final2006sbpp2

ppr maths nbk

1449/2 Matematik Kertas 2 Okt 2006 2 jam

SEKTOR SEKOLAH BERASRAMA PENUH BAHAGIAN SEKOLAH

KEMENTERIAN PELAJARAN MALAYSIA

PEPERIKSAAN AKHIR TAHUN TINGKATAN 4 2006

Kertas soalan ini mengandungi 22 halaman bercetak 1449/2 @ 2006 Hak Cipta SBP

MATEMATIK Kertas 2

Dua jam tiga puluh minit

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU

Pemeriksa

Bahagian Soalan Markah Penuh Markah

Diperoleh 1 4

2 4

3 4

4 5

5 5

6 4

7 7

8 5

9 3

10 5

A

11 6

12 12

13 12

14 12

15 12

B

16 12

1. Kertas soalan ini mengandungi dua bahagian : Bahagian A dan Bahagian B.

2. Jawab semua soalan Bahagian A dan empat soalan daripada Bahagian B.

3. Jawapan hendaklah ditulis dengan jelas dalam ruang yang disediakan dalam kertas soalan.

4. Tunjukkan langkah-langkah penting dalam kerja mengira anda. Ini boleh membantu anda untuk mendapatkan markah.

5. Rajah yang mengiringi soalan tidak dilukis mengikut skala kecuali dinyatakan.

6. Satu senarai rumus disediakan di halaman 2 dan 3.

7. Anda dibenarkan menggunakan kalkulator saintifik yang tidak boleh diprogram.

Jumlah

NAMA : TINGKATAN :

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

RELATIONS 1 am an = am + n 2 am an = am - n 3 (am)n = am n

4 A-1 =

ac

bdbcad

1

5 P(A) = ( )( )SnAn

6 P(A) = 1 P(A)

7 Distance = ( )212212 )( yyxx + 8 Midpoint

(x, y) =

++2

,2

2121 yyxx

9 Average speed = 10 Mean = 11 Mean = 12 Pythagoras Theorem

c2 = a2 + b2

13 12

12xx

yym

=

14 erceptintxerceptintym

=

distance traveled time taken

sum of data number of data

Sum of (midpoint of interval frequency) Sum of frequencies

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SHAPE AND SPACE

1 Area of trapezium = 21 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 = 31 r2h

9 Volume of sphere = 34 r3

10 Volume of right pyramid = 31 base area height

11 Sum of interior angles of a polygon = (n 2) 180 12 13

14 Scale factor, k = PAPA'

15 Area of image = k2 area of object.

arc length angle subtended at centre

circumference of circle 360

area of sector angle subtended at centre

area of circle 360

=

=

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Section A

[52 marks]

Answer all questions in this section.

1 Solve the quadratic equation 21

352 2 =

pp .

[4 marks] Answer: 2 Calculate the value of p and of q that satisfy the following simultaneous linear

equations: 3p 4q = 2

13221 =+ qp

[4 marks] Answer:

For Examiners

Use

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3 Diagram 1 shows a right prism. Right angled triangle PQR is the uniform cross-

section of the prism.

DIAGRAM 1 (a) Name the angle between the plane STP and the plane STQR, (b) Calculate the angle between the plane STP and the plane STQR.

[4 marks]

Answer: (a) (b)

For Examiners

Use

[Lihat sebelah

P

Q R

S T

U

9 cm18 cm

12 cm

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4 A straight line ST is parallel to line y = 2x + 5 and passes through point (4, 2). Find

(a) the gradient of the straight line ST, (b) the equation of the straight line ST and hence, state its y-intercept.

. [5 marks]

Answer: (a) (b)

5 A bag contains 50 pens. 15 of them are blue and the rest are red and green. A pen is chosen at random from the bag.

(a) Find the probability that a blue pen is chosen.

(b) The probability of choosing a green pen is 51 . How many green pens

are there in the bag?

