 325 ml if the factory produced 5000 bottles daily. [5 marks]

ANSWER:

19.1 (a)(i) 0.0683 2. (a) 0.7881

(ii) 0.9803 (b) (i) 1 694

(b)(i) 0.3085

(ii) 142

(ii) 156

3. (a) (i) 0.3758

(ii) 1 000

(b) (i) 0.2266

(ii) 46.48

4. (a)(i) 0.04013

(ii) 0.1673

(b)(i) 0.1587

(ii) 152

118

20.0 MOTION ALONG A STRAIGHT LINE

20.1 A particle moves along a straight line and passing through a fixed point O. Its displacement, s m, from point O is given by s 3t 2 2t 1, where t is a time, in seconds, after motion has begun. ( Assume motion to the right is positive ) Find (a) the initial velocity in ms 1 of the particle. [2 marks] the velocity of the particle in ms 1 when the particle is 7 m to the right of (b) O. [3 marks] (c) the range of values of t during which the particle moves to the left. [2 marks] (d) the maximum distance in m, traveled by the particle to the left of O. [3 marks]

20.2 A particle moves in a straight line and passes through a fixed point O. Its velocity, v ms–1, is given by v t 2 5t 4 , where t is the time, in seconds, after leaving O. (Assume motion to the right is positive ) Find (a)

the time when the particle is at instantaneous rest.

(b)

the initial acceleration ms 2of the particle.

[ 2 marks ]

(c) (d)

[ 2 marks ] the time interval during which the particle moves towards the right. [ 3 marks ] the displacement in m, of the particle in 1 second. [ 3 marks ]

119

20.3 A particles moves along a straight line and passes through a fixed point O. Its velocity , v cms-1 , is given by v 2t 2 5t 3 , where t is the time, in seconds, after passing through O. (Assume motion to the right is positive ) Find (a) the initial velocity, in ms-1, of the particle, [1 mark] (b) the minimum velocity, in ms-1, of the particle, [3 marks] (c) the range of values of t during which the particle moves to the left, [2 marks] 3 the total distance, in m, travelled by the particle in the first seconds. (d) 2 [4 marks] 20.4 A particles moves along a straight line and passes through a fixed point O with velocity of 4 ms 1 . Its acceleration , a ms 2 , is given by a 3t 5 , where t is the time, in seconds, after passing through O. (Assume motion to the right is positive ) Find (a) the velocity 4 s after leaving O. [4 marks] (b) the time when the particle is at rest. [3 marks] the acceleration when the particle moves towards the right O with (c) velocity of 4 ms 1 . [3 marks] 20.5 A particle moves along a straight line passes through the fixed point A with v 10 ms 1 and a 3 ms 2 . Given a p qt and the particle stop momentarily at t 5 . (Assume motion to the right is positive ) Find, (a) (b) (c)

the value of p and of q . [4 marks] the displacement in m, of the particle from A when it is at stationary. [3 marks] 2 the uniform velocity ms of the particle. [3 marks] 120

ANSWER:

20.1 20.2

(b) 10 (c) 0 t

(a) -2

(a) 1 , 4 (b) 5

20.3 (a)3 20.4

20.5

(b)

(a) 8

(a) p

3 q

(d)

(c) 0 t 1 @ t

1 t 1 (c) 8 (b)

1 3

3 2

4 3 4 (d)

29 (d) 24

4 , 2 (c) 5 3

2 (b)

275 49 (c) 4 6

121

11 6

21.0 LINEAR PROGRAMMING

21.1 Use graph paper to answer this question. Gunakan kertas graf untuk menjawab soalan ini. Table 1 shows the fees charged for the computer course, “Excel” and “ Power Point” offered by ABC Computer Centre. The number of students enrolled for the“ Excel” course is x and for “Power Point” course is y. Jadual 1 menunjukkan yuran yang dikenakan bagi kursus komputer “Excel” dan “Power Point” yang ditawarkan oleh Pusat Komputer ABC. Bilangan pelajar yang daftar untuk kursus “ Excel” ialah x orang dan kursus “Power Point” ialah y orang. Course/Kursus

