MATHEMATICS. Unit 2. Relationships

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1 MATHEMATICS Unit 2 Relationships

2 The Straight Line Eercise 1 1) Given 3 find when: a) 2 b) 4 2) Given 4 find when: a) 3 b) 1 3) Given 2 find when: a) 1 b) 2 4) Given 3 find when: a) 5 b) 5) Given 2 7 find when: a) 3 b) 2 ) Given 2 3 find when: a) 4 b) 1 7) Given 3 5 find when: a) 4 b) 3 8) Given 3 2 find when: a) b) 2 9) Given 5 1 find when: a) 1 b) 0 10) Given 3 find when: a) 2 b) 3 11) Given 12) Given 13) Given 14) Given 15) Given 1) Given 1 find when: a) 10 b) find when: a) 3 b) find when: a) 4 b) find when: a) 2 b) find when: a) b) find when: a) 9 b) 3 17) Given find when: a) b) 9 Maths Department -2- National 4

3 Eercise 2 1) Write down the equation of the lines in these diagrams. a) P G K C M A E I -8-5½ ½ b) Q H L D N F J A M 4½ E 3 I 1 C -2 G -4 N F J D H P -5 Q K -½ L Maths Department -3- National 4

4 2) Write down the equation of the lines in these diagrams. a) I E C M A 4 K 1½ L -4-2½ 2 3 G -3 H P Q -5 J F D N b) G P A K (-4, 5) E F M (-1, 2) N -4 C (5, -2) D I (2, -5) H Q L J Maths Department -4- National 4

5 3) Draw, aes running from -8 to 8. On this diagram draw the lines with equations a) = 3 b) = 7 c) = 4 d) = 8 e) = 1 f) = 2 g) = 2½ h) = 2½ Eercise 3 Find the equation of the straight line shown in the diagram. 1) ) Maths Department -5- National 4

6 3) ) ) ) ) ) Maths Department -- National 4

7 9) ) ) ) Maths Department -7- National 4

8 Eercise 4 Find the equation of the straight line shown in the diagram. 1) 2) (0, 5) (-4, 0) (0, 2) (-3, 0) 3) 4) (, ) (4, 7) (0, -4) (0, -1) 5) ) (0, ) (-, 3) (3, 0) (0, -5) Maths Department -8- National 4

9 7) 8) (7, 1) (4, ) (0, 0) (0, -4) 9) 10) (0, 7) (5, 0) (0, 0) (3, -9) 11) 12) (, 10) (-5, 0) (0, -4) (0, -8) Maths Department -9- National 4

10 Scatter Graphs 1) The scattergraph shows the height and weight of si children. a) What height is Fred? b) What weight is Eric? c) Which two children are the same height? d) Who is the lightest? e) Who is the shortest? 2) This graph shows a relation between the temperature during the da and the sales of ice lollies on that da. a) Suggest a connection between the temperature and the sale of ice lollies. b) Use the chart to estimate the number of ice lollies which might be sold when the temperature reaches 35 degrees. c) Estimate what the temperature was when thirteen ice lollies were sold? 3) This scattergraph shows the prices which tai cabs charge for fairl short distances. a) Suggest a connection between the fare and the number of miles travelled. b) Use the chart to give a reasonable guess at how far ou could go in a tai for c) How much do ou think a mile journe would cost? Maths Department -10- National 4

11 4) Here is a table of eam marks (out of 50) from a modern languages department. Name Ali o Ed Dan Flo Hal Nan Pen Rab Sid German Mark French Mark a) Draw a diagram similar to the one shown, using the same scale for each ais. b) Plot the information from the table to make a scatter diagram c) Describe some connections between the French marks and the German marks. d) Draw a line of best fit through the points. e) Use the graph to estimate what the German mark would be if the French mark was 35. f) One person seems to go against the trend. Who is it, and what makes ou think that? 5) rad and his pals record the number of take awa meals the deliver each evening, and the time it takes them. Time (min) No. of meals a) Draw up a set of aes on squared paper, with time on the horizontal ais and number of meals on the vertical ais. Using suitable scales draw a scatter diagram. b) Estimate the time it would take rad & Co. to deliver 28 meals. Maths Department -11- National 4

