[WGS] [2023] 4E AM 4049 Prelims P2_MS

Name Index Number Class O-LEVEL PRELIMINARY EXAMINATIONS 2023 LEVEL & STREAM : SECONDARY 4 EXPRESS/ 5 NORMAL ACADEMIC SUBJECT (CODE) : ADDITIONAL MATHEMATICS (4049) PAPER NO : 02 DATE (DAY) : 12 SEPTEMBER 2023 ( TUESDAY) DURATION : 2 HOURS 15 MINUTES READ THESE INSTRUCTIONS FIRST Write your name, index number and class in the spaces at the top of this page. 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 the 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 in this paper is 90. DO NOT TURN OVER THE QUESTION PAPER UNTIL YOU ARE TOLD TO DO SO. Student’s Signature Parent’s Signature Date Date This document consists of _18_ printed pages including this cover page Setter : Mr Eric Bay___ 90

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ,02 =++ cbxax x = 2 4 2 b b ac a −  − Binomial expansion ,......21)( 221 nrrnnnnn bbar nbanbanaba ++     ++     +     +=+ −−− where n is a positive integer and ! ( 1)...( 1) !( )! ! n n n n n r r r n r r  − − +== − 2. TRIGONOMETRY Identities 1cossin 22 =+ AA AA 22 tan1sec += cosec 2 A = 1 + cot 2 A BABABA sincoscossin)sin( = BABABA sinsincoscos)cos( = BA BABA tantan1 tantan)tan(  = AAA cossin22sin = AAAAA 2222 sin211cos2sincos2cos −=−=−= A AA 2tan1 tan22tan −= Formulae for ABC C c B b A a sinsinsin == Abccba cos2222 −+= 1 sin2 bc A=

3 1 It is given that ( )f ( ) 2 sin cosxx e x x=− . (a) Show that f '( ) 4 sin xx e x= . [4] ( ) ( ) ( ) ( ) ( ) ( ) f ( ) 2 sin cos f'( ) 2 sin cos 2 cos sin -------------------- M1 2 sin cos cos sin -------------------- M1 =2 2sin =4 sin (shown)------------------- A1 x xx x x x x e x x x e x x e x x e x x x x ex ex =− = − + + = − + + (b) Hence evaluate π 0 sin dxe x x . [4] ( ) ( ) ( ) 4 sin d =2 sin cos -------------------- M1 4 sin d =2 sin cos 1sin d = sin cos ------------------- M12 xx xx xx e x x e x x c e x x e x x c e x x e x x c −+ −+ −+    ( ) ( ) ( ) 0 0 0 1sin d sin cos ------------------- M12 11= sin cos sin 0 cos 022 11 ------------------- A122 xxe x x e x x ee e     =−  − − − =+ 

4 2 (a) Prove that 3sin 3 3sin 4sinx x x=− . [3] ( ) ( ) ( ) 2 23 23 3 LHS sin 3 = sin 2 ------------------- M1 sin 2 cos cos 2 sin 2sin cos cos 1 2sin sinx ---