2021 NTSS Prelim 4E AM P1 MS v20210804a

This document consists of 17 printed pages. Setter: Mr Alan Cheng NEW TOWN SECONDARY SCHOOL Preliminary Examination Secondary 4 Express NAME Mark Scheme CLASS INDEX NUMBER Additional Mathematics Paper 1 4049/01 28 July 2021 10:35 – 12:50 2 hours 15 min READ THESE INSTRUCTIONS FIRST Write your name, register number and class in the spaces provided above and on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use 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. For Examiner’s Use

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02 =++ cbxax , 2( 4 ) 2 b b acx a −  −= . Binomial Expansion nrrnnnnn bbar nbanbanaba ++     ++     +     +=+ −−− 221 21)( , where n is a positive integer and ( ) ! ( 1) ( 1) ! ! ! n n n n n r r r n r r  − − +== − 2. TRIGONOMETRY Identities sin2 A + cos2 A = 1 sec2 A = 1 + tan2 A cosec2 A = 1 + cot2 A BABABA sincoscossin)sin( = BABABA sinsincoscos)cos( = BA BABA tantan1 tantan)tan(  = AAA cossin22sin = AAAAA 2222 sin211cos2sincos2cos −=−=−= A AA 2tan1 tan22tan −= Formulae for ABC A a sin = B b sin = C c sin a2 = b2 + c2 − 2bc cos A  = 2 1 bc sin A

3 1 The curve 2 4 xy xba− = + , where a and b are constants, intersects the y-axis at A and the x-axis at B and C. The coordinates of B are ( )2,0− . Given that the gradient of AB is 3− , find the value of a and of b. [5] Sub ( )2,0B − , ( ) 2 2 0 24 ba − − =− + [M1] 3b= [A1] Sub 0,x= 3b= , y ba−= 3ya=− --- (1) [M1] ( )0, 3Aa − Gradient of AB ( )03 320 a−−= =−−− [M1] 3 32 a =−− 2a= [A1]

4 2 The equation of a curve is 3 cos 22y c x=− where c is a constant. The curve passes through the point 1,64   . (a) Find the value of c. [2] Sub 1,64   13 cos 24 2 6c =−  [M1] 1 3 1 4 2 2c =−  1c= [A1] (b) Using the value of c found in part (a), sketch the graph of 3 cos 22y c x=− for 02 x  . [B1 – 2 complete cycles] [B1 – start and end at -0.5] [B1 – fully correct curve with correct Max value @ 2.5 or Min value @ -0.5] [3]

5 3 (a) Write down, and simplify, the first 4 terms in the expansion of 8 1 2 x− in ascending powers of x. [3] ( ) ( ) ( ) ( ) 8 1 2 3 8 7 6 58 8 81 1 1 1 1 ... 1 2 32 2 2 2 x x x x             − = + − + − + − +                         231 4 7 7 ...x x x= − + − + [B1 – 1st and 2nd term correct] [B1 – 3rd term correct] [B1 – 4th term correct] (b) Find the coefficient of x in the expansion of 82 213 2 x xx    −+       . [3] 2 2 2 24 3 12 9xxxx  + = + + [M1] ( ) 82 2 3 2 2 241 3 1 4 14 7 ... 12 92 x x x x x xxx      − + = − + − + + +           Terms with x ( )( ) ( ) 3 2 44 12 7xx x = − + −  [M1] 48 28 76x x x=− − =− Coefficient of x 76=− [A1]

6 4 (a) Given that 2 1 xey x= + , 1x , explain, with working, whether y is an increasing or decreasing function. [3] ( ) ( ) ( ) 2 22 12 1 xxx e x edy dx x +− = + [M1] ( ) ( ) 2 22 12 1 xe x x x +− = + ( ) ( ) 2 22 1 1 xex x −= + [M1] Since ( ) 22 10x + , ( ) 2 10x− and 0xe  , therefore 0dy dx  y is increasing function. [B1] (b) Air is escaping from a hole in a spherical balloon of radius r cm in such a way that the total volume, V cm3, is decreasing at a constant rate of 25 cm3/s. Assuming that the balloon retains its shape, calculate the rate of change of r when 5r = . [3] 34 3Vr = 24dV rdr = [M1] dV dV dr dt dr dt= ( ) 2 25 4 5 dr dt− =  [M1] 1 4 dr dt =− cm/s [A1]

7 5 The number of fishes, F in a fish farm after t days can be modelled by the formula 6000 ktF Ae=+ where A and k are constants. Initially, there were 2500 fishes in the fish farm and 5 days later, there were 3500 fishes. (a) Show that 3500A=− [1] Sub 0, 2500tF== ( )0 2500 6000 k Ae=+ [M1] 2500 4000 3500A= − =− (b) Find the number of days required for the fishes to increase its population by 80%. [3] ( )5 3500 6000 3500 k e=− [M1] 0.067294k =− 0.067294180 2500 6000 3500100 te− = − [M1] 12.6t = days [A1] (c) Explain why the number of fishes in the fish farm cannot be 6000. [2] Suppose the no. of fishes is 6000, then 0.06736000 6000 3500 te−=− 0.0673 0te− = [B1] But 0.0673 0te−  for all real values of t Hence the no. of fishes cannot be 6000. [B1]

8 6 A curve is such that 2 2 2sin 3cos 2dy xxdx =− and the point ( ),5A  lies on the curve. The gradient of the curve at A is 3− . Find the equation of the curve. [6] 2 2 2sin 3cos 2dy xxdx =− 32cos sin 22 dy x x cdx =− − + [M1] Sub ,3dyx dx= =− ( ) ( )33 2cos sin 2 2 c− =− − + [M1] ( )3 2 1 0 c− =− − − + 5c=− 32cos sin 2 52 dy xxdx =− − − [A1] 32sin cos 2 54y x x x d=− + − + [M1] Sub ,5xy == ( ) ( ) ( )35 2sin cos 2 5 4 d  =− + − + [M1] ( ) ( )35 0 1 5 4 d= + − + 17 54d =+ 3 172sin cos 2 5 544y x x x =− + − + + [A1]

9 7 (a) Express each of 284 xx−+ and 22 12 14xx− − − in the form ( ) 2 a x b c++ , where a, b and c are constants. [4] 228 4 4 8x x x x− + = − + ( ) 2 2 428 2x = − − +  [M1] ( ) 2 24x= − + [A1] ( ) 222 12 14 2 6 7x x x x− − − =− + + ( ) 2 2 62 3 7 2x  =− + − +  [M1] ( ) 2 2 3 2x=− + − ( ) 2 2 3 4x=− + + [A1] (b) Use your answers from part (a) to explain if the curves with equations 284y x x= − + and 22 12 14y x x=− − − will intersect. [3] Min point of 284y x x= − + is ( )2, 4 [B1] Max point of 22 12 14y x x=− − − is ( )3, 4− [B1] Since 28 4 4xx− +  and 22 12 14 4xx− − −  , the only possible intersection is at the turning point, however, the turning points are not the same and hence both curves will not intersect. [B1]

10 8 Without using a calculator, (a) show that 31sin15 22 −= , [3] ( )sin15 sin 60 45= −  sin 60 cos 45 cos 60 sin 45=  −   [M1] 3 2 1 2 2 2 2 2      =−       