MATHEMATICS. S2 Level 3/4 Course -1- Larkhall Maths Department Academy

MATHEMATICS S2 Level 3/4 Course -1- Larkhall Maths Department Academy

Equations Eercise 1(A) Solve the following equations 1) 3 = 5 2) 4 = 6 3) 2 = 11 4) + 5 = 8 5) + 7 = 12 6) + 15 = 21 7) 3 = 1 8) + 4 = 5 9) 8 = 0 10) + 4 = 2 11) + 6 = 3 12) + 8 = 3 13) y 7 = 5 14) y 8 = 10 15) y + 10 = 20 16) y + 9 = 4 17) y 7 = 6 18) y + 25 = 15 19) 4 + = 9 20) 5 + = 7 21) 8 + = 24 22) a + 6 = 2 23) a 7 = 3 24) a + 6 = 0 25) 7 = 5 26) + 11 = 20 27) + 12 = 30 28) 6 = 2 29) 8 = 9 30) + 5 = 0 31) 13 = 7 32) + 10 = 3 33) 5 + = 9 34) 9 + = 17 35) y 6 = 11 36) y + 8 = 3 37) 7 = + 2 38) 9 = 3 39) 15 = + 4 40) 12 = 7 41) 5 = + 11 42) 16 = 7 43) 18 = 9 + 44) 23 = 11 + 45) 10 = + 6 46) 7 = 6 + 47) 18 = 13 + 48) 5 = 7 + 49) 3 = 2 50) 14 = y + 11 Maths Department -2- S2 Level 3/4 Course

Eercise 2(A) Solve the following equations 1) 3 = 9 2) 2 = 12 3) 4 = 28 4) 5 = 30 5) 7 = 56 6) 4 = 36 7) 9 = 81 8) 9 = 90 9) 6 = 180 10) 12 = 60 11) 10 = 1000 12) 8 = 96 13) 5 = 2 14) 7 = 5 15) 8 = 3 16) 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 26) 6 = 24 27) 5 = 10 28) 4 = 36 29) 3 = 2 30) 12 = 1 31) 7 = 10 32) 5 = 1 33) 4 = 19 34) 10 = 10 35) 18 = 18 36) 17 = 68 37) 8 = 4 38) 10 = 2 39) 12 = 3y 40) 72 = 9a 41) 6 = 5a 42) 15 = 2z 43) 8 = 2y 44) 7 = 2 45) 9 = 3m 46) 15 = 5n 47) 20 = 2 48) 40 = 4y 49) 35 = 7a 50) 2 = 4 S2 Level 3/4 Course -3- Maths Department

Eercise 3(A) Solve the following equations 1) 2 3 = 3 2) 3 1 = 5 3) 4 3 = 5 4) 3 5 = 13 5) 5 7 = 3 6) 7 1 = 27 7) 2 1 = 5 8) 3 + 1 = 13 9) 4 + 2 = 5 10) 5 + 1 = 7 11) 5 3 = 10 12) 3 + 1 = 2 13) 2y + 10 = 11 14) 3y 6 = 3 15) 2y + 10 = 9 16) 3y + 10 = 7 17) 4y + 10 = 10 18) 3y 6 = 4 19) 5a + 6 = 10 20) 7a + 4 = 0 21) 9a 7 = 0 22) 10a 3 = 0 23) 5n 3 = 11 24) 6n + 3 = 2 25) 3 + 4 = 3 26) 8 2 = 10 27) 4t + 10 = 0 28) 6 + 3 = 12 29) 5 + 2 = 13 30) 7 + 4 = 20 31) 5 + 8 = 10 32) 8 + 5 = 2 33) 3 + 7 = 2 34) 8 + 5 = 0 35) 9 + 3 = 0 36) 5 + 3 = 1 37) 10 = 2 + 3 38) 2 = 3 4 39) 7 = 4 5 40) 4 = 5 1 41) 7 = 3 + 6 42) 11 = 3 5 43) 6 = 3 4 44) 2 = 2 + 1 45) 0 = 10 1 46) 0 = 11 + 2 47) 19 = 6 5 48) 7 = 3 + 7 49) 3 + 1 = 16 50) 4 + 3 = 27 51) 2 3 = 1 52) 5 3 = 1 53) 3 7 = 0 54) 2 + 5 = 20 55) 6 9 = 2 56) 7 + 6 = 6 57) 3 + 4 = 16 58) 5 + 2 = 17 59) 4y + 3 = 19 60) 2y + 5 = 15 61) 8z + 9 = 25 62) 6a + 7 = 25 63) 7b + 8 = 36 64) 5c + 7 = 32 65) 9m + 5 = 41 66) 8n + 5 = 53 67) 6 + 7 = 13 68) 3y + 8 = 29 Maths Department -4- S2 Level 3/4 Course

