Program Ma Yang Dal Ton 2009

SMK. KOTA KLIAS, BEAUFORT

MATEMATIK SPM 2009 PROGRAM MAYANG

SECTION A REGION FOR INEQUALITUES MATRICES LINEAR SIMULTANEOUS EQUATIONS SECTION B GRAPH OF FUNCTION STATISTIC PLANS AND ELEVATION

Disediakan oleh: MOHD NAZAN BIN KAMARUL ZAMAN

PROGRAM MATEMATIK SPM 5DALTON 2009 No. 1. 2 3.a 3.b 4. 5. 6. 7. 8.

9. 10. 11.

12.

13. 14. 15.

16.

Topic SECTION A Linear Simultaneous Equations Quadratic Equation Sets (Shade Venn diagrams) Region for Inequalities Mathematical Reasoning (Statement, implications, argument, mathematical induction, converse ) The Straight Line ( parallel, equation, y-intercept) Probability II Arc Length & Area of Sector Volume of Solids a. Pyramids and half cylinders b. Cones and Cylinders c. Cones and cuboids d. Pyramid and prism Matrices ( Inverse, matrix equation ) Gradient and Area Under a Graph a. Speed-Time Graph b. Distance-Time graph Lines & Planes in 3 Dimensions (angle between 2 planes) SECTION B Graphs of Functions II a. Quadratic b. Cubic c. Reciprocal Transformations III ( combined ) Earth as a Sphere Plans and elevations a. Prism and cuboids b. Cuboids and half cylinder, prism c. Prism and prism Statistics III a. Raw data, frequency table, mean, frequency polygon, modal class b. Ogive c. Histogram d. Frequency Polygon e. Communication K2 K1

K1

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K7

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75 25 100

6O 20 80

45 15 60 6C

35 15 50 7D

28 12 40 8P

28 12 40 8P

28 12 40 8P

No. 1. 2 3.a 3.b 4. 5. 6. 7. 8.

9. 10. 11.

12.

13. 14. 15.

16.

Topic SECTION A Linear Simultaneous Equations Quadratic Equation Sets (Shade Venn diagrams) Region for Inequalities Mathematical Reasoning (Statement, implications, argument, mathematical induction, converse ) The Straight Line ( parallel, equation, y-intercept) Probability II Arc Length & Area of Sector Volume of Solids a. Pyramids and half cylinders b. Cones and Cylinders c. Cones and cuboids d. Pyramid and prism Matrices ( Inverse, matrix equation ) Gradient and Area Under a Graph a. Speed-Time Graph b. Distance-Time graph Lines & Planes in 3 Dimensions (angle between 2 planes) SECTION B Graphs of Functions II a. Quadratic b. Cubic c. Reciprocal Transformations III ( combined ) Earth as a Sphere Plans and elevations a. Prism and cuboids b. Cuboids and half cylinder, prism c. Prism and prism Statistics III a. Raw data, frequency table, mean, frequency polygon, modal class b. Ogive c. Histogram d. Frequency Polygon e. Communication K2 K1

K1

K2

K3

K4

K5

K6

K7

4 4

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√ √

2 1

3 4

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√ √

3 2

1 2

1 2

1 2

4 2 3 3

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4 1

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1 1

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75 25 100 A/B

45 15 60 C

45 15 60 C

35 15 50 D

28 12 40 E

28 12 40 E

28 12 40 E

KUMPULAN 1 1. 2. 3. 4.

FAZRIN EZYANA MEME SAFINAH

KUMPULAN 2 1. 2. 3. 4.

NORITA SARINAH NURFITRI ZURIHA

KUMPULAN 3 1. 2. 3. 4.

SHAHRUL HADI MD. SHAHARLIH HALIM NUIN ANUAR

KUMPULAN 4 1. 2. 3. 4.

ANINA ASNANI SURAYA ROZELEEYANA

KUMPULAN 5 1. 2. 3. 4.

ROZANA RUMAINAH NORAFIZAH NORAZILAH

KUMPULAN 6 1. 2. 3. 4. 5.

JACKEY KADAPI ZULALIF ZARUL LIZWAN AZRUL

KUMPULAN 7 1. 2. 3. 4.