(c) Another 10 green pens are put into the bag. Find the probability that a green pen is chosen from the bag.

[5 marks] Answer:

(a) (b) (c)

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6 Diagram 2(a) shows a container in the form of a right pyramid fully filled with water. Diagram 2(b) shows an empty cylindrical container. The height of the pyramid is 15 cm.

DIAGRAM 2(a) DIAGRAM 2(b) All the water from the right pyramid container is poured into the cylindrical

container.

By using = 722 , calculate the height, in cm, of the water level in the cylinder.

[4 marks]

Answer:

For Examiners

Use

[Lihat sebelah

14 cm

A B

C D

E

12 cm

8 cm

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7 In Diagram 3, OPQS is a quadrant with the centre O and OSQR is a semicircle with the centre S.

DIAGRAM 3

Given that OP = 14 cm. Using = 722 , calculate

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

[6 marks] Answer: (a) (b)

O P

Q

R S 60

T

For Examiners

Use

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8 (a) Determine whether the following is a statement or not. Give a reason to your answer. (b) Rewrite the following statement by inserting the word not into the original statement. State the truth value of your new statement. (c) Construct a true statement using a suitable quantifier for the given object and the property.

[5 marks] Answer:

(a) ..

(b) ..

(c) .

For Examiners

Use

[Lihat sebelah

64 = 42

3 is the factor of 15

Object : Triangles Property : Have a right angle

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9 The Venn diagram in the answer space shows set P, set Q and set R with the universal set = P Q R.

On the diagrams provided in the answer space, shade (a) the set Q (P R), (b) the set Q R P.

. [3 marks] Answer:

(a) (b)

10 In Diagram 4 , AE is a tangent to the circle centre O, at A. CDE is a straight line.

DIAGRAM 4 Find the value of

(a) x (b) y (c) z

[5 marks]

Answer: (a) (b) (c)

For Examiners

Use

P

R Q

A

20

B

C D

70

z x

y O E

P

R Q

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11 (a) Diagram 5 shows a unit circle.

DIAGRAM 5

Find the value of (i) sin v (ii) cos v + sin 270

[3 marks]

(b) Diagram 6 shows a rhombus PQRS.

DIAGRAM 6

It is given that TSQ is a straight line and tan x = 125 .

Find (i) the length of RP (ii) sin x

[3 marks] Answer:

(a) (i) (ii)

(b) (i)

(ii)

For Examiners

Use

[Lihat sebelah

1 1

1

1

y

x v

( 0.42, 0.91)

O

P

Q

R

S T

x 26 cm

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Section B

[48 marks]

Answer four questions in this section. 12 (a) In Diagram 7, a straight line RS is parallel to the straight line PQ. The

equation of PQ is 2y = x 4.

DIAGRAM 7

Find (i) the y-intercept of PQ, (ii) the value of m, (iii) the equation of RS.

[6 marks]

Answer: (a) (i) (ii) (iii)

For Examiners

Use

y

x O

R Q(8, m)

P

S(4, 5)

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(b) In Diagram 8, EFGH is a parallelogram and O is the origin. The gradient of EF is 2.

DIAGRAM 8

Find (i) the equation of line EF, (ii) the coordinates of H, (iii) the gradient of HF.

[6 marks]

Answer: (b) (i) (ii) (iii)

[Lihat sebelah

For Examiners

Use

H

x O G

E

F(4, 10) y

12

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13 (a) Diagram 9 shows a right prism with a horizontal rectangular base PQRS.

DIAGRAM 9

Given that E and F are midpoints of WV and SR respectively.

(i) Find the length of PF, (ii) Calculate the angle between the line PE and the plane PQRS, (iii) Name the angle between the plane PQVE and the plane PQUT.

[6 marks]

Answer: (a) (i) (ii) (iii)

For Examiners

Use

8 cm

5 cm 7 cm

12 cm

P

Q R

U

W

T

E

F

S

V

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(b) In Diagram 10, MP and LQ are two vertical flagpoles standing on horizontal ground KLM .