12 Equations Eercise 1 Solve the following equations 1) 3 = 5 2) 4 = 3) 2 = 11 4) + 5 = 8 5) + 7 = 12 ) + 15 = 21 7) 3 = 1 8) + 4 = 5 9) 8 = 0 10) + 4 = 2 11) + = 3 12) + 8 = 3 13) 7 = 5 14) 8 = 10 15) + 10 = 20 1) + 9 = 4 17) 7 = 18) + 25 = 15 19) 4 + = 9 20) 5 + = 7 21) 8 + = 24 22) a + = 2 23) a 7 = 3 24) a + = 0 25) 7 = 5 2) + 11 = 20 27) + 12 = 30 28) = 2 29) 8 = 9 30) + 5 = 0 31) 13 = 7 32) + 10 = 3 33) 5 + = 9 34) 9 + = 17 35) = 11 3) + 8 = 3 37) 7 = ) 9 = 3 39) 15 = ) 12 = 7 41) 5 = ) 1 = 7 43) 18 = ) 23 = ) 10 = + 4) 7 = + 47) 18 = ) 5 = ) 3 = 2 50) 14 = + 11 Maths Department -12- National 4

13 Eercise 2 Solve the following equations 1) 3 = 9 2) 2 = 12 3) 4 = 28 4) 5 = 30 5) 7 = 5 ) 4 = 3 7) 9 = 81 8) 9 = 90 9) = ) 12 = 0 11) 10 = ) 8 = 9 13) 5 = 2 14) 7 = 5 15) 8 = 3 1) 4 = 1 17) 2 = 1 18) 9 = 5 19) 3 = 5 20) 4 = 7 21) 3 = 7 22) 2 = 9 23) 3 = 10 24) 5 = 11 25) 5 = 4 2) = 24 27) 5 = 10 28) 4 = 3 29) 3 = 2 30) 12 = 1 31) 7 = 10 32) 5 = 1 33) 4 = 19 34) 10 = 10 35) 18 = 18 3) 17 = 8 37) 8 = 4 38) 10 = 2 39) 12 = 3 40) 72 = 9a 41) = 5a 42) 15 = 2z 43) 8 = 2 44) 7 = 2 45) 9 = 3m 4) 15 = 5n 47) 20 = 2 48) 40 = 4 49) 35 = 7a 50) 2 = 4 Maths Department -13- National 4

14 Eercise 3 Solve the following equations 1) 2 3 = 3 2) 3 1 = 5 3) 4 3 = 5 4) 3 5 = 13 5) 5 7 = 3 ) 7 1 = 27 7) 2 1 = 5 8) = 13 9) = 5 10) = 7 11) 5 3 = 10 12) = 2 13) = 11 14) 3 = 3 15) = 9 1) = 7 17) = 10 18) 3 = 4 19) 5a + = 10 20) 7a + 4 = 0 21) 9a 7 = 0 22) 10a 3 = 0 23) 5n 3 = 11 24) n + 3 = 2 25) = 3 2) 8 2 = 10 27) 4t + 10 = 0 28) + 3 = 12 29) = 13 30) = 20 31) = 10 32) = 2 33) = 2 34) = 0 35) = 0 3) = 1 37) 10 = ) 2 = ) 7 = ) 4 = ) 7 = ) 11 = ) = ) 2 = ) 0 = ) 0 = ) 19 = 5 48) 7 = Maths Department -14- National 4