NAZRI ISMATHI HAFIZUL SALMANAN

SECTION A November 2003 1. Calculate the values of k and w that satisfy the simultaneous linear equations: 2k −3w =10 4k + w = −1

[4 marks] July 2004 2. Calculate the values of d and e that satisfy the following simultaneous linear equations: 4d +3e =6 2d −e =8

[4 marks] November 2004 3. Calculate the values of p and q that satisfy the following simultaneous linear equations: 1 p − 2q =13 2 3 p + 4 q = −2

[4 marks] July 2005 4. Calculate the values of d and e that satisfy the following simultaneous linear equations: 3d +e =12

d −2e =−10

[4 marks] November 2005 5. Calculate the values of p and q that satisfy the following simultaneous linear equations: 2 p − 3q =13 4 p + q =5

July 2009 6. Calculate the values of p and q that satisfy the following simultaneous linear equations: 2 x + 3 y =1 4 x − y =9

[4 marks]

On the graph provided, shade the region which satisfies the three inequalities. 1.

y ≥ 2x – 4, y ≥ -x + 4 y < 4

2.

y ≤ x – 2, x + y ≥ 4 and x < 6

3.

y ≤ x –3, y ≤ -x +4, y ≥ -3

4.

x + 2y ≥ 6 , y ≤ x , x < 5

July 2005

4.

1 − 2  1 0   P = P is a 2 x 2 matrix such that  3  0 1   4     a) Find the matrix P b) Write the following simultaneous linear equations as a matrix equation: x − 2 y =8 3 x + 4 y = −6

Hence, using matrices, calculate the value of x and of y. ( 6 marks ) November 2005 5.

2 − 5  3  It is given that matrix P =  and matrix Q = k  1  −1 3    a) Find the value of k and of h

b)

h  such that 2 

Using matrices, find the value of x and of y that satisfy the following simultaneous equations : 2x – 5y = -17 x + 3y = 8

1 PQ =  0 

0  1 

linear

( 6 marks )

July 2006 6. (a) (b)

3 − 4  . It is given that matrix M =  1 2    Find the inverse matrix of M Write the following simultaneous linear equations as matrix equation: 3x − 4 y =13 x + 2y = 6

Hence, using matrices, calculate the value of x and of y. ( 6 marks ) November 2006 7.

1

2  n  is inverse matrix of 

3  −1 

−4  . Find the value of n. 2  

(a)

It is given that  1

(b)

Write the following simultaneous linear equations as matrix equation:

 2

3u −4v =−5 −u +2v =2

Hence, using matrices, calculate the value of u and of v ( 6 marks )

SECTION B Graphs of Functions

Question 1 Complete the following table for the equation y = −3x 2 + 2 x + 5

a)

x -3 -2 -1 0 1 2 3 4 y 11 0 4 -16 -35 By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 5 unit on the y-axis, draw the graph of y = −3 x 2 + 2 x + 5 for − 3 ≤ x ≤ 4 From your graph, find i) the value of y, when x = -0.5 ii) the value of x, that satisfy the equation of 3 x 2 − 2 x = 5 Draw a suitable straight line on your graph to find the values of x which satisfy the equation 3 x 2 + 2 x − 25 = 0 for − 3 ≤ x ≤ 4 . State the values of x.

b) c)

d)

Question 2 Complete the following table for the equation y = 8 + 3 x − 2 x 2

a) x y

b) c)

d)

-3 -19

-2

-1 3

0 8

1 9

2 6

3

4 -12

By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 5 unit on the y-axis, draw the graph of y = 8 + 3 x − 2 x 2 for − 3 ≤ x ≤ 4 From your graph, find i) the value of y, when x = 1.35 ii) the values of x, when y = -10 Draw a suitable straight line on your graph to find the values of x which satisfy the equation − 2 x 2 − 3 x + 26 = 0 for − 3 ≤ x ≤ 4 . State the values of x.

Question 3 a)

Complete the following table for the equation y = x 3 −10 x + 5 x y

b) c) d)

-3.5 -2.9

-3 8

-2

-1 14

0 5

1

2 -7

3

By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 5 unit on the y-axis, draw the graph of y = x 3 −10 x + 5 for − 3.5 ≤ x ≤ 3.5 From your graph, find the value of y, when x = -2.5 Draw a suitable straight line on your graph to find the values of x which satisfy the equation x 3 − 10 x = 6 for − 3.5 ≤ x ≤ 3.5 . State the values of x.

STATISTICS Question 1

3.5 12.9

The data above shows the pocket money, in RM, per week of a group of 40 students.

