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Chapter 8: Exponential Functions 43

20.( )

3 4

2

a

a16

-=

3

4

a

a4= a4

13

4-

= a4 31

-=

3

1

a

4

21.3 2 2

1

a

b

a

3

-

a bk l= 3

32

1

aa

b

3eco

m=

22

1

a b

27

22.2-

( ) ( )a b a22 1

3

1

2

1

b-

=( )2

12

13

1

22a b

a b-

= 42

13

1

ab

a b

2

-

=2

3

11

a b4

1

23.

3

64

64

= 84

=12

24.4

5

161

-a k = 16 45 = 5( )4 16 = 25 = 32

25. ( )a b162 34a

4-

= (16 )a b41

2

1

4

3a4

-

=2

1

4

3

a b

4

2

=2

1

4

3

a b2

26.b3

1 1 2

1

b

a

8

2- -d n = 4 2

1

ab

1

4c m =

2a b2

1

2

1

27. n

n

2

2

3

( )

a

a3-

-

= - n

n n

2

2 2

a

a

3

3-

=1-a3

n n

2 2

n

2--- e o

= a3n

n

2

2

--

28.4

62581

- =53-

29.3

1

1258 -a k = 3

1

8125a k = 3 8125 = 25

30.5

12581

-b l = 5125

9-c m

=5

9125b l

= .2 96, cor. to 3 sig. fig.

31.128 7

1

729a k = 7

729128

= 7729

2

= .0 780, cor. to 3 sig. fig.

32.6 4

3

3 3

b

a

a

b

-=

6 43

1

32

3

( )a b

a b

-

=2

3

4

3 2

3

a b

a b-

= a b( )3 2

2

3

3

4- - -

= a b5

6

1

33. ( )b a b31

2 5-= ( )b a b

51

12

1

3-

= ( )a b1 5

6

5-

=6

5

25

b

a

34.0

( )

a

3 21

321 2( ) ( )

a b

ab

3

-

=3

2

23

ab

a-

=1

2

3

3

2

ba

1

+=

5

2 3

2

a b

1

35.( )

a

13

3

3

1

3

22

( )

a b

b

2-

=3

2

9

3

2

3

4

a b

a b

-= a b

( )3

23

3

4

2

9- - -

=6

19

3

11

a

b

36.

4

3

2

1

ab

af p =4

3

1

3

1

2

1

a b

af p =4

3

1

6

1

b

af p =3

4

3

2

b

a

37. ( )ab32

31-: a9

23

= ( )a b3 31

3

1

3

2-

:( )a9 31

3

2

= a b3 31

3

2 1

3

2

3

2

3+ + -

= ab3 32

-

=3

2

b

a3

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44 Solutions

38.3

a #( )a3 23

-#( )a2

23

1

= 31

a # a3 23

-# a4 3

2

= a12 31

2

3

3

2- +

= a12 21

-

=2

1

a

12

39.

2

3

2

a

a

3f p ' a 24 - = 3

4

aa3

'a 21

-

= a31 1

3

4

2

1- - -a k

=6

5

a3

40.3

93

b

a8-

# 44

3

a

bc m = 39 31

b

a8-

d n # 43a

b4c m

=3

b

a21- # 3

4

3

a

b

= a b2( )3 3 1

4

3- - -

= b2 47

41.a4

33

b

27#

31

3a

b8d n = 34 31

b

a27e o #3

31

a

b8d n = 4

3b

a3#

2

1

3

1

a

b2

= a b61

2

1

3

1

3

4- -

=2

1

ba6

42.-

a b3

3

5

1

21

51

( )b

a -c m = a b5

3

10

1

( )

b

a101

53-f p = 1

43. a164

a25

2

1

= a a16 254

4

1

= a16 4114

= ( )a16 41

4

11

= a2 1611

44. m

m

1 23

9- =

( )1 2m

m

3

32

- = 1 2

2

m

m

3

3- = 3

( )m m2 1 2- -= 3

m4 1-

45. 3n# 9

n#

-( )27

n3 31

= 3

n

#

9

n

#

27

-n

= 3

n# (3

2)

n# (3

3)-n

= 3n# 3

2n# 3

-3n

= 3n + 2n + (-3n)

= 1

46. 2( )

( )x

x x2

2

4 2 2-=

x

x

x

2

2 2( ) ( )

2

2 2 2-=

( )x

x x

2

2 2

2

2 2 2-= 1-

47. n

n n

1

1 1

16

16 16-

- +-

= n

n

1

1 2( )

16

16 1 16-

--

= 1 - 162

= 255-

48. a a

a a

2 1 2 1

2 1 2 2

5 5

5 5

-

-+ -

+ -

= a a

a a

2 2 1

2 2 2

5 5 5 5

5 5 5 5

-

-

-

$ $

$ $-

=a

a

2

2

5 5

5 5

51

251

-

-

aa

kk

=5

5

51

251

-

-

=3031

49. k

k

1

2 2

16

8

-

+

= k

k

4 1

3 2 2 2

1

( )

( )

2

2-

+> H

= k

k

4 4

6 6 2

1

2

2-

+d n = 2

( )k k6 6 4 4 21

+ - -6 @

= ( )2 21

k2 10+

= 25k +

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Exercise 8B (P.20)

1. C1:y = 2x,

C2:y = 8x.

