CGS 2025 AMATH PRELIM P1 MS

Name: Solution Register No.: Class: ADDITIONAL MATHEMATICS 4049/01 Paper 1 29 August 2025 Candidates answer on the Question Paper. 2 hours 15 minutes No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your name, register number and class in the spaces at the top of this page. Write in dark blue or black pen. 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 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. This document consists of 23 printed pages and 1 blank page. Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Marks Table of Penalties Qn. No. Presentation –1 Significant Figures / Units –1 Parent’s/ Guardian’s Signature For Examiner’s Use CRESCENT GIRLS’ SCHOOL SECONDARY FOUR 2025 PRELIMINARY EXAMINATION 90

2 Crescent Girls’ School 2025 Prelim Sec 4 A Math P1 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02 = + +c bx ax, a ac b bx 2 42 − ± −= Binomial expansion 1 22( ) ... ...12 n n n n nr r nnn nab a ab a b a b b r −− −  + = + + ++ ++     , where n is a positive integer and ! ( 1)...( 1) !( )! ! n n nn n r r rn r r  − −+= = − 2. TRIGONOMETRY Identities 1cos sin22 = +A A AA 22 tan1sec + = 22cosec 1 cotAA= + B A B A B Asin cos cos sin)sin( ±= ± cos( ) cos cos sin sinAB A B A B±=  tan tantan( ) 1 tan tan ABAB AB ±±=  sin 2 2sin cosA AA= AA A A A 2222 sin2 1 1cos2sin cos2cos − = − = − = 2 2 tantan 2 1 tan AA A= − Formulae for ∆ABC C c B b A a sin sin sin= = A bc c b acos22 2 2− + = ∆ = 1 sin2 bc A

3 Crescent Girls’ School 2025 Prelim Sec 4 A Math P1 1 The curve 22 6 90x xy y+ − += and line 1yx= + intersect at the points R and S. Find the x-coordinate of the midpoint of RS. [3] Sub 1yx= + into 22 6 90x xy y+ − += 22 ( 1) 6( 1) 9 0x xx x+ + − + += 22 2 6( 2 1) 9 0xxx x x+ +− + + += 222 6 12 6 9 0xxx x+− − −+= 24 11 3 0xx + −= ( )( )41 30xx− += 1 4x = or 3x = − x-coordinate of the midpoint of RS = 1 ( 3) 34 128 +− = − . 2 Integrate 2 43 sin 515 xxx π−+−− with respect to x. [5] = 2 34 si n 5 d15x xx x π− −+−−∫ = 14 3ln(1 5 ) cos5 155 x xx xcπ − −− − −+−− = 43 1 ln(1 5 ) cos555 x x xcx π−+ − − − + , c is an arbitrary constant [Turn over

4 Crescent Girls’ School 2025 Prelim Sec 4 A Math P1 3 A cuboid has base area ( )4 25+ cm2 and a volume of ( )9 55+ cm3. Find, without using a calculator, the height of the cuboid, in cm, giving your answer in the form ( ) 1 52 ab+ , where a and b are integers. [3] Height = 9 55 4 25. 4 25 4 25 +− +− = 36 18 5 20 5 50 16 20 −+ − − = 14 2 5 4 −+ − = 71 522− = ( ) 1 752 − cm 4 (a) Express 24 12 2xx−+ + in the form 2()ax b c++ and state the coordinates of

5 Crescent Girls’ School 2025 Prelim Sec 4 A Math P1 the turning point of the curve 24 12 2yx x= −+ + . [3] ( ) 224 12 2 4 3 2x x xx− + += − − + = 22 2 3343 222xx   − −+ − +    = 2 34 92 2x− − ++ = 2 34 11 2x−−+ The turning point is 11 ,112   . (b) Hence explain why the turning point is a maximum point. [1] For all real values of x, 2 3 02x −≥ , 2 340 2x−− ≤ , 2 34 11 112x− − +≤ . Since coefficient of x2 is negative, 11 ,112   is a maximum point. 5 Express 2 2 3 52 ( 1)( 4) xx xx +− ++ in partial fractions. [5] [Turn over

6 Crescent Girls’ School 2025 Prelim Sec 4 A Math P1 2 22 3 52 ( 1)( 4) 1 4 x x A Bx C xx x x +− + = +++ + + 223 5 2 ( 4) ( )( 1)x x A x Bx c x+ −= + + + + Sub x = −1, 352 5 A−−= 4 5A = − Compare constant: 24 AC−= + 424 5C = −− − = 6 5 Compare coeff of x: 5 BC= + 6 195 55B = −= ( ) ( ) 2 2 2 3 5 2 4 19 6 ( 1)( 4) 5 1 54 xx x xx x x +− += −+++ + + 6 It is given that 21 2 xy x −= + for .xq> (a) State the value of q. [1]

7 Crescent Girls’ School 2025 Prelim Sec 4 A Math P1 q = −2 (b) Find d d y x . [3] 21 2 xy x −= + ( ) ( ) 1 2 12 2 2 1 . 2 (1)d 2 d2 xxxy xx − +− − + = + = (2 1)22 22 2 xx x x −+− + + = ( ) 3 4( 2) (2 1) 22 xx x +− − + = ( ) 3 29 22 x x + + (c) State whether 21 2 xy x −= + is an increasing or decreasing function. Explain your answer clearly. [2] For 2x >− ( ) 3 20x +> , ( ) 3 2 20x +> and 2 90x +> ⇒ d 0d y x > 21 2 xy x −= + is an increasing function. [Turn over

8 Crescent Girls’ School 2025 Prelim Sec 4 A Math P1 7 The diagram shows the curve cosy a bx c= + for 30 2x π≤≤ radians. A minimum point ,22 π − and a maximum point 5 ,44 π  are indicated on the diagram. (a) Explain why 4b = . [2] Period = 2 π Period, 2 2b ππ= 24 2b ππ= ÷= 0 x y |

9 Crescent Girls’ School 2025 Prelim Sec 4 A Math P1 (b) Explain why c = 1. [2] yc= is the axis of curve maximum + minimum 2c = = 4 ( 2) 2 +− c = 1 (c) Hence find the equation of the curve. [2] By observation, a < 0. ( )41a = −− OR [ ]1 ( 2)a = − −− OR maximum minimum 2a −= − = 4 ( 2) 2 −−− = −3 Equation of curve is 3cos 4 1yx= −+ . [Turn over

10 Crescent Girls’ School 2025 Prelim Sec 4 A Math P1 8 (a) Prove the identity sin tan1 cos 2 θθ θ =+ . [3] sin tan1 cos 2 θθ θ =+ LHS = 2 2sin cos22 1 (2 cos 1)2 θθ θ+− = 2 sin cos22 cos 2 θθ θ = sin 2 cos 2 θ θ = tan 2 θ