Spm Percubaan 2008 Sabah Mathematics Paper 2

SULIT

1449/1 NAMA KELAS NO K.P A. GILIRAN

SULIT 1449/2 MATHEMATICS

: _____________________ : _____________________ : _____________________ : _____________________

PAPER 2 AUGUST 2008 2 HOURS 30 MINUTES

JABATAN PELAJARAN NEGERI SABAH SIJIL PELAJARAN MALAYSIA TAHUN 2008

EXCEL 2 ________________________________________________________________________

MATHEMATICS (MATEMATIK) PAPER 2 (KERTAS 2) TWO HOURS THIRTY MINUTES (DUA JAM LIMA TIGA PULUH MINIT)

________________________________________________________________________ JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU

1. Tulis nombor kad pengenalan dan angka giliran anda pada ruangan yang disediakan. 2. Kertas soalan ini adalah Bahasa Ingerís. 3. Calon dikehendaki membaca maklumat di halaman 2. ________________________________________________________________________ This question paper consists of 27 printed pages. (Kertas soalan ini terdiri daripada 27 halaman bercetak.)

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2 INFORMATION FOR CANDIDATES

1. This question paper consists of two sections : Section A and Section B. . 2. Answer all questions in Section A and four questions from Section B. 3. Write your answers clearly in the space provided in the question paper. 4. Show your working. It may help you to get marks. 5. If you wish to change your answer, neatly cross out the answer that you have done. Then write down the new answer. . 6. The diagram in the questions provided are not drawn to scale unless stated. 7. The marks allocated for each question and sub-part of a question are shown in brackets. 8. A list of formulae is provided on pages 3 to 4 . 9. A booklet of four-figure mathematical tables is provided. 10. You may use a non-programmable scientific calculator. 11. This question paper must be handed in at the end of the examination.

The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used. RELATIONS 1 1449/2

a m ×a n =a m +n

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2

a m ÷a n =a m −n

3

( a m ) n = a mn

4

A−1 =

5

P( A) =

6

P ( A' ) = 1 −P ( A)

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−b   a  

d 1  ad − bc  − c n( A) n( S )

7

Distance =

8

Midpoint,

9

Average speed =

10

Mean =

sum of data number of data

11

Mean =

sum of (class mark  frequency) sum of frequencies

12

Pythagoras Theorem

( x1 −x 2 ) 2 +( y1 − y 2 ) 2

 x + x 2 y1 + y 2  ( x, y ) =  1 ,  2  2 

distance travelled time taken

c 2 =a 2 +b 2

y 2 − y1 x 2 − x1

13

m=

14

m =

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y  intercept x  intercept

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4 SHAPES AND SPACE 1  sum of parallel sides  height 2

1

Area of trapezium =

2

Circumference of circle =  d  2 r

3

Area of circle

4

Curved surface area of cylinder

5

Surface area of sphere = 4 r 2

6

Volume of right prism = cross sectional area

7

Volume of cylinder =  r 2 h

8

Volume of cone =

9

Volume of sphere =

10

Volume of right pyramid =

11

Sum of interior angles of a polygon =

12

arc length angle subtended at centre  circumference of circle 3600

13

area of sector angle subtended at centre  area of circle 3600

14

Scale factor,

15

Area of image =

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=  r2

k=

= 2 rh

×

length

1 2 r h 3 4 3 r 3 1 3

×

base area  height ( n −2) ×180 

PA' PA

k2 ×

area of object

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5

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SECTION A (52 MARKS) Answer all questions in this section. 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)

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6

2. By using factorization, solve the quadratic equation

5n  2  3n . n [4 marks]

Answer:

3. Calculate the value of x and the value of y which satisfy the following simultaneous equations. x – 2 y = 14 1 x  y  2 3 [4 marks] Answer:

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7

4. Diagram 1 shows a right prism with a horizontal rectangular base PQRS. PQVU is a uniform cross-section of the prism.

Diagram 1 Calculate the angle between the plane TQR and the plane PQRS. [3 marks] Answer:

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8

5. In diagram 2, PQ and SR are parallel lines. The equation of the straight line QR is 3y = x + 9. y

R(6, t) Q(-3,2)

P S

-4

x

Diagram 2 Find (a) the value of t, (b) the equation of the straight line RS. (c) the y-intercept of RS. [5 marks] Answer: (a)

(b)

(c)

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6. Diagram 3 shows two sectors OTS and ORQ with centre O. It is given that OQ = 21 cm and OT = 14 cm.

Diagram 3 22 , calculate 7 (a) the perimeter of the whole diagram, (b) the area, in cm2, of the shaded region. By using  

[6 marks] Answer : (a)

(b)

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7. (a) Combine the following statements with the word “and” or “or” to form a true statement. I : 3 is an odd number. 14  (2)  7 II : (b) Complete the conclusion in the following argument. Premise 1 : If  is less than 180o , then sin  is positive. Premise 2

:

 is 135o.

Conclusion

:

………………………………………………………..

(c) Identify the antecedent and the consequent of the implication below: “If A  B  900 , then A and B are complementary angles.” [5 marks] Answer: (a) …………………………………………………………………………................ …………………………………………………………………………………… (b) Conclusion

: ……. ……………………………………………………….. ………………………………………………………………

(c) Antecedent Consequent

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: ……………………………………………………………… : ………………………………………………………………

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 1 3  .  2 4

8. The matrix G = 

1  4 n  , find the value of m and of n. mn  2  2 1  (b) Hence, using matrices, calculate the value of p and of q that satisfy the following equation. (a) Given that the inverse matrix of G is

 p  6     q   2

G

[7 marks]

Answer: (a)

(b)

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9. Diagram 4 shows 10 labeled cards in two boxes.