15 49) = 1 50) = 27 51) 2 3 = 1 52) 5 3 = 1 53) 3 7 = 0 54) = 20 55) 9 = 2 5) 7 + = 57) = 1 58) = 17 59) = 19 0) = 15 1) 8z + 9 = 25 2) a + 7 = 25 3) 7b + 8 = 3 4) 5c + 7 = 32 5) 9m + 5 = 41 ) 8n + 5 = 53 7) + 7 = 13 8) = 29 9) 8t + 9 = 41 70) 3z + 8 = 17 71) 4a + 11 = 19 72) 5b + 13 = 38 73) 7c + 11= 0 74) 9d + 1 = 34 75) 12p + 13 = 85 7) 11q + 1 = 0 77) 3 11 = 10 78) 4 9 = 3 79) 5 = 1 80) 7 2 = 12 81) z 7 = 11 82) 2t 3 = 15 83) 3a 5 = 1 84) 5b 8 = 17 85) 4c 5 = 19 8) 8d 9 = 15 87) 5m 7 = 18 88) 7n 8 = 27 89) 9p 8 = 28 90) q 7 = 47 91) 8 12 = 20 92) 7 10 = 25 93) 15 = 21 94) 9 20 = 34 95) 12z 14 = 22 9) 11t 15 = 40 97) 9 4 = 1 98) = 1 99) = 5 100) = 10 Maths Department -15- National 4

16 Eercise 4 Solve the following equations 1) = ) = ) 4 1 = + 2 4) 2 = 2 + 5) = ) 3 3 = + 3 7) = ) 7 8 = 2 9) 5 7 = ) = ) = 17 12) 5 3 = ) + 1 = ) 3 7 = ) 4 1 = 5 2 1) = ) 7 = ) = ) 10 8 = ) 3 12 = ) 5 2 = ) = ) = 8 24) 7 = ) 3 = ) 4 12 = 2 27) = 5 28) = 29) 3 2 = ) 5 4 = ) 7 3 = ) = ) 2 = ) = ) 7 10 = 3 8 3) 5 12 = 2 37) 4 23 = 7 38) 8 8 = ) = ) = 10 41) = ) = ) 8 = ) = ) 3 = 1 4) 3 10 = ) = 3 48) = Maths Department -1- National 4

17 49) + 2 = ) 7 + = ) 3 7 = ) = ) 7 5 = 2 54) 3z 1 = 5 4z 55) 8 = ) 10 = ) 13 = ) 8 = ) 5 + = 7 8 0) = 2 3 1) a + 2 = 2a ) 9b + 3 = b ) 12c + 9 = 7c ) 11d + 9 = 4d ) 5e + 8 = 4e + 15 ) 7f + 8 = f ) 8p 7 = p + 3 8) 9a 8 = 3a + 1 9) 11r 12 = 8r + 70) 20s 3 = 11s + 71) 15t 2 = 14t ) 5u 12 = u ) 5 9 = ) 7 10 = ) 9z 14 = z 5 7) 15t 5 = 7t 1 77) 10u 20 = 9u 1178) 9v 40 = v 24 79) 4m 5 = 7 2m 80) 5n 8 = 32 3n 81) 4s 13 = 5 2s 82) 2t 50 = 14 t 83) 3u 14 = u 84) v 8 = 1 3v 85) = ) = ) 2z + 7 = 25 4z 88) 4t + 15 = 25 t 89) 5a + 12 = 42 a 90) b + = 30 3b Maths Department -17- National 4

18 Eercise 5 Solve the following equations 1) 2( 1) = 4 2) 3( + 1) = 9 3) 4( 2) = 8 4) 5( 3) = 10 5) 3(2 1) = 9 ) 2(3 + 3) = 12 7) 5(3 2) = 5 8) 2(3 5) = 8 9) 10( 2) = 8 10) 3(4 + 1) = 15 11) 7( 3) = 10 12) 2( + 1) = ) 5( + 2) = ) 7( + 3) = ) 4( + 1) = ) 8(z 2) = 3z ) (t 3) = 2t ) 9(u 1) = 8u ) 5(v 4) = 2v 5 20) 7(m 2) = 5m 4 21) 4(n 5) = n 2 22) 3(a 2) = 9 2a 23) 8( + 2) = 3( + 7) 24) 4( 2) = 2( + 1) 25) 9(z + 1) = 5(z + 5) 2) 5( 3) = 3( + 2) 27) 7(u 3) = 3(u + 5) 28) 3( + 2) = 2( 1) 29) 9(p 3) = 7(p 1) 30) 5( 3) = 2( 7) Eercise Solve the following equations 1) 3( + 2) + 2( + 1) = 23 2) 5(a + 2) 2(a + 3) = 19 3) 4( + 3) + 3( + 2) = 32 4) 8(b + 3) 4(b + 4) = 12 5) 5( + 1) + 3( + 4) = 25 ) 4(c + 2) 2(c + 5) = 14 7) 3(z + 4) + 2(z 3) = 2 8) 5(t 1) 3(t + 2) = 1 Maths Department -18- National 4

