ACSI 2020 Y3EXP FYE AMath P2

Confidential – for internal circulation only INDEX NUMBER Anglo - Chinese School (Independent) FINAL EXAMINATION 2020 YEAR THREE EXPRESS ADDITIONAL MATHEMATIC PAPER 2 Thursday 8 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 an HD pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all questions. 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 an approved 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 11 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 The area of a square is 3 22 3 22 + − cm2. Find the perimeter of the square in the form of ab+ , where a and b are integers. [4] ________________________________________________________________________________________ 2 The number y, of E. coli bacteria after x hours of an experiment is given by ( )40 1.4 xy= . Find (i) the number of E. coli bacteria at the start of the experiment, [1] (ii) the time needed to grow the population to 1000, correcting your answer to the nearest hour. [3] [Turn over

4 3 Given that ( ) ( ) ( ) ( ) 232 23 2 16 2 1 1 2x x x b x x ax x c x+ − +− − −≡ −+ + for all values of x , find the value of each of the constants a, b and c. [4]

5 4 (i) Prove that 2 3 sec 1 sin costan tan θ θθθθ − =+ . [4] (ii) Hence, solve the equation 2 2 3 sec 1 2costan tan θ θθθ − =+ for 0 360oo θ≤≤ . [4] [Turn over

6 5 (a) Solve the equation 0.9 cos 30 02 ox− −= for 0 180oo x≤≤ [4] (b) Solve the equation 2 32cos 0secx co x+= for 02 x π<< . Leave your answer (s) in terms of π. [5]

7 6 (a) The equation of a curve is ( ) ( ) 223 4 2y px px=− +− + . Find the range of values of p for which y > 0. [5] (b) Find the largest positive integer k for which the line 7y kx= −− does not intersect the curve 24 32yx x= −+ . [5] [Turn over

8 7 In the diagram below, P and R are points on the y-axis and x-axis respectively. (a) Given that the equation of PR is 52 xy= −+ , find the coordinates of P and R. [2] (b) If ( )7,Qq lies on the perpendicular bisector of PR, find the value of q. [3] y Q P x R O

9 (c) Given the point 33, 2S − , explain why PQRS is a rhombus. [2] (d) Calculate the area of PQRS. [2] [Turn over

10 8 (a) Solve the following equations: (i) 2152xx −= [4] (ii) ( ) ( )2 16 2 5 4yy += [4]