times maths 09

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41

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SOALAN ULANGKAJI SPM 2009

Paper 1

2004 2005 2006 2007 2008

Form1-3

1 Polygon I, II 7 6, 7 7 6, 7 7

2 Transformations I, II 9, 10 9, 10 9, 10 9, 10 9,10

3 Trigonometry I 13 11

4 Algebraic Expressions I, II, III 20 19 19

5 Algebraic Formulae 21 21 21 20 21

6 Algebraic Fractions 19 20 20 21 20

7 Linear Equations 22 22 22 22 22

8 Indices 23, 24 23, 24 23 23, 24 23,24

9 Linear Inequalities 25, 26 25 24 25 25

10 Graph of Functions I

11 Solid Geometry I, II, III

12 Circles I, II

13 Statistics I, II 27, 28 27 25,26, 27 26, 27 26,27

1 Standard Form 1, 2, 3, 4 1, 2, 3 1, 2, 3, 4 1, 2, 3 1,2,3,4

2 Quadratic Expressions and Equations 19 19

3 Sets 30,31,32 29,30,31 29,30,31 29,30,31 29,30,31

4 Mathematics Reasoning

5 The Straight Lines 33,34 32,33 32,33 32,33 32,33

6 Statistics III 26

7 Probability I 34,35 34,35

8 Circles III 8 8 8 8 8

9 Trigonometry II 11,12 11,12,13 12,13 11,12,13 11,12,13

10 Angles of Elevation and Depression 15,16 15 15,16 15 15,16

11 Lines and Planes in 3-Dimensions 14 14 14 14 14

1 Number Base 5,6 4,5 5,6 4,5 5,6

2 Graph of Function II 28 28 28 28 28

3 Transformations III

4 Matrices 40 39,40 39,40 39,40 39,40

5 Variations 38,39 36,37,38 36,37,38 36,37,38 36,37,38

6 Gradient and the Area under a Graph

7 Probability II 35,36,37 34,35 34,35

8 Bearing 18 16 17 16 17

9 The Earth as a Spehere 17 17,18 18 17,18 18

10 Plan and Elevation

Total 40 40 40 40 40

SPM2009

[ 1449/1 ] [ 1449/2 ]

Mathematic


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PAPER 2

2004 2005 2006 2007 2008

Form1-3

1 Polygon I, II

2 Transformations I, II

3 Trigonometry I

4 Algebraic Expressions I, II, III

5 Algebraic Formulae

6 Algebraic Fractions

7 Linear Equations I, II 5 2 4 2 2

8 Indices

9 Linear Inequalities 3

10 Graph of Functions I

11 Solid Geometry I, II, III 2 6 5 11 4

12 Circles I, II 9 7 8 6 7

13 Statistics I, II

1 Standard Form

2 Quadratic Expressions and Equations 7 1 3 3 3

3 Sets 1 1 1

4 Mathematics Reasoning 4 8 6 7 5

5 The Straight Lines 6 5 10 5 6

6 Statistics III 14 14 14 16 14

7 Probability I 9

8 Circles III

9 Trigonometry II

10 Angles of Elevation and Depression

11 Lines and Planes in 3-Dimensions 3 4 2 4 11

1 Number Base

2 Graph of Function II 12 12 13 1, 12 12

3 Transformations III 13 13 12 15 13

4 Matrices 8 11 11 9 8

5 Variations

6 Gradient and the Area under a Graph 11 10 9 10 9

7 Probability II 10 7 8 10

8 Bearing

9 The Earth as a Spehere 16 16 16 14 16

10 Plan and Elevation 15 15 15 13 15

Total 16 16 16 16


3/15

SOALAN ULANGKAJI SPM 2009 43


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MATHEMATICAL FORMULAE

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

RELATIONS

1. am x an = a m+ n

2. am an = a m n

3. ( am )n = a mn

4. A-1 =d-b

5. Distance = (x2- x1)2+ (y2-y1)

2

6. Midpoint, ( x, y ) =( , )

7. Average speed =

8. Mean =

9. Mean =

10. Pythagoras Theorem

c2 = a2 + b2

11. P ( A ) =

12. P ( A') = 1 - P(A)

13. m =

14. m =

1 (d

-c

-b

a)

x1+ x22

y1+y2

2

distance travelled

time taken

sum of data

number of data

sum of (class mark x frequency)sum of frequencies

n (A)

n (S)

