SULIT 3472

SEKOLAH SULTAN ALAM SHAH PUTRAJAYA

PEPERIKSAAN PERTENGAHAN TAHUN 2010

TINGKATAN 4

Kertas soalan ini mengandungi 13 halaman bercetak

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3472 SULIT

Untuk Kegunaan Pemeriksa

Soalan Markah

Penuh

Markah

Diperoleh

Bahagian A

Jawab semua soalan

1 3

2 4

3 4

4 3

5 3

6 3

7 4

8 3

9 3

10 3

11 3

12 4

Bahagian B

Jawab semua soalan

13 5

14 7

15 7

16 7

17 7

18 7

Bahagian C

Jawab 2 soalan

19 10

20 10

21 10

Jumlah 100

Additional Mathematics

Dua jam dan tiga puluh minit

JANGAN BUKA KERTAS SOALAN INI

SEHINGGA DIBERITAHU

1. Kertas soalan ini mengandungi

Bahagian A,B dan C.

2. Jawab semua soalan dalam Bahagian A

dan B serta dua soalan dalam Bahagian

C.

3. Tuliskan jawapan anda dalam Bahagian

A di dalam ruangan yang disediakan

di dalam kertas soalan dan Bahagain B

dan C di atas kertas jawapan.

4. Gambarajah di dalam soalan yang

disediakan tidak dilukis mengikut skala

melainkan dinyatakan.

5. Senarai rumus disediakan di muka surat

2 dan 3.

6. Anda dibenarkan menggunakan

kalkulator saintifik tanpa

diprogramkan.

7. Kertas soalan Bahagian A perlu

diserahkan pada akhir waktu

peperiksaan.

Nama : ………………..……………

Tingkatan 4 : ………………………

3472

Additional Mathematics

Mei 2010

2 2

1 Jam

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2

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

commonly used.

ALGEBRA

1

2 4

2

b b acx

a

2 a

m a

n = a

m + n

3 a

m a

n = a

m - n

4 (am)

n = a

nm

5 loga mn = log am + loga n

6 loga n

m = log am - loga n

7 log a mn = n log a m

8 logab = a

b

c

c

log

log

9 Tn = a + (n-1)d

10 Sn = ])1(2[2

dnan

11 Tn = ar n-1

12 Sn = r

ra

r

ra nn

1

)1(

1

)1( , (r 1)

13 r

aS

1 , r <1

CALCULUS

1 y = uv , dx

duv

dx

dvu

dx

dy

2 v

uy ,

2v

dx

dvu

dx

duv

dx

dy

,

3 dx

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 =

)()(2

1312312133221 1

yxyxyxyxyxyx

1 Distance = 2

21

2

21 )()( yyxx

2 Midpoint

(x , y) =

2

21 xx ,

2

21 yy

3 22 yxr

4. 2 2

ˆxi yj

rx y

GEOMETRY

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STATISTICS

1 Arc length, s = r

2 Area of sector , L = 21

2r

3 sin 2A + cos

2A = 1

4 sec2A = 1 + tan

2A

5 cosec2 A = 1 + cot

2 A

6 sin 2A = 2 sinA cosA

7 cos 2A = cos2A – sin

2 A

= 2 cos2A - 1

= 1 - 2 sin2A

8 tan 2A = A

A2tan1

tan2

TRIGONOMETRY

9 sin (A B) = sinA cosB cosA sinB

10 cos (A B) = cosA cosB+ sinA sinB

11 tan (A B) = BA

BA

tantan1

tantan

12 C

c

B

b

A

a

sinsinsin

13 a2 = b

2 + c

2 - 2bc cosA

14 Area of triangle = Cabsin2

1

1 x = N

x

2 x =

f

fx

3 = N

xx 2)( =

2_2

xN

x

4 =

f

xxf 2)( =

22

xf

fx

5 m = Cf

FN

Lm

2

1

6 1

0

100Q

IQ

7 1

11

w

IwI

8 )!(

!

rn

nPr

n

9 !)!(

!

rrn

nCr

n

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

11 P (X = r) = rnr

r

n qpC , p + q = 1

12 Mean µ = np

13 npq

14 z =

x

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Section A

[40 marks]

Answer all questions

1. Given that f: x 5 – 8x , find

a) the image of −3 ,

b) the object of 5 under the function f.

[3 marks]

Answer : (a) ……………………..

(b) ……………………...

2. Given that g: x 1x

x , x ≠ r , find the value of

a) r ,

b) g ( – 2 )

c) g – 1

(x) [4 marks]

Answer : (a) ...............................

