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Secondary Three Additional Mathematics Chapter 1 Unit 1.1

Further Practice 1.1

1.(a) 1.(b)

2.(a) 2.(b)

4 3 8 -------------- (1)

2 3 11 -------------- (2)

(1) 2: 8 6 16 ------------- (3)

(2) 3: 6 9 33 ------------- (4)

x y

y x

x y

y x

(4) 3 :

6 9 8 6 33 16

17 17

1 sub. into (2)

2 3 1 11

2 8

y x x y

x

x

y

y

y

4

1, 4x y

3 4 7 ------------- (1)

5 7 39 ------------- (2)

(1) 5: 15 20 35 ------------ (3)

2 3: 15 21 117 ------------ (4)

x y

x y

x y

x y

(4) (3):

15 21 15 20 117 35

41 82

2 sub. into (1)

3 4 2 7

3

x y x y

y

y

x

x

15

5

5, 2

x

x y

2 2

3 2 1 ------------ (1)

( 2) (2 3) 26 ------------ (2)

From (1): 2 3 1 ------------ (3)

Sub. (3) into (2):

x y

x y

y x

2 2

2 2

2 2

2

2

2 3 1 3 26

2 3 2 26

4 4 9 12 4 26

10 8 18 0

5 4 9 0

5 9 1 0

x x

x x

x x x x

x x

x x

x x

9 or 1 sub. into (3)

5x

9

2 3 1 or 2 3 1 15

13 or 1

5

y y

y

4 11 , 3 or 1, 1

5 5x y x y

3 3 ------------- (1)

2 12 ------------- (2)

3

y x

y x

From (1): 3 3 ----------- (3)

From (2): 2 3 2 3 ------------- (4)

x y

x y y x

2

2

2

Sub. (3) into (4):2 3 3 3 6 3 3

6 6 3 18 18

18 21 6 0

6 7 2 0

3 2 2 1 0

2 1or sub. into (3)

3 2

y y y y

y y y y

y y

y y

y y

y

2 13 3 or 3 3

3 21

1 or 12

x x

x

2 1 11, or 1 ,

3 2 2x y x y

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Secondary Three Additional Mathematics Chapter 1 Unit 1.1

3.

2

2 3 1 ------------- (1)

9 8 ------------- (2)

From (1): 3 2 1 ------------ (3)

Sub. (3) into (2):

y x

y x

x y

2

2

2

2

3 8

2 1 8

4 4 1 8

4 3 7 0

4 7 1 0

71 or sub. into (3)

4

y x

y y

y y y

y y

y y

y

7

3 2 1 1 or 3 2 14

51 or

6

x x

x

5 3

The coordinates are , 1 and 1, 1 .6 4

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Secondary Three Additional Mathematics Chapter 1 Unit 1.1

Assignment 1

1. 2.

3. 4.

2 24 9( 1) 36 ------------ (1)

2 3 5 ------------ (2)

x y

x y

From (2): 2 5 3 ------------ (3)Subst. (3) into (1):

x y

2 2

2 2

2 2

2 9 1 36

5 3 9 2 1 36

25 30 9 9 18 9 36

12 20

2 1

3

Subst. y into (3)

x y

y y y

y y y y

y

y

22 3 1 5

3

5

x

x

2

5, 13

x y

2 22 1 -------- (1)

2 2 ------- (2)

x xy y

x y

From (2): 2 2 ------ (3)

Subst. (3) into (1):

x y

2 2

2 2 2

2

2 2 2 2 2 1

4 8 4 4 4 1

4 3 0

1 3 0

1 or 3

Subst. y into (3)

y y y y

y y y y y

y y

y y

y

2 2 1 or 2 2 3

0 or 4

x

x

The coordinates are 0, 1 and 4, 3 .

