MGS 2023 Sec 4 Prelim AM P2 Solutions

This question paper consists of 17 printed pages and 3 blank pages. Class Index Number Name : __________________________________________ METHODIST GIRLS’ SCHOOL Founded in 1887 PRELIMINARY EXAMINATION 2023 Secondary 4 Monday ADDITIONAL MATHEMATICS 4049/02 21 August 2023 PAPER 2 Solution 2 hours 15 minutes Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your class, index number and name in the spaces at the top of this page. 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 figure, 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. At the end of the examination, fasten all your work securely together. 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. 90

Page 2 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Examination 2023 1. ALGEBRA Quadratic Equation For the quadratic equation , Binomial Expansion , where n is a positive integer and . 2. TRIGONOMETRY Identities 22cosec 1 cotAA=+ AAAAA 2222 sin211cos2sincos2cos −=−=−= Formulae for ABC sin sin sin a b c A B C== a2 = b2 + c2 − 2bc cos A  = 2 1 bc sin A 02 =++ cbxax a acbbx 2 42 −−= ( ) nrrnnnnn bbar nbanbanaba ++     ++     +     +=+ −−− 221 21 ( ) ! ( 1)...( 1) ! ! ! n n n n n r r r n r r  − − +== − 1cossin 22 =+ AA AA 22 tan1sec += BABABA sincoscossin)sin( = BABABA sinsincoscos)cos( = BA BABA tantan1 tantan)tan(  = AAA cossin22sin = A AA 2tan1 tan22tan −=

Page 3 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Examination 2023 1. (i) A prism has a volume of ( ) 23 8 1xx++ cm3 and a cross-sectional area of ( ) 2 21xx++ cm2. Write down an expression for the height of the prism, in the form ( ) ( ) 2 1 1 BCA x x ++ + + . [4] 2 22 3 8 1 2 2Height, 3 2 1 2 1 x x xh x x x x + + −= = ++ + + + ( ) ( ) 22 22 2 1 1 1 x B C x x x x − =++ + + + ( ) 1 22 4 x C C =− −= =− 0 24 2 x B B = − = − = ( ) 2 24Height, 3 1 1 h x x = + −+ + (ii) Find ( ) ( ) 2 d1 1 BCAx x x ++ + + . [3] ( ) ( ) ( ) ( ) 2 2 243 d 1 1 43 2ln 1 1 xx x x x C x +− + + = + + + + +  where C is an arbitrary constant B1 B1 B1 M1 B1 B1 B1

Page 4 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Examination 2023 2. A circle C passes through the point )3 ,4( −P and has the same centre as the circle 012422 =−−++ yxyx . (i) Find the equation of the circle C. [3] ( ) ( ) ( ) ( ) 22 22 22 4 2 1 0 2 4 1 1 1 2 1 6 x y x y xy xy + + − − = + − + − − = + + − = Centre of circle = ( )2,1− Radius of circle = ( ) ( ) 22 4 2 3 1 52+ + − − = Equation of circle is ( ) ( ) 22 2 1 52xy+ + − = or 22 4 2 47 0x y x y+ + − − = (ii) Find the equation of the tangent to the circle C at the point )3 ,4( −P . [2] Gradient of P to the centre = 3 1 4 2 4 2 6 3 − − −= =−+ Gradient of tangent = 3 2 Equation of tangent at P is ( )334 2 3 632 3 92 yx yx yx + = − = − − =− B1 B1 A1 M1 A1

Page 5 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Examination 2023 (iii) Another point Q(4, q) which lies on the circle C, is the same distance from the y-axis as the point P. Find the coordinates of the point Q. [2] Subs x = 4 into equation of circle ( ) ( ) ( ) 22 2 4 2 1 52 1 52 36 14 5 or 3 y y y yy + + − = − = − − = = =− Q(4, 5) Alt Mtd : Since perp bisector of chord passes through centre of circle, 3 12 5 q q − = = Q(4, 5) 3. (i) Given that ( )1 4 1y x x= − + , show that dy 6 1 .dx 41 x x −= + [3] ( ) ( ) 11 4 4 1 2 4 1 21 41 4 1 4 1 61 41 dy xxdx x x x xx x x = −  + + + − +=+ ++ −= + A1 M1 M1 P (4, -3) Q (4, q) M1 A1

Page 6 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Examination 2023 (ii) Hence, evaluate 2 1 65 . 41 x dx x − + [4] ( ) 22 11 22 11 22 1 1 6 5 6 1 4 4 1 4 1 4 41 421 4 1 4 1 4 9 0 2 9 5 2 5 3 1.47 xx dx dx xx dy dx dxdx x x x x − − −= ++ =− + = − + − +      = − − −    =− =   4. (a) Express 2sin 1.5cos− in the form sin( )R − , where R > 0 and 0 2  . [3] 2sin 1.5cos sin cos cos sinRR     − = − 2 cos 1.5 sin R R   = = 2 2 2 2 (1.5) 5 2 R R =+ = 1.5tan 2 0.644   = = ( )52sin 1.5cos sin 0.6442  − = − M1 M1 A1 M1 B1 B1 A1

Page 7 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Examination 2023 (b) John models the height of sea water level, H metres, on a particular day by the equation 446 2sin 1.5cos25 25 ttH    = + −        , 0 12t , where t hours is the number of hours after midday. Using this model, calculate (i) the maximum possible height of sea water level and the value of t, to 2 decimal places, when it occurs. [3] 446 2sin 1.5cos25 25 54max 6 sin 0.64352 25 56 2 8.5 m ttH tH      = + −        = + −  =+ = It occurs when 4sin 0.643501 125 4 0.64350125 2 4.405204 4.41 t t t t    −= −= = = 4 (ii) the first time when the height of the sea water is 7 metres, leaving your answer correct to the nearest minute. [4] 547 6 sin 0.6435012 25 24 sin 0.6435015 25 2.09889 125.933 mins =126 mins t t t   = + −  =−  = = First time = 14 06 or 2.06 pm B1 A1 M1 M1 M1 A1 B1

Page 8 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Examination 2023 5. The temperature,  oC, of an oven t minutes after it was switched on is given by 0.05300 280 te −=− , 0t . (i) State the initial temperature of the oven. [1] Initial temperature = 300 280 20 C− =  (ii) Find the value of t when the temperature of the oven reaches 160 oC. [2] 0.05 0.05 0.05 300 280 160 280 140 1 2 10.05 ln 2 13.8629 13.9 mins t t t e e e t t − − − −= = = −=  = = (iii) Determine, with justification, the maximum temperature that this oven can approach. [2] As t increase, 0.05te− approaches zero. Hence, maximum temperature that the oven can reach = 300 C Alt Mtd : 0.05 0.05 0.05 0 280 0 300 280 300 300 t t t e e e  − − −  − −  or Since 300 = is an asymptote,  approaches 300 as t increases. B1 M1 A1 B1 A1 300 280

Page 9 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Examination 2023 5. The temperature,  oC, of another oven t minutes after it was switched on is given by 0.1250 230 te −=− , 0t . (iv) Assuming that both ovens are switched on at the same time, find the time when both ovens will have the same temperature since they were switched on. [5] 0.1 0.05 0.05 0.1 0.05 0.1 250 230 300 280 280 230 50 28 23 5 0 tt tt tt ee ee ee −− −− −− − = − −= − + + = ( )( ) 2 0.05 ( 0.05 ) 0.05 2 Let 23 28 5 0 23 28 5 0 23 5 1 0 t tt ep ee p