y2

- y1

x2- x

1

y - intercept

x - intercept

SHAPES AND SPACE

1. Area of trapezium = 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 = r2h

9. Volume of sphere = r3

10. Volume of right pyramid = base area height

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

12. =

13. =

14. Scale factor , k=

15. Area of image = k2

area of object

12

13

43

13

arc lengthcircumference of circle

angle subtended at centre360

area of sector

area of circle

angle subtended at centre

360

PA'PA

SOALAN ULANGKAJI SPM 2009MATHEMATIC

Kertas 1


4/15


7. In Diagram 2, PQRSUV is a regular hexagon . PUT, QUWand RST are straight lines.

Diagram 2Find the value x + y

A 120B 150C 180D 210

8. Diagram 3 shows a circle with centre O and TPQ as a tangentto the circle at P. Given that SRQ is a straight line.

Diagram 3Find the value of x.

A 20B 30C 40D 50

9. Diagram 4 shows five quadrilaterals.

Diagram 4

Answer all questions

1. Round off 0.060874 correct to two significant figures.

A 0.060B 0.061C 0.0608D 0.0609

2. Evaluate 7.52 x 10-5

4.8 x 10-6

=

A 2.72 x 10- 6

B 2.72 x 10- 5

C 7.04 x 10- 6

D 7.04 x 10- 5

3. Express 63480000 in standard form

A 6.348 x 10-8

B 6.348 x 10-7

C 6.348 x 107

D 6.348 x 108

4. Express 4368 as a number in base 5.

A 1215B 2125C 2121 5D 1221 5

5. Evaluate 100102 - 10112

A 1012B 1112C 10112D 11012

6. Diagram 1 shows a triangle PQR and quadrilateralRSTU. PRS and QRU are straight lines.

Diagram 1

Find the value of x.

A 95B 100C 105

D 110

80

P

U

Q

R

ST

160x

40

PV

W

Q

S

T

Q40

RS

P

T

xO

160

x (

U

y

P

A

B

C

D

y

6

4

2

0

-2

-4

2 4 6 8

R


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Which of the quadrilateral A, B, C or D is the image ofquadrilateral P under a rotation of anticlockwise aboutthe centre (4, 0) ?

10. Diagram 5 shows a triangle JKL is the image of triangle JKLunder a anticlockwise rotation of 90.

Diagram 5

Which of the points A, B, C or D, is the centre of the rotation

11.In Diagram 6, O is the centre of circle PQRST.

Diagram 6

Find the value of cos .

A 0.7B 0.2C 0.2D 0.7

12. Diagram 7 shows a rectangle JKLM.

Given that the ratio of LN : LM = 2 : 3. Calculate the valueof cos KNM.

A

B

C -

D -

13.Diagram 8 shows the graph of y = 6xn.

Find the value of s.

A 12B 6C 3D 2

14.Diagram 9 shows the position of points P, Q and R on amap. Given that bearing of R from P is 150.

Find the bearing of Q from R.

A 290B 310C 320D 330

15.Which of the following graphs represents y = - 5 x -1?

A

B

y

C

x0

Dy

0 x

o

TS

R

P

Q

(1,0)

(0.2 - 0.7)

L M

K J

N

5 cm

18 cm

(2,s)

O

y

x

1213

513

51312

13

140

Q

P

R

DIAGRAM 9

O

y

x

O

y

x

Diagram 7

Diagram 7


6/15


16. Diagram 10 shows a cuboid with a horizontal base PQRS.

Diagram 10

State the angle between the planes PWQ and PQRS.

A WQUB WQSC QPWD WPS

17.Diagram 11 shows a right prism with rectangular base

PQRS. Trapezium PQUT is the uniform cross-section ofthe prism.

Diagram 11

Identify the angle between the line RT and the planeTPSW.

A RTPB RTSC QRTD TRS

18.Diagram 12 shows Ahmad observing a kite which is flyingvertically above point L. The horizontal distance betweenhim and point L is 30 m.

Diagram 12

Given that the angle of elevation of the kite from Ahmad is25. Find the vertical height, h, in m, of the kite.

A 13.99B 15.59C 64.34D 65.94

13 cm

12 cm

5 cm

T

W V

U

R

QP

W

T

V

U

R

QP

S

h

L

1.6 m

30 m

19. Express - as a single fraction in its simplest from.

A

B

C

D

20.(x y)2 (x2 y2) =

A -2xyB -2y2C -2xy + 2y2

D -2xy + y2

21. Given that q = , thenp =

A C

B D

22. Given that = m-2 calculate the value ofh.

A 3B 6C 7D 13

23.Given that (53x)2 = (252)2. Find the value of x.

