Ats maths09

JABATAN PELAJARAN PERAK

ANSWER TO SCOREMATHEMATICS SPM

(SECONDARY SCHOOL)

TOPICAL EXERCISES

1

PART A

SETS

1. The Venn diagram in the answer space shows sets X, Y and Z such that the universal set,.X Y Z

On the diagrams in the answer space, shade(a) the set Y Z (b) the set ( ) .X Y Z

[3 marks]Answer:(a)

(b)

2. The Venn diagram in the answer space shows sets P, Q and R with = PQR.On the diagram provided in answer space, shade

a) P Q'b) Q ( P R )

[3 marks]Answer :

a) P Q

R

2

b)P Q

R

3. On the diagrams provided in the answer space, shade the region whichrepresent the following operation of set.

(a) Set A B'(b) (b) A (B C)'

[3 marks]

Answer:( a )

(b)

4 The Venn Diagram in the answer space shows sets A, B and C.On the diagram provided in the answer space, shade

a the set A B'

b the set A B C '

Answer:(a) (b)

CBA

C

BA

A

B CA

B

C

3

5 The Venn diagram in the answer space shows sets P,Q and R such that the universal set, = P Q R.

On the diagram in the answer space, shade

a the set P/ Q

b the set ( P Q / ) R.

Answer: (a)

(b)

P

QR

P

QR

4

LINEAR INEQUALITIES

1. On the graph in the answer space, shade the region which satisfies the threeinequalities 3 12,y x 2 4y x and 2x .

[3 marks]Answer :

2. On the graph provided, shade the region which satisfies the three inequalitiesy 2x + 8, y x and y < 8.

[3 marks]Answer :

y=x

y=

2x+

8

x

y

O

2 4y x

3 12y x

O x

y

5

3. On the graph in the answer space, shade the region which satisfies the three inequalitiesy – 2x + 10, x < 5 and y 10.

[3 marks]Answer :

x

y

O

y = 10

y=

-2x+

10

5

4. On the graph provided, shade the region which satisfies the three inequalitiesy ≥ -2x + 5, y ≥ x and y < 5.

[3 marks]Answer :

y = -2x + 5

y = x

y

xO

6

5. On the graph provided, shade the region which satisfies the three inequalitiesy ≥ x + 3, x > -3 and 3x + y ≤ 2.

[3 marks]Answer:

y = x + 3

3x + y = 2

y

x- 3 OI

7

SIMULTANEOUS LINEAR EQUATIONS

1. Calculate the values of m and n that satisfy the following simultaneouslinear equations.

54

1332

nm

nm [ 4 marks]

2. Calculate the values of p and q that satisfy the following simultaneouslinear equations.

243

1322

1

qp

qp [ 4 marks]

3. Calculate the values of r and s that satisfy the following simultaneouslinear equations.

72

3

62

sr

sr[ 4 marks]

8

4. Calculate the values of g and h that satisfy the following simultaneouslinear equations.

1834

12

hg

hg[ 4 marks]

5. Calculate the values of x and y that satisfy the following simultaneouslinear equations.

164

32

3

yx

yx [ 4 marks]

6. Calculate the values of a and b that satisfy the following simultaneouslinear equations.

93

53

12

ba

ba [ 4 marks]

9

7. Calculate the values of d and e that satisfy the following simultaneouslinear equations.

732

24

ed

ed[ 4 marks]

8. Calculate the values of h and k that satisfy the following simultaneouslinear equations.

42

1

434

kh

kh[ 4 marks]

9. Calculate the values of m and n that satisfy the simultaneous linearequations.

52

72

nm

nm[ 4 marks]

10

10. Calculate the values of u and v that satisfy the simultaneous linear equations

46

734

vu

vu[ 4 marks]

.

11

QUADRATIC EXPRESSIONS AND EQUATIONS

1. Solve the equation 21

52 2

x

xx[ 4 marks]

2. Solve the quadratic equation yy

33

52 2

[ 4 marks]

3. Solve the quadratic equation 62

)1(3

p

pp[ 4 marks]

12

4. Using factorization, solve the following quadratic equation.

qq 17154 2 [ 4 marks]

5. Solve the quadratic equationr

rr

2

361

[ 4 marks]

6. Solve the quadratic equation2

562

w

w

[ 4 marks]

13

7. Solve the quadratic equation 7)2)(12( aa [ 4 marks]


32 p

p [ 4 marks]

9. Solve the following quadratic equation

)3(5)3)(1( eee [ 4 marks]

14

10. Solve the equation 52)4( 2 yy [ 4 marks]

15

SOLID GEOMETRY

1.

DIAGRAM 1

Diagram 1 shows a composite solid formed by the combination of a cuboid and a half-

cylinder. Using π =7

22calculate the volume, in cm3, of the solid. [4 marks]

21 cm8 cm

P Q

RS

T

UV

W7 cm

16

2.

DIAGRAM 2

Diagram 2 shows a solid cylinder of height 15 cm and diameter 14 cm. A cone with height 6 cm is

taken out of the solid. Calculate the volume, in cm3 , of the remaining solid. (Use π =7

22).

` [4 marks]

15 cm

17

3.

DIAGRAM 3

Diagram 3 shows a solid cuboid ABCDEFGH. A right pyramid with its base EFGH is

taken out of the solid. The volume of the remaining solid is 480 cm 3 . Calculate the heightof the solid, h.

(Use π =7

22). [4 marks]

18

4. The diagram 4 shows a composite solid, formed by the combination of a cylinder to a rightprism. Trapezium ABEF is the uniform cross-section of the prism. AB = BC= 10 cm.

The diameter of the cylinder is 7 cm and the volume of the composite solid is 1075.5 cm3.

Using7

22 , calculate

(a) the volume, in cm3, of the right prism

(b) the length h in cm, of the cylinder.[5 marks]

DIAGRAM 4

19

5. The diagram 5 below shows a combined solid of a right prism and a right pyramid which arejoined at the plane PQRS. T is vertically above the base PQRS. Trapezium PQKJ is the uniformcross section of the prism.

The height of the pyramid is 6 cm and PQ = 18 cm.

(a) Calculate the volume, in cm3, of the right pyramid.(b) It is given that the volume of the combined solid is 448 cm3. Calculate the length, in cm, of

PJ.[5 marks]

DIAGRAM 5

20

MATHEMATICAL REASONING

1. a) State whether the following statement is true or false

b) Write down two implications based on the following sentence

c) State the converse of the following statement and hence determine whether itsconverse is true or false

if 2x > 8, then x > 4

Answer :(a)

(b) Implication 1 :Implication 2 :

(c)

2. (a) Complete each statement in the answer space with the quantifier “all” or“some” so that it will become a true statement.

(b) State the converse of the following statement and hence, determinewhether its converse is true or false.

(c) Write down Premise 2 to complete the following argument :Premise 1 : If m is less than zero, then m is a negative number.

Premise 2 : _________________________________________________

Conclusion : - 4 is a negative number. [5 marks]

Answer :(a) (i) of the multiples of 5 are even numbers.

(ii) hexagons have six sides.

(b)

(c) Premise 2 :

If p > 7, then p > 4

9 > 8 or 4² = 8

x³ = 64 if and only if x = 4

21

3. (a) State whether the sentence below a statement or non-statement ?

(b) Statement 1 : a 0 = 1 for any values of a except a = 0Statement 2 : All prime numbers are odd numbersCombine the above statement to form a new statement which is(i) true(ii) false

(c) Make a general conclusion by induction for the following number pattern:5 = 3(2 ) – 1

10 = 3(22) – 221 = 3(23) – 360 = 3(24) – 4……………… [5 marks]

Answer :(a)

(b) Conclusion :

(c) Conclusion :

4. (a) State whether each of the following statement is true or false.

(i) 23 = 6 or = 3.5

(ii) ( -4 ) x ( -5 ) = 20 and -4 > -2

(b) Complete the premise in the following argument:Premise 1 : If the determinant of a matrix = 0, then the matrix does not

have an inverse.

Premise 2 : _______________________________________________Conclusion :Matrix A does not have an inverse.

(c) Write down two implications based on the following sentence.

A B if and only if A B = A'

[5 marks]

Answer :(a) (i)

(ii)

p + q = 2

2

7

22

(b) Premise 2 :

(c) Implication 1 :

Implication 2 :

5. (a) State whether the following statement is true or false.

(b) Write down two implications based on the following statement:

(c) State the converse of the following statement and hence determine whetherits converse is true or false

if x > 9, then x > 5[5 marks ]

Answer :(a)


(c)

6. (a) State whether each of the following statement is true or false.(i) 42 = 8 or = -2(ii) a { a, b, c } and -3 > -7

(b) Write down premise 1 to complete the following argument.Premise 1 :_________________________________________________Premise 2 : 6 × p 42Conclusion : p 7

(c) Form a general conclusion by induction for the number sequence11, 23, 43, 71, … which follow the pattern11 = 4(1)2 + 723 = 4(2)2 + 743 = 4(3)2 + 771 = 4(4)2 + 7……………… [5 marks ]

-2 ( 3 ) = 6 or -4 > -5

p3 = 8 if and only if p = 2

3 8

23

Answer :(a) (i)

(ii)

(b) Premise 1 :

(c) Conclusion:

7. (a) Determine whether each of the following sentence is a statement.(i) 9 is a prime number.(ii) 2x(x – 2) = 0

(b) Fill in the suitable quantifier to make the following statement become true.

(c) Write down two implications based on the following sentence

3m > 15 if and only if m > 5 [5 marks]

Answer :(a) (i)

(ii)

(b)

(c)

8. (a) State whether the following statement is true or false.(i) 8 + 2 = 10 and 2 < -3(ii) All perfect square numbers are even numbers

.(b) Complete the following argument.

Premise 1 : If 3t = 0, then t = 0.Premise 2 : ________________________________________________Conclusion :3t 0

(c) State the converse of the following statement and hence, determinewhether its converse is true or false.

[5 marks]

…………. prime number are odd numbers

If x3 = 27 then x = 3

24

Answer :(a) (i)

(ii)

(b) Premise 2 :

(c)

9. (a) Is the sentence below a statement or non-statement?

(b) Write down two implications based on the following sentence:

(c) Make a general conclusion by induction for the following number pattern :2 = (0)2 + 23 = (1)2 + 26 = (2)2 + 2

11 = (3)2 + 2…. = ……….. [5 marks]

Answer :(a)


(c)

10. (a) State whether each of the following statement is true or false(i) (-2)2 = 4 or ( -3 )3 = -9

(ii)1 1

2 21

16 8 and 255

(b) Complete the following argument.Premise 1 :_________________________________________________Premise 2 : xn + x is not a quadratic expressions.Conclusion : n 2.


[5 marks]

3 and 4 are factors of 8

m is an even number If and only if m can be divided by 2.

1 – q > 2 if and only if q < -1

25

Answer :(a) (i)

(ii)

(b) Premise 1 :

(b) (c) Implication 1 :

Implication 2 :

26

THE STRAIGHT LINE

1. In Diagram 1, O is the origin, point R lies on the x-axis and point P lies on the y-axis. Straightline PU is parallel to x-axis and the straight line PR is parallel to straight line ST. The equationof straight line PR is x + 2y = 14.

R xO

T(2,-5)

S

U

x + 2y = 14

y

••

•

•

•

(a) State the equation of the straight line PU. [2 marks](b) Find the equation of the straight line ST and hence,

state its x-intercept. [3 marks]

Answer:

P

DIAGRAM 1

27

2. The diagram below shows that the straight line EF and GH are parallel.

DIAGRAM 2

Find(a) the equation of EF. [3 marks](b) the y - intercept and x - intercept of EF [2 marks]

Answer:

28

K O

N

y

x

P

M

L (4, 7)

DIAGRAM 3

3. In the following diagram, O is the origin, point K and point P lies on the x-axis and point N lieson the y-axis. Straight line KL is parallel to straight line NP and straight line MN is parallel to thex-axis. The equation of straight line NP is x – 2y – 18 = 0.

(a) State the equation of the straight line MN.(b) Find the equation of the straight KL and hence, state the coordinate of the point K.

[5 marks]Answer:

29

4. In the diagram 4, the gradient of the straight line KLM is2

1 . Find

(a) the value of p.(b) the x-intercept of the straight line MN.

Answer:

K(0, p)

0

y

x

N

M

L(2, 4)

DIAGRAM 4

30

5. The diagram 5 below shows STUV is a trapezium

i) Given that gradient of TU is -3, finda) the coordinates of point T. [2 marks]b) the equation of straight line TU. [1 marks]

ii) The equation of straight line TV [2 marks]

Answer:

DIAGRAM 5

31

PERIMETER AND AREA

1.

18 cm

Diagram 1

15 15

E

A

B

O F

C D

In Diagram 1, OAB, OCD and OFE are three sectors in a circle centre O. AOF, BCO and EDO arestraight lines. C and D are midpoints of OB and OE respectively.

Using22

7 , calculate

a) the perimeter , in cm, of the whole diagram,b) the area, in cm2, of the shaded region. [6 marks]

32

2.

Diagram 2

S

O

R

P Q120

Diagram 2 shows two sector ORS and OPQ with the same centre O. P and Q are midpoints of OR

and OS respectively and OS = 28 cm. Using22

7 , calculate

a) the perimeter, in cm, of the whole diagramb) the area, in cm2, of the shaded region. [6 marks]

33

3.

In Diagram 3, OFIE is a quadrant of a circle with centre O and GH is an arc of another circle withcentre O. OFG and OIH are straight lines.

OF = FG = 14 cm, and 35GOH .

Using22

7 , calculate


Diagram 3

35

H

E

GF

I

O

34

4.

14 cmDiagram 4

OP

Q

R

T

U

S

Diagram 4 shows two semicircles RSO and RTU with centre Q and O respectively. OPQ is aquadrant of a circle with centre O. RQ = QO. ROU is a straight line.

Using22

7 , calculate


35

5.

H

GF

EI

Diagram 5

J

K

In diagram 5, two arcs of two equal circles, GFE and GIJ, with centres H and K respectively. It isgiven that GH = 14 cm.

Using22

7 , calculate


36

6.

.

Diagram 6

O S

Q

RP

150

Diagram 6 shows two sectors OPQ and OQR with centre O. OS = SR = 7 cm. POSR is a straightline.

Using22

7 , calculate


37

MATRICES

1a) The inverse matrix of is4

5 3

pq

. Find the value of p and of q.

b) Using matrices, calculate the values of x and y that satisfy the following simultaneousequations :

3x – 2y = 6 [6 marks]5x – 4y = 4

Answer :1a) 1b)

2.a) Given that F = and the inverse matrix of F is find the valueof m and of n.

b) Hence, using matrices, calculate the value of p and of q that satisfies the followingequation

2

5

pF

q

[6 marks]

Answer :2a) 2b)

45

23

n

m

2

3,

2

34

14

1

m

38

3.a) Given that find matrix A.

b) Hence, using the matrix method, find the value of d and e which satisfy thesimultaneous equations below.

-d + 2e = 11-3d + 5e = 15 [6 marks]

Answer :3a) 3b)

4. Given matrix M = and matrix MN =a) Find the matrix N.b) Hence, calculate by using the matrix method, the values of r and s that satisfy the

following simultaneous equations :4r + 5s = 16r + 8s = 1 [6 marks]

Answer :4a) 4b)

,10

01

53

21

A

86

54

10

01

39

5. Given the matrix K is ,

a) Find the matrix L so that KL =

b) Hence, calculate the values of h and k, which satisfy the matrix equation:

[6 marks]Answer :5a) 5b)

6. Given matrix M =1 2

2 6

, find

a) the inverse matrix of M.b) hence, using matrix, find the values of x and y that satisfy the following

simultaneous equations :x 2y = 3

–2x + 6y = 4 [6 marks]Answer :6a) 6b)

11

7

58

34

k

h

10

01

58

34

40

7. It is given that matrix A =1

3 4

a

does not have an inverse matrix.

a) Find the value of a.b) If a = 2, find the inverse matrix of A and hence, using matrices, find the values of

x and y that satisfy the following simultaneous linear equations.x 2y = 63x + 4y = 2 [6 marks]

Answer :7a) 7b)

8a) Find matrix G such that3 1 3 1

4 2 4 2G

b) Using matrices, calculate the values of p and q that satisfy the following matrix

equation.2p + 4q = 4p + 3q = 8 [5 marks]

Answer :8 a) 8b)

41

9 a) Find the inverse of matrix4 5

1 2

b) Hence, using matrices, calculate the values of m and n that satisfy the followingsimultaneous equations :

– 4m + 5n = 1–m + 2n = 2 [6 marks]

Answer :9a) 9b)

42

10a) Given matrix S =6 3

4 m

, find the value of m if matrix S has no inverse.

b) Given the matrix equations

and

i) Find the value of h and kii) Hence, find the value of x and y. [6 marks]

Answer :10a) 10bi)

10bii)

11. It is given matrix U =1 2

3 2

and matrix V =2 2

3v

u

such that UV =1 0

0 1

a) find the values of u and v.b) using the matrix method, calculate the values of x and y that satisfy the following

simultaneous equationsx 2 y = 33x + 2y = 5 [6 marks]

Answer:11a)

11b)

1

4

85

67

y

x 8 41

5 7 1

x k

y h

43

GRADIENT AND AREA UNDER GRAPH

1. Diagram 1 shows a distance-time graph for the journey of a car and lorry from town Q.

Diagram 1Graph ABCD represents the journey of the car and graph ACE represents the journey of thelorry. Both vehicles depart from town Q at the same time along the same road.(a) State the length of time, in hours, during which the car is stationary.(b) Calculate the average speed in km/h for the total distance of the car.(c) At the certain time during the journey, both vehicles meet at the same location.

