PROMEJ - MATEMATIK SPM 2011

A+ Mathematics SPM 2011

SMK. KOTA KLIAS, BEAUFORTSABAH

PROMEJ - MATEMATIK

SPM 2011

FASA I

DISUSUN OLEH: MOHD NAZAN B. KAMARUL ZAMAN


CONTENTS

SECTION A

1. LENGTH OF ARC AND AREA OF SECTOR2. VOLUME SOLID GEOMETRY3. SIMULTANEOUS LINEAR EQUATIONS4. SETS5. MATHEMATICAL REASONING6. THE STRAIGHT LINE7. LINES AND PLANES IN 3-DIMENSIONS

SECTION B

1. STATISTICS


1. LENGTH OF ARC AND AREA OF SECTOR

1. In the diagram below, O is the centre of the arc of the circle MNPQ and RSM is a quadrant with centre P. MOP is a straight line.

Using , calculate

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

Answer :

a) Perimeter of the whole diagram

=

= 58 cm

b) Area of the shaded region =

= cm2

240o

R P

S

M

N

Q

O

14 cm


Exercise

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

Using , calculate

a. the area of the shaded region, in cm2,b. perimeter of the whole diagram, in cm.

[6 marks]

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

Using , 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


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

Given AOF, OCB and ODE are straight lines. Using , 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 line and TOUR is a square.

Using , 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

QP O

S

R

T

U7 cm

21 cm

4040

7 cm

O F

E

DC

A

B

DIAGRAM 4

P

Q

R

S

T

UO 14 cm


Using , calculate

a) ,

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. Use .

DIAGRAM 5

5 cm

5 cm

10 cm

10

7


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. Use .

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 , of the solid. [use ]

13 cm

5 cm.


Diagram 4

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

[use ] .

5 Diagram 5 shows a solid cuboid . A cylinder with radius 4 cm and height 7 cm is taken out of

the solid. Calculate the volume, in cm , of the remaining solid.[use ] .

3.SIMULTANEOUS LINEAR EQUATIONS

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

10 cm

15 cm

12 cm

Diagram 5


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

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

+ 7u = 1 (Ans : m = 3, u = )

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

2m = 5

3m n = 9 (Ans : m = 2, n = 3)

4.SETS


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

ξ = 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-x7-x

89-x7

Given that and that , find

(a) the value of x

(b)

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

(a)(b)

AB

x

C

PQ

R

PQ

R


(c)

4. Given the universal set, is an integer}Set P = {x : x is a prime number}Set Q = {x : x is a multiple of 4} and

Set 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( )’.

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

x > y if and only if x – y > 0


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 induction regarding 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.

Implication I: ……………………………………………………………………………………………

Implication II: ……………………………………………………………………………………………

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 or = 1.25

x = 4 if and only if x3 = 64


c) Complete the premise in the following argument.

Premise 1 : If 2y = 10, then y = 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

ii) 32 is a multiple of 8.

6. THE STRAIGHT LINE


1.In the diagram, PQRS is a parallelogram.

R (6, 5)

Q

P

y

x O

S

The gradient of RS is . Find

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

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

[5marks]

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

R ( 5, 8)

P

y

x O

S

Q (3, 8)

Given the gradient of RS is 2. Find

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

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

[5 marks]

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


y

x O

P (3, 6) Q

R (5, 3)

Find

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

(b) the equation of the straight line QR. (Ans : y = 2x 7)

[5 marks]

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

Q

R (4, 12)

x

y

O

P (3, 6)

Find

(a) the equation of the straight line PQ, (Ans : y = 3x 15)

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

[5 marks]

7. LINES AND PLANES IN 3-DIMENSIONS

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


U

R

P Q

V

W T

12 cm

9 cm

8 cm 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

M P S

Q R

A D

C B

Given PS = 16 cm, SR = 10 cm, and CR = 6 cm. Calculate the angle between the plane CMR and the plane CDSR.

(Ans : 38 40)

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

T S

U

P R

Q

5 cm 12 cm

18 cm

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

V

P

Q

R

12 cm 16 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.

C D

F

A B

E

P

5 cm

12 cm

Calculate the angle between the line BP and the base CDEF. (Ans : 21 2)

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

N

P

Q

R S M

T 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 Boundary

20 – 24

25 – 29

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

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

(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 the x-axis and 2 cm to one worker on the y-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.

178 176 159 171 160 166 164 171 174 154


174 154 177 179 158 168 167 174 169 164

172 162 175 153 167 167 155 168 173 169

173 151 176 156 178 152 163 160 172 154

(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),

(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 Frequency120 - 124 4


125 - 129 10130 - 134 26135 - 139 24140 - 144 17145 - 149 7150 - 154 3155 - 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

(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 the y-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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PROMEJ - MATEMATIK SPM 2011