IJC_H2_MATH_P2_(9740)_Solutions

1 (i)   5 i0 i0 2 π i2 π 243 243e 243e 243e k k w      2 πi 5 4π 2π 2π 4πii i i55 5 5 3e , where 2, 1, 0,1, 2 3e , 3e , 3, 3e , 3e k wk       (ii) 4πi 5 1 3ew   and 2πi 5 2 3ew   2 (i) 1f : 3 2x x  , ,2xx Let f( )yx . 13 2 12 3 12 3 y x x y x y       1 1 f ( ) 2 3x x    , ,3xx (ii)   f f D2 , R3 ,   Since ffRD , the composite function 2f exists. 3 Im(z) O Re(z)

(iii)  2 2 f 1f3 2 13 132 2 13 1 2 3( 1) ( 2) 1 45 (1 ) 55 0 xx xx x x xx x xx xx xx x xx                 Using GC, f 1.38 (rej 1.38 D ) or 3.62xx         11 1 ff ff f f ff xx x x xx      Therefore 3.62x  satisfies    1ff x x . 3 (i) When x = 0,  10 0 o r 1 1 1 or 2 tt t t yy      Coordinates are (0,1) and 10, 2   . x y O

(ii)     22 22 22 dd 2 21 , dd 1 d2 1 d2 1 1 2 12 1 xy tttt t yt xt t t tt         When tangent is parallel to y-axis,    2222 112 10 1 0 2tt t t         Equation of tangent: 1 4x (iii) Area of the required region   0 1/ 4 1 21/2 1 221/2 121 1/2 1 1 d 1 21 d 1 21 d 11 ln 1 tan 51ln 2 ln tan44 2 81ln tan54 2 yx tt t t t tt tt                            4 (a)(i) Area of unsown ploughed land  2 0.4 0.4 300 100 88 m    11When , 42 When 0, 1 xt xt  

(a)(ii) n Beginning of week End of week 1 300  0.4 300 2 0.4 300 100    2 0.4 0.4 300 100 0.4 300 0.4 100     3  20.4 300 0.4 100 100        2 32 0.4 0.4 300 0.4 100 100 0.4 300 0.4 100 0.4 100    .. … … n …     1 21 0.4 300 0.4 100 ... 0.4 100 0.4 100 nn    Area of land unsown ploughed land at the end of nth week     1 12 0.4 1 0.4 0.4 300 100 10 . 4 2000.4 300 1 0.4 m3 n n nn          the value of k is 200 3 . (a)(iii) Method 1      1 1 1 40 2000.4 300 1 0.4 703 200 2000.4 300 (0.4) 0.4 7033 400 100.433 10.4 40 ln ln 0.4 4.02588 nn nn n n n n           Hence the number of complete weeks required is 5. Method 2    12000.4 300 1 0.4 703 nn  

Using GC, when n = 4, unsown ploughed land = 70.08 (> 70) when n = 5, unsown ploughed land = 68.032 (< 70) when n = 6, unsown ploughed land = 67.213 (< 70) Hence the number of complete weeks required is 5. (b)(i) n Beginning of week End of week 1 300 300 80 2 300 100 80  300 100 80 100  3 300 2 100 80 100   300 2 100 80 100 120   .. … … n …     300 1 100 80 100 80 20 1 n n      Area of unsown ploughed land at the end of nth week      2 2 300 100 1 2 80 20 12 300 100 100 140 202 300 100 100