Woodgrove Secondary 4051/01 MS 2025

Name Index Number Class N LEVEL PRELIMINARY EXAMINATION 2025 LEVEL & STREAM : SECONDARY 4 NORMAL (ACADEMIC) SUBJECT (CODE) : ADDITIONAL MATHEMATICS (4051) PAPER NO. : 01 DATE (DAY) : 29 JULY 2025 (TUESDAY) DURATION : 1 HOUR 45 MINUTES READ THESE INSTRUCTIONS FIRST Write your name, index 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 70. DO NOT TURN OVER THE QUESTION PAPER UNTIL YOU ARE TOLD TO DO SO. Student’s Signature Parent’s Signature Date Date This document consists of 13 printed pages including this cover page. Setters: Mdm Zuraidah 70

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02  cbxax , a acbbx 2 42  2. TRIGONOMETRY Identities 1cossin 22  AA AA 22 tan1sec  2 2cosec 1 cotA A    BABABA sincoscossinsin    BABABA sinsincoscoscos    BA BABA tantan1 tantantan   AAAAA 2222 sin211cos2sincos2cos  A AA 2tan1 tan22tan  Formulae for ABC C c B b A a sinsinsin  Abccba cos2222   = Abc sin2 1 AAA cossin22sin 

3 Answer all questions. 1 Circle C has equation     2 2 1 3 36.x y    (a) Write down the radius and the coordinates of the centre of the circle. [2]            2 2 2 2 2 1 3 36 1 3 6 centre of ci ]rcle 1, 3 and radius 6 units [B2 x y x y             (b) Explain why any line that passes through the point  1, 1 will intersect circle C at two points. [1]          2 2 distance of 1, 1 from centre 1 1 1 3 4 2 6 units (radius of circle) As the point 1, 1 lies inside the c            ircle, any line that passes through it will intersect circle at two points. C 2 Given that 3cos 7A where π0 , 2A  find the exact value of (a) sin ,A [2] 2 3 2 2 2 [M1] [A1] 3 7 7 3 40 2 10 2 10sin 7 y y y y A        (b) sin 2 .A [2] ] sin 2 2sin cos 2 10 3 40 32 or 27 7 7 7 12 10 [M1] [A149 A A A                            3 7 A 2 10 [B1] or distance 3 1 2 6 units    

4 3 Do not use a calculator in answering this question. Express 1 7 5 7   in the form 7 ,a b c  stating the values of the integers , a b and .c [4]       2 22 ] 1 7 1 7 5 7 5 7 5 7 5 7 5 7 5 7 7 = 5 7 12 6 7 18 2 7 7 3 2, 1, 3 [M1] [M1] [A1] [B1 a b c a b c                    4 (a) It is given that 4 1 2 5 xy x   for 5 ,2x find d .d y x [3]             2 2 2 ] 4 1 [M2] Given: 2 5 2 5 4 4 1 2 2 5 8 20 8 2 2 5 22 2 5 [A1 xy x x xdy dx x x x x x             (b) Explain why y is an increasing function. [1]       2 2 2 d 22 d 2 5 5For , 2 5 0 2 22Hence 0 and is an increasing function. 2 5 y x x x x y x        M1: correct numerator M1: correct denominator [B1]

5 5 Express 2 2 6 21 25 2 5 x x x x    in the form .2 5 B CA x x   [5]       2 2 2 2 2 2 6 21 25Given: 2 5 2 5 6 21 25 25 6 3 2 5 2 5 25 6 25 6= = + 2 5 2 5 2 5 25 6 2 5 when 0, 25 5 [M1] [M1] x x B C Ax x x x x x x x x x x x x B C x x x x x x x B x Cx x B                          2 5 5when , 2 5 5 5 25 6 2 52 2 2 5 10 2 4 6 21 25 2 [M1] [M1] B x B C C C x x                                  2 2 5 435 2 5 5 4 3 2 5 Alternative method: 6 21 25 2 5 2 5 when 0, 5 25 [A1] [M1] x x x x x x x x Ax x B x Cx x B                   5 5 5 when , 102 2 4 when 1, [M1] [M ] 1 B x C C x      2 2 ] 0 1 3 15 4 [ M 1 ] 3 6 21 25 5 4 [A1 3 2 5 2 5 A A x x x x x x            3 26 21 25x x    26 15x x  6 25x  22 5x x

6 6 (a) Express 22 6 7x x  in the form   2 ,p x q r  where , p q and r are constants. [3]   2 2 2 2 2 2 2 2 2 2 2 ] 6 7 2 3 7 3 32 3 7 2 2 3 3 3 3= 2 2 7 2 72 r2 2 2 3 52 [M1] [M1] 2 2 o [A1 x x x x x x x x x                                                                     (b) Hence explain why the minimum point of the curve 22 6 7y x x   is 3 5, .2 2      [1] 2 2 2 2 3 52 6 7 2 2 2 At the minimum point, must be the least . 3 5 0, 2 6 72 2 5 2 5 3 when , 02 2 x x x y x x x y y x                       3 2 3 5Hence minimum point is , . 2 2 x      (c) State the range of values of k for which y k has no solutions. [1] ]12 2 [B1k  [B1]

7 7 The volume, V cm3, of a spherical balloon is given by 34 π ,3V r where r cm is the radius of the balloon. The radius is increasing at a rate of 2 cm/s. Calculate the rate at which the volume of the balloon is increasing when the radius is 10 cm. [4]   3 2 2 2 2 4Given: , 2 cm/s, 10 cm3 4 [M1] 4 2 8 When 10 cm, 8 10 [M1] M [ 1] drV r r dt dV rdr dV dV dr dt dr dt r r dVr dt                 3 3 [A1] ) or 0 8 0 cm /s 2510 cm /s (3 sig fi g 8 Show that cos 1 sin .sec tan      [5] 2 cosShow: 1 sinsec tan cos cosLHS 1 sinsec tan cos cos cos 1 sin cos cos 1 sin [M1] [M1]                               2 ] 1 sin 1 sin 1 sin 1 sin = 1 sin 1 s S [M1] [M1] R [Ai 1n H              

8 9 (a) Find 7 . 11 5 dx x [3]     1 2 1 2 ] 7 7 11 5 11 5 11 5 7 1 .1 [M2 12 14 11 5 [ 11 ] A1 dx x dx x x c x c            (b) Hence evaluate 4 1 14 . 11 5 dx x [3]        4 4 1 1 4 1 ] 14 7 2 11 5 11 5 142 11 5 11 28= 11 4 5 11 1 5 [M1] [M1] 7 11 28 7 411 84 11 7 11 [A1 dx dx x x x 