2023_CCHM_Prelim_AMath P2 - MS

2023 Preliminary Exam/CCHMS/Secondary 4/Additional Mathematics/4049/02 [Turn over Name: Class: Class Register Number: PRELIMINARY EXAMINATION 2023 SECONDARY 4 ADDITIONAL MATHEMATICS 4049/02 Paper 2 Tuesday 29 August 2023 Candidates answer on the Question Paper. 2 hours 15 minutes MARKS SCHEME This document consists of 20 printed pages and 2 blank pages.

2 2023 Preliminary Exam/CCHMS/Secondary 4/Additional Mathematics/4049/02 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2 0ax bx c+ + = , 2 4 2 b b acx a −  −= Binomial expansion nrrnnnnn bbar nbanbanaba ++     ++     +     +=+ −−− ......21)( 221 , 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 cos sin 2cos 1 1 2sinA A A 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=

3 2023 Preliminary Exam/CCHMS/Secondary 4/Additional Mathematics/4049/02 [Turn over 1 (a) The line 4 3 2xy=+ intersects the curve 2 50x xy− + = at the points A and B. Find the midpoint of AB. [5] 2 4 3 2 ...(1) 5 0 ...(2) xy x xy =+ − + = From (1): 3 4 2 42 ...(3)3 yx xy =− −= Sub. (3) into (2): ( )( ) 2 22 2 2 42 503 3 4 2 15 0 2 15 0 2 15 0 5 3 0 5 or 3 sub. into (3): 146 or 3 xxx x x x xx xx xx x y −− + = − + + = − + + = − − = − + = =− =−  143, 3A−− and B(5, 6) A1 M1 f.t. – substitution M1 f.t. – solving quadratic M1 – either correct x or y values 14 635 3Midpoint of , 22 21, 3 AB  −+−+=   = M1 f.t. – midpoint formula

4 2023 Preliminary Exam/CCHMS/Secondary 4/Additional Mathematics/4049/02 (b) Find the least value of the integer h for which 2 5hx x h++ is positive for all real values of x. [3] (c) Given that the line 3y x p=+ is tangent to the curve 2 5y x x q= + + , where p and q are integers, prove that p and q are consecutive numbers. [4] 2 3 ... (1) 5 ... (2) y x p y x x q =+ = + + Sub. (1) into (2): 2 2 2 35 5 3 0 20 x p x x q x x q x p x x q p + = + + + + − − = + + − = a = 1, b = 2, c = k – c Line is tangent to curve → 2 40b ac−= ( )( ) 22 4 1 0 4 4 4 0 4 4 4 1 qp qp qp qp − − = − + = =+ =+ Since q = 1 + p, q will always be the next number after p. Hence, p and q are consecutive numbers (proved). M1 – substitution M1 – forming quadratic M1 f.t. – any use of discriminant A1 – with explanati