2024 TPSS AMATH P2 PRELIM QP

This document consists of 17 printed pages TAMPINES SECONDARY SCHOOL Secondary Four Express / Five Normal Academic Preliminary Examination 2024 NAME CLASS REGISTER NUMBER ADDITIONAL MATHEMATICS 4049/02 Paper 2 9 September 2024 2 hours 15 minutes Candidates answer on the Question Paper. READ THESE INSTRUCTIONS FIRST For Examiner’s Use Write your name, class and register number on all the work you hand in. 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. [Turn over O

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02 = + +c bx ax, 2 4 2 b b acx a −± −= Binomial expansion ( ) nr r nnnnn b b ar nb anb ana b a + +     + +     +     + = + −−− ......21 2 21 , where n is a positive integer and ( ) .! ) 1 )...( 1 ( ! ! ! r r n n n r r n n r n + − −=−=      2. TRIGONOMETRY Identities A AA AA A A A A A A B A B AB A B A B A B A B A B A B A AA AA A A 2 2222 22 22 22 tan1 tan22tan sin2 1 1cos2sin cos2cos cos sin2 2sin tan tan1 tan tan)tan( sin sin cos cos)cos( sin cos cos sin)sin( cot 1 cosec tan 1 sec 1 cos sin − = − = − = − = = ±= ± = ± ±= ± + = + = = +   Formulae for ABC∆ C ab A bc c b a C c B b A a sin 2 1 cos 2 sin sin sin 2 2 2 = − + = = = ∆

3 1 The expression 3832 23 +−+ xxx leaves a remainder of p when divided by )1( +x . (a) Find the value of p. [2] (b) Solve the equation 03832 23 =+−+ xxx . [3] (c) Hence, solve the equation 01181)3(1)2( 23 =+−−+− xxx . [2]

4 2 Peter buys a new car. After t months, its value $C is given by 120000 atCe −= , where a is a constant. (a) Find the value of the car when Peter bought it. [1] (b) The value of the car after 12 months is expected to be $90000. (i) Show that 0.02397a= , correct to 4 significant figures. [3] (ii) Calculate the age of the car, to the nearest month, when its expected value will be $70000. [2] (iii) After 5 years, a car dealer offers to pay Peter $29000 for your car. Based on the equation above, would you agree to sell it? Explain your answer. [2]

5 3 (a) Without using a calculator, show that 4 2 675cos −= ° . [3] (b) Prove that x x x x x x sin cos 1 cosecsec cot tan +=+ + . Hence solve the equation tan cot 3 sec cosec 4cos 3sin xx x x xx + =+− for 0 2.x π≤≤ [6]

6 4 The diagram shows a trapezium OABC inscribed in a semi-circle, centre O and radius 14 cm. CB is parallel to the diameter, and AB is perpendicular to both CB and OA. Given that OA = x cm, (a) show that the area, A cm2, of the trapezium is given by ( ) 23 1962 xAx= − . [2]

7 Given that x may vary, (b) find the value of x for which A has a stationary value and determine whether this value of A is a maximum or a minimum, [5] (c) find this stationary value of A. [2]

8 5 (a) By using a suitable substitution, solve the equation 4222 523 +=+ ++ xxx . [5]

9 (b) Given that a x2log= and b y2log= , express 3log aba in terms of x and y. [3] (c) Solve 5 25 5log 50 4log log (2 4) 2yy+ = ++ . [4]

10 6 The diagram shows a plot of garden ABCD in which AB = 10 m and BC = 8 m, 90angle ABC angle ADC= = ° and .angle BAD θ= ° The garden will be used to grow sunflower plants. (a) Show that the length 10cos 8sinAD θθ= + . [2] (b) Express 10cos 8sinAD θθ= + in the form ),cos( α θ−R where R is a positive constant and α is acute. [3] °θ A D C B 8 m 10 m