19 9) 5(t + 2) + 3(t 1) = 31 10) (u 2) 2(u + 4) = 8 11) 5 2( 2) = 2 12) 3( 1) = 2( + 1) 2 13) 3( + 1) + 2( + 2) = 10 14) 4(2 1) = 3( + 1) 2 15) 4( + 3) + 2( 1) = 4 1) 5 + 2( + 1) = 5( 1) 17) 3( 2) 2( + 1) = 5 18) + 3( + 2) = 2( + 5) ) 5( 3) + 3( + 2) = 7 20) 5( + 1) = ) 3(2 + 1) 2(2 + 1) = 10 22) 4(2 2) = ) 4(3 1) 3(3 + 2) = 0 24) + 2( + 4) = 4 25) 7 (2 3) = 17 2) 3(t + 4) 1 = 3 (4t 1) 27) 5( + ) 10 = 9 (2 + 3) 28) 3 (2d 5) = (5d + 1) 29) 4(1 3) = 7 (4 5) 30) 5p (1 2p) = 9 (p 8) Inequalities Eercise 1 Solve the following inequalities 1) > 7 2) ) 3 2 > 10 4) ) > 3 ) ) > 17 8) 2 2 > 18 9) 3 1 < 11 10) ) ) Maths Department -19- National 4

20 13) 3t + 4 > 2 14) u + 14 < 3 15) 3v + 2 > 14 1) 5w ) ) ) > 10 20) 5 7 < 13 21) ) 2(5 4) 27 23) 4(3 + 2) < 38 24) 5( + 2) > 25 25) 8( ½) < 20 2) (2 ½) 15 27) 3(4 + 7) < 9 28) 9( + 2) > 3 29) ½( + 8) 10 30) ¼(8 12) > 1 Eercise 2 Solve the following inequalities 1) ) < 5 8 3) 5q + < 3q ) 7r 3 > 3r ) 3s s ) 3s s 7) 3t t 8) u u 9) (2p 1) 5 > 23 10) 5(2z 1) ) 2( + 1) + 3 > 15 12) 3( + 5) 4 < 29 13) 3( + 4) 3 5( 3) 14) 7(2a + 5) < 3(a + 3) + 5a 15) 2(4 + 7) < 4(3 + 3) 2( + 1) 1) 5(3 + 1) 7 4(5 4) 17) 4(3 + 2) 3(2 1) > ) 2(5 1) 4( 3) < ) 5( + 3) 2(2 + 5) 20) 3(2 + 1) + 2( 4) > 3( 5) Maths Department -20- National 4

21 Mathematical Models Eercise 1 1) In triangle AC, angle A =. Angle is three times larger than angle A. Angle C is twice the size of angle A. A a) Write down (in terms of ) the sizes of angles and C. b) Form an equation for and solve it. c) What are the sizes (in degrees) of angles and C? C M 2) In triangle LMN, angle N =. Angle M is 40 bigger than angle N Angle L is 10 smaller than angle N. L a) Write down (in terms of ) the sizes of angles L and M. b) Form an equation for and solve it. c) What are the sizes (in degrees) of angles L and M? N 3) I have pence in m pocket. John has 20 pence more than me. Ian has twice as much as I have. Altogether we have 80 pence. a) How much (in terms of ) have John and Ian? b) Write down an equation for and solve it. c) How man pence have John and Ian? Maths Department -21- National 4

