ACSI 2021 Y3EXP FYE AMath P2

INDEX NUMBER Anglo - Chinese School (Independent) FINAL EXAMINATION 2021 YEAR THREE EXPRESS ADDITIONAL MATHEMATICS PAPER 2 4049/02 Tuesday 12 October 2021 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 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 A right angled triangle has an area of 216 6 2 cm+ and a base length of 4 2 2 cm.− Without using a calculator, find the perpendicular height of the triangle in the form of 2 ab+ where a and b are integers. [4] [Turn over

4 2 Given that ( ) ( ) ( ) 2228 5 32 1 3x Ax x B x C x+ −≡ − + − − − for all real values of x, find the value of each of the constants A, B and C. [4]

5 3 (a) Prove that 2 1 sin(sec tan ) 1 sin θθθ θ −−≡ + . [4] (b) Hence, solve the equation 2(sec tan ) 3θθ−= for 0 360θ≤≤ . [3] [Turn over

6 4 (a) Solve the equation 10 sin 9 2cosec xx+= − for 0 360x≤≤ [5] (b) Solve the equation sin 2 cos 244 ππθθ += +   for 0 θπ≤≤ . Leave your answer (s) in terms of π. [3]

7 4 (c) On the same axes, sketch the graphs of 2cos 2yx= and 2sin 1yx= + for 0 360x≤≤ . Hence state the number of solutions for which 1 cos 2 sin2 xx−= − [5] [Turn over

8 5 A radioactive substance, Radium (Ra 226), of mass 100g is decaying such that after t years, the amount remaining, M, is given by the equation 0.0004279100 tMe −= . (a) Sketch the graph of M against t, indicating any axes intercepts. [1] (b) Find the amount of the Radium remaining after 20 000 years. [2] (c) Find the half-life of the Radium (i.e. the time taken to decay to half the initial mass). [3] M t 0

9 6 (a) Given that the curve 23yx= lies completely above the line 3.y px= − Find the range of values of p. [5] (b) It is given that 12 23 x−<< is the solution set to the quadratic inequality 2 20px qx+ −< where p and q are integers. Find the value of p and of q. [3] [Turn over

10 7 ( )1, 1Q − and ( )6, 1R − are two vertices of a rhombus PQRS as shown in the diagram below. PR and QS intersect at M. PR cuts the y – axis at ( )0, 2 . (a) Find the equation of PR and QS. [4] y x O S P M