SCSS 2023 4NA AM PRELIM P1 QP

Home of Thoughtful Leaders: Serve with Honour, Lead with Humility SWISS COTTAGE SECONDARY SCHOOL SECONDARY FOUR PRELIMINARY EXAMINATION Name: ________________________________( ) Class: __________ ADDITIONAL MATHEMATICS Paper 1 4051/01 Tuesday 1 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 11 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  = 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 −+= Abcsin2 1=

3 Answer all the questions. 1 Factorise 354 128x + completely. [3] 2 (i) Given that 2 4 1 3 0x kx k+ + − = has two real and distinct roots, show that k satisfies 24 3 1 0kk+ −  . [2] (ii) Solve the inequality 24 3 1 0kk+ −  . [2]

4 3 (i) Write down the period and amplitude of 4cos 12 xy =−  . [2] (ii) Sketch, on the axes below, the graph of 4cos 12 xy =−  for 02 x  radians. [3] x y 0  2

5 4 A curve has gradient ( ) 223x + at every point on the curve . It is given that the curve passes through the point (1, 12). (i) Find the equation of the curve. [4] (ii) Explain why this curve has no stationary point. [1] (iii) Given that the equation of the curve can be written in the form P( )yx= , explain why P( )x is an increasing function. [1]

6 5 (a) Without using a calculator, and showing all your working, find the exact value of sin 30 cos 45 cos30 sin 45 +   . [2] (b) Solve 2cos 2 32x  += for 0 x  radians giving your answers in terms of  . [4]

7 6 (i) Differentiate ( ) 32 1 1 x− . [2] (ii) Hence evaluate ( ) 2 40 2 d 1 x x x− . [4]

8 7 The following trigonometric function models the depth of water motion along Kallang River. ( )siny a bt c=+ y is the depth of water, t is time in hours and a, b and c are constants. At high tide, the depth of water is 2.7 m. Six hours later, at low tide, the depth of water is 0.1 m. This diagram shows part of the above trigonometric function. (i) Find the values of a, b and c. [4] Depth of water (m) Time (hours) 0 3 6 9 12 1 2 3

9 (ii) Find the depth of water after 16 hours. [2] (iii) Kayaking is permitted when the depth of water is at least 1.4 m. State the time interval(s) between 0 hour and 12 hours where kayaking is permitted. [1]

10 8 (a) Show that 32 5 3 1 + − can be written in the form of 3ab c + where a, b and c are integers. [4] (b) It is given that ( )7 2 1xx+ = − . Find x in the form 7pq+ , where p and q are rational numbers. [4]