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SULIT 1

3472/1 Hak Cipta Panitia Matematik Tambahan SMK Ti nggi Kajang 2011 [Lihat halaman sebelah

SULIT

UNIT PEPERIKSAAN

SEKOLAH MENENGAH KEBANGSAAN TINGGI KAJANG

PEPERIKSAAN PERTENGAHAN TAHUN 2011

TINGKATAN 5

Kertas soalan ini mengandungi 13 halaman bercetak

For examiner’s use only

Question Total MarksMarks

Obtained

1 32 4

3 3

4 3

5 3

6 3

7 3

8 3

9 3

10 311 4

12 3

13 4

14 3

15 3

16 4

17 3

18 3

19 3

20 3

21 4

22 3

23 3

24 3

25 4

TOTAL 80

MATEMATIK TAMBAHAN

Kertas 1

Dua jam

JANGAN BUKA KERTAS SOALAN INI

SEHINGGA DIBERITAHU

1 This question paper consists of 25 questions.

2. Answer all questions.

3. Give only one answer for each question.

4. Write your answers clearly in the spaces provided in

the question paper.

5. Show your working. It may help you to get marks.

6. If you wish to change your answer, cross out the work

that you have done. Then write down the new

answer.

7. The diagrams in the questions provided are not

drawn to scale unless stated.

8. The marks allocated for each question and sub-part

of a question are shown in brackets.

9. A list of formulae is provided on pages 2 to 3.

10. A booklet of four-figure mathematical tables is provided.

.11 You may use a non-programmable scientific

calculator.

12 This question paper must be handed in at the end of

the examination .

Name : ………………..……………

Form : ………………………..……

3472/1

Matematik Tambahan

Kertas 1

16 Mei 2011

2 Jam

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SULIT 3472/1

3472/1 Hak Cipta Panitia Matematik Tambahan SMK Tinggi Kajang 2011 SULIT

2

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

ones commonly used.

ALGEBRA

1 x =a

acbb2

42

2 am an = a m + n

3 am an = a m n

4 (am) n = a nm

5 log a mn = log a m + log a n

6 log a n

m = log a m log a n

7 log a mn = n log a m

8 log a b =ab

c

c

loglog

9 T n = a + (n 1)d

10 Sn = ])1(2[2

d nan

11 T n = ar n 1

12 Sn =r

r a

r

r a nn

1

)1(

1

)1( , (r 1)

13r

aS

1 , r < 1

CALCULUS

1 y = uv ,dx

duv

dx

dvu

dx

dy

2v

u y ,

2

du dvv u

dy dx dx

dx v

,

3dx

du

du

dy

dx

dy

4 Area under a curve

= b

a

y dx or

= b

a

x dy

5 Volume generated

= b

a

y 2 dx or

= b

a

x 2 dy

5 A point dividing a segment of a line

( x, y) = ,21

nm

mxnx

nm

myny 21

6. Area of triangle =

1 2 2 3 3 1 2 1 3 2 1 3

1( ) ( )

2

x y x y x y x y x y x y

1 Distance = 2 22 1 12( ) ( ) x x y y

2 Midpoint

( x , y) =

2

21 x x ,

2

21 y y

3 22 y xr

42 2

x i yjr

x y

GEOMETRY

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SULIT 3472/1

3472/1 Hak Cipta Panitia M atematik Tambahan SMK Tin ggi Kajang 2011 Lihat halaman sebelah

