2022 HS AMath Prelims P2

1 [Turn over _____________________________________________________________________________ ADDITIONAL MATHEMATICS 4049 / 02 Paper 2 Tuesday 23 August 2022 2 hours 15 minutes Candidates answer on the Question Paper ____________________________________________________________________________ Instructions to students: ▪ Write your name, centre number, index number and class in the spaces at the top of this page. ▪ Write in dark blue or black pen on spaces provided. ▪ You may use a HB pencil for any diagrams or graphs. ▪ Do not use staples, paper clips, glue or correction fluid. ▪ Answer all the questions in this paper. ▪ 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. ▪ The use of an approved scientific calculator is expected, where appropriate. ▪ 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: ______________________ This question paper consists of 17 printed pages (including this cover page). 90 HOUGANG SECONDARY SCHOOL PRELIMINARY EXAMINATION / 2022 SECONDARY FOUR (EXPRESS)

2 Mathematical Formulae 1. ALGEBRA 2For the equation 0, Quadratic Equation ax bx c+ + = a acbbx 2 42 −−= 1 2 2( ) .... .... 12 n n n n n r r n Binomial Theorem n n na b a a b a b a b b r − − −     + = + + + + + +           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 += += =+ sin ( ) sin cos cos sinA B A B A B =  cos( ) cos cos sin sinA B A B A B= BA BABA tantan1 tantan)tan(  = AAA cossin22sin = AAAAA 2222 sin211cos2sincos2cos −=−=−= A AA 2tan1 tan22tan −= Formulae for ABC C c B b A a sinsinsin == Abccba cos2222 −+= 1 sin2 ab C=

3 [Turn over 1 (a) Sketch the graph of 2xye=+ . [2] (b) Solve the equation 3 e 2exx−−= . [4]

4 2 The cubic polynomial f(x) is such that the coefficient of x3 is –1 and the roots of f(x) = 0 are 1, k and k2. It is given that f(x) has a remainder of –7 when divided by x – 2. (i) Show that 32 2 2 3 0k k k− − − = . [3] (ii) Hence find a value for k and explain that there are no other real values of k which satisfy this equation. [6]

5 [Turn over 3 (i) Given that ( 2) 1y x x= + − , show that d d 21 y kx x x = − where k is constant. [4] Hence (ii) find the rate of change of x when x = 2, given that y is changing at a constant rate of 2 units per second, [2]