2022 HS AMath Prelims P1

CANDIDATE CLASS: NAME: CENTRE INDEX NUMBER: NUMBER: S ADDITIONAL MATHEMATICS 4049/01 Paper 1 Friday 19 August 2022 2 hours 15mins Candidates answer on the Question Paper Instructions to students: • Write your name, index number and class clearly in the spaces at the top of this page. • Write in dark blue or black pen on spaces provided. • 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 in this paper. • The use of an approved scientific calculator is expected, where appropriate. ▪ Give non-exact numerical answers correct to 3 significant figures, or one decimal place in case of angles in degrees, unless a different level of accuracy is specified in the question. ▪ You are reminded of the need for clear presentation in your answers. Information for pupils • The number of marks is given in brackets [ ] at the end of each question or part question. • The total mark for this paper is 90. Calculator Model: ______________________ The Question Paper consists of 20 printed pages (including this cover page) HOUGANG SECONDARY SCHOOL PRELIMINARY EXAMINATION / 2022 SECONDARY FOUR (EXPRESS) 90

Mathematical Formulae 1. ALGEBRA ,0equationtheFor : 2 =++ cbxax EquationQuadratic a acbbx 2 42 −−= nrrnnnnn bbar nbanbanaba TheoremBinomial ++     ++     +     +=+ −−− ........21)( : 221 where n is appositive integer and !)!( ! rrn n r n −=      = ! )1.().........1( r rnnn +−− 2. TRIGONOMETRY Identities AAec AA AA 22 22 22 cot1cos tan1sec 1cossin += += =+ BABABA sincoscossin)sin( = cos(𝐴 ± 𝐵) = cos 𝐴 cos 𝐵 ∓ sin 𝐴 sin 𝐵 tan tantan( ) 1 tan tan ABAB AB = AAA cossin22sin = AAAAA 2222 sin211cos2sincos2cos −=−=−= A AA 2tan1 tan22tan −= Formulae for ABC C c B b A a sinsinsin == Abccba cos2222 −+= CabABC sin2 1=

3 2022 4E Additional Mathematics Prelim 4049/01 1 The line 2 3 7yx=− meets the curve 2 38y x x= + − at two points A and B. Find the distance between A and B. [4]

4 2022 4E Additional Mathematics Prelim 4049/01 2 Express ( )( ) 2 78 2 1 1 x xx + +− in partial fractions. [5]

5 2022 4E Additional Mathematics Prelim 4049/01 3 Without using the calculator, find the values of the integer a and b such that 4 3 2 3 3 4 3ab −+ = +− . [3] 4 The line of symmetry of a quadratic curve is 2x=− and the curve lies above the x-axis for all x. Given that the point ( )1, 4− lies on the curve, find a possible equation of the curve in the form ( ) 2 y a x h k= − + where a, h, and k are integers. [4]

6 2022 4E Additional Mathematics Prelim 4049/01 5 The diagram shows the curve sin xy p r q=+ for 08 x  radians. The curve has a minimum point at ( )2 , 3 − and a maximum point ( )6 ,1 . (a) Show that 1r=− . [1] (b) Find the values of p and q. [2] (c) Hence write down the equation of the curve. [1] y x (6π, 1) 8π (2π, ─3)

7 2022 4E Additional Mathematics Prelim 4049/01 6 The function f is defined for all real values of x and is such that ( )f '' 6 2xx=+ . The gradient to the curve ( )fyx= at the point ( )1,10− is 11. (a) Find an expression for ( )f' x . [3] (b) Hence find the equation of the curve ( )fyx= . [2] (c) Determine whether the curve ( )fyx= have stationary point(s). Explain with clear working. [3]

8 2022 4E Additional Mathematics Prelim 4049/01 7 When the hemispherical bowl above contains water to a depth of x cm, the volume, V cm3, of the water is given by ( )21 183V x x =− . The bowl is initially empty. After water has been poured into the bowl at a constant rate for 9 seconds, the depth of water is 4.5 cm. (a) Find the constant rate of change of volume in terms of π. [3] (b) Find the rate at which the water level is rising when the depth is 4.5cm. [4] x cm

9 2022 4E Additional Mathematics Prelim 4049/01 8 (a) Factorise 33sin cosxx+ completely. [1] (b) Show that 33sin cos 1 1 sin 2sin cos 2 xx xxx + =−+ . [3] (c) Hence solve the equation 33sin cos 5 sin cos 4 xx xx + =+ for oo0 360x . [4]

10 2022 4E Additional Mathematics Prelim 4049/01 9 The diagram shows a point A on the circle and XAY is a tangent to the circle. Points S, B and C lie on the circle. The chords AB and SC intersect at T and angle ACB = angle ATC. (a) Prove that triangles ABC and ACT are similar. [2] (b) Show that 22AC AT AT TB− =  . [3] X Y A B C S T