A

B

C 1

D

24.Simplify k (3k-1)2 (3k-4).

A 3kB 3k3

C 4kD 4k

3

25.List all the integers x that satisfy the inequality 4 < 2x 2.

A 0, 1, 2B 0, 1C 0, 1, 2D 1, 2

mn

1-m2n

3m-12n

m-12n

2m-12n2

13n

p-5p+5

5q-5

q-1

5+5qq-1

5q-5

1-q

5+5q1-q

3m+54

23

43

83


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29. Diagram 15 is a Venn diagram which shows the numberof element of set P,Q and R.

It is given that the universal set = P Q R and n(Q)=17. Find n[(P R) Q]

A 12B 13C 18D 21

30. Diagram 16 is a Venn diagram which shows theuniversal set = P Q R

The shaded region in the Venn Diagram represent the set

A R (P Q)B (Q' R) (P Q)

C (P Q) (Q P)'D (P Q) (Q' R)

31. In Diagram 17, PQ = 5 units and OQ = 2OR.

The equation of QR is

A y= x-3

B y= 2x-3

C y= x-2

D y= 2x-2

26.Diagram 13 is a bar chart which shows the scores of agroup of pupils

State the modal score.

A 6B 5C 3

D 2

27.Table 1 shows the number of pencils brought by a groupof students to school.

Table 1

The difference between the mode and the median of thedata is

A 1B 2C 5D 6

6

4

2

01 2 3 54

Number of pupils

ScoreDIAGRAM 13

Number of pencils 1 2 3 4 5

Frequency 10 4 9 5 2

28. Diagram 14 shows the universal set, = R S T, n( )= 35, n (S) = 17, n (R S)=9and n(S T) = 3

Find n(R' T)

A 17B 18C 21D 26

T

S

R

Diagram 14

PQ

4

R

6

23

1

5

Diagram 15

P Q R

Diagram 16

4Ox

y

R

P

Q

Diagram 17

12

12


8/15


32. The gradient of the straight line 3y= x = 6 is

A

B

C 2

D 6

33. A bag contains 5 yellow balls, 3 red balls and 9 blue balls.A ball is picked at random from the bag. What is theprobability that the ball picked is not blue?

A

B

C

D

34. A box contains 45 green marbles and a number of redmarbles. A marble is picked at random from the box.Probability of getting red marble is . Another 9 redmarbles were added into the box. If a marble is picked atrandom from the box, what is the probability of gettingred marble?

A C

B D

35. ( )- ( )

A ( )

B ( )

C ( )

D ( )

36. Given ( ) ( )= ( ), calculate the value ofp

A 1B 3C 6D 9

37. P(15N, 20E), Q and R are three points on the earthssurface. Q is due south ofP. The difference in latitudebetween P and Q is 40 . R is due west ofQ. Thedifferenc in longitude between Q and R is 60. Theposition ofR is

A (25 N, 40 W)B (25 N, 80 E)C (55 N, 40 W)

D (55 N, 80 E)

23

2923

3

175178

179

17

38

13

49

59

12

5

4

1

3

6

-3

113

-25

35

7-2

p

4-31

6p

927

38. In Diagram 18, Nis the North Pole, S is the South Poleand NOS is the axis of the earth. PRQ is the equatorand NGS is Greenwich Meridian. The position of point T is

39. P varies inversely as the square root ofQ. The relationbetween P and Q is

A P Q

B P Q2

C P

D P

40. Table 2 shows some values ofp, q, and r.t

Given thatp , calculate the value of m.

A 2B 4C 6D 8

END OF QUESTION PAPER

NFORMATION FOR CANDIDATES

1. This question paper consists of 40 questions.

2. Answer all questions.

3. Each question is followed by four alternative answers, A, B, C or D.For each question, choose one answer only. Blacken your answer onthe objective answer sheet provided.

4. If you wish to change your answer, erase the blackened mark thatyou have done. Then blacken the space for the new answer.

