ACSBR 2023 4E5N Add Maths Prelim P1 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 1 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 18 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 1 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 1 1 A cuboid has a square area ( ) 26 35 m .− The capacity of the cuboid is ( ) 345 5 38 7 m .− Find the height of the cuboid, in m, in the form ( )57ab − , where a and b are integers. [3]

Anglo-Chinese School (Barker Road) 4 Preliminary Examination 2023 Secondary 4 Express / 5 Normal (Academic) Additional Mathematics 4049 Paper 1 2 (a) Show that ( 1)x− is a factor of 32f ( ) (2 1) ( 2 )x x h x k h x k= − + − − + ,where h and k are constants. [1] (b) Explain with working why the equation f ( ) 0x = will have 3 real roots if 2 0hk+ . [4]

Anglo-Chinese School (Barker Road) 5 Preliminary Examination 2023 Secondary 4 Express / 5 Normal (Academic) Additional Mathematics 4049 Paper 1 3 The temperature, CT , of glass being heated in a kiln is given by the formula 2030 2000 ktTe −=− , where t is the time in hours since the glass was placed in the kiln. The temperature of the glass after 90 minutes is 1000 C . (a) Find the temperature of the glass before it was heated. [1] (b) The glass can be molded into different shapes when its temperature reaches at least 1300 C . What is the earliest time that the glass can be removed from the kiln and molded? [4] (c) The instruction manual of the kiln gives the range of the temperature of the kiln as CTy . State the value of y. [1]

Anglo-Chinese School (Barker Road) 6 Preliminary Examination 2023 Secondary 4 Express / 5 Normal (Academic) Additional Mathematics 4049 Paper 1 4 The curve f ( )yx= is such that 3 18f "( ) (1 2 )x x= − and 2f '( 1) 1.−= Find the equation of the curve given that it passes through the point (1,0) . [6]

Anglo-Chinese School (Barker Road) 7 Preliminary Examination 2023 Secondary 4 Express / 5 Normal (Academic) Additional Mathematics 4049 Paper 1 5 It is given that ( ) xp xx − += 2 2)(f , px  , where p is an integer. (a) Obtain and simplify an expression for )('f x in terms of p. [3] (b) The range of values of x for which f ( )x is an increasing function is 28 x−   . Show that the value of p is 3. [3] (c) Hence find the equation of the normal to the curve at the point where the curve crosses the x-axis. [3]

Anglo-Chinese School (Barker Road) 8 Preliminary Examination 2023 Secondary 4 Express / 5 Normal (Academic) Additional Mathematics 4049 Paper 1 6 Given that 225sin 3cos 7sin 2A A A−= where 0 90A . (a) Show that tan 3A= . [3] (b) Hence find the value of ( )cos 60 A+ , leaving your answer in the form 3 , 2 10 ab+ where a and b are integers. [3] (c) Without finding the value of A, explain whether 60 A+ is acute or obtuse. [1]

Anglo-Chinese School (Barker Road) 9 Preliminary Examination 2023 Secondary 4 Express / 5 Normal (Academic) Additional Mathematics 4049 Paper 1 7 (a) Given that ( )( )3 1 6 1y x x= − + , show that d 27 .d 61 yx x x = + [3] (b) Hence find the value of 4 0 93 d 61 x x x − + . [4]

Anglo-Chinese School (Barker Road) 10 Preliminary Examination 2023 Secondary 4 Express / 5 Normal (Academic) Additional Mathematics 4049 Paper 1 8 (a) In the binomial expansion of 32 n x x  − , where n is a positive integer, the coefficient of the third term is ( )270 28 n . Show that n = 6. [4] (b) Using the value of n in part (a), find the coefficient of 4x in the expansion of ( ) 2 312 n xx x +−  . [3]