22 4) Mar goes on holida with. Anne has three times as much as Mar Joanne has more than Mar. Altogether the have 41. a) Write down an equation for and solve it. b) How much each do Anne and Joanne have? 5) Three people go shopping in the cit. Mrs White spends Mrs James spends twice as much as Mrs White Mrs rown spends three times as much as Mrs White If the spend 12 altogether, what is the value of? ) Three bos go on a school trip Douglas takes in pocket mone Jim takes three times as much as Douglas Malcolm takes four times as much as Douglas If altogether the have 1, find the value of. 7) Sandra is ears old. Helen is 3 ears older than Sandra Karen is 2 ears ounger than Sandra. If all their ages added together give 43 ears, find the value of. 8) Alan is ears old. His elder brother is ears older than he is and his ounger brother is 8 ears ounger than Alan. If all their ages add up to 37 ears, find the value of. Maths Department -22- National 4

23 9) Julie is z ears old. Her father is 4 times older than Julie. Her mother is 7 ears ounger than her father. If their ages add up to 101 ears, find the value of z. Also find the ages of both Julie's parents. 10) I spent minutes doing m homework toda. Yesterda I spent twice as long on homework; tomorrow I shall spend 20 minutes more than toda on it. If I do 180 minutes over these 3 das, find the value of. 11) Three bos ccle to school. Peter takes minutes; rian takes 15 minutes longer than Peter; Stephen takes 5 minutes less than Peter. The total time taken for all three bos together is 4 minutes. Find the value of. Find also how long rian and Stephen take. 12) Alfred has and arr has more than Alfred. If the have 30 altogether, how much do the each have. 13) Mr Clegg is 3 ears older than his wife and 33 ears older than his son. Their ages add up to 72 ears. If Mr Clegg is ears old, find the value of and find all their ages. 14) 85 pence is shared between 2 bos so that one receives pence and the other receives 17 pence more than this. Find the value of. 15) I have a piece of string 3cm long. I use z cm of it so that the piece remaining is twice the length I have used. Find the value of z. 1) I am thinking of a whole number n. a) Write down the net whole number bigger than n. b) If these 2 numbers add up to 29, find the value of n. Maths Department -23- National 4

24 Eercise 2 1) p is a whole number. a) Write down the net two whole numbers bigger than p. b) If these 3 numbers add up to 99, find the value of p. 2) q is a whole number. a) Write down the whole number one less than q. b) If these 2 numbers add up to 49, find the value of q. 3) A bo has marbles. If he wins 20 more, he will have three times as man as when he started. Find the value of. 4) A class has 27 pupils of whom are bos. a) How man girls are there? b) If the number of girls is twice the number of bos, find the value of. 5) A outh club has 30 members of whom z are girls. a) How man bos are there? b) If the number of girls is four more than the number of bos, find the value of z. ) Mr and Mrs Harris have five children. a) If there are girls, how man bos are there? b) Each bo is given 5 for his holida, and each girl is given 8. If the children are given 34 altogether, find the value of. Maths Department -24- National 4

25 7) The area of each rectangle is given in cm². If the lengths of the sides are in centimetres, find the value of in each rectangle. a) 5 Area = 35 b) 4 Area = 22 c) 3 Area = e) d) 4 Area = Area = 12 3 f) ½ Area = ) The perimeter of each shape is given in cm. If the lengths of the sides are also in cm, find the values of. a) b) c) 3 P = P = P = d) 2-1 P = e) + 8 P = f) 9 5 P = g) P = 30 7 h) P = i) P = ) A square has sides of length + 2 cm. Its perimeter is 32cm. Find the value of. Maths Department -25- National 4

26 Changing the Subject of a Formula Eercise 1 Make the subject of these formulas. 1) 2) 3) 4) 5) ) 7) 8) 9) 10) 11) 12) 13) 14) 15) 1) 17) 18) 19) 20) 21) 22) 23) 24) 25) 2) Eercise 2 Make the subject of these formulas. 1) 2) 3) 4) 5) ) Maths Department -2- National 4