(b) ...............................

(c) ...............................

3

1

4

2

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3. Given the function f : x 4x – 1 and composite function fg : x 5x .

Find

(a) g(x) ,

(b) the value of x when gf (x) = 9 . [4 marks]

Answer : (a) ...............................

(b) ...............................

4. Given that one of the roots of the quadratic equation 5 x2 − 4 x + r = 0 is twice the

other root. Find the value of r . [3 marks]

Answer : . r = .............................

For

examiner’s

use only

4

3

3

4

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5. Given that the quadratic equation (x + 2) (x – 5) = p has only one root, calculate the

value of p. [3 marks]

Answer : . .................................

6. Solve the quadratic equation x(2x - 5) = 2x – 1. Give your answer correct to three decimal

places. [3 marks]

For

examiner’s

use only

3

5

3

6

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Answer : . .................................

7. f(x)

G •

•

(p,1)

O x

Diagram 1

In diagram 1, (p,1) is the minimum point of the function f(x) = (x − 3) 2 + q .

Find

a) the values of p and q .

b) the coordinates of point G . [3 marks]

Answer : . (a) p =................. q = .....................

(c) ...................................................

(d)

For

examiner’s

use only

3

7

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8. Given (2, − 3) is the maximum point of a quadratic function,

f(x) = 1 – 5k – (3p + x) 2 . Determine the values of p and k . [3 marks]

Answer : p = ............................

k = ............................

9. Find the range of values of x for which 12)4( xx [ 3 marks]

Answer ..................................

3

8

3

9

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10 Solve the equation 213 93 xx . [4 marks]

Answer : …………………..

11. Express 2 log3 2p – 5 log3 p + log3 4p as a single logarithm in its simplest form.

[3 marks]

Answer : ................................

3

11

4

10

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12. Solve the equation log3 (2x – 1) + log3 (x − 2) = 2 [4 marks]

Answer : . ................................

.

4

12

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Section B

[40 marks]

Answer all questions in this section

13. Solve the simultaneous equations x - y = 1 and x2 + 3x - 3y

2 = 7

[5 marks]

14. Diagram 1 shows part of the mapping pxnxm

xf

,,

36)( . Find

(a) the values of m and n [3 marks]

(b) the value of p [2 marks]

(c) the value of x when f(x) = 3 [2 marks]

x f(x)

12

9 9

8

Diagram 2

15. Functions f and g are defined by f : x 2x -3 and g : x 2,2

3

x

x.

Find

(a) )(1 xg [2 marks]

(b ) )(1 xfg [2 marks]

(c ) h(x) such that gh(x) = 3x + 4 [3 marks]

11

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16. y

(2,1)

khxy 2)(2

x

0

T

Diagram 3

In diagram 3, (2,1) is the maximum point of the quadratic function khxy 2)(2

which intersects the y-axis at point T.

Find

(a) the values of h and k, [3 marks]

(b) the coordinates of T, [1 mark]

(c) If a straight line y = c does not intersect the graph khxy 2)(2 , find

the range of the values of c. [3 marks]

17. For the equation 0443 2 xx ,

(a) determine the type of roots [2 marks]

(b) the sum and product of the roots of the equation [2 marks]

(c) the roots of the equation [3 marks]

18. (a) Simplify 3

1

2

28

n

n

[1 mark]

(b) Solve the equation

2

3

9

13

x

x [3 marks]

(c) Given that k2log3 and H5log3 , express 180log3 in terms of k and H

[3 marks]

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Section C

[20 marks]

Answer two questions from this section.

19.

(a) A wire of length 50 cm is bent to form a rectangle with length and breadth of x

cm and y cm respectively. Given that the area of of the rectangle is 100 2cm ,

find the values of x and y. [5 marks]

(b) Solve the simultaneous equations k – h = 3 and 1532 hk . Give your

answers correct to three decimal places. [ 5 marks]

20.

(a) Determine the minimum point of the function f(x) = 822 xx [3 marks]

(b) Hence, sketch the graph of the function f(x) = 822 xx [4 marks]

(c) Find the range of the values of x if f(x) = 822 xx < 0 [3 marks]

21.

(a) The quadratic equation 0652 xx has roots α and β where α > β. Find

(i) the value of α and β

(ii) the range of x if 0652 xx [5 marks]

(b) Using the values of α and β from 15(a)(i), form the quadratic equation which

has roots α + 2 and 3β -2 [ 2 marks]

(c) Solve the equation 2 x+4

– 2 x+3

= 1 [3 marks]

END OF QUESTION PAPER