40369470 Promej Matematik Spm 2011

8/6/2019 40369470 Promej Matematik Spm 2011

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A+ Mathematics SPM 2011

SMK. KOTA KLIAS, BEAUFORT

SABAH

PROMEJ - MATEMATIK

SPM 2011

FASA I

DISUSUN OLEH:MOHD NAZAN B. KAMARUL ZAMAN


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CONTENTS

SECTION A

1. LENGTH OF ARC AND AREA OF SECTOR

2. VOLUME SOLID GEOMETRY

3. SIMULTANEOUS LINEAR EQUATIONS

4.SETS

5.MATHEMATICAL REASONING

6.THE STRAIGHT LINE

7. LINES AND PLANES IN 3-DIMENSIONS

SECTION B

1.STATISTICS


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1. LENGTH OF ARC AND AREA OF SECTOR

1. In the diagram below, Ois the centre of the arc of the circle MNPQ and RSM is a

quadrant with centre P. MOP is a straight line.

Using7

22= , calculate

a) the perimeter of the whole diagram,b) the area of the shaded region. [ 6

marks]

Answer :

a) Perimeter of the whole diagram

= 1477

222

360

18014

7

222

360

90++

= 58 cm

b) Area of the shaded region =2 290 22 60 2214 - 7

360 7 360 7

=3

1128 cm2

240o

R P

S

M

N

Q

O

14

cm


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Exercise

1. In Diagram 1,JKL is arc of circle with centre M. NML is a straight line andJN= NM = 7 cm.

Using7

22= , calculate

a. the area of the shaded region, in cm2,

b. perimeter of the whole diagram, in cm.[6 marks]

2. In Diagram 2, Ois the centre of the arc of the circle PQR and a quadrant STU.OSR is a straight line.

Using

7

22= , calculate

a. perimeter of the whole shaded region,b. area of the whole shaded region.

[6 marks]

DIAGRAM 1

DIAGRAM 2

N M L

J

K

U

R

P

Q

S

T

O

14 cm

7 cm

45o


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3. In Diagram 3, OAB, OCD and OEFare three sectors with same centre O.

GivenAOF, OCB and ODE are straight lines. Using7

22= , calculate

a. the area of sector OCD,b. the perimeter of the whole diagram.

[6 marks ]

4. Diagram 4 shows three quadrants OPQ, TQR and URS. POUS is a straight lineand TOUR is a square.

Using7

22= , calculate

a) the perimeter of the whole diagram,

b) the area of the whole diagram.

[6 marks ]

5. In Diagram 5, QR and TU are two arc of circles with the same centre O.QPOU and RSTO are straight lines.

DIAGRAM 3

DIAGRAM 5

QP O

S

R

T

U7 cm

21 cm

4040

7 cm

OF

E

DC

A

B

DIAGRAM 4

P

Q

R

S

T

UO14 cm


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Using7

22= , calculate

a) ROQ ,

b) area of the shaded sector OTU,

c) perimeter of the whole diagram.

[6 marks]

2.VOLUME OF SOLIDS

DIAGRAM 1

1. Diagram 1 shows the tip of a cone touches the top of the cuboid and the base

rests on the base of the cuboid. If the cone is taken out of the solid.

Calculate the volume, in cm3, of the remaining solid. Use7

22= .

5 cm

5 cm

10 cm

10

7


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DIAGRAM 2

2 Diagram 2 shows a hemisphere resting on top of a cylinder, both having

bases of identical area. The height of cylinder is 10 cm and the diameter of the

cylinder is 7 cm. Find the volume, in cm3, of the composite object. Use7

22= .

Diagram 3

3 Diagram 3 shows a solid formed by combining a right prism with a half

cylinder on the rectangular ABCD. BF= CE = 10 cm , FG = EH = 8 cm and BC

= 13cm.

Calculate the volume,in cm 3 , of the solid.[use7

22= ]

13 cm

5 cm.


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Diagram 4

4 Diagram 4 shows a solid formed by combining a cone with ahemisphere.Find the volume of the composite in cm3.

[use7

22= ] .

5 Diagram 5 shows a solid cuboid . A cylinder with radius 4 cm and height 7cm is taken out of the solid. Calculate the volume, in cm3 , of the remaining solid.

[use7

22= ] .

