Q1. Fatima and Mohammed each buys a bike. FINAL EXAM PAPER 4 MAY 2014 (a) Fatima buys a city-bike which has a price of $120. She pays 50 % of this price and then pays $20 per month for 6 months. (i) How much does Fatima pay altogether? Answer(a) (i)... [2] (ii) Work out your answer to part (a)(i) as a percentage of the original price of $120. Answer(a) (ii).. [2] (b) Mohammed pays $159.10 for a mountain-bike in a sale. The original price had been reduced by 14 %. Calculate the original price of the mountain-bike. Answer(b) [2] (c) Mohammed s height is 169 cm and Fatima s height is 156 cm. The frame sizes of their bikes are in the same ratio as their heights. The frame size of Mohammed s bike is 104 cm. Calculate the frame size of Fatima s bike. 1
Answer(c). [2] (d) Fatima and Mohammed are members of a school team which takes part in a bike ride for charity. (i) Fatima and Mohammed ride a total distance of 40 km. The ratio distance Fatima rides: distance Mohammed rides is 5 : 3. Work out the distance Fatima rides. Answer(d) (i)... [2] (ii) The distance of 40 km is only of the total distance the team rides. Calculate this total distance. Answer(d) (ii).. [2] Q2. A shopkeeper sells cartons of milk and bottles of water. Each carton of milk costs $2.40, and each bottle of water costs $0.80. One day he sells x cartons of milk. On the same day, he sells 20 more bottles of water than cartons of milk. (i) Write down an expression, in terms of x, for the number of dollars he receives from the sale of these cartons and bottles. Simplify your answer. 2
Answer (i)... [2] (ii) The total amount he receives that day from the sale of these cartons and bottles is greater than $250. Form an inequality in x and solve it. Answer (ii)... [2] (iii) Hence write down the least number of cartons of milk that he sells that day. Q3. The route for the sponsored walk in winter is triangular Answer (iii). [1] (i) Senior students start at A, walk North to B, then walk on a bearing 110 to C. They then return to A. AB = BC. Calculate the bearing of A from C. 3
... [3] (ii) Answer (i). AB = BC = 6 km. Junior students follow a similar path but they only walk 4 km North from A, then 4 km on a bearing 110 before returning to A. Senior students walk a total of 18.9 km Calculate the distance walked by junior students. 4
Answer (ii)...[3] Q4. The depth, d centimeters, of a river was recorded each day during a period of one year (365 days). The results are shown by the cumulative frequency curve. (a) Use the cumulative frequency curve to find (i) the upper and lower quartile range, (ii) the median depth, Answer(a) (i)... [2] Answer(a) (ii).. [2] (iii) the depth at the 20th percentile, (iv) the number of days when the depth of the river was at least 30 cm. Answer(a) (iii).. [2] 5
(b) Answer(a) (iv).. [2] d 0 10 20 30 40 50 60 Number of days 17 98 85 41 62 p q (i) Show that p = 47 and q = 15 (ii) Use the information in the table and the values of p and q to calculate an estimate of the mean depth of the river. [2] 6
Answer(b) (ii)...[4] (c) The following information comes from the table in part (b). d 0 20 40 Number of days 115 126 124 A histogram was drawn to show this information. The height of the column for the interval 20 < d 40 was 8 cm. Calculate the height of each of the other two columns. [Do not draw the histogram.] Answer (c). [3] Q5. A solid metal cone has base radius 6 cm and vertical height 24 cm. (a) Calculate the volume of the cone. 7
Answer (a) [3] (b) A cone of height 8 cm is removed by cutting parallel to the base, leaving the solid shown above. Show that the volume of this solid rounds to 871 cm 3, correct to 3 significant figures. Answer (b) [4] (c) The 871 cm 3 of metal in the solid in part (b) is melted and made into 5 identical cylinders, each of length 10 cm. Show that the radius of each cylinder rounds to 2.4 cm, correct to 1 decimal place. 8
Answer (c) [3] Q6. (i) The diagram shows a box ABCDEFGH in the shape of a cuboid measuring 2 m by 1.5 m by 1.7 m. (a) Calculate the length of the diagonal EC. (b) Calculate the angle between EC and the base EFGH. Answer(i)(a).[4] 9
