AISS 2023 AM Prelim Paper 2 solutions

AHMAD IBRAHIM SECONDARY SCHOOL GCE O-LEVEL PRELIMINARY EXAMINATION 2023 SECONDARY 4 EXPRESS Name: Class: Register No.: MARKING SCHEME ADDITIONAL MATHEMATICS Paper 2 Candidates answer on the Question Paper. 4049/02 11 August 2023 2 hours 15 minutes READ THESE INSTRUCTIONS FIRST Write your name, class and index number on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all questions. Give non-exact numerical answers 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. At the end of the examination, fasten all your work securely together. 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 90. _________________________________________________________________________ This document consists of 20 printed pages. For Examiner’s Use /90

2 AISS PRELIIM/4E/4049/P2/2023 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ,02 =++ cbxax a acbbx 2 42 −−= Binomial expansion 1 2 2() 12 n n n n n r r n n n na b a a b a b a b b r − − −     + = + + + + + +           where n is a positive integer and ! ( 1) ( 1) !( )! ! n n n n n r r r n r r  − − +== − 2. TRIGONOMETRY Identities 22sin cos 1AA+= 22sec 1 tanAA=+ 22cosec 1 cotAA=+ sin( ) sin cos cos sinA B A B A B =  cos( ) cos cos sin sinA B A B A B= tan tantan( ) 1 tan tan ABAB AB = sin 2 2sin cosA A A= 2 2 2 2cos 2 cos sin 2cos 1 1 2sinA A A A A= − = − = − 2 2 tantan 2 1 tan AA A= − Formulae for ABC sin sin sin a b c A B C== 2 2 2 2 cosa b c bc A= + − 1 sin2 bc A=

3 AISS PRELIIM/4E/4049/P2/2023 [Turn over 1 (a) Find the range of values of x for which the expression 232 x− is negative. [2] (b) Find the set of values of the constant k for which the curve 2yx= lies entirely above the line ( )1.y k x=+ [3] ( ) 2 2 1 0 x k x x kx k + − −  For the quadratic expression to be always positive, ( ) 2 2 40 40 40 40 b ac kk kk k − + + −   2 2 3 2 0 2 3 0 33 or OR22 66 or 22 x x xx xx − − −  −  M1 A1 M1: finding roots or factorisng A1 M1

4 AISS PRELIIM/4E/4049/P2/2023 2 (a) Find the range of values of k such that the line 3xy+= intersects the curve 22 22x x y k− + = . [4] ( ) ( ) ( )( ) 22 22 22 2 2 2 3 3 --------(1) 2 2 3 2 2(9