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Mathematics Form 4 F4AM00401FLesson 01: Significant figures.

Learning Area: Understand and use the concept of significant figures 1 / 5

QUESTION 1Question:Determine the numbers of significant figures in each of the numbers given below:

a) 48.173 5b) 0.049 271c) 26 078d) 1.005

Solution:

QUESTION 2

Question:Express each of the following numbers correct to 3 significant figures:

a) 0.003 219b) 38.270 01c) 234 900d) 7 891

Solution:

ACTIVITY SHEETPupils’ copy



QUESTION 3


a) 0.040 862b) 51.008 9c) 349 781d) 19 983

Solution:

QUESTION 4

Question:Evaluate the following and state your answer correct to 2 significant figures.

a) 2.9 × 3.01b) 0.8 × 1.2c) 34 000 ÷ 17d) 9 ÷ 0.3

Solution:



QUESTION 5

Question:In an experiment, it is found that a light takes 0.46457 second to get from point A to point B in astraight line. State the time taken correct to

a) 2 significant figures.b) 3 significant figuresc) 4 significant figures

Solution:



QUESTION 1

Question:Determine the numbers of significant figures in each of the numbers given below:

e) 48.173 5f) 0.049 271g) 26 078h) 1.005

Solution:a) 6b) 5c) 5d) 4

QUESTION 2


e) 0.003 219f) 38.270 01g) 234 900h) 7 891

Solution:a) 0.003 22b) 38.3c) 235 000d) 7 890

QUESTION 3


e) 0.040 862f) 51.008 9g) 349 781h) 19 983

Solution:a) 0.040 86b) 51.01c) 349 800d) 19 990

ACTIVITY SHEETTeacher’s copy



QUESTION 4

Question:Evaluate the following and state your answer correct to 2 significant figures.

e) 2.9 x 3.01f) 0.8 x 1.2g) 34 000 ÷ 17h) 9 ÷ 0.3

Solution:a) 2.9 x 3.01 = 8.729 = 8.7 (correct to 2 significant figures)b) 0.8 x 1.21 = 0.968 = 0.97 (correct to 2 significant figures)c) 34 000 ÷ 16 = 2125 = 2100 ((correct to 2 significant figures)d) 6693 ÷ 3 = 2 231

= 2 200 (correct to 2 significant figures)

QUESTION 5

Question:In an experiment, it is found that a light takes 0.46457 second to get from point A to point B in astraight line. State the time taken correct to

d) 2 significant figures.e) 3 significant figuresf) 4 significant figures

Solution:a) 0.46 seconds (correct to 2 significant figures)b) 0.465 seconds (correct to 3 significant figures)c) 0.4646 second (correct to 4 significant figures)

Mathematics Form 4 Lesson 02: Standard form.

Learning Area: Understand and use the concept of standard form to solve problems.

QUESTION 1Question:Write each of the following in standard form:

a) 42b) 0.06c) 0.0804d) 82 000e) 7 000 000

Solution:

QUESTION 2

Question:Convert the numbers in standard form to single numbers.

a) 1.2 x 104

b) 9.7 x 108

c) 2.37 x 10-6

d) 1.45 x 10-7

e) 9.8 x 10-9

Solution:




QUESTION 3

Question:Evaluate the following and give your answer in standard form:

a) 6 x 0.004 75b) 3 000 x 7 000c) 15 000 x 2 000d) 90 000 ÷ 300e) 45 000 ÷ 90f) 750 000 ÷ 500

Solution:

QUESTION 4


a) (2.9 x 10-2) + (3.9 x 10-1)b) 8.53 x 103 + 6.7 x 102

c) 1.4 x 105 – 6.4 x 104

d) 10-6 – 2 x 10-2

e) 1.3 x 103 – 4.1 x 102

Solution:



QUESTION 5

Question:There are 127 242 ships calling at the port of Port Klang in1995.

a) Write the exact value of 127 242 in standard form.b) The total tonnage of these ships was 846 million tonnes. Calculate the average tonnage per

ship in 1995. Give your answer in standard form.Solution:



QUESTION 1

Question:Write each of the following in standard form:

a) 42b) 0.06c) 0.0804d) 82 000e) 7 000 000

Solution:a) 4.2 x 101

b) 6 x 10-2

c) 8.04 x 10-2

d) 8.2 x 104

e) 7.0 x 106

QUESTION 2

Question:Convert the numbers in standard form to single numbers.

a) 1.2 x 104

b) 9.7 x 108

c) 2.37 x 10-6

d) 1.45 x 10-7

e) 9.8 x 10-9

Solution:a) 12 000b) 970 000 000c) 0.000 002 37d) 0.000 000 145e) 0.000 000 009 8




QUESTION 3


a) 6 x 0.004 75b) 3 000 x 7 000c) 15 000 x 2 000d) 90 000 ÷ 300e) 45 000 ÷ 90f) 750 000 ÷ 500

Solution:a) 0.0285b) 2.1 x 107

c) 3.0 x 107

d) 5.0 x 102

e) 1.5 x 103

QUESTION 4


a) (2.9 x 10-2) + (3.9 x 10-1)b) 8.53 x 103 + 6.7 x 102

c) 1.4 x 105 – 6.4 x 104

d) 10-6 – 2 x 10-2

e) 1.3 x 103 – 4.1 x 102

Solution:a) 4.19 x 10-1

b) 9.2 x 103

c) 7.6 x 104

d) – 1.999 x 10-2

e) 8.9 x 102

QUESTION 5

Question:There are 127 242 ships calling at the port of Port Klang in1995.

c) Write the exact value of 127 242 in standard form.a) The total tonnage of these ships was 846 million tonnes. Calculate the average tonnage per

ship in 1995. Give your answer in standard form.Solution:

a) 1.272 42 x 105

b) 6.648748 x 103

Mathematics Form 4 F4AM00401FLesson 03: Quadratic expressions.

Learning Area: Identifying quadratic expressions. 1 / 4

ACTIVITY SHEET

QUESTION 1Question:State if the expression given is a quadratic expression in one variable.

x2 + 2x + 1Solution:

QUESTION 2

Question:State if the expression given is a quadratic expression in one variable. 4x2 – 2x + 1Solution:

QUESTION 3

Question:State if the expression given is a quadratic expression in one variable.

96

12 ++ xx

Solution:

QUESTION 4

Question:State if the expression given is a quadratic expression in one variable. p3 + 2p + 1Solution: .

QUESTION 5

Question:State if the expression given is a quadratic expression in one variable. s4 – 2s4 – 3Solution:

Pupil’s copy



QUESTION 6Question:State if the expression given is a quadratic expression in one variable. x – 2x2 + 3Solution:

QUESTION 7

Question:State if the expression given is a quadratic expression in one variable. w2 – 2w2 + 3Solution:

QUESTION 8

Question:State if the expression given is a quadratic expression in one variable. y – 2y3 + 9Solution:

QUESTION 9

Question:State if the expression given is a quadratic expression in one variable. 2x2 – 4x – 16Solution:

QUESTION 10

Question:State if the expression given is a quadratic expression in one variable. 3y2 – 2x2 + 4xSolution:



QUESTION 1Question:State if the expression given is a quadratic expression in one variable.

x2 + 2x + 1Solution:Yes.

QUESTION 2

Question:State if the expression given is a quadratic expression in one variable. 4x2 – 2x + 1Solution:Yes.

QUESTION 3

Question:State if the expression given is a quadratic expression in one variable.

96

12 ++ xx

Solution:No.

QUESTION 4

Question:State if the expression given is a quadratic expression in one variable. p3 + 2p + 1Solution:No.

QUESTION 5

Question:State if the expression given is a quadratic expression in one variable. s4 – 2s4 – 3Solution:No.




QUESTION 6Question:State if the expression given is a quadratic expression in one variable. x – 2x2 + 3Solution:Yes.

QUESTION 7

Question:State if the expression given is a quadratic expression in one variable. w2 – 2w2 + 3Solution:Yes.

QUESTION 8

Question:State if the expression given is a quadratic expression in one variable. y – 2y3 + 9Solution:No.

QUESTION 9

Question:State if the expression given is a quadratic expression in one variable. 2x2 – 4x – 16Solution:Yes.

QUESTION 10

Question:State if the expression given is a quadratic expression in one variable. 3y2 – 2x2 + 4xSolution:No.

Mathematics Form 4 Lesson 04: Quadratic expressions from two linear expressions..

Learning Area: Form quadratic expressions by multiplying any two linear expressions. 1 / 4

Activity SheetPupils’ copy

QUESTION 1Question:Form quadratic expressions by multipliying the following expressions:

a) (x + 2)(x + 3)b) (s + 3)(s + 9)c) (y + 2)(y + 5)d) (2y + 3)(y + 1)e) (3y + 1)(3y + 2)

Solution:

QUESTION 2

Question:Form quadratic expressions by multipliying the following expressions:

a) (x + 3)(x – 3)b) (2x + 1)(2x – 1)c) (2y + 2)(y – 2)d) (2r + 1)(r – 2)e) (2s + 3)(2s – 5)

Solution:



QUESTION 3


a) (x – 3)(x – 3)b) (3x – 1)(2x – 1)c) (4y – 2)(y – 2)d) (2r – 1)(6r – 2)e) (2s – 3)(9s – 5)

Solution:

QUESTION 4


a) (x – 3)(x + 3)b) (3r – 1)(r + 1)c) (4t – 2)(t + 2)d) (3a – 1)(a + 2)e) (4e – 2)(4e + 2)

Solution: .



QUESTION 1Question:Form quadratic expressions by multipliying the following expressions:

a) (x + 2)(x + 3)b) (s + 3)(s + 9)c) (y + 2)(y + 5)d) (2y + 3)(y + 1)e) (3y + 1)(3y + 2)

Solution:a) x2 + 5x + 6b) s2 + 12s + 27c) y2 + 7y + 10d) 2y2 + 5y + 3e) 9y2 + 9y + 2

QUESTION 2


a) (x + 3)(x – 3)b) (2x + 1)(2x – 1)c) (2y + 2)(y – 2)d) (2r + 1)(r – 2)e) (2s + 3)(2s – 5)

Solution:a) x2 – 9b) 4x2 – 1c) 2y2 – 2y – 4d) 2r2 – 3r – 2e) 4s2 – 4s – 15




QUESTION 3


a) (x – 3)(x – 3)b) (3x – 1)(2x – 1)c) (4y – 2)(y – 2)d) (2r – 1)(6r – 2)e) (2s – 3)(9s – 5)

Solution:a) x2 – 6x + 9b) 6x2 – 5x + 1c) 4y2 – 10y + 4d) 12r2 – 10r + 2e) 18s2 – 37s + 15

QUESTION 4


a) (x – 3)(x + 3)b) (3r – 1)(r + 1)c) (4t – 2)(t + 2)d) (3a – 1)(a + 2)e) (4e – 2)(4e + 2)

Solution:a) x2 – 9b) 3r2 + 2r – 1c) 4t2 + 6t – 4d) 3a2 + 5a – 2e) 16e2 – 4

Mathematics Form 4 Lesson 05: Quadratic expressions from everyday life situations.

Learning Area: Form quadratic expressions based on specific situations.

QUESTION 1Question:A rectangular shaped aquarium has the width of 3 cm more than its height and the length of 20 cm.If the height is x cm, can we form the quadratic expression for the volume of the aquarium?

Solution:

QUESTION 2

Question:Aishah has a piece of rectangular shaped carpet in her bedroom. Its measurements are as shown inthe diagram. Can we help her find the area of the carpet?

Solution:


2x m

(3x + 1) m



QUESTION 3

Question:Mr Ahmad drives to his office every morning. He drive at the speed of (s − 2) metre/minute for 4sminutes. Form the quadratic expression for the distance between his house and his office.

Solution:

QUESTION 4

Question:Yusof Music organise a concert in which some local artist to performing. There are 20n numbers ofVIP tickets sold for RM n and 5000 normal tickets sold at RM 7 each. Form a quadratic expressionfor the collection for the concert if all the tickets were sold.

Solution:

QUESTION 5

Question:Mustapha inherits two pieces of land from his grandfather. One piece of land is square in shapewith sides of 50x metres and the area of the other piece is 2000 square metres. Find the total area ofthe two pieces of lands.

Solution:



QUESTION 1

Question:A rectangular shaped aquarium has the length of 20 cm and the width of 3 cm more than its height.If the height is x cm, can we form the quadratic expression for the volume of the swimming pool?

Solution:Formula for volume is:

Volume = length × width × height= 20 cm × (x + 3) cm × x cm= [20 × (x2 + 3x)] cm3

= (20x2 + 60x) cm3

QUESTION 2

Question:Aishah has a piece of rectangular shaped carpet in her bedroom. Its measurements are as shown inthe diagram. Can we help her find the area of the carpet?

Solution:The formula for the area is:

Area = width × length= 2x × (3x + 1)= 6x2 + 2x

QUESTION 3

Question:Mr Ahmad drives to his office every morning. He drive at the speed of (s − 2) km/minute for 4sminutes. Form the quadratic expression for the distance between his house and his office.

Solution:The formula for distance is:

Distance = speed × time= (s – 2) km/minute × 4s minutes= 4s2 – 8s km


2x m

(3x + 1) m



QUESTION 4

Question:Yusof Music organise a concert in which some local artist to performing. There are 20m numbersof VIP tickets sold for RM m and 5 000 normal tickets sold at RM 7 each. Form a quadraticexpression for the collection for the concert if all the tickets were sold.

Solution:The collection for all the tickets = RM (20m × m) + RM (5 000 × 7)

= RM (20m2 + 35 000)

QUESTION 5

Question:Mustapha inherits two pieces of land from his grandfather. One piece of land is square in shapewith sides of 50t metres and the area of the other piece is 2000 square metres. Find the total area ofthe two pieces of lands.

Solution:The total area of the two pieces of land = (50t × 50t) m2 + 2000 m2

= (2500t2 + 2000) m2

Mathematics Form 4 Lesson 06: Quadratic expressions of the form ax2 + bx + c, where b = 0 or c = 0

Learning Area: Factorise quadratic expressions of the form ax2 + bx + c,where b = 0 or c = 0. 1 / 4

QUESTION 1Question:Factorise the given quadratic expression.

6x2 + 9Solution:

QUESTION 2

Question:Factorise the given quadratic expression.

12 – 3r2

Solution:

QUESTION 3


5w2 – 5Solution:

QUESTION 4


16y2 + 8Solution: .

QUESTION 5


6s2 + 18Solution:




QUESTION 6Question:Factorise 8x2 + 6y.

Solution:

QUESTION 7

Question:Factorise 12r2 – 2r.

Solution:

QUESTION 8

Question:Factorise 10w – 4w2

Solution:

QUESTION 9

Question:Factorise 7y2 + 28y.

Solution:

QUESTION 10

Question:Factorise 9s2 – 27s.

Solution:




6x2 + 9Solution:6x2 + 9 = 3(2x2 + 3)

QUESTION 2


12 – 3r2

Solution:12 – 3r2 = 3(4 – r2)

QUESTION 3


5w2 – 5Solution: 5w2 – 5 = 5(w2 – 1)

QUESTION 4


16y2 + 8Solution: . 16y2 + 8 = 8(2y2 + 1)

QUESTION 5


6s2 + 18Solution: 6s2 + 18 = 6(s2 + 3)




QUESTION 6Question:Factorise 8x2 + 6y.

Solution: 8x2 + 6y = 2(4x2 + 3y)

QUESTION 7

Question:Factorise 12r2 – 2r.

Solution: 12r2 – 2r = 2(6r2 – r)

QUESTION 8

Question:Factorise 10w – 4w2

Solution: 10w – 4w2 = 2(5w – 2w2)

QUESTION 9

Question:Factorise 7y2 + 28y.

Solution: 7y2 + 28y = 7(y2 + 4y)

QUESTION 10

Question:Factorise 9s2 – 27s.

Solution: 9s2 – 27s = 9(s2 – 3s)

Mathematics Form 4 Lesson 07: Quadratic expressions of the form px2 - q where p and q are perfect squares

Learning Area: Factorise quadratic expressions of the form px2 – q,where p and q are perfect squares. 1 / 5


4x2 – 16Solution:

QUESTION 2


25 – r2

Solution:

QUESTION 3


9w2 – 1Solution:

QUESTION 4


4y2 – 16Solution: .

QUESTION 5


64s2 – 25Solution:




QUESTION 6Question:

Factorise 9

4 – 49

2x

Solution:

QUESTION 7

Question:

Factorise 25

1 – x2

Solution:

QUESTION 8

Question:Factorise 100 – 4w2

Solution:

QUESTION 9

Question:Factorise 81y2 – 1.

Solution:

QUESTION 10

Question:Factorise 144 – 121s2.

Solution:




4x2 – 16Solution:4x2 – 16 = (2x)2 – 42

= (2x – 4)(2x + 4)

QUESTION 2


25 – r2

Solution: 25 – r2 = 52 – r2

= (5 – r)(5 + r)

QUESTION 3


9w2 – 1Solution: 9w2 – 1 = (3w)2 – 12

= (3w – 1)(3w + 1)

QUESTION 4


4y2 – 16Solution: 4y2 – 16 = (2y)2 – 42

= (2y – 4)(2y + 4)




QUESTION 5

Question:Factorise the given quadratic expression.64s2 – 25Solution: 64s2 – 25 = (8s)2 – 52

= (8s – 5)(8s + 5)

QUESTION 6

Question:

Factorise 9

4 - 49

2x

Solution:

9

4 - 49

2x=

22

73

2

−

x

=

−73

2 x

+73

2 x



QUESTION 7

Question:

Factorise 25

1 – r2

Solution:

−25

1 r_ =

2

5

1

– r2

=

+

− rr5

1

5

1

QUESTION 8

Question:Factorise 100 – 4w2

Solution: 100 – 4w2 = 102 – (2w)2

= (10 – 2w)(10 + 2w)

QUESTION 9

Question:Factorise 81y2 – 1.

Solution: 81y2 – 1 = (9y)2 – 12

= (9y – 1)(9y + 1)

QUESTION 10

Question:Factorise 144 – 121s2.

Solution: 144 – 121s2 = 122 – (11s)2

= (12 – 11s)(12 + 11s)

Mathematics Form 4 Lesson 08: Quadratic expressions of the form ax2 + bx + c, where a,b and c ≠ 0

Learning Area: Factorise quadratic expressions of the form ax2 + bx + c,where a, b and c are not equal to zero. 1 / 4


4x2 + 22x + 28Solution:

QUESTION 2


10 – 14r – 12r2

Solution:

QUESTION 3


2w2 + w – 1Solution:

QUESTION 4


2y2 + 5y – 12Solution: .

QUESTION 5


24s2 + 49s + 15Solution:




QUESTION 6Question:Factorise 9 – 9x – 4x2.

Solution:

QUESTION 7

Question:Factorise 16 – 6r – r2.

Solution:

QUESTION 8

Question:Factorise 10 – w – 3w2

Solution:

QUESTION 9

Question:Factorise 7y2 + 10y – 8.

Solution:

QUESTION 10

Question:Factorise 2s2 + 7s + 6.

Solution:




4x2 + 22x + 28Solution:4x2 + 22x + 28 = (2x + 4)(2x + 7)

QUESTION 2


10 – 14r – 12r2

Solution: 10 – 14r – 12r2 = (5 – 3r)(2 + 4r)

QUESTION 3


2w2 + w – 1Solution: 2w2 + w – 1 = (2w – 1)(w + 1)

QUESTION 4


2y2 + 5y – 12Solution: 2y2 + 5y – 12 = (2y – 3)( y + 4)

QUESTION 5


24s2 + 49s + 15Solution: 24s2 + 49s + 15 = (8s + 3)(3s + 5)




QUESTION 6Question:Factorise 9 – 9x – 4x2.

Solution: 9 – 9x – 4x2 = (3 – 4x)(3 + x)

QUESTION 7

Question:Factorise 16 – 6r – r2.

Solution: 16 – 6r – r2 = (2 – r)(8 + r)

QUESTION 8

Question:Factorise 10 – w – 3w2

Solution: 10 – w – 3w2 = (5 – 3w)(2 + w)

QUESTION 9

Question:Factorise 7y2 + 10y – 8.

Solution: 7y2 + 10y – 8 = (7y – 4)( y + 2)

QUESTION 10

Question:Factorise 2s2 + 7s + 6.

Solution: 2s2 + 7s + 6 = (2s + 3)(s + 2)

Mathematics Form 4 Lesson 09: Quadratic expressions with common factor coefficients

Learning Area: Factorise quadratic expressions containing coefficients with common factors. 1 / 5


4x2 + 28x + 24Solution:

QUESTION 2


12t2 – 14t – 10Solution:

QUESTION 3


18u2 + 6u – 24Solution:

QUESTION 4


21y2 − 36y – 12Solution: .

QUESTION 5


28r2 − 14r − 84Solution:




QUESTION 6Question:Factorise 30a2 + 55a − 50.

Solution:

QUESTION 7

Question:Factorise 8b2 − 32.

Solution:

QUESTION 8

Question:Factorise 30 – 3w – 9w2.

Solution:

QUESTION 9

Question:Factorise 18y2 – 6y – 24.

Solution:

QUESTION 10

Question:Factorise 24 − 56x − 16x2.

