MGS 2023 Sec 4 Prelim AM P1_Solutions

Class Index Number Name : ___Mark Scheme_________________ This question paper consists of 17 printed pages and 3 blank pages. METHODIST GIRLS’ SCHOOL Founded in 1887 PRELIMINARY EXAMINATION 2023 Secondary 4 Friday ADDITIONAL MATHEMATICS 4049/01 18 August 2023 Paper 1 2 h 15 min Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your class, index number and name in the spaces at the top of this page. Write in dark blue or black pen You may use a 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 correct to 3 significant figure, 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 90. 90

Page 2 of 20 Methodist Girls’ School Additional Mathematics Paper 1 Sec 4 Preliminary Examination 2023 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the quadratic equation , Binomial Expansion , where n is a positive integer and . 2. TRIGONOMETRY Identities 22sin cos 1AA+= 22sec 1 tanAA=+ 22cosec 1 cotAA=+ cos2A = cos2 A- sin2 A = 2cos2 A-1= 1- 2sin2 A Formulae for ABC sin sin sin a b c A B C== a2 = b2 + c2 − 2bc cos A  = 2 1 bc sin A 02 =++ cbxax a acbbx 2 42 −−= ( ) nrrnnnnn bbar nbanbanaba ++     ++     +     +=+ −−− 221 21 ( ) ! ( 1)...( 1) ! ! ! n n n n n r r r n r r  − − +== − BABABA sincoscossin)sin( = BABABA sinsincoscos)cos( = BA BABA tantan1 tantan)tan(  = AAA cossin22sin = A AA 2tan1 tan22tan −=

Page 3 of 20 Methodist Girls’ School Additional Mathematics Paper 1 Sec 4 Preliminary Examination 2023 1 A calculator must not be used for this question. In the diagram, AB is parallel to DC, and AC meets BD at X. Given that AB = ( )1 3 3+ cm, CX = ( )53− cm and CD = ( )6 2 3+ cm, find the exact length of AX. [5] A B D C X 1 3 3+ 6 2 3+ 53− ( ) 1 3 3 5 3 6 2 3 1 3 3 53 6 2 3 5 3 15 3 9 6 2 3 14 3 4 6 2 3 7 3 2 3 3 3 3 3 3 21 3 21 6 2 3 93 23 3 27 23 9 36 6 2 AX AX or += −+ +=  − + − + −= + −= + −−= +− − − += − −=−

Page 4 of 20 Methodist Girls’ School Additional Mathematics Paper 1 Sec 4 Preliminary Examination 2023 2 A curve has the equation 12 3 .x xy e − −= (i) Show that the curve is an increasing function of x for 5 .2x [5] (