2025 JVSS Sec 4 AM Prelim P1 MS

This document consists of 18 printed pages and 0 blank page. [Turn Over JURONGVILLE SECONDARY SCHOOL PRELIMINARY EXAMINATION 2025 Secondary 4 Express STUDENT NAME MARKING SCHEME CLASS INDEX NUMBER ADDITIONAL MATHEMATICS 4049/01 Paper 1 19 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 and index number 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. DO NOT WRITE ON ANY BARCODES. 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 marks for this paper is 90. DO NOT OPEN THE BOOKLET UNTIL YOU ARE TOLD TO DO SO Setter: Mr Shahul Hameed For Examiner’s Use 90

2 Jurongville Secondary School 4E PRELIM / 4049_01 / 2025 1. ALGEBRA Quadratic Equation For the equation ,02  cbxax a acbbx 2 42  Binomial expansion ,......21)( 221 nrrnnnnn bbar nbanbanaba                  w h e r e n is a positive integer and ! )1(...)1( )!(! ! r rnnn rnr n r n       2. TRIGONOMETRY Identities AAec AA AA 22 22 22 cot1cos tan1sec 1cossin    BA BABA BABABA BABABA tantan1 tantan)tan( sinsincoscos)cos( sincoscossin)sin(      A AA AAAAA AAA 2 2222 tan1 tan22tan sin211cos2sincos2cos cossin22sin    Formulae for ABC Cab Abccba C c B b A a sin2 1 cos2 sinsinsin 222   

3 Jurongville Secondary School 4E PRELIM / 4049_01 / 2024 [Turn Over 1 The curve has the equation 2 47yx x . A line with gradient 1 is a tangent to the curve. Find the equation of the tangent line. [3] 24 [ M 1 ] 24 1 2.5, 3.25 [M1] 3.25 1( 2.5) 0.75 [A1] dy xdx x xy yx yx       2 Find the set of values of the constant p for which the curve 2 (2 ) 5yp x p x  lies entirely above the x-axis. [5]  2 2 0 [B1] 24 ( ) ( 5 ) 0 [ M 1 ] 24 4 0 Use of quadratic formula [M1] Critical Points: 12 2 35 [M1] 12 2 35 12 2 35 [A1] 0.168 23.8 p pp pp p p p           0.168 + + െ 23.8

4 Jurongville Secondary School 4E PRELIM / 4049_01 / 2025 3 (a) Prove that   22 22 cot tan cos 2 1t a n 1c o t      . [5]         22 22 22 44 22 22 44 2 2 22 44 22 22 cos sin sin cos [M1] sec cosec cos sin sin cos= [ M 1 ]11 cos sin cos sin cos sin= [ M 1 ]sin cos 1 cos sin cos sin cos sin [M1] cos 2 1 cos 2 [A1] LHS                        

5 Jurongville Secondary School 4E PRELIM / 4049_01 / 2024 [Turn Over 3 (b) Hence, solve the equation      22 22 cot tan 2sin 2 0.75 1t a n 1c o t       for 90 90  . [5]       22 22 1 cot tan 2sin 2 0.75 1t a n 1c o t 2sin 2 cos 2 0.75 [M1] sin 4 0.75 [M1] sin 0.75 48.59037789 [M1] 4 48.5903, 311.4096,131.4096, 228.5903 [M1] 77.9 , 57.1 ,12.1 ,32.9 [A1]                       

6 Jurongville Secondary School 4E PRELIM / 4049_01 / 2025 4 The line 5yx intersects the curve 22 91xx y y  at two points. Find the coordinates of these two points. [5]     22 22 2 2 2 5 (5 ) 5 9 1 [ M 1 ] 51 0 2 5 9 1 0 31 5 6 6 0 [ M 1 ] 52 2 0 Using Quadratic Formula, [B1] 2.82 or 7.82 [M1] 2.82, 7.82 or 7.82, 2.82 [A1] yx xx x x xx x x x xx xx x           

7 Jurongville Secondary School 4E PRELIM / 4049_01 / 2024 [Turn Over 5 The function f is defined for x > 0 by 2 4() 1 xfx x   . (a) Find '( )fx in the form  2 2 1 ax bx c x   . [2]         2 ' 2 2 2 12 41 ( ) [M1] 1 24 [A1] 1 xx x fx x xx x       (b) Hence determine the values of x for which f is increasing. [4] F o r f to be increasing, '( )fx must be > 0. Since,  2 1x > 0 for all real values of x, then to solve 2 24xx > 0. [B1] Show use of quadratic formula [B1] Critical Points: 3.24 or 1.24 [M1] 3.24 or 1.24 [A1] x xx    െ3.24 + + െ 1.24

8 Jurongville Secondary School 4E PRELIM / 4049_01 / 2025 6 A balloon is being inflated such that its volume increases and the radius, r cm of the balloon, changes over time. The rate of change of the radius at time 𝑡 seconds is given by d4 d2 r tt  , 0t . Initially, at 0t , the radius is 3 cm. (a) Find an expression for r in terms of t. [2]    4 4 l n2 [ M 1 ]2 At 0, 3, 3 4ln 2 or 0.227 4ln 2 3 4ln 2 or 4ln 2 0.227 [A1] rd t t ct trc rt t            (b) The volume V of the balloon is given by 34 3Vr  . Find the rate of increase of the volume of the balloon at t = 2. [4] 2 3 At 2, 4 ln 4 3 4 ln 2 4 [ M 1 ] At 2, 418.7464104, [M1] 1 418.7464104 1 [M1] 419 / [A1] tr dv rdr t dv dr dr dt dv dv dr dt dr dt dv dt dv cm sdt           

9 Jurongville Secondary School 4E PRELIM / 4049_01 / 2024 [Turn Over 7 (a) Express   2 2 53 7 12 xx xx   in partial fractions. [6]          2 22 22 2 22 53 7 [M1]1212 2 53 7 2 1 2 1 [ M 1 ] When 2, 11 [M1] When 1, 1 [M1] When 0, 4 [M1] 53 7 1 4 1 1 [A1]1212 2 xx A B C xxxx x xx A x B x x C x xC xA xB xx xxxx x                  (b) Hence   2 2 53 7 d 12 xx x xx   . [3]       2 2 2 2 53 7 d 12 14 1 1 [M1]12 2 14 11 212 11ln 1 4ln 2 [A2][Minus 1 for each incorrect part]2 xx x xx dxxx x dx dx xxx xx c x                

10 Jurongville Secondary School 4E PRELIM / 4049_01 / 2025 8 A curve has equation 3yx x k  , where k is a positive constant. (a) Show that d3 2 d 2 yx k x x k   . [3]  1 2 1 [M1]2 2 22 [M1] 2 32 [A1] 2 dy xx k x kdx dy x xkdx xk dy x x k dx xk dy x k dx xk          