27 7) 8) 9) 10) 11) 12) Change the subject of each of the following formulas to the variable indicated. 13) 14) 15) 1) 17) 18) 19) 20) 21) 22) 23) 24) 25) 2) 27) The perimeter of a square is. Change the subject to. 28) The area of a rectangle is. Change the subject to. 29) The volume of a cuboid is. Change the subject to. 30) The speed of a train is. Change the subject to. 31) The current in a circuit is. Change the subject to. 32) The equation of a straight line is. Change the subject to. Maths Department -27- National 4

28 PYTHAGORAS Eercise 1 hpotenuse Calculate the length of, giving our answer where necessar to 2 decimal places (all sizes in centimetres). 1) 5 2) 7 3) 5 3 4) 4 5) 2 ) 8 5 7) 5 8) ) 3 10) ) ) ) ) ) ) ) 5 18) Maths Department -28- National 4

29 Eercise 2 shorter side Calculate the length of, giving our answer where necessar to 2 decimal places (all sizes in centimetres). 1) 9 2) ) 13 4) 8 5 5) 8 12 ) 5 9 7) 5 7 8) 2 3 9) 10) ) ) ) ) ) 3.4 1) ) ) Maths Department -29- National 4

30 Eercise 3 mied Calculate the length of, giving our answer where necessar to 2 decimal places (all sizes in centimetres). 1) 2) 8 5 3) 3 4) ) ) 7) 8) ) ) ) ) ) ) ) ) ) 18) 20) ) Maths Department -30- National 4

31 Eercise 4 questions in contet 1) A ladder of length 12 feet is leaning against a wall. It reaches to a height of 10 feet. How far is the foot of the ladder from the wall? 2) The foot of a ladder is 5 feet from a wall. The ladder is 14 feet long. How far up the wall does the ladder reach? 3) The foot of a ladder is 2 m from a wall. It reaches up to a height of 7 m. How long is the ladder? 4) A ladder 15 m long leans against a wall and reaches a window 14 m above the ground. Calculate the distance from the foot of the ladder to the wall. 5) If a ladder 41 feet long is placed with its foot 9 feet from the bottom of a wall 30 feet high, how much of the ladder etends beond the top of the wall? ) This diagram shows the gable end of a shed, with dimensions as shown in the diagram. Calculate the length of the roof. 3 5 m roof 3 m 2 m 7) The tops of two masts on a ship are joined b a wire 9 m long. If the masts are 1 m and 20 m high, how far apart are the? Maths Department -31- National 4

32 8) A barn has a sloping roof and is 14 m high at the front and 18 m high at the back. It is 12 m from front to back. Calculate the length of the sloping roof. roof 14 m 18 m 12 m 9) This is the diagram of a lawn. Kerb stones are put round the outside of the lawn. Calculate the total length of kerb stones required. m 8 m 12 m 10) AC is a lawn in the shape of an isosceles triangle. Kerb stones are put round the outside of the lawn. If the 'base' AC is m, and the 'height' is 10m, calculate the total length of kerb stones required. A C 11) Calculate the perimeter of this shape. 7 cm 10 cm 1 cm 12) In a rectangular garden which measures 38 m b 21 m, a path goes diagonall from one corner to the opposite corner. Calculate the length of the path. Maths Department -32- National 4

33 Pthagoras using co-ordinates Eercise 1 1) Calculate the length between each pair of points on the line. A 0 - C 0 - D - - F E 0 - G 0 - H - - Maths Department -33- National 4

34 K J 0 - M 0 - L - - 2) Calculate the length between each pair of points on the line. (a) (b) (c) - (d) Maths Department -34- National 4

35 (e) (f) ) Calculate the length between each pair of points on the line. (-1, ) (1, 1) (-2, -5) (3, -) (a) (b) (, 1) (1, -4) (2, -3) (c) (d) (4, -) Maths Department -35- National 4