(i) Find the distance, in km, between that location and town Q.(ii) State the time taken by the lorry to reach that location from town Q.

[ 5 marks ]Answer:

(a)

(b)

(c) (i)

(ii)

70

142

195

Time (hour)

Distance (km)

0 3.21.41.0

A

E

D

B C

44

2. The diagram shows a speed-time graph of the movement of a particle for a period of 35 s.

(a) State the uniform speed, in ms-1, of the particle.(b) Given the distance travelled by the particle at a uniform speed is 250m,

calculate(i) the value of t,(ii) the average of speed, in ms-1, of the particle for the period of 35 seconds.

[ 6 marks ]

Answer:(a)

(b) (i)

(ii)

Time (s)

Speed (m s-1)

20

12

10 t 35

45

3. Diagram 2 shows a distance-time graph for the journey of a bus and van.

Diagram 2

Graph AFB represents the journey of the van from town X to town Y. The graph CDEFGrepresents the journey of the bus from town Y to town X. The van leaves town X and thebus leaves town Y at 9.45 p.m. and they travel along the same road.(a) (i) At what time do the vehicles meet ?

(ii) Find the distance, in km, from town Y when the vehicles meet.(b) State the length of time, in minutes, during which the bus is stationary.(c) Calculate the average speed, in km h-1, of the bus for the whole journey.

[ 5 marks ]

Answer:(a) (i)

(ii)(b)

(c)

Time(minutes)0

10 50 70 150

B

Distance (km)

65

100

A

C

G

DTown Y E

F

Town X

46

4. Diagram 3 shows a speed-time graph for a particle, of 30 seconds.

Speed (m s-1)

(a) The distance travelled during the last seconds is 106 m, find the value of v.

(b) Calculate the acceleration, in m s-2, of the particle in the last 5 seconds.

(c) The total distance travelled before 10 seconds is 230 m. Calculate the averagespeed for the whole journey.

[ 6 marks ]Answer:

(a) (i)

(b)

(c)

10 25 30 Time (s)

21

0

Diagram 3

v

25

47

5. The diagram 4 shows the speed-time graphs of two particles M and N over a period of 100seconds. ABCD is the speed-time graph of particle M and EFG is the speed-time graph ofparticle N.

(a) State the length of time in seconds, during which the particle N is constant speed.(b) Calculate the rate of change of speed, in ms-2, of the particle M in the first 20s,(c) Given that the distance travelled by both particles over the 100 s period is the

same and the distance traveled by particles N is 8500 m. Calculate the value of v.[ 6 marks ]

Answer:(a)

(b) (i)

(ii)

Speed (ms-1)

20 40 60 10080Time (s)

0

v

80

145

G

D

F

E

BC

A

Diagram 4

48

PROBABILITY

1.

StudentsProbability of choosing:

Biology Physics Chemistry

Ahmad4

3

8

1

Ling5

1

5

2

The table 1 shows the probability of how Ahmad and Ling might choose their subjects inForm 6. The table is incomplete.Calculate the probability that(a) Ahmad will choose Chemistry and Ling will choose Biology,(b) Ahmad and Ling will choose the same subjects. [5 marks]

Answer:(a)

(b)

2. In bag A, there are 4 blue balls and 3 yellow balls while in bag B, there are 5 blue balls and6 yellow balls. A ball is drawn at random from bag A. After its colour is recorded, it is putinto beg B. Then a ball is drawn at random from beg B. Find the probability that(a) the colour of the two balls are blue,(b) two different colour balls are drawn. [5 marks]

Answer:

(a)

(b)

Table 1

49

3. A company van carries a group of workers consists of 6 males and 8 females. They aredropped off at random at various houses along the route as shown in the diagram below toserve customers.

(a) If two workers are dropped off at house A, calculate the probability that both are females,(b) Two male workers are dropped off at house A. If another two workers are then dropped

off at house B, calculate the probability that at least one of them is female.[5 marks]

Answer:

(a) (b)

4 In a lucky draw, there are three categories of gifts consisting of 4 hotel vouchers, 6 shoppingvouchers and 10 restaurant vouchers. All vouchers are placed inside similar envelopes andput inside a box. The honored guests of the lucky draw are requested to draw at random twoenvelopes from the box.

Calculate the probability that the first honored guest draw(a) the first envelope with a hotel voucher and the second with a shopping voucher,(b) two envelopes with vouchers of the same categories. [5 marks]

Answer:

(a)

(b)

Other houses●

Company House A

House B

●

●

●

50

5. The table below shows the number of participants in a quiz contest according to school andgender.

SchoolNumber of Participants

Male FemaleSMK Ria 4 4SMK ACS 5 2SMK Puteri 3 6

(a) If two participants are chosen at random from SMK ACS, calculate the probability thatboth of them are males.

(b) If two participants are chosen from the male participants, calculate the probability thatboth of them are from the same school. [5 marks]

Answer :

(a)

(b)

6. Bottle A contains three red marbles and two yellow marbles. Bottle B contains four red marblesand three yellow marbles. A marble is chosen at random from each bottle.Find the probability that(a) both marbles are in yellow,(b) both marbles are in different color. [5 marks]

Answer:a)

b)

Table 2

51

LINES AND PLANES IN 3-DIMENSIONS

1. Diagram 1 shows a pyramid with a horizontal rectangular base ABCD. M is themid-point of CD and apex V is 9 cm vertically above the point MIdentify and calculate the angle between the line VB and the base ABCD.

24 cm

14 cm

B

A

D

M

C

V(4 marks)

Answer :

2. Diagram 2 shows a right prism. Right angled triangle PQR is the uniform cross-section of the prism. Calculate the angle between the plane RTU and the planePQTU. (3 marks)

Answer :

DIAGRAM 1

T

P

5 cm

Q

DIAGRAM 2

R

S

U

18 cm

12 cm

52

3. Diagram 3 shows a prism with a horizontal square base HJKL. Trapezium EFLK isthe uniform cross-section of the prism. The rectangular surface DEKJ is vertical whilethe rectangular surface GFLH is inclined.Calculate the angle between the line DL and the base HJKL (4 marks)

Answer :

4. Diagram 4 show s a right prism . Plan ABFE, ADHE and BCGF are rectangularplanes . Planes ABCD and EFGH are parallel trapezium. It also given BAD and

ADC = 90, FG = 5 cm, AB = 2 cm and CG = 12 cm. Identify and calculate the angle

between the plane ADG and AEHD.

5 cm

4 cm

HG

FE

D C

BA

[4 marks]Answer :

8 cm

G

L

H

K

J

D

FE L

5 cm

6 cm DIAGRAM 3

DIAGRAM 4

53

5. Diagram 5 shows a right prism with horizontal rectangle base. Right triangleRSW are the uniform cross section of the prism.Calculate the angle between plane SRV and plane RSTU (3 marks)

Answer :

6. Diagram 6 shows a right prism. The base PQRS is a horizontal rectangle. Right angledtriangle QRU is the uniform cross section of the prism. V is the midpoint of PS.Identify and calculate the angle between the line UV and the plane PQRS.

(4 marks)Answer :

DIAGRAM 5

5 cm

10 cm12 cm

T

U

W

S

R

VV

T

U

W

S

R

DIAGRAM 6

S

Q

U

T

P

10 cm

R

12 cm

5 cmV

54

7. Diagram 7 shows a right prism. Right angled triangle PQR ia the uniform cross-section of the prism. Calculate the angle between the line TR and the planePQTU. (4 marks)

Answer :

18 cm

12 cm

5 cm S

T

U

Q

P

R

DIAGRAM 7

55

PART B

GRAPH OF FUNCTIONS

1 (a) Complete Table 1 in the answer space for the equation y =x

6by writing down

the values of y when x = -2 and x = 0.5 [2 marks](b) (For this part of the questions, use a graph paper)

By using a scale 2 cm to 1 unit on the x-axis and 2 cm to 1 unit on the

y-axis, draw the graph of y =x

6for -4 ≤ x ≤ 4. [4 marks]

(c) From your graph, find

i. the value of y when x = -2.7,ii. the value of x when y = 3.6. [2 marks]

(d) Draw a suitable straight line on the graph in (b) to find the values of x which

satisfy the equation 0236

xx

for -4 ≤ x ≤ 4. [4 marks]

Answer:(a)

X -4 -3 -2 -1 -0.5 0.5 0.8 1.5 2.5 4

Y -1.5 -2 -3 -6 -12 12 4 1.5

Table 1

(c) i) y =

ii) x =

(d) x =

2. (a) Complete Table 2 in the answer space for the equation y =x

72by writing

down the values of y when x = 2.4, x = 6 and x = 12.5 [3 marks]

(b) (For this part, use a graph paper)By using a scale 1 cm to 1 unit on the x-axis and 1 cm to 2 units on the y-axis, draw

the graph of y =x

72for 2 ≤ x ≤1 4. [4 marks]

(c) From your graph, finda. the value of y when x = 4.3,b. the value of x when y = 22. [2 marks]

(d) Draw a suitable straight line on your graph to find all the values of x which


36

x

xState these values of x. [3 marks]

56

Answer:

(a)X 2 2.4 3 4 6 8 10 11.6 12.5 14

Y 36 24 18 9 7.2 6.2 5.14

Table 2

(c) (i) y =

(ii) x =

(d) x =

3. a) Complete Table 3 in the answer space for the equation y = 2x2 – 5x – 3.[2 marks]

b) For this part, use a graph paper.By using a scale 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, drawthe graph of y = 2x2 – 5x – 3 for -3 ≤ x ≤ 5. [4 marks]

c) From your graph, findi) the value of y when x = -2.4,ii) the value of x when 2x2 – 5x – 3 = 0. [2 marks]

d) Draw a suitable straight line on your graph to find all the values of x which satisfythe equation 2x2 – 8x = 7 for -3 ≤ x ≤ 5.State these values of x. [4 marks]

Answer:a)

c) i) y =

ii) x =

d) x =

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

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

Table 3

57

4. a) Complete Table 4 in the answer space for the equation y = -2x² + 5x + 8 by writingdown the values of y when x = -2 and x = 3 [2 marks]

b) For this part of question , use the graph paper provided . You may use a flexiblecurve ruler.By using a scale 2 cm to 1 unit on the x axis and 2cm to 5 unit on they axis , drawthe graph of y = -2x² + 5x +8 for -3.5 ≤ x ≤ 4 [4 marks]

c) From the graph, find

(i) the value of y when x = 3.3

(ii) the value of x when y = -16[2 marks]

d) Find and draw a suitable straight line on your graph to determine the values of xwhich satisfy the equation 2x – 2x ² = -10 for -3.5 ≤ x ≤ 4State the values of x [4 marks]

Answer :

a)

x -3.5 -3 -2 -1 0 1 2 3 4y -34 -25 1 8 11 10 -4

c) i) y =

ii) x =

d) x =

5. a) Complete Table 5 in the answer space for the equation y = x3 – 13x + 18 .

[2 marks]b) For this part, use a graph paper.

By using a scale 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, draw thegraph of y = x3 – 13x + 18 for -4 ≤ x ≤ 4. [4 marks]

c) From your graph, findi. the value of y when x = -1.5,ii. the value of x when y = 25. [2 marks]

d) Draw a suitable straight line on your graph to find all the values of x which satisfy theequation x3 – 11x – 2 = 0 for -4 ≤ x ≤ 4. State these values of x. [4 marks]

Table 4

58

Answer:(a)

X -4 -3 -2 -1 0 1 2 3 4

Y 6 36 30 18 6 6 30

c) i) y =

ii) x =

d) x =

6. a) Complete Table 6 in the answer space for the equation y = x3 + x2 – 12x – 5.[2 marks]

b) For this part, use a graph paper.By using a scale 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, drawthe graph of y = x3 + x2 – 12x – 5 for -4 ≤ x ≤ 4. [4 marks]

c) From your graph, findi. the value of y when x = 0.5,ii. the value of x when y =11.9. [2 marks]

d) Draw a suitable straight line on your graph to find all the values of x whichsatisfy the equation x3 + x2 – 10x = 0 for -4 ≤ x ≤ 4.State these values of x. [4 marks]

Answer:

(a)X -4 -3 -2 -1 0 1 2 3 4

Y -5 13 7 -5 -15 -17 27

c) i) y =

ii) x =

d ) x =

Table 5

59

TRANSFORMATION

1. Diagram 3 shows three pentagons EFGHJ, EKLMN and KPQRS on a Cartesian

plane.

Diagram 3

(a)Transformation T is the translation

2

3

Transformation R is a reflection about the line KMN.

State the coordinates of the image of point J under the following

transformations:

(i) TR

(ii) RT

[4 marks]

(b) EKLMN is the image of EFGHJ under transformation V and EKLMN is the

image of KPQRS under transformation W.

Describe in full the transformation

(i) V

(ii) W

[5 marks]

60

(c) Given that the area of EFGHJ is 26.8cm2 , calculate the area of shaded region

[3 marks]

Answer:

(a)

(i)

(ii)

(b)

(i) V =

(ii) W=

(c)

2 Diagram 5 shows quadrilateral ABCD, PQRS and KLRM drawn on a Cartesian

plane.

Diagram 5


2

4.

Transformation V is a reflection in the line y = 1 .

State the coordinates of the image of point A under the following

transformations:

(i) Translation T,

(ii) Combined transformation VT

[3 marks]

61

(b) (i) KLRM is the image of ABCD under the combined transformations WU.

Describe in full, the transformation U and the transformation W.

(ii) Given that the shaded region KLQPSM represents a region of area 120 m2.

Calculate the area, in m2, of the region represented by PQRS.

[9 marks]

Answer:

(a)

(i)

(ii)

(b)

(i) U =V =

(ii)

3. (a) Diagram 3 shows the point F on a Cartesian plane.

DIAGRAM 3

x

y

2 4 6 8 10 12 14 16

2

4

6

8

10

0

F

62

Transformation S is a translation

2

5.

Transformation T is a reflection in the x = 9.

(i) State the coordinates of the image of point F under transformation S.

(ii) State the coordinates of image of point F under transformation TS. [3 marks]

Answer:

(a) (i)

(ii)

(b) Diagram 4 shows three triangle PQR, ACG and EFG on a Cartesianplane.

DIAGRAM 4

Triangle ACG is the image of triangle PQR under transformation V.Triangle EFG is the image of triangle ACG under transformation W.

(i) Describe in full transformation :

(a) V(b) W [6 marks]

(ii) Given that the area of triangle EFG represents a region of area 72 unit2.

x

y

E

2 4 6 8 10 12 14 16

2

4

6

8

10

O

F

C

A

GP

RQ

63

Calculate the area, in unit2, of the region represented by triangle PQR.

[3 marks]Answer:

(b) (i) (a)

(b)

(ii)

4. (a) Transformation R is a 90° clockwise rotation about the centre (2, 2).

Transformation T is a translation

3

4.

State the coordinates of the image for coordinate (6 , 4) under the followingtransformations:

(i) R2.

(ii) TR. [4 marks]

Answer:

(a) (i)

(ii)

(b) Diagram 5 shows quadrilateral , ABCD, PQRS and EFGH, drawn on a Cartesian

plane.

DIAGRAM 5

-12 -10 -8 -6 -4 -2 2 4 6 810 12

F

E

GH

S

QP

DC

BA

y

xO

R

6

4

2

- 2

- 4

64

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

under transformation Q.


(a)Transformation S

(b)Transformation Q [5 marks]

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

[3 marks]

Answer:

65

STATISTICS

1. The data in Diagram 1 shows the monthly pocket money, in RM, received by 40 students.

56 44 60 64 52 53 55 3536 47 54 59 34 54 52 4849 51 62 58 38 63 49 4345 38 48 57 44 49 46 4032 41 46 56 42 48 51 39

Diagram 1

(a) Based on the data in Diagram 1 and using a class interval of RM5, completeTable 1 in the answer space. [4 marks]

(b) From the table in a),(i) state the modal class,(ii) calculate the mean monthly pocket money of the students. [4 marks]

(c) By using a scale of 2 cm to RM5 on the x-axis and 2 cm to 1 student on the y-axis,draw a histogram based on the data.