DIAGRAM 10 Given that the angle of depression of P from Q is 38, MP = 5 m, LQ = h m and KM = 10 m. Calculate

(i) the angle of elevation of P from K, (ii) the value of h.

[6 marks] Answer:

(b) (i) (ii)

For Examiners

Use

400

5 m

P

70 M

Q

h m

K

L

13

[Lihat sebelah

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14 (a) (i) Fill the blanks with and or or in order to form a true statement.

(a) sin 30 = 21 cos 30 =

21

(b) All the multiples of 3 are multiples of 6 all multiples of 6 are multiples of 3.

(ii) Complete the argument below. Premise 1 : If a A, then a A B. Premise 2 : __________________________________ Conclusion : 5 A

(iii) Make a general conclusion by induction for the sequence 13, 28, 49, 76, which follows the following pattern: 13 = 3 (2)2 + 1 28 = 3 (3)2 + 1 49 = 3 (4)2 + 1 76 = 3 (5)2 + 1

[6 marks] Answer:

(a) (i) (a) . (b) .

(ii) ..........................................................................................................

(iii) ..........................................................................................................

For Examiners

Use

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(b) Diagram 11 is an incomplete Venn diagram. A group of 45 students involved in their favourite games.

DIAGRAM 11

Given that = B S T and the number of students who involve in badminton only is the same as the students who involve in tennis only.

Find the number of students who involve in (i) all the three games, (ii) neither badminton nor tennis, (iii) two games only. (iv) tennis only.

[6 marks]

Answer:

(b) (i) (ii) (iii) (iv)

For Examiners

Use

[Lihat sebelah

B = Badminton 2

5 3

6

11

T = Tennis

S = Squasy

14

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15 The daily distances, in km, travelled in 40 days by a salesman are shown in Diagram 12.

DIAGRAM 12

(a) Using data in Diagram 12 and a class interval of 10 km, complete Table 1 in the answer space.

[4 marks] Answer: (a)

TABLE 1

(b) Determine the range of the given data. [1 mark] (c) Based on your table in (a), calculate the estimated mean distance travelled.

[3 marks] Answer: (b) (c) (i)

(ii)

(d) For this part of the question, use the graph paper on page 19. By using a scale of 2 cm to 10 km on the x-axis and a scale of 2 cm to 1 day on the y-axis, draw a frequency polygon for the data above.

[4 marks] Answer:

(d) Refer graph on page 19.

42 62 51 37 70 14 12 93 53 45 54 27 70 76 91 72 68 58 57 65 21 18 22 87 64 63 57 59 70 19 41 55 36 56 50 57 73 75 13 39

Distance (km) Midpoint Frequency

11 20

21 30

31 40

For Examiners

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Graph for Question 15(d)

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16 The histogram in Diagram 13 shows the marks achieved in a test by a group of students in a particular school.

DIAGRAM 13

(a) Find the total number of students who sat for the test. [1 mark] (b) Find the modal class. [1 mark] (c) Based on the histogram, complete the Table 3 below. [3 marks] Answer: (a)

(b)

(c) Marks Upper boundary Cumulative

frequency 20 29 29.5 0 30 39 39.5 4

TABLE 3 (d) For this part of the question, use the graph paper on page 22.

By using the scale of 2 cm to 10 marks on the x-axis, and 2 cm to 10 students on the y-axis, draw an ogive for the data.

[4 marks] (e) From the ogive, find (i) the median, (ii) the number of students who passed the test if the passing mark is 45%. [3 marks]

For Examiners

Use

29.5 39.5 49.5 59.5 69.5 79.5 89.5 99.5

4

8

12

16

20

Marks

Frequency

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Answer: (d) Refer graph on page 22. (e) (i) (ii)

For Examiners

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Graph for Question 16(d)

END OF QUESTION PAPER