2022 MGS AMath Prelims P2

This question paper consists of 19 printed pages and 1 blank page. Class Index Number Name : __________________________________________ METHODIST GIRLS’ SCHOOL Founded in 1887 PRELIMINARY EXAMINATION 2022 Secondary 4 Thursday ADDITIONAL MATHEMATICS 4049/02 18 August 2022 PAPER 2 2 hours 15 minutes 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. 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. 90

Page 2 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Exam 2022 1. ALGEBRA Quadratic Equation For the quadratic equation , Binomial Expansion , where n is a positive integer and . 2. TRIGONOMETRY Identities Formulae for DABC a2 = b2 + c2 - 2bc cos A D = bc sin A 02 =++ cbxax a acbbx 2 42-±-= ( ) nrrnnnnn bbar nbanbanaba ++÷÷ ø ö çç è æ++÷÷ ø ö çç è æ+÷÷ ø ö çç è æ+=+ --- !!221 21 ( ) ! ( 1)...( 1) !! ! n nn n n r r rnr r æö -- +==ç÷ -èø 1cossin 22 =+ AA AA 22 tan1sec += 22 co sec 1 cotAA=+ BABABA sincoscossin)sin( ±=± BABABA sinsincoscos)cos( !=± BA BABA tantan1 tantan)tan( ! ±=± AAAcossin22sin=AAAAA 2222 sin211cos2sincos2cos -=-=-= A AA 2tan1 tan22tan - = sin sin sin abc ABC== 21

Page 3 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Exam 2022 1 Given that . (i) Show that [2] (ii) Given that x is increasing at radians per second, find the rate of change of y when . [2] 4 sin 2yx x= )2sin2cos2(4d d xxxx y += 522 p=x

Page 4 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Exam 2022 2 (i) Show that . [2] (ii) Hence evaluate . [4] 2 2 dl n 2 2 l n d xx xx x æö -=ç÷èø 2 1 2l n d e x xx æö ç÷èøò

Page 5 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Exam 2022 3 (a) Find the coefficient of x3 in the expansion of . [2] (b) (i) Write down the first three terms, in ascending powers of x, in the binomial expansion of , where a is a constant. [2] (ii) Given that the expansion of in ascending powers of x is , calculate the value of a, and of b. [3] ( )6 23x- ( )8 1ax+ ( )82(1 3 ) 1xa x-+ 2 11 0 . . .xb x+++

Page 6 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Exam 2022 4 In the diagram, AD is a tangent to the circle at A. BCD is a straight line that intersects the tangent at D. KD bisects and it cuts AC at H and meets AB at K. Prove that (i) is similar to . [3] (ii) , [2] ADCÐAHDDBKDD HD BD KD AD´=´ C B A D H K

Page 7 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Exam 2022 (iii) is isosceles. [3] AHKD

Page 8 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Exam 2022 5 (a) Show that . [4] (b) Hence solve the equation for . Leave your answers in terms of . [3] cos ec sec tan cotxx x xº+ cosec sec 2xx=03xp<<p

Page 9 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Exam 2022 6 Express in partial fractions. 2 74 (2 ) ( 5 ) x xx + -+ [5]

Page 10 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Exam 2022 7 (a) Express in the form , where a, b and c are constants. Without solving, explain why the curve will not intersct the line . [4] (b) By using a suitable substitution, solve the equation . [4] 2 26 7xx-+ 2 ()ax b c++ 2 26 7yx x=- + 2y= 3 1log 3 2 log 812 xx=+