[4 marks]

Answer:(a)

Pocket money(RM) Frequency Midpoint

31-35

36-40

Table 1(b) (i)

(ii)

(c)

66

2. The data in Diagram 2 shows the marks obtained by 42 students in a Mathematics finalexam.

51 20 45 31 26 40 3048 32 37 41 25 36 3846 38 28 37 39 23 3933 35 42 29 38 31 4942 34 26 35 43 42 2226 47 40 48 44 34 54

Diagram 2

(a) Based on the data in Diagram 2 and using a class interval 5 marks, completeTable 2 in the answer space. [4 marks]

(b) From the table in a),(i) state the modal class,(ii) calculate the mean mark for the Mathematics final exam and give your

answer correct to 2 decimal places. [4 marks](c) By using a scale of 2 cm to 5 marks on the horizontal axis and 2 cm to 1 student on

the vertical axis, draw a histogram based on the data. [4 marks]

Answer:(a)

Marks Frequency Midpoint

20-24

25-29

Table 2(b) (i)

(ii)

(c)

67

3. The data below shows the height, in cm for a group of 40 students in a class.

132 141 146 156 142 148 151 139136 147 154 159 134 154 152 148140 153 162 155 138 163 149 143156 144 160 164 152 151 158 135145 138 148 157 144 149 146 149

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

b) From the table in a),i) State the modal class,ii) Calculate the mean height of the students. [4 marks]

c) By using a scale of 2 cm to 5cm on the x-axis and 2cm to 1 student onthe y-axis, draw a frequency polygon based on the data. [5 marks]

Answer:a)

Height (cm) Midpoint Frequency

131-135

136-140

Table 3

b i)

ii)

c)

68

4. The data shows the time taken by 42 participants to complete a road race.

51 20 45 31 26 40 3025 32 37 41 21 36 3846 38 28 37 39 23 3933 35 42 29 38 31 2342 34 26 35 43 28 2225 47 31 48 44 34 54

a) Using the data above and a class interval of 5 minutes, complete Table 4 in the answerspace. [3marks]

Answer:

Minutes Midpoint Frequency20-2425 -29

Table 4

b) Based on your table in (a),

Calculate the mean for the time taken and give your answer correct to two decimal places.

c) By using a scale of 2 cm to 5 minutes on the horizontal axis and 2 cm to 1 participant on thevertical axis, draw a frequency polygon based on the data. [5 marks]

d) Based on the polygon in (c), state one piece of information. [1 mark]

69

5. The data shows the length, in cm, of 50 pieces of wood.

18 42 23 13 29 33 30 38 30 277 25 35 11 27 35 27 34 16 2616 21 41 31 20 33 27 28 17 3226 20 30 31 26 32 23 19 22 3721 24 25 28 35 12 29 32 33 31

a) Using the data and a class interval of 5 cm, complete Table 5 in the answer space.[5 marks]

b) For this part of the question, use the graph paper.By using a scale of 2 cm to 5 cm on the x-axis and 2 cm to 5 pieces of wood on the y-axis, drawan ogive based on the data. [5 marks]

c) (i) From your ogive in (b), find the first quartile,(ii) Hence, explain briefly the meaning of the first quartile. [2 marks]

Answer:5a)

Length(cm) Upperboundary

Frequency Cumulative Frequency

0 4

5 9

6. A donation drive for the School Building Fund accumulated a substantial amount of money from400 people as shown in Table 6.

Collection(RM)

1-10 11-20 21-30 31-40 41-50 51-60 61-70 71-80 81-90

Frequency 10 20 40 60 85 90 50 35 10Table 6

a) State the modal class for the above data. [1 mark]b) Construct a cumulative frequency table. [2 marks]c) By using a scale of 2 cm to RM10 on the x-axis and 2 cm to 40 people on the y-axis, plot an

ogive for the data. [5 marks]d) From the ogive, find the

(i) median, [1 mark](ii) percentage of donors who donated less than RM30, [1 mark](iii) number of donors who donated at least RM65. [2 marks]

Table 5

70

PLAN AND ELEVATION

1 (a) Diagram 1(i) shows a solid right prism. The base BCKJ is on horizontal plane.EFGM and CDLK are vertical planes whereas EDLM is a horizontal plane.The plane AFGH is inclined. Hexagon ABCDEF is the uniform cross section of the prism.The sides AB, FE and DC are vertical.

Draw in full scale, the plan of the solid . [3 marks]

(b) A half-cylinder is joined to in Diagram 1(i) at the vertical plane BCQP to form a combined solid asshown in Diagram 1(ii)The height of the half-cylinder is 2 cm

b ii)

Draw in full scale ,i. the elevation of the combined solid on a vertical plane parallel to BC as viewed from X [4 marks]ii. the elevation of the combined solid on a vertical plane parallel to CK as viewed from Y. [5 marks]

C

3 cm

K

L

M

ED

G

H

F

A

B

J

8 cm

3 cm

4 cm

6

6

DIAGRAM 1(i)

X C

K

L

M

ED

G

H

FA

B

J

8 cm

3 cm

4 cm

DIAGRAM 1(ii)

P

QY

3 cm

71

2. Diagram 2(i) shows a solid right prism with a rectangular base ABCD. ABMKGF is theuniform cross- section of the prism. Rectangle GHJK is an inclined plane. EFGH andJKMN is a horizontal plane.The edges AF, BM, CN and DE are verticals. BM = CN = 2cm, AF = DE = 4cmAB =DC = 7cm, KM = JN = 5cm and FG = EH = 3cm.

S 2 cm

5 cm

6 cm

7 cm

3 cm

4 cm

J N

MK

H

G

E

F

DC

BA

a) Draw full scale, the elevation of the solid on a vertical plane parallel to AB asviewed from X. [3 marks]

b) A prism is joined to the solid in Diagram 2(i) at the vertical plane CDEHJN.Thecombined solid is as shown in Diagram 2(ii). Right angle triangle PQR and DSC is theuniform cross-section of the prism.QPR = SDC = 900 , DP = CR = SQ = 6cm and PQ = 3 cm.

S

3 cm

R

Q

P

2 cm

5 cm

6 cm

7 cm

3 cm

4 cm

J N

MK

H

G

E

F

DC

BA

Draw to full scale,(i) The elevation of the combined solid on a

vertical plane parallel to BC as viewedfrom Y. [4 marks]

(ii) The plan of the combined solid. [5 marks]

DIAGRAM 2(i)

DIAGRAM 2ii)

Y

X

72

3. (a) Diagram 3(i) shows a solid prism.Hexagon ABCDEF is the uniform cross section of the prism. The base ALGF is on thehorizontal plane. The sides BA, CD and EF are vertical whereas the sides BC and DE arehorizontal.

Draw in full scale, the plan of the solid prism. [3 marks]

(b) A solid prism with triangle AFM as its uniform cross section is joined at the vertical planeABCDEF to form a combined solid as shown in Diagram 3(ii).

Draw in full scale,i. the elevation of the combined solid on a vertical plane parallel to GF as viewed from X.

[4 marks]ii. the elevation of the combined solid on a vertical plane parallel to AF as viewed from Y.

[5 marks]

P

FG

HE

DI

J

K B

C

AL

2 cm

4 cm

4 cm

4 cm

8 cm

5 cm

DIAGRAM 3(i)

DIAGRAM 2(ii)

6 cm

Y

M

FG

HE

DI

J

K B

C

AL

2 cm

4 cm

4 cm

X

Q N

4 cm

73

4. Diagram shows a solid right prism with rectangular base ABCD on a horizontal plane. Thesurface BFGJKC is its uniform cross-section. The rectangle LKJI is an inclined plane and thesquare EFGH is a horizontal plane. The edges GJ and HI are vertical and HI = GJ = 2 cm.

(a) Draw full scale, the elevation of the combined solid on a vertical plane parallel to JI asviewed from X. [3 marks]

(b) A half-cylinder of diameter 4 cm is joined to the prism in diagram 4(i) at the vertical planeLDPM. The combined solid is as shown in Diagram 4 (ii)

Draw full scale,(i) the plan of the combined solid. [4 marks](ii) the elevation of the combined solid on a vertical plane parallel to AB as

viewed from Y.[5marks]

C

B

E

A

DF

G

L

H

X

I J

K

4

8

86

DIAGRAM 4(i)

M

P

Q

N

C

B

E

A

DF

G

L

H

Y

I J

K

4 cm

8 cm

8 cm6 cm

DIAGRAM 4(ii)

74

5. Diagram 5(i) shows a solid right prism with square base ABCD on a horizontal plane.ABKIGF is the uniform cross-section of the prism. AF, LC and BK are vertical edges.Rectangle IJLK is a horizontal plane and rectangle EFGH is an inclined plane.

Answer :

(a) Draw full- scale, the plan of the solid. [3 marks]

(b) A prism with isosceles triangle as its uniform cross-section such that Q is 2 cm above themidpoint of JL is joint to the prism at the plane MPLJ. The length of PL is 2 cm. Thecombined solid is shown in the diagram 5(ii)

cm

Draw full-scale,i. the elevation of the combined solid on a vertical plane parallel to AB as viewed

from R [4 marks]ii. the elevation of the combined solid on a vertical plane parallel to BC as viewed

from S. [5 marks]

S

R

P

N

M

Q

2 cm

4 cm

5 cmL

K

J

I

H

G

F

E

D

C

B

A

DIAGRAM 5 (ii)

2 cm

2 cm

2 cm

4 cm

5 cmL

K

J

I

H

G

F

E

D

C

B

AA

DIAGRAM 5 (i)

75

EARTH AS A SPHERE

1. Table 1 shows the latitudes and longitudes of four points A, B, C and D, on the surface ofthe earth.

Point Latitude LongitudeABCD

200 Sx0 N200 S300 N

250 W250 Wy0 Ey0 E

(a) Q is a point on the surface of the earth such that AQ is the diameter of the earth.State the position of Q. [2 marks]

(b) Calculate(i) the value of x, if the distance from A to B measured along the meridian is

4800 nautical mile,(ii) the value of y, if the distance from A due east to C measured along the

common parallel of latitude is 3664.8 nautical mile. [7 marks](c) An aero plane took off from A and flew due east to C along the common parallel of

latitude and then due north to D.If the average speed for the whole flight is 500 knots, calculate the time taken for thewhole flight. [3 marks]

2. P(600 N, 600 W), Q, R and S are four points on the surface of the earth. PQ is the diameterof the parallel of latitude 600 N. S lies 4800 nautical mile due to south of P and R lies 450

due east of P.(a) State the longitude of Q. [2 marks](b) Find the position of R. [3 marks](c) Calculate the distance, in nautical mile, from P to Q measured along the parallellatitude. [3 marks](d) An aero plane took off from Q and flew towards P using the shortest distance, asmeasured along the surface of the earth, and then flew due south to S, with an averagespeed of 600 knots.Calculate the time, in hours, taken for the flight. [4 marks]

Table 1

JABATAN PELAJARAN NEGERI PERAKKERTAS MODEL PEPERIKSAAN SEBENAR SPM

MATEMATIK

Kertas 2

Dua jam tiga puluh minit

Kod Pemeriksa

Bahagian SoalanMarkahPenuh

MarkahDiperoleh

A

1 3

2 4

JANGAN BUKA KERTAS SOALAN INISEHINGGA DIBERITAHU

1. Kertas soalan ini mengandungi dua bahagian :Bahagian A dan Bahagian B. Jawab semuasoalan daripada Bahagian A dan empat soalandalam Bahagian B.

2. Jawapan hendaklah ditulis dengan jelas dalamruang yang disediakan dalam kertas soalan.Tunjukkan langkah-langkah penting. Ini bolehmembantu anda untuk mendapatkan markah.

3. Rajah yang mengiringi soalan tidak dilukismengikut skala kecuali dinyatakan.

4. Satu senarai rumus disediakan di halaman 2 & 3

5. Anda dibenarkan menggunakan kalkulatorsaintifik yang tidak boleh diprogram.

3 4

4 4

5 4

6 5

7 5

8 6

9 6

10 5

11 6

B

12 12

13 12

14 12

15 12

16 12

Jumlah

Kertas soalan ini mengandungi 15 halaman bercetak.

SULIT1449/2MatematikKertas 22009

2

12 jam

1449/2

1449/2 2009 Hak Cipta JPN Perak [Lihat sebelahSULIT

NAMA :

TINGKATAN :

76

SULIT1449/2

77

UntukKegunaanPemeriksa

MATHEMATICAL FORMULAE

The following formulae may be helpful in answering the questions. The symbols given are theones commonly used.

RELATIONS

1. nmnm aaa .

2. nmnm aaa

3. mnnm aa

4.

ac

bd

bcadA

11

5.)(

)()(

Sn

AnAP

6. )(1)'( APAP

7.2 2

1 2 1 2tan ( ) ( )dis ce x x y y

8. Midpoint

2,

2, 2121 yyxxyx

9. Average speed =distance travelled

time taken

10. Mean =sum of data

number of data

12. Pythagoras Theorem222 bac

13 2 1

2 1

y ym

x x

14.intercept

intercept

x

ym

11. Mean =sfrequencieofsum

frequency)rkof(classmasum

SULIT1449/2


SHAPE AND SPACE

1. Area of trapezium =2

1 sum of parallel sides height

2. Circumference of circle= d = 2r

3. Area of circle = r2

4. Curved surface area of cylinder = 2rh

5. Surface area of sphere = 4r2

6. Volume of right prism = cross sectional area length

7. Volume of cylinder = r2h

8. Volume of cone =3

1r2h

9. Volume of sphere =3

4r3

10. Volume of right pyramid = 3

1base area height

11. Sum of interior angles of a polygon = ( n 2 ) 180 0

12.0

arc of length angle subtended at centre

circumference of circle 360

13.0

area of sector angle subtended at centre

area of circle 360

14. Scale factor, k =PA

PA '

15. Area of image= k 2 area of object

78

SULIT1449/2

79


SECTION A[ 52 marks]

Answer all question in this section.

1. On the graph provided, shade the region which satisfies the three inequalitiesy ≤ x + 5, x + y ≤ 5 and y > 1. [3 marks]

Answer:

2. Calculate the values of e and f that satisfy the simultaneous linear equations.

104

1625

fe

fe[4 marks]

Answer:

x

y

y = x + 5

x + y = 5

O

1 -

SULIT1449/2

UntukKegunaanPemeriksa 3. Diagram 1 below shows a right prism with PST is the uniform cross section of the prism.

Diagram 1

A solid cylinder of diameter 6 cm is taken out from the solid prism. Using7

22 ,

calculate the volume in cm3 of the remaining solid.[4 marks]

Answer:


)52(

xx

Answer:[4 marks]

SULIT1449/2

81


5. Diagram 2 shows a right prism with an isosceles triangle base, STU.The isosceles triangle STU is the uniform cross-section of the prism.ST = SU and W is the midpoint of TU.Calculate the angle between the line PW and the base STU. [4 marks]

Answer:

6. (a) State whether the following sentence is a statement or a non-statement.Give a reason for your answer.

(b) State whether each of the following statements is true or false.(i) { 0 } is an empty set or is an empty set.

(ii){ } is and empty set and is also an empty set.

(c ) Complete the following argument.

Premise 1 : If a > 3, then 5 a > 15.Premise 2 : 5 a < 15.Conclusion :________________________________________________

[5 marks]Answer:(a) _____________________________________

(b) i.___________________ii. _________________

(c) Conclusion: ____________________________________

13

P

W

1012

Q

R

S T

U

DIAGRAM 2

2 + 7 = 1 + 6

SULIT1449/2


7. Bottle A contains three red marbles and two yellow marbles. Bottle B contains four redmarbles and three yellow marbles. A marble is chosen at random from each bottle.Find the probability that(a) both marbles are in yellow,(b) both marbles are in different color. [5 marks]

Answer:

8.

.

T

Diagram 6

O S

Q

R

P

150

Diagram 3 shows two sectors OPQ and ORS with centre O. OP = 8 cm, QR = 6 cm andOS = 2 OT. POTS is a straight line.

Using22

7 , calculate

a) the perimeter, in cm, of the whole diagramb) the area, in cm2, of the shaded region. [6 marks]Answer :

3

SULIT1449/2

83


9. The following diagram shows the distance-time graph of the journeys of two particles Sand T.

Graph ABCD represents the journey of the particle S from station X to Y. The graph EFrepresents the journey of the particle T from station Y to X.Both particles leave station X and station Y respectively at the same time along the sameroad.(a) State the length of time, in minute, during which the particle S is

stationary.(b) If the journey starts at 1500, at what time do the particles meet?(c) Given that the distance travelled by particle S over the 60 minutes period is 186km.

Calculate the value of v.(d) Find the distance, in kilometres, from station Y when the particles meet.(e) Calculate the average speed in m/s of particle S for the whole journey.

[ 6 marks ]

Answer:

A

D

F

CB

600

v

145 E

Speed (ms-1)

Time (minute)20 40 100

80

StationX

StationY

SULIT1449/2


10. In Diagram 4, KL, LN and MN are straight lines. L is on y-axis. LN is parallel tox-axis and KL is parallel to MN.

Diagram 4

The equation of straight line KL is 2x + y + 4 = 0, find(a) The equation of straight line LN,(b) The equation of straight line MN hence state its x-intercept.

[ 5 marks ]Answer :


3 2

and matrix V =2 2

3v

u

such that UV =1 0

0 1


simultaneous equations32 yx

523 yx [6 marks]

Answer :

NL

M(1,6)

KO

x

y

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85


SECTION B[ 48 marks]

Answer any four questions from this section.

12. (a) Table 1 shows values of x and y for equation 823 xxy .

Find the values of v and w.[ 2 marks ]

(b) For this part of the question, use graph paper. You may use a flexible curveruler.By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to10 unit on the y-axis,

Draw the graph of 823 xxy for 43 x .

[ 4 marks ](c) From your graph, find

(i) the value of y when x = 15,(ii) the value of x when y = 20.

[ 2 marks](a) Draw a suitable straight line on your graph to find the values of x that satisfy the

equation 0493 xx for 43 x .

State the values of x.[ 4 marks]

Answer :

x 3 2 1 0 1 2 3 4

y 29 v 7 8 9 4 w 48

TABLE 1

SULIT1449/2

UntukKegunaanPemeriksa 13. Diagram 5 shows four quadrilateral ABCD, ADEF, GHJK and LPNM drawn on

Cartesian plane.

Diagram 5

(a) Transformation M is a reflection about the line x = 3.

Transformation N is the translation

3

5

State the coordinates of the image of point (2,4) under the following transformations:

i) MNii) NM

(b) ADEF is the image of ABCD under transformation V and GHJK is the image of ADEF

under transformation W.