SULIT

3

STATISTICS

TRIGONOMETRY

1 Arc length, s = r

2 Area of sector , A = 21

2r

3 sin 2 A + cos 2 A = 1

4 sec2 A = 1 + tan2 A

5 cosec2 A = 1 + cot2 A

6 sin 2 A = 2 sin Acos A

7 cos 2 A = cos2 A – sin2 A

= 2 cos2 A 1

= 1 2 sin2 A

8 tan2 A = A

A2tan1

tan2

9 sin ( A B) = sin Acos B cos Asin B

10 cos ( A B) = cos Acos B sin Asin B

11 tan ( A B) = B A

B A

tantan1

tantan

12C

c

B

b

A

a

sinsinsin

13 a2 = b2 +c2 2bc cos A

14 Area of triangle = C absin2

1

1 x = N

x

2 x =

f

fx

3 =

2( ) x x

N

=

22 x

x N

4 =

2( ) f x x

f

=

22 fx

x f

5 m = C f

F N

Lm

2

1

6 1000

1 P

P I

7 i

i

iw I I

w

8)!(

!r n

n P r

n

9 !)!(

!

r r n

nC r

n

10 P ( A B) = P ( A) + P ( B) P ( A B)

11 P ( X = r ) = r nr

r

nq pC , p + q = 1

12 Mean , = np

13 npq

14 z =

x

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3472/1 Hak Cipta Panitia Matematik Tambahan SMK Tinggi Kajang 2011 [Lihat halaman sebelah

SULIT

Answer all questions.

1 Diagram 1 shows a function f with its objects in x and its images in f(x). Rajah 1 menunjukkan satu fungsi f dengan objek-objek dalam x dan imej-imejnya dalam f(x).

(a) State the value of p . Nyatakan nilai bagi p.

(b) State the type of relation of the function. Nyatakan jenis hubungan bagi fungsi itu

(b) Using the function notation, express f in terms of x.

Menggunakan tata tanda fungsi, ungkapkan f dalam sbutan x. [ 3 marks ] [ 3 markah ]

2. Two functions are defined by f : x 3x + 4 and2: 3 1 g x x x , find

Dua fungsi ditakrifkan sebagai f : x 3x + 4 dan g : x x2 + 3x + 1, cari

(a) f

-1

(-5)

(b) fg(x) [ 4 marks ][ 4 markah ]

4

2

For

examiner’s

use only

1

9

p

1

3

5

−3

3

1

x f(x)

DIAGRAM 1 Rajah 1

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3472/1 Hak Cipta Panitia M atematik Tambahan SMK Tinggi Kajang 2011 [Lihat halaman sebelah

SULIT

8

x3

3. Function g is defined as8

x3 x: g and the composite function

x35 x: fg , find f (x ) . [ 3 marks ]

Fungsi g ditakrif sebagai g : x dan fungsi gubahan fg : x 5 – 3x, cari f(x). [ 3 markah ]

4. Find the range of values of k if the quadratic equation (k + 2)x 2 − 8x + 2 = 0 has two

different roots. [ 3 marks ] Cari julat nilai k jika persamaan kuadratik (k + 2)x

2 – 8 + 2 = 0 mempunyai dua punca yang berberza.

[ 3 markah ]

5. Given that a and b are the roots of the quadratic equation 2x2 + 3x – 7 = 0. Form the

quadratic equation whose roots are 2a and 2b . [ 3 marks ] Diberi a dan b adalah punca-punca bagi persamaan kuadratik 2x

2 + 3x – 7 = 0. Bentuk persamaan

kuadratik yang mempunyai punca-punca 2a dan 2b. [ 3 markah ]

For

xaminer’s

use only

3

4

3

3

3

5

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6. Diagram 2 shows the graph of a quadratic function y = f(x). The straight line

y = 16 is a tangent to the curve y = f(x). Rajah 2 menunjukkan graf fungsi kuadratik y = f(x). Garis lurus y = 16 ialah tangen kepada

lengkung y = f(x).

y = 16

(a) Write the equation of the axis of symmetry of the curve.Tulis persamaan paksi simetri lengkung itu.

(b) Express f(x) in the form of −(x + p) 2

+ q , where p and q are constant. [ 3 marks ]Ungkapkan f(x) dalam bentuk −(x + p)

2 + q, yang mana p dan q adalah pemalar. [ 3 markah ]

7. A set of data comprising nine numbers has a mean of 12.Satu set yang terdiri daripada sembilan nombor mempunyai min 12.