2. C3:y = 6x,

C4:y = 0.4x.

3. C5:y = 0.2x,

C6:y = 0.8x.

4. (a)

x

y

1

10

2 3 41

y = 3x

234

2

3

4

5

6

7

8

9

(b) y-intercept = 1

(c) (i) Whenx1 0, 0 1y1 1.

(ii) WhenxH 0,yH 1.

5. (a)

x

y

1

10 2 3 41

y = 0.3x

234

2

3

4

5

6

7

8

9

(b) The coordinates of the pointA are (0 , 1).

(c) (i) Whenx1 0,y2 1.

(ii) WhenxH 0, 0 1yG 1.

6. (a) When t= 0, R = 30,

` 30 = a# 1.20

a = 30

(b) When t= 10,

R = 30 # 1.210

= 186, cor. to the nearest integer

` The number of hamsters after

10 weeks is 186.

7. (a) When n = 0 (i.e. at the beginning of 2008),

V= 4 000 000,

` 4 000 000 = k# 1.030

k = 4000000

(b) At the beginning of 2015,

t= 2 015 - 2 008 = 7,

` V= 4 000 000 # 1.037

= 4 920 000, cor. to the nearest

10 000 The increase in the value of the flat from

the time of purchase to the beginning of

2015

= $(4 920 000 - 4 000 000)

= $920 000, cor. to the nearest $10 000

8. (a) When t= 0, S= 6,

` 6 = m# 1.150

m = 6

(b) When t= 14,

S= 6 # 1.1514


` The number of people suffering from

that infection after 2 weeks is 42.

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46 Solutions

9. (a) When n = 0, M= 0,

` 0 = 250 000 # 1.020

- k

k = 250 000

(b) When n = 17,

M= 250 000 # 1.0217 - 250 000

= 100 060, cor. to the nearest integer

` The amount of bonus he will receive

when he retires is $100 060.

10. (a) When n = 0 (i.e. at the beginning of 2007),

L = 0,

` 0 = 8 000 - k# 0.850

k = 8000


n = 2 012 - 2 007 = 5,

` L = 8 000 - 8 000 # 0.855

= 4 450, cor. to the nearest integer

` The depreciation of the camera is

$4 450.

11. (a) When n = 0, Y= 1 000,

` 1 000 = a# 0.8-1

a = 800

(b) When n = 6,

Y= 800 # 0.86 - 1


The number of bacteria destroyed by the

germicide in 6 hours

= 1 000 - 262


12. y

xO

y = 10xy = 10x

13. (a) C1:y = 0.4x, C2:y = 4

x, C3:y = 1.4

x.

(b) The coordinates of the pointA are (0 , 1).

(c) y

x

y = ` jx

O

1

4

C2

C3

C1

A

14. (a) When t= 2, V= 2 450,

` 2 450 = 5 800 #r2

r= .0 65, cor. to 2 sig. fig.

0.649 9


t= 2 013 - 2 009 = 4,

` V= 5 800 # 0.649 94

= 1 035, cor. to 4 sig. fig.

` The percentage decrease

=5800

5 800 1035-# 100%

= %82 , cor. to 2 sig. fig.

15. (a) When n = 3, A = 14 000,

` 14 000 = 10 000 #r3


1.118 7

(b) When n = 8,

A = 10 000 # 1.118 78

= 24 531, cor. to 5 sig. fig.

` The percentage increase

=10000

24 531 10 000-# 100%

= %145 , cor. to 3 sig. fig.

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16. (a) At the beginning of 2007,

t= 2 007 - 2 000 = 7, R = 1.011,

` 1.011 = 0.964 #a7

a = .1 007, cor. to 4 sig. fig.

1.006 823 7


t= 2 030 - 2 000 = 30,

` R = 0.964 # 1.006 823 730

= 1.182, cor. to 4 sig. fig.

1.182 17

` The exchange rate of the currency to

Hong Kong dollar at the beginning of

2030 is 1.182.

(c) The percentage increase

=.

. .1 011

1 182 17 1 011-# 100%

= . %16 93 , cor. to 4 sig. fig.