Describe in fulli) Transformation Vii) A single transformation which is equivalent to transformation WV.

(c) LPNM is the image of ABCD under an enlargement.i) State the coordinates of the centre of the enlargement.ii) Given that the area LPNM is 72.8 unit2, calculate the area of ABCD.

[12 marks]

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87


Answer:

a. i) ____________________________________________________

ii) ____________________________________________________

b. i) _____________________________________________________

ii) ____________________________________________________

c. i) _____________________________________________________

ii) ________________________________

SULIT1449/2

UntukKegunaanPemeriksa 14. The data in Diagram 6 shows the mass in kg of 50 students in a class.

40 52 47 64 52 35 54 43 39 5443 39 54 57 53 39 55 38 46 4245 53 60 38 54 62 47 45 52 5338 55 62 54 45 54 36 58 44 4252 48 49 46 49 48 45 54 41 46

Diagram 6

(a) Based on the data in Diagram 7 and using a class interval of 5 kg, completeTable 2 in the answer space. [4 marks]

(b) From the table in a),(i) state the modal class,(ii) calculate the mean mass of the students and give your answer correct to 2

significant figures. [4 marks](c) By using a scale of 2 cm to 5 kg on the x-axis and 2 cm to 1 student on the y-axis,

draw a histogram based on the data.[4 marks]

Answer:(a)

Mass Frequency Midpoint

35-39

40-44

Table 2(b)

(c)

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89


15. Diagram 8 shows a right prism with a horizontal rectangular base ABCD. BCHGF is theuniform cross -section of the prism. GHIJ is a horizontal plane. ABFE and DCHI arevertical planes.

(a) Draw to full scale, the elevation of the solid on a vertical plane parallel to BC as seenfrom X [3 marks]

Answer:

D

B

G

E

A

J

1 cm

C

I

F

H

X4 cm

3 cm

3 cm

2 cm

DIAGRAM8(i)

SULIT1449/2

UntukKegunaanPemeriksa (b) A solid right prism with uniform cross-section KLM is removed from the solid. The

remaining solid is shown in diagram 8(ii).

Draw at full scale

(i) the plan of the remaining solid . [4 marks](ii) the elevation of the remaining solid on a vertical plane parallel to AB as

seen from Y. [5 marks]

Answer:

NL

K M

1 cm

Y

D

B

G

E

A

J

1 cm

C

I

H

4 cm3 cm

3 cm

2 cm

DIAGRAM 8(ii)

SULIT1449/2

91


16. P(650 S, 400 E), Q(650 S, 600 W), R and S are four points on the surface of the earth. PRis the diameter of the parallel of latitude 650 S.(a) (i) State the longitude of R.

(ii) Calculate the shortest distance, in nautical mile, from P to R measured along thesurface of the earth. [4 marks]

(b) S lies north of Q and the distance of SQ measured along the surface of the earth is5100 nautical mile. Calculate the latitude of S. [3 marks]

(c) An aero plane took off from P and flew due west to Q and then flew due north to S.The average speed for the whole flight was 560 knots.Calculate(i) the distance, in nautical mile, taken by the aero plane from P to Q measured

along the common parallel of latitude,(ii) the total time, in hours, taken for the whole flight. [5 marks]

Answer:

END OF QUESTION PAPER

JABATAN PELAJARAN PERAK

ANSWER TO SCOREMATHEMATICS SPM

(SECONDARY SCHOOL)

TOPICAL EXERCISESANSWERS

92

PART A

SETS

1. The Venn diagram in the answer space shows sets X, Y and Z such that the universal set,.X Y Z

On the diagrams in the answer space, shade(a) the set Y Z (b) the set ( ) .X Y Z

[3 marks]Answer:

(a)

[ 1 ](b)

[ 2]

93

2. The Venn diagram in the answer space shows sets P, Q and R with = PQR.On the diagrams provided in answer space, shade

a) P Q´b) Q ( P R )

[3 marks]

Answer :

a) P Q

[ 1 ]

R

b) P Q

[ 2 ]

R

3. On the diagrams provided in the answer space, shade the region whichrepresent the following operation of set.

(a) Set A B’(b) (b) A (B C)’

[3 marks]

Answer / Jawapan:( a ) (b)

[ 1 ] [ 2 ]

94

4 The Venn Diagram in the answer space shows sets A, B and C.On the diagrams provided in the answer space, shade(a) The set A B´(b) The set A B C ´ [ 3 marks ]

Answer:(a) (b)

5 The Venn diagram in the answer space shows sets P,Q and R such that the universal set, = P Q R.

On the diagrams in the answer space, shade

(a) the set P´ Q

(b) the set ( P Q´) R. [ 3 marks ]

Answer:(a) (b)

R

Q

P

A B C AB

C

P

Q

R

[ 1 ] [ 2 ]

[ 1 ] [ 2 ]

95

LINEAR INEQUALITIES

1. On the graph in the answer space, shade the region which satisfies the threeinequalities 3 12,y x 2 4y x and 2x . [3 marks]

Answer :

Straight line 2x drawn correctly. [ 1 ]

Region shaded correctly. [ 2 ]

2. On the graph provided, shade the region which satisfies the three inequalitiesy 2x + 8, y x and y < 8. [3 marks]

Answer :

y = 8

y = 2x + 8

y = x

y

x

Straight line y = 8 drawn correctly. [ 1 ]


42 xy

R

123 xy

2x

y

x

96

3. On the graph in the answer space, shade the region which satisfies the three inequalitiesy – 2x + 10, x < 5 and y 10. [3 marks]

Answer :

x = 5

y = - 2x + 10

y = 10

y

x

Straight line x=5 drawn correctly. [ 1 ]


4. On the graph provided, shade the region which satisfies the three inequalitiesy ≥ -2x + 5, y ≥ x and y < 5. [3

marks]

Answer :

Straight line y=5 drawn correctly. [ 1 ]


y = 5

y = xy = -2x + 5

97

5. On the graph provided, shade the region which satisfies the three inequalitiesy ≥ x + 3, x > -3 and 3x + y ≤ 2. [3 marks]

Straight line x = -3 drawn correctly. [ 1 ]


x = -3

y = x + 3

3x + y = 2

98

SIMULTANEOUS LINEAR EQUATIONS

1 . Calculate the values of m and n that satisfy the following simultaneous linearequations.

54

1332

nm

nm [ 4 marks]

Answer:

2

2814

13)45(32

45

m

m

mm

mn

3

)2(45

n

n

2. Calculate the values of p and q that satisfy the following simultaneous linearequations.

243

1322

1

qp

qp [ 4 marks]

Answer:

6

244

243

264

p

p

qp

qp

5

204

264

q

q

qp

3. Calculate the values of r and s that satisfy the following simultaneous linearequations.

72

3

62

sr

sr[ 4 marks]

Answer:

2

84

1423

62

r

r

sr

sr

4

622

s

s

4. Calculate the values of g and h that satisfy the following simultaneous linearequations.

1834

12

hg

hg[ 4 marks]

Answer:

2

2211

183)21(4

21

h

h

hh

hg

3

)2(21

g

g

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

99

5. Calculate the values of x and y that satisfy the following simultaneous linearequations.

164

32

3

yx

yx [ 4 marks]

Answer:

4

287

164

1264

y

y

yx

yx

3

124

16)4(4

x

x

x

6. Calculate the values of a and b that satisfy the following simultaneous linearequations.

93

53

12

ba

ba [ 4 marks]

Answer:

2

63

93

156

a

a

ba

ba

3

9)2(3

b

b

7. Calculate the values of d and e that satisfy the following simultaneous linearequations.

732

24

ed

ed[ 4 marks]

Answer:

1

1111

73)42(2

42

e

e

ee

ed

2

)1(42

d

d

8. Calculate the values of h and k that satisfy the following simultaneous linearequations.

42

1

434

kh

kh[ 4 marks]

Answer:

4

205

1624

434

k

k

kh

kh

2

42

4)4(2

1

h

h

h

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

100

9. Calculate the values of m and n that satisfy the simultaneous linear equations.

52

72

nm

nm[ 4 marks]

Answer:

1

33

1042

72

n

n

nm

nm

3

62

7)1(2

m

m

m

10. Calculate the values of u and v that satisfy the simultaneous linear equations.

46

734

vu

vu[ 4 marks]

Answer:

3

1

927

73)64(4

64

v

v

vv

vu

2

)3

1(64

u

u

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ] [ 1 ]

[ 1 ]

101

QUADRATIC EXPRESSIONS AND EQUATIONS

1. Solve the equation 21

52 2

x

xx

[ 4 marks]Answer:

2,2

1

0)2)(12(

0232

22522

2

x

xx

xx

xxx

2. Solve the quadratic equation yy

33

52 2

[ 4 marks]

Answer:

5,2

1

0)5)(12(

0592

9522

2

y

yy

yy

yy


)1(3

p

pp

[ 4 marks]Answer:

3,3

4

0)3)(43(

01253

122332

2

p

pp

pp

ppp

4. Using factorization, solve the following quadratic equation.

qq 17154 2

[ 4 marks]Answer:

5,4

3

0)5)(34(

015174 2

q

qq

qq

5. Solve the quadratic equationr

rr

2

361

[ 4 marks]

Answer:

2,2

3

0)2)(32(

062 2

r

rr

rr

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ] [ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 1 ]

102

6. Solve the quadratic equation2

562

w

w

[ 4 marks]Aswer:

4,2

3

0)4)(32(

01252 2

w

ww

ww

7. Solve the quadratic equation 7)2)(12( aa

[ 4 marks]

Answer:

3,2

3

0)3)(32(

0932 2

a

aa

aa


32 p

p

[ 4 marks]

Answer:

2,2

1

0)2)(12(

0232 2

p

pp

pp

9. Solve the following quadratic equation

)3(5)3)(1( eee

[ 4 marks]

Answer:

3,6

0)3)(6(

01832

e

ee

ee

10. Solve the equation 52)4( 2 yy

[ 4 marks]Answer:

7,3

0)3)(7(

021102

y

yy

yy[ 1 ]

[ 1 ]

[ 1 ] [ 1 ]

[ 1 ]

[ 1 ]

[ 1 ] [ 1 ]

[ 1 ]

[ 1 ]

[ 1 ] [ 1 ]

[ 1 ][ 1 ]

[ 1 ] [ 1 ]

[ 1 ]

[ 1 ]

[ 1 ] [ 1 ]

103

SOLID GEOMETRY

1.

DIAGRAM 1Diagram 1 shows a composite solid formed by the combination of a cuboid and

a half-cylinder.Using π =7

22,

calculate the volume, in cm3, of the solid. [4 marks]

Solution:= 7821 [ 1 ]

2

1x

7

22x 42 x 21 [ 1 ]

= 7821 +2

1x

7

22x 42 x 21 [ 1 ]

= 1704 [ 1 ]

2.

DIAGRAM 2

Diagram 2 shows a solid cylinder of height 15 cm and diameter 14 cm. A cone withheight 6 cm is taken out of the solid. Calculate the volume in cm3 of the remaining solid.

(Use π =7

22). `[4 marks]

Solution : 1577

22 2 [ 1 ]

677

22

3

1 2 [ 1 ]

[ 1 ]= 2002 [ 1 ]

P Q21 cm8 cm

RS

T

UV

W7 cm

15 cm

1577

22 2 - 677

22

3

1 2

104

3.

Diagram 3 shows a solid cuboid ABCDEFGH. A right pyramid with its base EFGH istaken out of the solid. The volume of the remaining solid is 480 cm3. Calculate the heightof the solid, h.

(Use π =7

22). `[4 marks]

Solution : 89h [ 1 ]

h 893

1[ 1 ]

hh 893

189 = 480 [ 1 ]

h = 10 [ 1 ]

4. The diagram 4 shows a composite solid, formed by the combination of a cylinder to a rightprism. Trapezium

ABEF is the uniform cross-section of the prism. AB = BC= 10 cm.

The diameter of the cylinder is 7 cm and the volume of the composite solid is 1075.5 cm3.

Using7

22 , calculate

(a) the volume, in cm3, of the right prism

(b) the length, h, in cm, of the cylinder. [5 marks]

Diagram 3

Diagram 4

105

Solution :

(a) 10810142

1 [ 1 ]

= 960 [ 1 ]

(b) h

2

2

7

7

22[ 1 ]

10810142

1 + h

2

2

7

7

22= 1075.5 [ 1 ]

h = 3 [ 1 ]

5. The diagram 5 below shows a combined solid consists of a right prism and a right pyramidwhich are joined at the plane PQRS. T is vertically above the base PQRS. TrapeziumPQKJ is the uniform cross section of the prism.

The height of the pyramid is 6 cm and PQ = 18 cm.

(a) Calculate the volume, in cm3, of the right pyramid.(b) It is given that the volume of the combined solid is 448 cm3. Calculate the length, in cm, of

PJ.[5 marks]

Solution :

(a) 68183

1 [ 1 ]

= 288 [ 1 ]

(b) PJ18142

1 [ 1 ]

68183

1 + PJ1814

2

1 = 448 [ 1 ]

PJ = 10 [ 1 ]

Diagram 5

106

MATHEMATICAL REASONING

1. a) State whether the following statement is true or false

b) Write down two implications based on the following sentence

c) State the converse of the following statement and hence, determine whether itsconverse is true or false.

“ If 2x > 8, then x > 4 “[5 marks]

Answer :(a) True [ 1 ]

(b) Implication 1 : If x³ = 64 then x = 4 [ 1 ]Implication 2 : If x = 4 then x³ = 64 [ 1 ]

(c) If x > 4 , then 2x > 8.True. [ 2 ]

2. (a) Complete each statement in the answer space with the quantifier “all” or“some” so that it will become a true statement.

(b) State the converse of the following statement and hence, determinewhether its converse is true or false.

(c) Write down Premise 2 to complete the following argument :Premise 1 : If m is less than zero, then m is a negative number.

Premise 2 : _________________________________________________

Conclusion : - 4 is a negative number. [5 marks]

Answer :(a) (i) Some of the multiples of 5 are even numbers. [ 1 ]

(ii) All hexagons have six sides. [ 1 ]

(b) If p > 4 then p > 7 [ 1 ]False [ 1 ]

(c) Premise 2 : - 4 is less than zero [ 1 ]

If p > 7, then p > 4

9 > 8 or 4² = 8

x³ = 64 if and only if x = 4

107

3. (a) State whether the sentence below a statement or non-statement ?

(b) Statement 1 : a0 = 1 for any values of a except a = 0.Statement 2 : All prime numbers are odd numbers.

Combine the above statements to form a new statement which isi) trueii) false

(c) Make a general conclusion by induction for the following number pattern:5 = 3(2 ) – 1

10 = 3(22) – 221 = 3(23) – 360 = 3(24) – 4……………… [5 marks]

Answer :(a) Non statement [ 1 ]

(b) i) a0 = 1 for any values of a except a = 0 or all prime numbers are odd numbers.ii) a0 = 1 for any values of a except a = 0 and all prime numbers are odd

numbers.[ 2 ]

(c) Conclusion : 3(2n) – n , n = 1,2,3,4… [ 2 ]

4. (a) State whether each of the following statement is true or false.

(i) 23 = 6 or = 3.5

(ii) ( -4 ) x ( -5 ) = 20 and -4 > -2

(b) Complete the premise in the following argument:Premise 1 : If the determinant of a matrix = 0, then the matrix does not

have an inverse.Premise 2 : _______________________________________________Conclusion :Matrix A does not have an inverse.


A B if and only if A B = A' [5 marks]

Answer :(a) (i) True [ 1 ]

(ii) False [ 1 ]

(b) Premise 2 : The determinant of matrix A = 0 [ 1 ]

(c) Implication 1 : If A B then A B = A' [ 1 ]

Implication 2 : If A B = A' then A B [ 1 ]

p + q = 2

2

7

108

5. (a) State whether the following statement is true or false.

(b) Write down two implications based on the following statement:

c) State the converse of the following statement and hence, determine whether itsconverse is true.

“ If x > 9, then x > 5 “

Answer :(a) True [ 1 ]

(b) Implication 1 : If p3= 8 then p = 2 [ 1 ]Implication 2 : If p = 2 then p3 = 8 [ 1 ]

(c) If x > 5, then x > 9 [ 1 ]False [ 1 ]

6 (a) State whether each of the following statement is true or false.(i) 42 = 8 or = -2(ii) a { a, b, c } and -3 > -7

(b) Write down premise 1 to complete the following argument.Premise 1 :_________________________________________________Premise 2 : 6 × p 42Conclusion : p 7

(c) Form a general conclusion by induction for the number sequence11, 23, 43, 71 , … which follow the pattern11 = 4(12) + 723 = 4(22) + 743 = 4(32) + 771 = 4(42) + 7……………… [5 marks ]


(ii) False [ 1 ]

(b) Premise 1 : If p = 7, then 6 × p = 42 [ 1 ]

(c) Conclusion: 4n2 + 7, n = 1,2,3,4… [ 2 ]

-2 ( 3 ) = 6 or -4 > -5

p3 = 8 if and only if p = 2

3 8

109

7. (a) Determine whether each of the following sentence is a statement.(i) 9 is a prime number.(ii) 2x(x – 2) = 0

(b) Fill in the suitable quantifier to make the following statement become true.

