ACSI 2016 Y3 Core Mathematics Paper 1 Solution

ACS(Independent)/Y3IPCoreMathsP1/2016/Final Year 1 Anglo - Chinese School (Independent) FINAL EXAMINATIONS 2016 YEAR 3 INTEGRATED PROGRAMME CORE MATHEMATICS PAPER 1 WEDNESDAY 5th OCTOBER 2016 1 h 30 min Additional Material Graph Paper (1 sheet) INSTRUCTIONS TO CANDIDATES  Write your index number in the boxes above.  Do not open this examination paper until instructed to do so.  You are not permitted access to any calculator for this paper.  Answer all questions in the spaces provided.  Unless otherwise stated in the question, all nu merical answers must be given exactly or correct to three significant figures.  The maximum mark for this paper is 80. For Examiner’s Use _______________________________________________________________ This paper consists of 13 printed pages. [Turn over Candidate Index Number

ACS(Independent)/Y3IPCoreMathsP1/2016/Final Year 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 a correct method, provided this is shown by writte n working. You are therefore advised to show all working. Answer all the questions in the spaces provided. 1 [Maximum mark: 8] (a) Simplify ac ac bc bc ab ba  . [2 marks] (b) Simplify 42 333 yx xyx  , giving your answer in the from mn yx , where n and m are rational numbers. [3 marks] (c) Factorise completely 6)()( 2  yxyx . [3 marks] ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… 𝑐(𝑎−𝑏)−𝑎(𝑐−𝑏)+𝑏(𝑐−𝑎) 𝑎𝑏𝑐 = 𝑐𝑎−𝑐𝑏−𝑎𝑐+𝑎𝑏+𝑏𝑐−𝑏𝑎 𝑎𝑏𝑐 = 0 (𝑥 + 𝑦 − 3)(𝑥 + 𝑦 + 2) 𝑥31 3𝑦3 𝑥𝑦2 = 𝑥 7 3𝑦

ACS(Independent)/Y3IPCoreMathsP1/2016/Final Year 3 2 [Maximum mark: 8] Mrs. Lim imported some olive oil for $500. She paid x$ for each liter of the olive oil. (a) Find, in terms of x , an expression for the amount of olive oil she bought. [1 mark] During transportation, 30 liters of olive oil was spilled. She sold the remaining olive oil for $1 more per liter than what she paid initially. (b) Write down an expression, in terms of x , for the sum of money she received. [2 marks] Mrs. Lim made a loss of $25. (c) Write down an equation in x to represent this information, and show that it can be reduced to 01006 2  xx . [2 marks] (d) Solve the equation 01006 2  xx , and hence, find the amount of olive oil she bought. [2 marks] ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… (a) Amount = 500 𝑥 (b) Amount left = 500 𝑥 − 30 Amount received = ( 500 𝑥 − 30) (𝑥 + 1) (c) 500 − ( 500 𝑥 − 30) (𝑥 + 1) = 25 500 − 500 − 500 𝑥 + 30𝑥 + 30 = 25 −500 + 30𝑥2 + 5𝑥 = 0 6𝑥2 + 𝑥 − 100 = 0 (d) (6𝑥 + 25)(𝑥 − 4) = 0 𝑥 = 4 Amount = 500 4 = 125

ACS(Independent)/Y3IPCoreMathsP1/2016/Final Year 4 3 [Maximum mark: 8] (a) Given that baln is a solution to 9225  xe , find the value of a and of b , where a and b are integers. [4 marks] (b) Solve )213(log)1(log 42 xx  [4 marks] ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… 25 − 2𝑒−𝑥 = 9 16 = 2𝑒−𝑥 16𝑒 𝑥 = 2 𝑒 𝑥 = 1 8 𝑥 = ln 1 8 𝑥 = −3 ln 2 log2(𝑥 + 1) = log4(13 − 2𝑥) log2(𝑥 + 1) = log2(13−2𝑥) log2 4 log2(𝑥 + 1) = log2(13 − 2𝑥) 2 2 log2(𝑥 + 1) = log2(13 − 2𝑥) log2(𝑥 + 1)2 = log2(13 − 2𝑥) (𝑥 + 1)2 = 13 − 2𝑥 𝑥2 + 4𝑥 − 12 = 0 (𝑥 + 6)(𝑥 − 2) = 0 𝑥 = −6 (NA) or 𝑥 = 2

ACS(Independent)/Y3IPCoreMathsP1/2016/Final Year 5 4 [Maximum mark: 5] A cannonball was fired and is modelled by a quadratic graph as shown in the diagram below. The cannonball was fired from a height of 3 m. The trajectory of the cannonball reached a maximum point at (2, 8). S (a) Express the equation of the trajectory in the form   khxay  2 , where a , h and k are constants. [3 marks] (b) Find the range of values of x for which   3 2  khxa . [2 marks] ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… (2, 8) 3 𝑦 𝑥 𝑦 = 𝑎(𝑥 − 2)2 + 8 3 = 4𝑎 + 8 4𝑎 = −5 𝑎 = − 5 4 Using symmetry property of quadratic curve, 0 < 𝑥 < 4

ACS(Independent)/Y3IPCoreMathsP1/2016/Final Year 6 5 [Maximum mark: 5] Given that 105  a and 16  b , where a and b are integers. Find (a) the largest possible value of ba . [1 mark] (b) the smallest possible value of ab 2 . [1 mark] (c) the smallest possible value of b aa 31122  . [3 marks] ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… (a) 10 − (−6) = 16 (b) (−1)2 − 10 = −9 (c) −𝑎2+12𝑎−31 𝑏 = −(𝑎−6)2+5 𝑏 = 5 −1 = −5

ACS(Independent)/Y3IPCoreMathsP1/2016/Final Year 7 6 [Maximum mark: 12] The diagram shows