A

4

B

F

I

9

Box P

G

8

J

K

Box Q Diagram 4 A card is picked at random from each of the boxes. By listing the outcomes, find the probability that (a) both cards are labeled with a number, (b) one card is labeled with a number and the other card is labeled with a consonant. [5 marks] Answer: (a)

(b)

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10. Diagram 5 shows the speed-time graph of a particle for a period of t seconds. Speed (ms-1) 22

10

3 0

4

16

t

Time (s)

Diagram 5 (a) State the length of time, in s, when the particle moves with uniform speed. (b) Calculate the rate of change of speed, in ms-2, in the first 4 seconds. (c) Calculate the value of t , if the total distance travelled in t seconds is 306 m. [6 marks] Answer: (a)

(b)

(c)

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11. Diagram 6 shows a solid cylinder with hemisphere PQR removed form one end. Both the cylinder and the hemisphere have the same radius of 3 cm.

Diagram 6 22 , calculate the length, t cm, of the cylinder if the volume of the 7 6 remaining solid is 282 cm3. 7 Using  

[4 marks] Answer:

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15 Section B [48 marks] Answer any four questions from this section.

12

Diagram 7 shows quadrilateral JIGH, EFGH and ABCD drawn on a Cartesian plane. y D 10 8 C6

J

H

A -6

B -4

-2

2

0

I

G

4

F

E

2

4

6

8

x

-2 Diagram 7 12

(a)

 3   . N is a reflection in the line x = 3. State the M is a translation    2 coordinates of the image of point H under each transformation below: (i) MN (ii) NM

(b)

[ 4 marks] EFGH is the image of JIGH under a transformation U. ABCD is the image of EFGH under a transformation V. Describe in full (i) (ii)

the transformation U the transformation V [ 5 marks]

(c)

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Given that the area of ABCD is 336 unit2, calculate the area, in unit2. of JIGH.

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Answer : (a) (i)

(ii)

(b)

(i)

U:

(ii) V:

(c)

13

(a)

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Complete Table 1 in the answer space for the equation y  4 x 2  5 x  7 by writing down the values of y when x = –1 and x = 2.5. [ 2 marks] For this part of the question, use the graph paper provided on page [Lihat sebelah SULIT

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18. You may 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 of y  4 x 2  5 x  7 for − 2.5 ≤ x ≤ 3. [ 4 marks] (c)

From your graph, find (i) the value of y when x = -1.5, (ii) the value of x when y = 22. [ 2 marks]

(d)

Draw a suitable straight line on your graph to find a value of x which satisfies the equation 4 x 2 + 2 x − 2 = 0 for − 2.5 ≤ x ≤ 3. State these values of x. [ 4 marks] Answer : (a)

x

–2.5

–2

y

30.5

19

–1

0

1

2

-7

-8

-1

2.5

3 14

TABLE 1 (b)

Refer graph on page 18.

(c)

(i)

y = ……………………………………..

(ii)

x = …………………………………….

(d)

x = ……………………………………………. Graph for Question 13 Graph for Question 13

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14 (a)

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The table 2 shows the masses, in kilogram, of waste paper collected by 40 students. 5

12

7

3

32

17

22

29

35

34

29

4

9

14

20

18

15

23

21

8

26

38

25

13

19

24

27

1

30

21

10

25

27

24

16

11

4

26

23

17

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TABLE 2 (a)

Based on the data above and by using a size class of 5, complete the table on page 20. [ 4 marks]

(b)

Based on the table 2 in (a), find the estimated mean for the mass of waste paper collected by each student. [ 3 marks]

(c)

For this part of the question, use the graph paper provided on page 21. 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 for the data.

(d)

[ 4 marks] Based on the histogram in (c), state one information related to the mass of waste paper collected by the students. [ 1 mark]

Answer : (a)

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Mass (g) Frequency Midpoint 1- 5 5 3

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(b)

(c)

Refer graph on page 21.

(d)

Graph for Question 14 Graph for Question 14

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22

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You are not allowed to use graph paper to answer this question. (a)

The diagram shows two prisms joined together at the vertical plane BMQG. The base AFGNKB is a rectangle lying on a horizontal table. ABCDE and BKLM are the uniform cross sections of the respective prisms. AE and BMC are vertical edges of the same length. EJID and IHCD are inclined planes. MLPQ is a horizontal plane. ED = CD and DI is 6cm vertically above the base of the solid. J

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I

H

E C

5cm

D

Q

P

F

7cm

G N M A

4cm

B

L

3cm

Draw to full scale, the plan of the solid.

6cm

K

[ 3 marks]

Answer : (a)

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(b)

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A triangular prism is joined to the solid in (a) at the horizontal plane PQML. The combined solid is as shown in the diagram below. LMR is the uniform cross section of the triangular prism. RM = RL and RS is 2 cm vertically above the plane PQML.

J 1449/2

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H

I

E C

5cm

D

S

Q F

7cm

R

P G N

M A

4cm

B

L

3cm

6cm Y

K

X

Draw to full scale, (i)

The elevation on a vertical plane parallel to ABK as viewed from X. [4 marks]

(ii)

The elevation on a vertical plane parallel to KN as viewed from Y.

[5 marks]

Answer : (b)(i)

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(ii)

16

A (54oS, 5oW), B (54oS, 25oE), C and D are four points on the earth’s surface. AC is a diameter of a parallel of latitude and D is due north of A. (a)

Find the longitude of C. [ 2 marks]

(b)

Calculate the shortest distance, in nautical miles, from A to C. [ 3 marks]

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parallel of latitude. [ 3 marks] (d)

A plane took off from A due north at a speed of 550 knots and took 6 hours to reach D. Find the latitude of D. [ 4 marks]

Answer : (a)

(b)

(c)

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(d)

END OF QUESTION PAPER

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