Solution:




4x2 + 28x + 24Solution:4x2 + 28x + 24 = 4(x2 + 7x + 6)

= 4(x + 1)(x + 6)

QUESTION 2


12t2 – 14t – 10Solution: 12t2 – 14t – 10 = 2(6t2 − 7t − 5)

= 2(2t + 1)(3t −5)

QUESTION 3


18u2 + 6u – 24Solution: 18u2 + 6u – 24 = 6(3u2 + u − 4)

= 6(3u + 4)(u − 1)

QUESTION 4


21y2 − 36y – 12Solution: 21y2 − 36y – 12 = 3(7y2 − 12y − 4)

= 3(7y + 2)(y − 2)




QUESTION 5


28r2 −14r − 84Solution: 28r2 −14r − 84 = 14(2r2 − r − 6)

= 14(2r − 3)(r + 2)



QUESTION 6Question:Factorise 30a2 + 55a − 50.

Solution: 30a2 + 55a − 50 = 5(6a2 + 11a _ 10)

= 5(2a + 5)(3a − 2)

QUESTION 7

Question:Factorise 8b2 − 32.

Solution: 8b2 − 32 = 8(b2 − 4)

= 8(b − 2)(b + 2)

QUESTION 8

Question:Factorise 30 – 3w – 9w2.

Solution: 30 – 3w – 9w2 = 3(10 − w − 3w2)

= 3(5 – 3w)(2 + w)

QUESTION 9

Question:Factorise 18y2 − 6y – 24.

Solution: 18y2 − 6y – 24 = 3(6y2 − 2y − 8)

= 3(3y – 4)( 2y + 2)

QUESTION 10

Question:Factorise 24 − 56x − 16x2.

Solution: 24 − 56x − 16x2 = 4(6 − 14x − 4x2)

= 4(3 − x)(2 − 4x)

Mathematics Form 4 Lesson 10: Quadratic equations and their general forms

Learning Area: 1. Identify quadratic equations with one unknown;2. Write quadratic equations in general form i.e. ax2 + bx + c 1 / 5

QUESTION 1


Question:Is “12k2 + k + 2 = 0” a quadratic equation with one unknown? State the reason for your answer.

Solution:

QUESTION 2

Question:Is “4 + 12t + t2 = t – 2” a quadratic equation with one unknown? State the reason for your answer.

Solution:

QUESTION 3

Question:Identify if the equation given is a quadratic equation with one unknown.

3u – 5 = 0

Solution:

QUESTION 4


3e – 4 = 5f2

Solution: .

QUESTION 5


2p2 + 9p = 15

Solution:



QUESTION 6Question:Write the given quadratic equation in general form i.e. ax2 + bx + c.

5c2 − 2c = 25

Solution: Write the given quadratic equation in general form i.e. ax2 + bx + c.

3 + 2e+ 3e2 = e – 2

QUESTION 7

Question:Write the given quadratic equation in general form i.e. ax2 + bx + c.

x2 – 5x = 3x2 + 14

Solution:

QUESTION 8


1 + 9t = 12t2 – 5

Solution:

QUESTION 9

Question:State the values of a, b and c when 5 + 4p – p2 = –3p + 17 is written in general form.

Solution:

QUESTION 10

Question:Write –2a + 8 = 2a2 – 1 in the general form.

Solution:



QUESTION 1Question:Is “12k2 + k + 2 = 0” a quadratic equation with one unknown? State the reason for your answer.

Solution:Yes, because it has one unknown and the highest power of the unknown is equal to 2.

QUESTION 2

Question:Is “4 + 12t + t2 = t – 2” a quadratic equation with one unknown? State the reason for your answer.

Solution: Yes, because it has one unknown and the highest power of the unknown is equal to 2.

QUESTION 3


3u – 5 = 0

Solution: No.

QUESTION 4


3e – 4 = 5f2

Solution:No




QUESTION 5


2p2 + 9p = 15

Solution: Yes.



QUESTION 6Question:Write the given quadratic equation in general form i.e. ax2 + bx + c.

5c2 – 2c = 25

Solution:5c2 – 2c – 25 = 0

QUESTION 7


x2 – 5x = 3x2 + 14

Solution:2x2 + 5x + 14 = 0

QUESTION 8


1 + 9t = 12t2 – 5

Solution:12t2– 4 – 9t = 0

QUESTION 9

Question:State the values of a, b and c when 5 + 4p – p2 = –3p + 17 is written in general form.

Solution: p2 – 7p + 12 = 0, a = 1, b = –7 and c = 12.

QUESTION 10

Question:Write –2a + 8 = 2a2 – 1 in the general form.

Solution:2a2 + 2a – 9 = 0

Mathematics Form 4 Lesson 11: Quadratic equations based on specific situations.

Learning Area: Form quadratic equations based on specific situations. 1 / 6

QUESTION 1Question:The area of a school field is 200 m2. Its length is 5 m longer than the width. Form a quadraticequation based on the situation.

Solution:

QUESTION 2

Question:Abu Bakar builds a rectangular shaped dining table himself. The measurements of the table top areas shown in the diagram. Can we help him find the quadratic equation to represent the situation?

Solution:


Area = 2m3a m

(2a + 4) m



QUESTION 3

Question:Ali and Din went for a cycling trip. Ali cycles at the speed of (r − 2) metre/minute for 4r minuteswhile the distance travelled by Din is 2 670 m. The total distance travelled by both of them is 4 km.Form the quadratic equation to represent the situation above.

Solution:

QUESTION 4

Question:Siew Leng brother is 3 years younger than her and her grandfather’s age is 5 times the product ofboth their ages. Her grandfather’s age is 65. Form a quadratic equation to represent the situationgiven.

Solution:



QUESTION 5

Question:The number of girls in a class is half of the number of boys. If the product of multiplicationbetween the number of girls and the number of boys is equal to 6 times the sum of the girls andboys, form a quadratic equation to represent the situation.

Solution:



QUESTION 1

Question:The area of a school field is 200 m2. Its length is 5 m longer than the width. Form a quadraticequation based on the situation.


Area = width x lengthLet the width be x, therefore the length is x +5

200 = x (x + 5)200 = x2 + 5x

x2 + 5x – 200 = 0

QUESTION 2

Question:Abu Bakar builds a rectangular shaped dining table himself. The measurements of the table top areas shown in the diagram. Can we help him find the quadratic equation to represent the situation?


Area = width x length2 = 3a (2a + 4)2 = 6a2 + 12a

6a2 + 12a – 2 = 0


Area = 2m3a m

(2a + 4) m



QUESTION 3

Question:Ali and Din went for a cycling trip. Ali cycles at the speed of (r _ 2) metre/minute for 4r minuteswhile the distance travelled by Din is 2 670 m. The total distance travelled by both of them is 4 km.Form the quadratic equation to represent the situation above.

Solution:The distance travelled by Ali + The distance travelled by Din = The total distance travelled.[(r – 2) _ 4r] m + 2 670 = 4 0004r2 – 8r + 2 670 = 4 0004r2 – 8r – 1330 = 0

QUESTION 4

Question:Siew Leng brother is 3 years younger than her and her grandfather’s age is 5 times the product ofboth their ages. Her grandfather’s age is 65. Form a quadratic equation to represent the situationgiven.

Solution:Let Siew Leng age be y.5 × (Siew Leng’s age) x (Her brother’s age) = Her grandfather’s age5 × (y) × (y – 3) = 655y (y – 3) = 655y2 – 15y = 655y2 – 15y – 65 = 0



QUESTION 5

Question:The number of girls is a class is half of the number of boys. If the product of multiplication betweenthe number of girls and the number of boys is equal to 6 times the sum of the girls and boys, form aquadratic equation to represent the situation.

Solution:Let the number of girls be g and the number of boys be 2g.The number of girls x The number of boys = 6 (The number of girls + The number of boys)

Substitute the values into the formulag x 2g = 6(g + 2g) 2g2 = 6g + 12g 2g2 = 18g

Form a quadratic equation2g2 – 18g = 0

Mathematics Form 4 Lesson 12: Roots of quadratic equation

Learning Area: Determine whether a given value is the root of a specific quadratic equation. 1 / 7


QUESTION 2

Question:Determine if r = 4 and r = −2 are the roots of the quadratic equation,

r2 − 2r − s8 = 0

Solution:

QUESTION 3

Question:Determine if w = 2 and w = −2 are the roots of the quadratic equation,

2w2 − w −1 = 0

Solution:

QUESTION 4

Question:Determine if y = 1 is a root of the quadratic equation, 2y2 + 4y – 6 = 0.

Solution: .

QUESTION 5

Question:Determine if p = 0 and p = 5 are the roots of the quadratic equation,

p2 – 5p = 0.

Solution:



QUESTION 6Question:Determine if x = 3 and x = −3 are the roots of the quadratic equation,

9 – 9x – 4x2 = 0.

Solution:

QUESTION 7

Question:

014xequation quadratic theof roots theare 2

1 xand ,

2

1 xif Determine 2 =−−==

Solution:

QUESTION 8

Question:

0116yequation quadratic theof roots theare 4

1 y and ,

4

1y if Determine 2 =−−==

Solution:

QUESTION 9

Question:

013sequation quadratic theof roots theare 3

1 s and ,

3

1s if Determine 2 =−−==

Solution:

QUESTION 10

Question:

019tequation quadratic theof roots theare 3

1 tand ,

3

1 tif Determine 2 =−−==

Solution:



QUESTION 1Question:Determine if x = 2 and x = −3 is the root of the equation, x2 + x – 6 = 0.

Solution:x2 + x – 6 = (2)2 + (2) − 6

= (4) + 2 − 6= 0

x2 + x – 6 = (−3)2 + (−3) − 6= (9) − 3 − 6= 0

Both x = 2 and x = −3 are the roots of the quadratic equation x2 + x – 6 = 0.

QUESTION 2

Question:Determine if r = 4 and r = _2 are the roots of the quadratic equation,

r2 − 2r − 8 = 0

Solution:r2 −2r − 8 = (4)2 − 2(4) − 8

= 16 − 8 − 8= 0

r2 − 2r − 8 = (−2)2 − 2(−2) − 8= 4 + 4 − 8= 0

Both r = 4 and r = −2 are the roots of the quadratic equation r2 − 2r − 8 = 0



QUESTION 3

Question:Determine if w = 2 and w = −2 are the roots of the quadratic equation,

2w2 − w −1 = 0

Solution:2w2 − w – 1 = 2(2)2 − (2) − 1

= 8 − 2 − 1= 5

2w2− w – 1 = 2(−2)2 − (−2) − 1= 8 + 2 − 1= 9

Both w = 2 and w = −2 are not the roots of the quadratic equation 2w2 − w − 1 = 0.

QUESTION 4

Question:Determine if y = 1 is a root of the quadratic equation, 2y2 + 4y – 6 = 0.

Solution: 2y2 + 4y – 6 = 2(1)2 + 4(1) − 6

= 2 + 4 − 6= 0

y = 1 is a root of the quadratic equation, 2y2 + 4y – 6 = 0.

QUESTION 5

Question:Determine if p = 0 and p = 5 are the roots of the quadratic equation,

p2 – 5p = 0.

Solution: p2 – 5p = (0)2 − 5(0)

= 0 p2 – 5p = (5)2 − 5(5)

= 25 − 25= 0

Both p = 0 and p = 5 are the roots of the quadratic equation p2 – 5p = 0.



QUESTION 6

Question:Determine if x = 3 and x = − 3 are the roots of the quadratic equation,

9 – 9x – 4x2 = 0.

Solution: 9 – 9x – 4x2 = 9 − 9(3) − 4(3)2

= 9 − 27 − 36= − 54

9 – 9x – 4x2 = 9 − 9(−3) − 4(−3)2

= 9 + 27 − 36= 0

x = 3 is not the roots of the quadratic equation 9 – 9x – 4x2 = 0, while x = −3 is the root of thequadratic equation.



QUESTION 7Question:

014xequation quadratic theof roots theare 2

1 xand ,

2

1 xif Determine 2 =−−==

Solution:

0

1-1

1)4

14(

1)2

1(414 22

=

=

−=

−=−x

0

1-1

1)4

14(

1)2

1(414 22

=

=

−=

−−=−x

014x equation, quadratic theof roots theare 2

1 xand ,

2

1Both x 2 =−−==

QUESTION 8

Question:

0116yequation quadratic theof roots theare 4

1 y and ,

4

1y if Determine 2 =−−==

Solution:

0

1-1

1)16

116(

1)4

1(16116 22

=

=

−=

−=−y

0

1-1

1)16

116(

1)4

1(16116 22

=

=

−=

−−=−y

0116y equation, quadratic theof roots theare 4

1 y and ,

4

1yBoth 2 =−−==



QUESTION 9Question:

013sequation quadratic theof roots theare 3

1 s and ,

3

1s if Determine 2 =−−==

Solution:

3

2

13

1

1)9

13(

1)3

1(313 22

−=

−=

−=

−=−s

3

2

13

1

1)9

13(

1)3

1(313 22

−=

−=

−=

−−=−s

013s equation, quadratic theof roots not the are 3

1 s and ,

3

1sBoth 2 =−−==

QUESTION 10

Question:

019tequation quadratic theof roots theare 3

1 tand ,

3

1 tif Determine 2 =−−==

Solution:

0

1-1

1)9

19(

1)3

1(919 22

=

=

−=

−=−t

0

1-1

1)9

19(

1)3

1(919 22

=

=

−=

−−=−t

019t equation, quadratic theof roots theare 3

1 tand ,

3

1Both t 2 =−−==

Mathematics Form 4 Lesson 13: Solutions for quadratic equations by trial and error.

Learning Area: Determine the solutions for quadratic equations by trial and error.1 / 7

QUESTION 1Question:Determine the solution for h2 + 17h + 16 = 0 by trial and error method.

Solution:

QUESTION 2

Question:Determine the solution for a2 − 9a + 14 = 0 by trial and error method.

Solution:

QUESTION 3

Question:Determine the solution for z2 + 7n + 10 = 0 by trial and error method.

Solution:

QUESTION 4

Question:Determine the solution for p2 + 8p − 9 = 0 by trial and error method.

Solution: .

QUESTION 5

Question:Determine the solution for b2 + 6b + 8 = 0 by trial and error method.

Solution:




QUESTION 6Question:Determine the solution for 2x2 + 7x + 3 = 0 by trial and error method.

Solution:

QUESTION 7

Question:Determine the solution for 2y2 + 8y + 6 = 0 by trial and error method.

Solution:

QUESTION 8

Question:Determine the solution for 9x2 − 12x + 4 = 0 by trial and error method.

Solution:

QUESTION 9

Question:Determine the solution for 3 − n − 2n2 = 0 by trial and error method.

Solution:

QUESTION 10

Question:Determine the solution for 12q2 + 18q + 6 = 0 by trial and error method.

Solution:



QUESTION 1Question:Determine the solution for h2 + 17h + 16 = 0 by trial and error method.

Solution:h2 + 17h + 16 = (−16)2 + 17(−16) + 16

= 256 − 272 + 16= 0

h2 + 17h + 16 = (−1)2 + 17(−1) + 16= 1−17 + 16= 0

h = −1 or h = −16

QUESTION 2

Question:Determine the solution for a2 − 9a + 14 = 0 by trial and error method.

Solution:a2 − 9a + 14 = (2)2 − 9(2) + 14

= 4 − 18 + 14= 0

a2 − 9a + 14 = (7)2 − 9(7) + 14= 49 − 63 + 14= 0

a = 2 or a = 7




QUESTION 3

Question:Determine the solution for n2 + 7n + 10 = 0 by trial and error method.

Solution:n2 + 7n + 10 = (− 2)2 + 7(−2) + 10

= 4 − 14 + 10= 0

n2 + 7n + 10 = (−5)2 + 7(−5) + 10= 25 − 35 + 10= 0

n = −2 or n = −5

QUESTION 4

Question:Determine the solution for p2 + 8p − 9 = 0 by trial and error method.

Solution:p2 + 8p − 9 = (1)2 + 8(1) − 9

= 1 + 8 − 9= 0

p2 + 8p − 9 = (− 9)2 + 8(−9) − 9= 81 − 72 − 9= 0

p = 1 or p = −9



QUESTION 5

Question:Determine the solution for b2 + 6b + 8 = 0 by trial and error method.Solution:b2 + 6b + 8 = (− 2)2 + 6(−2) + 8

= 4 − 12 + 8= 0

b2 + 6b + 8 = (− 4)2 + 6(−4) + 8= 16 − 24 + 8= 0

b = −2 or b = −4

QUESTION 6

Question:Determine the solution for 2x2 + 7x + 3 = 0 by trial and error method.

Solution:2x2 + 7x + 3 = 2(−1/2)2 + 7(−1/2) +3

= 1/2 − 7/2 + 3= 0

2x2 + 7x + 3 = 2(−3)2 + 7(−3) +3= 18 − 21 + 3= 0

x = −1/2 or x = −3



QUESTION 7

Question:Determine the solution for 2y2 + 8y + 6 = 0 by trial and error method.

Solution:2y2 + 8y + 6 = 2(−1)2 + 8(−1) + 6

= 2 − 8 + 6= 0

2y2 + 8y + 6 = 2(−3)2 + 8(−3) + 6= 18 − 24 + 6= 0

y = −1 or y = −3

QUESTION 8

Question:Determine the solution for 9x2 − 12x + 4 = 0 by trial and error method.

Solution:9x2 − 12x + 4 = 9(2/3)2 − 12(2/3) + 4

= 4 − 8 + 4= 0

x = 2/3QUESTION 9

Question:Determine the solution for 3 − n − 2n2 = 0 by trial and error method.

Solution:3 − n − 2n2 = 3 − (1) − 2(1)2

= 3 − 1 − 2= 0

3 − n − 2n2 = 3 − (−3/2) − 2(_3/2)2

= 3 + 3/2 − 9/2= 0

n = 1 or n = −3/2

Question:Determine the solution for 12q2 + 18q + 6 = 0 by trial and error method.Solution:12q2 + 18q + 6 = 12(−1/2)2 + 18(−1/2) + 6

= 3 − 9 + 6= 0

12q2 + 18q + 6 = 12(−1)2 + 18(−1) + 6= 12 − 18 + 6= 0

q = −1/2 or −1

Mathematics Form 4 Lesson 14: Solutions for quadratic equations by factorisation.

Learning Area: Determine the solutions for quadratic equations by factorisation.1 / 6

QUESTION 1Question:Determine the solution for p2 − 5p + 6 = 0 by factorisation.

Solution:

QUESTION 2

Question:Determine the solution for 6m2 + 5m +1 = 0 by factorisation.

Solution:

QUESTION 3

Question:Determine the solution for 3m2 + 3m − 6 = 0.

Solution:

QUESTION 4

Question:Determine the solution for b2 + 3b − 10 = 0.

Solution: .

QUESTION 5

Question:Factorise j2 + 7j + 12 and determine the solution for j2 + 7j + 12 = 0.

Solution:




QUESTION 6Question:Factorise x2 + 5x + 6 and determine the solution for x2 + 5x + 6 = 0.

Solution:

QUESTION 7

Question:Determine the solution for w2 − 13w + 36 = 0 by factorisation.

Solution:

QUESTION 8

Question:Determine the solution for x2 − 4x − 5 = 0 by factorisation.

Solution:

QUESTION 9

Question:What is the solution for t2 − 10t + 25 = 0 by factorisation?

Solution:

QUESTION 10

Question:What is the solution for 12r2 − 12r + 3 = 0 by factorisation?

Solution:



QUESTION 1Question:Determine the solution for p2 − 5p − 6 = 0 by factorisation.

Solution:p2 − 5p − 6 = (p + 1)(p − 6)

p + 1 = 0p = −1

p − 6 = 0p = 6

p = −1 or p = 6

QUESTION 2

Question:Determine the solution for 6m2 + 5m +1 = 0 by factorisation.

Solution:6m2 + 5m +1 = (2m + 1)(3m + 1)

2m + 1 = 02m = −1

m = −1/23m + 1 = 0

3m = −1m = _1/3

m = −1/2 or m = −1/3

QUESTION 3

Question:Determine the solution for 3m2 + 3m − 6 = 0.

Solution:3m2 + 3m − 6 = 3(m2 + m − 2)

= 3(m − 1)(m + 2)m − 1 = 0

m = 1m + 2 = 0

m = −2m = 1or m = −2




QUESTION 4

Question:Determine the solution for b2 + 3b − 10 = 0.

Solution:b2 + 3b − 10 = (b + 5)(b − 2)

b + 5 = 0b = −5

b − 2 = 0b = 2

b = 2 or b = −5

QUESTION 5

Question:Factorise j2 + 7j + 12 and determine the solution for j2 + 7j + 12 = 0.

Solution:j2 + 8j + 12 = (j + 2)(j +6)

j + 2 = 0j = −2

j + 6 = 0j = −6

j = −2 or j = −6

QUESTION 6

Question:Factorise x2 + 5x + 6 and determine the solution for x2 + 5x + 6 = 0.

Solution:x2 + 5x + 6 = (x + 3)(x + 2)

x + 3 = 0x = −3

x + 2 = 0x = −2

x = −3 or x = −2



QUESTION 7

Question:Determine the solution for w2 15w + 36 = 0 by factorisation.

Solution:w2 − 15w + 36 = (w − 3)(w − 12)

w − 3 = 0w = 3

w − 12 = 0w = 12

w = 3 or w = 12

QUESTION 8

Question:Determine the solution for x2 − 4x − 5 = 0 by factorisation.

