[FFM] [2023] 4E AM 4049 Prelims P2 QP

NAME: ____________________________ ( ) CLASS: _______ FAIRFIELD METHODIST SCHOOL (SECONDARY) PRELIMINARY EXAMINATION 2023 SECONDARY 4 EXPRESS ADDITIONAL MATHEMATICS 4049/02 Paper 2 Date: 25 August 2023 Duration: 2 hours 15 minutes ______________________________________________________________________ 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 a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all questions. Write your answers on the space provided. 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 a 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. For Examiner’s Use Table of Penalties Question Number 90 Presentation  1  2 Rounding off  1 Parent’s/Guardian’s Signature Setter : Mr Wilson Ho This paper consists of 19 printed pages including this cover page.

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2023 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 2023 3 Additional Mathematics Paper 2 1 (a) By using long division, divide 322 6 3x x x+ + + by 3x+ . [1] (b) Express 2 32 9 10 16 2 6 3 xx x x x −− + + + in partial fractions. [5]

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2023 4 Additional Mathematics Paper 2 2 (a) Given that ( ) 2 54xy e x=− , show that ( ) 2d 10 3d xy exx =− . [3] (b) Hence find 24 xxe dx and evaluate 3 2 0 4 xxe dx . [5]

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2023 5 Additional Mathematics Paper 2 3 The graph of log pyx= passes through the points (27, 3) and (q, − 2). (a) Find the value of p and of q. [2] (b) Sketch, on the same diagram, the graphs of log pyx= and 3 xy −= . [3] (c) State the number of solutions for log 3 x p x −= . [1] y x

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2023 6 Additional Mathematics Paper 2 4 (a) Solve the equation ( ) ( )66log 2 1 log 2 4 1yy+ − − = . [4] (b) Given that ( )( ) 6log log 8 0xyxy x −= , express y in terms of x. [4]

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2023 7 Additional Mathematics Paper 2 5 It is given that f( )x is such that f "( ) 4cos 4 2sin 2x x x=+ . Given also that f(0) 0= and 3f( )44  = . (a) Find f( )x . [5]

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2023 8 Additional Mathematics Paper 2 5 (b) Show that 7 2 3f( )68  −= . [3]

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2023 9 Additional Mathematics Paper 2 6 A circle, 1C , has equation 22 4 6 12.x y x y+ − + = The equation of the normal to this circle at a point is 3 4 .y x k−= (a) Find the value of the constant k and the radius of 1C . [4] A second circle, 2C , centre (14, 2), just touches 1C . (b) Find the equation of 2C . [3]

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2023 10 Additional Mathematics Paper 2 7 (a) Prove that 2 2 2tan sec cos sin 1 tan cos sin x x x x x x x ++ =−− . [4]