2023 CCHM Prelim AMath P2 - Questions

2023 Preliminary Exam/CCHMS/Secondary 4/Additional Mathematics/4049/02 [Turn over For Examiner’s Use Question Number Marks Obtained 1 2 3 4 5 6 7 8 9 10 Total Marks 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 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 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. 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 20 printed pages and 2 blank pages. _________________________ Parent’s Signature

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]

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]

5 2023 Preliminary Exam/CCHMS/Secondary 4/Additional Mathematics/4049/02 [Turn over 2 (a) By considering the general term in the binomial expansion of 7 3 2x x  − , explain why there are only odd powers of x in this expansion. [3] (b) Find the term independent of x in the expansion of 7 32 25 2xx xx    −−       . [3]

6 2023 Preliminary Exam/CCHMS/Secondary 4/Additional Mathematics/4049/02 3 The expression 7sin 3cos+ is defined for 0° ≤ θ ≤ 360°. (a) Using ( )sinR + , where R > 0 and 0° < α < 90°, solve the equation 7sin 5 3cos=− . [5]

7 2023 Preliminary Exam/CCHMS/Secondary 4/Additional Mathematics/4049/02 [Turn over (b) State the largest and smallest values of ( ) 2 7sin 3cos 12+− and find the corresponding values of θ. [4]

8 2023 Preliminary Exam/CCHMS/Secondary 4/Additional Mathematics/4049/02 4 The diagram shows a circle passing through the points A, B, C, D and E. The straight line FDG is tangent to the circle at D while FAB and FEC are secant lines. Given that angle FDA = angle ADB, (a) show that triangle ABD is an isosceles triangle, [2] (b) prove that AF BF EF CF =  . [3] A D C B E G F

9 2023 Preliminary Exam/CCHMS/Secondary 4/Additional Mathematics/4049/02 [Turn over 5 An object is heated in an oven until it reaches a temperature of X C. It is then allowed to cool under room temperature. Its temperature, T C, can be modelled by 18 62e ktT −=+ , where t is the time in minutes since the object starts cooling. (a) Find the value of X. [1] When t = 10, the temperature of the object is 65 C. (b) Find the temperature of the object an hour later, giving your answer to one decimal place. [5] (c) What does the model suggest about the room temperature? Explain your answer. [2]

10 2023 Preliminary Exam/CCHMS/Secondary 4/Additional Mathematics/4049/02 6 The equation of a curve is 21ln 31 xy x −=  − , where 1 2x . (a) Find d d y x , expressing it as a single fraction. [3] (b) Explain why the curve will be almost parallel to the x-axis as x becomes very large. [2]