Solution:x2 − 4x − 5 = (x − 5)(x + 1)

x − 5 = 0x = 5x + 1 = 0x = −1

x = 5 or x = −1QUESTION 9

Question:What is the solution for t2 − 10t + 25 = 0 by factorisation?

Solution:

t2 + t − 30 = (t − 5)(t + 6)t − 5 = 0

t = 5t + 6 = 0

t = −6t = 5 or t = −6

QUESTION 10

Question:What is the solution for 12r2 − 12r + 3 = 0 by factorisation?

Solution:12r2 − 12r + 3 = 3(4r2 − 4r +1)

= 3(2r − 1)(2r − 1)2r − 1 = 0

2r = 1r = 1/2

Mathematics Form 4 Lesson 15: Quadratic equations in everyday life

Learning Area: Solve problems involving quadratic equations. 1 / 6

QUESTION 1

Question:A rectangular shaped playing field has a length which is 13 m more than its width. If the area is30 m2, find the length and width of the field.

Solution:

QUESTION 2

Question:A piece of wood was used to form a square. If the area of the square is 25 m2, determine the lengthof one side of the square.

Solution:




QUESTION 3

Question:Ali owns a few houses in the rural area. He wishes to sell some of them to invest in one of hisinvestment plans. He offers to sell n number of houses for RM600 000 and if the number of housesincreases by 1, then the price for the price of each house will be reduce by RM50 000. Determinethe number of houses he offers for sale in the first offer.

Solution:

QUESTION 4

Question:Encik Ahmad arranged 48 flower pots in a few rows in his garden. He re-arranges the flower potsand found that if he adds 4 flower pots to every row he will have 1 row less than the firstarrangement. Determine the number of flower pots per row in the first arrangement.

Solution:



QUESTION 5

Question:A swimming pool has the length of 12 m and the width of 8 m. A fence is build at the distance of xmetre around it. If the area of the swimming pool is equal to the area of the land bounded by thefence, determine the value of x.

Solution:



QUESTION 1

Question:A rectangular shaped playing field has a length which is 13 m more than its width. If the area is 30m2, find the length and width of the room.

Solution:Formula Area = width x lengthLet the width = w, therefore the length = w + 13Substitute the given values into the formulaArea = width _ length30 = w x (w + 13)30 = w2 + 13wW2 + 13w − 30 = 0(w − 2)(w + 15) = 0w − 2 = 0 w + 15 = 0 w = 2 w = −15

Therefore we can say the width is equal to 2m and the length is equal to 15m.


QUESTION 2

Question:A piece of wood was used to form a square. If the area of the square is 25 m2, determine the lengthof one side of the square.

Solution:Formula Area of a square = length x widthLength = width = sTherefore,Area = s2

25 = s2

s =√25s = 5The length of one side of the square is 5m.



QUESTION 3

Question:Ali owns a few houses in the rural area. He wishes to sell some of them to invest in one of hisinvestment plans. He offers to sell n number of houses for RM600 000 and if the number of housesincreases by 1, then the price for the price of each house will be reduce by RM50 000. Determinethe number of houses he offers for sale in the first offer.

Solution:The difference of price of a house of the 2 offers is RM50000.Therefore,

50000n2 + 50000n – 600000

We can conclude that the number of houses offer for sale in the first offer is 3.

QUESTION 4

Question:Encik Ahmad arranged 48 flower pots in a few rows in his garden. He re-arranges the flower potsand found that if he adds 4 flower pots to every row he will have 1 row less than the firstarrangement. Determine the number of flower pots per row in the first arrangement.

Solution:The difference of rows of the two arrangements is 3.Therefore,

We can conclude that the number of flower pots per row in the first arrangement is 12.

500001

000 600000 600 =−+nn

50000)1()1()1(1

000 600000 600 +=+−++

nnnnnnnn

)1(50000600000)600000600000( +=−+ nnnn

nn 5000050000600000 2 +=

0)12(50000 2 =−+ nn

0)4)(3(50000 =+− nn

4or 3 −=n

14

4848 =−+nn

1)4()4()4(4

4848 +=+−++

nnnnnnnn

)4(48)19248( +=−+ nnnn

nn 4192 2 +=

019242 =−+ nn

0)16)(12( =+− nn16or 21 −=n



QUESTION 5

Question:A swimming pool has the length of 12 m and the width of 8 m. A fence is build at the distance of xmetre around it. If the area of the swimming pool is equal to the area of the land bounded by thefence, determine the value of x.

Solution:If the area of the swimming pool is equal to the area of the land bounded by the fence,(2x + 12)(2x +8) = 96(2)4x2 + 40x + 96 = 1924x2 + 40x − 96 = 04(x2 + 10x − 24) = 04(x − 2)(x + 12) = 0x − 2 = 0 x = −12Therefore x = 2 or x = −12.

Mathematics Form 4 Lesson 16: Set

Learning Area:1. Sort given objects into groups 2.Define sets by: i) description ii) using set notation 1 / 5

QUESTION 1Question:Sort the given objects into three groups.Dog, eagle, cat, house fly, cow, mosquito, ant, sparrow and kingfisher.

Solution:

QUESTION 2

Question:The numbers of pupils in each form 4 class in S.M. Taman Puteri are 37, 35, 39, 37, 36, 38 and 35.Write the set above by:

a) Descriptionb) Using set notation.

Solution:




QUESTION 3

Question:December, April, March, May, January, June, July and February.Write the set above by:


Solution:

QUESTION 4

Question:Use set notation to write the sets given below.

a) Set of even numbers from 10 to 20b) Set of odd numbers from 20 to 30c) Set of positive integers that is less than 6.

Solution:



QUESTION 5

Question:Write the following sets in the form of A = {x : a ≤ x ≤ b, x is a whole number}

a) P = {4, 5, 6, …, 12}b) Q = {11, 12, 13, …, 20}c) R = {2, 3, 5, 7, 11, 13}d) S = {1, 3, 5, …, 13}e) T = {2, 4, 6, …, 12}

Solution:



QUESTION 1

Question:Sort the given objects into three groups.Dog, eagle, cat, house fly, cow, mosquito, ant, sparrow and kingfisher.

Solution:a) Set of animals = {dog, cat, cow}b) Set of insects = {house fly, mosquito, ant}c) Set of birds = {eagle, sparrow, kingfisher}

QUESTION 2

Question:The numbers of pupils in each form 4 class in S.M. Taman Puteri are 37, 35, 39, 37, 36, 38 and 35.Write the set above by:

c) Descriptiond) Using set notation.

Solution:a) Set of numbers of pupils in form 4 classes in S.M. Taman Puterib) A = {37, 35, 39, 37, 36, 38, 35}

QUESTION 3

Question:December, April, March, May, January, June, July and February.Write the set above by:


Solution:a) Set of months in a year.b) B = {December, April, March, May, January, June, July, February}




QUESTION 4

Question:Use set notation to write the sets given below.

a) Set of even numbers from 10 to 20b) Set of odd numbers from 20 to 30c) Set of positive integers that is less than 6.

Solution:a) E = {10, 12, 14, 16, 18, 20}b) O = {21, 23, 25, 27, 29}c) P = {1, 2, 3, 4, 5}

QUESTION 5

Question:Write the following sets in the form of A = {x : a ≤ x ≤ b, x is a whole number}

a) P = {4, 5, 6, …, 12}b) Q = {11, 12, 13, …, 20}c) R = {2, 3, 5, 7, 11, 13}d) S = {1, 3, 5, …, 13}e) T = {2, 4, 6, …, 12}

Solution:a) P = {x : 4 ≤ x ≤ b12, x is a whole number}b) Q = {x : 11 ≤ x ≤ 20, x is a whole number}c) R = {x : 2 ≤ x ≤ 13, x is a prime number}d) S = {x : 1 ≤ x ≤ 13, x is a odd number}e) T = {x : 2 ≤ x ≤ 12, x is a even number}

Mathematics Form 4 Lesson 17: Elements of a set.

Learning Area: Identify whether a given object is a element of a set and use the symbol∈ or ∉

1 / 6

QUESTION 1Question:Determine if the sports given are the element of A = {sports competed in the MSSM}

a) Footballb) Basketballc) Badmintond) Table tennise) Tennisf) Squash

Solution:




2 / 6

QUESTION 2

Question:Determine if the months given are the element of Q = {Months in a year which have 31 days}

a) Januaryb) Februaryc) Marchd) Aprile) Mayf) June

Solution:

QUESTION 3

Question:Determine if the numbers given are the element of Y = {x : 1 ≤ x ≤ 20, x is a multiple of 3}Write the set above by:

a) 1b) 6c) 15d) 17e) 19f) 21

Solution:



3 / 6

QUESTION 4

Question:Determine if the numbers given are the element of R = {x : −10 ≤ x ≤ 10, x is a positive integer}

a) −10b) −5c) 0d) 5e) 10

Solution:

QUESTION 5

Question:Determine if the statement given below is True or False.

a) R ∈ {alphabets of the word ROUGH}b) 7 ∉ {prime numbers}c) 0 ∉ {positive integers}d) Malaysia ∈ {countries in Asean}e) 20 ∈ {multiple of 4}

Solution:



4 / 6

QUESTION 1

Question:Determine if the sports given are the element of A = {sports competed in the MSSM}

a) Footballb) Basketballc) Badmintond) Table tennise) Tennisf) Squash

Solution:a) Football is an element of set A.b) Basketball is an element of set A.c) Badminton is an element of set A.d) Table tennis is an element of set A.e) Tennis is an element of set A.f) Squash is an element of set A.

QUESTION 2

Question:Determine if the months given are the element of Q = {Months in a year which have 31 days}

a) Januaryb) Februaryc) Marchd) Aprile) Mayf) June

Solution:a) January is an element of set Q.b) February is not an element of set Q.c) March is an element of set Q.d) April is not an element of set Q.e) May is an element of set Q.f) June is not an element of set Q.




5 / 6

QUESTION 3

Question:Determine if the numbers given are the element of Y = {x : 1 ≤ x ≤ 20, x is a multiple of 3}by usingthe symbol ∈ or ∉.

a) 1b) 6c) 15d) 17e) 19f) 21

Solution:a) 1 ∉ Yb) 6 ∈ Yc) 15 ∈ Yd) 17 ∉ Ye) 19 ∉ Yf) 21 ∉ Y

QUESTION 4

Question:Determine if the numbers given are the element of R = {x: −10 ≤ x ≤ 10, x is a positive integer} byusing the symbol ∈ or ∉.

a) −10b) −5c) 0d) 5e) 10

Solution:a) −10 ∉ Rb) −5 ∉ Rc) 0 ∉ Rd) 5 ∈ Re) 10 ∈ R



6 / 6

QUESTION 5

Question:Determine if the statement given below is True or False.

a) R ∈{alphabets of the word ROUGH}b) 7 ∉ {prime numbers}c) 0 ∉ {positive integers}d) Malaysia ∈ {countries in Asean}e) 20 ∈ {multiple of 4}

Solution:a) R ∈ {alphabets of the word ROUGH}(True)b) 7 ∉ {prime numbers}(False)c) 0 ∉ {positive integers}(True)d) Malaysia ∈ {countries in Asean}(True)e) 20 ∈ {multiple of 4}(True)

Mathematics Form 4 Lesson 18: Sets representations using Venn diagrams.

Learning Area: Represent sets by using Venn diagrams. 1 / 6

QUESTION 1Question:Given that P = {1, 2, 3, 4, 5}. Represent set P by using a Venn diagram

Solution:

QUESTION 2

Question:If D = {x : 1 ≤ x ≤ 10, x is an odd number}, can you draw a Venn diagram to represent the elementsin this set?

Solution:




QUESTION 3

Question:If R = {x : 30 ≤ x ≤ 40, x is a multiple of 3}, can you draw a Venn diagram to represent the elementsin this set?

Solution:

QUESTION 4

Question:If T = {x : 30 ≤ x ≤ 50, x is a multiple of 5 and 10}, can you draw a Venn diagram to represent theelements in this set?

Solution:



QUESTION 5

Question:If M = {states of Malaysia}, can you draw a Venn diagram to represent the number of elements inthis set?

Solution:



QUESTION 1

Question:Given that P = {1, 2, 3, 4, 5}. Represent set P by using a Venn diagram

Solution:

QUESTION 2

Question:If D = {x : 1 ≤ x ≤ 10, x is an odd number}, can you draw a Venn diagram to represent the elementsin this set?

Solution:


P• 1

• 5

• 2

• 3

• 4

D• 1

• 9

• 3

• 5

• 7



QUESTION 3

Question:If R = {x : 30 ≤ x ≤ 40, x is a multiple of 3}, can you draw a Venn diagram to represent the elementsin this set?

Solution:

QUESTION 4

Question:If T = {x : 30 ≤ x ≤ 50, x is a multiple of 5 and 10}, can you draw a Venn diagram to represent theelements in this set?

Solution:

R• 30 • 33

• 36• 39

T• 30 • 40

• 50



QUESTION 5

Question:If M = {states of Malaysia}, can you draw a Venn diagram to represent the number of elements inthis set?

Solution:

M14

Mathematics Form 4 Lesson 19: Number of elements of a set and an empty set.

Learning Area: 1. List the elements and state the number of elements of a set,2. Determine whether a set is an empty set. 1 / 5

QUESTION 1


Question:A is the set of letters in the word “TELECOMUNICATION”. List the elements of set A and state thenumber of elements.

Solution:

QUESTION 2

Question:S is the set of multiples of 4 less than 20 that can be divided by 11. List the elements of number of set S.How many elements in it?

Solution:

QUESTION 3

Question:Given that Y is the Number of elements of a set and empty set of letters in the word “ACTIVITIES”.

(a) List the elements of set Y.(a) State the number of elements of set Y.

Solution:

QUESTION 4

Question:M is the set of the months in a year with 31 days. List the elements of set M and state the number of itselements.

Solution:

QUESTION 5

Question:Given that set A is the set of prime numbers that are less than 0. Determine whether Number of elementsof a set and empty set A is an empty set.

Solution:



QUESTION 6Question:Given that set B is the set of the factors of 12. Determine whether Number of elements of a set andempty set B is an empty set.

Solution:

QUESTION 7

Question:Given that set D is the set of even numbers less than 10. Determine whether number of elements ofa set and empty set D is an empty set.

Solution:

QUESTION 8

Question:Determine whether Q is an empty set given that set Q = {x: x is prime number and 11 < x < 20}.

Solution:

QUESTION 9

Question:X is the set of common multiples of 2 and 3 which are less than 5. Is set X an empty set?

Solution:

QUESTION 10

Question:Determine whether T is an empty set given that set T = {x: x is factor of 3 and 1 < x < 10}

Solution:



QUESTION 1

Question:A is the set of letters in the word “TELECOMUNICATION”. List the elements of set A and state thenumber of elements.

Solution:A = {T, E, L, C, O, M, U, N, I, A, T,}n(A) = 11

QUESTION 2

Question:S is the set of multiples of 4 less than 20 that can be divided by 11. List the elements set S. Howmany elements in it?

Solution: S = {}n(S) = 0

QUESTION 3

Question:Given that Y is the Number of elements of a set and empty set of letters in the word “ACTIVITIES”.

(b) List the elements of set Y.(b) State the number of elements of set Y.

Solution:Y = {A, C, T, I, V, E, S}n(Y) = 7

QUESTION 4

Question:M is the set of the months in a year with 31 days. List the elements of set M and state the number ofits elements.

Solution:M = {January, March, May, July, August, October, December}n(M) = 7




QUESTION 5

Question:Given that set A is the set of prime numbers that are less than 0. Determine whether Number ofelements of a set and empty set A is an empty set.

Solution:A = { }Set A is an empty set.

QUESTION 6Question:Given that set B is the set of the factors of 12. Determine whether Number of elements of a set andempty set B is an empty set.

Solution:B = {1, 2, 3, 4, 6, 12}Set B is no an empty set.

QUESTION 7Question:Given that set D is the set of even numbers less than 10. Determine whether Number of elements of aset and empty set D is an empty set.

Solution:D = {2, 4, 6, 8}Set D is not an empty set.

QUESTION 8Question:Determine whether Q is an empty set given that set Q = {x: x is prime number and 11 < x < 20}.

Solution:Q = {13, 17, 19}Set Q is not an empty set.

QUESTION 9

Question:X is the set of common multiples of 2 and 3 which are less than 5. Is set X an empty set?

Solution:X = { }Set X is an empty set.



QUESTION 10

Question:Determine whether T is an empty set given that set T = {x: x is factor of 3 and 1 < x < 10}

Solution:T = {3}Set T is not an empty set.

Mathematics Form 4 Lesson 20: Equal sets.

Learning Area: Determine whether two sets are equal. 1 / 6

QUESTION 1


Question:A = { , , , }B = { , , , }Determine if sets A and B are equal.

Solution:

QUESTION 2

Question:C = { , , , }D = { , , , }Determine if sets C and D are equal.

Solution:

QUESTION 3

Question:E = { , , , }F = { , , , }Determine if sets E and F are equal.

Solution:

QUESTION 4

Question:P = {x : x is an odd number, 11 ≤ x ≤ 20}Q = {11, 13, 15, 17, 19}Are sets P and Q equal?

Solution: .



QUESTION 5

Question:X = {multiples of 2 which are between 10 and 20}Y = {Even numbers between 10 and 20}Are sets X and Y equal?

Solution:

QUESTION 6

Question:S = {x : x is a prime number, 1 ≤ x ≤ 10}T = {x : x is an odd number, 1 ≤ x ≤ 10}Are sets S and T equal?

Solution:

QUESTION 7

Question:V = {x : x is a prime number, 11 ≤ x ≤ 20}U = {11, 13, 17, 19}Are sets V and U equal?

Solution:

QUESTION 8

Question:Determine whether sets G and H are equal if set G is equal to common factors of 2 while H is evennumbers which are less than 10.

Solution:



QUESTION 9

Question:W = {vowels in the word ‘STRAIGHT’}V = {vowels in the word ‘BAIT’}Are sets W and V equal?

Solution:

QUESTION 10

Question:K = {A, B, O, R, T}L = {A, B, U, R, T}Are sets K and L equal?

Solution:



QUESTION 1


Question:A = { , , , }B = { , , , }Determine if sets A and B are equal.

Solution: A = B

QUESTION 2Question:C = { , , , }D = { , , , }Determine if sets C and D are equal.

Solution: C ≠ D

QUESTION 3Question:

E = { , , , }F = { , , , }Determine if sets E and F are equal.

Solution: E = F

QUESTION 4Question:P = {x : x is an odd number, 11 ≤ x ≤ 20}Q = {11, 13, 15, 17, 19}Are sets P and Q equal?

Solution:P = {11, 13, 15, 17, 19}Q = {11, 13, 15, 17, 19}P = Q



QUESTION 5Question:X = {multiples of 2 which are between 10 and 20}Y = {Even numbers between 10 and 20}Are sets X and Y equal?

Solution:X = {12, 14, 16, 18}Y = {12, 14, 16, 18}X = Y

QUESTION 6Question:S = {x : x is a prime number, 1 ≤ x ≤ 10}T = {x : x is an odd number, 1 ≤ x ≤ 10}Are sets S and T equal?

Solution:S = {2, 3, 5, 7}T = {1, 3, 5, 7, 9}S ≠ T

QUESTION 7Question:V = {x : x is a prime number, 11 ≤ x ≤ 20}U = {11, 13, 17, 19}Are sets V and U equal?

Solution:V = {11, 13, 17, 19}U = {11, 13, 17, 19}V = W

QUESTION 8Question:Determine whether sets G and H are equal if set G is equal to common factors of 2 while H is evennumbers which are less than 10.

Solution:G = {1, 2}H = {2, 4, 6, 8}G ≠ H



QUESTION 9Question:W = {vowels in the word ‘STRAIGHT’}V = {vowels in the word ‘BAIT’}Are sets W and V equal?

Solution:W = {A, I}V = {A, I}W = V

QUESTION 10Question:K = {A, B, O, R, T}L = {A, B, U, R, T}Are sets K and L equal?

Solution:K≠ L

Mathematics Form 4 Lesson 21: Subset.

Learning Area: Determine whether a given set is a subset of a specific set and use the symbol ⊂ or ⊄. 1 / 7

QUESTION 1


Question:A = { , , , }B = { , , }Determine if set B is a subset of set A.

Solution:

QUESTION 2

Question:C = { , , , }D = { , , , }Determine if set D is a subset of set C.

Solution:

QUESTION 3

Question:E = { , , , }F = { , , , }Determine if set F is a subset of set E

QUESTION 4

Question:S = {a : a is an odd number, 1 ≤ a ≤ 10}T = {1,5, 7}Is set T a subset of set S?

Solution: .



QUESTION 5

Question:G = {multiples of 5 which are between 10 and 30}H = {Even numbers between 10 and 20}Is set H a subset of set G?

Solution:

QUESTION 6

Question:P = {x : x is a prime number, 1 ≤ x ≤ 10}Q = {3, 7}Is set Q a subset of set P?

Solution:

QUESTION 7

Question:M = {x : x is a prime number, 11 ≤ x ≤ 20}N = {11, 13, 17}Is set N a subset of set M?

Solution:

QUESTION 8

Question:Determine whether set R is a subset of set S if set R is equal to common factors of 2 while set S iseven numbers which are less than 10.

Solution:



QUESTION 9

Question:X = {vowels in the word ‘TROUGH’}Y = {vowels in the word ‘TOUGH’}Is set Y a subset of set X?

Solution:

QUESTION 10

Question:V = {SECONDARY}U = {DRY}Is set U a subset of set V?