5. The diagrams in the questions provided are not drawn to scale unlessstated.

6. A list of formulae is provided on page 2 to 3.

7. You may use a non-programmable scientificcalculator.

40

80O Q

G

NT

P

S

R

Diagram 18

A (40N, 80W)B (40N, 100W)C (50N, 80W)D (50N, 100W)

12

1

Q12

1Q2

p 2 9

q 3 m

r 4 1 7

9Table 2

q

3r


9/15



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MATHEMATICAL FORMULAE

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

RELATIONS

1. am x an = a m+ n

2. am an = a m n

3. ( am

)n

= amn

4. A-1 =d-b

5. Distance = (x2- x1)2+ (y2-y1)

2

6. Midpoint, ( x, y ) =( , )

7. Average speed =

8. Mean =

9. Mean =

10. Pythagoras Theorem

c2 = a2 + b2

11. P ( A ) =

12. P ( A') = 1 - P(A)

13. m =

14. m =

1 (d

-c

-b

a)

x1+ x22

y1+y2

2

distance travelled

time taken

sum of data

number of data

sum of (class mark x frequency)

sum of frequencies

n (A)

n (S)

y2

- y1

x2- x

1

y - intercept

x - intercept

SHAPES AND SPACE

1. Area of trapezium = 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 = r2h

9. Volume of sphere = r3

10. Volume of right pyramid = base area height

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

12. =

13. =

14. Scale factor , k=

15. Area of image = k2

area of object

12

13

43

13

arc lengthcircumference of circle

angle subtended at centre360

area of sector

area of circle

angle subtended at centre

360

PA'PA

SOALAN ULANGKAJI SPM 2009MATHEMATIC

Kertas 2


10/15


Section A[52 marks]

Answer all questions in this section.

1. The Venn diagram below shows sets P, Q, and R. Given that the universal set = P O R. Shade the region representing

(i) A B' (ii) A (B C)

[3 marks]

2. Using factorisation, solve the following quadratic equation x -6 = (x+9) [4 marks]

3. Calculate the value of m and n that satisfy the following simultaneous linear equations: [4 marks]

4. Identify and calculate the angle between the line AH and the plane ABCD. [4 marks]

5. (a) State whether the following statement is true or false. [1 marks]

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

Premise 1: If a quadrilateral is a rhombus, then it has two parallel sides.

Premise 2 : . ........

Conclusion : KLMN is not a rhombus [2 marks]

(c) Make a general conclusion by induction for the number sequence 7, 13, 23, 37, which follows the following pattern.

7 = 2(1) + 513 = 2(2) + 523 = 2(3) + 537 = 2(4) + 5 [2 marks]

1

2

2m - 3n = 7

m - n = 212

A B

CD

E

F G

H

5 cm

12 cm

9 cm

DIAGRAM 2

r {p, q, r, s} or -4 > -5

A B

C

A B

C


11/15



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o150 o

30

T

P U S

Q

R

6. Diagram 1 shows a solid that was formed by joining a right prism and a right pyramid.

The right angled triangle PST is the uniform cross-section of the prism. PQRS is a square and the heightof the pyramid is 6 cm. Calculate the volume, in , of the solid. [4 marks]

7. In diagram 3, O is the origin and PQRS is a trapezium. PS is parallel to QR. The straight line RS is parallel to the y-axis.The points Q and S lie on the x-axis.

Find(a) the equation of the straight line QR,(b) the x-intercept of the straight line QR. [5 marks]

8. (a) P is a matrix 2 x 2 such that [ ] P = [ ]. [6 marks]Find matrix P.

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

Hence using matrices, calculate the value of x and of y.

9. In diagram 4, QRS and UT are arcs of two circles, centres P and S respectively.

It is given that PUS is a straight line, PQ = 28 cm and US = 14 cm. Using, = , calculate

(a) the area, in , of the shaded region,

(b) the perimeter, in cm, of the shaded region. [6 marks]

P

Q

V

R

U

T

S 7cm

4cm

DIAGRAM 1

Q

( )10,3

P

S

( )11,2 R

Ox

y

DIAGRAM 3

24

-31

10

01

2x-3y=64x+y=5

DIAGRAM 4

227


12/15


10. Diagram 5 shows the speed-time graph for the movement of a particle for a period of 30 seconds.

(a) State the length of time, in s, for which the particle moves with uniform speed.

(b) Calculate the rate of change of speed, in ms-2 , in the last 8 seconds.

(c) Calculate the value ofu, if the distance traveled in the first 22 seconds is 303 m.

11. Table 1 shows the number of students who attend a motivation camp.

Two students from the group are required to take part in a show.

(a) If one student is chosen at random from School A and another one from school B, calculate the probability that both are girls.

