2023 CGS Sec 4 AM Prelim P1 with Ans Key

Name ___________________________________ ( ) Class 4 ______ ADDITIONAL MATHEMATICS 4049/01 Paper 1 24 August 2023 2 hours 15 minutes Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your name, class and index number in the spaces at the top of this page. 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 three 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. At the end of the examination, fasten all your work securely together. 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. Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Marks Table of Penalties Qn. No. Presentation –1 Accuracy/ Units –1 Parent’s/ Guardian’s Signature For Examiner’s Use CRESCENT GIRLS’ SCHOOL SECONDARY FOUR PRELIMINARY EXAMINATION 90

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02 =++ cbxax , a acbbx 2 42 −−= Binomial expansion 1 2 2( ) ... ... 12 n n n n n r r n n n na b a a b a b a b b r − − −     + = + + + + + +           , where n is a positive integer and ! ( )...( 1) !( )! ! n n n n r n r r r n r r  − − +== − 2. TRIGONOMETRY Identities 1cossin 22 =+ AA AA 22 tan1sec += 22cosec 1 cotAA=+ BABABA sincoscossin)sin( = cos( ) cos cos sin sinA B A B A B= tan tantan( ) 1 tan tan ABAB AB = sin 2 2sin cosA A A= AAAAA 2222 sin211cos2sincos2cos −=−=−= A AA 2tan1 tan22tan −= Formulae for ABC C c B b A a sinsinsin == Abccba cos2222 −+=  = 1 sin2 bc A

3 1 Given that 2 2 3k =− , without using the calculator, express 23k k− in the form 23ab c − , where a, b and c are integers. [3] 2 The straight line 20y kx=+ intersects the curve 23 2 21y kx=− at the points A and B whose x-coordinates are ‒3 and 4.5 respectively. Find the value of k. [4]

4 3 Express 23 12 4xx− + − in the form 2()a x h k−+ , where a, h and k are integers. Hence state the coordinates of the turning point of the curve 23 12 4y x x=− + − . [4] 4 Integrate 2tan 2 x with respect to x. [3]

5 5 Express ( ) ( ) 2 2 6 5 5 1 2 xx xx −+ −+ in partial fractions. [4]

6 6 ( ) ( ) pxpxx n ++−= 22 1f , where n and p are positive integers. (a) Show that ( )1+x is a factor of ( )xf for all values of p.