SCGS 2025 AM Prelim Paper 2

[Turn over SINGAPORE CHINESE GIRLS’ SCHOOL PRELIMINARY EXAMINATION 2025 SECONDARY FOUR O-LEVEL PROGRAMME CANDIDATE NAME CLASS 4 REGISTER NUMBER CENTRE NUMBER INDEX NUMBER ADDITIONAL MATHEMATICS 4049/02 Paper 2 Friday 29 August 2025 2 hours 15 minutes Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your name, class, register number, centre number and index number at the top of this page. 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 electronic 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 90. FOR EXAMINERS USE Q1 Q5 Q9 Q2 Q6 Q10 Q3 Q7 Q11 90 Q4 Q8 The Question Paper consists of 19 printed pages and 3 blank pages.

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the quadratic equation ax 2 + bx + c = 0 , Binomial Expansion , where n is a positive integer and 2. TRIGONOMETRY Identities sin 2 A + cos 2 A = 1 sec 2 A = 1 + tan 2 A cosec 2 A = 1 + cot 2 A sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B sin A sin B tan ( A ± B ) = sin 2A = 2 sin A cos A cos 2A = cos2 A – sin2 A = 2cos2 A – 1 = 1 – 2 sin2 A tan 2A = Formulae for D ABC a 2 = b 2 + c 2 - 2bc cos A D = ab sin C ! !"##$ ! "!!±!= ( ) !" " !" ! !! ! ! " " !!! !#$ # #$ # $ # $ $ " !! !"# "# "#+= + + + + + +$% $% $%&' &' &' !! ( ) ( )( )!!" "" " ! !! ! "! " "!" " !! +"#==$% !&' ! !!"# !"# $ !"# !"# !" !" ± ! ! ! "#$ % "#$ ! !! !"# !"# !"# !"# ABC == !"

3 [Turn over 1. In an experiment, 5 million cells were stored in a container. It was observed that 40% of the cells are dying every minute. The number of cells remaining, N, after t minutes, is given by , where A and k are constants. (a) Explain why . [2] (b) Find the time taken when the number of cells decreases to 2000. [3] !"#A %= !"#$!=

4 2. (a) Use the substitution , solve the equation and show that the solution may be written in the form where a and b are integers to be determined. [4] (b) Solve the equation . [5] !!"= ( )!! "#$ #% &!!++!! = !"# $!"! ! " # !$%&' (! )* %&' ( !* %&' $!!!! + =

5 [Turn over 3. A curve has the equation . (a) Find the coordinates of the stationary point on the curve. [3] (b) Observe the changes in the sign of the gradient of the curve and determine the nature of this stationary point. [2] !" !! #!" " "=! +!

6 4. A tank is firing rounds during a target practice. The height, h metres, of a projectile above the ground during flight is governed by the equation , where t is the time in seconds for the projectile in flight. (a) Express h in the form , where a, b and c are constants to be determined. [2] (b) Hence state the maximum height attained by the projectile and the time at which this occurs. [2] (c) Find the length of time for which the height of the projectile is at least 32 metres above the ground. [3] !"#!$! $ %!" "=+ ! ( )! !" #$+!

7 [Turn over 5. (a) When the expression is divided by x + 2, the remainder is 3 less than the remainder obtained when the expression is divided by x - 1. Find the value of k. [3] (b) Solve the equation , expressing non-integer roots in exact form. [4] ! ""!" !++ !""#! $ % %!!!!! + =

8 6. (a) By considering the general term in the binomial expansion of , where k is a positive constant, explain why there are only even powers of x in this expansion. [3] (b) Given that the term independent of x in the binomial expansion of is , show that k is 3. [2] ! " # $! "! !"#$%&' ! " # $! "! !"#$%&' ! "#

9 [Turn over (c) Hence find the term independent of x in . [4] ( ) ! !" # $#%!! "! !"## $%&'

10 7. A particle, P, travels in a straight line so that at time t seconds, its velocity, v m/s, is given by . The initial displacement of P from a fixed point O is m. (a) Find the initial velocity of P, giving your answer in exact value. [1] (b) Find the first two values of t when P comes to instantaneous rest. [3] !"#$ %&'!" !!=+ !!