SCSS 2024 AM Prelim P2 QP

SWISS COTTAGE SECONDARY SCHOOL SECONDARY FOUR AND FIVE PRELIMINARY EXAMINATION Name: _____________________________________ ( ) Class: _________ ADDITIONAL MATHEMATICS Paper 2 4049/02 Tuesday 10 September 2024 2 hours 15 minutes Candidates answer on the Question Paper. READ THESE INSTRUCTIONS FIRST Write your name, class and index 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. Questions 1 2 3 4 5 6 Marks This document consists of 18 printed pages and 2 blank pages. Setter: Mrs Chen Yen Wah Vetter: Mdm Zoe Pow [Turn over Home of Thoughtful Leaders: Serve with Honour, Lead with Humility O

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2 0ax bx c+ + = , 2 4 2 b b acx a −  −= . Binomial Theorem 1 2 2 12 n n n n n r r n n n n(a b) a a b a b ..... a b .... b r − − −     + = + + + + + +           where n is a positive integer and ( ) ( ) ( )1 ........ 1! ! ! ! n n n n rn r n r 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=

3 Answer all the questions. Section A (41 marks) 1 A man bought a new car. The value of the car depreciated with time so that its value, $,P after t months’ use is given by 175 000 , ktPe −= where k is a constant. (a) Find the value of the car, $,P when the man bought it. [1] The value of the car is expected to be $162 000 after eight months’ use. (b) Show that 0.01.k = [2] (c) Use the result from part (b) to determine the age of the car correct to the nearest month, when its value reached half of the original value when the man bought it. [2]

4 2 A calculator must not be used in this question. (a) Show that cot15 3 2.= + [4] (b) Use the result from part (a) to find an expression for 2cosec 15 , in the form 3pq+ where p and q are integers. [2]

5 3 Given that 22 0 532 d , 24 m xxe e x − −= where m is a positive constant, (a) show that 424 12 5 0.mmee − + = [3] (b) Use the result from part (a) and a suitable substitution to find the value of .m [4]

6 4 The point A lies on the curve ln .y x x= The tangent to the curve at A is parallel to the line 2 3.yx=+ (a) Find the exact coordinates of .A [4] The normal to the curve lny x x= at A meets the line 23yx=+ at the point .B (b) Show that the -coordinatex of B is ( )2,ke − where k is a constant to be found. [3]

7 5 (a) Prove the identity 1 cos 2 sin 2 tan .1 cos 2 sin 2 xx xxx −+ =++ [3] (b) Hence solve the equation 1 cos 2 sin 2 3cot 21 cos 2 sin 2 xx xxx −+ =++ for 0 180 .x   [4]

8 6 The diagram shows two rods AB and BC rigidly hinged at B so that angle 90 .ABC= The lengths of AB and BC are 2 cm and 8 cm respectively. The point C is fixed on horizontal ground and the rod BC rotates in a vertical plane with the rod BC inclined at an angle  to the ground. (a) Show that the height, cm,h of A above the ground is given by sin cos ,h a b =− where a and b are integers to be found. [2]

9 (b) Using the values of a and b found in part (a), express h in the form ( )sin ,R − where 0R and 0. 2  [4] (c) Hence, state the maximum value of h and find the corresponding value of . [3]

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