Northland Sec Prelim 2024 AM P1

NAME INDEX NO. CLASS NORTHLAND SECONDARY SCHOOL PRELIMINARY EXAMINATION Secondary 4 Express / 5 Normal Academic ADDITIONAL MATHEMATICS 4049/01 Paper 1 27 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 22 printed pages. [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 (a) Express each of 22 4 1xx+− and 2 8xx− + − in the form 2()a x b c++ , where a, b and c are constants. [4] (b) Use your answers from part (a) to explain why the curves with equations 22 4 1y x x= + − and 2 8y x x=− + − will not intersect. [2]

4 Learners who Share, Citizens who Care, Leaders who Dare 2 Use the substitution 2e xu= to s how that the solution to the equation 229e 14 8exx −+= can be expressed in the form ex k= , where k is a constant to be found. [3]

5 Learners who Share, Citizens who Care, Leaders who Dare [Turn over 3 A right circular cylinder tank has a radius of ( 5 3)− m and a volume of (26 3 20 5) − m3. Without using a calculator , express the height of the cylinder in the form () ab− m, where a and b are integers. [6]

6 Learners who Share, Citizens who Care, Leaders who Dare 4 f ( )x is such that f ''( ) 3sin 4cos 2x x x=− . Given that f8 6  = and f '( ) 9 = , find f ( )x . [6]

7 Learners who Share, Citizens who Care, Leaders who Dare [Turn over 5 (a) (i) State, in terms of π, the principal value of ( ) 1tan 3− − . [1] (ii) Explain why the principal value of 1 1cos 2 − − cannot be 4 − . [1] (b) The acute angles A and B are such that 2sin( ) 1 2sin( )A B A B+ = − − and 1cos 3B= . Without using a calculator, find the exact value of tan A . [4]

8 Learners who Share, Citizens who Care, Leaders who Dare 6 The diagram shows an isosceles triangle ABC in which AC BC= . Point A is (3, 3) , B is (6, 3) and C lies below the x-axis. The line CB is extended to the point D with coordinates (7.5, 7) . A line is drawn from D, parallel to AC, to the point E (10, )k and C is joined to E. (a) Given that the area of the triangle ABC is 6 square units, find the value of k. [5] y x O D A E C B

9 Learners who Share, Citizens who Care, Leaders who Dare [Turn over (b) Using this value of k, find the area of triangle CDE. [2]

10 Learners who Share, Citizens who Care, Leaders who Dare 7 (a) A particle moves along the curve tanyx= in such a way that the y-coordinate of the particle is decreasing at a rate of 0.12 units per second. Find the rate of change of the x-coordinate of the particle at the instant when 1y= . [4]