ACSI 2016 Y3 Core Mathematics Paper 2

Anglo - Chinese School (Independent) FINAL EXAMINATION 2016 YEAR 3 INTEGRATED PROGRAMME CORE MATHEMATICS PAPER 2 MONDAY 10th October 2016 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. Answers in degrees are to be given to one decimal place. INFORMATION FOR STUDENTS The maximum mark for this paper is 80. _____________________________________________________ This question paper consists of 4 printed pages. [Turn over

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2016/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. Begin each question on a new page. 1 [Maximum mark: 9] (a) Simplify 183 36 2   p p . [2] (b) Solve for x, given that 023 4 5 2  xx [2] (c) Make k the subject of the formula h khm 22  . [2] (d) Simplify            xx 2 1 3 2 2 123 1 as a single fraction. [3] 2 [Maximum mark: 7] (a) Solve the inequality 212 12386  xx . [5] (b) Hence, write down (i) the least value of x, [1] (ii) the greatest integer value of x. [1] 3 [Maximum mark: 6] (i) Sketch the curve ( 1 )( 6)y x x   , indicating clearly the x- and y-intercepts and the coordinates of the turning point of the curve. [3] (ii) The tangent to the curve ( 1 )( 6)y x x   at the point where 1x has a gradient of 3. Find the equation of the line perpendicular to the tangent at this point. [3] 4 [Maximum mark: 7] (a) If 6416 1 x , find the value of 11 3 x . [3] (b) Solve  22 1345  xxx [4]

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2016/FinalExamination 3 5 [Maximum mark: 5] (a) Find the range of values of x for which 2124 xx  . [2] (b) Find the range of values of m for which the expression 3)2(22  mxmmx is negative for all real values of x. [3] 6 [Maximum mark: 6] (i) The points of intersection of the graphs of 3xy and xy 2 1 are A and B. Find the coordinates of A and B. [3] (ii) Sketch the graphs of 3xy and xy 2 1 on the same axes for 33  x . Indicate clearly the points of intersection. [3] 7 [Maximum mark: 9] The diagram represents the roof of a house and EF is parallel to the rectangular base ABCD. EG is perpendicular to the base ABCD and M is the midpoint of AD. EA = ED = FB = FC. Given that EF = 12m, BC = 14m, EG = 6m and AB = 32m, calculate (i) the length of AG, [3] (ii) EAG , [2] (iii) the length of BE. [4] 8 [Maximum mark: 8] (a) Given that the line pxy 2 does not intersect the curve 432 2  xxy , find the range of values of p. [4] (b) Given that 213 2  qxxy and that 0y only when rx or 7x , find the value of q and of r. [4] 14 m 32 m A B C D M E 6 m G 12 m F

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2016/FinalExamination 4 9 [Maximum mark: 9] Solve the equations (i)   4log2log272log 93 2 3  xx , [5] (ii) )2(7)5(3 2xx  . [4] 10 [Maximum mark: 14] The points A, B, C and D represent the foot of four blocks of flats on ground level. It is given that B is due east of A, the bearing of A from C is 220, ,60ˆ CBA BC = 47 m, BD = 51 m and CD = 32 m. (a) Prove that  50BAC . [2] (b) Calculate (i) the distance between A and C, [3] (ii)  CBD, [3] (iii) the bearing of D from B. [2] (c) A boy walks directly from C to A. Calculate the distance he has walked when he is closest to B. [2] (d) If the block of flats at C is 65 m high, find the angle of depression of B from the top of the building at C. [2] End of Paper 51 m N D C B A 220 60 32 m 47 m

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2016/FinalExamination 5 ANSWER KEY 1 (a) 3 6 p (b) -12 (c) )( hmhk  (d) 6 37 x 2 (a) 7 195 46  x (b) (i) 5 46 (c) 9 3 (i) (ii) 313  xy 4 (a) 27 (b) 23.4 5 (a) 62  x (b) 4m 6 (i) (- 0.841, - 0.595) , (0.841, 0.595) (ii) 7 (i) 12.2 m (ii) 2.26 (iii) 23.9 m 8 (a) 16 111p (b) 1r , q = -24 9 (i) 10.3 (ii) 3.80 10 (b) (i) 53.1 m (ii) 37.8° (iii) 007.8° (c) 16.1 m (d) 54.1° –10 –5 5 10 –15 –10 –5 x y -6 -6 1 (-2.5, -12.25)