2023 CCHM Prelim AMath P1 - Questions

2023 Preliminary Exam/CCHMS/Secondary 4/Additional Mathematics/4049/01 Name: Class: Class Register Number: PRELIMINARY EXAMINATION 2023 SECONDARY 4 ADDITIONAL MATHEMATICS 4049/01 Paper 1 Thursday 24 August 2023 2 hours 15 minutes Candidates answer on the Question Paper. READ THESE INSTRUCTIONS FIRST Write your name, class and index number 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 paper clips, glue or correction fluid. Answer all the questions. Give non -exact numerical ans wers 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. 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. This document consists of 19 printed pages and 1 blank page. [Turn over For Examiner’s Use Question Number Marks Obtained 1 2 3 4 5 6 7 8 9 10 11 12 Total Marks

2 2023 Preliminary Exam/CCHMS/Secondary 4/Additional Mathematics/4049/01 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/404/01 [Turn over 1 The points R and S have coordinates ( )3, 2 3 and ( )5, 4 5 respectively. Show that the gradient of RS can be expressed in the form 15ab+ , where a and b are integers to be found. [4]

4 2023 Preliminary Exam/CCHMS/Secondary 4/Additional Mathematics/4049/01 2 Given that cos p = and that  is acute, express in terms of p, (a) sin , [1] (b) ( )tan 90 , − [2] (c) cos 2 . [2]

5 2023 Preliminary Exam/CCHMS/Secondary 4/Additional Mathematics/404/01 [Turn over 3 Express ( )( ) 2 2 12 32 31 2 1 2 xx xx ++ −+ in partial fractions. [5]

6 2023 Preliminary Exam/CCHMS/Secondary 4/Additional Mathematics/4049/01 4 The expression 32ax bx b++ leaves a remainder of R when divided by ( )1x+ and a remainder of 52R− when divided by ( )2x+ . (a) Show that 23 5 ab −= . [4] (b) Given further that 8ab=− and ab , find the value of a and of b. [3]

7 2023 Preliminary Exam/CCHMS/Secondary 4/Additional Mathematics/404/01 [Turn over (c) Using the values of a and b found in part (b), explain why the equation 32 0ax bx b+ + = has only one real root and state its value. [4]

8 2023 Preliminary Exam/CCHMS/Secondary 4/Additional Mathematics/4049/01 5 Find the coordinates of the stationary points of the curve ( ) 2 3xy x −= and determine the nature of each stationary point. [7]

9 2023 Preliminary Exam/CCHMS/Secondary 4/Additional Mathematics/404/01 [Turn over 6 The height of a ball above ground, h metres, released by a machine can be modelled by the equation 20.2 6 3h x x=− + + where x is the horizontal distance travelled by the ball in metres. (a) State the height above ground at which the ball is being released. [1] (b) Express 20.2 6 3h x x=− + + in the form ( ) 2 a x b c−+ , where a, b and c are constants to be found. [2] (c) Using your result from part (b), explain why the height of the ball can never be more than 48 metres. [2] (d) Hence, explain if this machine is safe for use in an indoor stadium with a ceiling height of 45 metres. [1]

10 2023 Preliminary Exam/CCHMS/Secondary 4/Additional Mathematics/4049/01 7 (a) Solve the equation ( )3 18 3 40xx += . [3] (b) Solve the equation ( )22log 3 log 6yy= + + . [5]