1 ------------- (1)

8 ------------- (2)

Sub. 2, 3 into (1) and (2):

ax by

ay bx

x y

2 3 1 -------------- (3)

3 2 8 -------------- (4)

a b

a b

3 3: 6 9 3 ------------- (5)

4 2 : 6 4 16 ------------ (6)

5 6 :

6 9 6 4 3 16

a b

a b

a b a b

13 13

1

Subst. b into (3)

2 3 1

2

b

b

a

a

2, 1a b

Let the radii of the circles be cm and cm, , 0.x y x y

2 2 38 ------------ (1)

From (1): 2 2 38

19

19 ------------ (2)

x y

x y

x y

x y

2 2

2 2

193 ------------- (3)

193 ------------- (4)

x y

x y

2 2

2 2

2

2

Sub. (2) into (4):

19 193

361 38 193

2 38 168 0

19 84 0

7 12 0

7 or 12

Subst. y into (2)

y y

y y y

y y

y y

y y

y

19 7 or 19 12

12 or 7

x

x

The radii of the circles are 7 cm and 12 cm.

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Secondary Three Additional Mathematics Chapter 1 Unit 1.1

Revision Questions

1. 2.

3.(a) 3.(b)

2 6 5 ------------ (1)

3 4 15 ------------ (2)

(1) 3 : 6 18 15 ------------ (3)

(2) 2: 6 8 30 ------------ (4)

x y

x y

x y

x y

(4) (3):

6 8 6 18 30 15

10 15

1 1 sub. into (1)

2

x y x y

y

y

12 6 1 5

2

7

x

x

1

7, 12

x y

5 4 7 -------------- (1)

2 6 18 -------------- (2)

x y

x y

(1) 3: 15 12 21 ----------- (3)(2) 2: 4 12 36 ----------- (4)

(3) (4):

15 12 4 12 21 36

19 57

3 sub. into (2)

x yx y

x y x y

x

x

2 3 6 18

6 12

2

y

y

y

3, 2x y

2 2

3 5 ----------- (1)

5 ----------- (2)

x y

x y

From (1): 5 3 ------------ (3)

Sub. (3) into (2):

x y

2 2

2 2

2

2

5 3 5

25 30 9 5

10 30 20 0

3 2 0

2 1 0

1 or 2 sub. into (3)

y y

y y y

y y

y y

y y

y

5 3 1 or 5 3 2

2 or 1

x x

x

2, 1 or 1, 2x y x y

2 5 13 ------------ (1)

2

3 25 ------------ (2)

x y

x y

6 15 39(1) 3: ----------- (3)2

6 4(2) 2: 10 ----------- (4)

x y

x y

(3) (4):

6 15 6 4 3910

2

19 19

2

2 sub. into (1)

x y x y

y

y

2 5 13

2 2

24

1

2

x

x

x

1, 2

2

x y

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Secondary Three Additional Mathematics Chapter 1 Unit 1.1

3.(c) 4.

5.

20 ----------- (1)

2 11 2 ----------- (2)

xy

x y

From (2): 2 2 11

2 11 --------- (3)

2

Sub. (3) into (1):

x y

yx

2

2 1120

2

2 11 40

2 11 40 0

2 5 8 0

58 or sub. into (3)

2

yy

y y

y y

y y

y

5

2 112 8 11 2

or2 2

12 or 8

2

x x

x

1 12 , 8 or 8, 2

2 2x y x y

Let John's present age be and his father's be .x y

2 12 ------------ (1)

11 1 ------ (2)

6

x y

x y

From (2): 6 6 1

6 5 ----------- (3)

Sub. (3) into (1):

x y

y x

2

2

6 5 12

6 7 0

7 1 0

7 or 1 (N.A) sub. into (3)

x x

x x

x x

x

John's present age is 7 years old.

1 2

1 2

500 --------------- (1)1 1 1

------------- (2)120

R R

R R

1 2

2

2

21

2

1 1 1From (2):

120

120

120

120 ----------- (3)

120

Sub. (3) into (1):

R R

R

R

RR

R

22

2

2 2 2 2

2

2 2

2

2 2

2 2

2

1 1

1

120500

120

120 120 500 120

500 60000

500 60000 0

300 200 0

200 or 300 sub. into (1)

200 500 or 300 500

300 or 200

RR

R

R R R R

R R

R R

R R

R

R R

R

1 2 1 2300, 200 or 200, 300R R R R