SCSS 2023 4NA AM PRELIM P2 QP

Home of Thoughtful Leaders: Serve with Honour, Lead with Humility SWISS COTTAGE SECONDARY SCHOOL SECONDARY FOUR PRELIMINARY EXAMINATION Name: ________________________________( ) Class: __________ ADDITIONAL MATHEMATICS Paper 2 4051/02 Monday 7 August 2023 1 hour 45 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 on both sides of the paper. 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 70. Questions 1 2 3 4 5 6 7 8 9 10 Marks This document consists of 14 printed pages. Setter: Mdm Zoe Pow Vetter: Mr Ang Hanping [Turn over NA

2 1. ALGEBRA Quadratic Equation For the equation 0cbxax =++2 , a acbbx 2 42 −−= . 2. TRIGONOMETRY Identities 1cossin 22 =+ AA AA 22 tan1sec += AAec 22 cot1cos += ( ) BABABA sincoscossinsin = ( ) BABABA sinsincoscoscos = ( ) BA BABA tantan1 tantantan  = AAA cossin22sin = AAAAA 2222 sin211cos2sincos2cos −=−=−= A AA 2tan1 tan22tan −= Formulae for ABC C c B b A a sinsinsin == Abccba cos2222 −+= Abcsin2 1=

3 Answer all the questions. 1 Express 43 2 3 50 24 26 x x x xx + + − −+ in the form 2ax bx c++ , where a, b and c are integers. [3] 2 The volume, V cm3, of water in a tank may be modelled by the equation 31 3Vx = , where x cm is the depth of the water in the tank. When the tap is turned on, the volume of water in the tank increases at a rate of 0.2 cm3/min. Find the rate at which the depth of water in the tank is increasing when 0.9x= . [5]

4 3 (i) Express 2 6 14xx−+ in the form ( ) 2 x a b++ where a and b are constants. [3] (ii) Hence state the coordinates of the vertex of the curve 2 6 14y x x= − + . [2]

5 4 The equation of a curve is 32 4 3 8y x x x= + − − . Find the equation of the tangent to the curve at the point 2x=− . [5]

6 5 The polynomial p( )x is given by 32p( ) 6 18x ax x bx= + + − , where a and b are constants. It is given that 2x+ is a factor of p( )x and when p( )x is divided by 3x− the remainder is 75. (i) Show that 2a= and find the value of b. [4] (ii) Using the values from part (i), find the remainder when p( )x is divided by 21x− . [2]

7 6 (a) Given that 3 2 13 xy x −= + , find d d y x . [3] (b) Given that ( ) 423 2 1y x x=+ , find d d y x , giving your answer in the form ( ) 3 2 1 f ( )xx+ where f ( )x is a quadratic expression. [4]

8 7 It is given that ( )( ) 5f( ) 3 2 1 xx xx= −+ , where 2 and 13xx − . (i) Express f( )x in partial fractions. [4]

9 (ii) Hence find f '( )x . [3] (iii) Given that a curve has equation f( )yx= , explain why the gradient of every point on the curve is negative. [2]

10 8 (a) Prove that 2 2 cot 1 cos 2cosec   − = . [3] (b) (i) Write 5cos 2sinxx− in the form ( )cosRx + , where 0R and 0 90   . [4]