69) 8t + 9 = 41 70) 3z + 8 = 17 71) 4a + 11 = 19 72) 5b + 13 = 38 73) 7c + 11= 60 74) 9d + 16 = 34 75) 12p + 13 = 85 76) 11q + 16 = 60 77) 3 11 = 10 78) 4 9 = 3 79) 6y 5 = 1 80) 7y 2 = 12 81) 6z 7 = 11 82) 2t 3 = 15 83) 3a 5 = 16 84) 5b 8 = 17 85) 4c 5 = 19 86) 8d 9 = 15 87) 5m 7 = 18 88) 7n 8 = 27 89) 9p 8 = 28 90) 6q 7 = 47 91) 8 12 = 20 92) 7 10 = 25 93) 6y 15 = 21 94) 9y 20 = 34 95) 12z 14 = 22 96) 11t 15 = 40 97) 9 4 = 1 98) 11 10 = 1 99) 15y + 2 = 5 100) 7y + 8 = 10 Eercise 3(B) Solve the following equations 1) 3 + 1 = 2 + 3 2) 5 + 3 = 2 + 12 3) 4 1 = + 2 4) 6 2 = 2 + 6 5) 5 + 7 = 4 + 11 6) 3 3 = + 3 7) 10 + 1 = 4 + 4 8) 7 8 = 2 9) 5 7 = 3 3 10) 11 20 = 6 + 5 11) 4 + 2 = 17 12) 5 3 = 11 2 13) 6 + 1 = 33 2 14) 3 7 = 1 5 15) 4 1 = 5 2 16) 8 + 2 = 7 2 17) 6 7 = 2 4 18) 3 + 9 = 17 + 2 19) 10 8 = 20 + 6 20) 3 12 = 4 3 21) 5 2 = 6 + 4 22) 10 + 7 = 12 2 23) 3 + 5 = 8 24) 7 = 8 2 S2 Level 3/4 Course -5- Maths Department

25) 3 = 10 + 26) 4 12 = 2 27) 7 + 1 = 5 28) 2 + 1 = 6 29) 3 2 = 2 10 30) 5 4 = 2 10 31) 7 3 = 3 + 8 32) 5 + 4 = 2 + 9 33) 6 2 = + 8 34) 8 + 1 = 3 + 2 35) 7 10 = 3 8 36) 5 12 = 2 6 37) 4 23 = 7 38) 8 8 = 3 2 39) 11 + 7 = 6 + 7 40) 9 + 8 = 10 41) 5 + 3 = + 8 42) 4 + 7 = + 5 43) 6 8 = 4 3 44) 5 + 1 = 7 2 45) 6 3 = 1 46) 3 10 = 2 3 47) 5 + 1 = 6 3 48) 11 20 = 10 15 49) 6 + 2 = 8 3 50) 7 + = 9 5 51) 3y 7 = y + 1 52) 8y + 9 = 7y + 8 53) 7y 5 = 2y 54) 3z 1 = 5 4z 55) 8 = 13 4 56) 10 = 12 2 57) 13 = 20 9 58) 8 = 5 2 59) 5 + = 7 8 60) 3 + 11 = 2 3 61) 6a + 2 = 2a + 10 62) 9b + 3 = 6b + 18 63) 12c + 9 = 7c + 14 64) 11d + 9 = 4d + 30 65) 5e + 8 = 4e + 15 66) 7f + 8 = f + 20 67) 8p 7 = 6p + 3 68) 9a 8 = 3a + 16 69) 11r 12 = 8r + 6 70) 20s 3 = 11s + 6 71) 15t 2 = 14t + 5 72) 5u 12 = u + 20 73) 5 9 = 2 3 74) 7y 10 = 5y 2 75) 9z 14 = 6z 5 76) 15t 56 = 7t 16 77) 10u 20 = 9u 11 78) 9v 40 = v 24 79) 4m 5 = 7 2m 80) 5n 8 = 32 3n 81) 4s 13 = 5 2s Maths Department -6- S2 Level 3/4 Course

82) 2t 50 = 14 6t 83) 3u 14 = 6 u 84) v 8 = 16 3v 85) 3 + 1 = 11 2 86) 4y + 3 = 31 3y 87) 2z + 7 = 25 4z 88) 4t + 15 = 25 6t 89) 5a + 12 = 42 a 90) b + 6 = 30 3b Eercise 4(B) 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 6) 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 13) 5( + 2) = 2 + 16 14) 7( + 3) = 5 + 29 15) 4(y + 1) = y + 13 16) 8(z 2) = 3z + 9 17) 6(t 3) = 2t + 10 18) 9(u 1) = 8u + 3 19) 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) 26) 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) 31) 8(r 6) = 5(r 3) 32) 6( + 2) = 2( 3) 33) 7(y + 2) = 2(y + 12) 34) 10( 3) = 35) 5(t 2) = 2(t + 4) 36) 3(2 1) = 4( + 1) 37) 9(v 1) = 7(v + 1) 38) 4(2 + 1) = 5( + 3) S2 Level 3/4 Course -7- Maths Department

