2017 SNGS Sec 4 AM Prelim P1 Solutions

1 The mass of a radioactive substance reduces by half every 4 hours. It is given that m0 is the mass of the substance at a particular time and that m is the mass of the substance t hours later. Calculate the value of the constant k in the relationship ktemm  0 . [3] ktemm  0 i.e when 4t sub 0 2 1 mm kemm 4 00 2 1  ek ln42 1ln  2 1ln4 1k or 2ln4 1 or 0.173 (3.s.f.) Note: m0 is a constant When t = 0, initial 0mm 2 Solve the equation    xxx 212 3227363  . [4]    xxx 22 32273633  Let xu 3 , 22 22763 uuu  02765 2  uu    0953  uu 3u , 5 9u 33 x , 5 93 x (no solution) 1x 3 The function f is defined, for all values of x, by xexx 2)(f  . Find the range of values of x for which f is a decreasing function. [6] xx exxexf 22)('  )2( 2xxex  0)(' xf Since 0xe for all real values of x, 02 2  xx 0)2(  xx 02  x 0 –2 + + – x

2 CHIJ SNGS Preliminary Examinations 2017 - Additional Mathematics 4047/01 4 (i) On the same diagram, sketch the curves 3 1 2xy and 3 1 8   xy . [2] (ii) Find the coordinates of the intersection points of the two curves. [3] (i) (ii) 3 1 2x 3 1 8   x 4 3 1 3 1   x x 43 2 x Take cube 64432 x 8x When x = 8, 4y When x = –8, 4y Coordinates: (8, 4) and (–8, –4) Note: 2 3 2 3 3 2 4)( x 2 3 4x =8 Only 1 answer 5 The equation of a curve is x xy 25 8   . A particle moves along the curve in such a way that the y-coordinate of the particle is decreasing at a constant rate of 2 units per second. Find the possible y-coordinates of the particle at the instant when the x-coordinate of the particle is increasing at 11 71 units per second. [6]        2 25 28)1(25 x xx dx dy     2 25 16225 x xx     2 25 11 x  dt dx dx dy dt dy    11 18 25 112 2    x   2 25 11 9 11 x  Note: mistake in expansion

3 CHIJ SNGS Preliminary Examinations 2017 - Additional Mathematics 4047/01   2 259 x 325  x 1x , 4x 3 12y , 3 11y 6 A curve has equation xy 42  and a line has equation )1(  xmy . Find the range of values of m for which the line )1(  xmy intersects the curve xy 42  at two distinct points. [5]   2 4 mmxx  2222 24 mxmxmx  0)24( 2222  mmxxm 042  acb      0424 2222  mmm 0441616 442  mmm 01 2  m    011  mm 11  m 7 The diagram shows part of the graph of cb xay    sin . (i) State the amplitude and period. [2] (ii) Find the value of each of the constants a, b and c. [3] (i) Amplitude = 3 Period = 8 Period 4 1 (ii) 3a       b 1 28  4 11 b Note: a is negative based on the inverted sine curve y x 4 –2 –4π –2π 0 2π 4π 6π 8π 10π 12π 1 –1