Solution:



QUESTION 1


Question:A = { , , , }B = { , , }Determine if set B is a subset of set A.

Solution: B ⊂ A

QUESTION 2Question:C = { , , , }D = { , , , }Determine if set D is a subset of set C.

Solution: D ⊄ C

QUESTION 3Question:E = { , , , }F = { , , , }Determine if set F is a subset of set E

Solution: F ⊂ E

QUESTION 4Question:S = {a : a is an odd number, 1 ≤ a ≤ 10}T = {1,5, 7}Is set T a subset of set S?

Solution:S = {1, 3, 5, 7, 9}T = {1, 5, 7}T ⊂ S



QUESTION 5Question:G = {multiples of 5 which are between 10 and 30}H = {Even numbers between 10 and 20}Is set H a subset of set G?

Solution:G = {15, 20 25}H = {12, 14, 16, 18}H ⊄ G

QUESTION 6Question:P = {x : x is a prime number, 1 ≤ x ≤ 10}Q = {3, 7}Is set Q a subset of set P?

Solution:P = {2, 3, 5, 7}Q = {3, 7}Q ⊂ P

QUESTION 7Question:M = {x : x is a prime number, 11 ≤ x ≤ 20}N = {11, 13, 17}Is set N a subset of set M?

Solution:M = {11, 13, 17, 19}N = {11, 13, 17}N ⊂ M

QUESTION 8Question:Determine whether set R is a subset of set S if set R is equal to common factors of 2 while set S iseven numbers, which are less than 10.

Solution:R = {1, 2}S = {2, 4, 6, 8}R ⊄ S



QUESTION 9Question:X = {vowels in the word ‘TROUGH’}Y = {vowels in the word ‘TOUGH’}Is set Y a subset of set X?

Solution:X = {O, U}Y = {O, U}Y ⊂ X

QUESTION 10Question:V = {SECONDARY}U = {DRY}Is set U a subset of set V?

Solution:U ⊂ V

Mathematics Form 4 Lesson 22: Subsets representations using Venn diagrams.

Learning Area: 1. Represent subsets by using Venn diagrams.2. List the subsets for specific set. 1 / 4

QUESTION 1Question:Given that P = {1, 2, 3, 4, 5} and Q = {1, 3, 5}. Show the relationship between sets P and Q using aVenn diagram

Solution:

QUESTION 2

Question:If A = {x : 1 ≤ x ≤ 10, x is an even number} and B = {2, 4, 6}, can you draw a Venn diagram torepresent the relationship between these two sets?

Solution:




QUESTION 3

Question:List all the subsets of set M in the diagram shown below.

Solution:

QUESTION 4

Question:If R = {x : 30 ≤ x ≤ 50, x is a multiple of 5 and 10}, can you list all the subsets for this set?

Solution:

QUESTION 5

Question:If T = {1, 2, 3, 4, 5}, can you list all the subsets of set T which contain three elements?

Solution:

M

• 1

• 5• 3

• 6

• 4

• 2



QUESTION 1

Question:Given that P = {1, 2, 3, 4, 5} and Q = {1, 3, 5}. Show the relationship between sets P and Q using aVenn diagram

Solution:

QUESTION 2

Question:If A = {x : 1 ≤ x ≤ 10, x is an even number} and B = {2, 4, 6}, can you draw a Venn diagram torepresent the relationship between these two sets?

Solution:


P

• 1

• 5

• 2

• 3

• 4

A

• 2

• 8

• 4• 6

• 10

Q

B



QUESTION 3

Question:List all the subsets of set M in the diagram shown below.

Solution:M = {1, 2, 3, 4}The subsets of set M are:{ } {1} {2} {3} {4} {1, 2} {1, 3} {1, 4} {2, 3} {2, 4} {3, 4}{1, 2, 3} {1, 2, 4} {2, 3, 4} {1, 3, 4}{1,2,3,4}

QUESTION 4

Question:If R = {x : 30 ≤ x ≤ 50, x is a multiple of 5 and 10}, can you list all the subsets for this set?

Solution:R = {30, 40, 50}The subsets of set R are:{ } {30} {40} {50}{30, 40} {30, 50} {40, 50}{30,40,50}

QUESTION 5

Question:If T = {1, 2, 3, 4, 5}, can you list all the subsets of set T which contain three elements?

Solution:The subsets of set T which contain three elements are:{1, 2, 3} {1, 2, 4} {1, 2, 5} {2, 3, 4} {2, 3, 5}{3, 4, 5} {1, 3, 4} {1, 3, 5} {1, 4, 5} {2, 4, 5}

M

• 1

• 3• 4

• 2

Mathematics Form 4 Lesson 23: Set, universal set and the compliment of a given set.

Learning Area: 1. Ilustrate the relationship between set and universal set using Venndiagram.

2. Determine the complement of a given set. 1 / 4

QUESTION 1

Question:Given that ξ = {pupils of S.M. Puteri} and Q = {male pupils of S.M. Puteri}. Show the relationshipbetween sets ξ and Q using a Venn diagram

Solution:

QUESTION 2

Question:If ξ = {x : 1 ≤ x ≤ 10, x is a prime number} and B = {2, 3}, can you draw a Venn diagram torepresent the relationship between these two sets?

Solution:





QUESTION 3

Question:Determine the complement of set M in the diagram shown below.

Solution:

QUESTION 4

Question:If ξ = {x : 1 ≤ x ≤ 10, x is a multiple of 2} and A = {4, 6}, can you determine the complement of setA?

Solution:

QUESTION 5

Question:If ξ = {A, B, C, D, E, F} and R = {D, E, F}, can you determine the complement of set R?

Solution:

M

• 1

• 5• 3

• 6

• 4

• 2

• 9

• 10

• 7

• 8

ξ




QUESTION 1

Question:Given that ξ = {pupils of S.M. Puteri} and Q = {male pupils of S.M. Puteri}. Show the relationshipbetween sets ξ and Q using a Venn diagram

Solution:

QUESTION 2

Question:If _ = {x : 1 ≤ x ≤ 10, x is a prime number} and B = {2, 3}, can you draw a Venn diagram torepresent the relationship between these two sets?

Solution:


ξ

ξ

• 2

• 7

• 3

• 5

Q

B• 1




QUESTION 3

Question:Determine the complement of set M in the diagram shown below.

Solution:M‘= {7, 8, 9, 10}

QUESTION 4

Question:If ξ = {x : 1 ≤ x ≤ 10, x is a multiple of 2} and A = {4, 6}, can you determine the complement of setA?

Solution:ξ = {2, 4, 6, 8, 10}A = {4, 6}A’ = {2, 8, 10}

QUESTION 5

Question:If ξ = {A, B, C, D, E, F} and R = {D, E, F}, can you determine the complement of set R?

Solution:R’ = {A, B, C}

M

• 1

• 5• 3

• 6

• 4

• 2

• 9

• 10

• 7

• 8

ξ

1

Determine the Relationship between Set, Subset, Universal Set and theComplement of a Set

Class Activity 1

Lesson:Class: Date:

Instructions:1. In this activity, there are two doors with two different Venn diagrams on them.2. First, choose one of the Venn diagrams.3. Then, cut the elements of the sets by using a pair of scissors.4. Decide on the place where you should stick the elements.5. Take the statement sheet after you have done the placement correctly.6. Mark “T” if you think that the statement is true7. Mark “F” if you think that the statement is false.

2

SHEET 1

{ }{ }{ }44,41,37,35,33,25,22,19,17,15,11

44,33,25,22,19,17,15,11

44,33,22,11

=

=

=

ξ

W

X

SHEET 2

4441373533252219171511

SHEET 3

ξ

ξ

⊂′

⊂′

′⊂′

⊂

X

W

WX

XW T F

T F

T F

T F

X

W

_ _

__

_

_

_

_

_

?_

_

The Concepts and the Intersection of Sets Class Activity

Lesson:Class: Date:

Instruction:1. In this activity, there are two types of problems, a problem with 2 sets

and a problem with 3 sets.2. First, choose one of the problem types.3. Then, write the sets based on your problem type.4. Determine whether they have any elements in common.5. Finally write the intersection of these sets based on the common

elements of the sets.

SHEET 1 FOR PROBLEM WITH 2 SETS

1) What is the intersection of set A and set B?A = {………………………………………………}B = {………………………………………...........}

A∩B =……………………………………………


A∩B =……………………………………………


A∩B =……………………………………………


A∩B =……………………………………………


A∩B =……………………………………………


A∩B =……………………………………………


A∩B =……………………………………………

SHEET 1 FOR PROBLEM WITH 3 SETS

1)What is the intersection of set A, set B and set C?A = {………………………………………………}B = {………………………………………...........}C = {………………………………………...........}

A ∩ B ∩ C =………………………………………

2)What is the intersection of set A, set B and set C?A = {………………………………………………}B = {………………………………………...........}C = {………………………………………...........}

A ∩ B ∩ C =………………………………………

3)What is the intersection of set A, set B and set C ?A = {………………………………………………}B = {………………………………………...........}C = {………………………………………...........}

A ∩ B ∩ C =………………………………………

4)What is the intersection of set A , set B and set C?A = {………………………………………………}B = {………………………………………...........}C = {………………………………………...........}

A ∩ B ∩ C =………………………………………


A ∩ B ∩ C =………………………………………


A ∩ B ∩ C =………………………………………

Mathematics Form 4 Lesson 26: The intersection of sets representations and A ∩ B and A and B.

Learning Area: 1. Represent the intersection of sets using Venn diagram.2. State the relationship between i) A∩B and A, ii) A∩B and B 1 /6

QUESTION 1

Question:Given that A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}. Show the relationship between sets A and Busing a Venn diagram

Solution:




QUESTION 2

Question:If P = {x : 20 ≤ x ≤ 50, x is a multiple of 10} and Q = {10, 20, 30}, can you represent theintersection of the two sets using a Venn diagram?

Solution:

QUESTION 3

Question:Given the Venn diagram below, determine the relationship between A ∩ B and A.

Solution:

A

• 1

• 5• 3• 6

• 4

• 2• 9

• 10

• 7• 8

B



QUESTION 4

Question:If R = {x : 1 ≤ x ≤ 10, x is a prime number} and S = {2, 3, 7}, can you represent the intersection ofthe two sets using a Venn diagram?

Solution:

QUESTION 5

Question:Given that C = {a, b, c, x, y, z} and D = {x, y, z}. Show the relationship between sets A and B usinga Venn diagram

Solution:



QUESTION 1

Question:Given that A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}. Show the relationship between sets A and Busing a Venn diagram

Solution:


Q• 2

• 1• 3

• 4

• 5

• 6

• 7

• 8



QUESTION 2

Question:If P = {x : 20 ≤ x ≤ 50, x is a multiple of 10} and Q = {10, 20, 30}, can you represent theintersection of the two sets using a Venn diagram?

Solution:

QUESTION 3

Question:Given the Venn diagram below, determine the relationship between A∩B and A.

Solution:The relationship between A∩B and A is A∩B is a subset of set A.

Q

• 40• 30

• 50

P• 20

A• 1

• 5• 3• 6

• 4

• 2• 9

• 10

• 7• 8

B

• 10



QUESTION 4

Question:If R = {x : 1 ≤ x ≤ 10, x is a prime number} and S = {2, 3, 7}, can you represent the intersection ofthe two sets using a Venn diagram?

Solution:

QUESTION 5

Question:Given that C = {a, b, c, x, y, z} and D = {x, y, z}. Show the relationship between sets A and Busing a Venn diagram

Solution:

R

• 1

• 3

• 2• 5

• 7

S

C

• a

• y

• x

• b • z

D

• c

C

• a

• y

• x

• b • z

D

• c

Or

The Complement of the Intersection of SetsClass Activity

Lesson: 27

Instructions:1. In this activity, there are 4 questions related with sets.2. Begin with one of the questions.3. Then, find the elements of the complement of the intersection of sets.

(M ∩ N ∩ Q)’ = ? (M ∩ N)’ = ? (N ∩ Q)’ = ? (M ∩ Q)’ = ?

M N

Q

_a_2

_f

_h

_1

_7

_c

_6_z

_9

_g

ξ

Problems on the intersection of setsClass Activity

Lesson: 28

Instructions:1. In this activity, there are 5 questions related with the same givens in the activity.2. Try to answer each question and write your answers into the boxes.

W : Set of applicants good at MS WordE : Set of applicants good at MS Excel

P : Set of applicants good at MS PowerPoint

P ⊂ Wn(P) = 10n(W ∩ E) = 6n(E′) = 16n(ξ ) = 29

1. What could be the maximum value of n(W ∩ E ∩ P)?

2. If n(W ∩ E ∩ P) = 2, then n(E) = ?

3. Can W ∩ E ∩ P be an empty set?

4. n(P') = ?

5. Is n(E ∩ P) equal to n(W ∩ E ∩ P)?

Mathematics Form 4 Lesson 29: Union of two sets and three sets.

Learning Area: 1. Determine the union of: i) two sets ; ii) three sets;and use the symbol ∪. 1 / 4

QUESTION 1

Question:Given that A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}. Can you determine the elements of the unionof set A and set B?

QUESTION 2

Question:If E = {x : 1 ≤ x ≤ 20, x is a prime number} and F = { y : 1 ≤ y ≤ 20, y is an odd number }, can youdetermine the elements of the union of set E and set F?

Solution:

QUESTION 3

Question:Given the Venn diagram below, determine the elements of the union of set X and set Y.

Χ Υ

Solution:

ACTIVITY SHEETPupil’s copy

X• a

• g• d• m

• f

• s• p

• k

• j• h

ΥΥΥΥΥΥΥΥΥΥ



QUESTION 4

Question:If U = {a : 1 ≤ a ≤ 30, a is a multiple of 6}, V = {b : 1 ≤ b ≤ 30, b is a multiple of 5 } andW = {c : 1 ≤ a ≤ 30, a is a multiple of 4}, can you the determine elements of the union of set U, setV and set W?

Solution:

QUESTION 5

Question:Given that P = {a, b, c}, Q = {1, 2, 3} and R = {x, y, z}. Determine the elements of the union ofset P, set Q and set R?

Solution:



QUESTION 1

Question:Given that A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}. Can you determine the elements of the unionof set A and set B?

Solution:A∪B = {1, 2, 3, 4, 5, 6, 7, 8}

QUESTION 2

Question:If E = {x : 1 ≤ x ≤ 20, x is a prime number} and F = { y : 1 ≤ y ≤ 20, y is an odd number }, can youdetermine the elements of the union of set E and set F?

Solution:E = {1, 2, 3, 5, 7, 11, 13, 17, 19}F = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}E∪F = {1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19}

QUESTION 3

Question:Given the Venn diagram below, determine the elements of the union of set X and set Y.

Χ Υ

Solution:The relationship between AB and A is AB is a subset of set A.


X• a

• g• d• m

• f

• s• p

• k

• j• h

Y



QUESTION 4

Question:If U = {a : 1 ≤ a ≤ 30, a is a multiple of 6}, V = {b : 1 ≤ b ≤ 30, b is a multiple of 5}andW = {c : 1 ≤ a ≤ 30, a is a multiple of 4}, can you the elements of the union of set U, set V andset W?

Solution:U = {4, 8, 12, 16, 20, 24, 28}V = {6, 12, 18, 24, 30}W = {5, 10, 15, 20, 25, 30} U∪V∪W = {4, 5, 6, 8, 10, 12, 15, 16, 18, 20, 24, 25, 28, 30}

QUESTION 5

Question:Given that P = {a, b, c}, Q = {1, 2, 3} and R = {x, y, z}. Determine the elements of the union ofset P, set Q and set R?

Solution:P∪Q∪R = {a, b, c, 1, 2, 3, x, y, z}

Union of sets using Venn diagram

Class Activity 1

Lesson 30Class: Date:

Instruction:1. In this activity, you will be given a Venn diagram containing three sets and some statements

regarding the sets.2. Read the statement carefully.3. Shade the sections on the Venn diagram which match the given statement.

Question 1

Show A ∪ B ∪ C on the Venn diagram.

Question 2

Show A ∪ B ∪ C on the Venn diagram.Question 3


Question 4


Relationship Between A ∪ B and A and A ∪ B and B

Class Activity 1

Lesson:Class: Date:

Instruction:1. In this activity, you will be given a Venn diagram containing three sets and

some statements regarding the sets.2. Read the statement carefully.3. Shade the sections on the Venn diagram which match the given statement.

• What is the relation between A ∪ C and A?Answer:

• What is the relation between (C ∪ B)’ and C’ ∩ B’?Answer:

Show A ∪ C on the Venn diagrams. Show (A ∪ C) ∩ A on the Venn diagram.

Show (C ∪ B)’ on the Venn diagram. Show C’ ∩ B’ on the Venn diagram.

Mathematics Form 4 Lesson 32: Problem solving involving union of sets.

Learning Area: 1. Determine the union of: i) two sets ; ii) three sets;and use the symbol ∪ and ∩ 1 / 4

QUESTION 1

Question:In a class, there are 16 pupils play chess and 25 pupils play monopoly. Given that there are only 10pupils plays both games, how many pupils are there in the class?

Solution:

QUESTION 2

Question:In a class of 35 pupils, 20 of them enjoy Mathematics, 25 of them enjoy Science. If 15 of the pupilsin that class enjoy both Mathematics and Science, how many pupils of that class do not enjoy bothsubjects?

Solution:




QUESTION 3

Question:Given that Mr Hisham sells newspaper to 50 families in Taman Wawasan. Twenty six of thesefamilies buy the English dailies, 23 buys the Malay dailies while 20 of them do not buy either theEnglish or Malay dailies. Determine the number of families that buy both the English and Malaydailies.Solution:

QUESTION 4

Question:Given that the numbers of candidate for the English oral examination in a class is 45. 20 pupilsscored A, 10 pupils scored B and the others scored C. Determine the number of pupils that scoredC, using the concept of union of sets.

Solution:



QUESTION 1

Question:In a class, there are 16 pupils play chess and 25 pupils play monopoly. Given that there are only 10pupils plays both games, how many pupils are there in the class?

Solution:The number of pupils in the class is equal to the number of elements in A∪B.n(A∪B) = n(A) + n(B) – n(A∩B) = 16 + 25 – 10 = 31

QUESTION 2

Question:In a class of 35 pupils, 20 of them enjoy Mathematics, 25 of them enjoy Science. If 15 of the pupilsin that class enjoy both Mathematics and Science, how many pupils of that class do not enjoy bothsubjects?

Solution:Let F be the set of pupils in the class, M be the set of pupils that likes Mathematics, S be the set ofpupils that likes Science.Therefore, n(F) = n(M) + n(S) – n(M ∩ S) + n(M ∪ S)’ 35 = 20 + 25 – 15 + n(M ∪ S)’ n(M ∪ S)’ = 35 – 20 – 25 + 15 = 5There are 5 pupils that do not like both the subject.




QUESTION 3

Question:Given that Mr Hisham sells news paper to 50 families in Taman Wawasan. Twenty six of thesefamilies buy the English dailies, 23 families buy the Malay dailies while 20 of them do not buyeither the English or Malay dailies. Determine the number of families that buy both the English andMalay dailies.

Solution:The number of families in Taman Wawasan that buy news paper [n(N)] from Mr Hisham is 50.The number of families in Taman Wawasan that buy the English dailies [n(E)] from Mr Hisham is26.The number of families in Taman Wawasan that buy the Malay dailies [n(M)] from Mr Hisham is26.The number of families in Taman Wawasan that buy the neither the dailies [n(O)] from Mr Hishamis 20. n(E) = n(N) + n(M) – n(N ∩ M) + n(N∪M)’ 50 = 26 + 23 – n(N ∩ M) + 20n(N ∩ M) = 69 – 50 = 19.

There are 19 families that buy both the enclish and Malay dailies from Mr Hisham.

QUESTION 4

Question:Given that the numbers of candidate for the English oral examination in a class is 45. 20 pupils gotA, 10 got B and the others got C. Determine the number of pupils that got C using the concept ofunion of sets.

Solution:Let P be the set of pupils in the class, A be the set of pupils that got A, B be the set of pupils thatgot B and C be the set of pupils that got C.n(P) = n(A ∪ B ∪ C) 45 = n(A) + n(B) + n(C) – n(A ∩ B) – n(A ∩ C) – n(B ∩ C) + 2n(A ∩ B ∩ C)because a pupil can only have one grade therefore all the intersections are empty sets. 45 = 20 + 10 + n(C)n(C)= 45 – 30 = 15Therefore there are 15 pupils that have obtained C.

Relationship Between A ∪ B and A and A ∪ B and B

Class Activity


Instruction:1. In this activity, you will be given sets with combined operations and asked to

draw the representation in a Venn diagram.2. Read the statement carefully.3. Draw the sets in a Venn diagram which match the given sets.

( )

( )

( )[ ]( )[ ]( )[ ] ?

?

?

?

?

=′

=

=′

=

′

=

′

MCWn

WCMn

WCMn

WCMn

WMCn

IIUIIU

IU

IU

Problems on combined operations on setsClass Activity


Instruction:In this activity, you will find the corresponding representation of a given phrase using a Venn diagram.1. Shade the suitable regions in the Venn diagrams, to construct the representation set of the given

phrase.2. Write the mathematical expression of the phrase.

Phrase:Pupils in a class studying Math and Science.Show the set of students studying only Math oronly Science.

1.