Eercise 4(C) 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(y + 1) + 3(y + 4) = 25 6) 4(c + 2) 2(c + 5) = 14 7) 3(z + 4) + 2(z 3) = 26 8) 5(t 1) 3(t + 2) = 1 9) 5(t + 2) + 3(t 1) = 31 10) 6(u 2) 2(u + 4) = 8 11) 4(a + 1) + 3(a 4) = 13 12) 4( 2) 3( + 4) = 4 13) 2(b + 1) + 4(b 3) = 20 14) 5(t + 3) (t 6) = 29 15) 3 + 2( + 1) = 3 + 12 16) 3( 1) = 2 2 17) 4 2( + 4) = + 1 18) 4( + 2) = 3 + 10 19) 2 3( + 2) = 2 + 1 20) 2(2 1) = + 4 21) 5 2( 2) = 6 2 22) 3( 1) = 2( + 1) 2 23) 3( + 1) + 2( + 2) = 10 24) 4(2 1) = 3( + 1) 2 25) 4( + 3) + 2( 1) = 4 26) 5 + 2( + 1) = 5( 1) 27) 3( 2) 2( + 1) = 5 28) 6 + 3( + 2) = 2( + 5) + 4 29) 5( 3) + 3( + 2) = 7 30) 5( + 1) = 2 + 3 + 31) 3(2 + 1) 2(2 + 1) = 10 32) 4(2 2) = 5 17 33) 4(3 1) 3(3 + 2) = 0 34) + 2( + 4) = 4 Maths Department -8- S2 Level 3/4 Course

35) 7 (2 3) = 17 36) 3(t + 4) 1 = 3 (4t 1) 37) 5(6 + y) 10 = 9 (2y + 3) 38) 3 (2d 5) = (5d + 1) 39) 4(1 3y) = 7 (4y 5) 40) 5p (1 2p) = 9 (p 8) 41) 7(2 + 1) = 5 4(2 3) 42) 5( 2) = 6 3( + 2) INEQUALITIES Eercise 5(B) Solve the following inequalities 1) 2 + 1 > 7 2) 2 + 1 7 3) 3 2 > 10 4) y + 5 1 5) 2y + 3 > 3 6) 5y 1 11 7) 4 + 5 > 17 8) 2 2 > 18 9) 3 1 < 11 10) 5y 3 27 11) 7y + 4 4 12) 8y + 3 59 13) 3t + 4 > 1 14) 6u + 14 < 2 15) 3v + 2 > 16 16) 5w + 1 34 17) 2 + 7 3 18) 4y 1 25 19) 2 + 3 > 11 20) 5 7 < 13 21) 7 8 13 22) 6 + 5 35 23) 3 + 4 16 24) 2 + 1 19 25) 5 + 7 < 32 26) 3 2 < 10 27) 4 3 > 29 28) 8 + 6 38 29) 7 3 > 39 30) 3 + 12 30 31) 2 + 6 < 15 32) 2 7 > 8 33) 3p + 1 > p + 7 S2 Level 3/4 Course -9- Maths Department

34) 5q + 6 < 3q + 24 35) 7r 3 > 3r + 13 36) 3s + 1 13 s 37) 3t 2 13 2t 38) u + 4 24 3u 39) 3(2 3) > 6 40) 2(5 4) 27 41) 4(3 + 2) < 38 42) 5( + 2) > 25 43) 8( ½) < 20 44) 6(2 ½) 15 45) 3(4 + 7) < 69 46) 9( + 2) > 63 47) ½(6 + 8) 10 48) ¼(8 12) > 1 49) 3(2 + 5) < 21 50) 5(2 7) > 15 51) 8 + 7 > 6 + 15 52) 4 1 + 8 53) 3 + 5 + 1 54) 9 + 4 < 5 8 55) 5( + 3) 2(2 + 5) 56) 3(2 + 1) + 2( 4) > 3 5 57) 4(3 + 2) 3(2 1) > 3 + 2 58) 2(5 1) 4( 3) < + 5 59) 2( + 1) + 3 > 15 60) 3(y + 5) 4 < 29 61) 6(2p 1) 5 > 23 62) 5(2z 1) + 4 29 EQUATIONS WITH FRACTIONS Eercise 6(B) Solve the following equations 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) Maths Department -10- S2 Level 3/4 Course

PROBLEMS Eercise 7(B) B 1 In triangle ABC, angle A =. Angle B 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 B and C. b) Form an equation for and solve it. c) What are the sizes (in degrees) of angles B and 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 C 3 I have pence in my 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 many pence have John and Ian? 4 Mary goes on holiday with. Anne has three times as much as Mary Joanne has 6 more than Mary. Altogether they have 41. a) Write down an equation for and solve it. b) How much each do Anne and Joanne have? S2 Level 3/4 Course -11- Maths Department

