ACSI 2022 Y3EXP FYE AMath P1

INDEX NUMBER Anglo - Chinese School (Independent) FINAL EXAMINATION 2022 YEAR THREE EXPRESS ADDITIONAL MATHEMATICS PAPER 1 4049/02 Tuesday 4 October 2022 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 Solve the following pair of simultaneous equations [4] 22 25 7 12 5 7 xy xy −= −= [Turn over

4 2 Two variables x and y are related in such a way that when 1 y is plotted against 2x , a straight line is obtained. Given that points ( 10, 7)A − and (5,1)B lie on the line, (i) express y in terms of x, [4] (ii) find the value of k, given that (0, )Ck also lies on the line. [1]

5 3 (a) The graph cosy a xb= + , where 0a> , has a maximum value of 7 and a minimum value of 3. Find the value of a and of b . [2] (b) Using the value of a and of b from part (a), sketch the graph of cosy a xb= + for 0 360x°≤ ≤ ° . Hence find the number of solutions for which 1cos 4x=− . [4] [Turn over

6 4 An equilateral triangle has sides of 10 cm 31− . Find the height of the triangle, expressing it in the form ( 3) cmab+ . [6] 10 cm 31−

7 5 (a) Given that the curve 2 2( 3) 25yx m x= + −+ has a minimum value greater than 0, calculate the range of values of m. [4] (b) Find the range of values of x for which 2 5 056xx >−+− . [3] [Turn over

8 6 The population of a certain species of microorganisms , P, after t days can be modelled by 0 ntP Pe= , where n and 0P are constants. (a) After 3 days, the population of the microorganisms is doubled. Show, with clear working, that the value of n is 0.231. [3] (b) Find the percentage increase in the population of the microorganisms after 7 days. [2]

9 7 Given that 1tan 2A= and 0 A π<< , find the exact value of (i) sin A , [2] (ii) cos( )A− , [1] (iii) sec( )2 Aπ − . [2] [Turn over

10 8 (a) Given that 324 3 25 1 ( 1)( )( 2)x x x Ax x B x C− − −≡ + + + + for all real values of x , find the constants A, B and C. [5] (b) Express 2 35 ( 1) ( 3) x xx − +− in partial fractions. [5]