Fees/yuran

Excel

RM 420

Power Point

RM 240 Table 1 Jadual 1

Enrolment of the students for the courses above follows the following constraints: Pendaftaran pelajar untuk kursus di atas adalah berdasarkan kekangan berikut: The total number of students enrolled for both courses is not more than 80 I Jumlah pelajar yang mendaftar bagi kedua-dua kursus adalah tidak melebihi 80 orang. The number of students enrolled for” Power Point” is at least half the number of students enrolled for “Excel”. II Bilangan pelajar yang mendaftar kursus “Power Point” adalah sekurang-kurangnya separuh daripadabilangan pelajar yang mendaftar untuk “Excel”. The minimum amount of fees collected is RM 20 000. III Yuran minimum yang dikumpulkan ialah RM 20 000. (a)

Write down three inequalities, other than x satisfy all the above constraints. Tuliskan tiga ketaksamaan, selain daripada x memenuhi semua kekangan di atas. 122

0

and y

0 , which

[3 marks] 0 dan y 0 , yang

(b)

(c)

2

[3 markah] Using a scale of 2 cm to 10 students for both axes, construct and shade the region R which satisfies all the above constraints. [3 marks] Dengan menggunakan skala 2 cm kepada 10 orang pelajar pada keduadua paksi, bina dan lorek rantau R yang memenuhi semua kekangan di atas. [3 markah] Using the graph constructed in 14(b), find Menggunakan graf yang dibina di 14(b), cari (i) the range of students enrolled for the “Excel” course if 30 students enrolled the “Power Point” course, Julat pelajar yang mendaftar kursus “ Excel” jika 30 orang pelajar mendaftar kursus “Power Point”, [2 marks] (ii) the maximum fees collected by ABC Computer Centre for the courses offered. yuran maksimum yang dapat dikumpul oleh Pusat Komputer ABC untuk kursus- kursus yang ditawarkan [2 markah]

Use the graph paper to answer this question Gunakan kertas graf untuk menjawab soalan ini. On a particular ferry to trip Langkawi, the passengers in the ferry comprise x adults and y childrens. Each adult paying a fare of RM 40 and each children paying a fare of RM 30. Pada suatu perjalanan feri pergi ke Langkawi, penumpang feri itu mengandungi x orang dewasa dan y orang kanak-kanak. Setiap orang dewasa perlu membayar RM 40 dan kanak-kanak perlu membayar RM 30. This particular ferry trips is based on the following constrains: Perjalanan feri itu perlu memenuhi syarat-syarat berikut : I

The ferry can accommodate up to 40 passengers only. Feri itu boleh muat sebanyak 40 orang penumpang sahaja.

II

The amount collected from the passenger’s must at least RM 360. Jumlah tambang yang dikutip daripada penumpang mesti sekurangkurangnya RM 360.

III

The number of adult passengers is not more than twice the number of children. Bilangan penumpang dewasa tidak lebih daripada dua kali ganda bilangan penumpang kanak-kanak. 123

(a) Write three inequalities, other than x 0 and y 0 , which satisfy the above constrains. [3 marks] Tulis tiga ketaksamaan, selain x 0 dan y 0 , yang memenuhi semua kekangan di atas.