5 Three people go shopping in the city. Mrs White spends Mrs James spends twice as much as Mrs White Mrs Brown spends three times as much as Mrs White If they spend 126 altogether, what is the value of? 6 Three boys go on a school trip Douglas takes in pocket money Jim takes three times as much as Douglas Malcolm takes four times as much as Douglas If altogether they have 16, find the value of. 7 Sandra is years old. Helen is 3 years older than Sandra Karen is 2 years younger than Sandra. If all their ages added together give 43 years, find the value of. 8 Alan is y years old. His elder brother is 6 years older than he is and his younger brother is 8 years younger than Alan. If all their ages add up to 37 years, find the value of y. 9 Julie is z years old. Her father is 4 times older than Julie. Her mother is 7 years younger than her father. If their ages add up to 101 years, find the value of z. Also find the ages of both Julie's parents. 10 I spent minutes doing my homework today. Yesterday I spent twice as long on homework; tomorrow I shall spend 20 minutes more than today on it. If I do 180 minutes over these 3 days, find the value of. 11 Three boys cycle to school. Peter takes y minutes; Brian takes 15 minutes longer than Peter; Stephen takes 5 minutes less than Peter. The total time taken for all three boys together is 46 minutes. Find the value of y. Find also how long Brian and Stephen take. Maths Department -12- S2 Level 3/4 Course

12 Alfred has and Barry has 6 more than Alfred. If they have 30 altogether, how much do they each have. 13 Mr Clegg is 3 years older than his wife and 33 years older than his son. Their ages add up to 72 years. If Mr Clegg is years old, find the value of and find all their ages. 14 85 pence is shared between 2 boys 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 36cm long. I use z cm of it so that the piece remaining is twice the length I have used. Find the value of z. 16 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. Eercise 7(C) 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 boy has marbles. If he wins 20 more, he will have three times as many as when he started. Find the value of. S2 Level 3/4 Course -13- Maths Department

4 A class has 27 pupils of whom are boys. a) How many girls are there? b) If the number of girls is twice the number of boys, find the value of. 5 A youth club has 30 members of whom z are girls. a) How many boys are there? b) If the number of girls is four more than the number of boys, find the value of z. 6 Mr and Mrs Harris have five children. a) If there are y girls, how many boys are there? b) Each boy is given 5 for his holiday, and each girl is given 8. If the children are given 34 altogether, find the value of y. 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 = 18 + 2 + 3-3 e) d) 4 Area = 24 2-1 4 + 2 Area = 12 3 f) ½ Area = 7 3 + 2 Maths Department -14- S2 Level 3/4 Course

8 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 = 15 + 4 P = 28 3-2 P = 32 + 1 3 + 2 3-2 d) 2-1 P = 12 + 2 e) + 8 P = 21 2 2 + 4 f) 9 5 P = 38 2-4 - 3 2 d) 3 + 2 e) 4 P = 30 7 4 + 3 P = 29 2 2 f) P = 46 6 3 3 2 + 5 9 A square has sides of length + 2 cm. Its perimeter is 32cm. Find the value of. 10 An equilateral triangle has sides of length y + 4 cm. Its perimeter is 16cm. Find the value of y. 11 A regular pentagon has sides 2 + 7 cm long and its perimeter is 65cm. Find the value of. S2 Level 3/4 Course -15- Maths Department

PYTHAGORAS Eercise 1(A) Calculate the length of, giving your answer where necessary to 2 decimal places (all sizes in centimetres). 1) 6 5 2) 6 7 3) 5 3 4) 4 6 5) 2 6) 8 5 7) 5 8) 2 1 8 7 9) 3 10) 9 8 11) 7.1 12) 7.4 4 9.2 12.3 13) 1.4 1.1 14) 1.3 1.5 15) 12 13 16) 8.2 6.1 17) 5 18) 4.7 5 2.9 Maths Department -16- S2 Level 3/4 Course

Eercise 2(A) Calculate the length of, giving your answer where necessary to 2 decimal places (all sizes in centimetres). 1) 9 2) 10 7 3) 13 4) 6 8 6 5 5) 8 12 6) 5 9 7) 5 7 8) 2 3 9) 6 10) 9 13 11) 11.1 12) 7.4 12.7 4 9.2 13) 1.4 1.1 14) 3.3 1.5 15) 3.4 16) 8.2 1.3 6.1 17) 8.3 5 18) 4.7 9.6 S2 Level 3/4 Course -17- Maths Department

Eercise 3(A) Calculate the length of, giving your answer where necessary to 2 decimal places (all sizes in centimetres). 1) 6 2) 8 5 3) 3 4) 4 8 6 12 5) 6) 7) 8) 10 9 8 6 2 7 4 4 9) 5 7 10) 13 16 11) 16 13 12) 10 19 13) 3.3 14) 3.3 1.5 15) 8.8 4.1 16) 1.2 8.22 6.2 17) 18) 20) 3.72 7.33 9.2 19) 12.2 16.2 14.3 5.26 9.8 Maths Department -18- S2 Level 3/4 Course

