ACSI 2013 Y3 Core Mathematics Paper 2

Anglo - Chinese School (Independent) FINAL EXAMINATIONS 2013 YEAR 3 INTEGRATED PROGRAMME CORE MATHEMATICS PAPER 2 With Answer Key Tuesday 8th October 2013 1 hour 30 minutes 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 4 printed pages. [Turn over

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2013/FinalExamination 2 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: 5] Solve 1 5 4 3 3 2  xxx , giving your answer correct to 2 decimal places. 2 [Maximum mark: 8] The line xy 31 intersects the curve 1122 22  xyxy at P and Q. Find (i) the coordinates of P and Q, [6 marks] (ii) the distance PQ [2 marks] 3 [Maximum mark: 9] (a) Simplify              2 3log45 9log2 25 8log 2222 aa a . [4 marks] (b) Solve the equation 3log21log)32(log 22 2 4  xx . [5 marks] 4 [Maximum mark: 11] (a) Calculate the values of k for which the equation 0)1(29 2  kkxx has equal roots. [4 marks] (b) (i) Express 588 2  xxy in the form rqxpy  2)( . [2 marks] (ii) Hence, sketch the curve, indicating clearly on your sketch the axes intercept (s), if any, as well as the turning point. [3 marks] (c) Sketch the curve xy 3 , indicating the point (1, 1) on your sketch. [2 marks]

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2013/FinalExamination 3 5 [Maximum mark: 7] (a) The base of a triangle is        3 233 cm and its area is  412 cm². Find the height of the triangle, giving your answer in the surd form. [3 marks] (b) Given that 230)232)(81( ba  , where a and b are integers, find the value of a and of b. [4 marks] 6 [Maximum mark: 13] In the diagram, A, B, C and D are four points on level ground, such that AC = 4 m, AD = 5 m, CD = 7 m,  48BAC and  70ABC . (a) Find (i) the length of BC, [2 marks] (ii) the angle ACD, [3 marks] (iii) the area of ACD , [2 marks] (iv) the shortest distance from D to AC. [2 marks] (b) A vertical tower, whose tip is T, is erected at D. Given that the angle of elevation of T from C is 55o, find the (i) height of the tower, [2 marks] (ii) the greatest angle of elevation of T, when viewed from a point along AC. [2 marks]

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2013/FinalExamination 4 7 [Maximum mark: 9] (a) Solve the equation yy 5)3(4 2  [4 marks] (b) Given the equation 02 3212   eee xx and that ax ln is a solution, find the value of a . [5 marks] 8 [Maximum mark: 8] Given the points A(4 – 2p, 5), B(p, –1) and C(2, 3), find the positive value of p for which (i) the area of ΔABC is 13 square units, [3 marks] (ii) the midpoint of AB is equidistant from C and the point (-2, -3). [5 marks] 9 [Maximum mark: 10] Answer the whole of Question 9 on a sheet of graph paper. It is known that variables x and y are related by the equation 5 bya x , where a and b are unknown constants. Observed values of the two variables are shown in the following table. x - 1 0 1 2 3 4 y 22.5 7.5 2.5 0.83 0.28 0.09 (i) Using the scale of 2 cm to one unit on the horizontal axis and 4 cm to one unit on the vertical axis, plot the graph of lg y against x and draw a straight line graph. [4 marks] (ii) Use your graph to estimate the value of a and of b. [4 marks] (iii) From the graph, estimate the value of y when x = 1.2. [2 marks] End of Paper

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2013/FinalExamination 5 Answer Key 1. 15.62 or 0.13 2. (i)     2 12,2 1 and  2,1 (ii) 4.74 units 3. (a) -1 (b) 0.473 4. (a) 3 or 6 (b) (i) 3)2 1(8 2  xy (ii) (c) 5. (a) 7 )323(4  cm (b) 2a , 4b 6. (a) (i) 3.16 m (ii) 44.4˚ (iii) 9.80 m² (iv) 4.90 m (b) (i) 10.0 m (ii) 63.9˚ 7. (a) -1.59 (b) 2e or 5.44 8. (i) 5 (ii) 10 –6 –4 –2 2 4 6 –4 –3 –2 –1 1 2 3 4 x y (1, 1)

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2013/FinalExamination 6 9. (i) (ii) a = 3.06; b = 2.50 (iii) 2.00