Mathematical expression:

Phrase:Pupils in a class take Biology, Physics andChemistry. All students in the class take at least oneof these courses.Show the set of students taking only one course ortaking three of them.

2.

Mathematical expression:

Mathematics Form 4

1

ACTIVITY SHEET

Pupil’s copy

LESSON 80: Possible Outcome of an Experiment.

1 Khatijah throws a dice. Determine whether the following are possible outcomes.

(a) 5 (b) 3 (c) 8 (d) an even number (e) a multiple of 7

2 Noor and Vijaya are playing Chongkak which has seven holes in each row. Siti is

watching the game. Determine if the following outcomes are possible. (a) Noor finally has 8 empty (‘burnt’) holes. (b) Vijaya wins the game. (c) Siti loses the game. (d) After one round of the game, both Noor and Vijaya have the same number of

empty or ‘burnt’ holes. 3 There are two bags. The first bag contains five cards with different numbers. The

numbers are 4, 6, 9, 12 and 15. The second bag contains three cards with different letters, which are A, C and E. If one card is taken randomly from each bag, determine whether the following (in the form of ordered pairs) are possible outcomes. (a) (4, E) (b) (4, 15) (c) (a prime number, C) (d) (an even number, A) (e) (9, D)

4 A spinner with the numbers which are all the factors of 36 is spun. List all the

possible outcomes. 5 A coin is tossed three times. Draw a tree diagram to list all the possible outcomes

of this experiment.

Mathematics Form 4

2

ACTIVITY SHEET

Teacher’s copy

LESSON 80: Possible Outcome of an Experiment.

1 Khatijah throws a dice. Determine whether the following are possible outcomes.

(a) 5 Possible (b) 3 Possible (c) 8 Impossible (d) an even number Possible (e) a multiple of 7 Impossible

2 Noor and Vijaya are playing Chongkak which has seven holes in each row. Siti is

watching the game. Determine if the following outcomes are possible. (a) Noor finally has 8 empty (‘burnt’) holes. Impossible (b) Vijaya wins the game. Possible (c) Siti loses the game. Impossible (d) After one round of the game, both Noor and Vijaya have the same number of

empty or ‘burnt’ holes. Impossible 3 There are two bags. The first bag contains five cards with different numbers. The

numbers are 4, 6, 9, 12 and 15. The second bag contains three cards with different letters, which are A, C and E. If one card is taken randomly from each bag, determine whether the following (in the form of ordered pairs) are possible outcomes. (a) (4, E) Possible (b) (4, 15) Impossible (c) (a prime number, C) Impossible (d) (an even number, A) Possible (e) (9, D) Impossible

4 A spinner with the numbers which are all the factors of 24 is spun. List all the

possible outcomes. The possible outcomes are 1, 2, 3, 4, 6, 8, 12, and 24. 5 A coin is tossed three times. Draw a tree diagram to list all the possible outcomes

of this experiment.

H

T

H

T

H

TH

TH

TH

T

H

T

(H, H, H)

(H, H, T) (H, T, H)

(H, T, T)

(T, H, H)

(T, H, T) (T, T, H)

(T, T, T)

Mathematics Form 4

1

ACTIVITY SHEET

Pupil’s Copy

LESSON 81: Sample Space of an Experiment.

For each of the following,

i ) List all the possible outcomes ii ) Write the sample space by using set notations. 1. The experiment of tossing two fair coins simultaneously.

(i) …………………………………………………………………………………...

(ii) ……………………………………………………………………………………

2. The experiment of tossing a fair coin twice.

(i) …………………………………………………………………………………...

(ii) ……………………………………………………………………………………

3. The experiment of picking an alphabet from the word ‘COURSEWARE’.

(i) …………………………………………………………………………………...

(ii) ……………………………………………………………………………………

4. A spinner with the numbers 7, 8, 9 is spun twice.

(i) …………………………………………………………………………………...

(ii) ……………………………………………………………………………………

5. A ball is selected at random from a bag containing 3 red balls, 2 blue balls and 4 green balls.

(i) …………………………………………………………………………………...

(ii) ……………………………………………………………………………………

6. An experiment of rolling a fair dice.

(i) …………………………………………………………………………………...

(ii) ……………………………………………………………………………………

7. Randomly select a marble from a box containing one blue marble and three yellow marbles.

(i) …………………………………………………………………………………...

(ii) ……………………………………………………………………………………

8. Randomly select a digit from the number 4894.

(i) …………………………………………………………………………………...

(ii) ……………………………………………………………………………………

Mathematics Form 4

2

ACTIVITY SHEET

Teacher’s Copy

LESSON 81: Sample Space of an Experiment.

For each of the following, i ) List all the possible outcomes ii ) Write the sample space by using set notations. 1. The experiment of tossing two fair coins simultaneously.

(i) HH, HT, TH, TT or (H, H), (H, T), (T, H), (T, T) (ii) S = {HH, HT, TH, TT} or S = {(H, H), (H, T), (T, H), (T, T)}

2. The experiment of tossing a fair coin twice.

(i) HH, HT, TH, TT or (H, H), (H, T), (T, H), (T, T) (ii) S = {HH, HT, TH, TT} or S = {(H, H), (H, T), (T, H), (T, T)}

3. The experiment of picking an alphabet from the word ‘COURSEWARE’.

(i) C, O, U, R, S, E, W, A, R, E (ii) S = {C, O, U, R, S, E, W, A, R, E}

4. A spinner with the numbers 7, 8, 9 is spun twice.

(i) (7,7), (7,8), (7,9), (8,7), (8,8), (8,9), (9,7), (9,8), (9,9) (ii) S = {(7,7), (7,8), (7,9), (8,7), (8,8), (8,9), (9,7), (9,8), (9,9)}

5. A ball is selected at random from a bag containing 3 red balls, 2 blue balls and 4

green balls.

(i) R1, R 2, R3, B1, B2, G1, G2, G3, G4 (ii) S = { R1, R2, R3, B1, B2, G1, G2, G3, G4 }

where R = red, B = blue, G = green

6. An experiment of rolling a fair dice.

(i) 1, 2, 3, 4, 5, 6 (ii) S = {1, 2, 3, 4, 5, 6}

7. Randomly select a marble from a box containing one blue marble and three

yellow marbles.

(i) B1, Y 1, Y2, Y 3 (ii) S = {B1, Y 1, Y2, Y3}

where Y = yellow, B = blue

8. Randomly select a digit from the number 4894.

(i) 4,8,9,6 (ii) S = {4, 8, 9, 6}

Mathematics Form 4

1

ACTIVITY SHEET

Pupil’s Copy

LESSON 82: Elements of a Sample.

An aquarium contains two gold fishes and a flower-horn fish. If Dina randomly catches 2 fishes consequently from the aquarium, identify the elements of the sample space and the following events (Let G = gold fish, F = flower-horn fish)

Answer: a. Sample space using set notations

Answer: b. One gold fish and one flower-horn fish are caught.

Answer:

c. Two gold fishes are caught.

A drawer has five number card; each labeled 3, 4, 5, 6 or 7. Two cards are picked randomly from the drawer. State the elements that satisfy the following conditions.

Answer: a. The sum of the two numbers is equal to 10.

Answer:

b. The product of the two numbers is greater than 14.

Answer:

c. Getting odd numbers.

Two balls are drawn simultaneously from a bag containing a blue ball, a green ball and a yellow ball.

Answer: a. Write the sample space of this experiment using set notations.

Answer:

b. List the elements of ‘getting a yellow ball’.

Answer:

c. List the elements of ‘getting a blue ball and a yellow ball’.

A dice is rolled, list the following events.

Answer: a. P = the event of getting an even number.

Answer:

b. Q = the event of getting a prime number.

Five number cards; 3, 4, 5, 6, 7

Mathematics Form 4

2

ACTIVITY SHEET

Teacher’s Copy

LESSON 82: Elements of a Sample.

An aquarium contains two gold fishes and a flower-horn fish. If Dina randomly catches 2 fishes consequently from the aquarium, identify the elements of the sample space and the following events ( Let G = gold fish, F = flower-horn fish)

Answer: a. Sample space using set notations

S = { G1G2, G1F,G2G1, G2F, FG1, FG2 }

Answer: b. One gold fish and one flower-horn fish are caught.

B = { G1F,G2F, FG1, FG2 }

Answer:

c. Two gold fishes are caught. C = { G1G2, G2G1 }

A drawer has five number card; each labeled 3, 4, 5, 6 or 7. Two cards are picked randomly from the drawer. State the elements that satisfy the following conditions.

Answer: a. The sum of the two numbers is equal to 10.

Q = { (3,7) (4,6) } Answer:

b. The product of the two numbers is greater than 14. F = {(3,5), (3,6), (3,7), (4,5), (4,6), (4,7), (5,6), (5,7), (6,7)} Answer:

c. Getting odd numbers. H = {(3,5), (3,7), (5, 3), (5,7), (7,3), (7,5)}

Two balls are drawn simultaneously from a bag containing a blue ball, a green ball and a yellow ball.

Answer: a. Write the sample space of this experiment using set notations. S = {BG, BY, GB, GY,

YB, YG} Answer:

b. List the elements of ‘getting a yellow ball’. Q = {BY, GY, YB, YG}

Answer:

c. List the elements of ‘getting a blue ball and a yellow ball’. T = {BY, YB}

A dice is rolled, list the following events.

Answer: a. P = the event of getting an even number.

P = {2, 4, 6}

Answer:

b. Q = the event of getting a prime number. T = {2, 3, 5}

Five number cards; 3, 4, 5, 6, 7

Mathematics Form 4

1

ACTIVITY SHEET

Pupil’s Copy

LESSON 83: Probability.

An aquarium contains two gold fishes, three flower-horn fishes and four carp fishes. If Siti randomly catches 2 fishes consequently from the aquarium, determine whether the following events are possible or impossible:

Answer: a. One gold fish and one flower-horn fish are caught.

Answer: b. One carp fish and one talapia fish are caught.

Answer:

c. Two gold fishes are caught.

A glove factory produces 1200 pairs of gloves, and 300 pairs are defected. Each pair of gloves are tied together with a string. A pair of gloves is picked randomly.

Answer: a. Find the ratio that the gloves are not defected.

Answer:

b. Find the ratio that the gloves are defected.

A drawer contains a pair of blue socks, two pairs of purple socks and a pair of red socks. If a sock is randomly selected from the drawer, what is the probability that:

Answer: a. the sock is purple

Answer:

b. the sock is blue or red Answer:

c. the sock is red or purple

In a National Day celebration, 10 000 peoples participate in the national march. 4000 of them wear soldier uniforms, 2500 wear police uniforms, 500 wear fireman uniforms and the rest are a variety of other uniforms. If the best uniform wi ll be rewarded among the participants, what is probability that:

Answer: a. a soldier will win the reward?

Answer:

b. a policeman will win the reward?

Answer:

c. a fireman will win the reward?

A dice is rolled. Determine the probability for the following outcomes:

Answer: a. An odd number that is greater than 3 is obtained.

Answer: b. A prime number is obtained.

Answer:

c. A number less than 4 is obtained.

A pair of blue socks, two pairs of purple socks,a pair of red socks.

Mathematics Form 4

2

ACTIVITY SHEET

Teacher’s Copy

LESSON 83: Probability.

An aquarium contains two gold fishes, three flower-horn fishes and four carp fishes. If Siti randomly catches 2 fishes consequently from the aquarium, determine whether the following events are possible or impossible:

Answer: a. One gold fish and one flower-horn fish are caught.

Possible

Answer: b. One carp fish and one talapia fish are caught.

Impossible

Answer:

c. Two gold fishes are caught. Possible

A glove factory produces 1200 pairs of gloves, and 300 pairs are defected. Each pair of gloves are tied together with a string. A pair of gloves is picked randomly.

Answer: a. Find the ratio that the gloves are not defected.

900/1200 = 3/4

Answer:

b. Find the ratio that the gloves are defected. 300/1200 = 1/4

A drawer contains a pair of blue sock, two pairs of purple socks and a pair of red sock. If a sock is randomly selected from the drawer, what is the probability that:

Answer: a. the sock is purple.

4/8 = 1/2 Answer:

b. the sock is blue or red. 4/8 = 1/2 Answer:

c. the sock is red or purple. 6/8 = 3/4

In a National Day celebration, 10 000 peoples participate in the national march. 4000 of them wear soldier uniforms, 2500 wear police uniforms, 500 wear fireman uniforms and the rest are a variety of other uniforms. If the best uniform will be rewarded among the participants, what is probability that:

Answer: a. a soldier will win the reward?

4000/10000 = 2/5 Answer:

b. a policeman will win the reward? 2500/10000 = 1/4

Answer:

c. a fireman will win the reward? 500/10000 = 1/20

A dice is rolled. Determine the probability for the following outcomes:

Answer: a. An odd number that is greater than 3 is obtained.

1/6

Answer: b. A prime number is obtained.

4/6 = 2/3

Answer:

c. A number less than 4 is obtained. 3/6 = 1/2

A pair of blue socks, two pairs of purple socks,a pair of red socks.

Mathematics Form 4

1

ACTIVITY SHEET Pupil’s Copy LESSON 84: Problems on probability

Answer all the questions. 1 The probability of getting a red card from one box is 0.45. If there are 100

coloured cards in the box, how many can you expect to be red cards?

2 A group of students were selected from a class to attend a meeting. If the

probability of choosing a student who is wearing a pair of spectacles is 41

estimate the number of students wearing spectacles from a group of 120 students.

3 In a class, the probability of choosing a girl is 53 . If there are 15 girls in the class,

(a) Calculate the number of students in the class.

(b) If the number of students is increased, estimate the number of girls in a group of 200 students.

Mathematics Form 4

2

4 In a survey on 200 consumers in an area, the following information was obtained.

Type of detergent Brand A Brand B Brand C Brand D Brand E

Frequency 40 55 10 50 45

(a) If a consumer is selected at random, find the probability that the consumer

(i) uses Brand B

(ii) uses Brand A or Brand D

(b) If 1500 consumers were selected from that area, how many would you predict to use Brand C?

Mathematics Form 4

3

ACTIVITY SHEET Teacher’s Copy LESSON 84: Problems on probability

Answer all the questions. 1 The probability of getting a red card from one box is 0.45. If there are 100

coloured cards in the box, how many can you expect to be red cards?

Expected number of red cards = Probability of getting a red card × number of cards

= 0.45 × 100 = 45

2 A group of students were selected from a class to attend a meeting. If the

probability of choosing a student who is wearing a pair of spectacles is 41

estimate the number of students wearing spectacles from a group of 120 students.

Expected number of students = Probability of choosing a student who is who are wearing spectacles wearing spectacles × 120 students

= 41

× 120

= 30

3 In a class, the probability of choosing a girl is 53 . If there are 15 girls in the class,

(a) Calculate the number of students in the class. Let x = number of students. Using ratio,

53

= x

15

35

=x × 15

=x 25 Therefore, the number of students in the class is 25.

(b) If the number of students is increased, estimate the number of girls in a group of 200 students.

Expected number of girls = the probability of choosing a girl × 200

= 53

× 200

= 120

Mathematics Form 4

4

4 In a survey on 200 consumers in an area, the following information was obtained.

Type of detergent Brand A Brand B Brand C Brand D Brand E

Frequency 40 55 10 50 45

(a) If a consumer is selected at random, find the probability that the consumer

(i) uses Brand B

Let B = event of using Brand B detergent

P(B) = 20055

= 4011

(ii) uses Brand A or Brand D

Let F = event of using Brand A or Brand D

= 200

5040 +

= 20090

= 209

(b) If 1500 consumers were selected from that area, how many would you predict to use Brand C?

Let C = event of using Brand C detergent

P(C) = 20010

= 201

Number of consumers using Brand C detergent = P(C) × 1500

= 201

× 1500

= 75

Mathematics Form 4

1

ACTIVITY SHEET

Pupil’s Copy

LESSON 85: Occurrence of an Outcome.

Fill in the empty boxes with the correct answers.

1 A big company wants to give out free advertisement stickers for vehicle owners who drive through one road near its office. To save cost and time, the company wants to estimate the number of cars, lorries and busses that will pass through the road from 9am to 1pm so that the number of stickers to be printed can be determined. So, a survey was done for 1 hour at the road. The following is the result of the survey:

Types of vehicles Cars Lorries Buses Frequency 105 30 15

(a) What is the total number of cars, lorries and busses that passed through the road in that 1 hour?

(b) Estimate the total number of those three vehicles that will pass through that road from 9am to 1pm.

(c) If a vehicle going through the road is selected at random during that 1-hour’s time, find the probability that the vehicle is

(i) a car. (ii) a bus. (d) If the company is only interested in giving the stickers to owners of cars and

buses only, predict the total number of cars and buses that will pass through the road from 9am to 1pm in a certain day.

2 Mrs. Vijay has invited 500 guests for an evening function. When she ordered 20

samples of cookies for the function to be sent to her, she found that for every 10 cookies, 1 cookie is burnt and hence cannot be served. From 50 invited guests that have replied, 4 of them said that they will not be able to come and 3 of them asked if they can bring a partner, which she consented. (a) If she ordered 1000 of those cookies, predict how many of those cookies

cannot be served because they are burnt. (b) If a guest is randomly chosen from the 50 that has replied, calculate the

probability that the guest will not be able to come to her function. (c) Predict the number of guests that may not be able to come to her function. (d) If a guest is randomly chosen from the 50 that has replied, calculate the

probability that the guest will come with a partner. (e) Predict the number of guests that will come to her function. (f) Hence, determine the minimum number of cookies that she should order so

that every guest who comes to her function will get at least 2 of those cookies. 3 From 15 students that are involved in a science project, 2 of them did not succeed

in completing the project as required. Predict how many would have successfully completed the project if 300 students were involved.

Mathematics Form 4

2

ACTIVITY SHEET

Teacher’s Copy

LESSON 85: Occurrence of an Outcome.

Fill in the empty boxes with the correct answers.

1 A big company wants to give out free advertisement stickers for vehicle owners who drive through one road near its office. To save cost and time, the company wants to estimate the number of cars, lorries and busses that will pass through the road from 9am to 1pm so that the number of stickers to be printed can be determined. So, a survey was done for 1 hour at the road. The following is the result of the survey:

Types of vehicles Cars Lorries Buses Frequency 105 30 15

(a) What is the total number of cars, lorries and busses that passed through the road in that 1 hour?

(b) Estimate the total number of those three vehicles that will pass through that road from 9am to 1pm.

(c) If a vehicle going through the road is selected at random during that 1-hour’s time, find the probability that the vehicle is

(i) a car. (ii) a bus. (d) If the company is only interested in giving the stickers to owners of cars and

buses only, predict the total number of cars and buses that will pass through the road from 9am to 1pm in a certain day.

2 Mrs. Vijay has invited 500 guests for an evening function. When she ordered 20

samples of cookies for the function to be sent to her, she found that for every 10 cookies, 1 cookie is burnt and hence cannot be served. From 50 invited guests that have replied, 4 of them said that they will not be able to come and 3 of them asked if they can bring a partner, which she consented. (a) If she ordered 1000 of those cookies, predict how many of those cookies

cannot be served because they are burnt. (b) If a guest is randomly chosen from the 50 that has replied, calculate the

probability that the guest will not be able to come to her function. (c) Predict the number of guests that may not be able to come to her function.

(d) If a guest is randomly chosen from the 50 that has replied, calculate the probability that the guest will come with a partner.

(e) Predict the number of guests that will come to her function. (f) Hence, determine the minimum number of cookies that she should order so

that every guest who comes to her function will get at least 2 of those cookies. 3 From 15 students that are involved in a science project, 2 of them did not succeed

in completing the project as required. Predict how many would have successfully completed the project if 300 students were involved.

150

520

0.7

0.1

416

100

0.08

40

0.06

490

1089

260

Mathematics Form 4

1

ACTIVITY SHEET

Pupil’s Copy

LESSON 86: Tangent to a Circle.

1 Determine which lines are the tangents to each of the circle given. Then, state the contact points of these tangents. O is the centre of each circle.

(a) (b)

Tangents to the

circle Contact points Tangents to

the circle Contact points

2 Given that the straight lines PQ and RS are tangents to the circle in the diagram

below, list all the angles that are marked and arranged them under less than 90°, equal to 90° or more than 90° in the table provided.

< 90° 90° > 90°

AO

B

C

D

E F

O

G

H

I J

K

L

M

N

O

Z

P Q R

S

T

U

V

W

X

Mathematics Form 4

2

ACTIVITY SHEET

Teacher’s Copy

LESSON 86: Tangent to a Circle.

1 Determine which lines are the tangents to each of the circle given. Then, state

the contact points of these tangents. O is the centre of each circle. (a) (b)

Tangents to the circle

Contact points Tangents to the circle

Contact points

AC

CE

B

D

GK

IK

L

J

2 Given that the straight lines PQ and RS are tangents to the circle in the diagram

below, list all the angles that are marked and arranged them under less than 90°, equal to 90° or more than 90° in the table provided.

< 90° 90° > 90°

∠OUT ∠VXO

∠PTO ∠OWS ∠UVX

∠PUV ∠TRW

AO

B

C

D

E F

O

G

H

I J

K

L

M

N

O

Z

P Q R

S

T

U

V

W

X

Mathematics Form 4

1

ACTIVITY SHEET

Pupil’s Copy

LESSON 87: Constructing tangent to a circle.