Eercise 4(B) 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? 10 12 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 beyond the top of the wall? 6 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 41 9 2 m 9 30 7 The tops of two masts on a ship are joined by a wire 9 m long. If the masts are 16 m and 20 m high, how far apart are they? 20 16 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. 14 m roof 18 m 12 m S2 Level 3/4 Course -19- Maths Department

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. 6 m 8 m 10 ABC 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 6 m, and the 'height' is 10m, calculate the total length of kerb stones required. 10 cm 12 m B 11 Calculate the perimeter of this shape. 7 cm A C 16 cm 12 In a rectangular garden which measures 38 m by 21 m, a path goes diagonally from one corner to the opposite corner. Calculate the length of the path. Eercise 4(C) 1 Calculate the length of the shorter side in each of the following right-angled isosceles triangles. a) 8 b) 12 c) 15 d) 4.8 2 Calculate the length of a, b, c etc in each of the following diagrams a) b) 5 29 c) d) c b 13 f h 32 d a 6 20 41 28 e g 26 8 40 26 Maths Department -20- S2 Level 3/4 Course

e) k j 18 h) 6 f) 3.4 n m 8.6 2.4 g) 3 p 8 5 9 4 7 r 3 ABCD is a rectangular plot 35m by 12m. AC is a diagonal path. Find how much further it is to go from A to C by way of B than to go directly from A to C. 4 A flagpole PS, 16 m high is supported by ropes PQ and PR each 20 m long. If PQR forms an isosceles triangle, find the distance QR. 5 AB and AC are tent rope attached to the central pole at a height of 8 m from the ground. The ropes are pegged at B and C at distances of 15 m from the pole. What lengths of ropes are required? 6 AB is a sloping roof of a farm shed. The front A is 12 ft high, the back B is 19 ft high, and the shed measures 24 ft from front to back. A ladder is placed against the shed at A with its foot 5 ft from the front. a What length of ladder is required, allowing for 3 ft above the shed? b How far is it from A to B up the slope of the roof? S2 Level 3/4 Course -21- Maths Department

7 This diagram shows a hoarding DC supported by strut BA fied to the ground at B and the hoarding at A. The strut AB = 6.5 m, AD = 1 m and BC = 2.5 m. Calculate the height of the hoarding. 8 This diagram shows a vertical pole AB, 18 ft tall, standing on horizontal ground AD. BD and CD are stay-wires 20 ft and 15 ft long respectively. Calculate the length of: a) AD b) AC c) CB. Eercise 5(B) Find the length of the line AB where a) A(1, 2) B(5, 5) b) A(1, 2) B(7, 6) c) A(1, 8) B(7, 3) d) A(-2, 8) B(7, 0) e) A(1, 4) B(5, -2) f) A(8, 3) B(2, -2) g) A(6, 1) B(4, -4) h) A(-3, 5) B(2, 1) i) A(-4, -5) B(1, -2) Eercise 5(C) 1 If A(3, 1), B(7, 6) and C(10, -3) are the three corners of triangle ABC, find the length of all 3 sides. 2 If A(4, -1), B(-3, 3) and C(9, 10) are the three corners of triangle ABC, find the length of all 3 sides. 3 If A(-2, -4), B(-2, 8) and C(6, 0) are the three corners of triangle ABC, find the length of all 3 sides. Maths Department -22- S2 Level 3/4 Course

4 If A(1, 0), B(-5, 6) and C(3, 6) are the three corners of triangle ABC, find: a) the length of all 3 sides of triangle ABC b) the area of triangle ABC c) the shortest distance from A to the line BC. 5 If A(-3, 1), B(5, -3) and C(1, 5) are the three corners of triangle ABC, find: a) the length of all 3 sides of triangle ABC b) the area of triangle ABC c) the shortest distance from A to the line BC. Eercise 6(C) 1 Use the Converse of Pythagoras to determine which of the following triangles are right-angled. S2 Level 3/4 Course -23- Maths Department

2 Triangles ABD and ADC are right angled with dimensions shown in the diagram. Prove that triangle ABC is right-angled at A. 3 Quadrilateral ABCD is divided into 2 triangles as shown in the diagram. The dimensions are also shown in the diagram. Triangle ABD is right-angled at A. Prove that triangle BCD is right-angled at C. 4 Use the information apparent in this diagram to prove that triangle KMN is right-angled at M. [NOTE: Leave your one intermediate calculation in square root form] 5 In each of the following parts, find by calculation which angle in triangle ABC is the right-angle. a) A(-1, 4) B(2, 1) C(8, 7) b) A(0, -2) B(10, 3) C(-2, 2) c) A(-4, -1) B(6, 9) C(2, -3) d) A(1, 4) B(-2, 1) C(4, -5) 6 This question refers to your working for Question 1 in this eercise. For each of the triangles in Question 1 which you found NOT to be right-angled use your previous working to decide whether the largest angle is acute or obtuse. Maths Department -24- S2 Level 3/4 Course

