presbyterian high school 2023 A math paper 2

Name: Index No.: Class: PRESBYTERIAN HIGH SCHOOL ADDITIONAL MATHEMATICS 4049/02 Paper 2 21 August 2023 Monday 2 hours 15 mins PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL 2023 SECONDARY FOUR EXPRESS / FIVE NORMAL (ACADEMIC) PRELIMINARY EXAMINATIONS DO NOT OPEN THIS QUESTION PAPER UNTIL YOU ARE TOLD TO DO SO. INSTRUCTIONS TO CANDIDATES Write your name, index number and class in the spaces provided above. 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. Write your answers in the spaces provided below 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. For Examiner’s Use Qn 1 2 3 4 5 6 7 8 9 10 Marks Deducted Marks Setter: Tan Chee Wee Vetter: Tan Lip Sing This question paper consists of 21 printed pages and 1 blank page. Category Accuracy Units Symbols Others Question No. TOTAL MARKS 90

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 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 2. TRIGONOMETRY Identities Formulae for 1 sin2 bc A= 2 0,ax bx c+ + = 2 4 2 b b acx a −  −= ! ( 1)...( 1) ( )! ! ! n n n n n r r n r r r  − − +== − 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 2cos2 cos sin 2cos 1 1 2sinA A A A A= − = − = − 2 2tantan 2 1 tan AA A = − ABC sin sin sin a b c A B C== 2 2 2 2 cosa b c bc A= + −

3 1 An object is heated in an oven until it reaches a temperature of X C. It is then allowed to cool. Its temperature,  C, when it has cooled for time t minutes, is given by 630 100(0.8) t  =+ . (a) Find the value of X. [1] (b) Find the value of  when t = 8. [1] (c) Find the value of t when 95.= [3] (d) A sketch of the graph of  against t is given below. State the value of p. [1] C t mins p 0

4 A calculator must not be used in this question. 2 (a) In the diagram, triangle ABC has an area of ( )8 2 4+ cm2, angle 4BAC = radian and AB = ( )2 2 2+ cm. Find the length of AC, leaving your answer in the form ( )2pq + cm, where p and q are integers. [5] A B C

5 (b) Find cos75 , giving your answer in the form 4 ab− , where a and b are integers. [3]

6 3 (a) Prove that cosec2 cot 2 tanx x x−= . [3]

7 (b) Hence solve 2cosec2 cot 2 2sec 3x x x− = − for 0 360x   . [5]

8 4 (a) Solve ( ) 19 5 2 3xx ++= . [5]

9 (b) Solve 2 4 100 1002log log ( 9) log 1xx + − =− . [5]

10 5 The diagram shows a quadrilateral ABCDE where triangle ABC is similar to triangle DEC. AB = 15 cm, DE = 9 cm, angle ACD = 90 o and angle ABC is a variable angle , where 0 o <  < 90 o. (a) Show that the perimeter, P cm, of the quadrilateral is given by 24 24sin 6cosP = + + . [4] B D E 9 cm 15 cm  C A