The diagrams below show a circle centre O. Construct the tangent based on the point given. 1

ANSWER: 2

ANSWER:

O P

O

Mathematics Form 4

2

3 ANSWER: 4 ANSWER:

O

O

Mathematics Form 4

3

5 ANSWER: 6 ANSWER:

O

O ?

Mathematics Form 4

4

ACTIVITY SHEET

Teacher’s Copy

LESSON 87: Constructing tangent to a circle.

The diagrams below show a circle centre O. Construct the tangent based on the point given. 1

ANSWER: 2

ANSWER:

O P

O P

O

P

Mathematics Form 4

5

3 ANSWER: 4 ANSWER:

O

O

P

O

O

Mathematics Form 4

6

5 ANSWER: 6 ANSWER:

O

O

O

O

Mathematics Form 4

1

ACTIVITY SHEET Pupil’s Copy LESSON 88: Properties of Two Tangents to a Circle

1 Based on the diagram, complete the table given with the right answer.

No. Questions Answers

1 Length of BC

2 Length of OA

3 ∠BOC

4 ∠ACO

5 ∠BCO

6 Length of OC

A

OC

B

12 cm

5 cm

60°

Mathematics Form 4

2

2 In the diagram above, PR and PQ are tangents to the circles with the centre O, OR

and OQ are radii of the circle. Given that ∠RPQ = 58°, calculate,

1. ∠POQ

b) ∠ROQ

c) The length of PR d) The length of OP

P O

Q

R

8 cm

6 cm 58°

Mathematics Form 4

3

e) ∠OQR

f) ∠RQP

Mathematics Form 4

4

ACTIVITY SHEET Teacher’s Copy LESSON 88: Properties of Two Tangents to a Circle

1 Based on the diagram, complete the table given with the right answer.

No. Questions Answers

1 Length of BC BC = AC = 12 cm

2 Length of OA OA = OB = 5 cm

3 ∠BOC ∠BOC = ∠AOC = 60°

4 ∠ACO ∠ACO = 180° – 60° – 90° = 30°

5 ∠BCO ∠ACO = 30°

6 Length of OC OC = 22 ACOA +

= 22 125 + = 13 cm

A

OC

B

12 cm

5 cm

60°

Mathematics Form 4

5

2 In the diagra m above, PR and PQ are tangents to the circles with the centre O, OR

and OQ are radii of the circle. Given that ∠RPQ = 58°, calculate,

2. ∠POQ

∠QPO = 58° ÷ 2 = 29°

∠POQ = (180° – 29° – 90°) = 61°

b) ∠ROQ

∠QPO = 29° ∠POQ = 61°

∠ROQ = 61° × 2 = 122°

or ∠ROQ = 180° − 58°

= 122°

c) The length of PR PR = PQ = 8 cm d) The length of OP

OP2 = PQ2 + OQ2 OP2 = 82 + 62 OP2 = 64 + 39 OP2 = 100 OP = 100 OP = 10 cm.

P O

Q

R

8 cm

6 cm 58°

Mathematics Form 4

6

e) ∠OQR

∠OQR = ( 180° − 122° ) ÷ 2 = 29°

f) ∠RQP ∠RQP = 90° − 29° = 61°

Mathematics Form 4

1

ACTIVITY SHEET

Pupil’s Copy

LESSON 89: Angle formed by the Tangent and the Chord.

1. Identify and shade the alternate segment for ∠JKM in each of the following figures. Note that for each figure, JKL is the tangent to the circle at point K.

(a) (b) (c) (d)

2. In each of the figures below, XYZ is a tangent to the circle at point Y. State the angle in the alternate segment for the angle labeled t°.

(a) (b) ∠ ∠ (c) (d) ∠ ∠

J

K

L

M J

K

L

M

J

K

L

M J

K

L

M

X

Y

Z

t° A

B

C

X

Y

Z

A

B

C D

X

Y

Z

A

B

C D

X Y Z

A

B

C

t°

t°

t°

Mathematics Form 4

2

ACTIVITY SHEET

Teacher’s Copy

LESSON 89: Angle formed by the Tangent and the Chord.

1. Identify and shade the alternate segment for ∠JKM in each of the following figures. Note that for each figure, JKL is the tangent to the circle at point K.

(a) (b) (c) (d)

2. In each of the figures below, XYZ is a tangent to the circle at point Y. State the angle in the alternate segment for the angle labeled t°.

(a) (b) ∠ ∠ (c) (d) ∠ ∠

J

K

L

M J

K

L

M

J

K

L

M J

K

L

M

X

Y

Z

t° A

B

C

Y

Z

A

B

C D

X

Y

Z

A

B

C D

X Y Z

A

B

C

t°

t°

t°

ABY (or YBA) DCY (or YCD)

BCY (or YCB) CBY (or YBC)

X

Mathematics Form 4

1

ACTIVITY SHEET Pupil’s Copy LESSON 90 : Calculations on the An gle in Alternate Segment.

1. The diagram below shows a circle, centre O. PQ is a diameter of the circle and

LPS is a tangent to the circle . Find (a) ∠ POR, (b) ∠ RPS .

ANSWER : 2.

In diagram above, QS is parallel to MNP. Find the values of a and b .

ANSWER :

O ?

O ?

14o

P

Q

S

L

R

b a 45o

65o

M N P

Q

R

S

Mathematics Form 4

2

3.

In diagram above, find the values of v and u. ANSWER : 4 In the diagram, XYZ is a tangent. Find the values of ∠YLN and ∠YNM. ANSWER :

O

?

O ?

A B 25o

D

v 30o

u

C

E

X Y Z

L

M

N

50°

20°

Mathematics Form 4

3

ACTIVITY SHEET Teacher’s Copy LESSON 90 : Calculations on the Angle in Alternate Segment.

2. The diagram below shows a circle, centre O. PQ is a diameter of the circle and

LPS is a tangent to the circle . Find (a) ∠ POR, (b) ∠ RPS .

ANSWER : 2.

In diagram above, QS is parallel to MNP. Find the values of a and b .

ANSWER :

O ?

O ?

14o

P

Q

S

L

R

∠ PQR = 14o (angle at circumference)

(a) ∠ POR = 14o × 2

= 28o ( angle at centre )

(b) ∠ RPS = ∠ PQR ( angle in the alternate segment )

Hence, ∠ RPS = 14o

b a 45o

65o

M N P

Q

R

S

∠ RNP = 45o ( alternate angle ) Hence b = 45o ( alternate segment )

∠ QNM = ∠ QRN = 65o

So, a = 180o – 65o = 115o

Mathematics Form 4

4

3.

In diagram above, find the values of v and u. ANSWER : 4 In the diagram, XYZ is a tangent. Find the values of ∠YLN and ∠YNM. ANSWER :

O

?

O ?

A B 25o

D

v

30o

u

∠ EBC = ∠ OEB u = 180o − ∠DEO − ∠OEB

∠ EBC = 30o

So, v = 180o – 30o – 30o

= 120o

Thus, u = 180° − 72.5° − 30°

= 77.5°

C

E

X Y Z

L

M

N

50°

20°

∠ YLN = 90o ( angle in semicircle )

∠ YNM = 180o – 20o − ∠YMN

∠ YMN = 90° ( angle in semicircle )

Hence, ∠ YNM = 180o – 20o – 90o

= 70o

°=

°+°=

−°−°=∠

5.72212025

2180(180

DEO

Mathematics Form 4

1

ACTIVITY SHEET Pupil’s Copy LESSON 91: Common Tangent to T wo Circles

1. Complete the table given with the number of common tangents for each case.

No. Circles Same size Different size

1 Intersect at two points

2 Intersect at one point ( external contact point )

3 Intersect at one point ( internal contact point )

4 Do not intersect

2. Based on the diagrams given, determine the properties related to the common

tangents to two circles.

Intersect at two point

Circles of the same size Circles of different sizes

A B

C D

M N

C

A

B

D

M

N

O

Mathematics Form 4

2

Intersect at one point

Circles of the same size Circles of different size s Internal contact point

Do not intersect

Circles of the same size Circles of different size

A B

C D

M N O A

B

C D

M

N OM N

O

RB

C D

M O

P Q

S

N

A

A

B

C D

OP

QR

S

Mathematics Form 4

3

ACTIVITY SHEET Teacher’s Copy LESSON 91: Common Tangent to T wo Circles 1. Complete the table given with the number of common tangents for each case.

No. Circles Same size Different size

1 Intersect at two points

2 2

2 Intersect at one point ( external contact point )

3 3

3 Intersect at one point ( internal contact point )

1

4 Do not intersect

4 4

2. Based on the diagrams given, determine the properties related to the common

tangents to two circles.

Intersect at two point

Circles of the same size Circles of different sizes

1. AB = CD 2. AB // MN // CD 3. The radii are perpendicular to the tangents .

1. AB = CD 2. OA = OC 3. The radii are perpendicular to the tangents. 4. O, M, and N are collinear points.

A B

C D

M N

C

A

B

D

M

N

O

Mathematics Form 4

4

Intersect at one point

1. AB = CD 2. AB // MN // CD 3. The radii are perpendicular to the tangents. 4. M, O and N are collinear points.

1. AB = CD 2. AP = PC 3. The radii are perpendicular to the tangents. 4. P , M, O and N are collinear points.

Circles of the same size Circles of different size s Internal contact point

1. ∠ABC = 90°

2. A, M, N and B are collinear points.

A B

C D

M N O A

B

C D

M

N O

M N O

Do not intersect

Circles of the same size Circles of different size

1. AB = CD 2. AB // MN // CD 3. PS = RQ 4. O, M and N are collinear points. 5. RO = SO = PO = QO 6. The radii are perpendicular to the tangents.

1. AB = CD 2. OQ = OP 3. PS = RQ 4. L, O, M and N are collinear points 5. OR = OS 6. The radii are perpendicular to the tangents.

RB

C D

M O

P Q

S

N

A

A

B

C D

OP

QR

S

Mathematics Form 4

1

ACTIVITY SHEET

Pupil’s Copy

LESSON 92: Problems on Common Tangents to two Circles.

1 In the figure on the right, MN is a common

tangent to two circles with centres O and P. M and N are the contact points. Given that OP = 17.5 cm and that the radii of the circles with centres O and P are 2.6 cm and 7.5 cm respectively, calculate

(a) the perimeter of the quadrilateral MNOP,

(b) the area of the quadrilateral MNOP .

Solution:

1. Understand The Problem

2. Devise A Plan

3. Carry Out The Plan

4. Check The Solution

M N

O P

Mathematics Form 4

2

2. PQ is a common tangent to the two

circles shown in the figure on the right. ACF and BCD are straight lines. AB is the diameter of the circle on the left. ∠ACP = 70° and ∠PCE = 65°. Calculate the values of x, y and z.

Solution:


2. Devise A Plan 3. Carry Out The Plan


P

Q

A

B

C

D E

F

x° y° z°

Mathematics Form 4

3

ACTIVITY SHEET

Teacher’s Copy

LESSON 92: Problems on Common Tangents to two Circles.

1 In the figure on the right, MN is a common tangent to two circles with centres O and P. M and N are the contact points. Given that OP = 17.5 cm and that the radii of the circles with centres O and P are 2.6 cm and 7.5 cm respectively, calculate

(a) the perimeter of the quadrilateral MNOP,

(b) the area of the quadrilateral MNOP .

Solution:


MN is perpendicular to PM and ON. OP = 17.5 cm PM = 7.5 cm ON = 2.6 cm Calculate the perimeter and the area of MNOP .

2. Devise A Plan

Perimeter of MNOP = MN + NO + OP + PM Use Pythagoras Theorem to calculate MN. MNOP is a trapezium.

Area of MNOP = MNONPM ×+ )(21

3. Carry Out The Plan

(a) 22 9.45.17 −=MN

24.282= = 16.8 cm So, perimeter of MNOP = 16.8 + 2.6 + 17.5 + 7.5 = 44.4 cm

(b) Area of MNOP = 8.16)6.25.7(21

×+

= 84.84 cm2


22 PMMN + = 22 9.48.16 + Area of MNOP = (21 × 16.8 × 4.9)

= 25.306 + (2.6 × 16.8)

= 17.5 cm = 84.84 cm2 = OP

M N

O P

P

M N

O Q 2.6 cm

4.9 cm 17.5 cm

2.6 cm

Mathematics Form 4

4

2. PQ is a common tangent to the two circles shown in the figure on the right. ACF and BCD are straight lines. AB is the diameter of the circle on the left. ∠ACP = 70° and ∠PCE = 65°. Calculate the values of x, y and z.

Solution:


MN is the diameter. ∠ACP = 70° ∠PCE = 65° CDEF is a cyclic quadrilateral. Calculate the values of x, y and z.

2. Devise A Plan Use angle properties of circles and properties of opposite angles to calculate x. Use properties of angle in the alternate segment to calculate y and z. 3. Carry Out The Plan

MN is the diameter. This means that ∠ACB = 90°. ∠ACB and ∠DCF are opposite angles. So, ∠DCF = ∠ACB = 90° CDEF is a cyclic quadrilateral. Opposite angles in a cyclic quadrilateral add up to 180°. So, x° + ∠DCF = 180° x + 90 = 180 x = 180 − 90 x = 90

y = ∠QCE = 180 − 65 (∠PCE = 65°) y = 115

z° + ∠ABC = 90° z + 70 = 90 (∠ABC = ∠ACP = 70°) z = 90 − 70 z = 20

4. Check The Solution ∠CFE = ∠PCE = 65° The sum of the interior angles of CDEF should be 360°. x° + y° + ∠DCF + ∠CFE = 90° + 115° + 90° + 65 = 360° The sum of the interior angles of ∆ABC should be 180°. z° + ∠ABC + ∠ACB = 20° + 70° + 90° = 180°

P

Q

A

B

C

D E

F

x° y° z°

Mathematics Form 4

1

ACTIVITY SHEET Pupil’s copy Lesson 93: Quadrants and Angles in the Unit Circle

Complete all the answer boxes with the correct angles.

For each of the following, state the quadrant the angle lies in:

a) 88o Answer : b) 156o

Answer :

c) 320o Answer : d) 190o

Answer :

Question 1.

≤ θ ≤ 90o

0o 360o 180o O

≤ θ ≤ o

180o ≤ θ ≤ 270o 270o ≤ θ ≤ 360o

Question 2.

Mathematics Form 4

2

The above diagram shows a unit circle with four points L, M, N and P on it’s circumference. Find the x–coordinate, y–coordinate and the ratio of y–coordinate to x–coordinate for each point. a) For point L,

The x–coordinate =

The y–coordinates =

The ratio = b) For point M,


The y–coordinate =

The ratio =

Question 3.

O 1

1

−1

−1

L M

N

P

x

y

Mathematics Form 4

3

c) For point N,



The ratio = d) For point P,



The ratio =

Mathematics Form 4

4

ACTIVITY SHEET Teacher’s copy Lesson 93: Quadrants and Angles in the Unit Circle

Complete all the answer boxes with the correct angles.

For each of the following, state the quadrant the angle lies in:

e) 88o Answer : Quadrant I f) 156o

Answer : Quadrant II

g) 320o Answer : Quadrant IV h) 190o

Answer : Quadrant III

Question 1.

0o ≤ θ ≤ 90o

0o 360o 180o O

90o ≤ θ ≤ 180o

180o ≤ θ ≤ 270o 270o ≤ θ ≤ 360o

Question 2.

Mathematics Form 4

5

The above diagram shows a unit circle with four points L, M, N and P on it’s circumference. Find the x–coordinate, y–coordinate and the ratio of y–coordinate to x–coordinate for each point. e) For point L,

The x–coordinate = 0.7

The y–coordinates = 0.7

The ratio = 1 f) For point M,

The x–coordinate = −0.4

The y–coordinate = 0.9

The ratio = −2.25

Question 3.

O 1

1

−1

−1

L

M

N

P

x

y

Mathematics Form 4

6

g) For point N,

The x–coordinate = −0.8

The y–coordinate = −0.6

The ratio = 0.75 h) For point P,

The x–coordinate = 0.5

The y–coordinate = −0.85

The ratio = −1.7

Mathematics Form 4

1

ACTIVITY SHEET Pupil’s Copy LESSON 94: Angle in Quadrant I

1. Based on the unit circle given, complete the table given by determining the values of sine,

cosine and tangent of the following angles . Your answers should be rounded off to two decimal places .

No. Trigonometrical ratios Answer

1 sin 20°

2 sin 50°

3 sin 70°

4 cos 20°

5 cos 50°

6 cos 70°

7 tan 20°

8 tan 50°

9 tan 70°

y2

0.2

0.4

0.6

0.8

1.0

A2

B2

C

20°

50° 70°

x0.2 0.4 0.6 0.8 1.0

o

Mathematics Form 4

2

2. From the answers above, make an inference about the values of sine, cosine and

tangent as θ increases from 0° to 90°, by completing the table below.

sine ? ( 0° to 90° )

cosine ? ( 0° to 90° )

tangent ? ( 0° to 90° )

Mathematics Form 4

3

ACTIVITY SHEET Teacher’s Copy LESSON 94: Angle in Quadrant I

1. Based on the unit circle given, complete the table given by determining the values of sine,

cosine and tangent of the following angles . Your answers should be rounded off to two decimal places.

No. Trigonometrical ratios Answer

1 sin 20° 0.34

2 sin 50° 0.77

3 sin 70° 0.94

4 cos 20° 0.94

5 cos 50° 0.64

6 cos 70° 0.35

7 tan 20° 0.36

8 tan 50° 1.20

9 tan 70° 2.69

y2

0.2

0.4

0.6

0.8

1.0

A2

B2

C2

20°

50° 70°

x0.2 0.4 0.6 0.8 1.0

o

Mathematics Form 4

4

2. From the answers above, make an inference about the values of sine, cosine and tangent as θ increases from 0° to 90°, by completing the table below.

sine ? ( 0° to 90° )

The value of sin ? increases as ?

increases from 0° to 90°.

cosine ? ( 0° to 90° )

The value of cos ? decreases as ?


tangent ? ( 0° to 90° )

The value of tan ? increases as ?


Mathematics Form 4

1

ACTIVITY SHEET Pupil’s Copy LESSON 95: Angle in Quadrant II, III and IV

1. Based on the unit circle given, complete the following table by determining the values of

sine, cosine and tangent for the stated angles. Your answers should be rounded off to two decimal places.

No. Trigonometrical Ratios Answer

1 sin 120°

2 sin 220°

3 sin 310°

4 cos 120°

5 cos 220°

6 cos 310°

7 tan 120°

8 tan 220°

9 tan 310°

1

1

- 1

- 1

y

x

A

B C

0.5

−0.5

−0.5

0.5

120° 220°

310°

Mathematics Form 4

2

2.

Based on the unit circle shown, determine the values of

(a) sin 135° Answer:

(b) tan 285° Answer:

(c) cos 210° Answer:

(d) cos 135° Answer:

(e) y if tan 210° = 0.57 Answer:

O x

y

135°

M(−0.71, 0.71)

210°

N(−0.87, y)

75°

P(0.23, −0.97)

Mathematics Form 4

3

ACTIVITY SHEET Teacher’s Copy LESSON 95: Angle in Quadrant II, III and IV

1. Based on the unit circle given, complete the following table by determining the values of

sine, cosine and tangent for the stated angles. Your answers should be rounded off to two decimal places.

No. Trigonometrical Ratios Answer

1 sin 120° 0.87 or 0.86

2 sin 220° −0.64 or −0.65

3 sin 310° −0.77 or −0.76

4 cos 120° −0.5

5 cos 220° −0.77 or −0.76

6 cos 310° 0.64 or 0.65

7 tan 120° −1.73 or −1.74

8 tan 220° 0.84 or 0.83

9 tan 310° −1.19 or –1.2

1

1

- 1

- 1

y

x

A

B C

0.5

−0.5

−0.5

0.5

120° 220°

310°

Mathematics Form 4

4

22.423.097.0

285tan

−=

−=°

5.0)87.0(57.0

)87.0(210tan87.0

210tan

−=−×=

−×°=−

=°

y

y

2.

Based on the unit circle shown, determine the values of

(a) sin 135° Answer: sin 135° = 0.71

(b) tan 285° Answer:

(c) cos 210° Answer: cos 210° = −0.87

(d) cos 135° Answer: cos 135° = −0.71

(e) y if tan 210° = 0.57 Answer:

O x

y

135°

M(−0.71, 0.71)

210°

N(−0.87, y)

75°

P(0.23, −0.97)

Mathematics Form 4

Page 1 of 1

ACTIVITY SHEET

Pupil’s Copy

LESSON 96: Values of Sine, Cosine and Tangent.

1 State the quadrant of each angle and the determine whether the value of each of

the following is positive or negative.

Angle Quadrant Trigonometric ratio Positive or Negative

56° cos 56°

145° sin 145°

280° tan 280°

96° cos 96°

265° sin 265°

175° sin 175°

315° tan 315°

185° tan 185°

323° cos 323°

2 Calculate the value of each of the following:

(a) 4 sin 30° − 3 cos 60° =

(b) 6 tan 45 + 2 cos 180° =

(c) 5 sin 270° − 7 cos 90° =

(d) 3 tan 180° + 9 cos 360° =

(e) 10 cos 60° − 6 sin 90° =

Mathematics Form 4

Page 2 of 2

(a) Given cos θ = 1 and 0 ° ≤ θ ≤ 360°, state the values of θ.

(b) Given tan θ = 0 and 180° ≤ θ ≤ 360°, state the values of θ.

