2022 AMath Prelims P2 w Ans

1 Solutions 1 Given the function 1 2y x  , where 1 2x . Find d d y x and show that y has no turning points. 2 3 2 d 1 d 1 2d d 12 2 1 1 12 2 1 2 2 1 y x x x x x x x x                When d 0d y x  , 3 2 1 0 2 2 1x x     has no solution.  y has no turning points. [4] Fraction over a fraction is not complete.

2 (i) Express   3 2 2 4 10 2 2 x x x x x      in partial fractions. [5] 4 3 2 2x x  3 24 10 2x x x    3 2( 4 8 )x x   22 2x x      2 23 2 2 4 10 2 2 2 4 2 2x x x x x x x x x             22 2 2 2 2 2 x x x x x A B C x x           2 22 2 22x x Ax Cxx B x      Sub 0x , 2 2 1 B B    Sub 2x , 8 4 2 C C   Sub 1x , 1 1 2 0 A A         3 2 2 2 4 10 2 2 1 42 2 x x x x x x x      

(ii) Hence find   3 2 2 4 10 2 d2 x x x xx x      . [3]   3 2 2 4 10 2 d2 x x x xx x        2 2 14 2 14 2ln 2 , where is an arbitrary constant. dxx x x x c c x           Deduct mark if did not put + c

3 The diagram shows the graph of 4.1 2.3sinh bt  , for 0 12t  . (i) State the value of b. [1] 2 8 4b    (ii) The height, h m, of the tidal wave at a harbour t hours after 4am is given by 4.1 2.3sinh bt  . A yacht at the harbour ca n only leave and return to the harbour when the height at the wave is at least 2 metres. Calculate the range of time in hours and minutes for the yacht to leave and return in the next 10 hours. [4] 4.1 2.3sin 2 4 2.3sin 2.14 t t              Solve for t, 21sin 4 23 1.1507 t        1.1507, 1.9909, 7.4339 4 1.4651, 2.5348, 9.4651 t t    Yacht to sail out between 4 am to 5.27 am and return between 6.33 am to 1.27 pm. 4.1 2.3sinh bt  t h 2 4 6 8 10 12 0 1.8 6.4 2 4 6 8 0 2 1.4651 2.5348 9.4651 1.4651 h = 1 hour 27.9 min 2.5348 h = 2 hour 32.1 min 9.4651 h = 9 hour 27.9 min h Must be in rad mode.

4 A curve is defined by 2 2 xy x   . Find an expression for d d y x and obtain the exact x – coordinates of the point(s) on the curve at which the