(c) Write down two implications based on the following sentence:“ 3m > 15 if and only if m > 5 “

[6 marks]

Answer :(a) (i) Statement [ 1 ]

(ii) Non Statement [ 1 ]

(b) Some [ 1 ](c) Implication 1 : If 3m > 15 then m > 5 [ 1 ]

Implication 2 : If m > 5 then 3m > 15 [ 1 ]

8. (a) State whether the following statement is true or false.(i) 8 + 2 = 10 and 2 < -3(ii) All perfect square numbers are even numbers

.(b) Complete the following argument.

Premise 1 : If 3t = 0, then t = 0.Premise 2 : ________________________________________________Conclusion : 3t 0

(c) State the converse of the following statement and hence, determinewhether its converse is true or false.

[5 marks]Answer :

(a) (i) False [ 1 ](ii) False [ 1 ]

(b) Premise 2 : t 0 [ 1 ]

(c) If x = 3, then x 3 = 27 [ 1 ]

True [ 1 ]

…………. prime numbers are odd numbers

If x3 = 27 then x = 3

110

9. (a) Is the sentence below a statement or non-statement?

(b) Write down two implications based on the following sentence:

(c) Make a general conclusion by induction for the following number pattern :2 = (0)2 + 23 = (1)2 + 26 = (2)2 + 2

11 = (3)2 + 2 [5 marks]Answer :

(a) Statement [ 1 ]

(b) Implication 1 : If m is an even number then m can be divided by 2. [ 1 ]Implication 2 : If m can be divided by 2 then m is an even number. [ 1 ]

(c) n 2 + 2 , n = 0, 1, 2,3 … [ 2 ]

10. (a) State whether each of the following statement is true or false(i) (-2)2 = 4 or ( -3 )3 = -9

(ii) and

(b) Complete the following argument.Premise 1 :_________________________________________________Premise 2 : xn + x is not a quadratic expression.Conclusion : n 2.


[5 marks]


(ii) False [ 1 ]

(b) Premise 1 : If n = 2, then x n + x is a quadratic expression [ 1 ]

(b) (c) Implication 1 : If 1 – q > 2 then q < -1 [ 1 ]

Implication 2 : If q < -1 then 1 – q > 2 [ 1 ]

3 and 4 are factors of 8

m is an even number If and only if m can be divided by 2.

816 2

1

5

125 2

1

1 – q > 2 if and only if q < -1

111

THE STRAIGHT LINES

1. In Diagram 1, O is the origin, point R lies on the x-axis and point P lies on the y-axis.Straightline PU is parallel to x-axis and the straight line PR is parallel to straight line ST. Theequation of straight line PR is x + 2y = 14.

R xO

T(2,-5)

S

U

x + 2y = 14

y

••

•

•

•

(a) State the equation of the straight line PU. [2 marks](b) Find the equation of the straight line ST and hence,

state its x-intercept. [3 marks]Answer:

a) Equation PR, y = 72

1 x [ 1 ] (b) mST =

2

1

Point P = (0, 7) y – (-5) =2

1 (x – 2) [ 1 ]

Equation PU, y = 7 [ 1 ] y =2

1 x – 4 [ 1 ]

x- intercept = -8 [ 1 ]

2. The diagram below shows that the straight line EF and GH are parallel.

DIAGRAM 2Find

(a) the equation of EF. [3 marks](b) the y - intercept and x - intercept of EF [2 marks]

P

DIAGRAM 1

112

K O

N

y

x

P

M

L (4, 7)

Answer:

(a) mGH =)1(4

)5(2

=

5

7[ 1 ] (b) y – intercept = 5 [ 1 ]

-2 = c )5(5

7[ 1 ] x –intercept = -

5/7

5

c = 5

y = 55

7x [ 1 ] =

7

25[ 1 ]

3. In the following diagram, O is the origin, point K and point P lies on the x-axis and point N lieson the y-axis. Straight line KL is parallel to straight line NP and straight line MN is parallel to thex-axis. The equation of straight line NP is x – 2y – 18 = 0.

(a) State the equation of the straight line MN.(b) Find the equation of the straight KL and hence, state the coordinate of the point K.

[5 marks]Answer:

(a) Equation NP, y = 92

1x [ 1 ] (b) mKL =

2

1

Point N = (0, -9) y – 7 =2

1(x – 4) [ 1 ]

Equation MN, y = - 9 [ 1 ] y =2

1x + 5

0 =2

1x + 5

K = ( -10 , 0 ) [ 1 ]

Diagram 3

113

4. In the diagram 4, the gradient of the straight line KLM is2

1 . Find

(a) the value of p.(b) the x-intercept of the straight line MN.

[5 marks]

Answer:

(a) mKL =2

1

20

4

p[ 1 ] (b) mLM =

2

1

2

40

x[ 1 ]

p = 5 [ 1 ] x = 10x-intercept = 10 [ 1 ]

5. The diagram 5 below shows STUV is a trapezium

Given that gradient of TU is -3, find(a) the coordinates of point T. [2 marks](b) the equation of straight line TU. [1 marks](c) the equation of straight line TV [2 marks]

Answer:

K(0, p)

0

y

x

N

M

L(2, 4)

Diagram 4

Diagram 5

114

a) The gradient =06

2

p

-3 =6

2 p[ 1 ]

-18 = 2 – pp= 20

Coordinates of point T = (0 , 20) [ 1 ]

b) m = -3, c = 20

y = -3x + 20 [ 1 ]

c)3

10

60

020

[ 1 ]

203

10 xy [ 1 ]

115

PERIMETER AND AREA

1.

18 cm

Diagram 1

15 15

E

A

B

O F

C D

In Diagram 1, OAB, OCD and OFE are three sectors in a circle centred O. AOF, BCO and EDOare straight lines. C and D are midpoints of OB and OE respectively.

Using22

7 , calculate

a) the perimeter , in cm, of the whole diagram,b) the area, in cm2, of the shaded region. [6 marks]

Answer:

a)15 22 150 22

2 18 2 9360 7 360 7

or

[ 1 ]

=15 22 150 22

2(18) 2(9) 2 2 18 2 9360 7 360 7

[ 1 ]

= 87 cm [ 1 ]

b)15 22 150 22

18 18 9 9360 7 360 7

or

[ 1 ]

=15 22 150 22

18 18 9 9360 7 360 7

[ 1 ]

=1 297

1482 2

or or 148.5 cm2 [ 1 ]

116

2.

Diagram 2

S

O

R

P Q120

Diagram 2 shows a sector two sectors ORS and OPQ with the same centre O. P and Q aremidpoints of OR and OS respectively and OS = 28 cm.

Using22

7 , calculate


Answer:

a)120 22 240 22

2 28 2 14360 7 360 7

or

[ 1 ]

=120 22 240 22

2(14) 2 28 2 14360 7 360 7

[ 1 ]

=1 436

1453 3

or or 145.33 cm [ 1 ]

b)120 22 120 22

28 28 14 14360 7 360 7

or

or240 22

14 14360 7

[ 1 ]

=120 22 120 22 240 22

28 28 14 14 14 14360 7 360 7 360 7

[ 1 ]

=2 3080

10263 3

or or 1026.67 cm2 [ 1 ]

117

3.

In Diagram 3, OFIE is a quadrant of a circle with centre O and GH is an arc of another circle withcentre O. OFG and OIH are straight lines.

OF = FG = 14 cm, and 35GOH

Using22

7 , calculate


Answer :

a)

28

7

222

360

3514

7

222

360

55or [ 1 ]

=

28

7

222

360

3514

7

222

360

55)14(4 [ 1 ]

= 56.869

779

9

586 oror [ 1 ]

b)

2828

7

22

360

351414

7

22

360

55or [ 1 ]

=

1414

7

22

360

352828

7

22

360

351414

7

22

360

55[ 1 ]

= 78.2739

2464

9

7273 oror cm2 [ 1 ]

Diagram 3

35

H

E

GF

I

O

118

4.

14 cmDiagram 4

OP

Q

R

T

U

S

Diagram 4 shows two semicircles RSO and RTU with centre Q and O respectively. OPQ is aquadrant of a circle with centre O. RQ = QO. ROU is a straight line.

Using22

7 , calculate


Answer :

45 222 7

360 7

or

180 222 14

360 7

[ 1 ]

a) =45 22 180 22

2(7) 14 2 7 2 14360 7 360 7

[ 1 ]

=1 155

772 2

or or 77.5 cm [ 1 ]

45 227 7

360 7

or

180 2214 14

360 7

or

180 227 7

360 7

[ 1 ]

b) =45 22 180 22 180 22

7 7 14 14 7 7360 7 360 7 360 7

[ 1 ]

=1 1001

2504 4

or or 250.25 cm2 [ 1 ]

119

5.

H

GF

EI

Diagram 5

J

K

In diagram 5, two arcs of two equal circles, GFE and GIJ, with centres H and K respectively. It isgiven that GH = 14 cm.

Using22

7 , calculate


Answer :

a)120 22

2 14360 7

or

240 222 14

360 7

[ 1 ]

=120 22 240 22

2(14) 2 14 2 14360 7 360 7

[ 1 ]

= 116 cm [ 1 ]

b)120 22

14 14360 7

or

240 2214 14

360 7

[ 1 ]

=120 22 240 22

14 14 14 14360 7 360 7

[ 1 ]

= 616 cm2 [ 1 ]

120

MATRICES

1a) The inverse matrix of is4

5 3

pq

. Find the value of p and of q.

b) Using matrices, calculate the values of x and y that satisfy the following simultaneousequations :

3x – 2y = 6 [6 marks]5x – 4y = 4

1a)

4 21

5 312 ( 10)

4 21

5 32

p = 2 [ 1 ]

q =1

2 [ 1 ]

1b)3 2

5 4

=

6

4

[ 1 ]

4 2 61

5 3 42

x

y

[ 1 ]

16

21

x

y

x = 16 [ 1 ]y = 21 [ 1 ]

2.a) Given that F =3

2

m

n

and the inverse matrix of F is4 31

214 m

, find the value of m and

of n.b) Hence, using matrices, calculate the value of p and of q that satisfies the following

equation2

5

pF

q

[6 marks]

Answer :

2a) F 1 31

26

n

mmn

mn 6 = 14n = 4 [ 1 ]

4m 6 = 14m = 5 [ 1 ]

2b)5 3

2 4

p

q

=2

5

[ 1 ]

4 3 21

2 5 514

p

q

[ 1 ]

p =1

2 [ 1 ]

q =3

2[ 1 ]

3.a) Given that find matrix A.

b) Hence, using the matrix method, find the value of d and e which satisfy thesimultaneous equations below.

-d + 2e = 11-3d + 5e = 15 [6 marks]

Answer :

3a) A =5 21

3 15 6

[ 1 ] 3b)

1 2

3 5

11

15

d

e

[ 1 ]

=5 2

3 1

[ 1 ]

5 2 11

3 1 15

d

e

=

25

18

[ 1 ]

d = 25 [ 1 ]e = 18 [ 1 ]

45

23

y

x

,10

01

53

21

A

121

4. Given matrix M = and matrix MN =

a) Find the matrix N.b) Hence, calculate by using the matrix method, the values of r and s that satisfy the

following simultaneous equations :4r + 5s = 16r + 8s = 1 [6 marks]

4a) N =8 51

6 432 30

(1)

= (1)

4b)4 5 1

6 8 1

r

s

(1)

8 5 11

6 4 12

r

s

(1)

=

13

2

5

r =13

2(1)

s = 5 (1)5. Given the matrix K is ,

a) Find the matrix L so that KL =

b) Hence, calculate the values of h and k, which satisfy the matrix equation:

[6 marks]

Answer :

5a) L =5 31

8 420 24

=5 31

8 44

(1)+(1)

5b)4 3 7

8 5 11

h

k

(1)

5 3 71

8 4 114

h

k

(1)

=21

124

h =1

2(1)

k = 3 (1)

6. Given matrix M =1 2

2 6

, find

a) the inverse matrix of M.b) hence, using matrices,find the values of x and y that satisfy the following

simultaneous equations :x 2y = 3

–2x + 6y = 4 (6 marks)

86

54

10

01

46

58

2

1

58

34

11

7

58

34

k

h

10

01

122

Answer :

6a) M 1 =6 21

2 16 4

=6 21

2 12

[ 1 ] + [ 1 ]

6b)1 2 3

2 6 4

x

y

[ 1 ]

6 2 31

2 1 42

x

y

[ 1 ]

x = 5 [ 1 ]y = 1 [ 1 ]

7. It is given that matrix A =1

3 4

a

does not have an inverse matrix.

a) Find the value of a.b) If a = 2, find the inverse matrix of A and hence, using matrices, find the values of

x and y that satisfy the following simultaneous linear equations.x 2y = 63x + 4y = 2 [6 marks]

Answer :7a) 4 – 3a = 0

3a = 4

a = 4

3[ 1 ]

7b) A 1 =4 21

3 14 6

=4 21

3 12

[ 1 ]

1 2 6

3 4 2

x

y

[ 1 ]

4 2 61

3 1 22

x

y

[ 1 ]

=201

162

x = 10 [ 1 ]y = 8 [ 1 ]

8a) i) Find matrix G such that3 1 3 1

4 2 4 2G

ii) Find the determinant of matrix C

22

14

b) Using matrices, calculate the values of p and q that satisfy the following matrixequation.

2p + 4q = 4p + 3q = 8 (6 marks)

Answer :8 a) i) G = [ 1 ]

ii) = 6)1)(2()2)(4( [ 1 ]

8b2 4 4

1 3 8

p

q

[ 1 ]

3 4 41

1 2 86 4

p

q

[ 1 ]

=201

122

p = 10 [ 1 ]q = 6 [ 1 ]

10

01

123

9.a) Find the inverse of matrix4 5

1 2

b) Hence, using matrices, calculate the values of m and n that satisfy the followingsimultaneous equations :

– 4m + 5n = 1–m + 2n = 2 (6 marks)

Answer :

9a)2 51

1 48 5

[ 1

]

=

41

52

3

1[ 1

]

9b)4 5 1

1 2 2

m

n

[ 1 ]

2

1

41

52

3

1

m

m[ 1 ]

m = 4 [ 1 ]n = 3 [ 1 ]

10.a) Given matrix S =

m4

36, find the value of m if matrix S has no inverse.

b) Given the matrix equations

and8 41

5 7 1

x k

y h

i) Find the value of h and k.ii) Hence, find the value of x and y. [6 marks]

Answer :10a) 6m + 12 = 0

m = 2 [ 1 ]

10b i) h = 5630= 26 [ 1 ]

k = 6 [ 1 ]

ii) =8 6 41

5 7 126

[ 1 ]

=

1

1

2

x = 1 [ 1 ]

y = [ 1 ]

1

4

85

67

y

x

y

x

2

1

124

GRADIENT AND AREA UNDER THE GRAPH

1. Diagram 1 shows a distance-time graph for the journey of a car and lorry from town Q.

Diagram 1Graph ABCD represents the journey of the car and graph ACE represents the journey ofthe lorry. Both vehicles depart from town Q at the same time along the same road.(a) State the length of time, in hours, during which the car is stationary.(b) Calculate the average speed in km/h for the total distance of the car.(c) At the certain time during the journey, both vehicles meet at the same location.

(i) Find the distance, in km, between that location and town Q.(ii) State the time taken by the lorry to reach that location from town Q.

[ 5 marks ]Answer:

(a) 0.14.1 BC tt

= 0.4 hours [ 1 ]

(b) Average speed =takentimetotal

traveleddistancetotal

=2.3

142[ 1 ]

= 44.38 km/h [ 1 ](c) (i) 70km [ 1 ]

(ii) 1.4 hours [ 1 ]

2. The diagram shows a speed-time graph of the movement of a particle for a period of 35 s.

0

A

Distance (km)

Time (hour)1.0 1.4 3.2

E

D142

70

195

B C

20

12

10 t Time (s)

Speed (m s-1)

35

125

(a) State the uniform speed, in ms-1, of the particle.(b) Given the distance travelled by the particle at a uniform speed is 250m,

calculate(i) the value of t,(ii) the average of speed, in ms-1, of the particle for the period of 35 seconds.

[ 6 marks ]

Answer:(a) 20 m s-1 [ 1 ](b) (i) At uniform speed,

Distance = 250 m20 )10( t = 250 [ 1 ]

10t = 12.5t = 22.5 s [ 1 ]

(ii)

=35

25)5.1225(2

110)2012(

2

1

[ 1 ]

= 15.29 m/s [ 1 ]

3. Diagram 2 shows a distance-time graph for the journey of a bus and van.

Diagram 2

Graph AFB represents the journey of the van from town X to town Y. The graph CDEFGrepresents the journey of the bus from town Y to town X. The van leaves town X and thebus leaves town Y at 9.45 p.m. and they travel along the same road.(a) (i) At what time do the vehicles meet ?

(ii) Find the distance, in km, from town Y when the vehicles meet.(b) State the length of time, in minutes, during which the bus is stationary.(c) Calculate the average speed, in km h-1, of the bus for the whole journey.

