[HGS] [2023] 4E AM 4049 Prelims P1 MS

HGV Sec 4E Preliminary Examination Additional Mathematics Paper 1 2023 [MARK SCHEME] 1 HILLGROVE SECONDARY SCHOOL PRELIMINARY EXAMINATION 2023 SECONDARY FOUR (EXPRESS) [MARK SCHEME] CANDIDATE NAME ( ) CLASS - CENTRE NUMBER S INDEX NUMBER Additional Mathematics Paper 1 Candidates answer on the Question Paper. No Additional Materials are required. 4049/01 23 August 2023 2 hours 15 minutes 10.45 a.m. – 1.00 p.m. READ THESE INSTRUCTIONS FIRST Write your Centre number, index number and name 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 for this paper is 90. For Examiner’s Use Parent’s/ Guardian’s Signature: ___________________ TOTAL 90 Setter: Mdm Lee Li Lian This document consists of 18 printed pages, including this page.

HGV Sec 4E Preliminary Examination Additional Mathematics Paper 1 2023 [MARK SCHEME] 2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2 0ax bx c+ + = , 2 4 2 b b acx a −  −= Binomial expansion ( ) 1 2 2 ... ... ,12 n n n n n r r nn n na b a a b a b a b b r − − −     + = + + + + + +           where n is a positive integer and ( ) ! ( 1)...( 1) ! ! ! n n n n n r r r n r r  − − +== − 2. TRIGONOMETRY Identities 22sin cos 1AA+= 22sec 1 tanAA=+ 22cosec 1 cotAA=+ sin( ) sin cos cos sinA B A B A B =  cos( ) cos cos sin sinA B A B A B= tan tantan( ) 1 tan tan ABAB AB = sin 2 2sin cosA A A= 2 2 2 2cos 2 os sin 2 os 1 1 2sinA c A A c A A= − = − = − 2 2 tantan 2 1 tan AA A= − Formulae for ABC sin sin sin a b c A B C== 2 2 2 2 cosa b c bc A= + − 1 sin2 ab C=

HGV Sec 4E Preliminary Examination Additional Mathematics Paper 1 2023 [MARK SCHEME] 3 1 The equation of a curve is 24 16 19y x x= − + . (a) By expressing 24 16 19xx−+ in the form 2()a x b c++ , where a, b and c are constants, find the coordinates of the stationary point on the curve. [2] 22 2 2 2 4 16 19 4( 4 ) 19 4 ( 2) 4 19 4( 2) 16 19 4( 2) 3 [A1] Coordinates of the stationary point is (2, 3). [A1] x x x x x x x − + = − + = − − + = − − + = − +  (b) The line 43yx=+ intersects the curve at the points A and B. Find the value of k for which the distance AB can be expressed as 3 k . [4] 2 2 2 2 4 16 19 ............................(1) 4 3 ..................................... ....(2) Substitute (1) into (2): 4 16 19 4 3 4 20 16 0 5 4 0 ( 1)( 4) 0 [M1] 1 or 4 7 or 19 Coordi y x x yx x x x xx xx xx xx yx = − + =+ − + = + − + = − + = − − = == == 22 nates of and are (1, 7) and (4, 19). [ A1] Distance of (4 1) (19 7) 153 [A1] 3 17 17 [A1] AB AB k = − + − = = =

HGV Sec 4E Preliminary Examination Additional Mathematics Paper 1 2023 [MARK SCHEME] 4 2 It is given that ( ) ( ) ( ) ( ) ( ) k551055555 3579 =+−+− . By factorisation, find the value of k. [4] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 9 7 5 3 8 6 4 2 4 3 2 1 42 142 5 5 5 5 105 5 5 5 5 5 5 105 [M1] 5(5 5 5 5 105) [M1] 5(625 125 25 5 105) 5(625) 55 5 [A1] − + − + = − + − +  = − + − + = − + − + = = = 2 14=k [A1]

