ACSBR 2023 4E5N Add Maths Prelim P2 updated

Full Name Class Index No Class Anglo-Chinese School (Barker Road) PRELIMINARY EXAMINATION 2023 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 on all the work you hand in. Write in dark blue or black pen. Answer all questions. If working is needed for any question it must be shown with the answer. Omission of essential working will result in loss of marks. The use of an approved scientific calculator is expected, where appropriate. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For  , use either your calculator value or 3.142, unless the question requires the answer in t erms of  . The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks for this paper is 90. This question paper consists of 20 printed pages and 2 blank pages. For Examiner’s Use

Anglo-Chinese School (Barker Road) 2 Preliminary Examination 2023 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 2023 Secondary 4 Express/ 5 Normal (Academic) Additional Mathematics 4049 Paper 2 1 Mr Tan invested some money in stock in May. The value of a stock, $y, varies with time, x, can be modelled by the equation 21 213y x x=− + + , where y is in thousands and x is in months. (a) State the initial value of the stock. [1] (b) Express y in the form 2()a x b c−+ . [2] (c) Determine the best time to sell off his stocks to make a maximum profit. Find the value of the stock at this time. [2]

Anglo-Chinese School (Barker Road) 4 Preliminary Examination 2023 Secondary 4 Express/ 5 Normal (Academic) Additional Mathematics 4049 Paper 2 2 It is given that 32f ( ) 2 34 12x ax bx x= + − + , where a and b are constants, has a factor of 3x− and leaves a remainder of 32 when divided by 1x+ . (a) Find the value of a and of b. [4] (b) Find the range of values of k for which the line kxy =+ 9 intersects the curve xyx 3−= at two distinct points. [4]

Anglo-Chinese School (Barker Road) 5 Preliminary Examination 2023 Secondary 4 Express/ 5 Normal (Academic) Additional Mathematics 4049 Paper 2 3 (a) Show that 2 2 2 2cos sin cos 1 3cot cot 1sin     −+ = − + . [3] (b) Hence solve the equation 222cos sin cos 1 sin   − + = for   −   . [4]

Anglo-Chinese School (Barker Road) 6 Preliminary Examination 2023 Secondary 4 Express/ 5 Normal (Academic) Additional Mathematics 4049 Paper 2 4 A coffee filter is in the shape of a right circular cone. The cone has a base radius of 3 cm and a height of 12 cm. The coffee powder, as seen in the cross-section of the filter below, has a depth of 4 cm. Let h be the height of the hot water in the filter at time t seconds. Assume that the amount of water in the coffee powder is negligible. (a) Show that the volume of hot water, V, in the filter at the time t seconds is given by ( ) 3 4448 3Vh = + − . [3] [The volume of a cone of height h and base radius r is 21 3 rh ] Coffee powder 4 cm Hot water h cm 12 cm 3 cm

Anglo-Chinese School (Barker Road) 7 Preliminary Examination 2023 Secondary 4 Express/ 5 Normal (Academic) Additional Mathematics 4049 Paper 2 Initially, the filter contains only coffee powder. A machine dispenses hot water into the filter at a constant rate of 5 cm3 per second and hot water drips out of the filter at a constant rate of 2 cm3 per second. (b) Find the exact rate of change of the depth of water when h = 8. [4]

Anglo-Chinese School (Barker Road) 8 Preliminary Examination 2023 Secondary 4 Express/ 5 Normal (Academic) Additional Mathematics 4049 Paper 2 5 The diagram shows an isosceles triangle PQR with vertices at P(0, k), Q(3, 9) and R(h, 6), where h and k are positive constants and h>k. (a) Show that h + k = 12. [3] Q(3,9) 0 P(0,k) R(h,6) x y

Anglo-Chinese School (Barker Road) 9 Preliminary Examination 2023 Secondary 4 Express/ 5 Normal (Academic) Additional Mathematics 4049 Paper 2 (b) It is now given that h = 7. The point S is such that PQRS is a kite. Given that S lies on the line 21 7 8yx=− , find the coordinates of S. [4]

Anglo-Chinese School (Barker Road) 10 Preliminary Examination 2023 Secondary 4 Express/ 5 Normal (Academic) Additional Mathematics 4049 Paper 2 (c) Find the area of the kite. [2]