[ 5 marks ]

Answer:(a) (i) 9.45 pm + 70 minutes = 10.55 pm [ 1 ]

(ii) 100km km65 = 35km [ 1 ](b) 50 4010 minutes [ 1 ]

(c) =60150

100

[ 1 ]

= 40 km/h [ 1 ]

150

C

Time(minutes)

Town XA

G

100

050

B

F

ED

10

Town Y

70

65

Distance (km)

126

4. Diagram 3 shows a speed-time graph for a particle, of 30 seconds.

Speed (m s-1)

(a) The distance traveled during the 5 last second is 106m. Find the value of v.(b) Calculate the acceleration, in m s-2, of the particle in the last 5 seconds.(c) The total distance traveled before 10 seconds is 230m. Calculate the average

speed for the whole journey.[ 6 marks ]

Answer:

(a) 10652

1 v [ 1 ]

v = 42.4 m s-1 [ 1 ]

(b)5

4.420 [ 1 ]

= - 8.48 m s-2 [ 1 ](c)

30

106]15)214.42(2

1[230

[ 1 ]

= 27.05 m/s [ 1 ]

10 25 30 Time (s)

Diagram 3

v

21

0

25

127

5. The diagram 4 shows the speed-time graphs of two particles M and N over a period of 100seconds. ABCD is the speed-time graph of particle M and EFG is the speed-time graph ofparticle N.

(a) State the length of time, in seconds, during which the particle N is constant speed.(b) Calculate the rate of change of speed, in ms-2, of the particle M in the first 20s.(c) Given that the distance travelled by both particles over the 100 s period is the same

and the distance traveled by particle N is 8500m. Calculate the value of v.

[ 6 marks ]

Answer:

(a) 100 – 60 = 40 s [ 1 ]

(b)020

080

[ 1 ]

= 4 ms-2 [ 1 ]

(c) 850060401452

1 vv [ 2 ]

v = 70 ms-1 [ 1 ]

E

B

G

D

0

v

80

145

40 60 10020 80

F

Speed (ms-1)

Time (s)

C

A

Diagram 4

128

PROBABILITY

1.

StudentsProbability of choosing:

Biology Physics Chemistry

Ahmad4

3

8

1

Ling5

1

5

2

The table 1 shows the probability of how Ahmad and Ling might choose their subjects in Form6. The table is incomplete.Calculate the probability that(a) Ahmad will choose Chemistry and Ling will choose Biology,(b) Ahmad and Ling will choose the same subjects. [5 marks]

Answer:

(a)8

1

8

1

4

31 and

5

2

5

2

5

11

20

1

40

2

5

2

8

1

(b)

8

3

40

15

5

2

8

1

5

1

8

1

5

2

4

3

2 In bag A, there are 4 blue balls and 3 yellow balls while in bag B, there are 5 blue balls and6 yellow balls. A ball is drawn at random from bag A. After its colour is recorded, it is putinto beg B. Then a ball is drawn at random from beg B. Find the probability that(a) the colour of the two balls are blue,(b) two different colour balls are drawn. [5 marks]

Answer:

(a)

7

2

12

6

7

4

[ 1 ]

[ 1 ]

[ 2 ]

[ 1 ]

[ 1 ]

[ 1 ]

Table 1

129

(b)

84

39

12

5

7

3

12

6

7

4

3 A company van carries a group of workers consists of 6 males and 8 females. They aredropped off at random at various houses along the route as shown in the diagram belowto serve customers.

(a) If two workers are dropped off at house A, calculate the probability that both are females,(b) Two male workers are dropped off at house A. If another two workers are then dropped

off at house B, calculate the probability that at least one of them is female.[5 marks]

Answer:(a) (b)

13

4

182

56

13

7

14

8

11

10

132

120

11

4

12

8

11

7

12

8

11

8

12

4

4 In a lucky draw, there are three categories of gifts consisting of 4 hotel vouchers, 6 shoppingvouchers and 10 restaurant vouchers. All vouchers are placed inside similar envelopes andput inside a box.The honored guests of the lucky draw are requested to draw at random two envelopes fromthe box.Calculate the probability that the first honored guest draw(a) the first envelope with a hotel voucher and the second with a shopping voucher,(b) two envelopes with vouchers of the same categories. [5 marks]

Answer:

(a)

95

6

19

6

20

4

Other houses●

Company House A

House B

●

●

●

[ 1 ]

[ 1 ]

[ 2 ]

[ 1 ]

[ 2 ]

[ 1 ]

[ 1 ]

[ 1 ]

130

(b)

95

33

380

32

19

9

20

10

19

5

20

6

19

3

20

4

5. The table below shows the number of participants in a quiz contest according to school andgender.

SchoolNumber of Participants

Male FemaleSMK Ria 4 4SMK ACS 5 2SMK Puteri 3 6

(a) If two participants are chosen at random from SMK ACS, calculate the probability thatboth of them are males.

(b) If two participants are chosen from the male participants, calculate the probability thatboth of them are from the same school. [5 marks]

Answer:

(a)

42

20

42

20

6

4

7

5

(b)

72

19

144

38

144

62012

12

2

12

3

12

4

12

5

12

3

12

4

[ 2 ]

[ 1 ]

[ 1 ]

[ 1 ]

[ 2 ]

[ 1 ]

131

LINES AND PLANES IN 3-DIMENSIONS

1. Diagram 1 shows a pyramid with a horizontal rectangular base ABCD. M is themid-point of CD and apex V is 9 cm vertically above the point MIdentify and calculate the angle between the line VB and the base ABCD. [ 4 marks ]

24 cm

14 cm

B

A

D

M

C

V Answer :

Identify VBM [ 1 ]

Length of BM = 25 [ 1 ]

tan VBM =25

9[ 1 ]

VST 19.80 or 19 48’ [ 1 ]

2. Diagram 2 shows a right prism. Right angled triangle PQR is the uniform cross-section of the prism. Calculate the angle between the plane RTU and the planePQTU. [ 3 marks ]

Answer :

Identify RTQ [ 1 ]

tan RTQ =18

12[ 1 ]

RTQ 33.69 or 33 41’ [ 1 ]

DIAGRAM 1

T

P

5 cm

Q

DIAGRAM 2

R

S

U

18 cm

12 cm

132

3. Diagram 3 shows a prism with a horizontal square base HJKL. Trapezium EFLK isthe uniform cross-section of the prism. The rectangular surface DEKJ is vertical whilethe rectangular surface GFLH is inclined.Calculate the angle between the line DL and the base HJKL [ 4 marks ]

Answer :Identify DLJ [ 1 ]

Length of LJ = 10 cm [ 1 ]

tan DLJ =10

5[ 1 ]

DLJ 26.57 or 2634’ [ 1 ]

4. Diagram 4 shows a right prism . Plane ABFE, ADHE and BCGF are rectangularplanes .Planes ABCD and EFGH are parallel trapezium. It also given BAD and

ADC = 90, FG = 5 cm, AB = 2cm and CG = 12 cm.

Identify and calculate the angle between the plane ADG and AEHD.[ 3 marks ]

5 cm

4 cm

HG

FE

D C

BA

Answer :Identify GDH [ 1 ]

tan GDH =12

5[ 1 ]

GDH 22.62 or 22 37’ [ 1 ]

5. Diagram 5 shows a right prism with horizontal rectangle base URST. Righttriangle RSW are the uniform cross section of the prism.Calculate the angle between plane SRV and plane RSTU [ 3 marks]

Answer :

Identify VST [ 1 ]

tan VST =12

5[ 1 ]

VST 22.62 or 2237’ [ 1 ]

8 cm

G

L

H

K

J

D

FE L

5 cm

6 cm DIAGRAM 3

DIAGRAM 4

DIAGRAM 5

5 cm

10 cm12 cm

T

U

W

S

R

VV

T

U

W

S

R

133

7. Diagram 6 shows a right prism. The base PQRS is a horizontal rectangle. Right angledtriangle QRU is the uniform cross section of the prism. V is the midpoint of PS.Identify and calculate the angle between the line UV and the plane PQRS.

[ 4 MARKS]

Answer :

Identify UVR [ 1 ]

Length of VR = 13 [ 1 ]

tan UVR =13

5[ 1 ]

UVR 21.04 or 212’ [ 1 ]

8. Diagram 7 shows a right prism. Right angled triangle PQR ia the uniform cross-section of the prism. Calculate the angle between the line TR and the plane PQTU.

[ 3 marks]

Answer :

Identify RTQ [ 1 ]

tan RTQ =18

12[ 1 ]

UVR 33.69 or 3341’ [ 1 ]

DIAGRAM 6

S

Q

U

T

P

10 cm

R

12 cm

5 cmV

18 cm

12 cm

5 cm S

T

U

Q

P

R

DIAGRAM 7

134

12

10

8

6

4

2

-2

-4

-6

-8

-10

-12

y

-4 -2 2 4

y =3x-2

PART B

GRAPH OF FUNCTIONS

1 (a) Complete Table 1 in the answer space for the equation y =x

6by writing down

the values of y when x = -2 and x = 0.5 [2 marks](b) For this part of the question, use a graph paper.

By using a scale 2 cm to 1 unit on the x-axis and 2 cm to 1 unit on the

y-axis, draw the graph of y =x

6for -4 ≤ x ≤ 4. [4 marks]

(c) From your graph, find

a. the value of y when x = -2.7,b. the value of x when y = 3.6. [2 marks]

(d) Draw a suitable straight line on the graph in (b) to find the values of x which


xx

for -4 ≤ x ≤ 4. [4 marks]

Answer:(a)

X -4 -3 -2 -1 -0.5 0.5 0.8 1.5 2.5 4

Y -1.5 -2 -3 -6 -12 12 7.5 4 2.4 1.5

Table 1 [2](b)

xy

6

[4]

y=3x-2

135

(c) i) y = -2.2 [1]

ii) x = 1.67 [1]

(d) Identify equation of y = 3x – 2 [1]Straight line y = 3x – 2 correctly drawn [1]x = -1.2, 1.7 [2]

2. (a) Complete Table 2 in the answer space for the equation y =x

72by writing

down the values of y when x = 2.4, x = 6 and x = 12.5 [3 marks]

(b) For this part, use a graph paper.

By using a scale 1 cm to 1 unit on the x-axis and 1 cm to 2 units on the y-axis, draw

the graph of y =x

72for 2 ≤ x ≤1 4. [4 marks]

(c) From your graph, finda. the value of y when x = 4.3,b. the value of x when y = 22. [2 marks]

(d) Draw a suitable straight line on your graph to find all the values of x which


36

x

xState these values of x. [3 marks]

Answer:

(a)X 2 2.4 3 4 6 8 10 11.6 12.5 14

Y 36 30 24 18 12 9 7.2 6.2 5.76 5.14

Table 2[3]

136

(b)

35

30

25

20

15

10

5

2 4 6 8 10 12 14

[4](c) (i) y = 16.7 [1]

(ii) x = 3.3 [1]

(d) Identify equation of y = - x + 30 [1]Straight line y = - x + 30 correctly drawn [1]x = 2.7 [1]

3. a) Complete Table 3 in the answer space for the equation y = 2x2 – 5x – 3.[2 marks]

b) For this part, use a graph paper.By using a scale 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, drawthe graph of y = 2x2 – 5x – 3 for -3 ≤ x ≤ 5. [4 marks]

c) From your graph, findi) the value of y when x = -2.4,ii) the value of x when 2x2 – 5x – 3 = 0. [2 marks]

d) Draw a suitable straight line on your graph to find all the values of x which satisfythe equation 2x2 – 8x = 7 for -3 ≤ x ≤ 5.State these values of x. [4 marks]

xy

72

y = - x + 30

137

Answer:a)

[2]

b) graph

[4 ]

c) i) 20.4 ≤ y ≤ 20.6 [1]

ii) -0.6 ≤ x ≤ -0.4 [1]2.9 ≤ x ≤ 3.1

d) Identify equation y = 3x + 4 [1]Straight line y = 3x + 4 correctly drawn [1]

- 0.80 ≤ x ≤ -0.70 [1]

- 4.80 ≤ x ≤ - 4.70 [1]

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

y 30 15 4 -3 -5 -6 -5 0 9 22

40

35

30

25

20

15

10

5

-5

-10

-10 -5 5 10

-0.75 4.75

20.5

-2.4

y x = 3x+4

y x = 2x2-5x-3

Table 3

138

4. a) Complete Table 4 in the answer space for the equation y = -2x² + 5x + 8 by writingdown the values of y when x = -2 and x = 3 [2 marks]

b) For this part of question , use the graph paper provided . You may use a flexiblecurve ruler.By using a scale 2 cm to 1 unit on the x axis and 2cm to 5 unit on they axis , drawthe graph of y = -2x² + 5x +8 for -3.5 ≤ x ≤ 4 [4 marks]

c) From the graph, find

(i) the value of y when x = 3.3

(ii) the value of x when y = -16[ 2 marks]

d) Find and draw a suitable straight line on your graph to determine the values of xwhich satisfy the equation 2x – 2x ² = -10 for -3.5 ≤ x ≤ 4State the values of x [4 marks]

Answer :

a) y = -2x² + 5x + 8

x -3.5 -3 -2 -1 0 1 2 3 4y -34 -25 -10 1 8 11 10 5 -4

[2]

b) Graph

15

10

5

-5

-10

-15

-20

-5 -4 -3 -2 -1 1 2 3 4 5

y= 3x -2

y=-2x2+5x+8

[4]c) i) 2.5 ≤ y ≤ 2.7 [1]

ii) -2.45 ≤ x ≤ -2.40 [1]

Table 4

139

d) Identify equation y = 3x - 2 [1]Straight line y = 3x -2 correctly drawn [1]

-1.8 ≤ x ≤ - 1.7 [1]2.7 ≤ x ≤ 2.8 [1]

5. a) Complete Table 5 in the answer space for the equation y = x3 – 13x + 18 .

[2 marks]b) For this part, use a graph paper.

By using a scale 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, draw thegraph of y = x3 – 13x + 18 for -4 ≤ x ≤ 4. [4 marks]

c) From your graph, findi. the value of y when x = -1.5,ii. the value of x when y = 25. [2 marks]

d) Draw a suitable straight line on your graph to find all the values of x which satisfy theequation x3 – 11x – 2 = 0 for -4 ≤ x ≤ 4. State these values of x. [4 marks]

e)Answer:

(a)X -4 -3 -2 -1 0 1 2 3 4

Y 6 30 36 30 18 6 0 6 30

[2]

c) i) y=34, 33 ≤ y ≤ 35 [1]ii) x=3.85 , 3.80 ≤ x ≤ 3.90 [1]

45

40

35

30

25

20

15

10

5

-5

-6 -4 -2 2 4 6

y=x^3-13x+18

y=-2x+20

3.85

34

3.35

-0.25-3.2

[4]

Table 5

140

d) y=x3-13x+180=x3-11x-2 (-)--------------------------y = -2x+20 [1]

3.3 ≤ x ≤ 3.5 [1]-0.3 ≤ x ≤ -0.1 [1]-3.3 ≤ x ≤ -3.1 [1]

6. a) Complete Table 6 in the answer space for the equation y = x3 + x2 – 12x – 5.[2 marks]

b) For this part, use a graph paper.By using a scale 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, drawthe graph of y = x3 + x2 – 12x – 5 for -4 ≤ x ≤ 4. [4 marks]

c) From your graph, findi. the value of y when x = 0.5,ii. the value of x when y =11.9. [2 marks]

d) Draw a suitable straight line on your graph to find all the values of x whichsatisfy the equation x3 + x2 – 10x = 0 for -4 ≤ x ≤ 4.State these values of x. [3 marks]

Answer:

(a)X -4 -3 -2 -1 0 1 2 3 4

Y -5 13 15 7 -5 -15 -17 -5 27

[2]

20

15

10

5

-5

-10

-15

-20

-25

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6

h x = -2x-5

g x = x3+x2 -12x-5

A

[4]

141

c) i) -10.6 ≤ y ≤ -10.4 [1]ii) -1.6 ≤ x ≤ -1.4, -3.2 ≤ x ≤ -3.0 [1]

e) Identify equation y = -2x-5 [1]Straight line y = -2x-5 correctjy drawn. [1]

-3.8 ≤ x ≤ -3.62.6 ≤ x ≤ 2.7x = 0 [2]

142

TRANSFORMATION1. Diagram 3 shows three pentagons EFGHJ, EKLMN and KPQRS on a Cartesian

plane.

Diagram 3


2

3

Transformation R is a reflection about the line KMN.

State the coordinates of the image of point J under the following

transformations:

(i) TR

(ii) RT

[4 marks]

(b) EKLMN is the image of EFGHJ under transformation V and EKLMN is the

image of KPQRS under transformation W.

Describe in full the transformation

(i) V

(ii) W

[5 marks]

143

(c) Given that the area of EFGHJ is 26.8cm2 , calculate the area of shaded region

[3 marks]

Answer:

(a)

(i) (4, 8), (1,10) [2]

(ii) (4, 7), (6, 5) [2]

(b)

(i) V = enlargement of scale factor 2 with centre E [2]

(ii) W= clockwise rotation of 90o about the centre K [3]

(c) Area of EKLMN [1]= k 2 x area of EFGHJ= 2 2 x 26.8= 107.2 cm 2 [1]Area of the shaded region =107.2 – 26.8 = 80.4 cm 2 [1]

2 Diagram 5 shows quadrilateral ABCD, PQRS and KLRM drawn on a Cartesian

plane.

Diagram 5


2

4.

Transformation V is a reflection in the line y = 1 .

State the coordinates of the image of point A under the following

transformations:

144

(i) Translation T,

(ii) Combined transformation VT [ 3 marks]

(b) (i) KLRM is the image of ABCD under the combined transformations WU.

Describe in full, the transformation U and the transformation W.

(ii) Given that the shaded region KLQPSM represents a region of area 120 m2.

Calculate the area, in m2, of the region represented by PQRS.