[3 markah] (b) Using a scale of 2 cm to 5 passengers on both axes, construct and shade the region R which satisfies all of the above constrains. [3 marks] Dengan menggunakan skala 2 cm kepada 5 penumpang pada kedua-dua paksi, bina dan lorek rantau R yang memenuhi semua kekangan di atas. [3 markah] (c) By using the graph constructed in 2(b), find Dengan menggunakan graf yang dibina di 2(b), cari (i) the minimum number of pasengers on this ferry trip if there are five adult passengers only, bilangan minimum penumpang dalam perjalanan feri ini jika hanya ada lima orang penumpang dewasa sahaja, (ii) the maximum amount collected from the fares of the passengers. Amaun maksimum tambang penumpang. [4 marks] [4 markah]

3

Use graph paper to answer this question. Gunakan kertas graf untuk menjawab soalan ini. A whole sale company plans to sell cakes and biscuits in packets. Every packet A contains 100 pieces of cakes and 80 pieces of biscuits. Whereas every packet B contains 60 pieces of cakes and 120 pieces of biscuits. In a particular day, the company sell x unit packet A and y unit packet B. To gain profit, the company has to fulfill the following conditions: Sebuah syarikat pemborong merancang untuk menjual kek dan biskut dalam bungkusan. Setiap bungkusan A mengandungi 100 biji kek dan 80 keping biskut. Manakala setiap bungkusan B mengandungi 60 biji kek dan 120 keping biskut. Pada suatu hari tertentu, syarikat itu menjual x unit bungkusan A dan y unit bungkusan B. Untuk memperoleh keuntungan, syarikat itu perlu memenuhi syarat-syarat berikut: I:

Cakes being sold must not exceed 6600 pieces. Kek yang dijual tidak boleh melebihi 6600 biji. 124

II:

Biscuits being sold must not less than 4800 pieces. Biskut yang dijual tidak boleh kurang daripada 4800 keping.

III:

The total number of packet B must be at least 2 times of the total number of packet A. Jumlah bungkusan B mesti sekurang-kurangnya 2 kali jumlah bungkusan A.

(a) Write down three inequalities, other than x 0 and y 0 , which satisfy the above constrains. [3 marks] Tulis tiga ketaksamaan, selain x 0 dan y 0 , yang memenuhi semua kekangan di atas. [3 markah] (b) By using a scale of 2 cm to 10 units on both axes, draw and shade the region R that satisfies the conditions above. [3 marks] Dengan menggunakan skala 2 cm kepada 10 unit pada kedua-dua paksi, lukis dan lorekkan rantau R yang memenuhi syarat-syarat di atas. [3 markah] (c) Based on your graph, answer the following questions: Berdasarkan graph anda, jawab soalan-soalan yang berikut: (i) Find the range of the number of packet B that is sold if the number of packet A being sold is 15. Cari julat bilangan bungkusan B yang dijual jika 15 bungkusan A telah dijual. (ii) For each packet A and packet B being sold, the company will make a profit of RM 50 and RM 25 respectively. Calculate the maximum profit that the company will gain. Untuk setiap bungkusan A dan bungkusan B yang dijual, syarikat itu akan mendapat keuntungan sebanyak RM 50 dan RM 25 masing-masing. Hitungkan keuntungan maksimum yang akan diperoleh oleh syarikat itu. [4 marks] [4 markah]

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4

Use graph paper to answer this question. Gunakan kertas graf untuk menjawab soalan ini. School Recreation Club plans to takes member of age 13 and age 16 each year. The society comprised of x number of age 13 and y number of age 16. Kelab Rekreasi Sekolah merancang untuk mengambil ahli setiap tahun peringkat umur 13 tahun dan 16 tahun. The enrolment of the members is based on the following constrains : Pengambilan peserta adalah berdasarkan kekangan berikut : I

:

