2024 FMSS AMATH P2 PRELIM QP

NAME: ____________________________ ( ) CLASS: _______ FAIRFIELD METHODIST SCHOOL (SECONDARY) PRELIMINARY EXAMINATION 2024 SECONDARY 4 EXPRESS ADDITIONAL MATHEMATICS 4049/02 Paper 2 Date: 22 August 2024 Duration: 2 hours 15 minutes Candidates answer on the Question Paper. READ THESE INSTRUCTIONS FIRST Write your name, index number and class 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 the questions. The number of marks is given in brackets [ ] at the end of each question or part question. If working is needed for any question it must be shown with the answer. Omission of essential working will result in loss of marks. The total of the marks for this paper is 90. The use of an approved scientific calculator is expected, where appropriate. If the degree of accuracy is not specified in the question and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For , use either your calculator value or 3.142. For Examiner’s Use Table of Penalties Question Number Presentation  1  2 Rounding off  1 Parent’s/Guardian’s Signature Setters: Mr Joel Li This question paper consists of 23 printed pages  90

NAME: ____________________________ ( ) CLASS: _______ FMS(S) Sec 4 Express Preliminary Examination 2024 2 Additional Mathematics Paper 2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0 , x = -b ± b2 - 4ac 2a Binomial expansion a + b( ) n = an + n 1 æ èç ö ø÷ an-1b + n 2 æ èç ö ø÷ an-2b2 + ...+ n r æ èç ö ø÷ an-rbr + ...+ bn , where n is a positive integer and n r æ èç ö ø÷ = n! r!(n - r)! = n(n -1)...(n - r +1) r! 2. TRIGONOMETRY Identities sin2 A+ cos2 A = 1 sec2 A = 1+ tan2 A cosec2 A = 1+ cot2 A sin( A ± B) = sin Acos B ± cos Asin B cos( ) cos cos sin sinA B A B A B= tan tantan( ) 1 tan tan ABAB AB = sin2A = 2sin Acos A cos2A = cos2 A- sin2 A = 2cos2 A-1= 1- 2sin2 A tan 2 A = 2 tan A 1- tan2 A Formulae for DABC a sin A = b sin B = c sinC a2 = b2 + c2 - 2bccos A D = 1 2 absinC

NAME: ____________________________ ( ) CLASS: _______ FMS(S) Sec 4 Express Preliminary Examination 2024 3 Additional Mathematics Paper 2 Answer all the questions. 1 Show that the equation 1 22( 3) xxee−= has only one solution and find this value correct to 3 significant figures. [4]

NAME: ____________________________ ( ) CLASS: _______ FMS(S) Sec 4 Express Preliminary Examination 2024 4 Additional Mathematics Paper 2 2 The equation of a curve is 3y ax b=+ , where a and b are constants. The equation of the normal to the curve at the point where x = 1 is 5 2 12yx+= . Find the value of a and of b. [6]

NAME: ____________________________ ( ) CLASS: _______ FMS(S) Sec 4 Express Preliminary Examination 2024 5 Additional Mathematics Paper 2 3 Given that 2 3ln 2= xy x for x > 0. (a) Show that ).2ln21(3 d d 3 x xx y −= [3]

NAME: ____________________________ ( ) CLASS: _______ FMS(S) Sec 4 Express Preliminary Examination 2024 6 Additional Mathematics Paper 2 (b) Hence, find x x x d2ln 3 . [3]

NAME: ____________________________ ( ) CLASS: _______ FMS(S) Sec 4 Express Preliminary Examination 2024 7 Additional Mathematics Paper 2 4 Given that 2f ( ) 3,x x ax= − + where a is a constant, (a) find the range of values of a for which ,1)(f − xx for all real values of x, [4] (b) find the value(s) of a for which the line 4ya=+ is a tangent to the curve ).(f xy= [3]

NAME: ____________________________ ( ) CLASS: _______ FMS(S) Sec 4 Express Preliminary Examination 2024 8 Additional Mathematics Paper 2 5 (a) A and B are acute angles such that 3sin( ) 8AB−= and 5sin cos 8AB = . Without using a calculator, find the value of cos sinAB . [2] (b) Express 2sin 2 (sec tan )   − as a quadratic expression in sin . [3]

NAME: ____________________________ ( ) CLASS: _______ FMS(S) Sec 4 Express Preliminary Examination 2024 9 Additional Mathematics Paper 2 (c) Use your answer to part (b) to find, for 02  , the exact solutions of the equation 2sin 2 (sec tan ) 3 0   − + = . [3]

NAME: ____________________________ ( ) CLASS: _______ FMS(S) Sec 4 Express Preliminary Examination 2024 10 Additional Mathematics Paper 2 6 A curve is such that 2 8 2dy dx x=− . (i) Given that the curve passes through the point (1, 5), find the equation of the curve. [3] (ii) Find the x–coordinates of the stationary points of the curve. [2]