(b) If two students are chosen at random from the group of boys, calculate the probability that both are from the same school.

u

O16 22 30

25

( )1msSpeed

Time (s)

DIAGRAM 5

School Number of students

Boys Girls

School A 9 4

School B 7 8

TABLE 1


13/15



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Section B[48 marks]

Answer any four questions in this section.

12. (a) Table 2 shows the values of x and y which satisfy the equation y=x3-4x+11.

Calculate the value of k and of m. [2 marks]

(b) For this part of the question, use the graph paper provided.You can use a flexible curve rule.

By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, draw the graph ofy=x3-4x+11 for-3x3 . [4 marks]

(c) From your graph, find(i) the value ofy when x=1.4 ,(ii) the value ofx when x3-4x+11=0 . [2 marks]

(d) Draw a suitable straight line on your graph to find all the values ofx which satisfy the equationx3-6x+3=0 for -3x3.

State these values ofx. [4 marks]

13. Diagram 6 shows quadrilateralsABCD, PQRS and PTUVdrawn on a Cartesian plane.

(a) Transformation M is a translation ( )

Transformation N is a reflection in the liney=x .State the coordinates of the image of point S under each of the following transformations :

(i) M,(ii) NM.

(b) (i) PTUVis the image ofABCD under the combined transformationJK.Describe in full, transformation K and transformationJ. [6 marks]

(ii) Given the shaded region QTUVSR represents a region with an area of 87 cm2, calculate the area,in cm2, of the region represented by PQRS. [3 marks]

x -3 -2.5 -2 -1 0 1 1.7 2 3

y -4 5.38 k 14 11 8 m 11 26

TABLE 2

2 O 2 4 6 8

10

8

6

4

2 C

D

B

A

SR

Q

P

V U

T

Diagram 6

-43


14/15


14. The data in Diagram 7 shows the time, in minute, taken by 40 students to solve some geometry problems.

(a) Using data in Diagram 7 and a class interval of 5 minutes, complete Table 3 in the answer space. [3 marks]

(b) Based on table 3,

(i) find the modal class,(ii) calculate the mean time taken by the students. [4 marks]

(c) For this part of the question, use the graph paper provided on page 20.By using a scale of 2 cm to 10 minutes on the horizontal axis and 2 cm to 1 students on the vertical axis, draw ahistogram based on Table 3. [5marks]

15. You are not allowed to use graph paper to answer this question.

(a) Diagram 8(i) shows a right prism with rectangular baseABCD on a horizontal table. The surfaceAEHILD is itsuniform cross-section. The rectangle IJKL is an inclined plane. The rectangle EFGHis a horizontal plane. The edgesAE , BF, CK, DL, GJand HIare vertical edges. Given EH= FG = 2 cm.

Draw in full scale, the elevation of the solid on a vertical plane parallel to DC as viewed from P. [4 marks]

42

48

47

61

74

44

72

58

60

58

51

41

44

53

69

50

35

55

59

54

47

58

53

40

58

54

64

39

39

36

60

68

43

58

66

65

57

49

63

52

Diagram 7

Time(minutes) frequency Midpoint

35 39

40 44

F

G

H

E

L

A

D

C

K

J

B

6cm

6cm

5cm

5cm

7cm

P

I

Diagram 8(i)


15/15



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(b) A half-cylinder is joined to the solid in the Diagram 8(i) at the plane ABFE to form a combined solid as shown in Diagram 8(ii).

Draw to full scale,

(i) the plan of the combined solid, [4 marks](ii) the elevation of the combined solid on a vertical plane parallel to AD as viewed from Q. [4 marks]

16. P (67 N, 65 W) and Q are two points on the surface of the earth such that PQ is thediameter of the earth and PR is the diameter of a parallel of latitude.

(a) Mark the position of Q and R in Diagram 9. [2 marks]

(b) State the position of Q. [2 marks]

(c) Calculate the shortest distance, in nautical miles, from P to R measured along the surface of the earth. [3 marks]

(d) A jet plane took off from P due east to R and then flew due south to Q. The average speed for thewhole flight was 1500 knots. Calculate

(i) the distance , in nautical miles, from P to R measured along the parallel of latitude,(ii) the total time, in hours, taken for the whole flight. [5 marks]

END OF QUESTION PAPER

F

G

E

H

IB

C

J

D

A

K

L

6cm

5cm 6cm

5cm 7cm

Q Diagram 8(ii)

Diagram 9

O

N

P

S

times maths 09

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