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] (iii) evaluate 5 2 d 1 x x x− . [3]

6 4 The diagram above shows the side view of a bus stop shelter BCD such that BC = 4 m, CD = 1 m, angle 90BCD= and angle CBA = . AB is a concrete pavement under the shelter such that DA is perpendicular to AB. (i) Show that  sincos4 +=AB . [2] (ii) Express AB in the form )cos(  −R , where R > 0 and 0 90oo  . [3] B C D A 1 m 4 m

7 [Turn over (iii) State the maximum value of AB and find the corresponding value of  when AB is maximum. [2] (iv) Find the value of  when AB = 3 m. [2]

8 5 (a) A curve has the equation 226y x x c= − + , where c is a constant. Find the value of c for which the line 28yx+= is a tangent to the curve. [3] (b) Represent the solution set of 23( 5) 1xx−  − on the number line. [3] (c) Find the greatest value of integer p for which 22x x p− + − has real roots for all real values of x. [3]

9 [Turn over 6 (i) Expand and simplify 5 1 22 x − in ascending powers of x, up to the first 4 terms. [2] (ii) Hence find the value of a if the coefficient of 2x in the expansion of ( ) 5 2 11 3 2 2ax x x + + −  is 13 2 . [4] (iii) Using the answer from part (i), evaluate ( ) 5 0.47 correct to 5 decimal places. [3]

10 7 The table shows experimental values of two variables, x and y. x 0.5 1.0 1.5 2.0 y 15.9 19.1 23.4 30.2 It is known that x and y are related by the equation 10 xy Ab=+ . (i) On Pg 11, draw the graph of lg( 10)y− against x. [2] (ii) Use your graph in (i) to estimate the value of A and of b. [4] (ii) By drawing a suitable line on your graph, solve the equation 210xxAb = . [3]