ACSI 2011 Y3 Core Mathematics Paper 2

Anglo - Chinese School (Independent) FINAL EXAMINATIONS 2011 YEAR 3 INTEGRATED PROGRAMME CORE MATHEMATICS PAPER 2 Monday 10 October 2011 1 h 30 min INSTRUCTIONS TO STUDENTS Do not open this examination paper until instructed to do so. A calculator is required for this paper. Answer all the questions on the answer sheets provided. At the end of the examination, fasten the answer sheets together. Unless otherwise stated in the question, all numerical answers must be given exactly or correct to three significant figures. INFORMATION FOR STUDENTS The maximum mark for this paper is 80. ___________________________________________________________ __________ This question paper consists of 07 printed pages. [Turn over

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2011/FinalExamination 2 A Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. Where an answer is incorrect, some marks may be given for correct method, provided this is shown by written working. You are therefore advised to show all working. Answer all the questions on the answer sheets provided. Please start each question on a new page. 1 [Maximum mark: 7] (a) Solve 2 500323 xx  , giving your answer correct to 2 decimal places. [4 marks] (b) Find the range of values of x for which x (1 – x) < x² – 6. [3 marks] 2 [Maximum mark: 4] A cuboid has a square base of side )51(  m and a volume of )152)(25(  m3 . Find the height of the cuboid in the form ).5( ba 3 [Maximum mark: 4] The figure shows a right circular cone inscribed in a sphere with centre O and radius of 1 cm. The cone has a height AB and a base radius AC. Given that AOC radians, express the volume of the cone in terms of . [Volume of a Sphere= ,3 4 3r Volume of a cone= ]3 1 2hr C O B ѳ A 1 cm

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2011/FinalExamination 3 4 [Maximum mark: 7] Singapore Airlines has a fleet of aircraft consisting of 50 aircraft of type A, 80 of type B and 70 of type C. Each type has 3 classes of seat known as Economy, Business and First Class. The table below shows the number of these seats in each of the 3 types of aircraft. Type of Aircraft Economy Business First Class A 350 80 50 B 200 50 30 C 120 25 10 (a) Write down the two matrices whose product shows the total number of seats in each class. [2 marks] (b) Evaluate the product of the two matrices. [2 marks] On a typical day, each aircraft made one flight. It was found that 5%, 10% and 20% of the Economy, Business and First Class seats respectively were empty. (c) Write down a matrix whose product with the matrix found in part (b) will give the total number of empty seats on that typical day. [1 mark] (d) Hence, find the total number of empty seats. [2 marks] 5 [Maximum mark: 7] (a) Express 942  xxy in the form qpxy  2)( . [2 marks] (b) Hence, sketch the curve of 942  xxy . Indicate clearly, on your diagram the coordinates of the intercept(s) and the turning point. [3 marks] (c) The line joining the origin O to the turning point makes an angle  with the x-axis. Calculate the angle  . [2 marks]

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2011/FinalExamination 4 6 [Maximum mark: 10] During an experiment, the number n , of bacteria after t hours is given by 3)55.1(1400 t n . (i) Find the initial number of bacteria. [1 mark] (ii) Find the number of bacteria after 6 hours. [1 mark] (iii) Find the time taken, in hour and minute (correct to the nearest minute), for the number of bacteria to reach 25000. [4 marks] In another experiment, the number of bacteria is given by 3 1500 t en   (iv) Find the time taken for the bacteria to reach the same amount in both experiments. [4 marks] 7 [Maximum mark: 12] (a) Solve each of the following equations: (i)   125 1255 13 xx ; [3 marks] ).45(log)2(log (ii) 22 xx  [5 marks] (b) By using an appropriate substitution, solve 239 11   xx . [4 marks]

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2011/FinalExamination 5 8 [Maximum mark: 9] Solutions to this question by accurate drawing will not be accepted. The diagram, which is not drawn to scale, shows a right-angled triangle ABC in which  BAC = 90⁰ . The coordinates of A and B are (4, 5) and )1 ,0(  respectively. Given that the gradient of BC is 2 1 and D is the foot of the perpendicular from A to BC, find (i) the equation of BC , [1 mark] (ii) the equation of AC , [3 marks] (iii) the coordinates of C, [2 marks] (iv) the coordinates of D. [3 marks] D y x B( 0,-1) C A(4, 5) O

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2011/FinalExamination 6 9 [Maximum mark: 12] In the following diagram, ABC represents a horizontal triangular garden and AT represents a vertical pole . A walking path runs along the edge of the garden BC. It is also given that AB = 84 m, AC = 47 m and angle BAC = 65o. (a) The angle of elevation of the top of the pole when viewed from C is 15o. Calculate the height of the pole. [2 marks] (b) Calculate the length of the walking path BC [3 marks] (c) Calculate the area of the garden ABC. [2 marks] (d) Calculate the shortest distance from A to the walking path BC. [2 marks] (e) Calculate the greatest angle of elevation of the top of the pole when viewed from any point on the walking path. [3 marks] A 84 47 B C T 65o 15o

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2011/FinalExamination 7 10 [Maximum mark: 8] Answer the whole of this question on a sheet of graph paper. The table below gives some values of x and the corresponding values of y, correct to one decimal place, where x xy 4 105 2  . x 0.5 1 2 3 4 5 7 9 y -3.0 0.9 2.6 2.8 p 1.7 -0.5 -3.5 (a) Find the value of p. [1 mark] (b) Using a scale of 2cm to represent 1 unit on each axis, draw a horizontal x-axis for 95.0  x and a vertical y-axis for 35  y . On your axes, plot the points given in the table and join them with a smooth curve. [3 marks] (c) By drawing a tangent, find the gradient of the curve at the point where x = 2. [2 marks] (d) By drawing a suitable straight line, find the range of values of x in the interval 95.0  x for which .22 54 10 2 x x x  [2 marks] End of Paper

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2011/FinalExamination 8 Answers 1a 9.63, -17.30 1b 2,2 3  xx 2 58 1 8 9  4a           10 25 120 30 50 200 50 80 350 70) 80 50( 4b (41900 9750 5600) 4c           0.20 0.10 0.05 5600) 9750 41900( 4d (4190) 5a 5)2( 2  xy 5b Diagram on right side 5c 68.2⁰ 6a 1400 6b 3363 or 3360 –3sf 6c 19 hr 44 min 6d 0.144 hr 7ai 5 1x 7aii 1, -1 7b -1 8i ...........12 1 LineBCxy  8ii ..3 23 3 2-y  xso 8iii )7 19,7 52(C 8iv )8.1,6.5()5 9,5 28( orD 9a 12.6 m 9b 77.0 m 9c 1790 m2 9d 46.5 m 9e 15.2⁰ 10a 2.4 10c 0.60 10d 7.60x1.34 : Range

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