Northland Sec Prelim 2024 AM P2

NAME INDEX NO. CLASS NORTHLAND SECONDARY SCHOOL PRELIMINARY EXAMINATION Secondary 4 Express / 5 Normal Academic ADDITIONAL MATHEMATICS 4049/02 Paper 2 28 August 2024 2 hours 15 minutes Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your class, index number and name 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 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. Setter: Mr Chen Weizhong Vetter: Ms Joanne Yap For Examiner's Use 90 This document consists of 19 printed pages and 1 blank page. [Turn over

2 Learners who Share, Citizens who Care, Leaders who Dare 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 nn n na b a a b a b a b b r − − −     + = + + + + + +           where n is a positive integer and ! )1)...(1( )!(! ! r rnnn rnr n r n +−−=−=      2. TRIGONOMETRY Identities AAec AA AA 22 22 22 cot1cos tan1sec 1cossin += += =+ BA BABA BABABA BABABA tantan1 tantan)tan( sinsincoscos)cos( sincoscossin)sin(   = = = A AA AAAAA AAA 2 2222 tan1 tan22tan sin211cos2sincos2cos cossin22sin − = −=−=−= = Formulae for ABC 2 2 2 sin sin sin 2 cos 1 sin2 a b c A B C a b c bc A bc A == = + − =

3 Learners who Share, Citizens who Care, Leaders who Dare [Turn over 1 Express 2 2 1 8 3 ( 1)(2 3) xx xx −− −+ in partial fractions. [5]

4 Learners who Share, Citizens who Care, Leaders who Dare 2 (a) Find the smallest integer c for which the line 7y x c=+ intersects the curve 224yx=− at two distinct points. [4]

5 Learners who Share, Citizens who Care, Leaders who Dare [Turn over (b) Find the range of values of k for which the curve 2 2(2 5) 9y kx k x k= + − + lies below the x-axis. [4]

6 Learners who Share, Citizens who Care, Leaders who Dare 3 (a) Prove the identity 1 cos tansin cosec cot x xx x x − =−+ . [4]

7 Learners who Share, Citizens who Care, Leaders who Dare [Turn over (b) Hence solve the equation 1 cos 2 3sin 2 cosec2 cot 2 x x x x − =−−+ for 0 180x   . [3] (c) Show that there are no solutions to the equation 1 cos tan 2sin cosec cot x xx x x − =−+ for 0 180x   . [2]

8 Learners who Share, Citizens who Care, Leaders who Dare 4 (a) The equation 2 8 4log log log 2xx+= has the solution 2mx= . Find the value of m. [4] (b) Sketch the graph of 3logyx= . [2]

9 Learners who Share, Citizens who Care, Leaders who Dare [Turn over (c) Explain why the equation 55log (2 11) log ( 4) 1xx− − − = has no real solutions. [4] 5 (a) The function f is defined by 2 f ( ) e xxx += where 0x . Explain, with working, whether f is an increasing or a decreasing function. [3]

10 Learners who Share, Citizens who Care, Leaders who Dare (b) The equation of a curve is 2 3 xy x= + . (i) Find an expression for d d y x and obtain the coordinates of the stationary points of the curve. [5]