10 18 4 8 19

13 16 14 10 15

18 18 12 16 17

7 10 6 11 12

19 15 9 13 14

5 19 20 17 11

21 20 20 9 16

21 8 15 11 17

(a) Based on the data, complete the table below. Pocket money (RM) Frequency 1–3 0 4–6

Midpoint

(b) Based on the table, (i) state the modal class, (ii) calculate the mean, of the data. (c) For this part of the question, use a graph paper. By using a scale of 2 cm to RM 3 on the x-axis and 2 cm to 1 student on the y-axis, draw a histogram and a frequency polygon on the same graph paper. Question 2 The frequency table above shows the masses, in kg, of watermelons in a lorry. Mass (kg) 1.5 – 1.9 2.0 – 2.4 2.5 – 2.9 3.0 – 3.4 3.5 – 3.9 4.0 – 4.4 4.5 – 4.9 5.0 – 5.4

Frequency 4 10 26 84 50 15 8 3

(a) From the table, state (i) the size of the class interval, (ii) the midpoint of the modal class. (b)

(i)

Based on the information, complete the table below.

Upper boundary Cumulative Frequency (ii)

1.45

5.45

0

200

For this part of question, use a graph paper. By using a scale of 2 cm to 0.5 kg on the x-axis and 2cm to 20 watermelons on the y-axis, draw an ogive for the data.

(iii)

From the ogive, a) state the median of the data, b) find the number of watermelons with the weight less than 3.8 kg.

Question 3 The data shows the ages of a group of 40 workers in a factory.

(a) (b)

38

41

32

39

32

50

31

38

36

51

31

27

44

30

49

28

26

31

22

36

32

33

34

43

31

27

46

41

35

35

34

31

40

35

25

27

50

37

33

45

Find the range of the data. Based on the data and by using a class interval of 5 years, complete the table below. Ages (years) 21 – 25 26 – 30

Frequency

Midpoint

(c)

From the table, (i) state the modal class, (ii) calculate the mean, of the data.

(d)

For this part of question, use a graph paper. By using a scale of 2 cm to 5 years on the x-axis and 2 cm to 2 workers on the y-axis, draw a frequency polygon for the data.

PLAN AND ELEVATION November 2004

3(a) Diagram 2(i) shows a solid consisting of a cuboid and a half cylinder which are joined at the plane HJMN. The base GDEF is on a horizontal plane and HJ=3 cm

Diagram 2(i)

Draw to full scale, the elevation of the solid on a vertical plane parallel to DG as viewed from X. [3 marks ] (b) A solid right prism is joined to the solid in Diagram 2(i) at the vertical plane ELMW. The combined solid is as shown in diagram 2(ii). Trapezium PMWU is the uniform cross-section of the prism and PQRM is an inclined plane. The base DEUTSWFG is on a horizontal plane and EU= 2cm

Diagram 2(ii) Draw to full scale, (i) the plan of the combined solid (ii) the elevation of the combined solid on a vertical plane parallel to UT as view from Y

[4 marks] [5 marks]

2. V

T

U

H

7 cm

5

cm

G

Y

W

E F

D

C

4 cm

R

P 3

cm

Q

A

1 cm

B

Diagram 1(i)

The above diagram shows a solid which consists of two prisms whose base ABPQRCD lies on a horizontal table. PU = CV = 6 cm and W is the mid-point of PU. The length of PQ is 7 cm. Draw to full scale, the elevation of the solid on a vertical plane parallel to QR as viewed from Y. 1b S

V

T

U

H

7 cm

5

cm

G

W

E F

D

C

2 cm

P

Q

3

cm

4 cm

R

A 1 cm

B

X

Diagram 1(ii) A semi-cylinder is joined to the solid. Draw to full scale, i. the plan of the solid ii. the elevation the solid on a vertical plane parallel to DR as viewed from x

3(a) Diagram 3(i) shows a solid prism with its rectangular base, PQRS, on a horizontal table. The surface, FGKLRQ, is the uniform cross-section of the prism. Rectangle EFGH is an inclined plane and rectangle JKLM is a horizontal plane. FQ, KG and LR are vertical edges.

Diagram 3(i) Draw the full scale elevation of the solid on a vertical plane parallel to QR as viewed from X. 3(b) A solid right prism with the uniform cross-section, ITU, is removed from the solid in Diagram 3(i) . The remaining solid is shown in Diagram 3(ii). Rectangle TFVU is a horizontal plane. IU is a vertical edge. FT = 3 cm and IU = 2 cm.

Diagram 3(ii) Draw the full scale (i) plan of the remaining solid (ii) elevation of the remaining solid on a vertical plane parallel to PQ as viewed from Y.