7 In this diagram angle PQR is 90. Is angle PRS acute or 90 or obtuse? 8 In this diagram angle ABC is 90. Is angle ACD acute or 90 or obtuse? 9 In each of the following parts, name the largest angle in triangle PQR and by calculation state whether it is acute, right or obtuse. a) P(-2, 2) Q(2, 5) R(5, -1) b) P(-2, 4) Q(1, 7) R(4, -2) c) P(-3, 4) Q(-1, -1) R(7, 1) d) P(-3, -2) Q(0, 6) R(5, -1) e) P(-2, 4) Q(0, -3) R(6, 7) S2 Level 3/4 Course -25- Maths Department

PROBABILITY Eercise 1(A) 1 If a letter is chosen at random from the word SUCCESS, what is the probability that it will be: a) the letter S? b) the letter C? 2 If a letter is chosen at random from the word PEPPER, what is the probability that it will be: a) the letter P? b) the letter E? 3 If a letter is chosen at random from the word GEORGE, what is the probability that it will be: a) the letter E? b) the letter G? c) a vowel? d) a consonant? 4 If a letter is chosen at random from the word PENELOPE, what is the probability that it will be: a) the letter E? b) the letter P? c) a vowel? d) a consonant? 5 If a letter is chosen at random from the word WOODWORK, what is the probability that it will be: a) the letter O? b) the letter W? c) a consonant? 6 If a letter is chosen at random from the word NEEDLEWORK, what is the probability that it will be: a) the letter E? b) a vowel? c) a consonant? 7 On a supermarket shelf there are 16 bags of sugar, 12 of which contain white sugar and 4 of which contain brown sugar. If a bag is taken at random, what is the probability that it will contain: a) white sugar? b) brown sugar? Maths Department -26- S2 Level 3/4 Course

8 In class 3A there are 12 boys and 8 girls. If the pupils leave their classroom and walk to the assembly hall in any random order, what is the probability that the first pupil to enter the hall will be: a) a boy? b) a girl? 9 A farmer has 25 white sheep and 5 black sheep. If they are rounded up for shearing in any random order, what is the probability that the first one to be sheared will be: a) white? b) black? 10 A bo of sweets contains 15 chocolates, 9 toffees and 6 nougats. If a sweet is taken from the bo at random, what is the probability that it will be: a) a chocolate? b) a toffee? c) a nougat? 11 In class 2B there are 18 girls with dark hair, 10 girls with fair hair and 2 girls with red hair. If their teacher ask one girl at random to give out some books, what is the probability that she will have: a) dark hair? b) fair hair? c) red hair? 12 A 1 cash bag contains si 10p coins, four 5p coins, si 2p coins and eight 1p coins. If a coin is removed from the bag, what is the probability that it will be a: a) 10p coin? b) 5p coin? c) 2p coin? d) 1p coin? e) silver coin? f) copper coin? 13 On a supermarket shelf there are 8 packets of plain crisps, 5 packets of cheese and onion crisps, 3 packets of salt and vinegar crisps and 4 packets of smokey bacon crisps. If a bag is removed from the shelf at random, what is the probability that it will contain: a) plain crisps? b) cheese and onion? c) salt and vinegar? d) smokey bacon? e) any kind of flavoured crisps? S2 Level 3/4 Course -27- Maths Department

14 Each month of the year is written on a card and the twelve cards are then placed in a bag. If one of the cards is then removed from the bag, what is the probability that: a) the first letter on the card is J? b) the first letter on the card is M? c) the first letter on the card is A? d) the last letter on the card is R? e) the last letter on the card is Y? 15 A bag contains 40 counters, 8 of which are red, 12 of which are yellow, 4 of which are green and 16 of which are blue. If a counter is removed from the bag, what is the probability that it is: a) red? b) yellow? c) green? d) blue? e) red or yellow? f) red or green? Eercise 1(B) 1 If a dice is thrown, what is the probability that the score will be: a) a si? b) an odd number? c) an even number? d) a multiple of 3? e) a prime number? f) a square number? 2 Twelve counters numbered 1 to 12 are placed in a bag. If a counter is removed from the bag, what is the probability that the number on it will be: a) a prime number? b) a square number? c) an even number? d) a multiple of 3? e) a multiple of 5? f) a multiple of 4? 3 Twelve counters lettered A, B, C, D, E, F, G, H, I, J, K and L are placed in a bag. If a counter is removed from the bag, what is the probability that the letter on it will be: a) a vowel? b) a consonant? c) any letter of the word CAGE? d) any letter of the word BLEACH? Maths Department -28- S2 Level 3/4 Course