36 (-, 5) (5, 3) (e) (2, -) (f) (0, -3) Eercise 2 Find the length of the line A where 1) A(1, 2) (5, 5) 2) A(1, 2) (7, ) 3) A(1, 8) (7, 3) 4) A( 2, 8) (7, 0) 5) A(1, 4) (5, 2) ) A(8, 3) (2, 2) 7) A(, 1) (4, 4) 8) A( 3, 5) (2, 1) 9) A( 4, 5) (1, 2) Eercise 3 1) If A(3, 1), (7, ) and C(10, 3) are the three corners of triangle AC, find the length of all 3 sides. 2) If A(4, 1), ( 3, 3) and C(9, 10) are the three corners of triangle AC, find the length of all 3 sides. 3) If A( 2, 4), ( 2, 8) and C(, 0) are the three corners of triangle AC, find the length of all 3 sides. 4) If A(1, 0), (-5, ) and C(3, ) are the three corners of triangle AC, find the length of all 3 sides of triangle AC 5) If A( 3, 1), (5, 3) and C(1, 5) are the three corners of triangle AC, find the length of all 3 sides of triangle AC Maths Department -3- National 4

37 Enlargement and Reduction Eercise 1 scale factor of 2 Draw each of the following shapes TWICE as large in our jotter. A C D E F G H I K L J M N O Maths Department -37- National 4

38 Eercise 2 scale factor of 3 Draw each of the following shapes THREE times larger in our jotter. A C D E F G H I K L J M N O Maths Department -38- National 4

39 Eercise 3 scale factor of Draw each of the following shapes HALF the size in our jotter. A C D E F G H I K L J M N O Maths Department -39- National 4

40 Eercise 4 scale factor of Draw each of the following shapes a THIRD the size in our jotter. A C D E F G H I J K L M N O Maths Department -40- National 4

41 Eercise 5 scale factor of Draw each of the following shapes a QUARTER the size in our jotter. A C D E F G M Maths Department -41- National 4

42 Eercise scale factor of Draw each of the following shapes using a scale factor of A C D E F G H I K L J M N O Maths Department -42- National 4

43 Eercise 7 scale factor of Draw each of the following shapes using a scale factor of A C D E F G H I K L J M N O Maths Department -43- National 4

44 Eercise 8 scale factor of Draw each of the following shapes using a scale factor of A C D E F G H I J K L M N O Maths Department -44- National 4

45 Eercise 9 scale factor of Draw each of the following shapes using a scale factor of A C D E F G H I J K L M N O Maths Department -45- National 4

46 Circles and Angle Properties Eercise 1 Angle in a Semi-Circle Cop each diagram and fill in the sizes of all the angles. The line A is a diameter. 1) 2) A 30 A 10 3) 4) A 40 A 20 5) ) A A ) A 0 8) A 70 Maths Department -4- National 4

47 Eercise 2 Angle in a Semi-Circle Cop each diagram and fill in the sizes of all the angles. The line A is a diameter. 1) 2) A 2 A 13 3) 4) A 42 A 21 5) ) A 79 A 54 7) 8) A 3 A 82 Maths Department -47- National 4

48 Eercise 3 Tangent/Radius Cop each diagram and fill in the size of all the angles. The point O is the centre and TP is a tangent to the circle. 1) 2) P O 8 O T P 3) 4) 23 T O 5 P T O P 0 T 5) T P 25 ) O 7) 8) O P T O O 55 5 P 40 P 35 T T Maths Department -48- National 4

49 Eercise 4 Opposite Angles Cop the following diagrams into our jotter and fill in the sizes of all the angles. The first one has been done for ou. 1) 2) ) ) 11 5) ) 77 Eercise 5 Corresponding Angles Cop the following diagrams into our jotter and fill in the sizes of all the angles. The first one has been done for ou. 1) ) 113 3) 121 4) 5) ) ) 42 8) 9) ) 11) 12) Maths Department -49- National 4

50 Eercise Alternate Angles Cop the following diagrams into our jotter and fill in the sizes of all the angles. The first one has been done for ou:- 1) 2) 3) ) 110 5) 45 ) 7) 150 8) 11 Maths Department -50- National 4

51 Eercise 7 Angles in Shapes Cop the following diagrams into our jotter and fill in the sizes of all the angles:- 1) 50 2) ) 4) 5) 85 ) 7) ) 5 9) 70 Maths Department -51- National 4

