TJC_H2_Maths_P1_Solutions

1 (i) Find the derivative of 24 x with respect to x. [ 1 ] (ii) Given the differential equation 2 2 2 d41 d yx x  , find y in terms of x. [4] [Solution] (i)   1 2 2 2 d1 42d2 4 ux xxx x      (ii) 2 2 2 d41 d yx x 2 2 2 2 d1 d 4 d1 dd 4 y x x y xx x     1sin 2 x C 1sin d2 xyC x  1 2 1sin d22 1 2 xxx xC x x     1 2 sin d2 4 xxx xC x x    12sin 4 2 xx xC x D    from part (i) 2 (a) The point A has coordinates (3, a, b) where ,ab  . Given that A lies on the xy-plane and the magnitude of the position vector of A is 5, find the values of a and b. [ 3 ] (b) The real numbers c and d are such that the vectors dc mi j k and cdnij k are perpendicular to each other. Show that  2 1cmn . [3]

[Solution] (a) Since (3, a, b) lies on the x-y plane, b = 0. 5AB   22230 5a   2 16a 4 or 4a  (b) Since 1 and 1 c dd c         are perpendicular, 2 1 02 1 c dd d c c           sin90 o m n mn mn 2 2 22 2211 1dc cd cd      2212 1cc c    3 Given  2f zp z q z r  where p, q and r are complex numbers such that f1 2 i . The equation  f0 z  has roots 1i and 12 i . Find p, q and r. [6] [Solution] Since  f0 z  has roots 1i and 12 i ,     f1 i 1 2 i ,zp z z k      Since f1 2 i ,  11 i11 2 i 2 ip      i2 i 2 ip 22 ip  ip      fi 1 i 1 2 izz z          2i1 i + 1 2 i 1 i 1 2 izz        2i2 3 i 1 3 izz       2i3 2 i 3 izz     Therefore, ip , 32 iq  and 3ir

4 Without the use of a graphic ca lculator, solve the inequality 1025 2x x  . [3] Hence find the solution to the inequality 102cos 5 2c o s   , where 02   . [3] [Solution]   25 2 1 01025 0 22 xxx xx     2 2 2 02 2 02 xx x xx x     1 2x  or 02 x For 102cos 5 2c o s   Replace by cosx  , we have 1cos 2  or 0c o s 2 For 12 4cos 23 3      For 0c o s 2 0c o s 1    30 o r 2 22      24 30 or or 223 3 2         5 A souvenir company received an order to pr oduce a souvenir that must satisfy all of the following conditions: (1) The souvenir is a solid cuboid with a square base. (2) The souvenir is made using 1m3 of superior clay. (3) The external surface of the souvenir must be coated with a special-mixed glow paint. Find the dimensions of the souvenir, in m, such that the amount of special paint n e e d e d i s t h e m i n i m u m . [ 7 ]

[Solution] Let the length of the square base be x m and the height of the cuboid be y m. Volume of cuboid = 1 m3 2 2 11xy y x 