ACSI 2023 Y3EXP FYE AMath P1

INDEX NUMBER Anglo - Chinese School (Independent) FINAL EXAMINATION 2023 YEAR THREE EXPRESS ADDITIONAL MATHEMATICS PAPER 1 4049/01 Tuesday 3 October 2023 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 Mathematical Formulae For Examiner’s Use 60

2 1. ALGEBRA Quadratic Equation For the equation 02 = + +c bx ax, a ac b bx 2 42 − ± −= Binomial expansion ( ) , ......2 1 2 21 nr r nnnnn b b ar nb anb ana b a + +     + +     +     + = + −−− where n is a positive integer and ( ) ! ( 1)...( 1) !! ! n n nn n r r rnr r  − −+= = − 2. TRIGONOMETRY Identities 1cos sin22 = +A A AA 22 tan1sec + = AAec 22 cot1cos + = Formulae for ∆ABC C c B b A a sin sin sin= = A bc c b acos22 2 2− + = 1 sin2 ab C∆=

3 Answer all the questions. 1 Solve the following pair of simultaneous equations, [4] 35xy+= , 2225x xy y+−= . [Turn over

4 2 A straight line graph, obtained by plotting x y against 2 ,x passes through the points ( 2,5)− and (4,8). Prove that 2 2 .12 xy x= + [4]

5 3 Express 2 2 1 1 x x + − in partial fractions. [4] [Turn over

6 4 The population of bacteria, P, after t hours can be modelled by ( )0 2tPP= , where 0P is a constant. (i) Find the least number of hours (to the nearest hour) it will take for the population to reach 100 times its original size. [2] (ii) Sketch the graph of P against t. [2]

7 5 Solve the equation 3log 3log 9 1.xx−= [5] [Turn over

8 6 Do not use a calculator in this question. Given that 3tan 5θ = and that θ is non-acute, find the exact value of (i) 2cot sec ,θθ+ [2] (ii) sin ,θ [1] (iii) ( )cos 180 , θ°− [1] (iv) ( ) ( ) cosec .sec θ θ − [2]

9 7 (a) Given that ( ) ( )( ) 2325 27 1 1 3x x x A Bx x C x x++ +≡ − + − + for all real values of x , find the constants A, B and C. [4] (b) The polynomial f (x) leaves a remainder of 4 and 7− when divided by 21x− and 2x+ respectively. Find the remainder when f(x) is divided by 22 3 2.xx+− [4] [Turn over

10 8 (a) Find the range of values of x which satisfies 2 2 22 0.56 xx xx ++ ≥−+ [4] (b) Show that the line y xk= + always intersects the curve 2xy y= + at two distinct points for all real values of k. [4]