3.SIMULTANEOUS LINEAR EQUATIONS

1 Calculate the values of rand s that satisfy the following simultaneous linear equations :

5r+ 2s = 20 and 2r 3s = 11 (Ans : r= 2,

s = 5)

10

cm

15

cm

12

cm

Diagra

m 5


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2 Calculate the value of v and of w that satisfy the following simultaneous linear equations :

2v+ 3w = 8 and 3v+ w = 10 (Ans :

v= 2, w = 4)

3 Calculate the values of m and u that satisfy the following simultaneous linear equations :

m + 4u = 2

m4

1+ 7u = 1 (Ans : m =

3, u =41 )

4 Calculate the value of m and of n that satisfy the following simultaneous linear equations :

2m n3

1= 5

3m n = 9 (Ans : m =

2, n = 3)


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4.SETS

1. The Venn diagram in the answer space shows sets A, B and C. Given that

=A B C. On the diagram provided in the answer spaces, shade the area of ,

(a) C (A B)

(b) (A C) (B C)

Answer :

2.

9

8-x

7-x

89-x7

Given that CBA = and that 34)( =n , find

(a) the value ofx

(b) )'( CBAn

3. The Venn diagrams below show set P, Q and R. Given that the universal set,RQP = . On the diagrams, shade

(a)QP (b) 'RP

AB

x

C

PQ

R

PQ

R


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(c) '' RQP

4. Given the universal set, xxx

,21 0:{=

is an integer}

Set P = {x : x is a prime number}Set Q = {x : x is a multiple of 4} andSet R = {x: x is a number where one of its digits is more than 7}

(a) Find the elements of set P(b) Find the elements of set P R.(c) Find n( RQP ).

5. MATHEMATICAL REASONING

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

ii : Complete the premise in the following argument.

Premise 1 : If JKL is an equilateral triangle, then the value of its interior angle

is 60o

Premise 2 : ______________________________________________________

Conclusion : The value of the interior angle of JKL is 60o.

iii : Write down two implications based on the following sentence.

PQ

R

9 > 6 and 42 = 8


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Implication 1 :

.

Implication II :

.

2) i : Is the sentence below a statement or a non-statement ?

ii : Write down two implications based on the following sentence.

Implication 1 :

.

Implication II :

iii : Based on the information below, make a general conclusion by inductionregarding the sum of

the interior angles of a triangle.

General conclusion :

3. a) Determine whether the following statement is true or false.

b) Write two implications from the statement given below.

x>yif and only if x y > 0

5 is an even number

The sum of the interior angles of triangle ABC =

180o

The sum of the interior angles of triangle JKL =

180o

The sum of the interior angles of triangle PQR =

180o

PQR is a right-angled triangle if and only if PR2 = PQ2 +

QR2

34 = 12 or4

5= 1.25


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Implication I:

Implication II:

c) Complete the premise in the following argument.

Premise 1 : If 2y= 10, theny= 5.

Premise 2 : ..

Conclusion : 2y 10.

4. a) Complete the conclusion in the following argument.

Premise 1 : All regular hexagons have 6 equal sides.

Premise 2 : ABCDEF is a regular hexagon.

Conclusion :

b) Make a conclusion by induction for a list of numbers 9,29, 57, 93,that

follow the patterns

below :

9 = 4(2)2 7

29 = 4(3)2 7

57 = 4(4)2 7

93 = 4(5)2 7

c) Combine the two statements given below to form a true statement.

i) 15 ( 5) = 5

x= 4 if and only ifx3 = 64


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ii) 32 is a multiple of 8.

6. THE STRAIGHT LINE

1.In the diagram, PQRS is a parallelogram.

y

O

S

The gradient of RS is3

1. Find

(a) they-intercept of the straight line RS, (Ans : 3)

(b) the equation of the straight line PQ. (Ans : y= x31

2)

[5marks]

2. In the diagram, the straight line PQ is parallel to the straight line ST. O is the origin.

y

Q (

Given the gradient of RS is 2. Find

(a) the equation of the straight line PQ, (Ans : y= 2x+ 14)


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(b) thex-intercept of the straight line RS. (Ans : 1)

[5 marks]

3. In the diagram, OPQR is a parallelogram and O is the origin.

y

O

P (3, 6) Q

R (5,

Find

(a) the gradient of the straight line PQ, (Ans :5

3)

(b) the equation of the straight line QR. (Ans : y= 2x7)

[5 marks]

4. In the diagram, OPQR is a parallelogram and O is the origin.

R

y

O

Find

(a) the equation of the straight line PQ, (Ans : y= 3x15)

(b) they-intercept of the straight line QR. (Ans : 20)


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[5 marks]

7. LINES AND PLANES IN 3-DIMENSIONS

1.The diagram shows a cuboid with a horizontal rectangular base PQRS.