52 Trigonometr Eercise 1 1) Cop each of the triangles below into our jotter. On each triangle mark H for the hpotenuse and b looking at the 'marked' angle write O on the opposite side and A on the adjacent side. 2) For the following angles find correct to 3 decimal places: a) sin 20 b) sin 1 c) sin 9 d) sin 4 e) sin 27 f) cos 54 g) cos 5 h) cos 84 i) cos 7 j) cos 29 k) tan 43 l) tan 3 m) tan 59 n) tan 48 o) tan 71 p) sin 34 q) tan 89 r) cos 25 s) tan 18 t) sin 37 u) tan 24 v) cos 84 w) sin 35 ) tan 58 ) cos 47 Maths Department -52- National 4

53 3) Find the size of angle (correct to 1 decimal place) for a) tan = b) tan = c) tan = d) tan = e) tan = f) tan = g) tan = h) tan = i) tan = 1 27 j) tan = 014 4) Find the size of angle (correct to 1 decimal place) for a) cos = b) cos = c) cos = d) cos = e) cos = f) cos = g) cos = h) cos = i) cos = j) cos = ) Find the size of angle (correct to 1 decimal place) for a) sin = b) sin = c) sin = d) sin = e) sin = f) sin = g) sin = 0 58 h) sin = i) sin = j) sin = 0 12 Eercise 2 1) Find the length of the side marked. (TANGENT) Maths Department -53- National 4

54 2) Find the length of the side marked. (SINE) 3) Find the length of the side marked. (COSINE) Maths Department -54- National 4

55 Eercise 3 Find the length of the side marked. (MIXED) Maths Department -55- National 4

56 Eercise 4 Find the size of the angle marked in each triangle Maths Department -5- National 4

57 Eercise 5 Find the size of in each triangle. Maths Department -57- National 4

58 Eercise 1) A ramp is fitted at a school to allow disabled access to the second floor of the building. The ramp is 48 m long and is at an angle of 11 to the horizontal. What is the height of the second floor above the ground? 2) The diagram shows a shop s ramp for customers who are wheelchair users. It connects the pavement to the level of the shopping mall. The ramp is 14 metres long and slopes at an angle of 9, as shown. Calculate the difference in height, h metres, between the pavement and the shopping mall. Give our answer correct to the nearest metre. 3) The diagram shows a flagpole which is supported b a wire which is fied to the ground 8 2 metres from the base of the flagpole. The wire is 15 3 metres long. a) Calculate the angle marked between the wire and the ground. b) For safet reasons the angle should be less than m 8.2 m Can the angle of the wire be considered safe? Maths Department -58- National 4

59 4) Sam is fling a kite. The string is 48 metres long. How high is the kite above the ground? (marked in the diagram) 5) A triangular bracket is designed to support a shelf. Its length is 10 cm and its height is 7 5 cm. 10 cm a) Calculate the angle at the base of the bracket, angle. b) For safet reasons the angle should be less than cm Can the angle of the wire be considered safe? ) A ramp has been constructed at a bowling club. It is 3 5metres long and rises through 0 3metres. Calculate the angle,, that the ramp makes with the horizontal. VICTORIA OWLING CLU 0 3m 3 5m o 7) A bo fling a kite lets out 200 m of string which makes an angle of 72 with the horizontal. What is the height of the kite? 8) A ladder is 15 m long. The top rests against the wall of a house, and the foot rests on level ground 2 m from the wall. Calculate the angle between the ladder and the ground. Maths Department -59- National 4

60 9) A ladder 12 m long is set against the wall of a house and makes an angle of 75 with the ground. a) How far up the wall will the ladder reach? b) How far is the foot of the ladder from the wall? 10) A telegraph pole standing on horizontal ground is 9 m high, and is supported b a wire 10 m long fied to the top of the pole and to the ground. Calculate: a) the angle between the wire and the ground. b) the distance of the point on the ground from the foot of the pole. Maths Department -0- National 4

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