National 5 Maths Christmas Special Surds & Indices 1. Simplify the following: a) (b) (c) d) (e) 2. Express with a rational denominator: a) (b) (c) 3. Evaluate: (a) (b) 4. Find x when: (a) (b) 2 x = Algebra 5. Simplify a) (2a b) 2 + 4ab (b) (3x 6)(x 2 + 2x 1) (c) 3(2x y) 4(x y) d) (3a 5b) 2 (e) 6x 2 (2x 1)(3x 2) 6. Factorise a) 2x 2 8 (b) 5p 2 6p 8 (c) 2xy 8x d) 3u 2 12u (e) 2n 2 18m 2 (f) 6p 2 7p 3
7. Simplify: a) (b) (c) d) (e) (f) 8. Change the subject to x in each of the following: a) P = 3(x 2 y) (b) x 9. Complete the square for: (a) x 2 8x + 5 (b) x 2 + 12x 7 Volumes 10. Calculate the volumes of the following shapes: a) (b) 11. a) A cylindrical water container has radius 18 cm and height 42 cm. Calculate the volume of this container. b) When full, this container can be used to fill 200 cone shaped cups like the one shown. Calculate the radius of this cup. 12. These 2 cylinders have the same volume. Calculate the radius of the second cylinder:
Arcs and Sectors 13. a) Calculate the Area: (b) Calculate the Perimeter: 14. A pendulum is 45 centimetres long. When the pendulum swings it travels along the arc of a circle and covers a distance of 27.5 centimetres. Calculate the size of the angle through which the pendulum travels. Angles in Shapes 15. In the diagram MN is a tangent and PL is a diameter of the circle. Angle JLN = 31 0 and angle KPL = 62 0. Find the size of angle KLJ. 16. RP is a tangent to the circle, centre O, with a point of contact T. Angle PTQ = 27 0. Calculate the size of angle OPT.
17. The diagram opposite shows a tunnel in the shape of part of a circle with radius OB. The height of the tunnel is 10 metres. Calculate x. 18. The diagram opposite shows an elaborate entrance door to a stately home. The door is in the shape of a circle, centre O, with a straight edge. Given the information in the diagram, calculate the height of the door. 19. The diagram shows the side view of a bridge in the shape of an arc of a circle. O is the centre of the circle which has radius of 240 metres. AB is 320 metres. Calculate the distance h. 20. The diagram shows a segment of a circle with centre O. OB is a radius of the circle. Calculate the length of OB. Scientific Notation 21. A multinational car making company made a loss of 2.55 x 10 8 in 2005. Calculate the loss made by the company per minute. 22. In Astronomy, distances can be measured using different units. For example: 1 parsec = 3.08 x 10 13 kilometres 1 Astronomical Unit = 1.49599 x 10 8 kilometres Calculate the number of Astronomical Units in a parsec, giving your answer in Scientific Notation.
23. The pie chart shows the relative sizes of the continents of the world. The total land mass of the world is 1.47 x 10 8 square kilometres. a) Write this land mass as an ordinary number. b) Calculate the land mass of the continent of America. Give your answer to the nearest whole number. 24. Evaluate: Fractions and Percentages a) (b) (c) d) 25. There are 1260 pupils in a school. 5 / 9 of the pupils are boys. How many girls are in the school? 26. Michael earns 480 per week. He puts 35% of the money in the bank, gives his mother 1 / 10 in rent and spends the rest. a) How much does he put in the bank? (b) How much does he spend? 27. a) Calculate the simple interest on a sum of 2400 at 6.2% per annum for a period of 8 months. b) An antique cello is valued at 6500. If the value of the cello rises at a rate of 12% per annum, how much will it be worth in 4 years time? c) A piece of machinery costing 32 000 depreciates at a rate of 4.5% per annum. Find the value of the piece of machinery after 5 years. d) In a survey of 1250 people, 135 were self-employed. What percentage of the people surveyed is this? 28. The value of a boat decreased from 35 000 to 32 200 in one year. a) What was the percentage decrease? b) If the value of the boat continues to fall at this rate, what would its value be after a further 3 years? 29. A special offer bottle of shampoo contains 35% extra compared to a normal bottle. If the special offer bottle contains 405 ml, how much does a normal bottle contain? 30. A herd of buffalo contained 15000 individuals in 1985. In 1986 the number of buffalo in the herd dropped to 13200. If the number of buffalo in the herd continues to fall at the same percentage rate, How many will be in the herd after 5 more years?