The total intake is at least 150. Jumlah ahli sekurang-kurangnya 150 II : The number of age 13 must not be more than three times of age 16 Bilangan ahli yang berumur 13 tahun mestilah tidak melebihi tiga kali ahli yang berumur 16 tahun III : The total intake should not be more than 400 members. Jumlah pengambilan ahli tidak lebih daripada 400 ahli. (a) Write down three inequalities, other than x 0 and y 0 ,which satisfy the above constraints [3 marks] Tulis tiga ketaksamaan, selain x 0 dan y 0 , yang memenuhi semua kekangan di atas. [3 markah] (b) Using a scale of 2 cm to 50 members on both axes, construct and shade the region R that satisfies all the above constraints. [3 marks] Dengan menggunakan skala 2 cm untuk 50 ahli pada kedua-dua paksi, bina dan lorekkan rantau R yang memenuhi semua kekangan di atas. [3 markah] (c) By using your graph from (b), find Dengan menggunakan graf anda dari (b), cari (i) The minimum collection per month if the member fees for age 13 and age 16 are RM10 and RM20 respectively. Jumlah pungutan minimum dalam sebulan jika setiap ahli berumur 13 tahun dan 16 tahun dikenakan yuran masing-masing RM10 dan RM20 sebulan. (ii) The maximum and the minimum fees which can be collected if the society decides to take only 200 members for age 16 in one year. Jumlah pungutan yuran maksimum dan minimum yang dapat dipungut jika persatuan tersebut memutuskan hanya 200 ahli bagi peringkat umur 16 tahun dalam setahun.. 126

[4 marks] [4 markah] 5

Use graph paper to answer this question. Gunakan kertas graf untuk menjawab soalan ini. The Mathematics Club sells x packets of snack A and y packets snack B at a school funfair. Persatuan Matematik menjual x bungkus snek A dan y bungkus snek B pada hari pasaria sekolah. The number of packets of snacks is based on the following constrains: Bilangan bungkusan snek berdasarkan pada kekangan berikut: I

:

The number of packets of snack A is at least 150. Bilangan bungkus snek A sekurang-kurangnya 150.

II :

The number of packets of snack B is at least 250. Bilangan bungkus snek B sekurang-kurangnya 250.

III : The total number of snacks A and B ia at least 500 but not more than 800. Jumlah bilangan snek A dan B sekurang-kurangnya 500 tetapi tidak lebih daripada 800. (a) Write three inequalities, other than x 0 and y 0 , which satisfy the above constrains. [3 marks] Tulis tiga ketaksamaan, selain x 0 dan y 0 , yang memenuhi semua kekangan di atas. [3 markah] (b) Using a scale of 4 cm to 200 packets on both axes, construct and shade the region R which satisfies all of the above constrains. [3 marks] Dengan menggunakan skala 4 cm kepada 200 bungkus pada kedua-dua paksi, bina dan lorek rantau R yang memenuhi semua kekangan di atas. [3 markah] (c) The profit of a packet of snack A is RM 0.15 and the profit of snack B is RM 0.40. Keuntungan bagi sebungkus snek A ialah RM 0.15 dan keuntungan bagi snek B ialah RM 0.40. By using your graph from (b), find Dengan menggunakan graf anda dari (b), cari (i) the maximum profit frommm the sale of snack A and snack B, keuntungan maksimum jualan snek A dan snek B, (ii) the maximum profit frommm the sale of snack A and snack B. keuntungan maksimum jualan snek A dan snek B. [4 marks] 127

[4 markah] ANSWER:

1

(a)

2

(a) I x y 40 II 4x 3 y 36 III x 2y (b) Draw correctly at least one straight line* Draw correctly all the three straight lines The correct region,R shaded. (c) (i) min passangers = 11 (ii) max fares, k = 40*(26) + 30*(14)

x y 80 1 y x II 2 III 420x 240 y 20000 or 21x 12 y 1000 (b) Draw correctly at least one straight line* Draw correctly all the three straight lines The correct region,R shaded. (c) (i) 31 x 50 (ii) maximum fees, k = 420*(53) 240*(26) = RM 28 500 I

= RM 1 460 3

(a) I 100x + 60y ≤ 6600 or equivalent II 80x + 120y ≥ 4800 or equivalent III y ≥ 2x or equivalent (b) Draw correctly at least one straight line* Draw correctly all the three straight lines The correct region,R shaded. (c) (i) 30 ≤ y ≤ 85 (ii) 50x + 25y = k k = 50*(30) + 25*(60) Maximum profit = RM 3 000

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