4 A pack of 52 playing cards is shuffled thoroughly and a card is then removed. What is the probability that the card: a) is an ace? b) is any king, queen or jack? c) shows any number from 2 to 10? d) shows any even number? e) shows an odd number? 5 A pack of 52 playing cards is shuffled thoroughly and a card is then removed. What is the probability that the card: a) is a king? b) is the king of hearts? c) is a red king? d) is not a king? 6 A pack of 52 playing cards is shuffled thoroughly and a card is then removed. What is the probability that the card: a) is a red? b) is a heart? c) is an even heart? d) is the 4 of hearts? 7 A pack of 52 playing cards is shuffled thoroughly and a card is then removed. What is the probability that the card: a) is a 5, 6 or 7? b) is the 5, 6, or 7 of hearts? c) is a red 5, 6 or 7? d) is not the 5, 6 or 7 of hearts? 8 A bag contains 5 red counters and 12 green counters. a) If a counter is removed what is the probability that it is red? b) If the counter was red and it was not replaced what is the probability that the net counter to be picked out would also be red? 9 A bag contains 8 red counters and 5 green counters. a) If a counter is removed what is the probability that it is red? b) If the counter was red and it was not replaced what is the probability that the net counter to be picked out would also be red? S2 Level 3/4 Course -29- Maths Department

10 A bag contains 6 red counters and 10 green counters. a) If a counter is removed what is the probability that it is red? b) If the counter was red and it was not replaced what is the probability that the net counter to be picked out would also be red? 11 A bag contains 1 red counter and 5 green counters. a) If a counter is removed what is the probability that it is red? b) If the counter was red and it was not replaced what is the probability that the net counter to be picked out would also be red? 12 An ordinary die is thrown. a) What is the probability of obtaining a 5? b) What is the probability of not obtaining a 5? c) What do you notice about these two results? d) What is the probability of obtaining a number greater than 4? e) What is the probability of obtaining a whole number less than 7? f) What is the probability of obtaining a number more than 6? 13 A letter is selected at random from the word PROBABILITY. a) What is the probability that the letter is a vowel? b) What is the probability that the letter is one of the first 2 letters of the alphabet? c) What is the probability that the letter is a consonant not net to a vowel? 14 A card is selected at random from a normal pack of playing cards. a) What is the probability of obtaining a heart? b) What is the probability of obtaining a red 4? c) What is the probability of obtaining an ace? 15 A counter is drawn from a bo containing 10 red, 15 black, 5 green and 10 yellow counters. Find the probability that the counter is: a) black b) not yellow c) red, black or yellow. Maths Department -30- S2 Level 3/4 Course

Eercise 1(C) 1 There are 10 coloured beads in a bo. 1 is red, 2 are green, 3 are blue, and the rest yellow. a) A bead is taken from the bo and then replaced. This is repeated 60 times. How many times would you epect to get a blue bead? b) A bead is taken from the bo and then replaced. This is repeated 75 times. How many times would you epect to get a green bead? c) A bead is selected from the bo. It is blue. The bead is not put back. What is the probability that the net bead selected is also blue? 2 There are only 3 possible outcomes to an eperiment, namely A, B and C. If Pr(A) = 0 1 and Pr(B) = 0 2, what is Pr(C)? 3 A card is selected at random from a normal pack of playing cards. What is the probability of obtaining a face card? 4 There are 18 coloured beads in a bo. 2 are pink, 3 are black, 4 are cream and the rest purple. a) What is the probability of obtaining a purple bead when one is selected at random? b) A bead is selected at random and replaced 45 times. How many times would you epect to get a pink bead? c) A bead is selected at random and replaced 42 times. How many times would you epect to get a black bead? d) A bead is selected at random and replaced 54 times. How many times would you epect to get a cream bead? e) One bead of each colour is taken out. What is the probability now of selecting at random a black bead? f) Two beads of each colour are taken out. What is the probability now of selecting at random a pink bead? S2 Level 3/4 Course -31- Maths Department

5 An ordinary die is thrown 60 times. a) How many times would you epect to obtain a multiple of 3? b) How many times would you epect to obtain a prime number? 6 There are only 3 possible outcomes to an eperiment, namely A, B and C. a) If Pr(A) = 2 1 and Pr(B) = 3 1, what is Pr(C)? b) If all 3 outcomes are equally likely, what is Pr(C)? c) If Pr(A) = 2 1 and B and C are equally likely, what is Pr(C)? 7 A letter is selected at random from the word PARALLELOGRAM. What is the probability of selecting a letter which has a vowel net to it on both sides? 8 An eperiment has probability 0 3 of success. If the eperiment is repeated 40 times, how many times would you epect it to fail? 9 A regular icosahedron has 20 faces, numbered from 1 to 20. It is thrown 60 times. How many times would you epect to get: a) a multiple of 4 b) a square number c) a prime number? 10 In a class of 30 students, all study at least one of the subjects physics and chemistry. 18 attend the physics class and 21 attend the chemistry class. Calculate the probability that a student chosen at random attends both the physics and the chemistry class. 11 John s school sells 1200 raffle tickets. John buys 15 tickets. John s church sells 1800 raffle tickets. John buys 20 tickets. In which raffle has he a better chance of winning the first prize. Eplain your answer. Maths Department -32- S2 Level 3/4 Course

