2025 Fmss Prelim P1 with AnsKey

NAME: ____________________________ ( ) CLASS: _______ ADDITIONAL MATHEMATICS 4049/01 Paper 1 Date: 27 August 2025 Duration: 2 hours 15 minutes Candidates answer on the Question Paper. No Additional Materials are required. ____________________________________________________________________ READ THESE INSTRUCTIONS FIRST Write your name, index number and class on all the work you hand in. Write in dark blue or black pen. You may use a HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all 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 90. For Examiner’s Use Table of Penalties Question Number 90 Presentation  1  2 Rounding off  1 Parent’s/Guardian’s Signature Setter: Mdm Haliza This question paper consists of 22 printed pages including the cover page. FAIRFIELD METHODIST SCHOOL (SECONDARY) PRELIMINARY EXAMINATION 2025 SECONDARY 4 EXPRESS

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2025 2 Additional Mathematics Paper 1 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02 =++ cbxax , a acbbx 2 42 −−= Binomial expansion ( ) nrrnnnnn bbar nbanbanaba ++     ++     +     +=+ −−− ......21 221 , where n is a positive integer and ! )1)...(1( )!(! ! r rnnn rnr n r n +−−=−=      2. TRIGONOMETRY Identities 1cossin 22 =+ AA AA 22 tan1sec += cosec 2 A = 1 + cot 2 A BABABA sincoscossin)sin( = BABABA sinsincoscos)cos( = 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 −+= Cabsin2 1=

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2025 3 Additional Mathematics Paper 1 Answer all the questions. 1 The equation of a curve is 23 12 10 y x x= − + . (a) By expressing 23 12 10 y x x= − + in the form ( ) 2 a x b c++ , where a, b and c are constants, find the coordinates of the stationary point on the curve. [2] (b) If the curve is translated 3 units to the left and 2 units down, write the new equation in the form ( ) 2 y a x d e= + + , where a, d and e are constants. [1] (c) The line 3y x c=+ is a tangent to the curve. Find the value of c. [3]

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2025 4 Additional Mathematics Paper 1 2 A group of students investigated the cooling rate of a cup of hot chocolate in a caf e. The temperature of the cafe was 25°C. The temperature of the hot chocolate, T°C, after time t minutes is modelled by the formula 25ktT Ae −=+ where A and k are constants. Measurements of T and t are shown in the table below. t (min) 5 10 15 20 25 T (°C) 72 58 47 39 35 (a) On the next page, p lot ln ( 25)T− against t and draw a straight line graph to illustrate the information. [3]

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2025 5 Additional Mathematics Paper 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 5 10 15 20 25 30 t ln (T – 25)

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2025 6 Additional Mathematics Paper 1 2 (b) Use your graph to estimate the value of A and the value of k. [2] (c) If the ideal drinking temperature is 45°C, use your graph to estimate when to serve the hot chocolate. [1] (d) The hot chocolate was stirred constantly. Predict whether the value of k would increase or decrease. Justify your answer. [1]

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2025 7 Additional Mathematics Paper 1 3 (a) Divide 322 5 3 4x x x+ − + by 32 2x x x+− . [1] (b) Express 32 32 2 5 3 4 2 x x x x x x + − + +− in partial fractions. [5]

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2025 8 Additional Mathematics Paper 1 4 A swing follows a parabolic path, where its height, h metres, at time t seconds is given by the equation 2 6h t t=− + . The swing starts at rest at t = 0, reaches maximum height, and falls back down. At two moments during the swing’s motion, the heights are recorded at times k and 3k seconds, where 0 < k < 2. (a) Find the range of values of k for which the height at time 3k is greater than the height at time k. [3] (b) Hence, by referring to the graph or otherwise, explain why the speed of the swing at time k seconds is always greater than the speed at time 3k seconds. [2] t h 0

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2025 9 Additional Mathematics Paper 1 5 (a) Find the first 4 terms in the expansion of 8 2 2x x  − in descending powers of x, simplifying each term. [2] (b) Given that the coefficient of x10 in the expansion of 8 32 2( 5)kx x x +−  is 336,− find the value of k. [2] (c) Explain why there is no constant term in the expansion of 8 2 2x x  − . [2]

Name: __________________________________ ( ) Class: __________ FMS(S) Sec 4 Express Preliminary Examination 2025 10 Additional Mathematics Paper 1 6 (a) Find the amplitude and period of (i) 3 sin2 x , [1] (ii) 1 3cos 2 x− . [2] (b) Sketch, on the same diagram, the curves in3 s2yx= and 1 3cos 2yx=− for 02 x  . [3] 0 y x