SCSS 2024 AM Prelim P1 QP

SWISS COTTAGE SECONDARY SCHOOL SECONDARY FOUR AND FIVE PRELIMINARY EXAMINATION Name: _____________________________________ ( ) Class: _________ ADDITIONAL MATHEMATICS Paper 1 4049/01 Monday 9 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 Marks This document consists of 18 printed pages and 2 blank pages. Setter: Mr Heng Teng Boon 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 2ax bx c 0+ + = , 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 (17 marks) 1 The equation of a curve is 323y x ax b= + + , where a and b are constants. If 0a , find, in terms of a and/or b, the range of values of x for which y is increasing. [3] 2 The area of a rectangle is ( )72 b+ cm2. Given that the length of the rectangle is ( )42a+ cm and the breadth of the rectangle is ( )52− cm, find the value of a and of b. [4]

4 3 The equation of a curve is 22 12 11y x x= + + . (a) Express 22 12 11xx++ in the form 2()a x b c++ where a, b and c are constants. [2] (b) Find the range of values of p for which the line 11y px=+ intersects the curve at two distinct points. [3]

5 4 The diagram shows a triangle BCE whose vertices lie on the circumference of a circle. AD is a tangent to the circle at point C and AB is a tangent to the circle at point B. BED is a straight line. (a) Prove that angle ABC+ angle 180CED= . [3] (b) Show that there does not exist a circle that passes through points A, B, E and C. [2] A B C D E

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7 Name: ( ) Class: ___________ Section B (73 marks) 5 (a) Solve the equation 5log 2 3log 5 xx+= . [5] (b) Sketch the graph 0.5logyx= . [2] Questions 5 6 7 8 9 10 11 12 13 Marks

8 6 It is given that f ( ) 3sin 4 2 xx =+  . (a) State the least and greatest value of f ( )x . [2] (b) State the period of f ( )x . [1] (c) Sketch the graph of f ( )yx= for 04 x  . [2] (d) By drawing a suitable straight line on the same set of axes as the graph of f ( )yx= , state the number of solutions of the equation sin 24 xx   =− for 04 x  . [2]

9 7 A curve is such that 2 2 2 d 3 cos 2d xy exx −=+ . The curve passes through the point A (0, 3) and has a gradient of 5 at A. Find the equation of the curve. [7]

10 8 (a) The expansion of 2 23 n x x  − has a term independent of x. By considering the general term in the expansion, explain why n is a multiple of 3. [3] (b) It is given that 9n= . Find the value of 6 coefficient of term independent of 1coefficient of x x . [4]