ACSI 2020 Y3EXP FYE AMath P1

INDEX NUMBER Anglo - Chinese School (Independent) FINAL EXAMINATION 2020 YEAR THREE EXPRESS ADDITIONAL MATHEMATICS PAPER 1 Monday 5 October 2020 1 hour 30 minutes Candidates answer on the Question Paper. No additional materials are required. READ THESE INSTRUCTIONS FIRST Write your index number in the space at the top of this page. Write in dark blue or black pen. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all questions. Write your answers in the spaces provided under each question. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of a scientific calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 60. This document consists of 13 printed pages and 1 blank page. [Turn over For Examiner’s Use 60

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02 =++ cbxax , a acbbx 2 42 −±−= Binomial expansion ( ) ,......21 221 nrrnnnnn bbar nbanbanaba ++     ++     +     +=+ −−− where n is a positive integer and ( ) ! ( 1)...( 1) !! ! n n nn n r r rnr r  − −+= = − 2. TRIGONOMETRY Identities 1cossin 22 =+ AA AA 22 tan1sec += AAec 22 cot1cos += Formulae for ∆ABC C c B b A a sinsinsin == Abccba cos2222 −+= 1 sin2 ab C∆=

3 Answer all the questions. 1 Solve the following pair of simultaneous equations 2243 1x xy y+ += 1xy+= [4] [Turn Over

4 2 Given that 12tan 5θ = and that θ is acute, find the exact value of (i) cos( ),θ− [1] (ii) cos(90 ), θ°− [1] (iii) tan(180 ). θ°− [1]

5 3 (a) The graph of log ( 1)ay kx= − passes through the points with coordinates (1,0) and (5,2). (i) Determine the value of each of the constants a and .k [4] (ii) Write down the range of values of x such that y is defined. [1] (b) Sketch the graph of 4logyx= . [2] [Turn Over

6 4 It is given that 32( ) 4 16 21 9fx x x x=− +− . (a) Find the quotient when ()fx is divided by 2 1x + . [2] (b) Prove that 1x− is a factor of ()fx . [1] (c) Hence, factorise ()fx completely. [3]

7 (d) Express () x fx in partial fractions. [5] [Turn Over

8 5 The equation of a graph is 2sin 2 1yx= + for 0 x π≤≤ . (i) State the period and amplitude of .y [2] (ii) Solve 0y = for 0, x π≤≤ giving your answer in exact form. [3]

9 (iii) Sketch the graph of 2sin 2 1yx= + for 0 x π≤≤ . [3] (iv) By drawing a suitable straight line on the same axis in (iii), find the number of solutions to the equation 2sin 2 1x= . [3] [Turn Over

10 6 (i) Given that 3lg( ) 2 2lg lgxy x y= +− , express x in terms of .y [4] (ii) Solve the equation 4 16log ( 2) 4 log ( 1) 1xx+− −= . [4]