ACSBR 2024 Sec Prelim Add Math P2

Full Name Class Index No Class Anglo-Chinese School (Barker Road) PRELIMINARY EXAMINATION 2024 SECONDARY FOUR EXPRESS / FIVE NORMAL (ACADEMIC) ADDITIONAL MATHEMATICS 4049 PAPER 2 2 HOURS 15 MINUTES Candidates answer on the Question Paper. READ THESE INSTRUCTIONS FIRST Write your 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. Answer all 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. This document consists of 19 printed pages and 1 blank page. For Examiner’s Use

Anglo-Chinese School (Barker Road) 2 Preliminary Examination 2024 Secondary 4 Express / 5 Normal (Academic) Additional Mathematics 4049 Paper 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 ( ) ( ) ( )11! ! ! ! n n n n rn 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 bc A=

Anglo-Chinese School (Barker Road) 3 Preliminary Examination 2024 Secondary 4 Express / 5 Normal (Academic) Additional Mathematics 4049 Paper 2 1 A particle moves along the curve 3( 6) 4 xy x += + , where 4x− , in such a way that the y- coordinate of the particle is increasing at a constant rate of 4 27 units per second. Find the x-coordinates of the particle at the instant that the x-coordinate of the particle is decreasing at a rate of 2 units per second. [5]

Anglo-Chinese School (Barker Road) 4 Preliminary Examination 2024 Secondary 4 Express / 5 Normal (Academic) Additional Mathematics 4049 Paper 2 2 The number, v, of a certain virus present in a sample collected by a vaccine laboratory is given by 2tv me n=+ , where m and n are constants and t is measured in days. Initially, the number of virus present was 2000. It increased to 5000 after 1 day. (a) Find the value of m and of n. [4] (b) Find the number of days in which the number of virus present first reach 1 million. [2]

Anglo-Chinese School (Barker Road) 5 Preliminary Examination 2024 Secondary 4 Express / 5 Normal (Academic) Additional Mathematics 4049 Paper 2 3 (a) Show that the roots of the equation 2 (3 ) 3 0ax a b x b+ + + = are real for all real values of a and b. [3] (b) Find the range of values of m for which the line 3y mx=− will never cut the curve 2 4 6 34y x y= − − . [4]

Anglo-Chinese School (Barker Road) 6 Preliminary Examination 2024 Secondary 4 Express / 5 Normal (Academic) Additional Mathematics 4049 Paper 2 4 A tangent to a circle at the point (6, 10) passes through the point (9, 6). The centre of the circle lies on the line 3 4 13yx=+ . Showing all your working, find the equation of the circle. [7]

Anglo-Chinese School (Barker Road) 7 Preliminary Examination 2024 Secondary 4 Express / 5 Normal (Academic) Additional Mathematics 4049 Paper 2 5 Do not use a calculator in this question. (a) Express sin 22.5 in the form of aa a − , where a is an integer. [3] (b) Show that tan15 2 3= − . [5]

Anglo-Chinese School (Barker Road) 8 Preliminary Examination 2024 Secondary 4 Express / 5 Normal (Academic) Additional Mathematics 4049 Paper 2 6 (a) (i) Using the substitution 3ux= or otherwise, express 6 1x − as the product of two factors. [1] (ii) Hence express 6 1x − as the product of four factors with integer coefficients. [1]

Anglo-Chinese School (Barker Road) 9 Preliminary Examination 2024 Secondary 4 Express / 5 Normal (Academic) Additional Mathematics 4049 Paper 2 6 (b) (i) Find the remainder when 32f ( ) 3 5 7 4x x x x= − + − is divided by 1x− . [1] (ii) Hence show that h = −1 for which g( ) f ( )x x h=+ is divisible by 1x− . [2] (iii) Explain why the equation g( ) 0x = has only one real root. [4]

Anglo-Chinese School (Barker Road) 10 Preliminary Examination 2024 Secondary 4 Express / 5 Normal (Academic) Additional Mathematics 4049 Paper 2 7 (a) Two variables x and y are related by the equation 2xy ax by=+ . Explain how a straight line graph can be drawn to represent the given equation. [2] (b) The table shows experimental values of two variables x and y. It is known that x and y are related by the equation qxy pe −= where p and q are constants. x 1 3 5 7 9 y 98.2 65.9 44.1 29.6 19.8 (i) On the grid below plot ln y against x and draw a straight line graph. [2]