Answer(i)(b).[3] (ii) A, B, C and D are four points on level ground. BDC is a straight line.ad = 30 m and DC = 64 m. A 3 and A 5. (a) Calculate AB. (b) Calculate AC. Answer(ii)(a)..[3] (c) Calculate the area of triangle ADC. Answer(ii)(b)..[4] 10
Answer(ii)(c)..[4] (d) A vertical tower stands at A. P is the point on the line BC such that the angle of depression from the top of the tower to the line BC is greatest. Given that this angle of depression is 34, calculate the height of the tower. Answer(ii)(d).[3] Q7. A fuel tank is a cylinder of diameter 1.8 m. (a) The tank holds 25 000 litres when full. Given that 1 m 3 = 1000 litres, calculate the length of the cylinder. Give your answer in metres. 11
Answer(a)..[4] (b) The diagram shows the cross-section of the cylinder, centre O, containing some fuel. CD is horizontal and is the level of the fuel in the cylinder. AB is a vertical diameter and intersects CD at E. Given that E is the midpoint of OB, (i) Show that DOE = 60, (ii) calculate the area of the segment BCD, Answer(b)(i)..[2] (iii) calculate the number of litres of fuel in the cylinder. Answer(b)(ii)..[3] 12
Answer(b)(iii)..[2] (iv) [Volume of a sphere = ] A different fuel tank consists of a cylinder of diameter 1.5m. and a hemisphere of diameter 1.5 m at one end. The volume of the cylinder is 10 times the volume of the hemisphere. Calculate the length of the cylinder. Answer(b)(iv)..[3] (c) A rectangular card is 5 cm long and 4 cm wide. As shown in the diagram, a square of side x centimetres is cut off from each corner. The card is then folded to make an open box of height x centimetres. (i) Write down expressions, in terms of x, for the length and width of the box. 13
Answer(c)(i)..[2] (ii) Show that the volume, V cubic centimetres, of the box is given by the equation V = 4x 3 18x 2 + 20x. [3] Q8.(a) The diagram shows a toy boat. AC = 16.5 cm, AB = 19.5 cm and PR = 11 cm. Triangles ABC and PQR are similar. (i) Calculate PQ 14
(ii) Calculate BC. Answer(a)(i) [2] (iii) Calculate angle ABC Answer(a)(ii) [3] Answer(a)(iii) [3] (iv) The toy boat is mathematically similar to a real boat. The length of the real boat is 32 times the length of the toy boat. The fuel tank in the toy boat holds 0.02 litres of diesel. Calculate how many litres of diesel the fuel tank of the real boat holds. 15
Answer(a)(iv) [2] Q.9 (a) The diagram shows a sector AOB of a circle with centre O and radius 6 cm. The angle of the sector is 310. (i) Calculate the total perimeter of the sector. (ii) Calculate the area of the sector. Answer(a)(i)..[3] 16
(b) A, B, C, D and E lie on a circle, centre O. AOC is a diameter. Find the value of (i) p Answer(a)(ii)...[3] (ii) q. Answer(b)(i)...[3] 17
Answer(b)(ii)...[3] Q10. The diagram below shows a right angled triangle ABC, where AB = AC. The coordinates of A and B are given. (a)the equation of the line passing through the points A and B. Answer(a)..[3] (b) the equation of the line passing through the points A and C. 18
Answer(b)..[3] (c)the length of the line segment BC to 1 d.p Answer(c)..[3] Q11. (a) = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15} L = {x : x is an odd number} M = {x : x is a multiple of 3} Write down (i) ( L M) Answer(a)(i)...[2] (ii) L M. 19
Answer(a)(ii)..[2] (b) In a survey, a number of people were asked o you own a car? and o you own a bicycle?. The Venn diagram shows the set C of car owners and the set B of bicycle owners. The letters p, q and x are the numbers of people in each subset. 11 people owned neither a car nor a bicycle. A total of 66 people owned a car. The number of people who owned a car only is 4 times the number of people who owned a bicycle only. (i) Write down expressions, in terms of x, for (a) p, (b) q. Answer(a).[1] (ii) A total of 27 people owned a bicycle. Calculate (a) x, Answer(b)..[1] Answer(a)..[2] (b) the total number of people who were in the survey. 20
Answer(b)..[1] Rough Work 21