(a) Find ∑ x, Cari ∑ x,

(b) When a number p is removed from the set of data, the mean of the remaining

numbers is 11. Find the value of p. [ 3 marks ] Apabila satu nombor p dikeluarkan daripada set data itu, min bagi nombor-nombor yang masih ada

ialah 11. Cari nilai p. [ 3 markah ]

3

6

3

7

DIAGRAM 2

Rajah 2

x

y

0

Forexaminer’s

use only

−2 6

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8. Solve the equation 273

9

x

1x

[ 3 marks ]

Selesaikan persamaan 27 x3

9 1 x

[ 3 markah ]

9. Show that 2x + 2

+ 5(2x ) – 2

x = 2

x + 3 [ 3 marks ]Tunjukkan 2

x + 2+ 5(2

x ) – 2

x = 2

x + 3[ 3 markah ]

10. Given that log 10 2 = p and log 10 7 = q , express log 2 56 in terms of p and q . [ 3 marks ]

Diberi log 10 2 = p dan log 10 7 = q, express log 2 56 dalam sebutan p dan q. [ 3 markah ]

3

9

3

8

For

xaminer’s

use only

3

10

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11. The diagram 3 shows a semicircle of centre O and radius r cm. Rajah 3 menunjukkan sebuah separa bulatan berpusat O dan berjejari r cm.

C

A O B

The length of the arc AC is 7 2 cm and the angle of COB is 2 692 radians. Calculate Panjang lengkuk AC ialah 7.2 dan sudut COB ialah 2.692 radian. Kira

(a) the value of r ,nilai r,

(b) the area of the shaded region.

luas kawasan berlore.

[Use π = 3.142] [ 4 marks ]

[guna π = 3.142] [ 4 marahs ]

12. The first three terms of an arithmetic progression are 24, 20 and 16 .

The nth

term of this progression is negative. Find the least value of n . [ 3 marks ] Tiga sebutan pertama bagi suatu janjang arimetik ialah 24, 20 dan 16. Sebutan ke-n janjang ini adalah

negatif. Cari nilai n yang terkecil. [ 3 markah ]

4

11

For

examiner’s

use only

3

12

DIAGRAM 3Rajah 3

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13. Thethn term of an arithmetic progression is given by .1n3T 2

n Find

Sebutan ke n bagi suatu janjang arimetik diberi oleh T n = 3n2 – 1. Cari

(a) the first term, sebutan pertama,

(b) the common difference,beza sepunya,

(c) the sum of the first 10 terms of the progression. [ 4 marks ]

hasil tambah 10 sebutan pertama janjang tersebut. [ 4 markah ]

14. The first three terms of a geometric progression are x, 10 and 4x , where x is

positive. Find the first term and the common ratio of the progression. [ 3 marks ]Tiga sebutan pertama suatu janjang geometri ialah x, 10 dan 4x, dengan x bernilai positif. Cari sebutan

pertama dan nisbah sepunya janjang tersebut. [ 3 markah ]

15. Given the first three terms of a geometric progression are 42, 21 and 10.5. Find thesum to infinity of the progression. [ 3 marks ]

Diberi tiga sebutan pertama suatu janjang geometri ialah 42, 21 dan 10.5. Cari hasil tambah hingga

ketakterhinggaan janjang itu. [ 3 markah ]

3

13

3

15

3

14

For

xaminer’s

use only

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16. Diagram 4 shows the straight line obtained by plotting y10log against log 10 x .

Rajah 4 menunjukkan garis lurus yang diperolehi dengan memplotkan log 10 y terhadap log 10 x

The variables x and y are related by the equation ,kx y 3 where k is a constant.