Similarity 31. a) In the diagram opposite, triangles ABC and ADE are similar. BC = 7 cm, DE = 10 cm and CE = 6 cm. Calculate the length of AC. b) Triangles ACE and BCD are similar. Calculate, x, the length of AB. 32. a) The diagram shows two mathematically similar mirrors. The smaller mirror has area 2600 cm 2. Find the area of the larger mirror. b) The balloons opposite are similar in shape. The larger balloon has volume 1350 cm 3. Find the volume of the smaller balloon. Straight Line 33. a) Find the equation of the line opposite, which passes through the points (0, -2) & (8, 0). b) The point (-12,a) lies on this line. Find a. 34. A is the point (2,0) and B is (6,8). a) Find the gradient of the line joining A and B. b) Find the equation of this line. c) Does the point (-5,-12) lie on this line? d) The point (m,3m) lies on this line. Find m. 35. Solve: x + 5y = 16 5x 2y = -1
36. a) A group of teachers and pupils go to a concert. There are 20 people in the group altogether. Let x represent the number of teachers in the group and y the number of pupils. Write down an equation involving x and y. b) Tickets for the concert cost 8 for teachers an 3 for pupils. The total cost of the tickets is 80. Write down another equation in x and y. c) Use your equations to find the number of teachers and the number of pupils in the group. 37. John, David and Stephen are brothers. David is 6 years older than John and Stephen is twice as old as John. a) Using x to represent John s age, write down expressions for David and Stephen s ages. b) The sum of the ages of the three brothers is 50. Form an equation in x and use this equation to find John s age. Trigonometry 38. Triangle ABC has AB = 11 cm, BC = 14 cm and AC = 17 cm. a) Calculate the size of angle BAC. b) Hence find the area of the triangle 39. a) A regular octagon has area 407.29 cm 2. Calculate d. b) The diagram below shows a prism whose cross-section is a regular hexagon. The volume of this prism is 20 500cm 3 and its depth is 35 cm. Calculate the length x.
40. The diagram shows the directions walked by Amanda and Michael after they leave a lighthouse. Michael walks at a speed of 3.8 kmph and Amanda walks at a speed of 4.4 kmph. How far apart will Amanda and Michael be after 2 hours? 41. In the triangle opposite show that cos BAC = 5 21 42. In the diagram opposite ABD and BCD are right-angled triangles. Calculate the size of angle DCB. 43. Two ships are positioned 2000 metres apart. Each ship detects a wreck, W, on the sea bed, as shown below. Calculate the depth of the wreck, W, below the surface. 44. Bill and Ted leave checkpoint A on a cross-country trek. Bill walks on a bearing of 055 at a speed of 4kmph. Ted walks on a bearing of 115 at a speed of 3.5kmph. After 3 hours Bill stops walking but Ted walks for 4 hours before stopping. How far apart are Bill and Ted after 4 hours? 45. Two identical industrial chimneys are 65 metres apart. From a point between the chimneys the angles of elevation to the top of the chimneys are 60 0 and 75 0. Calculate h, the height of the chimneys. 46. From two points 20 metres apart the angles of elevation to the top of a tower are 28 0 and 50 0. Calculate x, the height of the tower.
47. A triangle PQR has area 150 cm 2. PQ = 35 cm and QR = 20 cm. Find the size of angle PQR given it is an obtuse angle. 48. Michael and Umair are standing 150 metres apart. Umair is due East of Michael. A flagpole is on a bearing of 040 from Michael and on a bearing of 316 from Umair. How far is Michael from the flagpole? Statistics 49. In a series of class tests Amanda scored the following 72 54 60 55 57 59 45 56 55 63 80 71 a) Write down the IQR of Amanda s marks. b) Show the information in a boxplot. 50. The costs of a can of diet coke in 6 different shops are 47p 49p 50p 44p 48p 44p a) Calculate the mean and standard deviation of these costs. b) Each shop increases the price of a can of diet coke by 4p. Write down, without working, the mean and standard deviation of the costs now. 51. Two classes, A and B, sat the same test. Their marks were: Class A: 32 24 45 46 44 31 26 18 19 24 38 40 Class B: 21 25 13 17 40 32 12 24 36 39 22 19 Draw an appropriate diagram and write 2 statements to compare these 2 sets of data. 52. The marks of a group of pupils in 2 different maths tests are shown on the scattergraph opposite. David scored 20 in the first test and 32 in the second test. Niruz scored 40 in the first test and 54 in the second test. A line of best fit has been drawn on the graph. a) Describe the relation between test 1 and test 2. b) Find the equation of this line. c) Amanda scored 30 in test 1. Use your equation to estimate her score in test 2.
Other Stuff 53. A bus leaves Aytoun at 1045 and travels to Beeville, 135 km away, where he arrives at 1315. Calculate the average speed of the bus. 54. a) Niruz goes to France on holiday. She changes 650 into euros at a rate of 1 = 1.42 Euros. How many Euros will she receive? b) While in France Niruz spends 700 Euros before changing the rest back to pounds at a rate of 1 = 1.46 Euros. How much will she have, to the nearest penny? 55. The diagram opposite shows a worktop. The worktop is rectangular in shape with semi-circular ends. Calculate the perimeter of the worktop. 56. In a survey of 550 children the ratio of those with a computer to those without is 8:3. How many owned a computer? 57. A bag contains 30 coloured counters 8 red, 14 blue, 5 green and 3 yellow. a) A counter is chosen from the bag at random, what is the probability it is red? Simplify your answer. b) A counter is chosen from the bag and replaced. This is done 150 times. How many times would you expect a green counter to be chosen? 58. A ladder 4 metres long is placed against a wall with the foot of the ladder 2.4 metres from the base of the wall. The ladder reaches the top of the wall which is 3.4 metres high. Is the wall perpendicular to the ground? Mr. Cameron would like to wish you all a Merry Christmas and a very Happy New Year!!!!!!!! See you all in January with your completed Homework!!