4 Find the value of x in each of the following:

(a)

(b)

(c) cos x = 0 and 180° ≤ x ≤ 360°.

(d) sin x = 0.5 and 0° ≤ x ≤ 180°.

4.5 cm 9 cm

10 cm

10 cm

x

x

Mathematics Form 4

Page 3 of 3

ACTIVITY SHEET

Teacher’s Copy

LESSON 96: Values of Sine, Cosine and Tangent.

1 State the quadrant of each angle and the determine whether the value of each of

the following is positive or negative.

Angle Quadrant Trigonometric ratio Positive or Negative

56° 1 cos 56° Positive

145° 2 sin 145° Positive

280° 4 tan 280° Negative

96° 2 cos 96° Negative

265° 3 sin 265° Negative

175° 2 sin 175° Positive

315° 4 tan 315° Negative

185° 3 tan 185° Positive

323° 4 cos 323° Positive

2 Calculate the value of each of the following:

(a) 4 sin 30° − 3 cos 60° = 4(0.5) − 3(0.5) = 2 − 1.5 = 0.5

(b) 6 tan 45 + 2 cos 180° = 6(1) + 2(− 1) = 6 − 2 = 4

(c) 5 sin 270° − 7 cos 90° = 5(−1) − 7(0) = − 5 − 0 = − 5

(d) 3 tan 180° + 9 cos 360° = 3(0) + 9(1) = 0 + 9 = 9

(e) 10 cos 60° − 6 sin 90° = 10(0.5) − 6(1) = 5 − 6 = − 1

Mathematics Form 4

Page 4 of 4

3 (a) Given cos θ = 1 and 0 ° ≤ θ ≤ 360°, state the values of θ.

If cos θ = 1 and 0° ≤ θ ≤ 360°, then θ = 0° or 360°.

(b) Given tan θ = 0 and 180° ≤ θ ≤ 360°, state the values of θ.

If tan θ = 0 and 180° ≤ θ ≤ 360°, then θ = 180° or 360°.

4 Find the value of x in each of the following:

(a) cos x = 95.4

= 21

x = 60°

(b) tan x = 1010

= 1 x = 45°

(c) cos x = 0 and 180° ≤ x ≤ 360°.

If cos x = 0, then x = 90° or 270°

but since 180° ≤ x ≤ 360°,

x = 270°

(d) sin x = 0.5 and 0° ≤ x ≤ 180°.

If sin x = 0.5, then x = 30° or 150°.

4.5 cm 9 cm

10 cm

10 cm

x

x

Mathematics Form 4

1

ACTIVITY SHEET Pupil’s copy Lesson 97: Relationships between the Values of Sine, Cosine and Tangent. Find the angle in quadrant I which corresponds to each of the following. a) 110o b) 94o c) 144o d) 168o Solution: a) Angle in quadrant I which corresponds to 110° =

b) Angle in quadrant I which corresponds to 94° =

c) Angle in quadrant I which corresponds to 144° =

d) Angle in quadrant I which corresponds to 168° =

Question 1.

x

y

x

y

x

y

x

y

O

O

O

O

Mathematics Form 4

2

Find the angle in quadrant I which corresponds to each of the following. a) 220o b) 186o c) 209o d) 256o Solution: a) Angle in quadrant I which corresponds to 220° =



d) Angle in quadrant I which corresponds to 256° =

Question 2.

x

y

y

x

O

O

x

y

O

x

y

O

Mathematics Form 4

3

Find the angle in quadrant I which corresponds to each of the following. a) 280o b) 345o c) 309o Solution: a) Angle in quadrant I which corresponds to 280° =



Question 3.

x

y

x

y

O

x

y

O

O

Mathematics Form 4

4

Express the following in terms of trigonometrical functions of acute angles.

a) tan 170o b) sin 133o

c) cos 190o

d) tan 340o

e) sin 200o

f) cos 120o

Solution: a) tan 170o =

b) sin 133o =

c) cos 190o =

d) tan 340o =

e) sin 200o =

f) cos 120o =

Question 4.

Mathematics Form 4

5

ACTIVITY SHEET Teacher’s copy Lesson 97: Relationships between the Values of Sine, Cosine and Tangent. Find the angle in quadrant I which corresponds to each of the following. a) 110o b) 94o c) 144o d) 168o Solution: a) Angle in quadrant I which corresponds to 110° = 180° − 110°

= 70o

b) Angle in quadrant I which corresponds to 94° = 180° − 94°

= 86o

c) Angle in quadrant I which corresponds to 144° = 180° − 144°

= 36o

d) Angle in quadrant I which corresponds to 168° = 180° − 168°

= 12o

Question 1.

x

y

110° 70°

x 94° 86°

y

x 144°

36°

y

x 168°

12°

y

O

O

O

O

Mathematics Form 4

6

Find the angle in quadrant I which corresponds to each of the following. a) 220o b) 186o c) 209o d) 256o Solution: a) Angle in quadrant I which corresponds to 220° = 220° − 180°

= 40o


= 6o


= 29°

d) Angle in quadrant I which corresponds to 256° = 256° − 180°

= 76°

Question 2.

x

y

220°

40°

y

x

186°

6°

O

O

x

y

209°

29° O

x

y

256°

76°

y

O

Mathematics Form 4

7

Find the angle in quadrant I which corresponds to each of the following. a) 280o b) 345o c) 309o Solution: a) Angle in quadrant I which corresponds to 280° = 360° − 280°

= 80°


= 15°


= 51°

Question 3.

x

y

280°

80° O

x

y

345°

15° O

x

y

309°

51° O

Mathematics Form 4

8

Express the following in terms of trigonometrical functions of acute angles.

a) tan 170o b) sin 133o

c) cos 190o

d) tan 340o

e) sin 200o

f) cos 120o

Solution: a) tan 170o = −tan ( 180o – 170o ) = −tan 10o b) sin 133o = sin ( 180o – 133o ) = sin 47o c) cos 190o = −cos ( 190o – 180o ) = −cos 10o d) tan 340o = −tan ( 360o – 340o ) = −tan 20o e) sin 200o = −sin ( 200o – 180o ) = −sin 20o f) cos 120o = −cos ( 180o – 120o ) = −cos 60o

Question 4.

Mathematics Form 4

1

ACTIVITY SHEET Pupil’s Copy Problems Solving on Sine, Cosine and Tangent.

1 Calculate each of the following values based on the table below:

θ 27° 46° 55° 64°

sin θ 0.45 0.72 0.82 0.90

cos θ 0.89 0.69 0.57 0.44

tan θ 0.51 1.04 1.43 2.05

(a) tan 226° =

(b) cos 116° =

(c) sin 305° =

(d) 2 cos 207° + 5 sin 116° =

(e) 3 tan 296° + 4 cos 125° =

2 (a) Given that cos θ = −0.1736 and 180°≤ θ ≤ 360°, find the value of θ . Solution:

(b) Given that sin x = 0.4695 and 0°≤ x ≤ 360°, find the values of x.

Solution:

Mathematics Form 4

2

(c) Given that tan y = −1.9626 and 90°≤ y ≤ 360°, find the values of y.

Solution:

3 In the diagram, KLM is a straight line. Find the values of cos θ and θ .

Solution:

4 In the diagram, EFG is a straight line. Find the values of z and tan z.

Solution:

θ K

L M

J

15 cm

11 cm

z

E

G

D

F

18.2 cm

7 cm

Mathematics Form 4

3

5 The above diagram shows the plan of a garden. PQT is an isosceles triangle.

Given that PQ = 2.25 m and QT = 3.6 m, calculate, (a) cos ∠TPQ. (b) the perpendicular distance between P and the line SR.

Solution:

(a)

(b)

P

Q

RS

T

2.5 m

Mathematics Form 4

4

ACTIVITY SHEET

Teacher’s Copy

LESSON 98: Problems Solving on Sine, Cosine and Tangent.

1 Calculate each of the following values based on the table below:

θ 27° 46° 55° 64°

sin θ 0.45 0.72 0.82 0.90

cos θ 0.89 0.69 0.57 0.44

tan θ 0.51 1.04 1.43 2.05

(a) tan 226° = tan(226° − 180°) = tan 46° = 1.04

(b) cos 116° = − cos(180° − 116°) = − cos 64° = − 0.44

(c) sin 305° = −sin(360° − 305°)

= −sin 55° = −0.82

(d) 2 cos 207° + 5 sin 116° = 2(− cos(207° − 180°)) + 5 sin(180° − 116°) = −2 cos 27° + 5 sin 64° = −2(0.89) + 5(0.90) = 2.72

(e) 3 tan 296° + 4 cos 125° = 3(− tan(360° − 296°)) + 4(−cos(180° − 125°)) = − 3 tan 64° − 4 cos 55° = − 3(2.05) − 4(0.57) = − 8.43

2 (a) Given that cos θ = −0.1736 and 180°≤ θ ≤ 360°, find the value of θ . Solution: cos θ = −cos 80° = cos(180° + 80°) = cos 260 ° Therefore, θ = 260°

(b) Given that sin x = 0.4695 and 0°≤ x ≤ 360°, find the values of x.

Solution: sin x = sin 28° or sin(180° − 28°) = sin 28° or sin 152° Therefore, x = 28° or 152°

Mathematics Form 4

5

(c) Given that tan y = −1.9626 and 90°≤ y ≤ 360°, find the values of y.

Solution: tan y = tan(180° − 63°) or tan(360° − 63°) = tan 117° or tan 297° Therefore, y = 117° or 297°

3 In the diagram, KLM is a straight line. Find the values of cos θ and θ .

Solution:

cos θ = −cos(180° − θ )

= −

= −

θ = 180° − 42° 50' = 137° 10'

4 In the diagram, EFG is a straight line. Find the values of z and tan z.

Solution:

tan z = −tan(180° − z )

= −

= 22

7

DEEF −−

= 22 72.18

7

−−

= 8.16

7−

= −0.4167

z = 180° − 22° 37'

= 157° 23'

θ K

L M

J

15 cm

11 cm

JL KL 11 15

DE DF z

E

G

D

F

18.2 cm

7 cm

Mathematics Form 4

6

5 The above diagram shows the plan of a garden. PQT is an isosceles triangle.

Given that PQ = 2.25 m and QT = 3.6 m, calculate,

(a) cos ∠TPQ. (b) the perpendicular distance between P and the line SR.

Solution:

(a) PQ

QTTPQ

×=∠ 2

1

21

sin

= 25.2

6.321

×

= 25.28.1

°=∠ 13.5321

TPQ

∠TPQ = 53.13° × 2

= 106.26°

Therefore, cos ∠ TPQ = cos 106.26°

= −cos(180° − 106.26°)

= −cos 73.74°

= −0.2800

(b) The height of ∆PQT = 22 )21

( QTPQ −

= 22 8.125.2 −

= 1.35 m Therefore, the perpendicular distance between P and the line SR = 1.35 + 2.5 = 3.85 m

P

Q

RS

T

2.5 m

Mathematics Form 4

1

?

00

300

600

900

120 0

1500

1800

2100

2400

2700

3000

3300

3600

y=sin?

0

-0.87

0

ACTIVITY SHEET

Pupil’s Copy

LESSON 99: Graphs of sine, cosine and tangent.

1. Based from the graph of sine given, complete the table of value below.

y

1.0

0.5

- 0.5

- 1.0

? 300 600 900 120 0 1500 180 0 210 0 2400 2700 300 0 3300 3600

×

×

× ×

×

×

×

×

×

× ×

×

×

0.87 1 0

Mathematics Form 4

2

?

300

600

900

1500

1800

240 0

270 0

3300

y=cos?

1

-0.5

-0.87

0.5

1

2. Based from the table of value, draw the graph of cosine ? where 00 < ? < 3600

y

1.0

0.5

- 0.5

- 1.0

? 900 1200 150 0 1800 2100 240 0 2700 3000 3300 3600

00 1200 210 0 3000 3600

0.87 0.5 0 - 0.87 - 1 - 0.5 0 0.87

Mathematics Form 4

3

? 00 300 450 750 900 105 0 135 0 150 0 180 0 2100 2250 2550 2700 y=tan? 0 0.58 1 3.73 8 -3.73 -1 -0.58 0 0.58 1 3.73 8 285 0 3150 3300 3600 -3.73 -1 -0.58 0

3. Based from the table of value given, dram the graph of tangent ? where

00 < ? < 360 0

y

1

? 900 1800 2700 3600

2

3

4

-4

-3

-2

-1

Mathematics Form 4

4

4. Complete the empty boxes with the properties of each graph given.

Sin ? 1

-1

1800 3600

Cos ? 1

-1

1800 3600

Tan ? 4

-4

1800 3600 900 2700

y

?

y

?

y

?

The graph of Tan ? The graph of sin ? The graph of cos ?

Mathematics Form 4

5

?

00

300

600

900

120 0

1500

1800

2100

2400

2700

3000

3300

3600

y=sin?

0

0.87

- 0.5

-0.87

0

ACTIVITY SHEET

Teacher’s Copy



y

1.0

0.5

- 0.5

- 1.0

? 300 600 900 120 0 1500 180 0 210 0 2400 2700 300 0 3300 3600

×

×

× ×

×

×

×

×

×

× ×

×

×

0.5 0.87 1 0.5 0 - 0.87 - 1 - 0.5

Mathematics Form 4

6

?

300

600

900

1500

1800

240 0

270 0

3300

y=cos?

1

-0.5

-0.87

0.5

1


y

1.0

0.5

- 0.5

- 1.0

? 900 1200 150 0 1800 2100 240 0 2700 3000 3300 3600

×

×

×

×

××

× ×

×

×

×

× ×

00 1200 210 0 3000 3600

0.87 0.5 0 - 0.87 - 1 - 0.5 0 0.87

Mathematics Form 4

7

? 00 300 450 750 900 105 0 135 0 150 0 180 0 2100 2250 2550 2700 y=tan? 0 0.58 1 3.73 8 -3.73 -1 -0.58 0 0.58 1 3.73 8 285 0 3150 3300 3600 -3.73 -1 -0.58 0

4. Based from the table of value given, dram the graph of tangent ? where 00 < ? < 360 0

y

1

? 900 1800 2700 3600

×

×

×

×

×

××

×

×

×

×

×

×

2

3

4

-4

×

-3

-2

-1

×

Mathematics Form 4

8


Sin ? 1

-1 1800 3600

Cos ? 1

-1 1800 3600

Tan ? 4

-4

1800 3600 900 2700

y

?

y

?

y

?

The graph of Tan ? • Increasing when 00 < ? < 900 and 900 < ? < 2700 • Decreasing at none of

the intervals • Tan ? is undefined

when? = 900 and 2700

• Tan ? are equal to zero when ? = 00, 1800 and 3600

• There are no minimum or maximum points on the graph

• dotted line drawn through these value of ? are called asymtotes.

The graph of sin ? • Increasing when

00 < ? < 900 and 2700 < ? < 3600

• Decreasing when 900 < ? < 2700

• Maximum point = ( 900, 1 ) • Minimum point = ( 2700 , -1 ) • Sin ? = 0 when ? = 00 , 1800, 3600 • The graph of sin ?

repeat itself at 3600 interval

The graph of cos ? • Increasing when 1800

< ? < 3600 • Decreasing when 00 < ? < 1800 • Maximum point

= ( 00, 1 ) and ( 3600, 1 )

• Minimum point = ( 1800 , -1 ) • Cos ? = 0 when ? = 900 , 2700 • The graph of cos ?


Mathematics Form 4

1

?

00

300

600

900

120 0

1500

1800

2100

2400

2700

3000

3300

3600

y=sin?

0

-0.87

0

ACTIVITY SHEET

Pupil’s Copy



y

1.0

0.5

- 0.5

- 1.0

? 300 600 900 120 0 1500 180 0 210 0 2400 2700 300 0 3300 3600

×

×

× ×

×

×

×

×

×

× ×

×

×

0.87 1 0

Mathematics Form 4

2

?

300

600

900

1500

1800

240 0

270 0

3300

y=cos?

1

-0.5

-0.87

0.5

1


y

1.0

0.5

- 0.5

- 1.0

? 900 1200 150 0 1800 2100 240 0 2700 3000 3300 3600

00 1200 210 0 3000 3600

0.87 0.5 0 - 0.87 - 1 - 0.5 0 0.87

Mathematics Form 4

3

? 00 300 450 750 900 105 0 135 0 150 0 180 0 2100 2250 2550 2700 y=tan? 0 0.58 1 3.73 8 -3.73 -1 -0.58 0 0.58 1 3.73 8 285 0 3150 3300 3600 -3.73 -1 -0.58 0

3. Based from the table of value given, dram the graph of tangent ? where

00 < ? < 360 0

y

1

? 900 1800 2700 3600

2

3

4

-4

-3

-2

-1

Mathematics Form 4

4


Sin ? 1

-1

1800 3600

Cos ? 1

-1

1800 3600

Tan ? 4

-4

1800 3600 900 2700

y

?

y

?

y

?

The graph of Tan ? The graph of sin ? The graph of cos ?

Mathematics Form 4

5

?

00

300

600

900

120 0

1500

1800

2100

2400

2700

3000

3300

3600

y=sin?

0

0.87

- 0.5

-0.87

0

ACTIVITY SHEET

Teacher’s Copy



y

1.0

0.5

- 0.5

- 1.0

? 300 600 900 120 0 1500 180 0 210 0 2400 2700 300 0 3300 3600

×

×

× ×

×

×

×

×

×

× ×

×

×

0.5 0.87 1 0.5 0 - 0.87 - 1 - 0.5

Mathematics Form 4

6

?

300

600

900

1500

1800

240 0

270 0

3300

y=cos?

1

-0.5

-0.87

0.5

1


y

1.0

0.5

- 0.5

- 1.0

? 900 1200 150 0 1800 2100 240 0 2700 3000 3300 3600

×

×

×

×

××

× ×

×

×

×

× ×

00 1200 210 0 3000 3600

0.87 0.5 0 - 0.87 - 1 - 0.5 0 0.87

Mathematics Form 4

7

? 00 300 450 750 900 105 0 135 0 150 0 180 0 2100 2250 2550 2700 y=tan? 0 0.58 1 3.73 8 -3.73 -1 -0.58 0 0.58 1 3.73 8 285 0 3150 3300 3600 -3.73 -1 -0.58 0

4. Based from the table of value given, dram the graph of tangent ? where 00 < ? < 360 0

y

1

? 900 1800 2700 3600

×

×

×

×

×

××

×

×

×

×

×

×

2

3

4

-4

×

-3

-2

-1

×

Mathematics Form 4

8


Sin ? 1

-1 1800 3600

Cos ? 1

-1 1800 3600

Tan ? 4

-4

1800 3600 900 2700

y

?

y

?

y

?

The graph of Tan ? • Increasing when 00 < ? < 900 and 900 < ? < 2700 • Decreasing at none of

the intervals • Tan ? is undefined

when? = 900 and 2700

• Tan ? are equal to zero when ? = 00, 1800 and 3600

• There are no minimum or maximum points on the graph

• dotted line drawn through these value of ? are called asymtotes.

The graph of sin ? • Increasing when

00 < ? < 900 and 2700 < ? < 3600

• Decreasing when 900 < ? < 2700

• Maximum point = ( 900, 1 ) • Minimum point = ( 2700 , -1 ) • Sin ? = 0 when ? = 00 , 1800, 3600 • The graph of sin ?


The graph of cos ? • Increasing when 1800

< ? < 3600 • Decreasing when 00 < ? < 1800 • Maximum point

= ( 00, 1 ) and ( 3600, 1 )

• Minimum point = ( 1800 , -1 ) • Cos ? = 0 when ? = 900 , 2700 • The graph of cos ?


Mathematics Form 4

1

ACTIVITY SHEET Pupils’ Copy Lesson 101: Horizontal Line, Angle of Elevation and Angle of Depression.

B

A E

F

D

C

Based on the figure given above, answer the questions below: 1. Identify the horizontal line through A. Answer: ___ ________ 2. Identify the horizontal line through E. Answer: ___________ 3. Identify the angle of elevation of A from B . Answer: ___________ 4. Identify the angle of depression of B from E. Answer: ___________ 5. Identify the angle of elevation of E from A. Answer: ___________

Question 1

Based on the figure given above, answer the questions below: 1. Identify the horizontal line through A. Answer: _____________ 2. Identify the horizontal line through B. Answer: _____________ 3. Identify the angle of elevation of B from A. Answer: _____________ 4. Identify the angle of depression of A from B . Answer: _____________

Question 2

A C

B D

Mathematics Form 4

2

ACTIVITY SHEET Teacher’s Copy Lesson 101: Horizontal Line, Angle of Elevation and Angle of Depression.

B

A E

F

D

C

Based on the figure given above, answer the questions below: 1. Identify the horizontal line through A. Answer: AC 2. Identify the horizontal line through E. Answer: EF 3. Identify the angle of elevation of A from B . Answer: ∠ DBA 4. Identify the angle of depression of B from E. Answer: ∠ FEB 5. Identify the angle of elevation of E from A. Answer: ∠ CAE

Question 1

Based on the figure given above, answer the questions below: 1. Identify the horizontal line through A. Answer: AC 2. Identify the horizontal line through B. Answer: BD 3. Identify the angle of elevation of B from A. Answer: ∠ CAB 4. Identify the angle of depression of A from B . Answer: ∠ DBA

Question 2

A C

B D

Mathematics Form 4

1

ACTIVITY SHEET Pupils’ Copy Lesson 101: Horizontal Line, Angle of Elevation and Angle of Depression.