[9 marks]

Answer:

(a)

(i) (2, 4) [1]

(ii) (2, – 2 ) [2]

(b)

(i) U = clockwise rotation of 90o about the centre (0, 1) [3]V = Enlargement about the centre R with scale factor of 3. [3]

(ii) Let Area of PQRS = A unit2

32 x A [1]32 x A – A= 120 [1]A = 15 unit2 [1]

3. (a) Diagram 3 shows the point F on a Cartesian plane.

DIAGRAM 3

x

y

2 4 6 8 10 12 14 16

2

4

6

8

10

0

F

145

Transformation S is a translation

2

5.

Transformation T is a reflection in the x = 9.

(i) State the coordinates of the image of point F under transformation S.

(ii) State the coordinates of image of point F under transformation TS. [3 marks]

Answer:

(a) (i) ( 12, 7 ) [1]

(ii) (12, 7 ) ( 6, 7) [2]

(b) Diagram 4 shows three triangle PQR, ACG and EFG on a Cartesianplane.

DIAGRAM 4

Triangle ACG is the image of triangle PQR under transformation V.Triangle EFG is the image of triangle ACG under transformation W.


(a) V(b) W [6 marks]

(ii) Given that the area of triangle EFG represents a region of area 72 unit2.Calculate the area, in unit2, of the region represented by triangle PQR.

x

y

E

2 4 6 8 10 12 14 16

2

4

6

8

10

O

F

C

A

GP

RQ

146

[3 marks]Answer:

(b) (i) (a) 90° clockwise rotation about the centre ( 7,0) [3]

(b) Enlargement at point G or ( 7, 3) with scale factor of 3. [3]

(ii) 72 unit2 = 32 x area of object [1]Area of object = 8 unit2 [2]

4. (a) Transformation R is a 90° clockwise rotation about the centre (2, 2).

Transformation T is a translation

3

4.

State the coordinates of the image for coordinate (6 , 4) under the followingtransformations:

(i) R2.

(ii) TR. [4 marks]

Answer:

(a) (i) (4, -2) (-2, 0) [2]

(ii) (4, -2 ) (8, -5) [2]

(b) Diagram 5 shows quadrilateral , ABCD, PQRS and EFGH, drawn on a Cartesian

plane.

DIAGRAM 5

-12 -10 -8 -6 -4 -2 2 4 6 810 12

F

E

GH

S

QP

DC

BA

y

xO

R

6

4

2

- 2

- 4

147

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

under transformation Q.


(a)Transformation S

(b)Transformation Q [5 marks]

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

[3 marks]

Answer:

(b) (i) (a) Reflection at x =1 [2]

(b) Enlargement at centre (-11,2) with scale factor of 2. [3]

(ii) Shaded region = (22 x 64 ) – 64 [2]= 192 units [1]

148

STATISTIC

1. The data in Diagram 1 shows the monthly pocket money, in RM, received by 40 students.

56 44 60 64 52 53 55 3536 47 54 59 34 54 52 4849 51 62 58 38 63 49 4345 38 48 57 44 49 46 4032 41 46 56 42 48 51 39

Diagram 1

(a) Based on the data in Diagram 1 and using a class interval of RM5, completeTable 1 in the answer space. [4 marks]

(b) From the table in a),(i) state the modal class,(ii) calculate the mean monthly pocket money of the students. [4 marks]

(c) By using a scale of 2 cm to RM5 on the x-axis and 2 cm to 1 student on the y-axis,draw a histogram based on the data.

[4 marks]

Answer:(a) [1] [2] [1]

Pocket money(RM) Frequency Midpoint

31-35 3 33

36-40 5 38

41-45 6 43

46-50 9 48

51-55 8 53

56-60 6 58

61-65 3 63

Table 1(b) (i) 46-50 [1]

(ii) mean = (33×3)+(38×5)+(43×6)+(48×9)+(53×8)+(58×6)+(63×3)40 [2]

= 1940/40= 48.5 [1]

(c) graphAxes [1]Scales [1]Bar [2]

149

Graph for Question 1Frequency

Pocket money(RM)30.5 60.555.550.545.540.535.5

1

2

4

3

5

9

6

10

7

8

65.50

150

2. The data in Diagram 2 shows the marks obtained by 42 students in a Mathematics finalexam.

51 20 45 31 26 40 3048 32 37 41 25 36 3846 38 28 37 39 23 3933 35 42 29 38 31 4942 34 26 35 43 42 2226 47 40 48 44 34 54

Diagram 2

(a) Based on the data in Diagram 2 and using a class interval 5 marks, completeTable 2 in the answer space. [4 marks]

(b) From the table in a),(i) state the modal class,(ii) calculate the mean mark for the Mathematics final exam and give your

answer correct to 2 decimal places. [4 marks](c) By using a scale of 2 cm to 5 marks on the horizontal axis and 2 cm to 1 student on

the vertical axis, draw a histogram based on the data. [4 marks]

Answer:(a) [1] [2] [1]

Marks Frequency Midpoint

20-24 3 22

25-29 6 27

30-34 7 32

35-39 10 37

40-44 8 42

45-49 6 47

50-54 2 52

Table 2(b) (i) 35-39 [1]

(ii) mean = (22×3)+(27×6)+(32×7)+(37×10)+(42×8)+(47×6)+(52×2)42 [2]

= 1544/42= 36.76 (2 dp) [1]

(c) graphAxes [1]Scales [1]Bar [2]

151


3. The data below shows the height, in cm for a group of 40 students in a class.

132 141 146 156 142 148 151 139136 147 154 159 134 154 152 148140 153 162 155 138 163 149 143156 144 160 164 152 151 158 135145 138 148 157 144 149 146 149

a) Based on the data, complete the Table 1 in the answer space. [3 marks]

b) From the table in a),i) State the modal class,ii) Calculate the mean height of the students. [4 marks]

c) By using a scale of 2 cm to 5cm on the x-axis and 2cm to 1 student onthe y-axis, draw a frequency polygon based on the data. [5 marks]

19.5 24.5 49.539.534.529.5 44.5Marks

1

2

3

4

10

5

6

7

8

9

54.50

152

Answer:a)

Height (cm) Midpoint Frequency

131-135 133 3

136-140 138 5

141-145 143 6

146-150 148 9

151-155 153 8

156-160 158 6

161-165 163 3

[1] [1] [1]

Table 1

bi) 146-150 [1]

ii) mean =(133 3) (138 5) (143 6) (148 9) (153 8) (158 6) (163 3)

(3 5 6 9 8 6 3)

[2]

148.5 cm [1]

c) frequency

height128 133 138 143 148 153 158 163 168

10

9

8

7

6

5

4

3

2

1

Axes 1]1 Plot Points [2] Connect points [1] Touch axes at two

ends [1]

153

4.. The data shows the time taken by 42 participants to complete a road race.

51 20 45 31 26 40 3025 32 37 41 21 36 3846 38 28 37 39 23 3933 35 42 29 38 31 2342 34 26 35 43 28 2225 47 31 48 44 34 54

a) Using the data above and a class interval of 5 minutes, complete Table 2 in the answerspace. [3marks]

Answer:

Minutes Midpoint Frequency20-24 22 525 -29 27 730-34 32 835-39 37 1040-44 42 645-49 47 450-54 52 2

[1] [1] [1]

Table 2

b) Based on your table in (a),

Calculate the mean for the time taken and give your answer correct to two decimal places.

Mean =(22 5) (27 7) (32 8) (37 10) (42 6) (47 4) (52 2)

(5 7 8 10 6 4 2)

[2]

= 34.98 minutes [1][3 marks]

c) By using a scale of 2 cm to 5 minutes on the horizontal axis and 2 cm to 1 participant on thevertical axis, draw a frequency polygon based on the data. [5 marks]

154

d) Based on the polygon in (c), state one piece of information. [1 mark]

- the modal class is 35-39 [1]

frequency

minutes17 22 27 32 37 42 47 52 57

10

9

8

7

6

5

4

3

2

1

Axes [1] Plot Points [2] Connect

points [1] Touch axes at

two ends [1]

155

5. The data shows the length, in cm, of 50 pieces of wood.

18 42 23 13 29 33 30 38 30 277 25 35 11 27 35 27 34 16 2616 21 41 31 20 33 27 28 17 32 Diagram 526 20 30 31 26 32 23 19 22 3721 24 25 28 35 12 29 32 33 31

a) Using the data in Diagram 5, and a class interval of 5 cm, complete Table 5 in the answer space.[5 marks]

b) For this part of the question, use the graph paper.By using a scale of 2 cm to 5 cm on the x-axis and 2 cm to 5 pieces of wood on the y-axis, drawan ogive based on the data. [5 marks]

c) (i) From your ogive in (b), find the first quartile,(ii) Hence, explain briefly the meaning of the first quartile. [2 marks]

Answer:5a)

Length(cm) Upperboundary

Frequency Cumulative Frequency

0 4 4.5 0 0

5 9 9.5 1 1

10 14 14.5 3 4

15 19 19.5 5 9

20 24 24.5 8 17

25 29 29.5 13 30

30 34 34.5 13 43

35 39 39.5 5 48

40 44 44.5 2 50

[1 ] [1] [2] [1]

L eng th of Wood

0

10

20

30

40

50

60

4.5 9.5 14.5 19.5 24.5 29.5 34.5 39.5 44.5

L eng th (c m)

Cu

mu

lati

ve

Fre

qu

en

cy

156

5b) Refer to graph- Uniform scales used for both axes [1]- 8 points plotted [2]- (4.5, 0) plotted [1]- A smooth and continuous curve passing through (4.5, 0) and the 8 points correct plotted

[1]

5 c) (i) 21.5 < first quartile < 22.5 [1](ii) 15 pieces of wood measured 23 cm and below (or equivalent) [1]

6. A donation drive for the School Building Fund accumulated a substantial amount of money from400 people as shown in Table 6.

Collection(RM)

1-10 11-20 21-30 31-40 41-50 51-60 61-70 71-80 81-90

Frequency 10 20 40 60 85 90 50 35 10Table 6

a) State the modal class for the above data. [1 mark]b) Construct a cumulative frequency table. [2 marks]c) By using a scale of 2 cm to RM10 on the x-axis and 2 cm to 40 people on the y-axis, plot an

ogive for the data. [5 marks]d) From the ogive, find the

(i) median, [1 mark](ii) percentage of donors who donated less than RM30, [1 mark](iii) number of donors who donated at least RM65. [2 marks]

Answer:6a) Modal class = 51 60 [1]

b)Upper boundary (RM) 0.5 10.5 20.5 30.5 40.5 50.5 60.5 70.5 80.5 90.5Cumulative Frequency 0 10 30 70 130 215 305 355 390 400- 8 upper boundary values correct [1]- 8 cumulative frequency values correct [1]

157

Amount of Money C ollec ted from 400

P eople

0

40

80

120

160

200

240

280

320

360

400

0.5 10.5 20.5 30.5 40.5 50.5 60.5 70.5 80.5 90.5

Amount of Money (R M)

Cu

mu

lati

ve

Fre

qu

en

cy

6c) Refer to graph- Uniform scales used for both axes [1]- 9 points plotted correctly [2]- (0.5, 0) plotted [1]- A smooth and continuous curve passing through (0.5, 0) and the 9 points plotted [1]

6d) (i) Median = 49.0 [1]

(ii) Percentage =68

400x 100% = 17% [1]

(iii) The number of donors donated less than RM65 = 332 [1] those donated at least RM65 = 400 332

= 68 [1]

158

PLAN AND ELEVATION

1 (a) Diagram 1(i) shows a solid right prism. The base BCKJ is on horizontal plane.EFGM and CDLK are vertical planes whereas EDLM is a horizontal plane.The plane AFGH is inclined. Hexagon ABCDEF is the uniform cross section of the prism.The sides AB, FE and DC are vertical.

Answer : 1a)

Draw in full scale, the plan of the solid . [3 marks]

(b) A half-cylinder is joined to in Diagram 1(i) at the vertical plane BCQP to form a combined solid asshown in Diagram 1(ii)

Answer : b i)

b ii)

C

3 cm

K

L

M

ED

G

H

F

A

B

J

8 cm

3 cm

4 cm

6

6

DIAGRAM 1(i)

X C

K

L

M

ED

G

H

FA

B

J

8 cm

3 cm

4 cm

DIAGRAM 1(ii)

P

QY

3 cm

F/E

A/B H/J

L/K

G/M

D/C

3cm

3cm

8cm

1 cmQ

F/G

A/H

E/M D/L

C/KB/J 6 cm

2 cm

6 cm

4 cm

3 cm

P

2 cm

C/B K/J

L/MD/E

Q/P

H

GF

A

3 cm 8 cm

3 cm

3 cm

1 cm

159

The height of the half-cylinder is 2 cmDraw in full scale ,i. the elevation of the combined solid on a vertical plane parallel to BC as viewed from X [4 marks]ii. the elevation of the combined solid on a vertical plane parallel to CK as viewed from Y. [5 marks]

2. Diagram 2(i) shows a solid right prism with a rectangular base ABCD. ABMKGF is theuniform cross- section of the prism. Rectangle GHJK is an inclined plane. EFGH andJKMN is a horizontal plane.The edges AF, BM, CN and DE are verticals. BM = CN = 2cm, AF = DE = 4cmAB =DC = 7cm, KM = JN = 5cm and FG = EH = 3cm.

Answer : 2a)

S 2 cm

5 cm

6 cm

7 cm

3 cm

4 cm

J N

MK

H

G

E

F

DC

BA

a) Draw full scale, the elevation of the solid on a vertical plane parallel to AB asviewed from X. [3 marks]

b) A prism is joined to the solid in Diagram 2(i) at the vertical plane CDEHJN.Thecombined solid is as shown in Diagram 2(ii). Right angle triangle PQR and DSC is theuniform cross-section of the prism.QPR = SDC = 900 , DP = CR = SQ = 6cm and PQ = 3 cm.

Answer : 2b i) & ii)

S

3 cm

R

Q

P

2 cm

5 cm

6 cm

7 cm

3 cm

4 cm

J N

MK

H

G

E

F

DC

BA

DIAGRAM 2(i)

DIAGRAM 2ii)

Y

X

5 cm

2 cm

7 cm

3 cm

6 cm K/J

B/C

M/N

A/D

G/HF/E

Q/S

HP/E/D

3

4

6

4 cm

JR/C

M/BGF/A

2

C/D

M/K

R/P Q

S36

2

2

2

G/F

B/A

H/E

N/J

160

Draw to full scale,(i) The elevation of the combined solid on a

vertical plane parallel to BC as viewedfrom Y. [4 marks]

(ii) The plan of the combined solid. [5 marks]

3. (a) Diagram 3(i) shows a solid prism.Hexagon ABCDEF is the uniform cross section of the prism. The base ALGF is on thehorizontal plane. The sides BA, CD and EF are vertical whereas the sides BC and DE arehorizontal.

Answer :3a)

Draw in full scale, the plan of the solid prism. [3 marks]

(b) A solid prism with triangle AFM as its uniform cross section is joined at the vertical planeABCDEF to form a combined solid as shown in Diagram 3(ii).

Answer : 3 b.i)

b.ii)

P

FG

HE

DI

J

K B

C

AL

2 cm

4 cm

4 cm

4 cm

8 cm

5 cm

DIAGRAM 3(i)

DIAGRAM 2(ii)

6 cm

Y

M

FG

HE

DI

J

K B

C

AL

2 cm

4 cm

4 cm

X

Q N

4 cm

J/I

C/D

H/G

E/F B/A

K/L4 cm

5 cm

4cm

N/O

B/K

M/A/L

E/H

F/D

P

4 cm

8 cm

2 cm

4 cm

2 cm

F/A M

NC/O

C/B

G/L

H/I

J/K

2 cm

2 cm 6 cm

4 cm5 cm

E/D

161

Draw in full scale,i. the elevation of the combined solid on a vertical plane parallel to GF as viewed from X.

[4 marks]ii. the elevation of the combined solid on a vertical plane parallel to AF as viewed from Y.

[5 marks]

4. Diagram shows a solid right prism with rectangular base ABCD on a horizontal plane. Thesurface BFGJKC is its uniform cross-section. The rectangle LKJI is an inclined plane and thesquare EFGH is a horizontal plane. The edges GJ and HI are vertical and HI = GJ = 2 cm.

Answer :4a)

(a) Draw full scale, the elevation of the combined solid on a vertical plane parallel to JI asviewed from X. [3 marks]

(b) A half-cylinder of diameter 4 cm is joined to the prism in diagram 4(i) at the vertica planeLDPM. The combined solid is as shown in Diagram 4 (ii)

Answer : 4b.i)

4b.ii

C

B

E

A

DF

G

L

H

X

I J

K

4

8

86

DIAGRAM 4(i)

M

P

Q

N

C

B

E

A

DF

G

L

H

Y

I J

K

4 cm

8 cm

8 cm6 cm

DIAGRAM 4(ii)

4 cm

G/JK/C

M/H/P

F/B

E/A

4 cm4 cm

L/D

J/I

G/H

K/L

C/DB/A

F/E 8 cm4 cm

4 cm

8 cm

2 cm

F/G

J

B/C

KL

Q

N

2 cm

4 cm2 cm

2 cm

4cm

E/H

I

A/D

162

Draw full scale,(i) the plan of the combined solid. [4 marks](ii) the elevation of the combined solid on a vertical plane parallel to AB as viewed

from Y.[5marks]

5. Diagram 5(i) shows a solid right prism with square base ABCD on a horizontal plane.ABKIGF is the uniform cross-section of the prism. AF, LC and BK are vertical edges.Rectangle IJLK is a horizontal plane and rectangle EFGH is an inclined plane.