Find the value of

Pemboleh ubah x dan y dihubungkan oleh persamaan y = kx3. Cari nilai

(a) log 10 k ,

(b) p [ 3 marks ][ 3 markah ]

___________________________________________________________________________

17 The coordinates of the vertices of a triangle KLM are K (3, h ), L (1, 0) and

M (5,−h). If the area of the the triangle KLM is 12 units2

, find the valuesof h . [ 3 marks ]

Koordinat bagi bucu-bucu sebuah segitiga KLM ialah K (3, h), L (1, 0) dan M (5, −h). Jika luas segitiga itu ialah 12 unit

2 , cari nilai h. [ 3 markah ]

y10log

x10log

0

(0, 2)

(4, p)

3

16

For

examiner’s

use only

DIAGRAM 4 Rajah 4

3

17

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1

3

y

a

x

18. Given the straight line 13

y

a

x is perpendicular to the straight line

,05 y3 x6 find the value of a. [ 3 marks ]

Diberi garis lurus adalah berserenjang kepada garis lurus 6x + 3y – 5 = 0, cari nilai a.

[ 3 markah ]

19. A straight line passes point P(−3, 4) and point R(5, 0). Point Q(x, y) lies on the

straight line PR such that it divides the line segment PR with the ratio of PQ : QR =3 : 1. Find the coordinate of point Q. [ 3 marks ]Satu gar is lurus melalui titik P(−3, 4) dan titik R(5, 0). Titik Q(x, y) terletak di atas garis lurus PR dalam

keadaan ia membahagi tembereng garis PR dengan nisbah PR : QR = 3 : 1. Cari koordinat titik Q.[ 3 markah ]

20. Point S moves such that its distance from point T (5, −2) is always 6 units. Find theequation of the locus of S . [ 3 marks ]Titik S bergerak dengan keadaan jaraknya dari titik T(5, −2,) sentiasa 6 unit. Cari persamaan lokus

bagi S. [ 3 markah ]

3

19

4

20

3

18

For

xaminer’s

use only

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s r

r s

r

s r

s

21 Given that y = 5u 2

and u = 3x + 2. Find Diberi bahawa y = 5u

2 dan u = 3x + 2. Cari

(a)dx

dy in terms of x ,

dx

dy dalam sebutan x,

(b) the small change in y when x increases from 2 to 2 01. [ 4 marks ]tokokan kecil dalam y apabila x bertambah dari 2 kepada 2.01. [ 4 markah ]

22. Find the coordinates of the turning points of the curve y = x( 3x – 2) . [ 3 marks ]

Cari koordinat titik pertukaranbagi l engkuk y = x(3x – 2). [ 3 markah]

23. The vectors and are non-zero and non-parallel. It is given that

(m + 2n − 2) = (3m – 2n − 1) , where m and n are constants. Find the value

Vektor-vektor dan adalah bukan sifar dan tidak selari. Diberi bahawa

(m + 2n − 2) = (3m – 2n − 1) , dengan keadaan m dan n i alah pemalar. Car i n il ai

(a) m, [ 3 marks ]

(b) n. [ 3 markah ]

3

23

3

22

4

21

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j2i3 AB j3i4 BC

BD

24. Given the vectors 3a i mj , 8b i j and 5 2c i j . If vector a b is parallel

to vector~c , find the value of the constant m . [ 3 marks ]

Diberi vektor-vektor 3a i mj , 8b i j dan 5 2c i j . Jika vektor a b adalah selar i

kepada vektor

~c , cari nilai pemalar m. [ 3 markah ]

25. The diagram 5 shows a parallelogram ABCD drawn on a Cartesian plane. Rajah 5 menunjukkan suatu segiempat selari ABCD dalam satah Cartesian

It is given that 3 2 AB i j

and 4 3 BC i j

. Find

Diberi bahawa dan . Cari

(a) BD

,

(b) unit vector in the direction of BD

. [ 4 marks ]

unit vektor dalam arah . [ 4 markah ]

END OF QUESTION PAPERKertas Soalan Tamat

For

xaminer’s

use only

3

24

4

25

y

O

A

B

C

D

x

DIAGRAM 5 Rajah 5

kajang 2011 (p1).pdf

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