ACSI 2020 Y3 Core Mathematics Paper 2

FINAL EXAMINATION 2020 YEAR 3 INTEGRATED PROGRAMME CORE MATHEMATICS PAPER 2 WEDNESDAY 7th October 2020 1 hour 30 minutes ADDITIONAL MATERIALS: Answer Paper (7 sheets) Graph Paper ( 1 sheet) 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/Y3IPCoreMathPaper2/2020/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: 6] (a) Evaluate 2 4 (0.3578) 7.647 43.96  , leaving your answer correct to 3 significant figures. [2] (b) Express 2 1 1 10 3 2 2 4 x x x x ++++ − − as a single fraction in its simplest form. [4] 2 [Maximum mark: 7] (a) Solve the equation 923 2x x−= , giving your answers correct to two decimal places. [4] (b) Expand and simplify 2 1 2 1 2 4 3 3 3 3 3 3p q q q p p  + − +      . [3] 3 [Maximum mark: 9] (a) Simplify ( ) ( ) 32 12 44 2 3 24 (4 ) xx xx − −  , expressing your answer in positive indices. [3] (b) Given that 1 2 2 21 16 2(2 ) 28 xx xx k + −+ + = , find the value of k . [3] (c) Find the value of 1 1 1 log log logpqrpqr pqr pqr++ . [3]

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2020/FinalExamination 3 4 [Maximum mark: 15] The diagram shows four towns A, B, C and D. Given that Town C and Town D lie west of B, AB = 8.2 km, BC = 8.8 km, CD = 9.3 km and 118oACD= , calculate (a) ABC , [4] (b) the bearing of B from A, [2] (c) the distance AD. [3] A cyclist starts from D at 1030 and travels towards A at a constant speed of 15 km/h. (d) Find the time, to the nearest minute, when he will be nearest to Town C. [6] 5 [Maximum mark: 13] (a) The equation of a curve is 23 3 12y qx px q= − + where p and q are positive integers. Show that the x− axis is the tangent to the curve if 22 p q = . [4] (b) Explain why the line 2y kx=− will always intersect the curve 3 3y x= − . [4] (c) The roots of ( 1)(3 )x x m− − = are  and  . (i) Find the value of + . [2] (ii) State the value of  in terms of m . [1] (iii) Given that ( ) 2 1 1 4   += , find the value(s) of m . [2] A B C D 8.2 8.8 9.3 118o N

ACS(Independent)MathDept/Y3IPCoreMathPaper2/2020/FinalExamination 4 6 [Maximum mark: 17] (a) Evaluate 32log 2 log 3+ . [2] (b) Find the value of k if 2 5 kee + = . [3] (c) 3 5 227xx−− = can be expressed as 7 49 32 x a  = where a is a positive integer. (i) Find the value of a . [3] (ii) Hence, find the value of x . [2] (d) When a cup of hot water is left in a room, its temperature T oC at time t minutes is given by 0.267 27 tTe −=+ . (i) Kenneth claimed that the initial temperature of the water is 100oC. Explain clearly whether Kenneth is correct. [2] (ii) Find the temperature when 5t= minutes. [2] (iii) How long will it take for the temperature of the water to drop to 30oC? [3] 7 [Maximum mark: 13] Answer the whole of this question on a sheet of graph paper. The variables 𝑥 and 𝑦 are connected by the equation 22 xy −= . Some corresponding values of x and y are given in the following table. x 2− 1− 0 1 2 3 4 5 y 16 a 4 2 1 0.50 b 0.125 (a) State the value of a and of b . [2] (b) Taking 2 cm to represent 1 unit on the horizontal axis and 1 cm to represent 1 unit on the vertical axis, draw the graph of 22 xy −= for 25 x−   . [4] (c) Use your graph to solve the equation 225 x− = . [2] (d) Find the gradient of the tangent at 0x= . [2] (e) By drawing a suitable straight line on the graph, find the solution of the equation ( ) 22 5 2 1x x− −= . [3] --------------------------------------------------------END OF PAPER---------------------------------------------------