Supp. Exercise 8 (P.26)

1. (a) 163

- = 163

2- = ( )4

2

3

2- = -26

= 64-

(b)1532

- = 532

1- =

21

-

(c) .0 00014

= .0 1

(d) ( )12523

- = 125 32

-] g = (-5)2

= 25

2. (a) 9 23

-= ( )9

3-= 3

-3= 3

3

1=

127

(b) 64 32

= ( )6423

= (4)2

= 16

(c)3

1

81a k = 3 1

8=

21

(d)-

( )3 32 2

1-

= 3(3) = 9

3. (a) 563

- = .3 83- , cor. to 2 d.p.

(b) 374

- = .2 47- , cor. to 2 d.p.

(c)3

54

-a k = .1 40, cor. to 2 d.p.(d) ( )15

35= .5 08, cor. to 2 d.p.

4. (a) 25 41

= .2 24, cor. to 2 d.p.

(b) 44 83

-= .0 24, cor. to 2 d.p.

(c) 30( )2

-3

= .9 65, cor. to 2 d.p.

(d)3

1

52a k = .0 74, cor. to 2 d.p.

5. (a)2

( )a

a

-

3

=3

1

aa

= a1

3

1-

=3

2

a

1

(b) a4'( )a a

3 4#

=3

12

4

a a

a

#

= a4 2

3

1- -

= a 35

(c) ( ) ( )b a a b3 2 2 33-

= ( )( )b a a b223

3

2

-

= a b2 1

3

2

2

3- + +

=3

2

5

4

a

b

(d)4

3

3 4

a

a b

81-

-

=4

1

4

3

4

31

a

a b

81-

-

=4

3

4

3

ba

3

- -c m

=2

3

ba3

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48 Solutions

6. (a)-

5

24

1

( )a

1= 1

10a

1

-= a 10

1

(b)3

2 3

2

1

a

a b39

-a ck m = 33 3 21

2a a b

3 9e co m

=3

2

3

a ab27 3

c dm n

=4

2

3

a b81

(c) ( ) ( )a a9 271 2

2

1

3

2

'- -

= 9 27( ) ( )a a21

2

1

3

2

3

4

'- - -

= 3 3( ) ( )a a2

2

1

3

4

'- - -

= a3( )1 2

2

1

3

4- - - - -a k

= a27 65

(d) n3

52

( )a

a

2

n

3

2

-= n

3

10

a

a

4

n

3

2

-

= an n

3

2

3

10

41 - -a k

=

n4a

4

7. (a) 3a: (3

3)3a

= 3a: 3

9a= 3

a + 9a= 3

a10

(b) xx x2

64

4 16-$

=

x x2

x6

2 4

( )

( ) ( )

2

2 2-

$

= 6x

x x44

2

2 2-

$

= 24x + 4x - (-6x)

= 2x14

(or47x

)

(c) 252 - n

: 51 + 2n

= (52)2 - n

: 51 + 2n

= 54 - 2n

: 51 + 2n

= 54 - 2n + 1 + 2n

= 55

(or3 125)

8. (a) Fig. I:y = 1.5x, Fig. II:y = 0.7

x.

(b) (i) y-intercept = 1

(ii) y-intercept = 1

(c) (i) When the value ofx increases, the

value ofy decreases.

(ii) When the value ofx increases, the

value ofy increases.

9. (a) When t= 0, N= 100,

` 100 = a# 1.260

a = 100

(b) When t= 7,

N= 100 # 1.267


The increase in the number of insects

= 504 - 100


10. (a) When n = 1, H= 10,

` 10 = c# 0.751 - 1

c = 10

(b) When n = 6,

H= 10 # 0.756 - 1

= 2.37, cor. to 3 sig. fig.

` The height from which the ball drops

before it hits the ground the 6th time

is 2.37 m.

11. (a) 64 643 3

- - = (-4) - 4 = 8-

(b) 3

44

1

( )3

125-

-

-=

3

3

1251

-=

315

-= 15-

12. (a)4

3

1681

-a k = 323

-a k = 332a k =

287

(b)271 1

813

2

4

3

+- -a ak k = 2 3

31

31

+- -a ak k

= 32

+ 33

= 36

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13. a a a a43

$ $ = a a a a3

4

1

$ $

= a a a3

4

5

$

= a a a5

12$

= a a 1217

= a a 2417

$

= a 2441

14.1

4

1 2

3

2

4

13

( )x y

x y

-

-

=2

2

1

9

2

12

1

x y

x y

-

-

=12

7

9

20

y

x

15. ( )a a64 8

93 93

1

#-

= 4a

3

#

3a2

1

= 2

16. (5 )m n mn253 6

2

1

3

1

'-

= 125 5( )m n m n3

2

3

2

1

'-

= 25m n1 3

2

3

2

1- - -

= 22m n

25

17. n n

n

1 1

1

27 27

27

-+ -

-

= n

n

1 2

1

( )27 27 1

27-

-

-

=12

27 1-

=7281

18. k k

k k

1 2

2 3 2( )

a b

a b+ +

- - -

=a b

k k

k k

1 2

2 24 6a b

+ +

- -

= a(2k - 4) - (k + 1)

b(2k - 6) - (k + 2)

