ACSI 2022 Y3EXP FYE AMath P2

1 INDEX NUMBER Anglo - Chinese School (Independent) FINAL EXAMINATION 2022 YEAR THREE EXPRESS ADDITIONAL MATHEMATICS PAPER 2 4049/02 Monday 10 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 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 function ( ) 322 12f x x ax bx= + +− is divisible by ( )3x− and leaves a reminder of 14− when divided by( )21x+ . Find the value of a and of b. [5] [Turn over

4 2 In the diagram, the circle with centre C passes through the point ( )3, 4A − and touches the line 35yx−= at the point ( )1, 2B . Find (a) the equation of the line BC, [3] (b) the coordinates of C, [5] (c) the equation of the circle. [2] [Turn over

5 3 Given that ( ) ( ) 324 16 13 3x x x x mgx+ + +≡ − , where ( )gx is a polynomial and m is an integer. (a) Find the value of m. [2] (b) Find ( )gx . [2] (c) Hence, or otherwise solve for 324 16 13 3 0xxx+ + += . [1] (d) Explain why the equation 6 424 16 13 3 0x xx+ + += has no real solutions. [2]

6 4 (a) Find the values of x and y which satisfy the equations, 93xy= , ( )2 1 72 2xy+= . [5] (b) Solve the equation ( )4241 log log 8 log 3aa++= + . [4]

7 5 (a) Given that ( ) 4 log 32 0log x y y x += , express y in terms of x. [4] (b) (i) Simplify ( )( ) ( )( ) 2 22 2 9 25 . 15 5 3 nn nn n ++ [4] (ii) Hence find the value of x if ( )( ) ( )( ) 2 22 215 5 3 31 9 25 nn n nn x ++ = − . [2] [Turn over

8 6 (a) Prove that cos 1 tan1 sin cos θ θθθ−=− . [3] (b) Hence solve the equation cos 1 3cot1 sin cos θ θθθ−=− for 0 360θ°≤ ≤ ° . [3]

9 7 (a) Obtain the first 3 terms in the expansion of ( ) 6 2 x+ in ascending powers of x. [2] (b) Find the term in x in ( ) 6112 xx ++ . [2] [Turn over

10 8 The table below shows experimental values of two variables x and y. The variables x and y are related by the equation qy px= , where p and q are constants. (a) On the grid opposite, draw a straight line graph of ln y against ln x . [2] (b) Use your graph to estimate the value of p and of q. [4] An equation of 8 2 ey x= was given. (c) Draw the graph of 8 2 ey x= on the same axes and find the value of x of the point of intersection. [3] x 10 50 300 800 3100 y 350 144.5 60.1 36.6 18.2