Answer :

5a)

(a) Draw a full- scale, the plan of the solid. [3 marks]

(b) A prism with isosceles triangle as its uniform cross-section such that Q is 2 cm above themidpoint of JL is joint to the prism at the plane MPLJ. The length of PL is 2 cm. Thecombined solid is shown in the diagram 5(ii)

Answer :5b i)

5b ii)

cm

K/I

G

F E

Q/H

L/J

C/DB/A 4 cm

1 cm

2 cm

2 cm

K/P/L

B/C

I/M/J

A

F

2 cm

5 cm

4 cm

N/QG/H

S

R

P

N

M

Q

2 cm

4 cm

5 cmL

K

J

I

H

G

F

E

D

C

B

A

DIAGRAM 5 (ii)

K/BG/I

F/A

H/JL/CE/D

2 cm

4 cm

2 cm

2 cm

2 cm

2 cm

4 cm

5 cmL

K

J

I

H

G

F

E

D

C

B

AA

DIAGRAM 5 (i)

163

Draw full-scale,i. the elevation of the combined solid on a vertical plane parallel to AB as viewed

from R [4 marks]ii. the elevation of the combined solid on vertical plane parallel to BC as viewed

from S. [5 marks]

164

EARTH AS A SPHERE

1. Table 1 shows the latitudes and longitudes of four points A, B, C and D, on the surface ofthe earth.

Point Latitude LongitudeABCD

200 Sx0 N200 S300 N

250 W250 Wy0 Ey0 E

(a) Q is a point on the surface of the earth such that AQ is the diameter of the earth.State the position of Q. [2 marks]

(b) Calculate(i) the value of x, if the distance from A to B measured along the meridian is

4800 nautical mile,(ii) the value of y, if the distance from A due east to C measured along the

common parallel of latitude is 3664.8 nautical mile. [7 marks](c) An aero plane took off from A and flew due east to C along the common parallel of

latitude and then due north to D.If the average speed for the whole flight is 500 knots, calculate the time taken for thewhole flight. [3 marks]

Answer:

(a) Q(200 N, 1550 E)

0

000

0

60

602080

8060

4800

x

ii.)0

0

0

40

65

8.366420cos60

y

(b) Distance CD = 50 X 60 = 300 n.m.

Distance of whole flight = 3664.8 + 300 = 3964.8

Time taken =500

8.3964

= 93.7 hrs

[2]

[1]

[1]

[1]

[2]

[1]

[1]

[1]

[1]

[1]

Table 1

(b) (i)

165

2. P(600 N, 600 W), Q, R and S are four points on the surface of the earth. PQ is the diameterof the parallel of latitude 600 N. S lies 4800 nautical mile due to south of P and R lies 450

due east of P.(a) State the longitude of Q. [2 marks](b) Find the position of R. [3 marks](c) Calculate the distance, in nautical mile, from P to Q measured along the parallellatitude. [3 marks](d) An aero plane took off from Q and flew towards P using the shortest distance, asmeasured along the surface of the earth, and then flew due south to S, with an averagespeed of 600 knots.Calculate the time, in hours, taken for the flight. [4 marks]

Answer:(a) Longitude of Q = 1100 E(b) R (600 N, 250 W)

(c) ..540060cos60180 00 mn(d) Total distance = 48006060

= ..8400 mn

Time =600

48006060

= 14 hours

[2][1][2]

[2] [1]

[1] [1]

[1]

[1]

JABATAN PELAJARAN NEGERI PERAKKERTAS MODEL PEPERIKSAAN SEBENAR SPM

MATEMATIK

Kertas 2

Dua jam tiga puluh minit

Kod Pemeriksa

Bahagian SoalanMarkahPenuh

MarkahDiperoleh

A

1 3

2 4

JANGAN BUKA KERTAS SOALAN INISEHINGGA DIBERITAHU

1. Kertas soalan ini mengandungi dua bahagian :Bahagian A dan Bahagian B. Jawab semuasoalan daripada Bahagian A dan empat soalandalam Bahagian B.

2. Jawapan hendaklah ditulis dengan jelas dalamruang yang disediakan dalam kertas soalan.Tunjukkan langkah-langkah penting. Ini bolehmembantu anda untuk mendapatkan markah.

3. Rajah yang mengiringi soalan tidak dilukismengikut skala kecuali dinyatakan.

4. Satu senarai rumus disediakan di halaman 2 & 3

5. Anda dibenarkan menggunakan kalkulatorsaintifik yang tidak boleh diprogram.

3 4

4 4

5 4

6 5

7 5

8 6

9 6

10 5

11 6

B

12 12

13 12

14 12

15 12

16 12

Jumlah

Kertas soalan ini mengandungi halaman bercetak.

SULIT1449/2MatematikKertas 22009

2

12 jam

1449/2

1449/2 2009 Hak Cipta JPN Perak [Lihat sebelahSULIT

NAMA :

TINGKATAN :

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MATHEMATICAL FORMULAE

The following formulae may be helpful in answering the questions. The symbols given are theones commonly used.

RELATIONS

1. nmnm aaa .

2. nmnm aaa

3. mnnm aa

4.

ac

bd

bcadA

11

5.)(

)()(

Sn

AnAP

6. )(1)'( APAP

7. 221

221 )()(distance yyxx

8. Midpoint

2,

2, 2121 yyxxyx

9. Average speed =distance travelled

time taken

10. Mean =sum of data

number of data

12. Pythagoras Theorem222 bac

13 2 1

2 1

y ym

x x

14.intercept

intercept

x

ym

11. Mean =sfrequencieofsum

frequency)(classmarkofsum

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UntukKegunaanPemeriksa SHAPE AND SPACE

1. Area of trapezium =2

1 sum of parallel sides height

2. Circumference of circle= d = 2r

3. Area of circle = r2

4. Curved surface area of cylinder = 2rh

5. Surface area of sphere = 4r2

6. Volume of right prism = cross sectional area length

7. Volume of cylinder = r2h

8. Volume of cone =3

1r2h

9. Volume of sphere =3

4r3

10. Volume of right pyramid = 3

1base area height

11. Sum of interior angles of a polygon = ( n 2 ) 180 0

12.0

arc of length angle subtended at centre

circumference of circle 360

13.0

area of sector angle subtended at centre

area of circle 360

14. Scale factor, k =PA

PA '

15. Area of image= k 2 area of object16.

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SECTION A[ 52 marks]

Answer all question in this section.

1. On the graph provided, shade the region which satisfies the three inequalitiesy ≤ x + 5, x + y ≤ 5 and y > 1. [3 marks]

Answer:

[2][1]

2. Calculate the values of e and f that satisfy the simultaneous linear equations.

104

1625

fe

fe[4 marks]

Answer:

[1][1]

[1]

[1]3

6622

50205

1625

f

f

fe

fe2

10)3(4

e

e

x+y=5

y=x+5

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3. Diagram 1 below shows a right prism with PST is the uniform cross section of the prism.

Diagram 1

A solid cylinder of diameter 6cm is taken out from the solid prism. Using7

22 ,

calculate the volume in cm3 of the remaining solid.[4 marks]

Answer:

91482

1 [1]

737

22 2 [1]

91482

1 - 73

7

22 2 [1]

= 306 [1]


)52(

xx

[4 marks]Answer:

4,2

3

0)4)(32(

01252 2

x

xx

xx [1][1]

[1] [1]

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5. Diagram 2 shows a right prism with an isosceles triangle base, STU. Theisosceles triangle STU is the uniform cross-section of the prism.ST = SU and W is the midpoint of TU.Calculate the angle between the line PW and the base STU. [4 marks]

Answer:

Identify PWS [1]

Length of SW = 8 [1]

tan PWS =8

13[1]

VST 58.39 or 5824’ [1]

6. (a) State whether the following sentence is a statement or a non-statement.Give a reason for your answer.

(b) State whether each of the following statements is true or false.(i) { 0 } is an empty set or is an empty set.

(ii){ } is and empty set and is also an empty set.

(c ) Complete the following argument.

Premise 1 : If a > 3, then 5 a > 15.Premise 2 : 5 a < 15.Conclusion :________________________________________________

[5 marks]Answer:(a) It is a statement because it has a truth value which is false. [2]

(b) (i) True [1](ii) True [1]

(c) Conclusion : a < 3 [1]

13

P

W

1012

Q

R

S T

U

DIAGRAM 2

2 + 7 = 1 + 6

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7. Bottle A contains three red marbles and two yellow marbles. Bottle B contains four redmarbles and three yellow marbles. A marble is chosen at random from each bottle.Find the probability that(a) both marbles are in yellow,(b) both marbles are in different color. [5 marks]

Answer:

a.)35

6

7

3

5

2 [2]

b.)35

17

7

3

5

3

7

4

5

2 [3]

8.

.

T

Diagram 6

O S

Q

R

P

150

Diagram 3 shows two sectors OPQ and ORS with centre O. OP = 8 cm, QR = 6 cm andOS = 2 OT. POTS is a straight line.

Using22

7 , calculate


Answer :

a)30 22 150 22

2 8 2 14360 7 360 7

or

[1]

30 22 150 226 8 14 2 8 2 14

360 7 360 7

[1]

=6 482

68 68.867 7

or or cm [1]

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b)

30 228 8

360 7

or

150 2214 14

360 7

or

150 227 7

360 7

[1]

30 22 150 22 150 228 8 14 14 7 7

360 7 360 7 360 7

[1]

=11 8789

209 209.2642 42

or or cm2 [1]

9. The following diagram shows the distance-time graph of the journeys of two particles Sand T.

Graph ABCD represents the journey of the particle S from station X to Y. The graph EFrepresents the journey of the particle T from station Y to X.Both particles leave station X and station Y respectively at the same time along the sameroad.(a) State the length of time, in minute, during which the particle S is

stationary.(b) If the journey starts at 1500, at what time do the particles meet?(c) Given that the distance travelled by particle S over the 60 minutes period is 186km.

Calculate the value of v.(d) Find the distance, in kilometres, from station Y when the particles meet.(e) Calculate the average speed in m/s of particle S for the whole journey.

[ 6 marks ]

Answer:(a) 402060 minutes [1](b) 1500 + 40 = 1540

3.40 pm [1]

(c) 18640602

1 v

v = 3.72 km/minute62 ms-1 [1]

(d) 6.9960401452

1 v km [1]

A

D

F

CB

600

v

145 E

Speed (ms-1)

Time (minute)20 40 100

80

StationX

StationY

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(e) Average speed =takentimetotal

travelleddistancetotal

=60100

]6040)145(2

1[186000

v[1]

= 72.4 m/s [1]

10. In Diagram 4, KL, LN and MN are straight lines. L is on y-axis. LN is parallel tox-axis and KL is parallel to MN.

Diagram 4

The equation of straight line KL is 2x + y + 4 = 0, find(a) The equation of straight line LN,(b) The equation of straight line MN hence state its x-intercept.

[ 5 marks ]Answer :

(a) y = 4 [1](b) Gradient, mMN = -2 [1]

2

42

4)1(26

x

xy

corc

[1]

NL

M(1,6)

KO

x

y

[1]

[1]

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3 2

and matrix V =2 2

3v

u

such that UV =1 0

0 1


simultaneous equations32 yx

523 yx [6 marks]

Answer :

a)2 2 2 21

3 1 32 ( 6)v

u

2 2 2 21

3 1 3 14v

u = 1 [1]

v =1

4[1]

b)1 2 3

3 2 5

x

y

(1)

2 2 31

3 1 54

x

y

(1)

x = 1 (1)y = 1 (1)

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SECTION B[ 48 marks]

Answer any four questions from this section.

12. (a) Table 1 shows values of x and y for equation 823 xxy .

Find the values of v and w.[ 2 marks ]

(b) For this part of the question, use graph paper. You may use a flexible curveruler.By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to10 unit on the y-axis,

Draw the graph of 823 xxy for 43 x .

[ 4 marks ](c) From your graph, find

(i) the value of y when x = 15,(ii) the value of x when y = 20.

[ 2 marks](a) Draw a suitable straight line on your graph to find the values of x that satisfy the

equation 0493 xx for 43 x .

State the values of x.[ 4 marks]

Answer :

(a) v = -12 [1]

w = 13 [1]

(b) (i) 0.70.8 y [1]

(ii) 5.26.2 x [1]

(d) Identify equation 127 xy [1]

Straight line 127 xy correctly drawn [1]

x = 5.04.0 x , x = 8.27.2 x [1] [1]

x 3 2 1 0 1 2 3 4

y 29 v 7 8 9 4 w 48

TABLE 1

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x

y

1 2 3 4- - -1

10

20

-

-

-

0

30

40

50

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13. Diagram 5 shows four quadrilateral ABCD, ADEF, GHJK and LPNM drawn onCartesian plane.

Diagram 5

(a) Transformation M is a reflection about the line x = 3.

Transformation N is the translation

3

5

State the coordinates of the image of point (2,4) under the following transformations:

i) MNii) NM

(b) ADEF is the image of ABCD under transformation V and GHJK is the image of ADEF

under transformation W.

Describe in fulli) Transformation Vii) A single transformation which is equivalent to transformation WV.

(c) LPNM is the image of ABCD under an enlargement.i) State the coordinates of the centre of the enlargement.ii) Given that the area LPNM is 72.8 unit2, calculate the area of ABCD.

[12 marks]

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Answer:

(a)

(i) (7, 1) (-1, 1) [1] [1]

(ii) (4, 4) ( 9, 1) [1] [1]

(b)

(i) V = reflection at line AD [2]

(ii) Anticlockwise rotation of 90° at centre (5, 0) [3]

(c)

(i)

(ii)

Centre of enlargement = (1,0) [1]

Area of ABCD = 8.722

1 [1]

= 18.2 unit2 [1]

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14. The data in Diagram 7 shows the mass in kg of 50 students in a class.

40 52 47 64 52 35 54 43 39 5443 39 54 57 53 39 55 38 46 4245 53 60 38 54 62 47 45 52 5338 55 62 54 45 54 36 58 44 4252 48 49 46 49 48 45 54 41 46

Diagram 6

(a) Based on the data in Diagram 6 and using a class interval of 5 kg, completeTable 2 in the answer space. [4 marks]

(b) From the table in a),(i) state the modal class,(ii) calculate the mean mass of the students and give your answer correct to 2

significant figures. [4 marks](c) By using a scale of 2 cm to 5 kg on the x-axis and 2 cm to 1 student on the y-axis,

draw a histogram based on the data.[4 marks]

Answer:

(a)Mass Frequency Midpoint

35-39 8 37

40-44 7 42

45-49 13 47

50-54 14 52

55-59 4 57

60-64 4 62

[1] [2] [1]Table 2

(b) (i) 50-54 [1](ii) mean = (37×8)+(42×7)+(47×13)+(52×14)+(57×4)+(62×4)

50 [2]= 2405/50= 48.1= 48 (2 s.f) [1]

(c) graph

Axes [1]Scales [1]Bar [2]

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64.559.554.534.5 39.5 44.5 49.5

1

2

3

4

5

6

7

8

9

10

11

12

13

Mass (kg)0

14

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15. Diagram 8 shows a right prism with a horizontal rectangular base ABCD. BCHGF is theuniform cross -section of the prism. GHIJ is a horizontal plane. ABFE and DCHI arevertical planes.

(a) Draw to full scale, the elevation of the solid on a vertical plane parallel to BC as seenfrom X [3 marks]

Answer:

[3]

(b) A solid right prism with uniform cross-section KLM is removed from the solid. Theremaining solid is shown in diagram 8(ii).

D

B

G

E

A

J

1 cm

C

I

F

H

X4 cm

3 cm

3 cm

2 cm

DIAGRAM8(i)

NL

K M

1 cm

Y

D

B

G

E

A

J

1 cm

C

I

H

4 cm3 cm

3 cm

2 cm

DIAGRAM 8(ii)

B/A C/D

H/I

F/E

2 cm3 cm

4 cm

1cmG/J

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Draw at full scale

(i) the plan of the remaining solid . [4 marks](ii) the elevation of the remaining solid on a vertical plane parallel to AB as

seen from Y. [5 marks]

Answer:

[4]

[5]

L/M

J

K

N/H

B/CA/D

E

3 cm

2 cm3 cm

GH/C

K/L

N/B

E/A

2 cm 3 cm

4 cm I/D

M

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16. P(650 S, 400 E), Q(650 S, 600 W), R and S are four points on the surface of the earth. PRis the diameter of the parallel of latitude 650 S.(a) (i) State the longitude of R.

(ii) Calculate the shortest distance, in nautical mile, from P to R measured along thesurface of the earth. [4 marks]

(b) S lies north of Q and the distance of SQ measured along the surface of the earth is5100 nautical mile. Calculate the latitude of S. [3 marks]

(c) An aero plane took off from P and flew due west to Q and then flew due north to S.The average speed for the whole flight was 560 knots.Calculate(i) the distance, in nautical mile, taken by the aero plane from P to Q measured

along the common parallel of latitude,(ii) the total time, in hours, taken for the whole flight. [5 marks]

Answer:

(a) (i) Longitude of R = 1400 W [2](ii) 500 X 60 = 3000 n.m. [2](b)

000

0

206585

8560

5100

N [1] [1] [1]

Latitude of S = 200 N [1]

(c) (i) 71.253565cos60100 00 [2]

(ii) 64.13560

71.7635 hours [2]

END OF QUESTION PAPER

Ats maths09

Technology

marks answer

y x o x y

x y o y

x y x o

marks diagram

inequalities y x

quadratic equation y

values of x