12 Which of these can represent a probability distribution for an eperiment with only 4 possible outcomes A, B, C or D. Eplain your answer each time. A B C D a 0 1 0 2 0 3 0 4 b 0 2 0 3 0 4 0 5 c 0 3 0 4 0 5 0 2 d 1 2 1 3 e 0 0 1 1 f 1 3 0 1 6 1 3 0 1 3 13 Two mumps vaccines X and Y were compared with a placebo P in a series of clinical trials. The children vaccinated were screened regularly during the following 2 years. The following results were obtained: Infection Level P X Y Totals Nil 42 53 50 Mild 31 17 20 Severe 17 14 13 Totals a) How many children were involved in the trials? b) Calculate the probabilities that: i a child selected at random was treated with vaccine X. ii a child selected at random was treated with placebo P and was severely infected. iii a mildly infected child has been treated with vaccine Y. iv a child vaccinated with vaccine Y was mildly infected. S2 Level 3/4 Course -33- Maths Department

14 In a study of deaf-mutism in three generations of certain affected families, the following results were obtained: Normal Affected Male 55 61 Female 49 46 a) How many individuals were studied altogether? b) Some of the person studied are to be selected for further studies. Epress as fractions the probabilities that: i one person chosen at random will be affected by deaf-mutism. ii one person chosen at random will be a male not affected by deaf-mutism. iii an affected person chosen at random will be female. iv a female chosen at random will be affected by deaf-mutism. 15 A study of literacy in a Third World village is summarised in the following table: Standard of Literacy Male Female A 51 20 B 63 18 C 7 0 A, B and C refer to attainment levels. Epressing your answer as an unsimplified fraction, calculate the probabilities that: a) a male selected at random is of standard B. b) a standard A person selected at random is a female. c) a person selected at random is a male of standard C. Maths Department -34- S2 Level 3/4 Course

16 Eyesight tests were given to a group of 299 eleven year old children after which the children were classified as follows: Children requiring glasses Glasses previously recommended Glasses not previously recommended Children not requiring glasses Boys 23 6 121 Girls 18 3 128 Use the information in the table to answer the following questions. a) How many children were classified as requiring glasses? b) Epress, as fractions, the probabilities for this group that: i a child selected at random previously required glasses. ii a boy chosen at random did not require glasses. iii a child not requiring glasses chosen at random is a girl. S2 Level 3/4 Course -35- Maths Department

ENLARGING & REDUCING Eercise 1(A) 1 Draw each of these shapes twice as big. 2 Draw each of these shapes half as big. 3 Repeat question 1 making each shape 3 times as big. 4 State the scale factor for each of the following from shape 1 to shape 2. a) b) c) d) e) f) g) Maths Department -36- S2 Level 3/4 Course

5 Re-draw each of these shapes using the scale factor given. scale factor 2 a) scale factor 3 b) scale factor 2 c) scale factor 2 d) scale factor 3 e) scale factor 2 f) scale factor 4 g) Eercise 1(B) 1 Re-draw this shape with all its sides twice as long. 2 Re-draw this shape with all its sides 3 times as long. 3 Re-draw this shape with all its sides half as long. S2 Level 3/4 Course -37- Maths Department

4 Re-draw this shape with all its sides half as long. 5 Re-draw this shape with all its sides twice as long. 6 Re-draw each of these shapes using the scale factor given. scale factor ½ a) scale factor ¾ b) 7 State the scale factor for each of the following from shape 1 to shape 2. a) b) d) c) Maths Department -38- S2 Level 3/4 Course

8 Re-draw each of these shapes using the scale factor given. scale factor 3 a) scale factor 3 b) scale factor 3 scale factor 1½ c) d) scale factor 2½ e) scale factor 2 f) scale factor 1½ g) 9 Re-draw each of these shapes using the scale factor given. scale factor 2 scale factor 2½ scale factor 3 Eercise 1(C) 1 Re-draw this shape with all its sides twice as long. 2 Re-draw this shape with all its sides four times as long. S2 Level 3/4 Course -39- Maths Department

3 Re-draw this logo using a scale factor of ½. 4 Re-draw this logo using a scale factor of ⅔. 5 Re-draw this logo using a scale factor of ½. 6 Re-draw this logo using a scale factor of 2. 7 Re-draw this logo using a scale factor of 4. 8 Re-draw this logo using a scale factor of 7. 9 Re-draw this logo using a scale factor of 1¼. Maths Department -40- S2 Level 3/4 Course