U

P

T

S

Calculate the angle between the plane PQW and the base PQRS. (Ans : 41

38 )

2.The diagram shows a cuboid resting on a horizontal plane PQRS. M is the

midpoint of PS

A

Given PS = 16 cm, SR = 10 cm, and CR = 6 cm. Calculate the angle

between the plane CMR and the plane CDSR.

(Ans : 3840 )

3.The diagram shows a right prism. Right angled triangle PQR is the uniform cross section of the prism.

T

U

512


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Calculate the angle between the plane RTU and the base PQTU. (Ans :

3341 )

[4 marks]

4.The diagram shows a right prism. The base PQRS is a horizontal rectangle. Right angled triangle QRU is the

uniform cross section of the prism. V is the midpoint of PS.

U

T

S

P

Q

R

12 cm16 cm

5 cm

Identify and calculate the angle between the line UV and the plane RSTU. (Ans :

31.61)

5. The diagram shows a right prism with a right-angled triangle BFC as its uniform cross section. P is the

midpoint of DC and DC = 10 cm.

E

Calculate the angle between the line BP and the base CDEF. (Ans : 212 )

6. The diagram shows a right prism with an equilateral triangle PRS as its uniform cross section. M and N arethe midpoints of RS and WT respectively.


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P

RSM

W

If RM = 5 cm and ST = 12 cm, calculate the angle between the line PT and the base

RSTW. (Ans : 3340 )

SECTION B

STATISTICS

1.The data shows the ages, in years, of 30 workers in a carpenter factory.

21 38 25 39 31 23 29 40 47 28

48 29 34 45 26 20 36 33 31 20

38 34 24 26 28 31 43 27 32 25

(a) State the range of the data. (Ans : 28)

[1 mark]

(b) Based on the data above and by using a class interval of 5 years,

complete the table in the answer space.

[4 marks]

Age (years) Frequency Mid-point Upper

20 24

25 29

(c) Based on the table in (b),

(i) state the modal class, (Ans : 25 29)


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(ii) calculate the estimated mean ages of the workers and given your

answer correct to 3 decimal places. (Ans : 31.667)

[3 marks]

(d) For this part of the question, use the graph paper provided on the next

page.

By using a scale of 2 cm to 5 years on thex-axis and 2 cm to one worker

on they-axis, draw a histogram for the data.

[4 marks]

2. The data shows the distribution of heights, in cm, of 40 students in a class.

17

8

17

6

15

9

17

1

160 16

6

16

4

17

1

17

4

154

17

4

15

4

17

7

17

9

158 16

8

16

7

17

4

16

9

164

17

2

16

2

17

5

15

3

167 16

7

15

5

16

8

17

3

169

17

3

15

1

17

6

15

6

178 15

2

16

3

16

0

17

2

154

(a) Using data above, complete the table in the answer space based on the

class interval of the same size.

[4 marks]

Height (cm) Frequency, f Mid-point,x

145 149

(b) Based on the table in (a),


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(i) state the modal class, (Ans : 170 174)

(ii) calculate the mean height of the students in the class and give

your answer correct to two decimal palces. (Ans : 165.88)

[4 marks]

(c) For this part of the question, use the graph paper provided on the next

page.

By using a scale of 2 cm to 5 cm on the horizontal axis and 2 cm to one

student on the

vertical axis, draw a frequency polygon for the data.

[4 marks]

3. The table shows the distribution of heights, in cm, of 92 students in a school.

Height (cm) Mid-point Frequency

120 - 124 4

125 - 129 10

130 - 134 26

135 - 139 24

140 - 144 17

145 - 149 7150 - 154 3

155 - 159 1

(a) (i) Based on the table above, complete the table (a) in the answer

space.

(ii) Hence, calculate the mean height of the students. (Ans :

136.239) [3 marks]

(b)

Upper

Boundrary

119.

5

159.

5

Cumulative

Frequency


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(i) Based on the table in (a), complete the table in (b) in the answer

space.

(ii) For this part of the question, use the graph paper provided on the

next page.

By using a scale of 2 cm to 5 cm on the x-axis and 2 cm to 10

students on they-axis, draw an ogive for the data.

(iii) From the ogive, find the interquartile range for the data. (Ans : 9)

(iv) The students whose height is above 152 cm are chosen as basketball

player. Find the number of students who are chosen. (Ans : 2)

[9 marks]

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40369470 Promej Matematik Spm 2011