HGV Sec 4E Preliminary Examination Additional Mathematics Paper 1 2023 [MARK SCHEME] 5 3 The loudness of a sound can be measured using the equation 0 10lg IL I= , where I is the intensity of sound to be measured and 0I is the intensity of sound that can barely be heard, also known as the threshold of hearing. The unit of L is the decibel (dB). (a) Given that the loudness of a scream is 110 dB, find the ratio of the intensity of the scream to the threshold of hearing. [2] 0 0 0 11 0 11 10lg When 110, 10lg 110 lg 11 10 [A1] Ratio is 10 :1 [A1] IL I L I I I I I I = = = = =  (b) 130 dB is the pain threshold (the maximum level of sound we can hear without feeling intense pain and instantly damaging our hearing). Explain the impact, on hearing, the loudness of the sound of an unknown object falling from the sky onto Earth, if it has a sound intensity of 10.510− units and threshold of hearing of 2510− units. [2] 10.5 25 0 10.5 25 14.5 Given that 10 and 10 1010lg 10 10lg10 145 [A1] Since the sound of the unknown object falling has a loudness of 145dB which exceeds the pain threshold, this can cause d II L −− − − == = = = amage to our hearing. [A1]

HGV Sec 4E Preliminary Examination Additional Mathematics Paper 1 2023 [MARK SCHEME] 6 4 Given that the range of values of x where 2 is 4 5x ax b x+  −   , find the value of a and of b. [4] From the diagram, 2 2 ( 4)( 5) 0 [M1] 5 4 20 0 20 [A1] xx x x x xx + −  − + −  − Comparing 2x ax b+ with 2 20xx− , 1a=− [A1] and 20b= [A1] + − 4− x + 5

HGV Sec 4E Preliminary Examination Additional Mathematics Paper 1 2023 [MARK SCHEME] 7 5 The line y mx c=+ is drawn on the same axes as the curve 242y x x=− . (a) Given that the line is a tangent to the curve when 1 2c= , find the possible values of m. [3] 2 2 2 2 1 .................................(1)2 4 2 .............................(2) Substitute (1) into (2): 1 422 12 4 0 2 12 ( 4) 0 [M1] 2 12, 4, 2 Since the line is a tangent to t y mx y x x mx x x x x mx x m x a b m c =+ =− + = − − + + = + − + = = = − = 2 2 he curve, Discriminant = 0 1( 4) 4(2) 0 [M1]2 ( 4) 4 4 2 or 4 2 6 or 2 [A1] m m mm mm − − =  −= − = − =− == (b) The line y mx c=+ has a negative y-intercept. Do the line and the curve have 0, 1 or 2 points of intersections? Show your working clearly. [3] 2 2 2 2 2 2 .................................(3) Substitute (3) into (2): 42 2 4 0 2 ( 4) 0 Discriminant = ( 4) 4(2) = ( 4) 8 [A1] Since ( 4) 0 and 0, discri y mx c mx c x x x x mx c x m x c mc mc mc =+ + = − − + + = + − + = −− −− −   minant 0. [A1] The line and the curve have 2 points of intersection. [A1]  

HGV Sec 4E Preliminary Examination Additional Mathematics Paper 1 2023 [MARK SCHEME] 8 6 A polynomial, P, is 3235x x x k+ − + , where k is a constant. (a) Find the value of k given that P leaves a remainder of 24 when divided by 2x− . [2] 32 32 Let f ( ) 3 5 Given that f (2) 24 3(2) 5(2) 2 24 [M1] 24 20 2 24 42 24 18 [A1] P x x x x k k k k k = = + − + = + − + = + − + = += =− (b) In the case where 3k =− , the quadratic expression 2233x ax+− is a factor of P. Find the value of the constant a. [4] 3 2 2 3 2 2 32 2 3 5 3 (3 3)( ) [M1] 3 3 3 3 3 (3 ) ( 3) 3 [M1] Equating constants, 33 1 [A1] Equating coefficients of , 3 x x x x ax x b x bx ax abx x b x b a x ab x b b b x ba + − − = + − + = + + + − − = + + + − − − =− = += 5 3(1) 5 2 [A1] a a += = Alternative Method Equating coefficients of : 31 31 2 x ab a a − =− − =− =

HGV Sec 4E Preliminary Examination Additional Mathematics Paper 1 2023 [MARK SCHEME] 9 7 (a) Divide 32 4 2xx+− by 3 2xx+ . [1] 3 3 2 3 3 33 2 2 2 0 4 2 (2 4 ) 2 2 4 2 2 2