= a bkk 5 8- -

19. n nn n3

43

9 2-

-

$

$= n n

n n

2

2 3

(2 )

( )

3

3 2

$-

-$

= n n

n n

2

2 3

23

3 2-

-

$

$

= 32n - (-n)

: 2-3n - 2n

= 33n

: 2-5n

= n

n

5

33

2

20. (a) 23x + 3

= 125

23x

: 23

= 125

(2x)3

=1258

2x

= 25

(b)x

41a k = x

4

1= x2

( )2

1= x 2

( )2

1= 2

1

25` j = 25

4

21. (a)

x

y

1

10 2 31

y = 5x

y = 0.2x

23

2

3

4

5

6

7

8

9

10

11

12

(b) The coordinates of the point Care (0 , 1).

(c) (i) Whenx2 0,y2 1.

(ii) Whenx2 0, 0 1y1 1.

22. (a) At the beginning of 2005,

n = 2 005 - 2 000 = 5, P= 2 805.

` 2 805 = 2 000 #r5


1.070 0


n = 2 018 - 2 000 = 18,

` P= 2 000 # 1.070 018

= 6 759.9, cor. to 5 sig. fig.

` The percentage increase

=.

2 000

6 759 9 2 000-# 100%

= %238 , cor. to 3 sig. fig.

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50 Solutions

23. (a) Given that m0 = 100 g,

wheny = 30, m = 98.71 g,

` 98.71 = 100#a30

a = .0 999 57, cor. to 5 sig. fig.

0.999 567 3

(b) Wheny = 1 000,

m = 100 # 0.999 567 31 000

g

= 64.869 43 g, cor. to 7 sig. fig.


=.

100100 64 869 43-

# 100%

= . %35 131 , cor. to 5 sig. fig.

24. (a) When n = 2, P= 48 020, ` 48 020 = 50 000 #r

2

r= .0 98

(b) In January 2009,

n = (2 009 - 2 006) # 12 = 36,

` P= 50 000 # 0.9836

= 24 161, cor. to 5 sig. fig.


=50000

50 000 24 161-# 100%

= . %51 7 , cor. to 3 sig. fig.

25. (a) S= 1100002n

%2

8+a k

S= 10 000(1 + 4%)2n

` S= 10 000(1.04)n2

(b) When n = 6,

S= 10 000(1.04)2(6)

= 16 010.322, cor. to 3 d.p. ` The percentage increase

=.10000

16 010 3 22 10 000-# 100%

= %.60 1 , cor. to 1 d.p.

Lively Maths Problem

26. (a) When h = 5.072,

P= .2 72 ..

8 55

5 702-

= 0.513 08, cor. to 5 sig. fig.

` The required percentage

=.

10 513 08

# 100%

= . %51 3 , cor. to 3 sig. fig.

(b) When h = 8.848,

P= .2 72 ..

8 55

8 848-

= 0.355 05, cor. to 5 sig. fig.

` The required percentage

=.

10 355 05

# 100%

= %.35 5 , cor. to 3 sig. fig.

Revision Test (P.33)

1. D 4x: 16

y= (2

2)x: (2

4)y

= 22 4x y+

2. D n

nn n

22

3

2

9 2$ $=

2

n

nn n

22

( )

2

2 33 $ $

= 3n + n

: 22n - n

= 32n

: 2n

= 18n

3. C 242

1

1

3

)( ba

a b-

- =

3

2

1

2 8

4

a b

a b-

-

= a b

( )2 8

4

3

2

1- - - -

= a b4 21711

-

=4

11

a

b 217

4. B 9x

= (32)x

= (3x)2

= y2

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5. C 8x

= 43

23x

= 4 31

23x

= 2 32

` 3x =32

x =92

6. B

7. B At the beginning of 2006,

n = 2 006 - 1 950 = 56, V= 360 000.

` 360 000 = 40 000 #a56

a = 1.04, cor. to 2 d.p.

At the beginning of 2030,

n = 2 030 - 1 950 = 80,

V= 40 000 # 1.0480

= 921 991.96, cor. to 2 d.p.

` The required number of times

=

the value of the vase at the

beginning of 2030

the value of the vase at the

beginning of 1950

=

.

40 000

921 991 96


HKCEE Question

8. E

sol_4b08

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