B

A E

F

D

C

Based on the figure given above, answer the questions below: 1. Identify the horizontal line through A. Answer: ___ ________ 2. Identify the horizontal line through E. Answer: ___________ 3. Identify the angle of elevation of A from B . Answer: ___________ 4. Identify the angle of depression of B from E. Answer: ___________ 5. Identify the angle of elevation of E from A. Answer: ___________

Question 1

Based on the figure given above, answer the questions below: 1. Identify the horizontal line through A. Answer: _____________ 2. Identify the horizontal line through B. Answer: _____________ 3. Identify the angle of elevation of B from A. Answer: _____________ 4. Identify the angle of depression of A from B . Answer: _____________

Question 2

A C

B D

Mathematics Form 4

2

ACTIVITY SHEET Teacher’s Copy Lesson 101: Horizontal Line, Angle of Elevation and Angle of Depression.

B

A E

F

D

C

Based on the figure given above, answer the questions below: 1. Identify the horizontal line through A. Answer: AC 2. Identify the horizontal line through E. Answer: EF 3. Identify the angle of elevation of A from B . Answer: ∠ DBA 4. Identify the angle of depression of B from E. Answer: ∠ FEB 5. Identify the angle of elevation of E from A. Answer: ∠ CAE

Question 1

Based on the figure given above, answer the questions below: 1. Identify the horizontal line through A. Answer: AC 2. Identify the horizontal line through B. Answer: BD 3. Identify the angle of elevation of B from A. Answer: ∠ CAB 4. Identify the angle of depression of A from B . Answer: ∠ DBA

Question 2

A C

B D

Mathematics Form 4

1

ACTIVITY SHEET Pupil’s Copy Lesson 102: Angle of Elevation and Angle of Depression using Diagrams.

Draw a diagram to represent the following situation. Question 1

Draw a diagram to represent the angle of elevation from the snake to the cat.

Draw the diagram to represent the angle of depression from the cat to the snake.

Mathematics Form 4

2

Question 2

Draw the diagrams to represent the angle of elevation and angle of depression for the following situation.

Zul

Haz.

Draw a diagram to represent the angle of elevation from Zul to the aeroplane.

Draw the diagram to represent the angle of depression from Haz to Zul.

Mathematics Form 4

3

Draw the diagram to represent the angle of depression from the aeroplane to Zul and Haz.

Mathematics Form 4

4

Question 3

Safura

Amir

Draw a diagram to represent the angle of elevation from Safura to the camera.

Mathematics Form 4

5

Draw the diagram to represent the angle of elevation from Amir to camera and angle of depression to Safura.

Mathematics Form 4

6

ACTIVITY SHEET Teacher’s Copy Lesson 102: Angle of Elevation and Angle of Depression using Diagrams.

Draw a diagram to represent the following situation. Question 1

Draw a diagram to represent the angle of elevation from the snake to the cat.

Snake

Cat

Angle of elevation

Draw the diagram to represent the angle of depression from the cat to the snake.

Snake

Cat Angle of depression

Mathematics Form 4

7

Question 2

Draw the diagrams to represent the angle of elevation and angle of depression for the following situation.

Zul

Haz.

Draw a diagram to represent the angle of elevation from Zul to the aeroplane.

Zul

Aeroplane

Angle of elevation

Zul

Haz Angle of depression

Draw the diagram to represent the angle of depression from Haz to Zul.

Mathematics Form 4

8

Draw the diagram to represent the angle of depression from the aeroplane to Zul and Haz.

• Aeroplane

Haz

Zul

Angle of depression.

Angle of depression.

Mathematics Form 4

9

Question 3

Safura

Amir

Draw a diagram to represent the angle of elevation from Safura to the camera.

Mathematics Form 4

10

Draw the diagram to represent the angle of elevation from Amir to camera and angle of depression to Safura.

Angle of elevation

Angle of depression

Mathematics Form 4

1

ACTIVITY SHEET Pupil’s Copy LESSON 103: Proble ms solving on angle of elevation and angle of depression

Solution: a) The angle of elevation from Q to R

b) The angle of depression of P from R

8m

12m

P

Q

R

S

1. The diagram below shows two vertical posts distanced from each other at 10m. The height of post RS and PQ are 12m and 8m respectively. Calculate :

a) The angle of elevation from Q to R b) The angle of depression of P from R

10m

?1

?2

Mathematics Form 4

2

Solution: a) The height of building AB in metres:

b) The angle of elevation of A from D:

2. In the diagram below, AB and CD are two vertical buildings on a horizontal surface. Given that BD = 16 m and CD = 20 m. The angle of depression of A from C is 40°. Find (a) the height of building AB in metres,

(b) the angle of elevation of A from D.

40°

A

C

20 m

?

40°

D B 16 m

Mathematics Form 4

3

Solution: a) The distance of AB:

b) The angle of elevation of Q from A

6m

3m

P

Q

3. In diagram below, A, B and C are three point on a horizontal surface. A is located to the south of B and C is located to the east of the A. AC is 3m and AB is 5m. The height of PA and QB are 6m and 8m respectively. Calculate: a) the distance of A, b) the angle of evelation of Q from A.

5m

8m

A

B

C

?

Mathematics Form 4

4

ACTIVITY SHEET Teacher’s Copy LESSON 103: Problems solving on angle of elevation and angle of depression

Solution: a) The angle of elevation from Q to R

tan ?1 = sideadjacentsideopposite

tan ?1 = 1012

tan ?1 = 1.2 ?1 = 50.19° or 50°11' b) The angle of depression of P from R

tan ?2 = sideadjacentsideopposite

tan ?2 = 10

812 −

tan ?2= 104

= 0.4 ?2 = 21.8° or 21°48'

8m

12m

P

Q

R

S

2. The diagram below shows two vertical posts distanced from each other at 10m. The height of post RS and PQ are 12m and 8m respectively. Calculate :

c) The angle of elevation from Q to R d) The angle of depression of P from R

10m

?1

?2

Mathematics Form 4

5

Solution: a) The height of building AB in metres:

tan 40° = 16

sideopposite

0.839 = 16

sideopposite

Opposite side = 16 × 0.839

= 13.43 m Thus, the height AB = 20 - 13.43 = 6.57 m

b) The angle of elevation of A from D:

tan ? = 1657.6

tan ? = 0.41 ? = 22.29° or 22°17'

2. In the diagram below, AB and CD are two vertical buildings on a horizontal surface. Given that BD = 16 m and CD = 20 m. The angle of depression of A from C is 40°. Find (a) the height of building AB in metres,

(b) the angle of elevation of A from D.

40°

A

C

20 m

?

40°

D B 16 m

Mathematics Form 4

6

Solution: a) The distance of AB:

AB2 = 52 - 32

AB2 = 25 - 9 AB2 = 16 AB = 16 AB = 4 m

b) The angle of elevation of Q from A

tan ? = 48

tan ? = 2 ? = 63.43° or 63°26'

6m

3m

P

Q

3. In diagram below, A, B and C are three point on a horizontal surface. A is located to the south of B and C is located to the east of the A. AC is 3m and AB is 5m. The height of PA and QB are 6m and 8m respectively. Calculate: c) the distance of A, d) the angle of evelation of Q from A.

5m

8m

A

B

C

?

Mathematics Form 4

1

ACTIVITY SHEET Pupil’s Copy Lesson 104: Horizontal Planes, Vertical Planes and Inclined Planes.

Plane Type of planes

EDIJ

ABGF

BGHC

ABCDE

CHID


ABCD

AEB

BCFE

B

G

H

I

J

E

AD

F

C

Based on the figure given above, identify the type of planes for:

Question 1

A

B

E

F C

D

Question 2


Mathematics Form 4

2


AFB

CFD

ABCD


ABFE

EFGH

ABCD

Question 3

A B

C D

F

E

Question 4

A B

F E

C

G H

D



Mathematics Form 4

3

ACTIVITY SHEET Teacher’s Copy Lesson 104: Horizontal Planes, Vertical Planes and Inclined Planes.


EDIJ Inclined plane

ABGF Horizontal plane

BGHC Vertical plane

ABCDE Vertical plane

CHID Horizontal plane


ABCD Vertical plane

AEB Horizontal plane

B

G

H

I

J

E

AD

F

C


Question 1

A

B

E

F C

D

Question 2


Mathematics Form 4

4

BCFE Vertical plane


AFB Inclined plane

CFD Inclined plane

ABCD Horizontal plane


ABFE Vertical plane

EFGH Horizontal plane

ABCD Horizontal plane

Question 3

A B

C D

F

E

Question 4

A B

F E

C

G H

D



Mathematics Form 4

1

ACTIVITY SHEET Pupils’ Copy Lesson 105: Lines on a Plane, Lines Intersect with a Plane and Normals.

A D

C B

F G

H E

1. Name all the lines that lie on the plane CDHG. Answer: 2. Name all the lines that intersect with the plane BCGF. Answer : 3. Name all the normals to the plane ABFE Answer :

Question 1

Q

R P

S U

T

V

W

1. Name all the lines that lie on the plane RQTU. Answer: 2. Name all the lines that intersect with the plane PQTS. Answer : 3. Name all the normals to the plane STU Answer :

Question 2

1. Name all the lines that lie on the plane ABHG. Answer: 2. Name all the lines that intersect with the plane FAGL. Answer : 3. Name all the normals to the plane ABCDEF Answer :

Question 3

1. Name all the lines that lie on the plane ABCD. Answer: 2. Name all the lines that intersect with the plane LKOP. Answer : 3. Name all the normals to the plane EFGH Answer :

Question 4

A B

C

D E

F G H

I

J K

L

A

B

C

D H

E

G

F I

J

K L

M

P O

N

Mathematics Form 4

2

ACTIVITY SHEET Teacher’s Copy Lesson 105: Lines on a Plane, Lines Intersect with a Plane and Normals.

A D

C B

F G

H E

1. Name all the lines that lie on the plane CDHG. Answer: 2. Name all the lines that intersect with the plane BCGF. Answer : 3. Name all the normals to the plane ABFE Answer :

Line: CD, DH, HG, GC, DG

Line: BA, CD, GH, FE, GD

Line: AD, BC, FG, EH

Question 1

Q

R P

S U

T

V

W

1. Name all the lines that lie on the plane RQTU. Answer: 2. Name all the lines that intersect with the plane PQTS. Answer : 3. Name all the normals to the plane STU Answer :

Line: RQ, QT, TU, UR, WV

Line: QR, VW, TU, PR, SU

Line: SP, UR, TQ

Question 2

1. Name all the lines that lie on the plane ABHG. Answer: 2. Name all the lines that intersect with the plane FAGL. Answer : 3. Name all the normals to the plane ABCDEF Answer :

Line: AB, BH, HG, GA

Line: AB, GH, LK, FE

Line: AG, BH, CI, DJ, EK, FL

Question 3

1. Name all the lines that lie on the plane ABCD. Answer: 2. Name all the lines that intersect with the plane LKOP. Answer : 3. Name all the normals to the plane EFGH Answer :

Line: AB, BC, CD, DA, IJ, JK, KL, LI

Line: LI, KJ, PM, ON

Line: EA, FB, GC, HD

Question 4

A B

C

D E

F G H

I

J K

L

A

B

C

D H

E

G

F I

J

K L

M

P O

N

Mathematics Form 4

1

ACTIVITY SHEET Pupil’s Copy LESSON 106: Orthogonal projection of a line on a plane.

1. The figure shows a triangular prism PQRSTU. Draw and name the orthogonal projection of each of the following lines on the plane PQRS.

T

P Q

R S

U

M

N

a) TQ

b) QU

Mathematics Form 4

2

2. The figure shows a cube ABCDEFGH . Based on the diagram given, complete the table below.

Line

Plane

Orthogonal Projection

Angle between line

and plane

1

AG

ABCD

2

HB

ABEF

3

CH

EFGH

4

EG

DCHG

5

HB

BCFG

A B

C D

E F

GH

Mathematics Form 4

3

3. The figure shows a pyramid ABCDEFG. Based from diagram given, complete the table below.

No

Line

Plane

Angle between line and plane

1

ABCD

EFG or GFE

2

ABCD

EAC or CAE

3

ABCD

EBD or DBE

4

ABCD

EGF or FGE

5

ABCD

EDB or BDE

B A

C D

E

F

G

Mathematics Form 4

4

ACTIVITY SHEET Teacher’s Copy LESSON 106: Orthogonal projection of a line on a plane.

1. The figure shows a triangular prism PQRSTU. Draw and name the orthogonal projection of each of the following lines on the plane PQRS.

T

P Q

R S

U

M

N

a) TQ

b) QU

T

M Q

U

MQ or QM

NQ or QN

N Q

Mathematics Form 4

5

2. The figure shows a cube ABCDEFGH . Based on the diagram given, complete the table below.

Line

Plane

Orthogonal Projection

Angle between line

and plane

1

AG

ABCD

AC

GAC

2

HB

ABEF

BF

HBF

3

CH

EFGH

HG

CHG

4

EG

DCHG

HG

EGH

5

HB

BCFG

FB

HBF

A B

C D

E F

GH

Mathematics Form 4

6


No

Line

Plane

Angle between line and plane

1

EF

ABCD

EFG or GFE

2

EA

ABCD

EAC or CAE

3

EB

ABCD

EBD or DBE

4

EG

ABCD

EGF or FGE

5

ED

ABCD

EDB or BDE

B A

C D

E

F

G

Mathematics Form 4

1

ACTIVITY SHEET

Pupil’s Copy

LESSON 107: Problems Solving on Angle between a Line and a Plane.

The diagram shows a cuboid. KG = 2HK.

DL = HK. By sketching an appropriate right-angled triangle, find the following:

(a) the length of FD.

(b) the length of KA.

(c) the angle between FD and BCGF.

(d) the angle between KA and EFGH.

A B

DE F

GH

C

9 cm

4 cm

6 cm

K

L

Mathematics Form 4

2

ACTIVITY SHEET

Teacher’s Copy

LESSON 107: Problems Solving on Angle between a Line and a Plane.

The diagram shows a cuboid. KG = 2HK. DL = HK. By sketching an appropriate right-angled triangle, find the following:

(a) the length of FD.

22 BDBFFD +=

)(4 222 ADAB ++=

22 6916 ++=

133= = 11.53 cm

(b) the length of KA.

22 ALKLKA +=

)(4 222 DLAD ++=

22 3616 ++= = 7.81 cm

(c) the angle between FD and BCGF.

The orthogonal projection of FD on BCGF is FC. So, the angle between FD and BCGF is ∠CFD.

DFCD

CFD =∠sin

53.11

9=

= 0.7806 ∠CFD = 51.32° or 51° 19'

(d) the angle between KA and EFGH.

The orthogonal projection of KA on EFGH is KE. So, the angle between KA and EFGH is ∠EKA.

AKAE

EKA =∠sin

81.74=

= 0.5122 ∠EKA = 30.81° or 30° 49'

A B

DE F

GH

C

9 cm

4 cm

6 cm

K

F

D B

K

A L

L

F

D C

11.53 cm

9 cm

A

E K

7.81 cm 4 cm

Mathematics Form 4

1

ACTIVITY SHEET Pupil’s Copy Lesson 108: Line of Intersection between Two Planes.

Plane A Plane B Line of intersection

EDIJ DCHI

BFIC ABGF

AGH BCHG

ABCDE DEJI

DCG BCHG

Plane A Plane B Line of intersection Pair of lines perpendicular to line of intersection

ABCD BCFE

AFE BCFE

BED ABE

B

G

H

I

J

E

AD

F

C

Based on figure given above, state the line of intersection between the following planes:

Question 1

A

B

E

F C

D

Question 2

Based on figure given above, state the line of intersection between the following planes and the pair of lines perpendicular to the line of intersection.

Mathematics Form 4

2


AFC ABCD

BEF BCE

BDF ABCD


AEGC ABFE

BHD ADHE

EFGH ABFE

Question 3

A B

C D

F

E


Question 4

Based on figure given above, state the line of intersection between the following planes and the pair of lines perpendicular to the line of intersection

A B

F E

C

G H

D

Mathematics Form 4

3

ACTIVITY SHEET Teacher’s Copy Lesson 108: Line of Intersection between Two Planes.

Plane A Plane B Line of intersection

EDIJ DCHI Line DI

BFIC ABGF Line BF

AGH BCHG Line GH

ABCDE DEJI Line ED

DCG BCHG Line CG


ABCD BCFE Line BC Line AB & Line BE or Line CD & Line CF

AFE BCFE Line EF Line EA & Line EB

BED ABE Line BE Line EA & Line ED

B

G

H

I

J

E

AD

F

C

Based on figure given above, state the line of intersection between the following planes:

Question 1

A

B

E

F C

D

Question 2


Mathematics Form 4

4


AFC ABCD Line AC Line EF & Line BD

BEF BCE Line BE Line EC & Line EF

BDF ABCD Line BD Line EF & Line AC


AEGC ABFE Line AE Line EF & Line EH or Line AB & Line AD

BHD ADHE Line DH Line AD & Line BD

EFGH ABFE Line EF Line EA & Eh or Line FB & Line FG

Question 3

A B

C D

F

E


Question 4

Based on figure given above, state the line of intersection between the following planes and the pair of lines perpendicular to the line of intersection

A B

F E

C

G H

D

Mathematics Form 4

1

ACTIVITY SHEET Pupil’s Copy LESSON 109: Angle between Two Planes.

1. Based on the diagram given, identify the angle between the two planes

given by completing the table below.

No

Plane

Plane

Angle between two p lanes

1

UON

MNPQ

2

ROQ

MNPQ

3

NORQ

MORP

4

RSU

MORP

5

TQU

NORQ

M N

O

P Q

R

S

T U

Mathematics Form 4

2

2. The figure shows a cuboid ABCDPQRS. Complete the table given in order to identify the the angle between two planes.

No

Plane

Plane

Angle between two planes

1

CDS

ABCD

2

PQB

PQRS

3

ABC

ABPQ

4

SRP

DCSR

5

ABPQ

CBRQ

Q P

R S

B A

C D

Mathematics Form 4

3


No

Plane

Plane


1

ABCD EFG or GFE

2

ABCD EGF or FGE

3

ABCD EHI or IHE

4

ABCD EIH or HIE

B A

C D

E

F

G

H I

Mathematics Form 4

4

ACTIVITY SHEET Teacher’s Copy LESSON 109: Angle between Two Planes.

1. Based on the diagram given, identify the angle between the two planes

given by completing the table below.

No

Plane

Plane


1

UON

MNPQ

ONM or MNO

2

ROQ

MNPQ

RQP or PQR

3

NORQ

MORP

UST or TSU, NOM or MON,

QRP or PRQ

4

RSU

MORP

UST or TSU

5

TQU

NORQ

SUT or TUS

M N

O

P Q

R

S

T U

Mathematics Form 4

5

2. The figure shows a cuboid ABCDPQRS. Complete the table given in

order to identify the the angle between two planes.

No

Plane

Plane


1

CDS

ABCD

SDA or ADS

2

PQB

PQRS

RQB or BQR

3

ABC

ABPQ

CBQ or QBC

4

SRP

DCSR

DSP or PSD

5

ABPQ

CBRQ

RQP or PQR, CBA or ABC

Q P

R S

B A

C D

Mathematics Form 4

6


No

plane

Plane


1

EAB

ABCD

EFG or GFE

2

ECD

ABCD

EGF or FGE

3

EAD

ABCD

EHI or IHE

4

ECB

ABCD

EIH or HIE

B A

C D

E

F

G

H I

Mathematics Form 4

1

ACTIVITY SHEET Pupil’s Copy Lesson 110: Problems Solving on Lines and Planes in 3-Dimensional Shapes.

The figure shows two wooden blocks attached together. Based on the figure, solve the questions given:

Question 1

Question 2

The figure shows a prism. M and N are the midpoints of AB and DE respectively.

A B

C D K

J

M

N

L

G I

F E

H 15cm

40cm

50cm

55cm

20cm

1. What is the length of the diagonal DI?

2. Determine the angle between planes ILNK and CILG.

3. Determine the angle between horizontal plane DCGH and the diagonal DF.

4. Determine the angle between planes LGKJ and CGFB.

A B

B

M

E N

F

D

16cm

16cm

8cm

1. What is the length of the diagonal AE?

2. Identify the angle between planes ACFD and ABED.

3. Determine the angle between the diagonal BF and the plane ABED.

Mathematics Form 4

2

ACTIVITY SHEET Teacher’s Copy Lesson 110: Problems Solving on Lines and Planes in 3-Dimensional Shapes.

The figure shows two wooden blocks attached together. Based on the figure, solve the questions given:

Question 1

Question 2

The figure shows a prism. M and N are the midpoints of AB and DE respectively.

A B

C D K

J

M

N

L

G I

F E

H 15cm

40cm

50cm

55cm

20cm

1. What is the length of the diagonal DI?

2. Determine the angle between planes ILNK and CILG.

3. Determine the angle between horizontal plane DCGH and the diagonal DF.

4. Determine the angle between planes LGKJ and CGFB.

25cm

69.44°

42.0°

36.02°

A B

B

M

E N

F

D

16cm

16cm

8cm

1. What is the length of the diagonal AE?

2. Identify the angle between planes ACFD and ABED.

3. Determine the angle between the diagonal BF and the plane ABED.

18.87cm

58.0°

25.52°

activity sheet - wordpress.com form 4 lesson 02: standard form. learning area: understand and use...

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